Experimental Fluid Mechanics

Experimental Fluid Mechanics

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http://www.springer.com/series/3837

L.P. Yarin

The Pi-Theorem

Applications to Fluid Mechanicsand Heat and Mass Transfer

L.P. YarinTechnion-Israel Institute of TechnologyDept. of Mechanical EngineeringTechnion City32000 HaifaIsrael

ISBN 978-3-642-19564-8 e-ISBN 978-3-642-19565-5DOI 10.1007/978-3-642-19565-5Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2011944650

# Springer-Verlag Berlin Heidelberg 2012This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protec-tive laws and regulations and therefore free for general use.

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Springer is part of Springer Science+Business Media (www.springer.com)

To the blessed memory of my parents Professor Peter Yarinand Mrs. Leah Aranovich

Preface

The book is devoted to the Buckingham Pi-theorem and its applications to various

phenomena in nature and engineering. The accent is made on problems character-

istic of heat and mass transfer in solid bodies, as well as in laminar and turbulent

flows of liquids and gases. Such choice is not accidental. It is dictated by the

requirements of modern technology and encompasses a vast majority of important

problems related with drag and heat transfer experienced by solid bodies moving in

viscous fluids. These problems involve the evaluation of temperature fields in

media with constant and temperature-dependent thermal diffusivity, heat and

mass transfer in boundary layers, pipe and jet flows, as well as thermal processes

occurring in reactive media. In all these cases a uniform approach to the

corresponding complex thermohydrodynamical problems is used. It is based on

the direct application of the Pi-theorem to the analysis of two types of problems:

those which admit a rigorous mathematical formulation, as well as those for which

such formulation is unavailable. For the former problems our attention will be

focused on the establishment of self-similarity which reduces the governing partial

differential equations to the ordinary ones by means of the Pi-theorem, whereas for

the latter problems the Pi-theorem will be used to reveal a set of the governing

dimensionless groups. To a certain degree the choice of the problems is subjective.

However, it allows the evaluation of the range of possible applications of the

Pi-theorem and the peculiarities characteristic of the complex thermohydrodyna-

mical processes in continuous media.

The book consists of nine chapters. They deal with the basics of the dimensional

analysis, the application of the Pi-theorem to find self-similarities and reduce partial

differential equations to the ordinary ones. Then, such interrelated topics as the drag

force, laminar flows in channels, pipes and jets are covered in detail. The discussion

also involves kindred heat and mass transfer in natural, forced and mixed convec-

tion and in situations with phase change and chemical reactions. Some problems of

turbulence theory are also covered in the framework of the Pi-theorem. In addition

to the in-depth exposition of the basic theory and the generic problems, a number of

worked examples of problems related to the application of the Pi-theorem to

different hydrodynamic, heat and mass transfer questions are presented in the end

of each chapter. They can be interest to the engineering and physics students.

vii

The book is intended to scientists and engineers interested in hydrodynamic and

heat and mass transfer problems. It could also be useful to graduate students

studying mechanical, civil and chemical engineering, as well as applied physics.

L.P. Yarin

viii Preface

Acknowledgment

I am especially grateful and deeply indebted to my son Professor Alexander Yarin

for some special consultations related to the applications of the dimensional

analysis to thermohydrodynamics problems, many insightful suggestions and dis-

cussions, as well as multiple comments on the contents of the book.

I am deeply obligated to my daughter Mrs. Elena Yarin and my granddaughter

Miss Inna Yarin. Without their help this book would not have materialized.

ix

Contents

1 The Overview and Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Basics of the Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Dimensional and Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . 3

2.2.2 The Principle of Dimensional Homogeneity . . . . . . . . . . . . . . . . . . . . . 7

2.3 Non-Dimensionalization of the Governing Equations . . . . . . . . . . . . . . . . 11

2.4 Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1 Characteristics of Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . 18

2.4.2 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 The Pi-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2 Choice of the Governing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Application of the Pi-Theorem to Establish Self-Similarity

and Reduce Partial Differential Equations to the Ordinary Ones . . . . 39

3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Flow over a Plane Wall Which Has Instantaneously Started

Moving from Rest (the Stokes Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem) . . . 47

3.4 Laminar Submerged Jet Issuing from a Thin Pipe

(the Landau Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Vorticity Diffusion in Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem) . . . . . 55

3.7 Capillary Waves after a Weak Impact of a Tiny Object onto

a Thin Liquid Film (the Yarin-Weiss Problem) . . . . . . . . . . . . . . . . . . . . . . . 58

3.8 Propagation of Viscous-Gravity Currents over a Solid Horizontal

Surface (the Huppert Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.9 Thermal Boundary Layer over a Flat Wall

(the Pohlhausen Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xi

3.10 Diffusion Boundary Layer over a Flat Reactive Plate

(the Levich Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Drag Force Acting on a Body Moving in Viscous Fluid . . . . . . . . . . . . . . . 71

4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Drag Action on a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.1 Motion with Constant Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.2 Oscillatory Motion of a Plate Parallel to Itself . . . . . . . . . . . . . . . . . 75

4.3 Drag Force Acting on Solid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.1 Drag Experienced by a Spherical Particle at Low,

Moderate and High Reynolds Numbers . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.2 The Effect of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.3 The Effect of Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.4 The Effect of the Free Stream Turbulence . . . . . . . . . . . . . . . . . . . . . 81

4.3.5 The Influence of the Particle-Fluid Temperature

Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Drag of Irregular Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Drag of Deformable Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6 Drag of Bodies Partially Submerged in Liquid . . . . . . . . . . . . . . . . . . . . . . . 86

4.7 Terminal Velocity of Small Spherical Particles Settling

in Viscous Liquid (the Stokes Problem for a Sphere) . . . . . . . . . . . . . . . . . 87

4.8 Sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.8.1 Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.8.2 Terminal Velocity of Heavy Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.8.3 The Critical State of a Fluidized Bed . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.9 Thin Liquid Film on a Plate Withdrawn Vertically from

a Pool Filled with Viscous Liquid (the Landau-Levich

Problem of Dip Coating) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 Laminar Flows in Channels and Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2 Flows in Straight Pipes of Circular Cross-Section . . . . . . . . . . . . . . . . . . . 106

5.2.1 The Entrance Flow Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.2 Fully Developed Region of Laminar Flows

in Smooth Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.3 Fully Developed Laminar and Turbulent

Flows in Rough Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3 Flows in Irregular Pipes and Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4 Microchannel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5 Non-Newtonian Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xii Contents

5.6 Flows in Curved Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.7 Unsteady Flows in Straight Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 Jet Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131


6.2 The Far Field of Submerged Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.3 The Dimensionless Groups of Jet Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.4 Plane Laminar Submerged Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.5 Laminar Wake of a Blunt Solid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.6 Wall Jets over Plane and Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.7 Buoyant Jets (Plumes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7 Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159


7.2 Conductive Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.2.1 Temperature Field Induced by Plane Instantaneous

Thermal Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.2.2 Temperature Field Induced by a Pointwise Instantaneous

Thermal Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.2.3 Evolution of Temperature Field in Medium with

Temperature-Dependent Thermal Diffusivity

(The Zel’dovich-Kompaneyets Problem) . . . . . . . . . . . . . . . . . . . . . 162

7.3 Heat and Mass Transfer Under Conditions of Forced Convection . . 165

7.3.1 Heat Transfer from a Hot Body Immersed in Fluid Flow . . . . 165

7.3.2 The Effect of Particle Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.3.3 The Effect of the Free Stream Turbulence . . . . . . . . . . . . . . . . . . . . 171

7.3.4 The Effect of Energy Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3.5 The Effect of Velocity Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.3.6 Mass Transfer to Solid Particles and Drops Immersed

in Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.4 Heat and Mass Transfer in Channel and Pipe Flows . . . . . . . . . . . . . . . . . 178

7.4.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.4.2 The Entrance Region of a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.4.3 Fully Developed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.5 Thermal Characteristics of Laminar Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.6 Heat and Mass Transfer in Natural Convection . . . . . . . . . . . . . . . . . . . . . . 186

7.6.1 Heat Transfer from a Spherical Particle Under the

Conditions of Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.6.2 Heat Transfer from Spinning Particle Under the

Condition of Mixed Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Contents xiii

7.6.3 Mass Transfer from a Spherical Particle Under the

Conditions of Natural and Mixed Convection . . . . . . . . . . . . . . . . 189

7.6.4 Heat Transfer From a Vertical Heated Wall . . . . . . . . . . . . . . . . . . 190

7.6.5 Mass Transfer to a Vertical Reactive Plate Under the

Conditions of Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.7 Heat Transfer From a Flat Plate in a Uniform Stream of Viscous,

High Speed Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.8 Heat Transfer Related to Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.8.1 Heat Transfer Due to Condensation of Saturated Vapor

on a Vertical Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.8.2 Freezing of a Pure Liquid (The Stefan Problem) . . . . . . . . . . . . . 202

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211


8.2 Decay of Isotropic Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.3 Turbulent Near-Wall Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.3.1 Plane-Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.3.2 Pipe Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8.3.3 Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

8.4 Friction in Pipes and Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.4.1 Friction in Smooth Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.4.2 Friction in Rough Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8.5 Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

8.5.1 Eddy Viscosity and Thermal Conductivity . . . . . . . . . . . . . . . . . . . . 224

8.5.2 Plane and Axisymmetric Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . 229

8.5.3 Inhomogeneous Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

8.5.4 Co-flowing Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

8.5.5 Turbulent Jets in Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

8.5.6 Turbulent Wall Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

8.5.7 Impinging Turbulent Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

9 Combustion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261


9.2 Thermal Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

9.3 Combustion Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

9.4 Combustion of Non-premixed Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

9.5 Diffusion Flame in the Mixing Layer of Parallel Streams

of Gaseous Fuel and Oxidizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

xiv Contents

9.6 Gas Torches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

9.7 Immersed Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Contents xv

Nomenclature

Chapter 2:

Ar Archimedes number

Bi Biot number

Bo Bond number

Br Brinkman number

C Speed of sound

Cd Drag coefficient

c Concentration

Ca Capillary number

cP Specific heat

Da Damkohler number

Da Darcy number

De Dean number

De Deborah number

D Diffusivity

D� Permeability coefficient of porous medium

Ec Eckert number

Ek Ekman number

Eu Euler number

Fd Drag force

Fr Froude number

Fg Gravity force

f Frequency

g Gravity acceleration

Gr Grashof number

h Heat transfer coefficient, or enthalpy

hm Mass transfer coefficient

Ja Jacob number

k Thermal conductivity

kB Boltzmann’s constant

Kn Knudsen number

Ku Kutateladze number

L Characteristic length scale

Lh Height of liquid layer

xvii

l Length of a pipe

Le Lewis number

m Mass of a particle

M Mach number

Nu Nusselt number

P Pressure

DP Pressure drop

Pe Peclet number

Ped Peclet number (for diffusion)

Pr Prandtl number

Qv Volumetric flow rate

q Heat of reaction

R Gas constant, or radius of curvature

r Cross-sectional radius of a pipe

Ra Rayleigh number

Re Reynolds number

Ri Richardson number

Ro Rossby number

rv Latent heat of vaporization

Sc Schmidt number

Se Semenov number

Sh Sherwood number

St Stanton number

St Strouhal number

Ta Taylor number

T Temperature, or torque

DT Temperature difference

t Time

tr Relaxation time

t0 Observation time

u Particle velocity

v Velocity vector with components u, v and w in projections to the Cartesian axes x, y and z

v Specific volume

Vm Mass flow rate

W Rate of chemical reaction rate, or power (Watt)

We Weber number

x, y, z Cartesian coordinates

Greek Symbols

b Coefficient of bulk expansion

g Ratio of the specific heat at constant pressure to the specific heat at

constant volume (the adiabatic index)

d Boundary layer thickness

dT Thermal boundary layer thickness

y Dimensionless temperature

L Angle between the axis of Earth rotation and the direction of fluid motion

l Mean free path

m Viscosity

n Kinematic viscosity

r Density

s Surface tension

xviii Nomenclature

t Time

tr Relaxation time

t0 Observation time

f Dissipation function

o Angular velocity

oe Angular velocity of Earth’s rotation

Subscripts

f Fluid

v Vapor

w Wall

1 Undisturbed fluid at infinity

Chapter 3:

a Thermal diffusivity

D Diffusivity


h Thickness of liquid layer

J Total momentum flux in jet

j Diffusion flux

P Pressure

Pr Prandtl number

Q Source strenght

r Radial coordinate

r; y; ’ Spherical coordinates

S Surface or surface area

Sc Schmidt number

t Time

U Plate or flow velocity in x directionu Fluid velocity

v Velocity vector with components vr ; vy; and v’ in spherical coordinate system

Greek Symbols

a Thermal diffusivity, exponent

G Strength of an infinitely thin vortex line

d Thickness of the boundary layer

� Dimensionless variable

# Dimensionless temperature


r Density

s Surface tension

t Shear stress

’ Polar angle, dimensionless function

O Vorticity component normal to the flow plane, or angular velocity

Nomenclature xix

Subscript

1 Undisturbed fluid

Chapter 4:

Ac Acceleration parameter

cd Drag coefficient

cl Lift coefficient

d Diameter

fd Drag force

fl Lift force


l Scale of turbulence, length of plate

P Pressure

Q Volumetric flow rate

R Radius

T Dimensionless turbulence intensity

u0 Root-mean square of turbulent fluctuations

u; v;w Velocity components

v Velocity vector

Fr Froud number

Re Reynolds number

We Weber number

Greak Symbols

a Angle

m Viscosity


r Density

s Surface tension

t Shear stress at the wall

g Dimensionless angular velocity

o Angular velocity

Subscripts

d Drag

l Lift

p Particle

1 Ambient

Chapter 5:

d Diameter

FI Inertial force

Fc Centrifugal force

xx Nomenclature

Fn Friction force

Fo Fouier number

k Dean number, or roughness

K Modified Dean number

l The entrance length of pipe

l� Characteristic length of pipe

P Pressure

DP Pressure drop

Po Poiseuille number

Q Volumetric flow rate

R Radius of curvature of a torus

Re Reynolds number

r0 Cross-sectional radius of a pipe

t Time

u; v;w Velocity components

u0 Initial velocity

umax Maximum velocity

v Velocity vector

w0 Mean velocity

x, y, z Cartesian coordinates

r; y; x Cylindrical coordinates

Greek Symbols

a Large semi-axis of an ellipse

b Small semi-axis of an ellipse

g Shear rate

d Ratio of pipe radius to its curvature

l Friction factor

m Viscosity

m0 Viscosity of Binham fluid

n Kinematic viscosityQBingham number

r Density

t Shear stress, geometric torsion

t0 Yield stress

Chapter 6:

h Enthalpy

Ix Kinematic momentum flux

Jx Momentum flux


Mx Total moment-of-momentum flux

P Pressure

Pr Prandtl number

Red Local Reynolds number

T Temperature

u Longitudinal velocity component

v Transversal velocity component

Nomenclature xxi

Greek Symbols

b Thermal expansion coefficient

d Jet thickness

m Viscosity


# Excessive temperature

r Density

Subscripts

1 Undisturbed fluid

m Jet axis

Chapter 7:

c Specific heat capacity, concentration

cP Specific heat at constant pressure

cv Specific heat at constant volume

D Diffusivity

d Diameter

E Pointwise energy release


H Channel height

h Heat transfer coefficient, rate of heat transfer, enthalpy

j Mechanical equivalent of heat


kB Boltzmann’s constant

l Turbulence scale

P Pressure

Q Strength of thermal source

q Heat flux

ql Latent heat of freezing

r Radius

T Temperature

Tu Turbulence intensity

v; u Velocityev0 Velocity fluctuation

Ec Eckert number

Gr Grashof number

M Mach number

Nu Nusselt number

Pe Peclet number

Pr Prandtl number

Ra Rayleigh number

Re Reynolds number

Reo Rotational Reynolds number

Sh Sherwood number

St Stephan number

xxii Nomenclature

Greek Symbols


b Thermal expansion coefficient

g Ratio of specific heat at constant pressure to specific heat at constant volume

(the adiabatic index)

d Delta function; boundary layer thickness

w Radiant thermal diffusivity

m Viscosity


r Density

Subscripts

en Entrance

f Front of thermal wave

P Pressure

T Thermal

W Wall

1 Undisturbed flow

Chapter 8:

A Cross-sectional area of a jet

C Concentration

d0 Nozzle diameter

dc Nozzle width

Fr Froude number

Gx Total mass flux

H Distance between the nozzle exit and the unperturbed liquid surface

hc Cavity depth

I0 The exit kinematic momentum flux

Jx Total momentum flux

l Characteristic length

Pr Prandtl number

P Pressure

Re Reynolds number

T Temperature

u,v Velocity components

um Centerline velocity

We Weber number

Greek Symbols

aT Eddy thermal diffusivity

d Jet half-width

� Dimensionless variable

m Viscosity

Nomenclature xxiii

mT Eddy viscosity


nT Eddy kinematic viscosity

r Density

s Surface tension

Subscripts

G Gas

L Liquid

Chapter 9:

c Reactant concentration

cP Specific heat

D Diffusivity

E Activation energy

h Enthalpy

k Chemical reaction constant; thermal conductivity

k0 Pre-exponential

Le Lewis number

lf Flame length

P Pressure

Pe Peclet number

Q1 Heat release

Q2 Heat losses

q Heat of reaction

R The universal gas constant

Re Reynolds number

T Temperature

uf Speed of combustion wave

u0 Speed of reactive mixture at the nozzle exit

W Rate of chemical reaction

Wj Rate of conversion of the j-th species

z Pre-exponential

Greek Symbols


d Frank-Kamenetskii parameter

m Viscosity


r Density

tk Characteristic kinetic time

tD Characteristic diffusion time

O Stoichiometric oxidizer-to-fuel mass ratio

xxiv Nomenclature

Subscripts

f Fuel

o Oxidizer

m Maximum; axis

0 Initial state

� Gas-liquid interface

Nomenclature xxv

Chapter 1

The Overview and Scope of the Book

The present book deals with the concepts and methods of the dimensional analysis

and their applications to various thermohydrodynamic phenomena in continuous

media. A comprehensive exposition of the results of systematic analysis of a

number of important problems in this area in the framework of the Pi-theorem is

given in nine chapters. In Chap. 2 the basics of the dimensional analysis are

discussed. In particular, the principle of dimensional homogeneity and non-

dimensionalization of the mass, momentum, energy and diffusion equations and

the corresponding initial and boundary conditions are described in this chapter. This

is complemented by the introduction of several dimensionless groups and similarity

criteria characteristic of hydrodynamic and heat and mass transfer problems. The

Buckingham Pi-theorem is also formulated in Chap. 2.

In Chap. 3 the Pi-theorem is used to establish self-similarity if it is admitted by a

particular problem and reduce the corresponding partial differential equations to the

ordinary ones. This approach to the search of self-similarity is illustrated with a

number of generic situations corresponding to the Stokes, Blasius, Landau, von

Karman, Yarin-Wess and Huppert hydrodynamic problems and the Pohlhausen and

Levich heat and mass transfer problems.

Chapter 4 deals with the drag force acting on a body moving in viscous fluid. The

attention is focused on drag experienced by spherical particles at low, moderate and

high Reynolds numbers. Such additional effects on the drag force as particle

rotation, free stream turbulence and particle-fluid temperature difference are also

analyzed. Then, some problems related to sedimentation are considered in the

framework of the dimensional analysis. Finally, the Landau-Levich withdrawal

problem on the thickness of thin liquid film on a vertical plate in dip coating process

is tackled. As before, the consideration is based on the Pi-theorem.

Chapter 5 is devoted to laminar channel and pipe flows. In this chapter the Pi-

theorem is applied to study stationary flows of Newtonian and non-Newtonian

fluids in straight smooth and rough pipes, as well as in curved channels and

pipes. In addition, some transient flows of Newtonian fluids are considered.

L.P. Yarin, The Pi-Theorem, Experimental Fluid Mechanics,

DOI 10.1007/978-3-642-19565-5_1, # Springer-Verlag Berlin Heidelberg 2012

1

The application of the Pi-theorem to the laminar submerged viscous jets is

discussed in Chap. 6. These include jets propagating in an infinite space, as well

as wall jets and wakes of solid bodies.

Chapter 7 deals with the heat and mass transfer phenomena. The results

presented in this chapter are related to the application of the Pi-theorem to conduc-

tive heat transfer in media with constant and temperature-dependent thermal con-

ductivity, convective heat and mass transfer in forced, natural and mixed

convection. The Pi-theorem is also used for the analysis of heat and mass transfer

associated with hot particles immersed in fluid flow, channel and pipe flows, high

speed gas flows, as well as flows with phase changes.

Chapter 8 is devoted to turbulence. Here the Pi-theorem is used to study

problems related to a decay of the uniform and isotropic turbulence, turbulent

near-wall flows and submerged and wall turbulent jets. The results are used to

interpret the wide range experimental data.

Chapter 9 is related with the application of the Pi-theorem to combustion

processes. A number of important problems of the combustion theory are consid-

ered in this chapter. These include the thermal explosion, propagation of combus-

tion waves and aerodynamics of gas torches.

2 1 The Overview and Scope of the Book

Chapter 2

Basics of the Dimensional Analysis

2.1 Preliminary Remarks

In this introductory chapter some basic ideas of the dimensional analysis are

outlined using a number of the instructive examples. They illustrate the applications

of the Pi-theorem in the field of hydrodynamics and heat and mass transfer.

The systems of units and dimensional and dimensionless quantities, as well as

the principle of dimensional homogeneity are discussed in Sect. 2.2. Section 2.3

deals with non-dimensionalization of the mass and momentum balance equations,

as well as the energy and diffusion equations. In Sect. 2.4 the dimensionless

groups characteristic of hydrodynamic and heat and mass transfer phenomena are

presented. Here the physical meaning of several dimensionless groups and simi-

larity criteria is discussed, In addition, similitude and modeling characteristic of the

experimental investigations of thermohydrodynamic processes are considered.

The Pi-theorem is formulated in Sect. 2.5.

2.2 Basic Definitions

2.2.1 Dimensional and Dimensionless Parameters

Momentum, heat and mass transfer in continuous media occur in processes

characterized by the interaction and coupling of the effects of hydrodynamic and

thermal nature. The intensity of these interactions and coupling is determined by

the magnitudes of physical quantities involved which characterize the physical

properties of the medium, its state, motion and interactions with the surrounding

boundaries and penetrating fields. The magnitudes of these quantities are deter-

mined experimentally by comparing the readings of the measuring devices

with some chosen scales, which are taken as units of the measured characteristics,



3

e.g. length, mass, time, etc. For example, an actual pipe diameter, fluid velocity or

temperature are expressed as

d ¼ nL�; v ¼ mV�; T ¼ kT� (2.1)

where n; m and k are some numbers, whereas L�;V� and T� are units of length,

velocity and temperature, respectively.

The quantities which characterize flow and heat and mass transfer of fluids

are related to each other by certain expressions based on the laws of nature. For

example, the volumetric flow rate Qv of viscous fluid through a round pipe of radius

r, and the drag force Fd acting on a small spherical particle slowly moving with

constant velocity in viscous fluid are expressed by the Poiseuille and Stokes laws

Qv ¼ pr4DP8ml

(2.2)

Fd ¼ 6pmur (2.3)

In (2.2) and (2.3) DP is the pressure drop on a length l; m is the fluid viscosity,

and u is the particle velocity. Equations 2.2 and 2.3 show that units of the

volumetric flow rate Qv and drag force Fd can be expressed as some combinations

of the units of length, velocity, viscosity and pressure drop. In particular, the unit of

r coincides with the unit of length L, of u is expressed through the units of length

and time as LT�1, the unit of m½ � ¼ L�1MT�1 in addition involves the unit of mass,

as well as the unit of the pressure drop DP½ � ¼ L�1MT�2 (cf. Table 2.1). Here and

hereinafter symbol A½ � denotes units of a dimensional quantity A.

It is emphasized that the units of numerous physical quantities can be expressed

via a few fundamental units. For example, we have just seen that the units of

volumetric flow rate and drag force are expressed via units of length, mass and time

only, as Qv½ � ¼ L3T�1; and Fd½ � ¼ LMT�2. A detailed information the units of

measurable quantities is available in the book by Ipsen (1960). The possibility to

express units of any physical quantities as a combination of some fundamental units

allows subdividing all physical quantities into two characteristic groups, namely (1)

primary or fundamental quantities, and (2) derivative (secondary or dependent)

ones. The set of the fundamental units of measurements that is sufficient for

expressing the other measurement quantities of a certain class of phenomena is

called the system of units. Historically, different systems of units were applied to

physical phenomena (Table 2.2).

In the present book we will use mainly the International System of Units

(Table 2.3).

In this system of units (hereinafter called SI Units) an amount of a substance is

measured with a special unit- mole (mol). Also, two additional dimensionless units:

one for a plane angle- radian (rad), and another one for a solid angle- steradian (sr),

are used. A detailed description of the SI Units can be found in the books of

Blackman (1969) and Ramaswamy and Rao (1971).

4 2 Basics of the Dimensional Analysis

The numerical values of the physical quantities expressed through fundamental

units depend on the scales of arbitrarily chosen for the latter in any given system of

units. For example, the velocity magnitude of a solid body moving in fluid, which is

1 m/s in SI units is 100 cm/s in the Gaussian CGS (centimeter, gram, second)

System of Units. The physical quantities whose numerical values depend on the

Table 2.1 Physical

quantitiesQuantity Dimensions Derived units

A. (Mechanical quantities)

Acceleration LT�2 m�s�2

Action ML2T�1 kg�m2� s

�1

Angle (plane) 1 rad:

Angle (solid) 1 sterad:

Angular acceleration T�2 rad�s�2

Angular momentum ML2T�1 kg�m2� s

�1

Area L2 m2

Curvature L�1 m�1

Surface tension MT�2 kg�s�2

Density ML�3 kg�m�3

Elastic modulus ML�1T�2 kg�m�1� s�2

Energy (work) ML2T2 J

Force MLT�2 N

Frequency T�1 s�1

Kinematic viscosity L2T�1 m2� s

�1

Mass M kg

Momentum MLT�1 kg�m�s�1

Power ML2T�3 W

Pressure ML�1T�2 N�m�2

Time T s

Velocity LT�1 m�s�1

Volume L3 m3

B. (Thermal quantities)

Enthalpy ML2T2 J

Entropy ML2T2y�1 J�K�1

Gas constant L2T�1y�1 J�kg��1K�1

Heat capacity per unit mass L2T�2y�1 J�kg�1� K�1

Heat capacity per unit volume ML�1T�2y�1 J�m��3K�1

Internal energy ML2T2 J

Latent heat of phase change L2T�2 J�kg�1

Quantity of heat ML2T�2 J

Temperature y K

Temperature gradient L�1y K�m�1

Thermal conductivity MT�3Ly�1 W�m�1� K�1

Thermal diffusivity L2T�1 m2� s

�1

Heat transfer coefficient MT3y�1 W�m2� K

�1

2.2 Basic Definitions 5

fundamental units are called dimensional. For such quantities, units are derivative

and are expressed through the fundamental unites according to the physical

expressions involved. For example, units of the gravity force Fg ¼ mg are

expressed through the fundamental units bearing in mind the previous expression

and the fact that m½ � ¼ M; and g½ � ¼ LT�2 as

Fg

� � ¼ LMT�2 (2.4)

In fact, units of any physical quantity can be expressed through a power law1

A½ � ¼ La1Ma2Ta3 (2.5)

where the exponents ai are found by using the principle of dimensional

homogeneity.

The quantities whose numerical values are independent of the chosen units of

measurements are called dimensionless. For example, the relative length of a pipe

l ¼ ld (where l and d are the length and diameter of the pipe, respectively) is

dimensionless. Formally this means that l� � ¼ 1:

In the general case, physical quantities can be characterized by their magnitude

and direction. Such quantities as, for example, temperature and concentration are

scalar and are characterized only by their magnitudes, whereas such quantities as

velocity and force are vectors and are characterized by their magnitudes and

directions. Vectors can also be characterized by introducing a so-called vector

length L (Williams 1892). Projections of the vector length L on, say, the axes of

Table 2.2 Systems of units

Quantity

Absolute Technical

CGS MKS FPS CGS MKS FPS

Mass Gram Kilogram Pound 9.81 g 9.81 kg Slug

Force Dyne Newton Poundal Gram-force Kilogram-force Pound-force

Length Centimeter Meter Foot Santimeter Meter Foot

Time Second Second Second Second Second Second

Table 2.3 International system of units-SI

Quantity Units Abbreviation

Mass Kilogram kg

Length Meter m

Time Second s

Temperature Kelvin K

Electric current Ampere A

Luminous intensity Candela cd

1A demonstration of this statement can be found in Sedov (1993).


a Cartesian coordinate system x; y and z are denoted as Lx; Ly and Lz, respectively.A number of instructive examples of application of vector length for studying different

problems of applied mechanics are presented in the monographs by Huntley (1967)

and Douglas (1969). The application of the idea of vector length in studying of

drag and heat transfer at a flat plate subjected to a uniform flow of the incompressible

fluid is discussed by Barenblatt (1996) and Madrid and Alhama (2005).

The expansion of a number of the fundamental units allows a significant

improvement of the results of the dimensional analysis. For this aim it is useful to

consider different properties the mass: (1) mass as the quantity of matter Mm, and

(2) mass as the quantity of the inertia Mi. Similarly, using projections of a vector Lon the Cartesian coordinate axes as the fundamental units it is possible to express

the units of such derivative (secondary) quantities as volume V and velocity vector

v as V½ � ¼ LxLyLz and u½ � ¼ LxT�1; v½ � ¼ LyT

�1; and w½ � ¼ LzT�1 where u,v and

w denote the projections of v on the coordinate axes as is traditionally done in fluid

mechanics. It is emphasized that using two different quantities of mass and

projections of a vector allows one to reveal more clearly the physical meaning

of the corresponding quantities. For example, the dimensions of work W in a

rectilinear motion and torque T in rotation system of units LMT are the same

L2MT�2;whereas in the system of unitsLxLyLzMT they are different, namely

W½ � ¼ L2xMT�2; whereas T½ � ¼ LxLyMT�2:

2.2.2 The Principle of Dimensional Homogeneity

Principle of dimensional homogeneity expresses the key requirements to a structure

of any meaningful algebraic and differential equations describing physical phe-

nomena, namely: all terms of these equations must to have the same dimensions.

To illustrate this principle, we consider first the expression for the drag force acting

on a spherical particle slowly moving in highly viscous fluid. The Stokes formula

describing Fd reads

Fd ¼ 6pmur (2.6)

Here Fd½ � ¼ LMT�2 is the drag force, m½ � ¼ L�1MT�1 is the viscosity of the

fluid, u½ � ¼ LT�1 and r½ � ¼ L are the particle velocity and its radius, respectively.

It is easy to see that (2.6) satisfies the principle dimensional homogeneity. Indeed,

substitution of the corresponding dimensions to the left hand side and the right hand

side of (2.6) results in the following identity

LMT�2 ¼ ðL�1MT�1ÞðLT�1ÞðLÞ ¼ LMT�2 (2.7)

As a second example, we consider the Navier–Stokes and continuity equations.

For flows of incompressible fluids they read


@v

@tþ ðv � rÞv ¼ � 1

rrPþ nr2v (2.8)

r � v ¼ 0 (2.9)

where v ¼ LT�1½ � is the velocity vector, r½ � ¼ L�3M, n½ � ¼ L2T�1 and P½ � ¼L�1MT�2 are the density, kinematic viscosity n and pressure, respectively.

It is seen that all the terms in (2.8) have dimensions LT�2 and in (2.9) have

dimensions T�1.

There are a number of important applications of the principle of the dimensional

homogeneity. For example, it can be used for correcting errors in formulas or

equations, which is advisable to students. Take the expression for the volumetric

rate of incompressible fluid through a round pipe of radius r as

Qv ¼ pr2

8mDPl

� �(2.10)

where Qv is the volumetric flow rate, DP is the pressure drop over an arbitrary

section of the pipe length of length l.The dimension of the term on the left hand side in (2.10) is L3T�1, whereas of the

one on the right hand side of this equation is LT�1. Thus, (2.10) does not satisfy

the principle of dimensional homogeneity. In order to find the correct form of the

dependence of the volumetric flow rate on the governing parameters, we present

(2.10) as follows

Qv ¼ p8ra1ma2

DPl

� �a3

(2.11)

where ai are unknown exponents.

Bearing in mind the dimensions of Qv; r; m and DPl

� �, we arrive at the following

system of algebraical equations for the exponents ai

a1 � a2 � 2a3 ¼ 3

a2 þ a3 ¼ 0

�a2 � 2a3 ¼ �1 (2.12)

From (2.12) it follows that the exponents ai are equal a1 ¼ 4; a2 ¼ �1;and a3 ¼ 1. Then, the correct form of (2.10) reads as

Qv ¼ pr4

8mDPl

� �(2.13)


The third example concerns the application the principle of dimensional homo-

geneity to determine the dimensionless groups from a set of dimensional

parameters. Consider a set of dimensional parameters

a1; a2 � � � ak; akþ1 � � � an (2.14)

Assume that k parameters have independent dimensions. Accordingly, the

dimensions of the other n� k parameters can be expressed as

akþ1½ � ¼ a1½ �a01 � � � � � ak½ �a

0k

� � � � � � � � � � � � � � � � � � � � � � ��an½ � ¼ a1½ �an�k

1 � � � � ak½ �an�kk (2.15)

Therefore, the ratios

akþ1

aa01

1 � � � aa0k

k

¼ P1

� � � � � � � � � � � � � � � � � � ��an

an�k1 � � � an�k

k

¼ Pn�k (2.16)

are dimensionless. Requiring that the dimensions of the numerator and denominator

in the ratios (2.16) will be the same, we arrive at the system of algebraical equations

for the unknown exponents.

In conclusion, we give one more instructive example of the application of the

principle of dimensional homogeneity for the description of the equation of state of

perfect gas. The general form of the equation of state reads (Kestin v.1 (1966) and

v.2 (1968)):

FðP; vs; TÞ ¼ 0 (2.17)

where P; vsand T are the pressure, specific volume and temperature, respectively.

Equation 2.17 can be solved (at least in principle), with respect to any one of the

three variables involved. In particular, it can be written as

P ¼ f ðvs; TÞ (2.18)

The set of the governing parameters involved in (2.18) is incomplete since the

dimension of pressure P½ � ¼ L�1MT�2 cannot be expressed in the form of any

combination of dimensions of specific volume vs½ � ¼ L3M�1 and temperature

T½ � ¼ y. Therefore, the function f on right hand side in (2.18) must include some

dimensional constant c


P ¼ f ðc; vs; TÞ (2.19)

It is reasonable to choose as such a constant the gas constant R that account

for the physical nature of the gas, but does not depend on its specific volume,

pressure and temperature. Assuming that c ¼ R g= (g is a dimensionless constant),

we write the dimension of this constant as c½ � ¼ L2T�2y�1: All the parameters

in (2.19) have independent dimensions. Then, according to the Pi-theorem (see

Sect. 2.5), (2.19) takes the form

P ¼ g1ca1va2s T

a3 (2.20)

where g1 is a dimensionless constant.

Using the principle of the dimensional homogeneity, we find the values of

the exponents ai as a1 ¼ 1; a2 ¼ �1; a3 ¼ 1: Assuming g ¼ g1, we arrive at the

Clapeyron equation

P ¼ RrT (2.21)

The equation of state of perfect gas can be also derived directly by applying the

Pi-theorem to solve the problems of the kinetic theory and accounting for the fact

pressure of perfect gas results from atom (molecule) impacts onto a solid wall.2

Considering perfect gas as an ensemble of rigid spherical atoms (or molecules)

moving chaotically in the space, we can assume that pressure of such gas is deter-

mined by atom (or molecule) mass m, their number per unit volume N and the

average velocity squared <v2>

P ¼ f ðm;N; <v2>Þ (2.22)

The dimensions of P and the governing parameters m;N and <v2> are

P½ � ¼ L�1MT�2; m½ � ¼ M; N½ � ¼ L�3; <v2>� � ¼ L2T�2 (2.23)

All the governing parameters have independent dimensions. Therefore, the

difference between the number of the governing parameters n and the number of

the parameters with independent dimensions k equals zero. In this case the pressurecan be expressed as Sedov (1993);

P ¼ gma1Na2<v2>a3 (2.24)

where g is a dimensionless constant.

2 This idea was expressed first by D. Bernoulli in 1727 who wrote that pressure of perfect gas is

related to molecule velocities squared.


Using the principle of dimensional homogeneity, we find the values of the

exponents in (2.24) as a1 ¼ a2 ¼ a3 ¼ 1: Then, (2.24) takes the form

P ¼ gmN<v2> (2.25)

Bearing in mind that m<v2> is directly proportional kBT (m<v2> ¼g1kBT; where g1 is a dimensionless constant), we arrive at the following equation

P ¼ ekBTN (2.26)

Here e ¼ gg1 is a dimensionless constant, kB½ � ¼ L2MT�2y�1 is Boltzmann’s

constant, T½ � ¼ y is the absolute temperature.

Applying (2.26) to a unit mole of a perfect gas, we can write the known

thermodynamic relations as

N ¼ Nm; kB ¼ mRNm

; mvs ¼ constant (2.27)

Here Nm is the Avogadro number, m is the molecular mass, vs is the specific

volume, and R½ � ¼ L2T�2y�1 is the gas constant. Then, (2.27) takes the form

P ¼ rRT (2.28)

Summarizing, we see that the pressure of perfect gas is directly proportional to

the product of the gas density, gas constant and the absolute temperature and does

not depend on the mass of individual atoms (molecules). Note that (2.28) can be

obtained directly from the functional equation P ¼ f ðm;N; T; kBÞ(Bridgman 1922).

2.3 Non-Dimensionalization of the Governing Equations

It is beneficial in the analysis complex thermohydrodynamic phenomena to trans-

form the system of mass, momentum, energy and species balance equations into a

dimensionless form. The motivation for such transformation comes from two

reasons. The first reason is related with the generalization of the results of theoreti-

cal and experimental investigations of hydrodynamics and heat and mass transfer in

laminar and turbulent flows by presentation the data of numerical calculation and

measurements in the form of dependences between dimensionless parameters.

The second reason is related to the problem of modeling thermohydrodynamic

processes by using similarity criteria that determine the actual conditions of

the problem. The procedure of non-dimensionalization of the continuity (mass

balance), momentum, energy and species balance equations is illustrated below

by transforming the following model equation

2.3 Non-Dimensionalization of the Governing Equations 11

Xnj¼1

AðiÞj ¼ 0 (2.29)

where AðiÞj includes differential operators, some independent variables, as well as

constants; superscript i refers to the momentum ði ¼ 1Þ; energy ði ¼ 2Þ; species

ði ¼ 3Þ and continuity ði ¼ 4Þ equations, n is the total number of terms in a given

equation.

The terms in (2.29) account for different factors that affect the velocity, temper-

ature and species fields: the inertia features of fluid, viscous friction, conductive and

convective heat transfer, etc. These terms are dimensional. The dimension of AðiÞj in

the system of units LMTy is

AðiÞj

h i¼ La

ðiÞj MbðiÞ

j TgðiÞj ye

ðiÞj (2.30)

where the values of the exponents a; b; g and e are determined by the magnitude of

i and j; all the terms that correspond to a given i have the same dimension:

AðiÞ1

h i¼ A

ðiÞ2

h i¼ � � � AðiÞ

j

h i¼ � � � AðiÞ

n

h i(2.31)

The variables and constants included in (2.29) may be rendered dimensionless

by using some characteristic scales of the density r�½ � ¼ L�3M; velocity v�½ � ¼LT�1; length l�½ � ¼ L; time t�½ � ¼ T, etc. Then, the dimensionless variables and

constants of the problem are expressed as

r ¼ rr�

; v ¼ v

v�;T ¼ T

T�; c ¼ c

c�; t ¼ t

t�;P ¼ P

P�; m ¼ m

m�; k ¼ k

k�;D

¼ D

D�; g ¼ g

g�(2.32)

where the asterisks denote the characteristic scales, and the dimensionless

parameters are denoted by bars. In addition, k� ¼ LMT�3y�1� �

;D� ¼ L2T�1½ �;and g� ¼ LT�2½ � are the characteristic scales of thermal conductivity, diffusivity

and gravity acceleration, respectively.

Taking into account (2.32), we can present all terms of (2.29) as follows

AðiÞj ¼ A

ðiÞj� A

ðiÞj (2.33)

where AðiÞj� is the corresponding dimensional multiplier comprised of the character-

istic scales, AðiÞj ¼ A

ðiÞj =A

ðiÞj� is the dimensionless form of the jth term in (2.29).

The exact form of the multipliers AðiÞj� is determined by the actual structure of the

terms AðiÞj . For example, the multiplier of the first term of the momentum balance

equation is found from


AðiÞ1 ¼ r

@v

@t¼ r�v�

t�

@ðv=v�Þ@ðt=t�Þ ¼ A

ðiÞ1�A

ðiÞ1 (2.34)

where AðiÞ1� ¼ r�v�

t�, A

ðiÞ1 ¼ @v

@t.

The substitution of the expression (2.33) into (2.29) yields

Xnj¼1

AðiÞj� A

ðiÞj ¼ 0 (2.35)

Dividing the left and right hand sides of (2.35) by a multiplier AðiÞk� ð1 � k � nÞ,

we arrive at the dimensionless form of the conservation equations

AðiÞk þ

Xk�1

j¼1

YðiÞj�

AðiÞj þ

Xnj¼kþ1

YðiÞj�

AjðiÞ

( )¼ 0 (2.36)

whereQðiÞ

j� ¼ Aj�=Ak� are the dimensionless groups.

To illustrate the general approach described above, we render dimensionless

the Navier–Stokes equations, the energy and species balance equations, as well

as the continuity equation. For incompressible fluids these equations read

r@v

@tþ r v � rð Þv ¼ �rPþ mr2vþ rg (2.37)

rcp@T

@tþ rcP v � rð ÞT ¼ kr2T þ f (2.38)

r@cx@t

þ r v � rð Þcx ¼ rDr2cx (2.39)

r � v ¼ 0 (2.40)

where r; v T; P and cx are the density, velocity vector, the temperature, pressure

and the concentration of the species x. In particular, let us use the Cartesian

coordinate system where vector v has components u; v and w in projections

to the x; y and z axes. In addition, m; k andD are the viscosity, thermal conductivity

and diffusivity which are assumed to be constant, g the magnitude of the gravity

acceleration g, f is the dissipation function f ¼ 2m ð@u=@xÞ2 þ ð@v=@yÞ2þh

ð@w=@zÞ2� þ mð@u=@yþ @v=@xÞ2 þ mð@v=@zþ @w=@yÞ2þ mð@w=@xþ @u=@zÞ2.The multipliers A

ðiÞj� in (2.37)–(2.40) are listed below

Að1Þ1� ¼ r�v�

t�;A

ð1Þ2� ¼ r�v2�

l�;A

ð1Þ3� ¼ P�

l�;A

ð1Þ4� ¼ r�g� (2.41)


Að2Þ1� ¼ r�cP�T�

t�; A

ð2Þ2� ¼ r�cP�v�T�

l�;A

ð2Þ3� ¼ k�T�

l�;A

ð2Þ4� ¼ m�v2�

l�

Að3Þ1� ¼ r�c�

t�;A

ð3Þ2� ¼ r�c�

l�;A

ð3Þ3� ¼ r�D�c�

l2�

Að4Þ1� ¼ v�

l�;A

ð4Þ2� ¼ v�

l�

Dividing the multipliers Að1Þj� by A

ð1Þ2� ; A

ð2Þj� by A

ð2Þ2� ; A

ð3Þj� by A

ð3Þ2� and A

ð4Þj� by A

ð4Þ2� ,

we arrive at the following system of dimensionless equations

St@v

@tþ v � rð Þv ¼ �EurPþ 1

Rer2vþ 1

Fr(2.42)

St@T

@tþ v � rð ÞT ¼ 1

Per2T þ Br

Ref (2.43)

St@cx@t

þ v � rð Þcx ¼ 1

Pedr2cx (2.44)

r � v ¼ 0 (2.45)

where St ¼ l�=v�t�; Eu ¼ P�=r�v2�; Re ¼ v�l�=n�; Pe ¼ v�l�=a�; Ped ¼ v�l�=D�,

Fr ¼ v2�=g�l�; Br ¼ m�v2�=k�T� are the Strouhal, Euler and Reynolds numbers,

as well as the thermal and diffusion Peclet numbers, and the Froude and Brinkman

numbers, respectively, n and a are the kinematic viscosity and thermal diffusivity,

and the dimensionless dissipation function f ¼ f= mðv�=l�Þ2h i

; v ¼ v v�= ;P ¼P rv2��

; T ¼ T T�= and cx ¼ c c�= are the dimensionless variables.

The non-dimensionalization of the initial and boundary conditions is similar to

the one described above. In that case each of the independent variables x; y; z and t,as well as the flow characteristics u; v; T and cx are also rendered dimensionless by

using some scales that have the same dimensions as the corresponding parameters.

For example, consider the non-dimensionalization of the initial and boundary

conditions for the following three problems of the theory of viscous fluid flows:

(1) steady flow in laminar boundary layer over a flat plate, (2) laminar flow about

a flat plate which instantaneous started to move in parallel to itself, and (3) sub-

merged laminar jet issued from a round nozzle.

In case (1), let the velocity and temperature of the undisturbed fluid far enough

from the plate be u1, T1, and the wall temperature be Tw ¼ const: Then, theboundary conditions read


x ¼ 0; 0 � y � 1; u ¼ u1; T ¼ T1 (2.46)

x> 0, y ¼ 0, u ¼ v ¼ 0; T ¼ Tw; y ! 1, u ! u1, T ! T1

Introducing as the scales of length some L, velocity u1 and temperature

Tw � T1, we rearrange (2.46) to the following dimensionless form3

x ¼ 0; 0 � y � 1 u ¼ 1; DT ¼ 1 (2.47)

x> 0, y ¼ 0 u ¼ v ¼ 0; DT ¼ 0; y ! 1 u ! 1; DT ! 1

where x ¼ x=L; y ¼ y=L; u ¼ u=u1; v ¼ v=u1; DT ¼ ðTw � TÞ=ðTw � T1Þ.The equation for the heat flux at the wall is used to introduce the heat transfer

coefficient h:

h Tw � T1ð Þ ¼ �k@T

@y

� �y¼0

(2.48)

Being rendered dimensionless, the heat transfer coefficient is expressed in the

following form

Nu ¼ @DT@y

� �y¼0

(2.49)

where Nu ¼ hL=k is the dimensionless heat transfer coefficient is called the Nusselt

number.

In case (2), the initial and boundary conditions of the problem on a plate starting

to move from rest with velocityU in the x-direction in contact with the viscous fluidread

t ¼ 0; 0 � y � 1 u ¼ 0 (2.50)

t> 0, y ¼ 0 u ¼ U; y ¼ 1, u ¼ 0

Since no time or length scales are given, we use as the characteristic time scale

t� ¼ n=U2 and as the characteristic length scale n=U. Then, (2.50) take the follow-ing dimensionless form

t ¼ 0; 0 � y � 1 u ¼ 0; t> 0; y ¼ 0 u ¼ 1; y ! 1 u ! 0 (2.51)

In case (3), the boundary conditions for a submerged laminar jet are

3 It is emphasized that in the problem on flow in the boundary layer over a semi-infinite plate,

a given characteristic scale L is absent. According to the self-similar Blasius solution of this

problem, the dimensionless coordinate y ¼ y=ðnx=u1Þ1=2 with ðnx=u1Þ1=2 playing the role of the

length scale (Sedov 1993).


x ¼ 0; 0 � y � r0; u ¼ u0; T ¼ T0; y> r0 u ¼ 0; T ¼ T1 (2.52)

x> 0; y ¼ 0,@u

@y¼ 0,

@T

@y¼ 0; y ! 1, u ! 0, T ! T1

where r0is the nozzle radius.The dimensionless form of the conditions (2.52) is

x ¼ 0; 0 � y � 1; u ¼ 1 DT ¼ 1; y>1; u ! 0; DT ! 0 (2.53)

x> 0, y ¼ 0,@u

@y¼ 0,

@DT@y

¼ 0; y ¼ 1, u ! 0, DT ! 0

where x ¼ x=r0; y ¼ y=r0; u ¼ u=u0; DT ¼ ðT1 � TÞ=ðT1 � T0Þ:At large enough distance from the jet origin at x=r0>> 1, it is possible to use the

integral conditionR10

u2ydy ¼ const; instead of the condition (2.52) at x ¼ 0. Note

that there is another way of rendering the system of fundamental equations of

hydrodynamics and heat and mass transfer theory dimensionless. It consists in

rendering dimensionless each quantity in these equations using for this aim the

scales of the density, velocity, temperature, etc. Requiring that the convective terms

of these equations do not contain any dimensional multipliers, it is not easy to arrive

at the equations identical to (2.42)–(2.45). To illustrate this approach to non-

dimensionalization of the mass, momentum, energy and species conservation

equations, consider, for example, the system of equations describing flows of

reactive gases

@r@t

þr � ðrvÞ ¼ 0 (2.54)

r@v

@tþ rðv � rÞv ¼ �rPþr � ðmrvÞ þ rg (2.55)

r@h

@tþ rðv � rÞh�r � ðkrTÞ ¼ qWk (2.56)

r@ck@t

þ rðv � rÞck �r � ðrDrckÞ ¼ �Wk (2.57)

P ¼ g� 1

grh (2.58)

where v is the velocity vector, r; P; h and T are the density, pressure, enthalpy

and temperature, ck ¼ rk=r is the relative concentration of the kth species,

r ¼ Srk; with rk being density of the kth species, Wkðck; TÞ and W are the chemi-

cal reaction rates, q is the heat of the overall reaction, and g ¼ cp=cvis the ratio of


specific heat at constant pressure to the one at constant volume (the adiabatic index).

Note that in the energy balance equation (2.56) the dissipation term is neglected.

Introducing dimensionless parameters as follows a ¼ aa�

(the asterisk denotes the

scale of a parameter a), we arrive at the following equations

r�t�

@r@t

þ r�v�L�

r � ðrvÞ ¼ 0 (2.59)

r�v�t�

@v

@tþ r�v�

L�rðv � rÞv ¼ �P�

L�rPþ m�v�

L2�r � ðmrvÞ þ r�g�rg (2.60)

r�h�t�

r@h

@tþ r�v�h�

L�rðv � rÞh� k�T�

L2�r � ðkrTÞ ¼ qWk:�Wk (2.61)

r�t�r@ck@t

þ r�v�L�

rðv � rÞck � r�D�L2�

r � ðrDrckÞ ¼ �Wk:��Wk (2.62)

P ¼ g� 1

gr�h�P�

rh (2.63)

where r�; v�; P�; T�; h� and L� are the scales of density, velocity, pressure,

temperature, enthalpy and length, respectively.

Requiring that the second terms on left hand sides in (2.59)–(2.62) do not contain

any dimensionless multipliers and also accounting for the fact that for perfect gas

r�h�=P� ¼ g=ðg� 1Þ, we obtain

St@r@t

þr � ðrvÞ ¼ 0 (2.64)

St@v

@tþ rðv � rÞv ¼ �EurPþ 1

Rer � ðmrvÞ þ 1

Frrg (2.65)

St@T

@tþ rðv � rÞT � 1

Per � ðkrTÞ ¼ Da3Wk (2.66)

St@ck@t

þ rðv � rÞck � 1

Pedr � ðrDrckÞ ¼ Da1Wk (2.67)

P ¼ rh (2.68)

where in addition to previously introduced Strouhal, Reynolds, Euler, the

thermal and diffusion Peclet numbers, and the Froude number, two Damkohler

numbers Da1 ¼ Wk:�L�=r�v�; and Da3 ¼ qWk:�L�=r�v�h� (defined according to

the Handbook of Chemistry and Physics,1968) appear.


2.4 Dimensionless Groups

2.4.1 Characteristics of Dimensionless Groups

As was shown in Sect. 2.3, the dimensionless momentum, energy and diffusion

equations contain a number of dimensionless groups, which represent themselves

some combinations of the physical properties of fluid, acting forces, heat fluxes, etc.

The physical meaning and number of these groups is determined by a specific

situation, as well as by a particular model used for description of the physical

phenomena characteristic of that situation (Table 2.4).4

Consider in detail some particular dimensionless groups. The Prandtl, Schmidt

and Lewis numbers belong to a subgroup of dimensional groups that incorporate

only quantities that account for the physical properties of fluid. They are expressed

as the following ratios (cf. Table 2.4)

Pr ¼ na; Sc ¼ n

D; Le ¼ a

D(2.69)

where n; a and D are the kinematic viscosity, thermal diffusivity and diffusivity,

respectively.

Consider, for example the Prandtl number. It represents itself the ratio of

kinematic viscosity to thermal diffusivity, i.e. of the characteristics of fluid respon-

sible for the intensity of momentum and heat transfer. Accordingly, the Prandtl

number can be considered as a parameter that characterizes the ratio of the extent of

propagation of the dynamic and thermal perturbations. Therefore, at very low

Prandtl numbers (for example, in flows of liquid metals), the thickness of the

thermal boundary layer dT is much larger than the thickness of the dynamical

one, d: In contrast, at Pr >> 1 (in flows of oils) the equality d>> dT is valid. The

Schmidt number is the diffusion analog of the Prandtl number. It determines the

ratio of the thicknesses of the dynamical and diffusion boundary layers.

The Reynolds number belongs to the subgroup of the dimensionless groups

which are ratios of the acting forces. It can be considered as the ratio of the inertia

force Fito the friction force Ff

4 Dimensionless groups can be also found directly by transformation of the functional equations of

a specific problem using the Pi-theorem (see Sect. 2.5). A detailed list of dimensionless groups

related to flows of incompressible and compressible fluids in adiabatic and diabatic conditions,

flows of non-Newtonian fluids and reactive mixtures can be found in Handbook of Chemistry and

Physics, 68th Edition, 1987–1988, CBC Inc. Boca Roton, Florida, and in Chart of Dimensionless

Numbers, OMEGA Technology Company. See also Lykov and Mikhailov (1963) and Kutateladze

(1986).


Table 2.4 Dimensionless groups

Name Symbol Definition Comparison ratio Field of use

Archimedes

number

Ar gL3rm2 ðr� rf Þ Gravity force to viscous

force

Motion of fluid due to

density

differences

(buoyancy)

Biot number Bi hLks

Convection heat transfer to

conduction heat transfer

Heat transfer

Bond

number

Bo rgL2

sGravitaty force to surface

tension

Motion of drops and

bubbles.

Atomization

Brinkman

number

Br mv2

kDTHeat dissipation to heat

transferred

Viscous flows

Capillary

number

Ca mvs Viscous force to surface

tension force

Two-phase flow.

Atomization.

Moving contact

lines

Damkohler

number

Da1Da3

WLVmqWL

rvcPDT

Chemical reaction rate to

bulk mass flow rate.

Heat released to

convected heat

Chemical reactions,

momentum, and

heat transfer

Darcy

number

Da2 vLD�

Inertia force to permeation

force

Flow in porous media

Dean

number

De vRrm

ffiffiffiRr

qCentrifugal force to inertial

force

Flow in curved

channels and

pipes

Deborah

number

De trt0

Relaxation time to the

characteristic

hydrodynamic time

Non-Newtonian

hydrodynamics.

Rheology

Eckert

number

Ec v21cPDT

Kinetic energy to thermal

energy

Compressible flows

Ekman

number

Ek m2roL2

�1=2 (Viscous force to Coriolis

force)1=2Rotating flows

Euler

number

Eu rv2

DPPressure drop to dynamic

pressure

Fluid friction in

conduits

Grashof

number

Gr r2gbL3DTm2

Buoyancy force to viscous

force

Natural convection

Jacob

number

Ja cPrfDTrrV

Heat transfer to heat of

evaporation

Boiling

Knudsen

number

Kn lL

Mean free path to

characteristic dimension

Rarefied gas flows

and flows in

micro- and nano-

capillaries

Kutateladze

number

K rvcPDT

Latent heat of phase change

to convective heat

transfer

Combined heat and

mass transfer in

evaporation

Lewis

number

Le krcPD

Thermal diffusivity to

diffusivity

Combined heat and

mass transfer

Mach

number

M vC Flow speed to local speed of

sound

Compressible flows

Nu hLk

Forced convection

(continued)

2.4 Dimensionless Groups 19

Re ¼ vL

n¼ rv2

m v L=ð Þ ¼rv2 L=

m v L2=ð Þ (2.70)

where r; m and L are the density, viscosity and the characteristic length.

The dimensions of the numerator and denominator in right hand side ratio in

(2.70) are rv2 L=½ � ¼ m v L2�� ¼ L�2MT�2, i.e. the same as the dimensions of the

terms r @v=@tþ v � rð Þv½ � and mr2v accounting for the inertia and viscous forces

in the momentum balance equation. The terms rv2=L and mv=L2 can be treated as

the specific inertia and viscous forces fi ¼ Fi V= and ff ¼ Ff V= , respectively, with

the dimensions Fi½ � ¼ LMT�2, Ff

� � ¼ LMT�2, and V½ � ¼ L3.At small Reynolds numbers when the influence of viscosity is dominant, any

chance perturbations of the flow field decay very quickly. At large Re such

perturbations increase and result in laminar-turbulent transition. Therefore, the

Table 2.4 (continued)

Name Symbol Definition Comparison ratio Field of use

Nusselt

number

Total heat transfer to

conductive heat transfer

Peclet

number

Pe Lrvcpk

Bulk heat transfer to

conductive heat transfer

Forced convection

Prandtl

number

Pr mcPk Momentum diffusivity to

thermal diffusivity

Heat transfer in fluid

flows

Rayleigh

number

Ra gbL3r2cPmk

Thermal expansion to

thermal diffusivity and

viscosity

Natural convection

Richardson

number

Ri � gr

@P@Lh

�@v@Lh

�w

.Gravity force to the inertia

force

Stratified flow of

multilayer

systems

Rossby

number

Ro voL sinL The inertia force to Coriolis

force

Geophysical flows.

Effect of earth’s

rotation on flow in

pipes

Schmidt

number

Sc mrD Kinematic viscosity to

molecular diffusivity

Diffusion in flow

Senenov

number

Se hmK

Intensity of heat transfer to

intensity of chemical

reaction

Reaction kinetics.

Convective heat

transfer.

Sherwood

number

Sh hmLD

Mass diffusivity to

molecular diffusitivy

Mass transfer

Stenton

number

St hrvcP

Heat transferred to thermal

capacity of fluid

Forced convection

Strouhal

number

St fLv

Time scale of flow to

oscillation period

Unsteady flow.

Vortex shedding

Taylor

number

Ta 2oL2rm

�2 (Coriolis force to viscous

force)2Effect of rotation on

natural convection

Weber

number

We v2rLs

The dynamic pressure to

capillary pressure

Bubble formation,

drop impact


Reynolds number is sensitive indicator of flow regimes. For example, in flows of an

incompressible fluid in a smooth pipe, three kinds of flow regime can be realized

depending on the value of the Reynolds number: (1) laminar (Re � 2300), transi-

tional (2300 � Re � 3500), and developed turbulent (Re > 3500).

The Peclet number is an example of a dimensionless group that is a ratio of heat

fluxes of different nature. It reads

Pe ¼ vL

a¼ rvcPDT

k DTL

� � (2.71)

where k and cP are the thermal conductivity and specific heat at constant pressure,

DT is the characteristic temperature difference.

The Peclet number is the ratio of the heat flux due to convection to the heat flux

due to conduction. It can be considered as a measure of the intensity of molar to

molecular mechanisms of heat transfer.

We mention also the Damkohler number that characterize the conditions of

chemical reaction which proceeds in a reactive mixture, i.e. in the process

accompanied by consumption of the initial reactants, formation of the combustion

products, as well as an intensive heat release. Under these conditions the evolution

of the temperature and concentration fields is determined by two factors: (1)

hydrodynamics of the flow of reacting mixture, and (2) the rate of chemical

reaction. The contribution of each of these factors can be estimated by the ratio

of the characteristic hydrodynamic time th � W�1 to the chemical reaction time

tr � V�1v i.e. by the Damkohler number

Da1 ¼ thtr

(2.72)

If the Damkohler number is much less than unity, the influence of the chemical

reaction on the temperature (concentration) field is negligible. At large values of

Da1 the effect of the chemical reaction and its heat release is dominant.

2.4.2 Similarity

Before closing the brief comments on the dimensionless groups, we outline how

such groups are used in modeling of hydrodynamic and thermal phenomena. For

this aim, we turn back to (2.64)–(2.68) that describe the mass, momentum, heat and

species transfer in flows of incompressible fluids with constant physical properties.

These equations contain eight dimensionless groups, namely, St; Re; Pe; Ped;Eu; Fr; Da1 andDa3: If the initial and boundary conditions of a particular problemdo not contain any additional dimensionless groups (as, for example, the conditions

y ¼ 0 v ¼ 0; T ¼ 0; ck ¼ 0, y ! 1 v ¼ 1; T ¼ 1; ck ¼ 1), the velocity,

2.4 Dimensionless Groups 21

temperature and concentration fields determined by (2.64)–(2.68) can be expressed

as follows

v ¼ fvðx; y; z; St;Re;Eu;FrÞ (2.73)

T ¼ ftðx; y; z; St;Pe;Da1Þ (2.74)

ck ¼ fcðx; y; z; St;Ped;Da3Þ (2.75)

In (2.73) and (2.75)T ¼ ðT�TwÞ=ðT1�TwÞ; and ck ¼ ðck� ck;wÞ=ðck;1� ck;wÞ;subscripts w; and1 correspond to the values at the wall and in undisturbed fluid.

The expressions (2.73)–(2.75) are universal in a sense that the fields of dimen-

sionless velocity, temperature and concentration determined by these expressions

do not depend on the absolute values of the characteristic scales. That means that in

geometrically similar systems (for example, cylindrical pipes of different diameter)

values of dimensionless velocity, temperature and concentration at any similar

point (with x1 ¼ x2 ¼ � � � ¼ xi; y1 ¼ y2 ¼ � � � ¼ yi; z1 ¼ z2 ¼ � � � ¼ zi) are the

same if the values of the corresponding dimensionless groups are the same. Thus,

the necessary conditions of the dynamic and thermal similarity in geometrically

similar systems consist in equality of dimensionless groups (similarity numbers)

relevant for the compared systems, i.e.

St ¼ idem; Re ¼ idem; Eu ¼ idem; Fr ¼ idem; Pe ¼ idem;

Ped ¼ idem; Da1 ¼ idem; Da3 ¼ idem(2.76)

for a considered class of flows. It is emphasized that in geometrically similar

systems the boundary conditions should also be identical in such comparisons.

The conditions (2.76) allow modeling the momentum, heat and mass transfer

processes in nature and technical applications by using the results of the

experiments with miniature geometrically similar models. Note that among the

totality of similarity numbers it is possible to select a family of dimensionless

groups that contain combinations of only scales of the considered flow family and

the physical parameters of a medium involved in a situation under consideration.

Such similarity numbers are called similarity criteria (Loitsyanskii 1966). A num-

ber of similarity criteria can be less than the number of similarity numbers. For

example, hydraulic resistance of cylindrical pipes with fully developed incompress-

ible viscous fluid flow with a given throughput is characterized by two similarly

numbers, namely, the Reynolds and Euler numbers. The first of them Re ¼ v0d=n isthe similarity criterion, since it contains known parameters: the average velocity of

fluid v0, its viscosity n and pipe diameter d. In contrast, the Euler number is not

a similarity criterion, since it contains an unknown pressure drop which has to be

found by solving the problem or measured experimentally (Loitsyanskii 1966).


2.5 The Pi-Theorem

2.5.1 General Remarks

This whole book is devoted to the Buckingham Pi-theorem (1914), which is widely

used in a number of important problems of modern physics and, in particular,

mechanics. The proof of this theorem, as well as numerous instructive examples

of its applications for the analysis of various scientific and technical problems are

contained in the monographs by Bridgman (1922), Sedov (1993), Spurk (1992)

and Barenblatt (1987). Referring the readers to these works, we restrict our consid-

eration by applications of the Pi-theorem to problems of hydrodynamics and the

heat and mass transfer only.

The study of thermohydrodynamical processes in continuous media consists in

establishing the relations between some characteristic quantities corresponding to

a particular phenomenon and different parameters accounting for the physical

properties of the matter, its motion and interaction with the surrounding medium.

Such relations can be expressed by the following functional equation

a ¼ f ða1; a2 � � � anÞ (2.77)

where a is the unknown quantities (for example, velocity, temperature, heat or mass

fluxes, etc.), a1; a2; � � �an are the governing parameters (the characteristics of an

undisturbed fluid, physical constants, time and coordinates of a considered point).

Equation 2.77 indicates only the existence of some relation between the unknown

quantities and the governing parameters. However, it does not express any particular

form of such relation. There are two approaches to determine an exact form of

a relation of the type of (2.77): one is experimental, and the other one theoretical.

The first approach is based on generalization of the results of measurements of

unknown quantities a while varying the values of the governing parameters

a1; a2; � � �an: The second, theoretical, approach relies on the analytical or numerical

solutions of the mass, momentum, energy and species balance equations. In both

cases the establishment of a particular exact form of (2.77) does not entail significant

difficulties while studying the simplest one-dimensional problemswhen (2.77) takes

the form a ¼ f ða1Þ: On the contrary, a comprehensive experimental and theoretical

analysis of amultiparametric equation a ¼ f ða1; a2 � � � anÞ is extremely complicated

and often represents itself an insoluble problem. The latter can be illustrated by the

problem on a drag force acting on a body moving with a constant velocity in an

infinite bulk of incompressible viscous fluid. In this case the drag force Fd acting

from the fluid to the body depends on four dimensional parameters, namely, the fluid

density r and viscosity m, a characteristic size of the body d, and its velocity v. Then,the functional equation (2.77) takes the form

Fd ¼ f ðr; m; d; vÞ (2.78)

2.5 The Pi-Theorem 23

In order to find experimentally the drag force, it is necessary to put the body into

a wind tunnel and measure the drag force at a given velocity by an aerodynamic

scale. That is the experimental way of solving the problem under consideration but

only for one point on the parametric plane drag force-velocity. To determine the

dependence of the drag force on velocity within a certain range of velocity v, it isnecessary to reiterate the measurement of Fd at N values of v to determine the

dependence Fd ¼ f ðvÞ within a range ½v1; v2� at fixed values of r; m and d. If wewant to find the dependence Fd on all four governing parameters, we have to

perform N4 measurement.5 Therefore, if the number of data points forFd at varying

one governing parameter is N ¼ 102; the total number of measurements that one

needs will be equal to 108! It is evident that such number of measurements is

practically impossible to perform. Moreover, even if we have an experimental

data bank with 108 measurement points, we cannot say anything about the behavior

of the function Fd ¼ f ðr; m; v; dÞ outside the studied range of the governing para-

meters. An analytical or numerical calculation of the dependence of drag force on

density, viscosity, velocity and size of the body is also an extremely complicated

problem in the general case (at the arbitrary values of r; m; v; and d) due to the

difficulties involved in integrating the system of nonlinear partial differential

equations of hydrodynamics.

Essentially both approaches to study the dependence of drag force on density,

viscosity, velocity and size of the body allow a significant simplification of the

problem by using the Pi-theorem. The latter points at the way of transformation

of the function of n dimensional variables into a function of m ðwith m < nÞdimensionless variables. As a matter of fact, the Pi-theorem suggests how many

dimensionless variables are needed for describing a given problem containing

n dimensional parameters.

The Pi-theorem can be stated as follows. Let some dimension physical quantities

a depend on n dimensional parameters a1; a2 � � � an; where k of them have an

independent dimension. Then the functional equation for the quantities a

a ¼ f ða1; a2 � � � ak; akþ1 � � � anÞ (2.79)

can be reorganized to the form of the dimensionless equation

P ¼ ’ðP1;P2 � � �Pn�kÞ (2.80)

that contain n� k dimensionless variables. The latter are expressed as

P1 ¼ a1

aa01

1 aa02

2 � � �aa0k

k

; P2¼ a2

aa001

1 aa002

2 � � �aa00k

k

� � � Pn�k ¼ an

aan�k1

1 aan�k2

2 � � �aan�kk

k

(2.81)

The dimensionless form of the unknown quantities a is

5With an equal number of data points for each one of the four governing parameters.


P ¼ a

aa11 aa22 � � � aakk

(2.82)

To illustrate the application of the Pi-theorem to hydrodynamic problems, return

to the drag force acting on a body moving in viscous fluid. The unknown quantities

and governing parameters of the corresponding problem have the following

dimensions

Fd½ � ¼ LMT�2; r½ � ¼ L�3M; m½ � ¼ L�1MT�1; d½ � ¼ L; v½ � ¼ LT�1 (2.83)

Three from the four governing parameters of this problem have independent

dimensions. That means that a dimension of any governing parameters in this case

can be expressed as a combination of dimensions of the three others. The dimension

of the unknown quantity is also expressed as a combination of the governing

parameters having independent dimensions Fd½ � ¼ LMT�2 ¼ rv2d2½ � ¼ m2=r½ � ¼mvd½ �:In accordance with the Pi-theorem, (2.78) takes the form

P ¼ ’ðP1Þ (2.84)

where P ¼ Fd

ra1 va2da3 ; and P1 ¼ m

ra01 v

a02d

a03

:

Taking into account the dimension of the drag forceFd and governing parameters

with independent dimension r; v and d and using the principle of the dimensional

homogeneity, we find the values of the exponents ai and a0i

a1 ¼ 1; a2 ¼ 2; a3 ¼ 2; a01 ¼ 1; a

02 ¼ 1; a

03 ¼ 1 (2.85)

Then (2.84) reads

Cd ¼ ’ðReÞ (2.86)

where Cd ¼ Fd=rv2d2 is the drag coefficient, and Re¼rvd=m is the Reynolds

number.

The exact form of the function ’ðReÞ cannot be determined by means of the

dimensional analysis. However, this fact does not diminish the importance of

the obtained result. Indeed, the dependence of the drag coefficient on only one

dimensionless group (the Reynolds number) allows generalization of the experi-

mental data on drag related to motions of bodies of different sizes moving with

different velocities in fluids with different densities and viscosities. All this data can

be presented in a collapsed form of a single curve CdðReÞ. Moreover, in some

limiting cases corresponding to motion with low velocities (the so-called, creeping

flows with Re<< 1) or high speeds when Re>> 1, it is possible to determine the

exact forms of the dependence of the drag coefficient on Re.


In particular, at Re << 1 the inertia effects become negligible. Returning to

(2.78), we can assume that the drag force depends on fluid viscosity, body size and

its velocity

Fd ¼ f ðm; v; dÞ (2.87)

All the governing parameters in (2.87) have independent dimensions

(n� k ¼ 0). Therefore, in this case (2.87) reduces to

Fd ¼ cma1va2da3 (2.88)

where c is a dimensionless constant and a1 ¼ 1; a2 ¼ 1; a3 ¼ 1.

Substituting the values of the exponents a1; a2 and a3 into (2.88) leads to the

following expression for the drag coefficient

Cd ¼ c

Re(2.89)

It is evident that to determine the dependence CdðReÞ at Re<< 1 it is sufficient to

perform only one measurement in order to establish the value of the constant c.It is emphasized that the efficiency of using the Pi-theorem in studies of physical

phenomena is determined by the value of the difference n� k, i.e. by the number of

the governing dimensionless groups. In all cases (excluding k ¼ 0) the transforma-

tion of the functional equation by the Pi-theorem allows one to decrease number of

variables. The most interesting two cases correspond to the difference n� k being

either 0 or 1. In the first case the functional equation takes the form

a ¼ caa11 aa22 � � � aann (2.90)

In the second one it becomes

P ¼ ’ðP1Þ (2.91)

where P represents itself the dimensionless group corresponding to the unknown

parameter. Decreasing the number of dimensionless variables in (2.91) to only one

is equivalent to the transformation of partial differential equations into the ordinary

ones. A number of examples of transformation of the functional equations similar to

(2.77) to a dimensionless form, as well as transformations of partial differential

equations into the ordinary ones is given in the following sections.

2.5.2 Choice of the Governing Parameters

The theoretical study of hydrodynamic and heat and mass transfer processes is

based on the system of partial differential equations that include the mass,


momentum, energy and species conservation balances. This system of equations is

supplemented by an equation of state and correlations determining the physical

properties of the medium. The exact and approximate solutions of hydrodynamic

and heat transfer problems in the framework of the continuum approach yield

comprehensive answers to different problems of the theory. In distinction the

dimensional analysis of hydrodynamic and heat and mass transfer problems

meets some difficulties that arise already at the first step of the investigation

when choosing the governing parameter of the problem. They stem from certain

vagueness in choosing the governing parameters beginning from a pure intuitive

evaluation of the features of a phenomenon under consideration. In addition,

such approach to choosing the governing parameters often involves a number of

parameters whose influence will appear to be negligible at the end. The latter makes

it difficult to foresee the results of the dimensional analysis from scratch in

generalizing hydrodynamic and heat and mass transfer. In order to improve the

procedure of choosing the governing parameters and simplify and the following

analysis, it is possible to use the system of the mass, momentum, energy and species

balance equations.

Let us illustrate such an approach by the following examples.6 We begin with the

drag force acting on a spherical particle moving with a constant velocity in an

infinite bulk of viscous incompressible fluid. It is reasonable to assume that the

force that acts on the particle depends on its size d; velocity v and physical

properties of the fluid, namely its density r and viscosity m: In this case the

functional equation for the drag force Fd reads

Fd ¼ f ðr; m; d; vÞ (2.92)

The dimensional analysis of (2.92) leads to the following transformation in the

form of the drag coefficient

Cd ¼ ’�ðReÞ (2.93)

where Cd ¼ Fd=rv2d2 is the drag coefficient, and Re ¼ vd=n is the Reynolds

number.

It is emphasized that the function ’�ðReÞon the right hand side of (2.93) can be

presented as c’ðReÞ, where c is a dimensionless constant with its value being

chosen according to the experimental data. For example, for creeple motion of

a small spherical particle when the drag force is given by the Stokes law

Fd ¼ 3pmvd, constant c ¼ 8 p= : Then the expression for the drag coefficient takes

the form Cd ¼ 24=Re:To determine the exact form of the dependence (2.93), one needs to integrate the

continuity and the Navier–Stokes equations subjected to the no-slip condition at the

particle surface. In some limiting cases corresponding to special conditions of

6A detailed analysis of these problems see in Chaps. 4 and 7


particle motion (say, very slow or fast), it is possible to find the exact form of the

function ’�ðReÞ in (2.93) using these equations only for determining the set of

the governing parameters. For example, in the case of slow motion (creeping flows)

the inertia terms on the left hand side of the Navier–Stokes equations rðv � rÞv is

much less than the terms in on the right hand side of these equations. That allows

one to omit the inertial term and thereby exclude density from the governing

parameters. As a result, the functional equation for the drag force reduces to the

form of (2.6). Such simplification of the problem formulation is a key element

which allows establishing an exact form of the dependence of the drag force on

viscosity, velocity and diameter of the particle as Fd mvd that coincide (up to a

numerical factor) with the exact result (the Stokes force) derived from the

Navier–Stokes equations.

In the second case corresponding to a rapid body motion (the case of a large

Reynolds number) the dominant role belongs to the turbulent transfer. The average

characteristics of fully developed turbulent flows are governed by the Reynolds

equations (Hinze 1975; Loitsyanskii 1966)

@vi@t

þ vj@vi@xj

¼ � 1

r@P

@xiþ nr2vi þ 1

r@

@xjð�rv0

iv0jÞ (2.94)

(bars over parameters denote the average values).

In high Reynolds number flows the term nr2vi associated with the effect of the

molecular momentum transfer through molecular viscosity mechanism can be

omitted. Then, assuming steady state average turbulent flow, the drag force which

does not depend on molecular viscosity and time is given by

Fd ¼ f ðr; v; dÞ (2.95)

Applying the Pi-theorem to (2.95), we can rearrange it to the following form

Fd r�v2d2 (2.96)

which agrees with the Newton law for drag.

Another example of employing the conservation equations to facilitate the

dimensional analysis of complicated hydrodynamic and heat transfer problems is

related to mass transfer to a vertical reactive plate in contact with a liquid solution

of a reactive species (a reagent) which is initially at rest. When the rate of

a heterogeneous reaction at the plate surface is much larger than the rate of

diffusion transport of the reagent toward the surface, its concentration there equals

zero, whereas far from the surface it is equal c1: The gradient of the reagent

concentration across the thickness of the diffusion boundary layer results in

a non-uniform density field. That, in turn, triggers buoyancy force which results

in liquid motion near the wall. It is reasonable to assume that the velocity

and concentration of the reactive species in the dynamic and diffusion boundary

layers are determined by four parameters r1; c1; n; g and two independent

variables x and y


u ¼ fuðr1; c1; n; g; x; yÞ (2.97)

c ¼ fcðr1; c1; n; g; x; yÞ (2.98)

where we consider for brevity only one component of the velocity vector u; nand D are the kinematic viscosity and diffusivity, and gis the acceleration due

to gravity.

The functional equations (2.97) and (2.98) contain six governing parameters.

Three of them have independent dimensions. Choosing r1; n and g or r1; D and gas parameters with the independent dimensions, we transform (2.97) and (2.98) to

the following form

Pu ¼ ’uðP1;P2;P3Þ (2.99)

Pc ¼ ’cðP1�;P2�;P3�Þ (2.100)

where Pu ¼ u= gnð Þ1=3; P1 ¼ c1=r1; P2 ¼ x=ðn2=gÞ1=3; P3 ¼ y=ðn2=gÞ1=3;and Pc ¼ c=r1; P1� ¼ c1=r1; P2� ¼ x=ðD2=gÞ1=3; P3� ¼ y=ðD2=gÞ1=3:

Equations (2.99) and (2.100) show that the dimensionless velocity and concen-

tration of the reactive species are the function of three dimensionless groups, which

makes the analysis of the problem under consideration difficult. Therefore, employ

also the conservation equations. The momentum and species balance equations that

describe flow in the boundary layer and mass transfer to the vertical reactive wall

read (Levich 1962) (see Sect. 3.10)

u@u

@xþ v

@u

@y¼ n

@2u

@y2þ g�c� (2.101)

u@c�@x

þ v@c�@y

¼ D@2c�@y2

(2.102)

where g� ¼ gðc1=rÞð@r=@cÞc¼c1 ; c� ¼ ðc1 � cÞ=c1; and r ¼ rðcÞ.The boundary conditions for (2.101) and (2.102) are

u ¼ v ¼ 0 c� ¼ 1 at y ¼ 0; u ¼ v ¼ 0 c� ¼ 0 at y ! 1 (2.103)

Equations (2.101) and (2.102) and the boundary conditions (2.103) contain four

parameters that determine the local velocity and concentration fields

u ¼ fuðx; y; n; g�Þ (2.104)

c� ¼ fcðx; y;D; g�Þ (2.105)


Applying the Pi-theorem to transform (2.104) and (2.105) to the dimensionless

form, we obtain

Pu ¼ cuðP1Þ (2.106)

Pc ¼ ccðP1�Þ (2.107)

or equivalently,

u ¼ xg�ð Þ1=2cuð�Þ (2.108)

c� ¼ ccð�ffiffiffiffiffiSc

pÞ (2.109)

where Pu ¼ u= xg�ð Þ1=2; P1 ¼ y g�=xn2ð Þ1=4; Pc ¼ c�; P1� ¼ y g�=xD2ð Þ1=4,� ¼ y g�=xn2ð Þ1=4; and Sc ¼ n/D is the Schmidt number.

A number of instructive examples of application of the mass, momentum, energy

and species conservation equations for dimensional analysis of the hydrodynamic

and heat and mass transfer problems can be found in Chap. 7.

Problems

P.2.1. Transform the van der Waals equation (Kestin 1966; Jones and Hawkis 1986)

to the dimensionless form. Show that such form is universal for any van der Waals

gas if one uses the critical values of the pressure, volume and temperature as the

characteristic scales.

In order to transform equation the van der Waals equation ðPþ a=V2ÞðV � bÞ ¼RT(where a and b are constants) to dimensionless form, we present this equation as

ðA1 þ A2ÞðA3 þ A4Þ ¼ A5 (P.2.1)

where A1 ¼ P; A2 ¼ a=V2; A3 ¼ V; A4 ¼ �b; A5 ¼ RT with P;V and Tbeing pressure, molar volume and temperature, respectively.

We can introduce some still undefined scales of pressure P0; volume V0 and

temperature T0 and write the expressions for scales of Aj as

A1� ¼ P�; A2� ¼ a

V2�; A3� ¼ V�; A4� ¼ �b; A5� ¼ RT� (P.2.2)

Then (P.2.1) reduces to the form

ðA1 þ aA2ÞðbA3 þ gAÞ ¼ eA5 (P.2.3)


where A1 ¼ A1=A1� ¼ P=P; A2 ¼ A2 A2�; ¼ V V�=ð Þ�2;.

A3 ¼ A3=A3� ¼ ðV=V�Þ;A4 ¼ A4 A4� ¼ 1= A5 ¼ A5=A5� ¼ T=T� are the dimensionless variables, and a ¼A2�=A1� ¼ a= V2

�P��

; b ¼ A3�=A1� ¼ V�=P�; g ¼ A4�=A1� ¼ �b=P�; and e ¼A5�=A1� ¼ RT�=P� are the dimensionless constants.

Equation (P.2.3) is the dimensionless van der Waals equation. For its further

transformation one should define the characteristic scales of pressure, volume and

temperature. For that purpose, take as the scales P�; V� and T� the critical values

of pressure, volume and temperature Pcr; Vcr and Tcr, respectively. Bearing

in mind that the critical point is the inflection point where ð@P=@VÞT ¼ 0;

and ð@2P=@V2ÞT ¼ 0 , we find

a ¼ 27

64

R2T2cr

Pcr; b ¼ Vcr

3(P.2.4)

Pcr ¼ 1

27

a

b2; Vcr ¼ 3b; Tcr ¼ 8a

27bR(P.2.5)

Using as the characteristic scales the critical values of pressure, temperature and

specific volume, we transform (P.2.3) to the following final form

ðpþ 3

o2Þð3o� 1Þ ¼ 8t (P.2.6)

where p ¼ P=Pcr; o ¼ V=Vcr; and t ¼ T=Tcr.Equation (P.2.6) does not contain any constants accounting for the physical

properties of any particular gas and, thus, is universal. It holds for any van der

Waals gas.

P.2.2. (i) Transform the momentum and continuity equations for laminar flow of

incompressible fluid over a plane plate in the boundary layer approximation to the

dimensionless form using the LMT and LxLyLzMT systems of units. (ii) Show that

the LxLyLzMT system of units cannot be used for transformation of the

Navier–Stokes equations to the dimensionless form.

(i)-A: The LMTsystem of units. The boundary layer and continuity equations

read

u@u

@xþ v

@u

@y¼ n

@2u

@y2(P.2.7)

@u

@xþ @v

@yþ 0 (P.2.8)

where the dimensions of u; v; n; x and yare as follows

u½ � ¼ LT�1; v½ � ¼ LT�1; n½ � ¼ L2T�1; x½ � ¼ L; y½ � ¼ L (P.2.9)

Problems 31

All the terms in (P.2.7) have the dimension LT�2; whereas the dimension of the

terms in (P.2.8) is T�1: That shows that (P.2.7) and (P.2.8) can be transformed to

the dimensionless form by using the multipliers N1½ � ¼ ðLT�2Þ�1and N2½ � ¼

ðT�1Þ�1; respectively. Introducing the scales of length L�; velocity V�and kine-

matic viscosity n� ¼ n, we write the expressions for the coefficients AðiÞj� as follows

Að1Þ1� ¼ A

ð1Þ2� ¼ V2

�L�

;Að1Þ3� ¼ n�

V�L2�

;Að2Þ1� ¼ A

ð2Þ2� ¼ V�

L�(P.2.10)

Bearing in mind the dimensions of AðiÞj� , we express the multipliers N1 and N2 as

N1 ¼ 1

Að1Þ1�

; N2 ¼ 1

Að2Þ1�

(P.2.11)

Then (P.2.7) and (P.2.8) reduce to the following form

u@u

@xþ v

@u

@y¼ 1

Re

@2u

@y2(P.2.12)

@u

@xþ @v

@y¼ 0 (P.2.13)

where u ¼ u=V�; v ¼ v=V�; x ¼ x=L�; y ¼ y=L�; andRe ¼ V�L�=n�.The coefficients A

ðiÞj� for the Navier–Stokes and continuity equations

u@u

@xþ v

@u

@y¼ n

@2u

@x2þ @2u

@y2

� �(P.2.14)

v@u

@xþ v

@v

@y¼ n

@2v

@x2þ @2v

@y2

� �(P.2.15)

@u

@xþ @v

@y¼ 0 (P.2.16)

are defined as follows

Að1Þ1� ¼A

ð1Þ2� ¼

V2�

L�;A

ð1Þ3� ¼n�

V�L�

;Að2Þ1� ¼A

ð2Þ2� ¼

V2�

L�;A

ð2Þ3� ¼n�

V�L�

;Að3Þ1� ¼A

ð3Þ2� ¼

V�L�

(P.2.17)

Using the multipliers N1 ¼ 1=Að1Þ1� ;N2 ¼ 1=A1

ð2Þ� ; and N3 ¼ 1=A

ð3Þ1� , we reduce

(P.2.14)–(P.2.16) to the following dimensionless form

u@u

@xþ v

@u

@y¼ 1

Re

@2u

@x2þ @2u

@y2

� �(P.2.18)


v@u

@xþ v

@v

@y¼ 1

Re

@2v

@x2þ @2v

@y2

� �(P.2.19)

@u

@xþ @v

@y¼ 0 (P.2.20)

(i)-B: The LxLyLzMT system of units. The dimensions of u; v; x and y are

u½ � ¼ LxT�1; v½ � ¼ LyT

�1; x½ � ¼ Lx; y½ � ¼ Ly (P.2.21)

where Lx and Ly are the scales of length in the x and ydirections.Introducing the characteristic scales of u; v; n; x and y as U�½ � ¼ LxT

�1; V�½ � ¼LyT

�1; and n�½ � ¼ n½ �, we transform first of all the boundary layer and continuity

equations (P.2.7) and (P.2.8). To this aim, we write the expressions for the coeffi-

cients AðiÞj� as

Að1Þ1� ¼ U2

�Lx

; Að1Þ2� ¼ U�V�

Ly; A

ð1Þ3� ¼ n�U�

L2y; A

ð2Þ1� ¼ U�

Lx; A

ð2Þ2� ¼ V�

Ly(P.2.22)

Then (P.2.7) and (P.2.8) are transformed to

u@u

@xþ V�Lx

U�Ly

� �v@u

@y¼ n�Lx

L2yU�

!@2u

@y2(P.2.23)

@u

@xþ V�Lx

U�Ly

� �@v

@y¼ 0 (P.2.24)

where u ¼ u=U�; v ¼ v=V�; x ¼ x=Lx; y ¼ y=Ly; and the multipliers before the

second terms on left hand side of the boundary layer and continuity equations are

dimensionless, i.e. V�Lx=U�Ly� � ¼ 1.

In planar viscous flows in the x-direction with shear in the y-direction an

important role is played by the shear component tyx of the stress tensor. The

shear stress tyx can be presented as the ratio of the force Fyx to the surface area

Szx which have the following dimensions: Fyx

� � ¼ MLxT�2; and Szx½ � ¼ LzLx:

Then the dimension of the shear stress is tyx� � ¼ ML�1

z T�2: For viscous Newtonianfluids tyx ¼ mdu=dy, where m is the viscosity. Then, we find the dimension of the

viscosity in the LxLyLzMT system of units as

m½ � ¼ tyx du dy=ð Þ=� � ¼ L�1

x LyL�1z MT�1 (P.2.25)

Problems 33

Bearing in mind that the dimension of density in the LxLyLzMT system of units is

r½ � ¼ L�1x L�1

y L�1z M, we determine the dimension of the kinematic viscosity n as

n½ � ¼ m r=½ � ¼ L2yT�1 (P.2.26)

Thus, the multiplier n�Lx L2yU�. �

on the right hand side of (P.2.23) is dimen-

sionless. It can be presented as Re�1� , where Re� ¼ U�L2y n�Lx=

�is a modified

Reynolds number. Taking into account that the characteristic scales Lx; Ly;U� andVx are arbitrary, it is possible to assume that the ratio U�Ly V�Lx=

� � ¼ 1. Then,

(P.2.23) and (P.2.24) take the following form

u@u

@xþ v

@u

@y¼ 1

Re�

@2u

@y2(P.2.27)

@u

@xþ @v

@y0 (P.2.28)

The Navier–Stokes and continuity equations (P.2.14) and (P.2.16) can be

presented as

Að1Þ1� A

ð1Þ1 þ A

ð1Þ2� A

ð1Þ2 ¼ A

ð1Þ3� A

ð1Þ3 þ A

ð1Þ4� A

ð1Þ4 (P.2.29)

Að2Þ1� A

ð2Þ1 þ A

ð2Þ2� A

ð2Þ2 ¼ A

ð2Þ3� A

ð2Þ3 þ A

ð2Þ4� A

ð2Þ4 (P.2.30)

Að3Þ1� A

ð3Þ1 þ A

ð3Þ2� A

ð3Þ2 ¼ 0 (P.2.31)

where Að1Þ1� ¼ U2

�=Lx; Að1Þ2� ¼ U�V�=Ly; A

ð1Þ3� ¼ n�U�=L2x; A

ð1Þ4� ¼ n�U�=L2y , A

ð2Þ1� ¼

U�V=Lx; Að2Þ2� ¼ V2

�=Ly; Að2Þ3� ¼ n�V�=L2x ; A

ð2Þ4� ¼ n�V�=L2y , A

ð3Þ1� ¼ U�=Lx;A

ð3Þ2� ¼

V�=Ly, Að1Þ1 ¼ u@u=@x; A

ð1Þ2 ¼ v@u=@y; A

ð1Þ3 ¼ @2u=@x2; A

ð1Þ4 ¼ @2u=@y2, A

ð2Þ1 ¼

v@u=@x, Að2Þ2 ¼ v@v=@y, A

ð2Þ3 ¼ @2v=@x2;A

ð2Þ4 ¼ @2v=@y2, A

ð3Þ1 ¼ @u=@x; and

Að3Þ2 ¼ @[email protected] (P.2.18)–(P.2.20) take the form

u@u

@xþ V�Lx

U�Ly

� �v@u

@y¼ n�Lx

U�L2y

!LyLx

� �2 @2u

@x2þ @2u

@y2

( )(P.2.32)

v@u

@xþ V�Lx

U�Ly

� �v@v

@y¼ n�Lx

U�L2y

!LyLx

� �2 @2v

@x2þ @2v

@y2

( )(P.2.33)

@u

@xþ V�Lx

U�Ly

� �@v

@y¼ 0 (P.2.34)


The system of Eqs. (P.2.32)–(P.2.34) can be written as

u@u

@xþ v

@u

@y¼ 1

Re

LyLx

� �2 @2u

@x2þ @2u

@y2

( )(P.2.35)

v@u

@xþ v

@v

@y¼ 1

Re�

LyLx

� �2 @2v

@x2þ @2u

@y2

( )(P.2.36)

@u

@xþ @v

@y¼ 0 (P.2.37)

if we account for the fact that the dimension of the kinematic viscosity n�½ � ¼L2yT

�1: However, even in this case (P.2.35) and (P.2.36) are not dimensionless,

since the dimension of the ratio Ly Lx= is not 1. Moreover, (P.2.35) and (P.2.36) do

not satisfy the principle of the dimensional homogeneity under any assumption on

the dimension of the kinematic viscosity. The latter shows that applying the

LxLyLzMT system of units to transformation of the Navier–Stokes is incorrect.

P.2.3. (Reynolds 1886) Determine the resistance force acting on each of two

circular disks of radii R which approach each other along the joint axis of symmetry

with a constant velocity u, while the gap between the disks and the surrounding

space are filled with incompressible viscous fluid. The pressure in the surrounding

fluid far from the disks is equal P�.The liquid flow in the gap is axisymmetric. Therefore, we use cylindrical

coordinates z; r; ’ with the origin at the center of the lower disk which is assumed

to be motionless (z and r correspond to the vertical and radial directions, respec-

tively). Consider the low velocity case when the inertial effects are negligible. The

effect of the gravity force we also will neglected. Then, it is possible to assume that

the pressure gradient DP=r ðDP ¼ P� P�Þ is determined by the speed of the upper

disk u; liquid viscosity m; the instantaneous height of the gap h, and the radial

position r

DPr

¼ f ðu; m; h; rÞ (P.2.38)

For analyzing the problem, we use two different systems of units with a single

(L) and two (Lz; Lr) length scales. In the first case the dimensions of the pressure

gradient and the governing parameters can be expressed as

DPr

� ¼ L�2MT�2; u½ � ¼ LT�1; m½ � ¼ L�1MT�1; h½ � ¼ L; r½ � ¼ L (P.2.39)

Three of the four governing parameters in (P.2.38) have independent

dimensions. Choosing u; m; and r as the parameters with the independent

dimensions, we reduce (P.2.38) according to the Pi-theorem to the following form

Problems 35

P ¼ ’ðP1Þ (P.2.40)

where P ¼ DP=rð Þ=ua1ma2ra3 and P1 ¼ h=ua01ma

02 ra

03 .

Using the principle of the dimensional homogeneity, we find the values of

the exponents ai and a0i: a1 ¼ 1; a2 ¼ 1; a3 ¼ �2; a

01 ¼ 0; a

02 ¼ 0; and a

03 ¼ 1.

Accordingly, we arrive at the following expression

DPr

¼ umr�2’h

r

� �(P.2.41)

The force acting at the disk is found as

Fd ¼ 2pZR

DPrdr (P.2.42)

Substituting the expression (P.2.41) into (P.2.42), we obtain

Fd ¼ 2pumRZ10

’ex

� �dx (P.2.43)

where e ¼ h=R; x ¼ r=R; andR10

’ e=xð Þdx ¼ c eð Þ.Equation P.2.43 shows that the resistance force acting on a disk is directly

proportional to its velocity, the radius of the disk, viscosity of the liquid, as well

as a function of the ratio of the gap to the disk radius.

Additionally we transform (P.2.38) using the system of units with the two length

scales Lz and Lr in the z and r directions, respectively. First, we determine the

dimensions of the governing parameters and pressure gradient. The dimensions of

the velocity u, gap thickness h and r are

u½ � ¼ LzT�1; h½ � ¼ Lz; r½ � ¼ Lr (P.2.44)

To determine the dimensions of viscosity m and pressure gradient DP=r, we takeinto account the fact that in flows of viscous fluids in a narrow gap the dominant role

is played by the radial velocity component, since the axial one is typically much

smaller, vz << vr: In this case the force acting in the r-direction is much larger than

in the z-direction, so that its dimension is Fr½ � ¼ MLrT�2: Accordingly, the dimen-

sion of the shear stress tzr ¼ Fr Srr= ð Srr½ � ¼ L2r Þ is tzr½ � ¼ ML�1r T�2: For Newtonian

viscous fluids tzr ¼ m dvr dz=ð Þ. As a result, we find the dimension of viscosity m½ � ¼ML�2

r LzT�1: The dimensions of pressure and its gradient are

DP½ � ¼ Fr

Srz¼ MLrT

�2

LrLz¼ ML�1

z T�2 (P.2.45)


DPr

� ¼ ML�1

z T�2

Lr¼ ML�1

z L�1r T�2 (P.2.46)

Thus, the dimensions of all the governing parameters are expressed in the system

of units with two length scales are independent. Then, according to the Pi-theorem,

(P.2.38) takes the form

DPr

¼ cua1ma2ha3ra4 (P.2.47)

where c is a dimensionless constant.

Determining the values of the exponents ai using the principle of the dimen-

sional homogeneity as a1 ¼ 1; a2 ¼ 1; a3 ¼ �3 and a4 ¼ 1, we obtain

DPr

¼ cumr

h3(P.2.48)

Then, the substitution of (P.2.48) into (P.2.42) yields

Fd ¼ c

2pumR

R

h

� �3

(P.2.49)

The exact solution of this problem reads (Landau and Lifshitz 1987)

Fd ¼ 3

2pmuR

R

h

� �3

(P.2.50)

The comparison of (P.2.49) and (P.2.50) shows that the exact solution and the

result of the dimensional analysis agree up to a dimensionless numerical factor. At

the same time, the dimensional analysis of the problem using the system of units

with a single length scale yields a less informative result, since (P.2.43) contains an

unknown function c h R=ð Þ:

References

Barenblatt GI (1987) Dimensional analysis. Gordon and Breach Science Publication, New York

Barenblatt GI (1996) Similarity, self-similarity, and intermediate asymptotics. Cambridge Uni-

versity Press, Cambridge

Blackman DR (1969) SI units in engineering. Macmillan, Melbourne

Bridgman PW (1922) Dimension analysis. Yale University Press, New Haven

Buckingham E (1914) On physically similar system: illustrations of the use of dimensional

equations. Phys Rev 4:345–376

Chart of Dimensionless Numbers (1991) OMEGA Technology Company.

References 37

Douglas JF (1969) An introduction to dimensional analysis for engineers. Isaac Pitman and Sons,

London

Hinze JO (1975) Turbulence, 2nd edn. McGraw-Hill, New York

Huntley HE (1967) Dimensional analysis. Dover Publications, New York

Ipsen DC (1960) Units, dimensions, and dimensionless numbers. McGraw-Hill, New York

Jones JB, Hawkis GA (1986) Engineering thermodynamics. An introductory textbook. John Wiley

& Sons, New York

Kestin J (v.1, 1966; v.2, 1968) A course in thermodynamics. Blaisdell Publishing Company, New

York

Kutateladze SS (1986) Similarity analysis and physical models. Nauka, Novosibirsk (in Russian)

Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, Oxford

Levich VG (1962) Physicochemical hydrodynamics. Prentice-Hill, Englewood Cliffs

Loitsyanskii LG (1966) Mechanics of liquid and gases. Pergamon, Oxford

Lykov AM, Mikhailov Yu A (1963) Theory of heat and mass transfer. Gosenergoizdat Moscow-

Leningrad (English translation 1965. Published by the Israel Program for Scientific Transla-

tion. Jerusalem)

Madrid CN, Alhama F (2005) Discriminated dimensional analysis of the energy equation:

application to laminar forced convection along a flat plate. Int J Thermal Sci 44:331–341

Ramaswamy GS, Rao VVL (1971) SI units. A source book. McGraw-Hill, Bombay

Reynolds O (1886) On the theory of lubrication. Philos Trans R Soc 177:157–233

Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC Press, Boca

Raton

Spurk JH (1992) Dimensionsanalyse in der Stromungslehre. Springer-Verland in Berlin, New

York

Weast RC Handbook of chemistry and physics, 68th edition. 1987–1988. CRC, Boca Raton, FL.

Williams W (1892) On the relation of the dimensions of physical quantities to directions in space.

Philos Mag 34:234–271


Chapter 3

Application of the Pi-Theorem to EstablishSelf-Similarity and Reduce Partial DifferentialEquations to the Ordinary Ones

3.1 General Remarks

Chapter 3 deals with the application of the Pi-theorem to reduce partial differential

equations (PDEs) of certain hydrodynamic and heat transfer problems to the

ordinary differential equations (ODEs). In the cases which allow for such transfor-

mation (including the initial and boundary conditions), solution of the problem

reduces to a much simpler problem posed for an ODE, i.e. depends on a single

compound variable. The latter represent itself a combination of variables and

dimensional constants involved in the problem formulation. Such solutions are

called self-similar, since a single fixed value of the compound single variable

corresponds to numerous combinations of, say, coordinates or coordinates and

time, which make them identical in the “space” of the single compound variable.

The general consideration in Chap. 3 is followed by a number of examples

illustrating the usage of the Pi-theorem for establishing self-similar solutions.

They include flows of viscous incompressible fluid (the Stokes, Landau, and von

Karman problems), hydrodynamic and thermal (diffusion) boundary layers (the

Blasius, Pohlhausen and Levich problems), as well as some special hydrodynamical

problems, e.g. propagation of viscous-gravity currents (the Huppert problem)

and capillary waves on a thin liquid after a weak impact of a tiny droplet (the

Yarin-Weiss problem).

Below we consider the applications of the Pi-theorem for solving the differential

equations of hydrodynamics and the theory of heat and mass transfer. In the frame

of continuum approach, fluid flows and heat and mass transfer are described by a set

of PDEs that include the continuity, momentum (the Navier–Stokes), energy and

species balance equations (Kays and Crawford, 1993; Baehr and Stephan 1998;

Schlichting 1979). Solving these equations is extremely difficult because of the

non-linearity of the inertial terms of the Navier–Stokes equations and dependences

of the velocity, temperature and species concentration fields on several variables: in

the general case on three spatial coordinates and time. An essential simplification of

the problem can be achieved by reducing the number of independent variables by



39

means of a transformation of the problem PDEs to a set of ODEs if the initial and

boundary conditional of a specific problem admit that as well. As a matter of fact,

such a reduction can be considered as a typical problem of the dimensional analysis,

namely, the transition from n dimensional variables to n� k dimensionless ones. In

the case where the number of dimensionless groups n� k ¼ 1, a multi-dimensional

problem reduces to a one-dimensional one.

In order to demonstrate the transformation of PDEs into ODEs by using the

Pi-theorem, we consider a process in which some unknown dimensional

characteristics a (say, velocity, temperature, etc.) is expected to be determined by

dimension parameters a1; a2; � � �an (say, viscosity, density, velocity of the undis-

turbed flow, a characteristic size of a body, etc.)

a ¼ f ða1; a2 � � � ak � � � anÞ (3.1)

where a1; a2; � � �ak are the parameters with independent dimensions.

Assume that two parameters ai and aj among the n governing parameters

are independent variables (say, a coordinate and time) and all the other

n� 2 parameters are constants (say, the undisturbed velocity, viscosity, etc.).

Then the unknown characteristic a is described by PDE and the boundary and

initial conditions which contain all the n� 2 constants of the problem, the indepen-

dent variables ai and aj, as well as the derivatives @a=@ai; @a=@aj;

@2a=@a2i ; @2a=@a2j , etc. In order to reduce the problem to the integration of a

set of ODEs with the appropriate boundary conditions, one should find such a

transformation of the unknown characteristics a and the governing parameters

a1;a2; � � �an that the dimensionless unknown characteristics (denoted as P) will

become a function of a single dimensionless group P1 formed from the governing

parameters of the problem. The latter is possible when the number of the governing

parameters equals k þ 1: Then (3.1) reduces to

P ¼ ’ðP1Þ (3.2)

where P ¼ a=aa11 aa22 � � � aakk , and P1 ¼ an=a

a01

1 aa02

2 � � � aa0k

k , with ai and a0i (i ¼ 1,

2,. . . kÞ being some exponents.

It is emphasized that P is described by an ODE (or a system of ODEs), since

it is a function of a single variableP1: The above consideration shows that

transformation of a PDE into an ODE is determined by the dimensions of the

governing parameters. For example, consider a particular case when the unknown

characteristics a depends on four dimension parameters: two constants a1; a2 andtwo independent variables a3; a4

a ¼ f ða1; a2; a3; a4Þ (3.3)

Let the dimensions of the unknown characteristics a and governing parameters

a1; a2; a3 and a4 be

40 3 Application of the Pi-Theorem to Establish Self-Similarity

a½ � ¼ Le1Me2Te3 ; a1½ � ¼ Le01Me

02Te

03 ; a2½ � ¼ Le

001Me

002 Te

003 ; a3½ �

¼ Le0001 Me

0002 Te

0003 ; a½ � ¼ Le

IV1 MeIV

2 TeIV3 (3.4)

where ei; e0i; e

00i ; e

000i and eIVi (i ¼ 1,2,3) are some known exponents.

When three governing parameters, for example, a1; a2 and a3 have independentdimensions, (3.3) takes the form of (3.2) with P ¼ a=aa11 a

a22 a

a33 and P1 ¼

a4=aa01

1 aa02

2 aa03

3 : Bearing in mind the dimensions of the parameters a; a1; a2; a3 anda4, we arrive at the following sets of the algebraic equations for the exponents

ai and a0i:

e01a1 þ e

001a2 þ e

0001 a3 ¼ e1

e02a1 þ e

002a2 þ e

0002 a3 ¼ e2

e03a1 þ e

003a2 þ e

0003 a3 ¼ e3

(3.5)

and

e01a

01 þ e

001a

02 þ e

0001 a

03 ¼ eIV1

e02a

01 þ e

002a

02 þ e

0002 a

03 ¼ eIV2

e03a

01 þ e

003a

02 þ e

0003 a

03 ¼ eIV3

(3.6)

Solution of the systems of (3.5) and (3.6) exists when

e01 e

001 e

0001

e02 e

002 e

0002

e03 e

003 e

0003

�� 6¼ 0 (3.7)

The inequality (3.7) is the condition under which transformation of a PDE into

an ODE is possible. This inequality determines only the necessary rather than the

sufficient condition of the existence of self-similar solutions. In addition to

satisfying the ODEs, such solutions should also satisfy the initial and boundary

conditions of the problem.

The transformation of a PDE into an ODE is also possible with n� k ¼ 2; 3; � � �lwhen all the constants in (3.1) have the same dimensions, which are different from

the dimensions of the independent variables ai and aj: Indeed, (3.1) can be recast intothe following form

a ¼ f ða1; a2 � � � ai � � � ak � � � aj � � � anÞ (3.8)

The latter equation is reduced to the dimensionless form when the Pi-theorem is

applied

P ¼ ’ðP1;P2 � � �Pj � � �Pn�kÞ (3.9)

3.1 General Remarks 41

where P ¼ a=aa11 aa22 � � � aaii � � � aakk ; P1 ¼ akþ1=a

a01

1 aa02

2 � � � aa0i

i � � � aa0k

k ; P2 ¼akþ2=a

a001

1 aa002

2 � � � aa00i

i � � � aa00k

k ;Pj ¼ aj=aaj1

1 aaj2

2 � � � aaji

i � � � aajk

k ; and Pn�k ¼ an�k=

aan�k1

1 aan�k2

2 � � � aan�ki

i � � � aan�kk

k .

Due to the fact that the dimensions of all the constants are different from the

dimensions of variable ai; we find that the exponents a0i, a

00i ; � � �an�k

i are equal to

zero. Then (3.9) takes the following form

P ¼ ’ðc1; c2 � � �Pj � � � cn�kÞ ¼ ’ðPjÞ (3.10)

where c1; c2; � � �cn�k are the dimensionless constants: c1 ¼akþ1=a

a01

1 aa02

2 � � � aa0i�1

i�1aa0iþ1

iþ1 � � � aa0k

k , c2 ¼ akþ2=aa001

1 aa002

2 � � � aa00i�1

i�1aa00iþ1

iþ1 � � �aa00k

k ; . . ..cn�k ¼an�k=a

an�k1

1 aan�k2

2 � � � aan�ki�1

i�1 aan�kiþ1

iþ1 � � � aan�kk

k :

It is emphasized that the possibility to transform the problem PDEs into ODEs

can be recognized by a simple analysis of the dimensions of constants involved in

the problem formulation, i.e. in the governing equations, the initial and boundary

conditions, as well as the additional characteristics which are given, such as the

invariant overall momentum flux in a submerged jet, etc. The rule that such a

transformation is possible (and thus, a self-similar solution of the PDE-based

problem exists) can be formulated as follows: the transition of PDEs into ODEs

is possible when scales of the independent variables cannot be constructed using the

scales of the problem constants (Loitsyanskii 1966). In order to illustrate this

statement, we consider the radial flow in a plane wedge. The radial component of

fluid velocity v in this case depends on the strength of a source Q located at the

wedge apex, kinematic viscosity n of fluid, as well as the radial coordinate r and thepolar angle ’ of the cylindrical coordinate system

v ¼ f�ðQ; n; r; ’Þ (3.11)

The dimensions of the given constants Q and n are: Q½ � ¼ L2T�1 (per unit length

of a source or sink normal to the wedge plane) and n½ � ¼ L2T�1: It is easy to see thatit is impossible to express the dimension of r via the dimensions of the constants

Q and n: This fact points out at the possibility of transformation of the PDE

describing the flow under consideration into an ODE. Indeed, accounting for the

fact that two from the four governing parameters have independent dimensions, we

can rewrite (3.11) as follows

v ¼ f1Q

n; ’

� �¼ f ð’Þ (3.12)

where v ¼ v= nr

� �, and Q=n are dimensionless constants.


Substituting the expression (3.12) into the set of the governing equations

vdw

dr¼ � 1

r@P

@rþ n

@2v

@r2þ 1

r2@2v

@’2þ 1

r

@v

@r� v

r2

� �(3.13)

� 1

rr@P

@’þ 2v

r2@v

@’¼ 0 (3.14)

@ðrvÞ@r

¼ 0 (3.15)

we arrive at the following ODE (Hamel 1917; see also Rosenhead 1963 and

Loitsyanskii 1966)

d2f

d’2þ 4f þ 6f 2 ¼ 2c1 (3.16)

where c1 is a constant.The constant c1, as well as the two additional constants c2 and c3 that appear

when integrating (3.16), are determined by the no-slip boundary conditions at the

wedge walls v �a=2ð Þ ¼ 0 and the condition that a constant mass flow rate is given.

It is emphasized that a similar analysis of a flow in a cone (instead of a wedge)

where Q has the dimension L3T�1 shows that transformation of the PDE describing

this flow into an ODE is impossible since the dimension of L is provided by the ratio

Q=n.Reduction of PDEs to ODEs is possible under a certain idealization of real

phenomena. The absence of a characteristic scale (for example, the length scale is

missing in flows about semi-infinite plates, in submerged jet flows issued from a

point wise source of momentum flux, etc.) is a sign of the existence a self-similar

solution of PDEs and a possibility of their transformation to ODEs (Sedov 1993).

Solving such problems in dimensionless form requires introduction of some arbi-

trary scales instead of the missing ones. Only then the governing equations and the

initial and boundary conditions can be transformed to the dimensionless form. As a

result, the unknown characteristics can be expressed as a dimensionless function of

dimensionless variables only. The requirement that the arbitrary length scale should

not be involved in the dimensional solution of an idealized problem allows reveal-

ing the form of this function. It is emphasized that the absence of a characteristic

length scale in the problem formulation is an essential but not a sufficient condition

for the existence of a self-similar solution of a set of the corresponding PDEs. For

example, the problem on flow in an axisymmetric diffuser with circle cross-section

has no self-similar solution.

Below we consider a number of examples of applications of the Pi-theorem for

reduction of PDEs describing incompressible fluid flows and heat transfer in media

at rest to ODEs, and thus finding self-similar solutions.

3.1 General Remarks 43

3.2 Flow over a Plane Wall Which Has Instantaneously StartedMoving from Rest (the Stokes Problem)

Consider an upper half-space filled with a viscous incompressible fluid in contact

with a flat plate corresponding to y ¼ 0 (Fig. 3.1). (Stokes 1851)

This wall has instantaneously started to move horizontally with a constant

velocity U. The wall motion is transmitted to the fluid due to the action of viscous

forces. As a result, the fluid is entrained into horizontal motion as well. The fluid

flow is subjected to the no-slip boundary condition at the wall surface and is

described as follows

@u

@t¼ n

@2u

@y2(3.17)

t ¼ 0 : 0 � y � 1 u ¼ 0; t>0 : y ¼ 0 u ¼ U; y ! 1 u ¼ 0 (3.18)

where u is the fluid velocity and n is the kinematic viscosity.

Equation (3.17) with the conditions (3.18) show that fluid velocity depends on

four parameters: two independent variables t and yand two constants n and U :

u ¼ f ðU; n; y; tÞ (3.19)

First of all, let us ascertain the possibility of reduction of (3.17) to an ODE. For

this aim we make use of the above-mentioned signs pointing at such a transforma-

tion. The lack of a given characteristic length in (3.17) and the conditions (3.18)

points at the possibility of reduction of (3.17) to an ODE.

To apply the Pi-theorem to transform (3.17) into an ODE, it is necessary to

consider the dimensions of the unknown characteristics u and the governing

parameters n; U; y and t: In principle, it is possible to use different systems of

fundamental units, in particular, the LMT system. Then, we have the dimensions as

u½ � ¼ LT�1; U½ � ¼ LT�1; n½ � ¼ L2T�1; y½ � ¼ L; t½ � ¼ T (3.20)

t > 0

U > 0

y

u0

t = 0

U = 0

y

u0

Fig. 3.1 Scheme of flow over

plane wall has instantaneous

by started moving from rest


Two from the four governing parameters have independent dimensions, so that

n� k ¼ 2: In accordance with that, we obtain

u ¼ ’ðt; yÞ (3.21)

where u ¼ u=U; t ¼ t= n=U2ð Þ; y ¼ y=ðn=UÞ.Equation (3.21) shows that u depends on two dimensionless groups that seem-

ingly shows that it is impossible of transform (3.17) into an ODE. This result

follows directly from the analysis of the dimensions of the parameters involved.

Indeed, we can construct the length and time scales, L� ¼ n=U, T� ¼ n=U2, that

shows that it is impossible to express u as a function of a single dimensionless

variable. At the first sight this result contradicts to the expectations based on

the absence of the characteristic length scale in the problem formulation as in

the present case. The apparent contradiction can be explained as follows. The

Pi-theorem determines only the number of dimensionless groups which can be

constructed from n governing parameters including k parameters with independent

dimensions. The number n is determined by the physical essence of the problem,

whereas the number k can be changed depending on the system of units used. Thus,

the difference n� k that determines the number of dimensionless variables depends

also on the system of units used.

Let us extend the system of units by introducing three different length scales Lxand Ly for x and y directions (along and normal to the wall in Fig. 3.1), and Lz forthe z direction normal to the xy plane. This means that the LxLyLzMT system of units

is used. Taking into account that the wall and the velocity component u, as well asthe velocity of the unperturbed flow U are directed along the x-axis, we define theirdimensions as

u½ � ¼ LxT�1; U½ � ¼ LxT

�1 (3.22)

where T is the time scale, t½ � ¼ T:The dimension of the kinematic viscosity, is

n½ � ¼ L2yT�1 (3.23)

since in the case under consideration viscosity transmits information about the

wall motion into the liquid bulk in the y direction. Indeed, the dimension of

viscosity m can be found directly from the rheological constitutive equation of the

Newtonian fluid tyx ¼ m du=dyð Þ as the ratio of the shear stress to the velocity

gradient (Huntley 1967; Douglas 1969). Bearing in mind that tyx ¼ Fyx=Sxz, wedetermine the dimension of tyx

tyx� � ¼ L�1

z MT�2 (3.24)

where Fyx and Sxz are the force in the x direction acting at the surface element in the

xz plane, respecticaly; Fyx

� � ¼ LxMT�2; Sxz½ � ¼ LxLz.

3.2 Flow over a Plane Wall Which Has Instantaneously Started Moving from Rest 45

Since the dimension of the velocity gradient du=dy is LxL�1y T�1, the dimension

of viscosity is expressed as

m½ � ¼¼ L�1x LyL

�1z MT�1 (3.25)

Then the dimension of the kinematic viscosity is

n½ � ¼ mr

¼ L2yT

�1 (3.26)

where r½ � ¼ L�1x L�1

y L�1z M is the fluid density.

As a result, we guarantee that the dimensions of all the terms in (3.17) are the

same: @u @t=½ � ¼ LxT�2; and n@2u @y2

�� ¼ LxT�2.

In the framework of the LxLyLzMT system among the four governing parameters

there are three parameters with independent dimensions

U½ � ¼ Le01x L

e02y T

e03 ; n½ � ¼ L

e001x L

e002y T

e003 ; t½ � ¼ L

e0001x L

e0002y Te

0003 (3.27)

where e01 ¼ 1; e

02 ¼ 0; e

03 ¼ �1; e

001 ¼ �1; e

002 ¼ 2; e

003 ¼ �1; e

0001 ¼ 0; e

0002 ¼ 0; and

e0003 ¼ 1.

At such values of the exponents e0i; e

00i and e

000i determinant (3.7) is not equal to

zero. In this case (3.19) takes the form of (3.2) with P ¼ u=Ua1na2 ta3 and P1 ¼y=Ua

01na

02 ta

03 : Taking into account the dimensions of u and U; n; t; y; we arrive

at the system of the algebraic equations for the exponents ai and a0i(i ¼ 1; 2; 3Þ

SLx1� a1 ¼ 0; a01 ¼ 0

SLy a2 ¼ 0; 1� 2a02 ¼ 0

STa1 � a3 � 1 ¼ 0; a02 � a

03 ¼ 0 (3.28)

where the symbols SLx ;SLy and ST refer to the summation of the exponents of

Lx; Ly and T; respectively.From (3.28) it follows

a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a01 ¼ 0; a

02 ¼

1

2; a

03 ¼

1

2(3.29)

Then (3.19) reduces to

u

U¼ ’

yffiffiffiffint

p� �

(3.30)

The substitution of the derivatives @u=@t ¼ �U�’0=2t and @2u=@y2 ¼ U’

00=nt

into (3.17) leads to the following ODE determining the function ’

’00 þ �

2’

0 ¼ 0 (3.31)

where ’ ¼ ’ð�Þ; ’0 ¼ d’=d�, and ’00 ¼ d2’=d�2, with � ¼ y=

ffiffiffiffint

p.


3.3 Laminar Boundary Layer over a Flat Plate(the Blasius Problem)

The previous example was related to flow development in fluid that has initially

been at rest and started moving being entrained by a plate. Below we consider an

application of the Pi-theorem for transformation of the boundary layers equations

into an ODE in the case of fluid flow about a motionless wall.

Consider a flow over a semi-infinite plate. The flow is assumed to be incom-

pressible and fluid velocity is considered to be uniform far away from the plate

surface (Fig. 3.2). (Blasius 1980)

The system of the governing equations in this case reads

u@u

@xþ v

@v

@y¼ n

@2u

@y2(3.32)

@u

@xþ @v

@y¼ 0 (3.33)

Equations (3.32) and (3.33) should be integrated subjected to the no-slip boundary

conditions at the plate surface, as well as and a given constant velocity of the stream

parallel to the plate is prescribed far away from the plate

y ¼ 0; u ¼ v ¼ 0; y ! 1; u ! U (3.34)

We use the Blasius problem to demonstrate the efficiency of using the ordinary

LMT and modified LxLyLzMT systems of units for dimensional analysis of

thermohydrodynamic problems. First of all, we consider the application of the

Pi-theorem to the Blasius problem using the LMTsystem of units. Equations

d (x)

y

x0

u

Fig. 3.2 The flow in the

boundary layer over a flat

plate

3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem) 47

(3.32–3.33) and the boundary conditions (3.34) show that flow velocity within the

boundary layer over a flat plate depends on four dimensional parameters: two

independent variables x; y and two constants n and U: Therefore, we can write the

following functional equation for the longitudinal velocity component u

u ¼ f ðx; y; n;UÞ (3.35)

where the dimensions of u; x; y; n and U are expressed as

u½ � ¼ LT�1; x½ � ¼ L; y½ � ¼ L; n½ � ¼ L2T�1; U½ � ¼ LT�1 (3.36)

It is seen that of the four governing parameters, two parameters possess indepensdent

dimensions. That means that the difference n� k ¼ 2, so that the dimensionless

velocity is a function of two dimensionless groups. Choosing n and U as the

parameters with independent dimensions, we tramsform (3.35) to the dimensionless

form using the Pi-theorem. As a result, we arrive at the following equation

u ¼ ’ðx; �Þ (3.37)

where u ¼ u=U; x ¼ xU=n; � ¼ yU=n:Consider (3.37) from the point of view of generalization of the experimental data

for flows over flat plates, as well as the theoretical analysis of the corresponding

problem. Assume that an experimental data bank for the velocity at a number of

points within the boundary layer is available. According to (3.37), these data

determine a surface in the parametric space u� x� �: A section of this surface

by a plane x ¼ const determines the velocity distribution in given cross-section of

the boundaty layer. The totality of the velocity profiles corresponding to different

values of x determines the flow field within the boundary layer. It is obvious that

usefulness of such an approach for the generalization of the experimental data

would be low, since it requires many diagrams corresponding to different cross-

sections of the boundary layer, which makes it extremely laborious.

On the other hand, we apply now (3.37) for the theoretical analysis of the Blasius

problem. For this aim we rewrite (3.32) and (3.33) and the boundary conditions

(3.34) using the variables u; x and �. Taking into account that

u@u

@x¼ U3

n

� �u@u

@x; v

@u

@y¼ U3

n

� �v@u

@�; n

@2u

@y2¼ U3

n

� �@2u

@�2(3.38)

and

@u

@x¼ U2

n

� �@u

@x;@v

@y¼ U2

n

� �@v

@�(3.39)

we arrive at the equations


u@u

@xþ v

@u

@�¼ @2u

@�2(3.40)

@u

@xþ @v

@�¼ 0 (3.41)

Their solutions are subject to the following boundary conditions

� ¼ 0; u ¼ v ¼ 0; � ! 1; u ! 1 (3.42)

were v ¼ v=U.

Is easy to see that the transformation of (3.32) and (3.33) and the boundary

conditions (3.34) using the LMTsystem of units does not lead to any simplification

of the theoretical analysis of the Blasius problem. The latter still reduces to

integrating the system of the partial differential equations (3.41) and (3.42).

Accordingly, for the analysis of the planar boundary layer problems, it is conve-

nient to use the modified LMT system that includes two different scales of length Lx,Ly and Lz for the x, y and z directions, respectively, where the x axis is parallel to theplate in the flow direction, while the y and z axes are normal to it (cf. Fig. 3.2). It is

easy to show that the introduction of the two additional length scales does not affect

the dimension uniformity of the terms of the boundary layer and continuity

equations. Indeed, assuming that the dimensions of x½ � ¼ Lx and t½ � ¼ T; we find

that the corresponding dimension of the longitudinal velocity component is

u½ � ¼ LxT�1 (3.43)

Requiring that both terms of the continuity equation (3.30) possess the same

dimensions @u @x=½ � ¼ T�1; and @v @y=½ � ¼ T�1; we find the dimensions of v and y as

n½ � ¼ LyT�1; y½ � ¼ Ly (3.44)

Then the dimensions of the terms in the momentum equation (3.32) become

u@u

@x

¼ LxT

�2; v@u

@y

¼ LxT

�2; n@2u

@y2

¼ LxT

�2 (3.45)

where the dimension of kinematic viscosity n is L2yT�1.

First we estimate the thickness of the boundary layers d: It is clear that

d can be a function of a single independent variable x, as well as of the two

constants of the problem: the kinematic viscosity n and the free stream velocity

U½ � ¼ LxT�1

d ¼ fdðU; n; xÞ (3.46)

3.3 Laminar Boundary Layer over a Flat Plate (the Blasius Problem) 49

Since all the governing parameters in (3.46) possess independent dimensions, the

difference n� k ¼ 0 and the thickness of the boundary layer can be expressed as

d ¼ cna1xa2Ua3 (3.47)

where d½ � ¼ Ly; c is a dimensionless constant and the exponents a1; a2 and a3 areequal to 1=2; 1=2 and � 1=2; respectively.

As a result, we obtain

d ¼ c

ffiffiffiffiffinxU

r(3.48)

The velocity at any point in the boundary layer depends on the variables x½ � ¼Lx; y½ � ¼ Ly and constants n and U

u ¼ fuðU; n; x; yÞ (3.49)

Three governing parameters in (3.49) possess independent dimensions. There-

fore, in accordance with the Pi-theorem, (3.49) can be reduced to the form of (3.2)

with P ¼ u=U and P1 ¼ y=ffiffiffiffiffiffiffiffiffiffiffinx=U

p, i.e.

u

U¼ ’

0uð�Þ (3.50)

where ’0u ¼ d’u=d�, � ¼ y=

ffiffiffiffiffiffiffiffiffiffiffinx=U

p.

Equation (3.50) shows that the dimensionless velocity u ¼ u U= is determined by

a single variable �: That allows one to generalize the experimental data for the

velocity distribution in different cross-sections of the boundary layer over a flat

plate in the form of a single curve uð�Þ. Naturally such presentation of the results ofexperimental investigations has a significant advantage compared to the presenta-

tion of the experimental data in the form of a surface in the parametrical space u�x� � discussed before. The theoretical analysis of the Blasius problem is also

significantly simplified by using the LxLyLzMT system of units, since the problem is

reduced in this case to integrating an ordinary differential equation. Indeed, the

substitution of the expression (3.50) into (3.32) and (3.33) results in the following

ODE for the unknown function ’uð�Þ

2’000u þ ’

0u’

00u ¼ 0 (3.51)

with the boundary conditions

� ¼ 0; ’u ¼ 0 ’0u ¼ 0; � ! 1’

0u ¼ 1 (3.52)

The shear stress at the wall tw ¼ mð@u=@yÞ0 ¼ mffiffiffiffiffiffiffiffiffiffiffiffiffiU3=nx

p’

0 ð0Þ;where’

0 ð0Þ ¼ du=d�j0.It is emphasized that there is another way of transforming (3.32) and (3.33) into

the ODE. It is based on the assumption that velocity at any point of the boundary

layer is determined by three governing parameters, namely, the free stream velocity


U½ � ¼ LxT�1; the thickness of the boundary layer d½ � ¼ Ly and the distance from the

plate to a point under consideration y½ � ¼ Ly

u ¼ fuðU; y; dÞ (3.53)

Since two of the three governing parameters in (3.53) possess independent

dimensions, we obtain

u

U¼ ’

0u

y

d

�(3.54)

where the dependence dðxÞ is given by (3.48).

The instructive examples of the applications of the Pi-theorem for the analysis of

the Stokes and Blasius problems allow one to evaluate the true value of the LMTand LxLyLzMT systems of units. The comparison of the results produced by both

systems of units shows that the expansion of the system of units by introducing

different length scales in the x; y and z directions allows one to reduce the number

of the dimensionless groups and significantly simplifies generalization of the

experimental data and theoretical analysis of these problems. As a matter of fact,

the rationale for choosing a system of units (LMT or LxLyLzMTÞ should be based

on the comparison of the number of parameters with independent dimensions in the

set of the governing parameters determining the problem. Indeed, since the total

number of the governing parameters n does not depend on the system of units, the

number of the dimensionless groups in any given problem, n� k, is fully deter-

mined by the number of parameters with independent dimensions k. Therefore, thechoice of the LxLyLzMT system of units is desirable when

k�� > k� (3.55)

where subscripts � and � � correspond to the LMT and LxLyLzMT systems of units,

respectively.

Thus, the LMT system of units should be used when ðn� kÞ� equals zero or

unity. In the case when ðn� kÞ� > 1, it is preferable to use the LxLyLzMT system of

units. In future we will use both systems of units without an additional discussion

of the reasons for choosing a given system.

3.4 Laminar Submerged Jet Issuing from a Thin Pipe(the Landau Problem)

Let an incompressible fluid be issued from a thin pipe into an infinite space filled

with the same medium (with the same physical properties as those of the jet).

As a result of the laminar jet flow, mixing of the issuing and the ambient fluids

takes place

3.4 Laminar Submerged Jet Issuing from a Thin Pipe (the Landau Problem) 51

The flow is described by the Navier–Stokes and continuity equations

ðr � vÞv ¼ � 1

rrPþ nr2v (3.56)

r � v ¼ 0 (3.57)

where v is the velocity vector with the components vr; vy; and v’, and r; y; and ’are the spherical coordinates with the y axis (the polar axis) in the direction of the

jet and centered at its origin; P is the pressure. The sketch of this flow is shown in

Fig. 3.3 (Landau 1944).

Let us assume that there is no swirl and v’ ¼ 0: In addition, due to the assumed

axial symmetry of the flow about the polar axis (y ¼ 0Þ, the velocity components vrand vy are the function of only two variables: r and y: The velocity components also

depend on viscosity, as well as on the kinematic momentum flux J ¼ I r= (I is thetotal momentum flux in the jet which is determined by the pipe flow and is given).

Thus, we can write the functional equations for the velocity components vr and v’and pressure P in the following form

vr ¼ f1ðr; y; n; JÞ (3.58)

vy ¼ f2ðr; y; n; JÞ (3.59)

P ¼ f3ðr; y; n; JÞ (3.60)

where r; n, y and J have the following dimensions

r½ � ¼ L; n½ � ¼ L2T�1; y½ � ¼ 1; J½ � ¼ L4T�2 (3.61)

It is seen that two of the four dimensional parameters in (3.58–3.60) possess

independent dimensions (n� k ¼ 2Þ: In this case the Pi-theorem yields

Pi ¼ ’iðP1i;P2iÞ (3.62)

thin pipe

Fig. 3.3 Stream lines in flow

is induced laminar jet issuing

from a thin pipe


where Pi ¼ Ni=ra1ina2i ; P1:i ¼ y=ra

01ina

02i ; P2;i ¼ J=ra

001ina

002i ; N1 ¼ vr; N2 ¼ vy; and

N3 ¼ P=r; with i ¼ 1; 2; 3.Bearing in mind the dimensions of vr, vy; P; J and y; we find the values of the

exponents in (3.62)

a11 ¼ �1; a21 ¼ 1; a12 ¼ �1; a22 ¼ 1; a13 ¼ �2; a23 ¼ 2

a011 ¼ 0; a

021 ¼ 0; a

012 ¼ 0; a

022 ¼ 0; a

013 ¼ 0; a

023 ¼ 0

a0011 ¼ 0; a

0021 ¼ 0; a

0012 ¼ 0; a

0022 ¼ 2; a

0013 ¼ 0; a

0023 ¼ 0 (3.63)

Then, the dimensionless groups in (3.62) become

P1i ¼ y; P21 ¼ J

n2¼ const: (3.64)

for i ¼ 1; 2; 3; and

P1 ¼ vrv�

; P2 ¼ vyv�

; P3 ¼ P

rv2�(3.65)

where v� ¼ n=r:Accordingly, we obtain the following expressions for the velocity components

and pressure

vr ¼ nr’1ðyÞ; vy ¼

nr’2ðyÞ;

P

r¼ n2

r2’3ðyÞ (3.66)

Substituting the expressions (3.66) into (3.56) and (3.57), we arrive at the

following system of ODEs

’001 þ ’

01ðctgy� ’2Þ þ ’2

1 þ ’22 � 2’3 ¼ 0 (3.67)

’2’02 � ’

01 � ’

03 ¼ 0 (3.68)

’1 þ ’02 � ’2ctgy ¼ 0 (3.69)

Excluding ’3 from (3.67–3.69), we obtain the following system of ODEs for the

unknown functions ’1 and ’2

’0001 þ ð’0

1ctgyÞ0 þ ð’2’

01Þ

0 þ 2’1’01 þ 2’

01 ¼ 0 (3.70)

’1 þ ’01 � ’2ctgy ¼ 0 (3.71)

A solution of (3.70) and (3.71) corresponding to the issuing viscous fluid from a

thin pipe (a point wise source of momentum flux) was found by Landau (1944)

3.4 Laminar Submerged Jet Issuing from a Thin Pipe (the Landau Problem) 53

’1 ¼ �2þ 2ðA2 � 1ÞðAþ cos yÞ2 ; ’2

2 sin yAþ cos y

(3.72)

where A is a constant of integration which is related to the total momentum flux of

the jet J by the following expression

J ¼ 16pn2A 1þ A

3ðA2 � 1Þ �A

2lnAþ 1

A� 1

(3.73)

A detailed analysis of the flow in a submerged jet issued from a thin pipe

following the original work of Landau (1944). It can be also found in the

monographs of Landau and Lifshitz (1987), Sedov(1993), Vulis and Kashkarov

(1965).

3.5 Vorticity Diffusion in Viscous Fluid

Consider transformation of the PDE into an ODE in the problem which describes

the evolution of an initially infinitely thin vortex line of strength G. Assume that the

vortex line is normal to the flow plane (Fig. 3.4 a).

The vorticity transport equation reads (Batchelor 1967)

@O@t

¼ nr

@

@rr@O@r

� �(3.74)

where O is the vorticity component (the only one which is non-zero and normal to

the flow plane).

The unknown characteristics O½ � ¼ T�1 depends on two variables-time t½ � ¼ Tand the radial coordinate reckoned from the location of the initial vortex line

r

ϕ

rΩ

a b

t0

Fig. 3.4 Diffusion of vorticity in viscous fluid. (a) Stream lines. (b) The dependence of vorticityon time for different values of radial coordinate


r½ � ¼ L, as well as on two constants of the problem-the vortex strength G½ � ¼ L2T�1

and kinematic viscosity n½ � ¼ L2T�1. From the initial condition of the problem

GR ¼ G at t ¼ 0 (with GR being the circulation over a circle of radius r ¼ Rwhere Ris arbitrary) and the fact that (3.74) is linear, it follows that O is directly propor-

tional to G (Sedov 1993)

O ¼ Gf1ðn; r; tÞ (3.75)

Two from the three governing parameters in (3.75) have independent

dimensions. Therefore, the difference n� k ¼ 1: Then, in accordance with the

Pi-theorem (3.75) takes the form

P ¼ ’ðP1Þ (3.76)

where P ¼ O=Gna1 ta2 ; and P1 ¼ r=na01 ta

02 .

Bearing in mind the dimensions of O; G; n; r; and t, we find the values of

the exponents ai and a0i as : a1 ¼ �1; a2 ¼ �1; a

01 ¼ 1=2; and a

02 ¼ 1=2: Then,

(3.76) takes the form

O ¼ Gnt’

rffiffiffiffint

p� �

(3.77)

Substituting the expression (3.77) into (3.74), we arrive at the ODE

2ð�’0 Þ0 þ �ð2’þ �’0 Þ ¼ 0 (3.78)

with � ¼ r=ffiffiffiffint

p.

Its solution with the account for the initial condition yields the following well-

known vorticity distribution (cf. Sherman 1990)

O ¼ G4pnt

exp � r2

4nt

� �(3.79)

depicted in Fig. 3.4 b.

3.6 Laminar Flow near a Rotating Disk (the Von KarmanProblem)

The flow sketch is presented in Fig. 3.5 (Karman 1921). The velocity vector of

flow over a rotating disk has three projections u; v and w on the radial, azimuthal

and axial axes of the cylindrical coordinate system associated with the center of

the disk.

3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem) 55

The system of the governing Navier–Stokes and continuity equations

corresponding to this flow takes the following form

u@u

@r� v2

rþ w

@u

@z¼ � 1

r@P

@rþ n

@2u

@r2þ @

@r

u

r

�þ @2u

@z

� �(3.80)

u@v

@rþ uv

rþ w

@v

@z¼ n

@2v

@r2þ @

@r

v

r

�þ @2v

@z2

� �(3.81)

u@w

@rþ w

@w

@z¼ � 1

r@P

@zþ n

@2w

@r2þ 1

r

@w

@rþ @2w

@z2

� �(3.82)

@u

@rþ u

rþ @w

@z¼ 0 (3.83)

The boundary conditions for (3.80–3.83) read

z ¼ 0; u ¼ 0 v ¼ rO w ¼ 0; z ¼ 1; u ¼ v ¼ 0 (3.84)

where it is assumed that the disk rotates with the angular velocity O.Assume that velocity components or pressure at any point of a thin liquid layer

over a rotating disk depend on some characteristic velocity (or pressure), the axial

distance from the disk z and the layer thickness d. Then, the functional equations forthe velocity components and pressure can be written as

u ¼ f1ðu�; z; dÞ (3.85)

v ¼ f2ðv�; z; dÞ (3.86)

z

Ω

w

u

v

0

ϕ

P

r

Fig.3.5 Flow over rotating

disk in liquid at rest


w ¼ f3ðw�; z; dÞ (3.87)

P ¼ f4ðP�; z; dÞ (3.88)

where the velocity component and pressure scales for a given radial position r are

denoted with the asterisks.

The dimensions of the governing parameters in (3.80–3.83) are

u�½ � ¼ LT�1; v�½ � ¼ LT�1; w�½ � ¼ LT�1; z½ � ¼ L; d½ � ¼ L; P½ � ¼ L�1MT�2 (3.89)

It is seen that two of the three governing parameters on the right hand side in

(3.85–3.88) possess independent dimensions. Accordingly, these equations can be

presented in the form

u

u�¼ ’1

z

d

�(3.90)

v

v�¼ ’2

z

d

�(3.91)

w

w�¼ ’3

z

d

�(3.92)

P

P�¼ ’4

z

d

�(3.93)

It is easy to show that the thickness of the fluid layer carried by the disk d is of

the order offfiffiffiffiffiffiffiffin=O

p. Then, taking as the characteristic scales of u; v; w and P as

u� ¼ rO; v� ¼ rO; w� ¼ffiffiffiffiffiffiffinO;

pP� ¼ rnO (3.94)

we arrive at the following expressions

u ¼ rO’1ð�Þ (3.95)

v ¼ rO’2ð�Þ (3.96)

w ¼ffiffiffiffiffiffinO

p’3ð�Þ (3.97)

P ¼ rnO’4ð�Þ (3.98)

where � ¼ ffiffiffiffiffiffiffiffin=O

p.

Using the expressions (3.95–3.98), we transform (3.80–3.83) into the following

ODEs

3.6 Laminar Flow near a Rotating Disk (the Von Karman Problem) 57

2’1 þ ’3 ¼ 0 (3.99)

’21 þ ’

01’3 � ’2

2 � ’001 ¼ 0 (3.100)

2’1’2 þ ’3’02 � ’

002 ¼ 0 (3.101)

’4 þ ’3’03 � ’

003 ¼ 0 (3.102)

where differentiation by � is denoted by prime.

The boundary conditions for (3.99–3.102) become

� ¼ 0; ’1 ¼ 0 ’2 ¼ 1 ’3 ¼ 0 ’4 ¼ 0; � ! 1; ’1 ¼ 0 ’2 ¼ 0 (3.103)

Note that above approach dealing with the flow over an infinite disk can also be

used for the evaluation of flow characteristics in the case of a finite radius disk if the

latter is much larger than the thickness of the liquid layer adjacent to the disk

surface (Schlichting 1979).

3.7 Capillary Waves after a Weak Impact of a Tiny Object ontoa Thin Liquid Film (the Yarin-Weiss Problem)

The flow in a planar thin liquid film on a solid surface after an impact of a tiny wire

(similarly to the axisymmetric case shown in Fig. 3.6) is governed by the following

system of PDEs (the beam equations; Yarin and Weiss 1995)

@2w@t2

¼ a2@4w@x4

(3.104)

@2v

@t2¼ a2

@4v

@x4(3.105)

where w ¼ Dh=h0 is the small dimensionless perturbation of the liquid layer

thickness, with h0 and h being the unperturbed and perturbed thicknesses,

Dh ¼ h� h0; v is the liquid velocity in the x-direction (along the surface),

a ¼ sh0=rð Þ1=2, where s is the surface tension and r the density, t is time.

Equations (3.104) and (3.105) correspond to the situations where gravity and

viscous effects are negligible and perturbations of the liquid layer thickness and

flow velocity are sufficiently small. All these assumptions are realized after impacts

of tiny wire as in Fig. 3.6. Moreover, these objects should be assumed to be point

wise. Then, a given length scale disappears from the problem, and there should exist

a self-similar solution, which we are searching for below.


These equations show that w and v depends on two variables x and t and one

constant a. Therefore, the functional equations for w and x read

w ¼ f1ða; x; tÞ (3.106)

v ¼ f2ða; x; tÞ (3.107)

The dimensions of w; v; a; x and t are

w½ � ¼ 1; n½ � ¼ L2T�1; a½ � ¼ L2T�1; x½ � ¼ L; t½ � ¼ T (3.108)

It is seen that (3.106) and (3.107) contain three governing parameters, whereas

two of them have independent dimensions. In accordance with the Pi-theorem,

(3.106) and (3.107) can be presented in the following dimensionless form

Pw ¼ ’wðP1wÞ (3.109)

Pv ¼ ’vðP1vÞ (3.110)

where Pw ¼ w=aa1 ta2 ; P1w ¼ x=aa01 ta

02 ; Pv ¼ v=aa1� ta2� ; and P1v ¼ x=aa

01� ta

02� .

The exponents ai; ai�; a0i and a

0i� found by applying the principle of the dimen-

sional homogeneity are equal to

a1 ¼ 0; a2 ¼ 0; a01 ¼

1

2; a

02 ¼

1

2; a1� ¼ 1

2; a2� ¼ � 1

2; a

01� ¼

1

2; (3.111)

Fig. 3.6 A system of concentric waves propagating over a thin liquid film on a solid surface from

the impact point of a thin stick seen at the center of the image Reprinted from Yarin and Weiss

(1995) with permission

3.7 Capillary Waves after a Weak Impact of a Tiny Object 59

Accordingly, (3.106) and (3.107) take the form

w ¼ ’wð�Þ (3.112)

v ¼ffiffiffia

t

r� ’vð�Þ (3.113)

where � ¼ x=ffiffiffiffiat

p:

Substituting the expressions (3.112) and (3.113) into (3.104) and (3.105) yields

the following ODEs for the functions ’wð�Þ and ’vð�Þ

’IVw þ 1

4�2’

00w þ

3

4�’

0w ¼ 0 (3.114)

’IVv þ 1

4�2’

00v þ

5

4�’

0v þ

3

4’v ¼ 0 (3.115)

Similarly, in the axisymmetric case corresponding to a weak impact of a tiny

droplet or a stick (Fig. 3.6) the equation for the surface perturbation w

@2w@t2

þ a2

r

@

@rr@3w@r3

� �(3.116)

with r being the radial coordinate can be transformed to the following ODE

’IVw þ 1

�’

000w þ �2

4’

00w þ

3

4�’

0w ¼ 0 (3.117)

where � ¼ r=ffiffiffiffiat

p.

It is emphasized that the solutions corresponding to the self-similar capillary

waves generated by impacts of poitwise objects in reality correspond to remote

asymptotics of capillary waves generated by weak impacts of small but finite

objects.

3.8 Propagation of Viscous-Gravity Currents over a SolidHorizontal Surface (the Huppert Problem)

Gravity currents belong to a wide class of flows in which one fluid with density r1 isintruding into another fluid with a different density r2. Such flows are characteristicof many natural phenomena and various engineering processes (Hoult 1972;

Simpson 1982). Below we consider one type of gravity currents, namely viscous-

gravity currents over a rigid surface (Fig. 3.7). (Huppert, 1982)


Let a layer of a denser fluid of density r invades into a thicker layer of another

fluid of a lower density r� Dr under the action of gravity. Assume that the volume

of the denser fluid increases as ta; where t is time and a½ � ¼ 1 is a constant. The

system of the governing equations that describes such flow in lubrication approxi-

mation reads (Huppert, 1982)

@h

@t� 1

3

g0

n@

@xh3

@h

@x

� �¼ 0 (3.118)

ðxN0

hdx ¼ qta (3.119)

where h is the thickness of the invading fluid layer, q is a constant, xN is the distance

from x ¼ 0 to the leading edge of the invading fluid layer, g0 ¼ Dr=rð Þg, and n is

the kinematic viscosity of the invading denser fluid (cf. Fig. 3.7).

The parameters that are involved in the problem formulation, (3.118) and

(3.119) have the following dimensions

h L½ �; t½ � ¼ T; g0

h i¼ LT�2; n½ � ¼ L2T�1; x½ � ¼ L; xN½ � ¼ L; q½ �

¼ L2T�a; a½ � ¼ 1 (3.120)

It is possible to reduce the number of parameters involved in (3.118) and (3.119)

by introducing new generalized parameters:

h� � ¼ h

q

¼ L�1Ta; A½ � ¼ 1

3

g0q3

n

¼ L5T�ð3aþ1Þ (3.121)

Then, (3.118) and (3.119) take the following form

z

P = P0

h(x, t)

xN(t) x0

r – Δr, na

r, νFig. 3.7 Scheme of viscous

gravity current over a solid

horizontal surface

3.8 Propagation of Viscous-Gravity Currents over a Solid Horizontal Surface 61

@h

@t� A

@

@xh3 @h

@x

� �¼ 0 (3.122)

ðxN0

hdx ¼ ta (3.123)

The position of the leading edge of the gravity-driven current xN depends on one

independent variable t and a generalized parameter A. Therefore, the functional

equation for xN has the form

xN ¼ f x;Að Þ (3.124)

The governing parameters in (3.124) have independent dimensions. In accor-

dance with the Pi-theorem, (3.124) can be reduced to the form

xN ¼ cAa1 ta2 (3.125)

where c½ � ¼ 1 is a constant, and the exponents a1 and a2 are equal to: a1 ¼ 1=5;and a2 ¼ 3aþ 1ð Þ=5, respectively.

Accordingly, the coordinate of the leading edge xN can be expressed as

xN ¼ cA1=5tð3aþ1Þ=5 (3.126)

The thickness h of the gravity-driven current is determined by two independent

variable x and t, as well as by the position of the leading edge of the denser layer

xN [the latter involves the constants A and a, as per (3.126)]. Accordingly, the

functional equation for h reads

h ¼ f ðx; xN; tÞ (3.127)

Two from the three governing parameters involved in (3.127) possess indepen-

dent dimensions. Applying the Pi-theorem to (3.127) we arrive at the following

dimensionless equation

P ¼ ’ðP1Þ (3.128)

where P ¼ h=xb1N tb2 ; and P1 ¼ x=xN; the exponents b1 and b2 are equal: b1 ¼ �1;and b2 ¼ a, respectively.

Bearing in mind the values of the exponents b1 and b2, as well as the expression(3.126), we rewrite (3.128) as

h ¼ x�1N ta’ð�Þ (3.129)


where � ¼ x=xN ¼ c0xA�1=5t�3ðaþ1Þ=5; and c

0 ¼ 1=c.Substitution of the expression (3.126) into (3.129) yields

h ¼ c0A�1=5tð2a�1Þ=5’ð�Þ (3.130)

Calculating the derivatives @h=@t and @h=@x, we transform PDE (3.122) into the

following ODE

c3c0�

�0

�þ 1

53aþ 1ð Þ�c0

� � 2a� 1ð Þcn o

¼ 0 (3.131)

where c ¼ c0� �5=3

’.Equation (3.23) takes the form

c0

�5=3 ð10

cð�Þ ¼ 1 (3.132)

Equations (3.131) and (3.132) manifest the fact that self-similar solutions of the

nonlinear partial differential equations (3.118) and (3.119) do exist. The solutions

of the ODEs corresponding to the plane and axisymmetric problems were found by

Huppert (1982).

Theoretical predictions were compared with the experimental data for the

axisymmetric spreading of silicon oil puddles into air for the release rates corres-

ponding to a ¼ 0 and a ¼ 1 in (2.219). Comparisons were also done between the

results of the theoretical analysis and data for the axisymmetric spreading of

salt water into sweet water in the experiments of Didden and Maxworthy (1982)

and Britter (1979). A good agreement of the theoretical predictions with the

experimental data was demonstrated.

3.9 Thermal Boundary Layer over a Flat Wall (the PohlhausenProblem)

Consider the thermal field over a hot or cold semi-infinite flat wall subjected to

a parallel uniform flow of an incompressible fluid of a different temperature T1far from the wall (Pohlhausen 1921). We assume that the difference between the

fluid and plate temperatures is sufficiently small, as well as neglect dissipation

kinetic energy. We also neglect dependence of the physical properties of the fluid

(the kinematic viscosity and thermal diffusivity) on temperature. In this case the

system of the governing equations reads

3.9 Thermal Boundary Layer over a Flat Wall (the Pohlhausen Problem) 63

u@u

@xþ v

@u

@y¼ n

@2u

@y2(3.133)

@u

@xþ @v

@y¼ 0 (3.134)

u@DT@x

þ v@DT@y

¼ a@2DT@y2

(3.135)

where DT ¼ T � T1:The boundary conditions for (3.133–3.134) are as follows

y ¼ 0; u ¼ v ¼ 0 DT ¼ DTw; y ! 1; u ! U DT ¼ 0 (3.136)

if the wall temperature Tw is given. Another type of the thermal boundary condition

at the plate might be that of thermal insulation. Then, the thermal boundary

condition at the wall in (3.136) is replaced by @DT=@y ¼ 0 at y ¼ 0:Under the assumptions made, the dynamic and thermal problems are uncoupled.

Then, the flow field is described by the self-similar Blasius solution of (3.133) and

(3.134) (see Sect. 3.3 and Schlichting 1979)

u ¼ U’0; v ¼ 1

2

ffiffiffiffiffiffinUx

rð�’0 � ’Þ (3.137)

where ’ ¼ ’ð�Þ is the function determined by (3.51), � ¼ yffiffiffiffiffiffiffiffiffiffiffiU=nx

p; and prime

denotes differentiation by �:The temperature at any point of the thermal boundary layer depends on the

temperature difference DTw ¼ Tw � T1; flow velocity, kinematic viscosity and

thermal diffusivity of fluid, as well as on the location

DT ¼ FðDTw;U; x; y; n; aÞ (3.138)

The dimensions of the governing parameters in (3.126) are

DT½ � ¼ y; U½ � ¼ LxT�1; x½ � ¼ Lx; y½ � ¼ Ly; n½ � ¼ L2yT

�1; a½ � ¼ L2yT�1 (3.139)

Since four of the six governing parameters in (3.138) have independent

dimensions, it can be reduced to the following dimensionless equation

P ¼ #ðP1;P2Þ (3.140)

where P ¼ DT=DTa1w Ua2xa3na4 ; P1 ¼ y=DT

a01

w Ua02xa

03na

04 ; and P2 ¼

a=DTa001

w Ua002 xa

003 na

004 .


Bearing in mind the dimensions of DT; DTw; U; x; y; n and a, we find the

exponents ai a0i and a

000 as

a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a4 ¼ 0

a01 ¼ 0; a

02 ¼ � 1

2; a

03 ¼

1

2; a

04 ¼

1

2

a001 ¼ 0; a

002 ¼ 0; a

003 ¼ 0; a

004 ¼ 1 (3.141)

Accordingly, (3.140) takes the form

DTDTw

¼ #ð�; PrÞ (3.142)

where Pr ¼ n=a is the Prandtl number.

Substituting the expression (3.142) into (3.135), we arrive at the following ODE

#00 þ 1

2Pr’#

0 ¼ 0 (3.143)

The boundary conditions for (3.143) read

# ¼ 1at � ¼ 0; # ¼ 0 at � ! 1 (3.144)

Then, the solution (3.143) is found in the following form

#ð�; PrÞ ¼

Ð1�

’00 ðxÞ� �Pr

dx

Ð10

’00 ðxÞ½ �Prdx(3.145)

where ’ð�Þ is determined Eq (3.51).

At Pr ¼1

#ð�Þ ¼ 1� ’0 ð�Þ ¼ 1� u

u1(3.146)

i.e. the dimensionless excess temperature and velocity fields coincide.

3.10 Diffusion Boundary Layer over a Flat Reactive Plate(the Levich Problem)

Consider distribution of liquid (or gaseous) reactant in the boundary layer over a flat

reactive plate (Levich, 1962). Assume that the rate of an exothermal hetorogenous

reaction at the plate surface exceeds significantly the diffusion flux toward the

3.10 Diffusion Boundary Layer over a Flat Reactive Plate (the Levich Problem) 65

surface, and also neglect the influence of the heat release due on the flow field.

Then, the field of the reactant concentration is described by the following problem

u@c

@xþ v

@c

@y¼ D

@2c

@c2(3.147)

y ¼ 0 c ¼ 0; y ! 1 c ! c0 (3.148)

where the velocity components u and v are determined by the Blasius solution (Sect.

3.3), and c0 is the concentration of the reagent in the undisturbed flow, and D is the

diffusion coefficient.

The governing parameters that determined the concentration field at any point

of the boundary layer are: the concentration c0, the velocity of the undisturbed

flow U, the kinematic viscosity of the liquid or gaseous carrier and diffusity n andD, respectively, as well as the coordinates of the point of interest x; and y: Accord-ingly, the functional equation for the reactant concentration c reads

c ¼ f ðc0;U; x; y; n;DÞ (3.149)

The dimensions of the governing parameters are as follows

c0½ � ¼ L�3M; U½ � ¼ LT�1; x½ � ¼ L; y½ � ¼ L; n½ � ¼ L2T�1; D½ �¼ L2T�1 (3.150)

Since four of the six governing parameters have independent dimensions, (3.49)

takes the following dimensionless form

P ¼ cðP1;P2Þ (3.151)

where P ¼ c=ca10 Ua2xa3na4 ; P1 ¼ y=c

a01

0 Ua02xa

03na

04 ; and P2 ¼ D=c

a001

0 Ua002 xa

003 na

004 :

Taking into account the principle of dimensional homogeneity, we find the

following values of the exponents ai; a0i and a

00i

a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a4 ¼ 0

a01 ¼ 0; a

02 ¼ � 1

2; a

03 ¼

1

2; a

04 ¼

1

2

a001 ¼ 0; a

002 ¼ 0; a

003 ¼ 0; a

004 ¼ 1

(3.152)

Then, (3.151) takes the following form

c

c0¼ cð�; ScÞ (3.153)


where � ¼ yffiffiffiffiffiffiffiffiffiffiffiU=nx

p; and Sc ¼ n=D is the Schmidt number.

Substituting the expression (3.153) into (3.47), we arrive at the following ODE

c00 þ 1

2Sc � c0 ¼ 0 (3.154)

The solution of (3.154) with the corresponding boundary conditions following

from (3.148) has the form

cð�; ScÞ ¼

Ð1�

’00ðxÞ½ �ScdxÐ10

’00 ðxÞ½ �Scdx(3.155)

where ’00 ð�Þis determined by Eq. (1.40).

The diffusion flux of the reactant at the wall is found as

j ¼ �D@c

@y

� �0

¼ Dc0 f ðScÞffiffiffiffiffiU

nx

r(3.156)

where the function f ðScÞ equals to: 0.332Sc1/3 for 0.6<Sc<10, 0:564Sc1=2 for

Sc ! 0, and 0:339Sc1=3 for Sc ! 1(Schlichting 1979).

Problems

P.3.1. Show that flow of an incompressible fluid in a cone does not correspond to

aself-similar solution.

The steady-state flow in a cone is described by the Navier–Stokes and continuity

equations

ðr � vÞv ¼ � 1

rrPþ nr2v (P.3.1)

r � v ¼ 0 (P.3.2)

where v is the velocity vector with the radial and two azimuthal components vr; vy;and v’, respectively, in the spherical coordinate system centered at the cone tip.

The flow in the cone is axially symmetric relative to the axis y ¼ 0: Let the swirlis absent, so that v’ ¼ 0. Under these conditions the velocity components vr and vydepend on two given constants of the problem, namely, the kinematic viscosity of

the fluid ½n� ¼ L2T�1 and the volumetric flow rate ½Q� ¼ L3T�1; as well as on two

Problems 67

coordinates r L½ � and y 1½ �: Then, the functional equation for the velocity components

vr and vy are

vr ¼ frðn;Q; r; yÞ (P.3.3)

vy ¼ fyðn;Q; r; yÞ (P.3.4)

From the three dimensional parameters n; Q and r one dimensionless group can

be constructed, namely, R ¼ r=ðQ=nÞ: Accordingly, (P.3.3) and (P.3.4) take the thefollowing form

vr ¼ ’rðR; yÞ (P.3.5)

vy ¼ ’yðR; yÞ (P.3.6)

where vr ¼ vr Q=n2ð Þ; and vy ¼ vy= Q=n2ð Þ.Thus, vr and vy are the functions of two dimensionless variables R and y that

does not allow to reduce PDEs (P.3.1) and (P.3.2) to ODEs.

P.3.2. Find self-similarity for the flow in the boundary layer of an incompress-

ible fluid near a solid wall in the case of a power-law velocity distribution far from

the plate (Flakner and Skan 1931).

In this case the velocity of the undisturbed flow far from the solid wall is given

by the power law U ¼ cxm; where c is a dimensional and m is a dimensionless

constants, respectively. The boundary layer and continuity equations read

u@u

@xþ v

@u

@y¼ n

@2u

@y2� U

dU

dx(P.3.7)

@u

@xþ @v

@y¼ 0 (P.3.8)

The boundary conditions are as following

y ¼ 0; u ¼ v ¼ 0; y ! 1; u ¼ UðxÞ ¼ cxm (P.3.9)

The governing parameters of the problem are: the dimensional constants c½ � ¼L1�mx T�1; and n½ � ¼ L2yT

�1 and the independent variables x½ � ¼ Lx and y½ � ¼ Ly:

Accordingly, the functional equation for u is

u ¼ fuðc; n; x; yÞ (P.3.10)

It is seen that three governing parameters have independent dimensions, so that

n-k ¼ 1. Then the dimensionless form of (P.3.10) is


P ¼ ’0 ðP1Þ (P.3.11)

where P ¼ u=ca1na2xa3 ; P1 ¼ y=ca01na

02xa

03 ; ’ ¼ ’ðP1Þ and ’

0 ¼ du=dP1.

Taking into account the dimensions of u; c; n; x and y; we find that

a1 ¼ 1; a2 ¼ 0; a3 ¼ m; a01 ¼ � 1

2; a

02 ¼

1

2; a

03 ¼

1� m

2(P.3.12)

Accordingly, (P.3.11) takes the form

u

U¼ ’

0y

ffiffiffiffiffiffiffiffiffiffiffifficxm�1

n

r !(P.3.13)

Substituting the expression (P.3.13) into (P.3.7) and (P.3.8), we obtain the follow-

ing ODE

’000 þ mþ 1

2’’

00 ¼ mð’02 � 1Þ (P.3.14)

where prime denotes differentiation by � ¼ yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficxm�1=n

p:

References

Baehr HD, Stephan K (1998) Heat and mass transfer. Springer, Heidelberg

Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge

Britter RE (1979) The spread of a negatively plum in calm environment. Atmos Environ

13:1241–1247

Blasius H (1908) Grenzschichten in Flussigkeiten mit kleiner Reibung. Z Math Phys 56:1–37

Didden N, Maxworthy T (1982) The viscous spreading of plane and axisymmetric gravity

currents. J Fluid Mech 121:27–42

Douglas JF (1969) An introduction to dimensional anlysis for engineers. Pitman, London

Flakner VM, Skan SW (1931) Some approximate solutions of the boundary layer equiation. Phil

Mag 12:865–896

Hamel G (1917) Spiralformige Bewegungen zahen Flussigkeiten. Jahr-Ber Dtsch Math Ver

25:34–60

Hoult DP (1972) Oil spreading on the sea. Ann Rev Fluid Mech 4:341–368


Huppert HE (1982) The propagation of two–dimensional and axisymmetric viscous- gravity

currents over a rigid horizontal surface. J Fluid Mech 121:43–58

Karman Th (1921) Uber laminare and turbulente Reibung. ZAMM 1:233–252

Kays WM, Crowford ME (1993) Convective heat and mass transfer, 3rd edn. McGraw-Hill, New

York

Landau LD (1944) New exact solution of the navie-stokes equation. DAN SSSR 44:311–314

Landau LD, Lifshitz EM (1987) Fluid mechnics, 2nd edn. Pergamon, Oxford

Levich VG (1962) Physicochemical hydrodynamics. Prentice Hall, Englewood Cliffs


References 69

Pohlhausen E (1921) Der Warmeaustaush zwichen festen Korpern and Flussigkeiten mit kleiner

Reibung and kleiner Warmeleitung. ZAMM 1:115

Rosenhead L (ed) (1963) Laminar boundary layers. Clarendon, Oxford

Sedov LI (1993) Similarity and dimensional methods in mechanics, 10th edn. CRC, Boca Raton

Schlichting H (1979) Boundary layer theory. McGraw-Hill, New York

Sherman FS (1990) Viscous flow. McGraw-Hill, New York

Simpson JE (1982) Gravity currents in the laboratory, atmosphere, and ocean. Ann Rev Fluid

Mech 14:213–234

Stokes GC (1851) On the effect of internal friction of fluids on the motion of pendulums. Trans

Cambridge Philos Soc 9:8–106

Vulis LA, Kashkarov VP (1965) Theory of viscous liquid jets. Nauka, Moscow (in Russian)

Yarin AL, Weiss DA (1995) Impact of drops on solid surfaces: Self-similar capillary waves, and

splashing as a new type of kinematic discontinuity. J Fluid Mech 283:141–173


Chapter 4

Drag Force Acting on a Body Movingin Viscous Fluid

4.1 Introductory Remarks

Drag force is one of the most important factors that determine dynamics of solid

bodies moving in viscous fluids. The knowledge of this force is essential for a

number of applications in engineering, in particular, for the evaluation the engine

power to ensure a desirable velocity of the airplanes, ships, etc., as well as for

analyzing the behavior of solid particles, droplets and bubbles in two-phase flows.

Numerous experimental and theoretical investigations dealing with drag of bodies

of different shapes moving with low and high velocities in viscous fluid were

performed during the last three centuries. The results of these researches are

generalized in monographs by Landau and Lifshitz (1987), Batchelor (1967),

Happel and Brenner (1983), and Soo (1990). The detailed data on drag of bubbles,

droplets and solid particles can be found in the monograph by Clift et al. (1978).

The effect of heating, evaporation and combustion on drag force of small particles

is discussed in the monograph by Yarin and Hetsroni (2004). The application of the

dimensional analysis in studies of drag force of bodies moving in viscous fluid are

discussed in Sedov (1993). The readers are referred to the above-mentioned works,

whereas we discuss briefly some principles that are essential for problems dealing

with drag force acting on bodies moving in viscous fluid and the related

applications of the Pi-theorem.

First we pose the problem on a steady-state linear translation of a solid body of a

given shape with velocity v1 in an infinite uniform incompressible fluid. This

problem is equivalent to the problem on the solid body subjected to a uniform

fluid flow with the undisturbed and constant velocity v1. The velocity and pressure

fields in such flow are determined by the Navier-Stokes and continuity equations

rðv � rÞv ¼ �rPþ mr2v (4.1)

r � v ¼ 0 (4.2)



71

subjected to the no-slip conditions at the body surface y�ðxÞ:v ¼ 0 at y ¼ y�ðxÞ (4.3)

v ¼ v1 at y ! 1 (4.4)

At low Reynolds number (the creeping flow with the Reynolds number

Re � 1) the convective terms in the momentum (4.1) can be omitted (the Stokes

approximation) and the system of (4.1) and (4.2) takes the following form

DP� mr2 v ¼ 0 (4.5)

r � v ¼ 0 (4.6)

Equations (4.1) and (4.2) and the conditions (4.3) and (4.4) show that the

situation under consideration is determined by the following four dimensional

parameters r; m; v1; and d�, which determine the velocity and pressure fields in

fluid, as well as the drag force.1 Then, the functional equation for the drag force

acting on a solid body moving in an unbounded fluid acquires the following form

Fd ¼ f ðr; m; v1; d�Þ (4.7)

where Fd½ � ¼ LMT�2 is the drag force.

The set of the governing parameters corresponding to solid body motion in this

case is

r½ � ¼ L�3M; m½ � ¼ L�1MT�1; v1½ � ¼ LT�1; d�½ � ¼ L (4.8)

Taking into account that three governing parameters in (4.8) have independent

dimensions, we can transform the functional equation for the drag force (4.7) into

the following dimensionless form

P� ¼ ’�ðP1Þ (4.9)

where P� ¼ Fd=ðrv21d2�Þ and P1 ¼ v1d�=n is the Reynolds number.

On the other hand, when a solid body moves in liquid near a gas-liquid interface,

it excites perturbations of the interface. The perturbed interface is wavy either due

to the effect of gravity or/and surface tension. In this case the set of the governing

parameters also includes gravity acceleration g or/and surface tension s. In this casethe set of the parameters determining the drag force reads

r½ � ¼ L�3M; m½ � ¼ L�1MT�1; v1½ � ¼ LT�1; d�½ � ¼ L; g½ �¼ LT�2; or=and, s½ � ¼ MT�2 (4.10)

1Here d� is characteristic size of the body which is fully determined by its shape.

72 4 Drag Force Acting on a Body Moving in Viscous Fluid

Then, the drag coefficient in the case of a body moving in liquid near a gas-liquid

interface becomes

P� ¼ ’1�ðP1;P2Þ; or=and P� ¼ ’2�ðP1;P2;P3Þ (4.11)

where P2 ¼ v1=ffiffiffiffiffiffiffigd�

pand P3 ¼ m=

ffiffiffiffiffiffiffiffiffiffirsd�

pare the Froude and Ohnesorge

numbers, respectively.

In the general case, the set of the governing parameters can also include a

number of parameters that account for roughness of the solid body surface, turbu-

lence of the free stream, etc. In accordance with that the dimensionless form of

the functional equation for the drag force (the functional equation for the drag

coefficient) takes the following form

P� ¼ ’ðn�1Þ�ðP1;P2; :::PnÞ (4.12)

where ’ðn�1Þ� accounts for the effect of the nth factor.It is emphasized that the unknown function ’i� in the expression for the drag

coefficient can be presented as ’i� ¼ a’i; where a is a dimensionless constant. The

numerical value of this constant is chosen in accordance with an accepted protocol

of processing of experimental data. As a rule, the value of the coefficient a is

assumed to be equal to 1=2. Then, (4.9), (4.11) and (4.12) determine the

dependences of the drag coefficient cd ¼ P� 1 2=ð Þ= ¼ Fd= ð1=2Þrv21d2��

on the

dimensionless groups P1;P2 � � �Pn characteristic of the considered problem.

For a spherical body moving with small velocity ( the creeping flow Stokes

approximation) when the drag force is expressed as Fd ¼ 3pmv1d�, it is assumed

that a ¼ p 8= : That leads to the generally accepted expression for the drag coeffi-

cient in the form cd ¼ P� p 8=ð Þ= ¼ 24 Re= where cd is the drag coefficient, Re ¼v1d�=n is the Reynolds number, v1 is the velocity of the undisturbed fluid (or a

particle moving in fluid at rest), and n is the kinematic viscosity.

4.2 Drag Action on a Flat Plate

4.2.1 Motion with Constant Speed

Consider viscous drag of a thin flat plate subjected to a parallel uniform fluid stream

(Fig. 4.1).

At sufficiently high values of the Reynolds number (Re>> 1), the boundary layers

are formed over both sides of the plate. The thicknesses of the boundary layers

increase downstream (cf. Fig. 4.1). The drag force that act on the plate is determined as

Fd ¼ 2b

ðl0

twdx (4.13)

4.2 Drag Action on a Flat Plate 73

where Fd is the drag force, tw is the shear stress at the plate surface, and b and l arethe width and length of the plate.

Bearing in mind the character of flow over the plate, we can assume that drag

force is determined by density and viscosity of the fluid, the undisturbed flow

velocity u1, as well as the length and width of the plate. Then, we can present

(4.13) as follows

Fd ¼ 2bf ðr; m; u1; lÞ (4.14)

where Fd½ � ¼ LMT�2; r½ � ¼ L�3M; m½ � ¼ L�1MT�1; u1½ � ¼ LT�1; l½ � ¼ L.Applying the Pi-theorem to (4.14), we arrive at the following expression for the

drag coefficient

cd ¼ ’ðReÞ (4.15)

where cd ¼ Fd=ru122bl, u1 is the velocity of the undisturbed fluid is the drag

coefficient, and Re ¼ u1l=n is the Reynolds number.

In order to reveal an explicit form of the dependence cdðReÞ, we use the

expression for the shear stress at the plate surface that was found in Chap. 3 by

via the dimensional analysis of laminar flow over a plate

tw ¼ m

ffiffiffiffiffiffiu31nx

r’

00 ð0Þ (4.16)

where ’ð�Þ is determined by solving the Blasius equation (3.51), and

� ¼ y=ffiffiffiffiffiffiffiffiffiffiffiffiffinx=u1

pis the dimensionless variable.

Substitution of the expression (4.16) into (4.13) yields

Fd ¼ 4b’00 ð0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffimrlu31

q(4.17)

y

0x

δ(x)

Plate

u∞Fig. 4.1 A thin flat plate

subjected to a uniform

parallel flow


and

cd ¼ 4’00 ð0ÞffiffiffiffiffiffiRe

p (4.18)

where ’00 ð0Þ is constant.

4.2.2 Oscillatory Motion of a Plate Parallel to Itself

The flow in the vicinity of an oscillating plate is determined by the unsteady boundary

layer equations. Transient and oscillatory flows in the boundary layers are discussed in

the monographs of Schlichting (1979) and Loitsyanskii (1967). In the framework

of the dimensional analysis the problem on a drag Fd experienced by an oscillating

plate involves choosing a set of the governing parameters and subsequent transforma-

tion of the functional equation for Fd to a dimensionless form using the Pi-theorem.

The set of the governing parameters in this case includes the parameters responsible

for the physical properties of the fluid (its density and viscosity r and mÞ; sizes of

the plate (l and bÞ, as well as such flow characteristics as its period t½ � ¼ T(or frequency) of the oscillations and the maximum velocity um that plays the role

of the velocity scale. Then, the functional equation for Fd takes the form

Fd ¼ 2bf1ðr; m; um; l; tÞ (4.19)

It is seen that the present problem contains five governing parameters, three of

which have independent dimensions. Therefore, according to the Pi-theorem (4.19)

can be reduced to the following form

P ¼ ’ðP1;P2Þ (4.20)

where P ¼ Fd=2bra1ua2m la3 ; P1 ¼ m=ra

01u

a02

m la03 ; and P2 ¼ t=ra

001 u

a002

m la003 .

Determining the values of the exponents ai; a0i and a

00i with the help of the

principle of dimensional homogeneity, we find that a1 ¼ 1; a2 ¼ 2; a3 ¼ 1;

a01 ¼ 1; a

02 ¼ 1; a

03 ¼ 1; a

001 ¼ 0; a

002 ¼ �1; and a

003 ¼ 1. Then, we arrive at the

following expression for the drag coefficient

cd ¼ ’1ðRe;KsÞ (4.21)

where cd ¼ Fd 2bru2ml�

is the drag coefficient, Re ¼ uml=n is the Reynolds number,

Ks ¼ St�1; where St is the Strouhal number determined by the maximum velocity

and length of the plate, while Ks ¼ Ks�b; with Ks� ¼ tum=b being the Keulegan-

Carpenter number, and b ¼ b=l. Shih and Buchanan (1971) studied experimentally

the dependence cd ¼ ’1ðRe;KsÞ. It was shown that the drag coefficient of an

4.2 Drag Action on a Flat Plate 75

oscillating plate decreases as the Reynolds number increases. An increase in Ks�also leads to decreasing cd. For the engineering applications the following empirical

correlation is useful

cd ¼ 15ðKsÞ exp 1:88

Re0:547�

� �(4.22)

where Re� ¼ umb=n.The forces acting on cylinders in viscous oscillatory flow are also determined

by Ks� at low values of the Keulegan-Carpenter numbers (Graham 1980; Bearman

et al 1985).

4.3 Drag Force Acting on Solid Particles

4.3.1 Drag Experienced by a Spherical Particle at Low, Moderateand High Reynolds Numbers

In Sect. 4.1 we discussed briefly the application of the Pi-theorem for evaluating

drag force experienced by a solid body moving in viscous fluid. In the present

section we consider this problem in more detail, in particular, dealing with the drag

force acting on a spherical particle at low, moderate and high Reynolds numbers.

The drag of a spherical particle moving in viscous fluid represents the total force

exerted by the surrounding fluid on the particle surface. This force depends on the

physical properties of the fluid, as well as on particle size and its velocity

fd ¼ f ðr; m; d; uÞ (4.23)

where fd is the drag force, d is the particle diameter, and uis the particle velocity

relative to fluid at infinity. The drag force fd and the governing parameters

r; m; d; and u have the following dimensions

fd½ � ¼ LMT�2; r½ � ¼ L�3M; m½ � ¼ L�1MT�1; d½ � ¼ L; u½ � ¼ LT�1

(4.24)

It is seen that three from the four governing parameters have independent

dimensions. Then, in accordance with the Pi-theorem we transform (4.23) into

the following dimensionless form

P� ¼ ’�ðP1Þ (4.25)

where P� ¼ fd=ra1ua2da3 and P1 ¼ m=ra01ua

02da

03 , and the exponents ai and a

0i are

determined from the principle of dimensional homogeneity. They are found as


follows: a1 ¼ 1; a2 ¼ 2; a3 ¼ 2; a01 ¼ 1; a

02 ¼ 1; and a

03 ¼ 1. Then, we obtain

thatP� ¼ fd=ru2d2; and P1 ¼ m=rud ¼ Re�1 and (4.25) takes the following form

cd ¼ ’ðReÞ (4.26)

where cd ¼ P� p 8=ð Þ= and Re are the drag coefficient and the Reynolds number,

respectively.

The explicit forms of the dependence (4.26) can be found in the framework of

the dimensional analysis for two limiting cases corresponding to very small and

very large Reynolds number (see Problems P.4.1 and P.4.2).

Equation 4.26 indicates that the drag coefficient of a spherical particle depends

on a single dimensionless group, namely, the Reynolds number. In order to deter-

mine an exact form of the dependence cdðReÞ, it is necessary to either solve the

hydrodynamic problem on flow of viscous fluid about the particle, or to study it

experimentally. The structure of such flow determines the normal and shear stresses

at the particle surface, i.e. the total drag force.

The flow about a spherical particle that moves rectilinearly with a constant

velocity in fluid is described by the system of the Navier-Stokes and continuity

equations

rðv � rÞv ¼ �rPþ mr2v (4.27)

r � v ¼ 0 (4.28)

which are subjected to the following boundary conditions

v ¼ 0; r ¼ R; v1 ¼ �u; r ¼ 1 (4.29)

where r and m are the density and viscosity of the fluid, R is the particle radius, v isfluid velocity relative the spherical coordinate system associated with the center of

the moving particle, u is the absolute particle velocity, r is the radial coordinate, P is

the pressure; the boldface symbols represent vector quantities.

The inertial term rðv � rÞv on the left-hand side of (4.27) is negligible at low

Reynolds numbers. The problem is thus can be simplified significantly and reduced

to the integration of the linear Stokes equations

rP� mr2v ¼ 0 (4.30)

r � v ¼ 0 (4.31)

subjected to the boundary conditions (4.29).

The solution of (4.30) and (4.31) results in the following Stokes expression for

the drag force

fd ¼ ff þ fp ¼ 3pmud (4.32)

4.3 Drag Force Acting on Solid Particles 77

where ff ¼ � Ðp0

tRy sin y � 2pa2 sin ydy ¼ 2pmud, fd ¼ � Ðp0

P cos y � 2pa2 sin ydy ¼pmud are the contributions to the total drag force from the viscous friction (the shear

stresses) and pressure, respectively, tRy; is the shear stress at the surface, P is the

pressure at the surface, a is the sphere radius, R and y are the radial and angular

coordinates in the spherical coordinate system (Stokes 1851)).

The drag coefficient for a spherical particle becomes accordingly

cd ¼ 24

Re(4.33)

The Stokes’ law (4.32) and (4.33) is valid only for low Reynolds numbers Re �0:1 (cf. Fig. 4.2). The deviation of the predicted values of cd from the experimental

data for the drag coefficient does not exceed 2% at Re � 0.24 and 20% at

Re � 0.75. The experimental data show that the dependence of cd ¼ cdðReÞ has arather complicated shape when a wider range of the Reynolds number values is

considered (Fig. 4.2). In the range of 1 < Re < 800 the drag coefficient is accu-

rately expressed by the empirical Schiller and Naumann law (Clift et al. 1978)

cD ¼ ð24=ReÞ 1þ 0:15Re0:687�

(4.34)

Significant deviations from Stokes’ law are related to the growth of the so-called

form drag component of the drag force at higher Reynolds numbers. It is associated

Fig. 4.2 Drag coefficient of a spherical particle: the solid line – the dependence of the drag

coefficient on the Reynolds number, the dotted line – the Stokes’ law


with the development of the boundary layer near the particle surface and its

separation at the rear part. The later results in a stagnation zone behind the particle

and a reduced pressure at the rear compared to the full dynamic pressure acting at

the front part of the particle.

In range of the Reynolds number 750 � Re � 3 � 105 the drag coefficient is closeto a constant value of 0.445 (the Newton law). At higher Re, the drag coefficient

reveals a dimple at about Re � 2 � 105. The latter is a result of the change in the flowstructure, when transition to turbulence happens in the boundary layer at the sphere

surface, which leads to the flow reattachment to the surface and diminishes the form

drag.

4.3.2 The Effect of Rotation

Particle rotation is a cause of lift force fl, which is directed normally to the plane

formed by the particle velocity and angular velocity vectors v and o, respectively.The magnitude of this force, which is the cause of the Magnus effect, depends on

the physical properties of the fluid and diameter of a spherical particle, as well as on

its velocity (relative to the fluid) u ¼ vj j and the magnitude of the angular velocity

o. Therefore, the functional equation for the lift force reads

fl ¼ f ðr; m; u; d;oÞ (4.35)

The problem at hand involves five governing parameters, three of them with

independent dimensions. Then, in accordance with the Pi-theorem, we find that the

lift force coefficient cl ¼ 4f= ru2pd2=2ð Þ is given by the following expression

cl ¼ ’ðRe; gÞ (4.36)

where g ¼ od=2u is the dimensionless angular velocity.

A lift force also acts at a spherical particle moving in a simple shear flow

characterized by velocity gradient du=dy (Saffman 1965, 1968). In this case the

functional equation for the Saffman lift force flS is

flS ¼ f r; m; u; d;du

dy

�(4.37)

Applying the Pi� theorem, we arrive at the following dimensionless expression

for the Saffman lift force coefficient clS normalized as cl before

clS ¼ ’ðRe; gÞ (4.38)

where g ¼ ðdu=dyÞd2=n.


The important results regarding the Saffman lift force were obtained by Dandy

and Dwyer (1990), McLaughlin (1991), Anton (1987) and Mei (1992). In particu-

lar, Dandy and Dwyer (1990) showed that at a fixed shear rate the lift and drag

coefficients for a spherical particle, normalized using the uniform flow velocity are

approximately constant over the range 40 � Re � 100. On the other hand, the drag

and lift coefficients cd and cl increase sharply as the Reynolds number decreases in

the range Re<10.

4.3.3 The Effect of Acceleration

Transient motions of particles result in additional forces imposed on them by the

surrounding fluid. These forces are related to the acceleration of the surrounding

fluid (the added mass force), as well as to the viscous effects due to delay in flow

development as the velocity changes with time (the Basset force). Transient

motions of spherical particles in an incompressible fluid at rest were studied by

Boussinesq (1903), Oseen (1910, 1927) and Basset (1961). Recently a number of

modified model equations describing transient particle motions in steady-state or

weakly fluctuating flows were proposed (Maxey and Riley 1983; Berlemont et al.

1990; Mei et al. 1991; Mei 1994; Chang and Maxey 1994, 1995). These

modifications were mostly dealing with the history term in the drag force. In

particular, they accounted for the effect of the initial velocity difference between

the fluid and a particle. At the same time, some other approaches to determine the

unsteady drag force are based on the empirical correlations constructed in accor-

dance with the Pi-theorem (Odar and Hamilton 1964; Karanfilian and Kotas 1978).

They account for the influence of particle acceleration du/dt on drag coefficient.

Accordingly, assuming that the transient drag force depends not only on particle

velocity but also on its acceleration, we present the functional equation for fd in thefollowing form

fd ¼ f r; m; u; d;du

dt

�(4.39)

It is easy to see that three from the five governing parameters involved have

independent dimensions. Then, in accordance with the Pi-theorem, we obtain the

drag coefficient based on the previous non-dimensionalization in the following

dimensionless form

cd ¼ ’ Re;Acð Þ (4.40)

where Ac ¼ u2=½ðdu=dtÞd� is the acceleration parameter.

An explicit expression for cd Re;Acð Þ reads (Karanfilian and Kotas 1978)


cd ¼ cd Reð Þ 1þ Ac�1� �n

(4.41)

where n ¼ 1:2� 0:03. The expression (4.41) gives a fairly good agreement with

experiments in the range 102 � Re � 104 and 0 � Ac<1:05.The forces exerted on an “infinite” cylindrical particle in an oscillatory flow

normal to its axis were studied by Kenlegan and Carpenter (1958), Graham (1980)

and Bearmen et al. (1985). There are three factors that determine the drag force: (1)

the inertia of the accelerating outer flow, (2) the influence of the viscous boundary

layers, and (3) separation of these boundary layers leading to vortex shedding. The

drag coefficient of such particles is determined by two dimensionless groups: the

Reynolds and Kenlegan-Carpenter numbers.

4.3.4 The Effect of the Free Stream Turbulence

There is a number of additional factors which affect particle drag. Turbulence of

free stream impinging on a particle is among them. In order to account for the effect

of turbulence intensity on particle drag, it is necessary to extend the set of the

governing parameters by including it in the set. For example, in the case when a

particle moves in a turbulent flow, the functional equation for the drag force fd reads

fd ¼ f ðr; m; u; u0; lÞ (4.42)

where u0 ¼

ffiffiffiffiffiffiu02

pis the root-mean square of turbulent fluctuations of the carrier

fluid, l is the turbulence length scale (it is implied that u0and l are some characteris-

tic values of the turbulent fluctuations and of the turbulence length scale).

Then, the dimensional analysis yields the following equation for the drag

coefficient

cd ¼ ’ðRe; Tu; lÞ (4.43)

where Tu ¼ u0=u is the dimensionless turbulence intensity, and l ¼ l=d is the

dimensionless turbulence scale.

At a given l , the effect of u0on the drag coefficient depends on the Reynolds

number. At sufficiently high Re, close to transition to turbulence in the particle

boundary layer, an increase in the turbulence intensity of the free stream is

accompanied by a decrease in cd. This is a result of a shift of the boundary layer

separation point towards the rear stagnation point. In the range of relatively low

Reynolds numbers, the drag coefficient slightly increase with u0. This effect is due

to the intensified viscous dissipation. The effect of the turbulence scale on the drag

coefficient depends on l . At l<<1 the effect of the turbulence scale is negligible,

whereas at l>1, the drag coefficient increases with l.


4.3.5 The Influence of the Particle-Fluid Temperature Difference

A difference between the particle and surrounding temperature can also affect the

drag force. For the flow of viscous incompressible fluid the functional equation for

the drag force fd in this case reads

fd ¼ f r1; m1; d; u1; T1; TPð Þ (4.44)

where subscripts P and 1 refer to the particle and the ambient parameters.

Applying the Pi-theorem to transform (4.44) to dimensionless form, we arrive at

the following equation

cd ¼ ’ðRe;cÞ (4.45)

where cd ¼ 4fd=p 1 2=ð Þr1u21d2 is the drag coefficient, and c ¼ TP=T1 is the

temperature ratio.

It is emphasized that when transforming (4.44) to the dimensionless form, we do

not account for the effect of temperature on the physical properties of the fluid. At

large enough values of the temperature difference TP � T1, a significant variation

of density and viscosity of fluid within the boundary layer can take place. In this

case the situation becomes complicated. In accordance with that, a set of the

governing parameters should includes thermal conductivity and heat capacity.

The effect of variation of the physical properties of the fluid within the boundary

layer on the drag force was studied by Fendell et al. (1966), Kassoy et al. 1966) and

Dwyer (1989). In particular, Kassoy et al. (1966) evaluated the drag coefficient of a

spherical particle moving in a high-velocity flow of a perfect gas. Assuming a linear

dependence of the viscosity and thermal conductivity on temperature, as well as

constant specific heat and Prandtl number, they derived the following relation for

the drag coefficient

cd ¼ 24

Re

16C

3K� K

3

�(4.46)

where K ¼ OðOþ 2Þ and C is tabulated function of the parameter O ¼ ðc� 1Þ.According to (4.46), the particle drag coefficient increases almost linearly with

O. When O is of the order of one, the drag coefficient at low Reynolds numbers

increases up to 70% over the isothermal value corresponding to the Stokes law.

4.4 Drag of Irregular Particles

Consider the factors which determine drag force of an irregular body moving in an

unbounded viscous incompressible fluid with a constant velocity. From the physical

point of view it is clear that the drag force of such a body should depend on the fluid


properties, the body center-of-mass velocity and orientation relatively to the veloc-

ity vector, as well as the body geometry. The latter (at fixed configuration of an

irregular body) can be determined by one characteristic length, for example, the

length of the wing chord (Fig. 4.3).

The pressure of the undisturbed fluid P1 does not affect the flow field in the

incompressible fluid. Indeed, in this case instead of the total pressure P it is

always possible to consider the difference P� P1: Thus, the value of P1 is

immaterial. It can be excluded from the set of the governing parameters of the

problem (Sedov 1993). Therefore, we can write the functional equation for the

drag force as follows

fd ¼ f ðr; m; u; d; aÞ (4.47)

where d is the characteristic length, and a is the angle of inclination (cf. Fig. 4.3).

By using the Pi-theorem, (4.47) can be transformed to into the following

dimensionless form

cd ¼ ’ðRe; aÞ (4.48)

where cd ¼ 4fd= 1 2=ð Þru2d2 is the drag coefficient.

The Reynolds number characterizes the ratio of the inertia and viscous forces.

The contribution of the viscous forces to the drag force decreases as the viscosity mdecreases, the Reynolds number grows. At large values of Re, the dominant role is

played by fluid inertia. In such cases it is possible to neglect the effect of viscosity

and to reduce the number of the governing parameters to four. Then we obtain

fd ¼ rd2u2’ðaÞ (4.49)

Equation 4.49 shows that in an ideal (inviscid) fluid the drag force is propor-

tional to the velocity squared.

Note that the effect of particle shape becomes important only at sufficiently high

Reynolds numbers when a vortex zone forms behind the particle. In creeping flow

this effect is less expressed. For example, for a thin disk oriented normally to the

flow, the factor in the Stokes’ law equals 8, whereas for a spherical particle it equals

3 p (Lamb 1959). In the general case, the drag coefficient of irregular particles cd

y

0x

α

d

Fig. 4.3 Orientation of an

irregular body in a uniform

flow directed along the x-axis

4.4 Drag of Irregular Particles 83

depends on two dimensionless groups: the Reynolds number and the shape factor

(Boothroyd 1971)

cd ¼ cdðRe;fÞ (4.50)

where f ¼ Seq=S is the shape factor, Seq is the surface area of a volume-equivalent

sphere, and S is the actual surface area. The surface area and diameter of the

volume-equivalent sphere are defined as Seq ¼ p1=3ð6VÞ2=3 and deq ¼ 6V=pð Þ1=3;respectively, where V is the particle volume. In (4.50) the Reynolds number is

based on the equivalent diameter deq.The dependence cdðRe;fÞ is presented in Fig. 4.4 as a family of curves cdðReÞ

corresponding to different values of f Boothroyd (1971).

It is seen that the drag coefficient of irregular particles is larger than the one for a

volume-equivalent spherical particle. The difference increases significantly with

the Reynolds number.

4.5 Drag of Deformable Particles

In distinction from rigid particles, the drag of drops and bubbles depends not only

on the outside velocity distribution but also on their deformation and the inside

flow. The particle deformation is controlled by the competition between the surface

tension and the hydrodynamic forces arising as a result of the particle-fluid interac-

tion. For sufficiently small Weber numbers (We ¼ ru2d=s; where s is the surface

tension), particles remain spherical for any finite Reynolds number. At Re 1 the

functional equation for the drag force of a droplet (a bubble) with viscosity m2moving in a fluid with viscosity m1 has the following form

fd ¼ f ðm1; m2; d; uÞ (4.51)

Fig. 4.4 The drag coefficient

of an irregular particle


Three governing parameters from the four in (4.51) have independent

dimensions. Therefore, (4.51) can be reduced to

P ¼ ’ðP1Þ (4.52)

where P ¼ fd=ma11 d

a2ua3 ; and P1 ¼ m2=ma01

1 da02ua

03 .

Using the principle of the dimensional homogeneity, we find the values of the

exponents in (4.52) as a1 ¼ a2 ¼ a3 ¼ 1; and a01 ¼ 1; a

02 ¼ a

03 ¼ 0. Then, we

arrive at the following equation for the drag force for a small deformable particle

(a drop or a bubble

fd ¼ m1du’ðm21Þ (4.53)

where m21 ¼ m2=m1.Equation 4.53 can be transformed to the following form for the drag coefficient

cd ¼ ’ðm21ÞRe

(4.54)

The function ’ðm21Þ is strongly dependent on the fluid circulation inside a drop,

which is determined by the ratio of fluid viscosities inside and outside it m21. An

increase in the value of m21 is accompanied by flow weakening inside the drop. In

the limiting case of an infinitely large m21, the flow of the inner fluid completely

stops and the drag coefficient approaches to the drag coefficient of a rigid spherical

particle. Therefore, ’ð1Þ ¼ 24. In the second limiting case corresponding to a very

small viscosity of the inner fluid (m21 � 0), the drag coefficient approaches to the

drag coefficient of a bubble and thus ’ð0Þ ! const:An explicit form of (4.54) can be found by integrating the following set of the

momentum balance (in the creeping flow, inertialess approximation) and continuity

equations

�rPþ mðiÞr2vðiÞ ¼ 0 (4.55)

r � vðiÞ ¼ 0; (4.56)

where i ¼ 1 and 2 for the outer and inner fluids, respectively.

The solution of (4.55) and (4.56) is subject to the following boundary conditions:

(1) a given uniform flow at infinity, (2) finite velocity everywhere inside the

particle. Also, at the drop interface with the surrounding fluid: (3) continuity of

tangential stress components, and a jump of the normal stresses due to surface

tension; (4) continuity of the velocity components of the inner and outer liquids. It

is also assumed that the drop keeps its spherical shape during its motion in an

immiscible fluid with different physical properties. This is possible when the drop

diameter is sufficiently small for the surface tension force being dominant com-

pared to the hydrodynamic tractions tending to distort the droplet shape.

The problem (4.55) and (4.56) was solved for a perfectly spherical drop by

Hadamard (1911) and Rybczynski (1911). They derived the following relation for

the drag force acting on a drop moving in uniform fluid

4.5 Drag of Deformable Particles 85

fd ¼ 3pm1u1d2m1 þ 3m23ðm1 þ m2Þ

(4.57)

Accordingly, the drag coefficient becomes

cd ¼ 16

Re

1þ 3m21=21þ m21

(4.58)

Equation 4.58 acquires the simplest forms corresponding to the drag coefficient

for a rigid particle as m2i ! 1 and cd ¼ 24=Re. Another important limit

corresponds to bubbles when m21 ! 0 and cd ¼ 16=Re. Note that the drag on a

spherical particle given by (4.57) and (4.58) does not depend on the density ratio of

the inner and outer fluids, because the creeping flow approximation was employed.

4.6 Drag of Bodies Partially Submerged in Liquid

Consider the drag of a spherical body moving with a constant velocity u along an

air-water interface. Let the density and viscosity of water and air be r1; r2 and

m1; m2, respectively, and the vertical size of the underwater part be l. The latter is

determined by the body weight G ¼ r1gD, where g and D are the gravity accelera-

tion and body displacement, respectively. Thus the set of the governing parameters

in this case includes seven dimensional parameters r1; m1; r2; m2; D; g, and u.The contribution to the drag force from the air-body interaction, as a rule, is

negligible in comparison with the effect of water-body interaction. This makes

the effect of the air density and viscosity r2; and m2 immaterial. Therefore, the

number of the governing parameters can be reduced to only five and, correspond-

ingly, the functional equation for the total drag force acting on a partially

submerged body moving along the air-water interface can be written as

fb ¼ f ðr; m; g;D; uÞ (4.59)

Here and hereinafter subscripts 1 for r and m are omitted. Three from the five

governing parameters in (4.59) have independent dimensions. Then, according to

the Pi-theorem the dimensionless form of (4.59) reads P ¼ ’ðP1;P2Þ, which is

equivalent to

fb ¼ ’ðFr;ReÞ (4.60)

where fb ¼ fb=L3=2mg1=2 is the dimensionless drag force, Fr ¼ u=

ffiffiffiffiffiffigL

pand Re ¼

uL=n are the Froude and Reynolds numbers, and L ¼ D1=3 is the characteristic size

of the body.


Equation 4.60 shows that the drag force of a partially submerged body is

determined by the dimensionless groups with different dependences on the charac-

teristic scale L. Namely, the Reynolds is proportional to L, whereas the Froude

number is inverse to L. This circumstance does not allow modeling of drag force

acting on a partially submerged body (for example, a ship) when using the same

liquid in laboratory experiments as in reality (Sedov 1993).

It is emphasized that drag acting on a solid body partially submerged in viscous

fluid is determined by two different phenomena. (1) Namely, by viscous friction

resulting from body-water interaction, and (2) generation of gravitational waves on

the account of the kinetic energy of motion. Accordingly, the total drag force can be

presented approximately as a sum of the friction ff and wave fw components, as

fb ¼ ff þ fw (4.61)

Introducing the corresponding drag coefficients as

cf ¼ ffð1=2Þru2S ; cw ¼ fw

rgL(4.62)

where S is the cross-sectional area of the body, one can present (4.61) as follows

fb ¼ cfru2S2

þ cwrgL (4.63)

where cf ¼ cf ðReÞ; and cw ¼ cwðFrÞ.A detailed consideration of the applications of the dimensional analysis to

motion of partially submerged solid bodies (in particular, ships) can be found in

the monograph by Sedov (1993).

4.7 Terminal Velocity of Small Spherical Particles Settling inViscous Liquid (the Stokes Problem for a Sphere)

The problem on a small heavy spherical particle settling in viscous liquid was

solved by Stokes (1851). Later on, Bridgman (1922) studied this problem in the

framework of the dimensional analysis. He considered the steady-state stage of

settling (which follows the initial transient stage) when the equilibrium between the

weight, buoyancy and viscous drag has already been established. Under such

conditions particles settle steadily with the so-called terminal velocity v, which is

determined by the particle and liquid densities, particle radius, liquid viscosity and

gravity acceleration g. Then, the functional equation for the terminal velocity reads

v ¼ f ðd; r1; r2; m; gÞ (4.64)

4.7 Terminal Velocity of Small Spherical Particles Settling 87

where r is the particle radius, r1 and r2 are the particle and liquid density,

respectively, and m is the liquid viscosity.

The terminal velocity v and the governing parameters d; r1; r2; m and g have

the following dimensions in the LMT System

v½ � ¼ LT�1; d½ � ¼ L; r1½ � ¼ L�3M; r2½ � ¼ L�1M; m½ � ¼ L�1MT�1;

g½ � ¼ LT�2(4.65)

It is seen that three of the five governing parameters in (4.64) have independent

dimensions, so that the difference n� k ¼ 2: Then, in accordance with the

Pi-theorem, (4.64) reduces to the following expression

P ¼ ’ðP1;P2Þ (4.66)

where P ¼ v= m=r1dð Þ; P1 ¼ r1=mð Þ2gd3; and P2 ¼ r2=r1.Regarding (4.66) Bridgman noticed that “we evidently can say nothing about the

effect on the velocity of any of the elements taken themselves, since they all occur

under the arbitrary functional symbol”. In order to further simplify the analysis,

Bridgman uses the system of units, which includes units of the length, mass, time

and force. At small Reynolds numbers when the effect of liquid inertia is negligible,

the force is treated as its own compensating dimensional constant. In this case

the dimensions of viscosity and gravity acceleration are expressed as it follows

from their definitions: m½ � ¼ F=S du dy=ð Þ0 ¼ FL�2T and g½ � ¼ F m=½ � ¼ FM�1,

where, m is the particle mass, S is its surface area, and du=dyð Þ0 is the velocity

gradient at the surface.

Then, the set of the governing parameters includes five parameters of which four

parameters have independent dimensions, i.e. the difference n� k ¼ 1: Accord-ingly, (4.64) takes the form

P ¼ ’ P1ð Þ (4.67)

where P ¼ v= r2r1g=mð Þ; and P1 ¼ r2=r1.Huntley (1967) obtained (4.67) by using the LxLyLzMT System of Units for the

dimensional analysis the present problem. If the z axis is in the direction of gravity,it is possible to say that the viscous drag depends on the particle cross-section

normal to the direction of its motion, i.e. on the scales Lx and Ly: Since, the diameter

with the dimension Lx has an equal status with the diameter with the dimension

Ly(owing to the axial symmetry of the problem), one can define the characteristic

particle size d½ � as L1=2x L1=2y . Bearing in mind that viscosity is defined as the ratio of

the shear stress to the corresponding velocity gradient, it is possible to determine the

expressions for the “directional” viscosities mx and my, the x and y directions,

respectively,

mx½ � ¼ LxL�1y L�1

z MT�1; my� � ¼ L�1

x LyL�1z MT�1 (4.68)


Taking into account the symmetry of the problem, we write the dimension of

viscosity as m½ � ¼ m1=2x m1=2y

h i¼ L�1

z MT�1� �

. Thus, the dimensions of the governing

parameters are

r1½ � ¼ L�1x L�1

y L�1z M; d½ � ¼ L

12xL

12y; r2½ � ¼ L�1

x L�1y L�1

z M; m½ � ¼ L�1z MT�1 g½ �

¼ LxT�2 (4.69)

Four of the five governing parameters in (4.69) have independent dimensions, so

that the difference n� k ¼ 1: In this case, in accordance with the Pi-theorem, (4.64)

reduces to the following one

P ¼ ’ðP1Þ (4.70)

where P ¼ v=ra11 da2ma3ga4 and P1 ¼ r2=r

a01

1 da02ma

03ga

04 .

The exponents ai and a0i are equal to: a1¼ 1; a2¼ 2; a3 ¼�1; a4¼ 1;

a01 ¼ 1; a

02 ¼ a

03 ¼ a

04¼ 0: Accordingly, (4.70) takes the form of (4.67)

Equation 4.67 determines the dimensionless settling velocity as a function of

only one dimensionless parameter, namely, the ratio of the liquid density to the

particle density. However, the explicit form of this dependence is unknown, which

is a drawback inherent in both (4.66) and (4.67). It should be also emphasized that

the previous consideration is related to a particular case of very-low-Reynolds

number particles when r1dv1=dt ¼ 0; and r2 @v2i=@tþ v2i@v2i=@xkð Þ ¼ 0: In

fact, in this case the influence of the inertial forces is completely neglected.

Obviously, in this case densities of the settling particle and the surrounding liquid

do not influence the process and have to be excluded from the set of the governing

parameters.

The above-mentioned drawbacks can be removed by a correct choice of the set

of the governing parameters beginning from the analysis of the actual physical

factors determining settling of particles in viscous liquid. The latter means that

particle and liquid densities should be accounted for even when the inertial effects

are expected to be small. In particular, settling of a particle in viscous fluid is due to

the excess of particle weight compared to its buoyancy force. Then, the force

responsible for the particle settling is the net weight-buoyancy force

Fnet ¼ 4

3pðr1 � r2Þgr3 (4.71)

Equation (4.71) shows that the difference of the particle and fluid densities

r1 � r2ð Þ, as a factor in the product r1 � r2ð Þg (but not separately r1 and r2 or

their ratio) is one of the governing parameters that affect the driving force and thus,

particle settling. In the set of the governing parameters one should also include fluid

viscosity and particle radius as the factors which the drag and driving forces.

Therefore, the functional equation for the settling velocity v takes the following

form

4.7 Terminal Velocity of Small Spherical Particles Settling 89

v ¼ f ðr; m; gDrÞ (4.72)

where Dr ¼ r1 � r2, and all the governing parameters of the problem have

independent dimensions in the LMT System of Units: r½ � ¼ L; m½ � ¼L�1MT�1; and gDr½ � ¼ L�2MT�2.

Applying the Pi-theorem to (4.72), we obtain

v ¼ cra1ma2ðgDrÞa3 (4.73)


Finding the values of the exponents using the principle of dimensional homoge-

neity, as a1 ¼ 2; a2 ¼ �1; and a3 ¼ 1, we arrive at the following expression

v ¼ cr2gðr1 � r2Þ

m(4.74)

It agrees (up to a dimensionless factor c) with the exact analytical solution of theproblem.

In conclusion, we mention an additional simple approach to determine terminal

velocity from the equilibrium of forces acting on a particle settling in fluid (Lamb

1959). The dimensional analysis shows that the drag force acting on a spherical

particle at Re < 1 can be expressed as Fd ¼ cmrv (with c being a dimensionless

constant). The driving force Fnet is determined by (4.71). The equality of Fd and

Fnet results in (4.74).

4.8 Sedimentation

4.8.1 Dimensionless Groups

Sedimentation of granular materials in fluid flows exerts considerable influence on

various physical processes in nature and engineering. In the present Section we

consider in accordance with Yalin (1972) only one aspect of sedimentation,

namely the application of the Pi-theorem in the sedimentation context. We begin

with the parameters that significantly affect flows carrying heavy granules. Namely,

the density and viscosity of the carrier fluid, the density of granules and their sizes,

the average flow depth h and its slope j ( j ¼ sin a; with a being the slope angle), aswell as gravity acceleration. Correspondingly, the set of the governing parameters

that determine characteristics of uniform two-phase stationary flows carrying

cohesionless granular material consists of the following list

r; m; rs; d; h; j; g (4.75)


where subscript s corresponds to granular material.

Sedimentation inevitably involves the compound parameter involved in weight

and buoyancy forces (cf. Sect. 4.7), gs ¼ gðrs � rÞ, as well as a characteristic

velocity v� ¼ ðgjhÞ1=2, which also accounts for the flow inclination. Then, the set

of the governing parameters takes the following form

r; m; rs; d; h; v�; gs (4.76)

In this case, the functional equation for a flow characteristics A reads

A ¼ f ðr; m; rs; d; h; v�; gsÞ (4.77)

where A is, for example, the average velocity of the fluid, or the rate of the granule

transport, etc.

The dimensions of the governing parameters in (4.77) are as follows

r½ � ¼ L�3M; m½ � ¼ L�1MT�1; rs½ � ¼ L¼3M; d½ � ¼ L; h½ � ¼ L; gs½ �¼ L�2MT�2 (4.78)

Three of the seven governing parameters have independent dimensions, so that

the difference n� k ¼ 4: In accordance with that, (4.77) can be reduced to the

following dimensionless form

P ¼ ’ðP1;P2;P3;P4Þ (4.79)

where P¼A=ra1va2� da3 ;P1¼dv�=n;P2¼rv2�=gsd;P3¼h=d; and P4¼rs=r, and

the exponents ai are determined for any given A by using the principle of dimen-

sional homogeneity.

The first dimensionless groupP1 represents itself the grain-size-based Reynolds

number, which expresses the ratio of the inertia and viscous forces acting on an

individual grain. The second group P2 characterizes the ratio of the hydrodynamic

and buoyancy forces acting on a grain. This group is referred to as the mobility

number and it plays an important role in studies grain motion. The limiting case

P2 ! 0(gs ! 1Þ corresponds to a rough rigid bed. The dimensionless groups P3

and P4 express the influence of the flow depth and the grain material density on

sedimentation, respectively.

4.8.2 Terminal Velocity of Heavy Grains

In the general case the sedimentation velocity of a heavy grain depends on

the physical properties of the carrier fluid (its density and viscosity r and m,respectively), the grain density rs, gravity acceleration g, as well as on time t:

4.8 Sedimentation 91

As it was established earlier, sedimentation is characterized rather by the product

gs ¼ gðrs � rÞ than by g. In this case the functional equation for the terminal

velocity of sedimentation vt takes the form

vt ¼ f ðr; m; rs; gs; tÞ (4.80)

At sufficiently large values of time t, the grain velocity reaches its stationary

value called the terminal velocity. In such a non-accelerating, steady-state motion,

the grain density rs and time tshould be excluded from (4.80). Then, it takes the

following form

vt ¼ f ðr; m; d; gsÞ (4.81)

Since, three of the four governing parameters in (4.81) have independent

dimensions, it can be reduced to the following dimensionless form

P ¼ ’ðP1Þ (4.82)

where P ¼ vtdr=m; P1 ¼ gsrd3=m2 ¼ ðgs=gÞ d3g=n2ð Þ; and g ¼ rg:

Measurements of terminal sedimentation velocity show that all the experimental

data for grains of different types ð1bgs=g<38Þ collapse onto a single curve P ¼’ P1ð Þ (Yalin 1972). For sedimentation of very small grains, when the effect of the

fluid inertia is negligible (in the Stokes creeping flow regime), (4.81) inevitably

takes the scaling form

vt ¼ cma1da2ga3s (4.83)

where the exponents ai are equal a1 ¼ �1; a2 ¼ 2; and a3 ¼ 1, as it follows from

the principle of the dimensional homogeneity.

As a result, we arrive at (4.74).

4.8.3 The Critical State of a Fluidized Bed

Consider detachment of an individual grain from a fluidized bed assuming that the

height of the sand roughness equals grain size and the lower surface of the fluidized

bed is approximately a horizontal plane. The critical state that corresponds to

the onset of the instability refers to a situation, in which grains are beginning

to settle down from the lower surface of the bed. In order to find a relation between

the critical parameters corresponding to onset of such settling, we use (4.79) written

for the sedimentation transport rate

Pqs ¼ fqsðP1;P2;P3;P4Þ (4.84)


where Pqs ¼ qs=rv2�; with qsbeing the total transport rate (the total load per unit

width of the flow).

The positive values of the parameter Pqs ðPqs>0Þ correspond to all possible

states of two-phase flow in the fluidized bed, whereas the casePqs ¼ 0 corresponds

to the critical state. Since the granular material does not settle from the lower side of

the fluidized bed in the critical state, the dimensionless group P4 ¼ rs=r can be

excluded from the set of the governing parameters. In addition, it is also possible to

omit the dimensionless group P3 ¼ h=d; since at the critical state grain motion is

entirely due to the action of the flow in the vicinity of the lower surface of the

fluidized bed. Accordingly, we arrive at the following equation

’ðP1;cr;P2;crÞ ¼ 0 (4.85)

Which is equivalent to

P2;cr ¼ ’ðP1;crÞ (4.86)

For small and large values of P1;cr, the following expressions are valid

P2;cr ¼ FðP1;crÞ ¼ const

P1;cr(4.87)

P2;cr ¼ FðP1;crÞ ¼ const (4.88)

Note, that the first of these equations follows directly from the assumption that

the critical stage at smallP1;cr corresponds to the equality of the hydrodynamic lift

force of Saffman (1965) and the grain weight, i.e to the condition Fl G ¼ 1= ; whereFl andG are the lift force and the grain weight, respectively. At large values ofP1;cr

the fluid viscosity, and accordingly, P1;cr, are no longer the characteristic

parameters. In this case corresponding to very small m, it is possible to assume

that P2;cr is constant.

The predictions of (4.86–4.88) are in a good agreement with the experimental

data (Yalin 1972).

4.9 Thin Liquid Film on a Plate Withdrawn Vertically from aPool Filled with Viscous Liquid (the Landau-LevichProblem of Dip Coating)

In dip coating process a thin film of viscous liquid is withdrawn by a flat plate from

a pool filled with that liquid, as shown in Fig. 4.5. The film is withdrawn by the

moving vertical plate due to the action of viscous forces, which overbear counter-

action of gravity and surface tension. As can be seen in Fig. 4.5, it is possible to

4.9 Thin Liquid Film on a Plate Withdrawn Vertically from a Pool Filled 93

distinguish two characteristic flow domains located at different distances from the

moving plate. In the first of these domains the free surface elevations are small and

the dominant role is played by the capillary force and gravity forces. On the other

hand, in the second domain close to the plate, viscous forces are dominant.

An asymptotic solution of the dip coating problem consists in the calculations

within each one of the above-mentioned domains 1 and 2 and matching the results,

which allows determination of the withdrawn film thickness (Landau and Levich

1942; Levich 1962). Referring the readers to the original work by Landau and

Levich, we restrict our discussion to the analysis of several particular cases of film

withdrawal in dip coating in the framework of the Pi-theorem.

We will consider plate motion with sufficiently small velocities, as specified

below, when the inertia effect is negligible. In this case the thickness of the

withdrawn liquid film h depends on plate velocity v, gravity acceleration g, liquidspecific weight, viscosity and surface tension rg (with r being density), m and s,respectively, as well as on the coordinate along the plate x. Accordingly, thefunctional equation for the thickness of the withdrawn liquid film reads

h ¼ f ðrg; m; v; s; xÞ (4.89)

It is emphasized that the set of the governing parameters does not contain liquid

density alone because the inertia effect is assumed to be negligibly small.

The dimensions of the governing parameters that determine the film thickness

are

ðrgÞ½ � ¼ L�2MT�2; m½ � ¼ L�1MT�1; v½ � ¼ LT�1; s½ � ¼ MT�2; x½ �¼ L (4.90)

Three of the five governing parameters have independent dimensions. Choosing

as these parameters (rgÞ; v and s, we transform (4.89) using the Pi-theorem to the


P ¼ ’ P1;P2ð Þ (4.91)

x

h(x)

3

1

y0

2

g

Fig. 4.5 Withdrawn liquid

film. 1-Liquid pool, 2-the free

surface, 3-plate moving

upwards


where P ¼ h=ðrgÞa1va2sa3 ; P1 ¼ m=ðrgÞa01va

02sa

03 ; and P2 ¼ x=ðrgÞa

001 va

002 sa

003 .

The corresponding exponents are found using the principle of the dimensional

homogeneity as a1 ¼ �1=2; a2 ¼ 0; a3 ¼ 1=2; a01 ¼ 0; a

02 ¼ �1; a

03 ¼ 1; and

a001 ¼ �1=2; a

002 ¼ 0; a

003 ¼ 1=2:

Then (4.91) takes the following form

h ¼ srg

�1=2

’mvs;

x2rgs

�1=2" #

(4.92)

In the limit x>>1; it is possible to assume that the effect of the dimensionless

group P2 ¼ x2rg=sð Þ1=2 has already been saturated and it does not affect the

withdrawn film thickness anymore. Then, (4.92) reduces to the following

expression

h ¼ srg

�1=2

’mvs

� (4.93)

It is convenient to present (4.93) as a dependence of the dimensionless with-

drawn film h ¼ h=l; where l ¼ s=rgð Þ1=2 is the capillary length, on the capillary

number Ca ¼ mv=s

h ¼ ’ðCaÞ (4.94)

The function ’ðCaÞ can be either found experimentally, or through the original

asymptotic solution of Landau and Levich. It has the following form (Levich 1962)

’ðCaÞ ¼ 0:93Ca1=6 (4.95)

at Ca<<1; and

’ðCaÞ � 1 (4.96)

at Ca � 1.

At a high withdrawal velocity the thickness of the withdrawn liquid film h shouldnot depend on the surface tension, since both the effects of gravity and viscous

forces become dominating. Therefore, the functional equation for the thickness of

the withdrawn liquid layer becomes

h ¼ f ðrg; m; vÞ (4.97)

All the parameters in (4.97) have independent dimensions. Therefore, it takes the

following form

h ¼ cðrgÞa1ma2va3 (4.98)


4.9 Thin Liquid Film on a Plate Withdrawn Vertically from a Pool Filled 95

The exponents ai are found from the principle of the dimensional homogeneity

as a1 ¼ �1=2; a2 ¼ 1=2; and a3 ¼ 1=2. Accordingly, (4.98) becomes

h ¼ cmvrg

�1=2

(4.99)

The numerical value of the constant c can be found experimentally and approxi-

mately equals to 2/3 (Derjaguin and Levi 1964).

Problems

P.4.1. Show that at low Reynolds numbers the drag coefficient of a spherical

particle moving with a constant velocity in an incompressible viscous fluid is

inversely proportional to the Reynolds number.

The terms rðv � rÞv and mr2v in the Navier-Stokes equation have the order

of rv2x=lx and mvx=l2x , respectively, whereas and their ratio is of the order of

rv2x=lx�

= mvx=l2x� ¼ Re<<1 (vx and lx are the characteristic scales of the velocity

and length in the direction of the particle motion). That allows to omit the term

rðv � rÞv from left hand side of the Navier-Stokes equation and thus eliminate

the fluid density from the set of the governing parameters. Then the functional

equation for the drag force takes the following form

fd ¼ f ðm; u; dÞ (P.4.1)

All the governing parameters in (P.4.1) have independent dimensions, so that the

difference n� k ¼ 0. Therefore, in accordance with the Pi-theorem we obtain

fd ¼ Cma1ua2da3 (P.4.2)

where C is a dimensionless constant.

Taking into account the dimensions of fd; m; u and d, we find the values of the

exponents ai from the principle of dimensional homogeneity as a1 ¼ a2 ¼ a3 ¼ 1.

Accordingly, the drag force and the drag coefficient are expressed as

fd ¼ Cmdu (P.4.3)

cd ¼ C1

Re(P.4.4)

Equations P.4.3 and P.4.4 coincide with the Stokes law up to a dimensionless

factor. The values of the constants C and C1, which can be either determined from


the exact solution of the Stokes equations or measured experimentally are 3 and 24,

respectively.

P.4.2. Show that the drag coefficient of a spherical particle does not depend on

Re at the high values of the Reynolds number.

At high values of the Reynolds number the inertia forces play dominant role.

Under these conditions it is possible to omit fluid viscosity from the set of the

governing parameters. That leads to (4.48) for the drag force. For a spherical

particle ’ðaÞ is constant. Then, we obtain from (4.48) the following expression

for the drag coefficient

cd ¼ fd1=2ð Þru2d2 ¼ const: (P.4.5)

P.4.3. Derive a dimensionless equation for the drag force acting on an infinitely

long cylinder in subjected to a uniform laminar flow of incompressible fluid normal

to its axis.

Let the fluid density and viscosity be r and m, respectively, and velocity of the

undisturbed flow and radius of cylinder be u and R, respectively. Then the drag

force fc is determined by the following fourth governing parameters

fc ¼ f ðr; m; u;RÞ (P.4.6)

That have the dimensions as listed: r½ � ¼ L�3M; m½ � ¼ L�1MT�1; u½ � ¼ LT�1

and R L½ �. The dimension of the drag force acting at a unit length of the cylinder is

fc½ � ¼ MT�2. Since three of the four governing parameters have independent

dimensions, (P.4.6) reduces to the following form

P ¼ ’ðP1Þ (P.4.7)

where P ¼ fc=ra1ma2ua3 ; and P1 ¼ R=ra01ma

02ua

03 .

Bearing in mind the dimensions of the drag force and the governing parameters

and applying the principle of the dimensional homogeneity, we find values of the

exponents ai and a0i as: a1 ¼ 0; a2 ¼ 1; a3 ¼ 1; a

01 ¼ �1; a

02 ¼ 1; and a

03 ¼ �1.

Then, the drag force fc can be expressed as

fc ¼ mu’ðReÞ (P.4.8)

where Re ¼ ruR=m is the Reynolds number.

Note, that the exact solution of the problem based on Oseen’s equation reads

(Lamb 1959)

fc ¼ 4pmu1=2� Cþ lnðReÞ ¼

4pmulnð37=ReÞ (P.4.9)

Problems 97

where C ¼ 0:577::::is the Euler constant.Comparing the expressions (P.4.8) and (P.4.9), we find that ’ðReÞis

’ðReÞ ¼ 4plnð37=ReÞ (P.4.10)

P.4.4. Show that in the frame of Stokes’ approximation it is impossible to

determine an steady-state drag force acting on a unit length of an infinitely long

cylinder.

Stokes’ approximation is based on the assumption that the inertial term rðv �rÞvin (P.4.6) can be neglected and, thus, the drug force acting on a unit length of a

cylinder is written as

fc ¼ f ðm; u;RÞ (P.4.11)

Applying the Pi-theorem to (P.4.11) leads to an unrealistic result: the drag force

does not depend on the body size, i.e. its radius (Landau and Lifshitz. 1987).

P.4.5. Find the drag force acting on two round coaxial disks submerged in a fluid

and moving towards each other.

Let the lower disk is motionless, whereas the upper one moves with a constant

velocity u towards the lower one. The approach of the disks inevitable squeezes thefluid from the gap in between. The force acting on the disks due to the difference of

pressure in the gap P and in the surrounding fluid P0 is

fdisk ¼ 2pðR0

ðP� P0Þrdr (P.4.12)

The force fc ¼ LMT�2 is determined by the following four governing

parameters: the fluid viscosity m½ � ¼ L�1MT�1, the velocity of the moving disk

u½ � ¼ LT�1, the disk radii R½ � ¼ L, and the current gap width between the disks

h½ � ¼ L. Therefore, it is given by the following equation

fc ¼ f ðm; u;R; hÞ (P.4.13)

Three of the four governing parameters have independent dimensions, which

allows us to reduce (P.4.13) to the following dimensionless form

P ¼ ’ðP1Þ (P.4.14)

where P ¼ fdisk=ma1ua2Ra3 ; and P1 ¼ h=ma01ua

02Ra

03 .

Taking into account the dimensions of fdisk and h; m; u; and R and applying the

principle of the dimensional homogeneity, we find values of the exponents ai and a0i

as: a1 ¼ a2 ¼ a3 ¼ 1; a01 ¼ a

02 ¼ 0; and a

03 ¼ 1. Then, we obtain the following

expression for the drag force


fdisk ¼ muR’h

R

�(P.4.15)

It is seen that the drag acting on a disk is directly proportional to the product of

the viscosity, velocity of the disk, its radius, as well as a function of the ratio h/R.This function cannot be established in the framework of the dimensional analysis.

An exact solution of the problem yields

fdisk ¼ 3p2muR

R

h

�3

(P.4.16)

P.4.6. Determine the dependence of the friction force acting on a sphere rotating

in an infinite viscous fluid.

The friction force ff� � ¼ LMT�2 depends on the following three governing

parameters: the fluid viscosity m½ � ¼ L�1MT�1, sphere radius R½ � ¼ L, and the

angular velocity of rotation o½ � ¼ T�1. These parameters have independent

dimensions. According to the Pi-theorem, we have the following functional equa-

tion for the drag force

ff ¼ Cma1Ra2oa3 (P.4.17)

where C is a constant.

Bearing in mind the dimensions of ff ; m; R and o, we arrive at the following

result

ff ¼ CmR2o (P.4.18)

P.4.7. Determine the dependence of the semi-axes ratio of a droplet moving in

air, which acquires an approximately spheroidal shape, on the physical properties of

liquid and velocity of motion.

The surface of a relatively small droplet moving in air approximately resembles

a spheroid. We adopt the Cartesian axes OX; OY; and OZ with the center O at the

droplet center and the axis OZ directed with the undisturbed flow experienced by an

observer moving with the droplet. The droplet surface is approximated as a spher-

oid with semi-axes a and b (with a>b)

x2 þ y2

a2þ z2

b2¼ 1 (P.4.19)

The deviation of the droplet surface from a sphere is characterized by the

difference a� b that depends on the characteristic size of the droplet, as well as

the density, velocity and surface tension of the liquid

a� b ¼ f ðb; r; u; sÞ (P.4.20)

Problems 99

The three governing parameters in (P.4.20) have independent dimensions.

Therefore, the dimensionless form of (P.4.20) reads

P ¼ ’ðP1Þ (P.4.21)

Where P ¼ ða� bÞ=ra1ua2sa3 ; and P1 ¼ b=ra01ua

02sa

03 .

Determining the values of the exponents ai and a0i, we arrive at the following

equation

w ¼ 1þ cðWeÞ (P.4.22)

where w ¼ a=b, and We ¼ ru2b=s is the Weber number. The latter can equally be

based on the volume-equivalent droplet diameter d.P.4.8.Determine the moment of force required for a slow steady-state rotation of

an infinitely long solid cylinder of radius R1 with the angular velocity o about its

axis. Consider two cases: (1) the cylinder rotates in an infinite viscous fluid; and (2)

the cylinder rotates inside a cylindrical shell of radius R2, which is sufficiently

larger than R1, filled with a viscous fluid.

1. The moment of force required for swirling of a cylinder in viscous fluid is given

by the expression

Mf ¼ð2p0

tSd’ (P.4.23)

where t is the surface traction, S ¼ 2pR1 is the area of the lateral surface of the

cylinder of unit length.

The functional equation for the moment of force in the inertialess case of slow

swirling considered here reads

Mf ¼ f ðm;o;RÞ (P.4.24)

All the governing parameters in (P.4.24) have independent dimensions,

namely m½ � ¼ L�1MT�1; o½ � ¼ T�1; and R½ � ¼ L. Then, in accordance with

the Pi-theorem, (P.4.24) reduces to the following one

Mf ¼ cma1oa2Ra31 (P.4.25)

where c is a constant.

Accounting for the dimension of the moment of force Mf

� � ¼ LMT�2, we find

the values of the exponents as a1 ¼ 1; a2 ¼ 1; and a3 ¼ 2: Then, (P.4.25) takesthe following form

Mf ¼ cmoR2 (P.4.26)


2. In the second case the functional equation for the moment of force reads

Mf ¼ f ðm;o;R1; e; eÞ (P.4.27)

where e ¼ R2 � R1 is the gap between the cylinder and the surrounding shell, and eis the eccentricity ðe L½ �; and e L½ �Þ:

Three of the five governing parameters in (P.4.27) possess independent

dimensions, so that the difference n� k ¼ 2: Therefore, the dimensionless

form of (P.4.27) reads

P ¼ ’ðP1;P2Þ (P.4.28)

where P ¼ Mf =moR21; P1 ¼ e=R1; and P2 ¼ e=R1:

In the particular case of concentric cylinder and shell ðe ¼ 0Þ, (P.4.28) takes theform

Mf ¼ moR21’

eR1

�(P.4.29)

In the general case the expression for Mf has the form (Loitsyanskii 1966)

Mf ¼ 4pmoR3

ecðlÞ (P.4.30)

where cðlÞ ¼ 1þ 2l2�

= 2þ l2� ffiffiffiffiffiffiffiffiffiffiffiffiffi

1� l2ph i

; and l ¼ e=e.

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Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge

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Bearman PW, Dowirie MJ, Graham JMP, Obasaju ED (1985) Forces on cylinders in viscous

oscillatory flow at low Kenlegan-Carpenter numbers. J Fluid Mech 154:337–356

Berlemont A, Desjonqueres P, Gouesbet G (1990) Particle Lagrangian simulation in turbulent

flow. Int J Multiphas Flow 16:19–34

Boothroyd RG (1971) Following gas-solids suspensions. Chapman and Hall, London

Boussinesq J (1903) Theorie Analytique de la Chaleur. L’ Ecole Polytechnique, Paris

Bridgman PW (1922) Dimensional analysis. Yale University Press, New Haven

Chang EJ, Maxey MR (1994) Unsteady flow about a sphere at low to moderate Reynolds number.

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17:42–54

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Chapter 5

Laminar Flows in Channels and Pipes


Fluid flow in pipes and ducts was a subject of numerous experimental and theoretical

investigations performed during the last two centuries. Beginning from the seminal

works of Hagen (1839) and Poiseuille (1840), a detailed data on flows of incom-

pressible viscous fluids in pipes and ducts of different geometry was obtained.

These results are presented in many review articles, monographs and textbooks.

A comprehensive analysis of problems related to laminar and turbulent flows in

pipes and ducts (the physical foundations of the theory and its mathematical

formulation) can be found in such widely known books as Schlichting (1979),

Landau and Lifshitz (1987), Loitsyanskii (1966) and Ward-Smith (1980). Refering

the readers to these monographs, we focus on the applications of the Pi-theorem for

the analysis of pipe and duct flows.

First we discuss some specific features of laminar flows of incompressible

viscous fluid outflowing from a large tank through a conical nozzle into a straight

pipe or duct. A sketch of such flow within the entrance and fully developed sections

of a pipe located, correspondingly, near and far from the inlet is shown in Fig. 5.1. It

is seen that the velocity profile which is uniform at the entrance cross-section of the

pipe becomes non-uniform at x>0 as a consequence of fluid-wall interaction at the

flow periphery.

Under the conditions corresponding to relatively large Reynolds numbers deter-

mined by the mean velocity, pipe diameter and fluid viscosity, laminar flows in

straight pipes can be schematically presented as follows. Near the pipe inlet, the

boundary layer forms over the wall. The thickness of the layer increases downstream

(cf. the left part of Fig. 5.1). The longitudinal velocity component u increases withinthe boundary layer from zero at the wall to the velocity of the undisturbed fluid flow

in the potential core. The velocity in the potential core gradually increases from an

initial velocity u0 at x ¼ 0 up to the maximum one, umax, in the cross-section x ¼ lenwhere the boundary layers from different sides merge. Approximately at x ¼ len thevelocity profile approaches to the parabolic Poiseuille profile, the flow becomes fully



103

developed, and does not vary with x anymore. Thus, the following two flow regions

in the pipe flows are distinguished. Namely, (1) the entrance flow region (x<len), andthe fully developed flow region (x>len). The flow characteristics within the entrance

region are mostly determined by the boundary layer, whereas within the fully

developed region all fluid elements move parallel to the pipe axis x with the

longitudinal velocity u ¼ uðyÞ; and v ¼ w ¼ 0, where x and y are the longitudinaland transversal coordinates, respectively. Schlichting (1979).

Stationary flows of incompressible viscous fluid in straight pipes are described

by the Navier–Stokes and continuity equations

v � rv ¼ � 1

rrPþ nr2v (5.1)

r � v ¼ 0 (5.2)

where r; P and v are the density, pressure and velocity vector, respectively, n is thekinematic viscosity.

Solutions of (5.1) and (5.2) should satisfy the no-slip boundary conditions at the

wall, as well as to correspond to given velocity distributions v0 and v00 at the inlet(x ¼ 0) and outlet (x ¼ L) of the pipe

v ¼ v0 at x ¼ 0; v ¼ v00 at x ¼ L; v ¼ 0 at the pipe wall (5.3)

The approximate analytical solutions of the problem on the flow in the entrance

region of a plane channel was obtained by Schlichting (1934) and Schiller

(1922). In this cases the length of the entrance region len is determined by the

expression

lenl�Re

¼ c (5.4)

Fig. 5.1 Velocity profiles in laminar flows in straight cylindrical pipes

104 5 Laminar Flows in Channels and Pipes

where l� is the characteristic size of a channel or a pipe, a semi-height or a radius,

respectively. Re is the Reynolds number based on the channel height or pipe

diameter, and c is a dimensionless constant.

The numerical solution of the Navier–Stokes equations (Friedmann et al. 1968),

as well as the experimental studies of Emery and Chen (1968) and Fargie and

Martin (1971) show that c in (5.4) depends on the Reynolds number at Re<500:Within this range of the Reynolds number, c decreases monotonously as Re

increases. In the range Re>500 c is practically constant (cf. Fig. 5.2).

Accordingly, in the framework of the two region model described above, the

problem on the laminar flow in a pipe of circular cross-section amounts to solving

the following system of equations (Loitsyanskii 1966)

ru@u

@xþ rv

@u

@y¼ � dP

dxþ m

1

yk@

@yyk@u

@y

� �(5.5)

@u

@xþ 1

yk@vyk

@y¼ 0 (5.6)

for the entrance region (in the boundary layer approximation), and

dP

dx¼ mr2u (5.7)

for the fully developed region. In (5.5) and (5.6) k ¼ 0 or 1 for the plane or

axisymmetric flows, respectively.

Fig. 5.2 The dependence of the length of the entrance section on the Reynolds number in laminar

flow in a straight cylindrical pipe. 1-Calculations by Friedman et al. (1968), 2-Data by Emery and

Chen (1968)

5.1 Introductory Remarks 105

The boundary conditions for (5.5) and (5.6) read

y ¼ 0 u ¼ v ¼ 0; y ¼ yd u ¼ ud (5.8)

(with yd being the boundary layer thickness and u ¼ udðxÞ the velocity of the

external flow velocity in the potential core), and for (5.7)

y ¼ 0@u

@y¼ 0; y ¼ r0 u ¼ 0 (5.9)

In (5.5), (5.6) and (5.8) y is understood as the distance from the wall to a point

within the boundary layer. On the other hand, in (5.7) and (5.9) y is reckoned from

the channel axis and r0 is the cross-sectional radius of the pipe.It is emphasized thatwithin the entrance region of the flow the pressure gradient is a

function on x, whereas within the fully developed flow region it is constant. That

allows us to understand, as usual for the Poiseuille flows, that in (5.7)

dP=dx ¼ �DP=L; where DP is the pressure drop over a pipe section of length l.As usual, two types of questions can be posed regarding flows in pipes (and

similarly, in planar channels). In the first one, a volumetric flow rate Q is given,

while the corresponding pressure gradient required to sustain such flow, DP=lshould be found. In the second one, a pressure drop DP=l is given, while the

corresponding volumetric flow rate Q is to be found.

5.2 Flows in Straight Pipes of Circular Cross-Section

5.2.1 The Entrance Flow Region

The flow of viscous fluid within the entrance region of a pipe is illustrated in

Fig. 5.1. As a result of the fluid friction at the wall, the boundary layer is formed.

The thickness of the boundary layer dðxÞ increases along the pipe, with the

longitudinal coordinate x. At some distance from the inlet x ¼ len, the boundary

layer fills the whole pipe cross-section and thus dðlenÞ ¼ d=2:Having in mind the two-region pattern of pipe flows, we evaluate their

characteristics using the dimensional analysis. First, we evaluate the dependence

of the length of the entrance region len on the flow parameters. It is natural to

assume that the thickness of the boundary layer depends on the fluid density and

viscosity r and m, velocity in the undisturbed core u, as well as the distance from

the inlet to a cross-section under consideration. Then, we have the functional

equation for dðxÞ as follows

dðxÞ ¼ f ðr; m; u; xÞ (5.10)

In the framework of the boundary layer theory expediently it is natural to use the

System of Units LxLyLzMT with three different length scales Lx,Ly and Lz in the x,y


and z directions, respectively. Using this system of units, we express the

dimensions of the governing parameters as r½ � ¼ L�1x L�1

y L�1z M; m½ � ¼

L�1x LyL

�1z MT�1; u½ � ¼ LxT

�1, x½ � ¼ L, and the thickness of the boundary layer as

d½ � ¼ Ly. Taking into account that all the governing parameters have independent

dimensions, we rewrite (5.10) as follows

dðxÞ ¼ cra1ma2ua3xa4 (5.11)

where c is a dimensionless constant and the exponents ai found using the

principle of the dimensional homogeneity, read a1 ¼ �1=2; a2¼ 1=2; a3 ¼ �1=2; and a4 ¼ 1=2.

Then (5.11) takes the form

dðxÞ ¼ cmxru

� �1=2

(5.12)

or

dðxÞd

¼ cnxd2u

� �1=2(5.13)

Bearing in mind that dðlenÞ ¼ d=2, we obtain

lend

¼ c�Red (5.14)

where1 Red ¼ ud=n; and c� ¼ ð2cÞ�21

.

The dependence (5.14) qualitatively agrees with the theoretically predicted

value of len ¼ 0:04dRe (Schlichting 1979). The results of the numerical investiga-

tion of the dependence lenðReÞ for flows in straight cylindrical pipes are presented inFig. 5.2. They show that the value of the ratio len=dRe is approximately constant at

Re > 500 when the boundary layer approximations are valid.

In addition, we address the pressure drop over the entrance region of the pipe

flow. Equation (5.14) shows that the length of the entrance region len depends on

pipe diameter and the Reynolds number. That allows us to use len as a generalizedparameter of the problem. Namely, assume that the ratio dP=dxð Þ=ru2 depends on asingle dimensional parameter len and one variable x. Then, the functional equationfor this ratio becomes

dP

dx

� �ru2� �� ¼ f ðlen; xÞ (5.15)

1 Re½ � ¼ 1 since it is defined by the parameter magnitudes.

5.2 Flows in Straight Pipes of Circular Cross-Section 107

In (5.15) one of the two governing parameters has an independent dimension.

Then, in accordance with the Pi-theorem, (5.15) reduces to the dimensionless

form

P ¼ ’ðP1Þ (5.16)

where P ¼ dP=dxð Þ= ru2ð Þ=laen; and P1 ¼ x=la0

en.

Bearing in mind the dimension of dP=dx½ � ¼ L�1y L�1

Z MT�2, we find the

values of the exponents ai and a0i as a ¼ �1; a

0 ¼ 1. Accordingly, (5.16) takes

the form

dP=dxð Þlenru2

¼ ’x

len

� �(5.17)

Substituting the expression (5.14) for len into (5.17), we arrive at the

dependence

Po ¼ ’ðXÞ (5.18)

where the Poiseuille numberPo¼ c�lRed; l¼ dP=dxð Þd=ru2; and X¼ x= c�dRedð Þ.Equation (5.18) shows that the Poiseuille number is a function of one dimen-

sionless variable X alone (Fig. 5.3).

Fig. 5.3 The dependence of the Poiseuille number on X


5.2.2 Fully Developed Region of Laminar Flows in Smooth Pipes

Consider pressure drop in the fully developed flow region. To choose the governing

parameters for this flow, it is necessary to account for such the specific features as

(1) constancy of the pressure gradient DP=l; and (2) fluid motion parallel to the pipe

axis with a constant mean velocity when density does not affect the flow. Under

these conditions the functional equation for the pressure drop takes the form

DPl

¼ f ðm; u0; dÞ (5.19)

where m½ � ¼ L�1MT�1 is the dimension of the viscosity, u0½ � ¼ LT�1 is dimension

of the mean velocity and d½ � ¼ L (with d ¼ 2r0 being the cross-sectional diameter),

i.e. they have independent dimensions.

Then, using the Pi-theorem, we can transform (5.19) to the following one

DPl

¼ cma1ua20 da3 (5.20)

where c is a dimensionless constant, and the exponents ai are equal to a1 ¼ 1;a2 ¼ 1 and a3 ¼ �2:

As a result, we arrive at the expression

DPl

¼ cmu0d2

(5.21)

Dividing both sides of (5.21) by ru2, we obtain the following expression for the

friction factor l

l ¼ c�Re

(5.22)

where l ¼ 2 DP=lð Þ d=ru20� �

is the friction factor, c� ¼ 2c is a dimensionless

constant.

As follows from the exact analytical solution of the Hagen-Poiseuille problem

on laminar flows in round smooth pipes, the factor c� in (5.22) equals 64.

5.2.3 Fully Developed Laminar and Turbulent Flowsin Rough Pipe

Turbulent pipe flows were studied experimentally in Nikuradse (1933) who used

uniform sand grains of different sizes glued to the inside walls of pipes to vary

roughness in a controlled manner. It is emphasized that in stable laminar pipe flows

5.2 Flows in Straight Pipes of Circular Cross-Section 109

at sufficiently low values of the Reynolds number (Re � 2000) the wall roughness

was assumed to have minor effect. In stable pipe flows perturbations introduced by

roughness elements were expected to fade sufficiently fast to affect flow field

globally (Schlichting 1979). Later on, Kandlicar (2005) revisited the results of

Nikuradse’s experiments in the light of the recent studies of laminar flows in

micro-channels with rough walls and concluded that surface roughness plays an

important role in fluid flow in micro and mini-channels. The numerical calculations

by Hezwig et al. (2008) showed that in laminar flows the resistance of rough-wall

channels with regular roughness significantly depends on the relative roughness

size. A number of experimental studies of laminar flows in micro-channels also

suggest some influence of wall roughness on resistance to flow in such channels

(Yarin et al. 2009). Nevertheless, in spite of the existence of a number of theoretical

and experimental works devoted to laminar flows in rough pipes and channels, the

role of roughness and mechanisms of its influence on the resistance of such

channels is not clear yet.

Below we attempt to reveal some special features characteristic of resistance of

rough straight pipes on fully developed laminar flow by using the dimensional

analysis. Suppose that a rough pipe is characterized by two dimensional parameters,

namely, its effective diameter def and the height of roughness elements k. Inaddition, the set of the governing parameters of such flows should include three

dimensional parameters that characterize the physical properties of fluid-its density

r and viscosity m; as well as the average flow velocity u: (A discussion concerning

the choice of the characteristic size of rough elements see in Hezwig et al. 2008).

Then, the functional equation for pressure drop over pipe section of length l reads

DPl

¼ f ðr; m; u; def ; kÞ (5.23)

The governing parameters in (5.23) have the following dimensions:

r½ � ¼ L�3M; m½ � ¼ L�1MT�1; u½ � ¼ LT�1; def ¼ L; k½ � ¼ L: Three of the five

governing parameters have independent dimensions. Therefore, according to the

Pi-theorem, (5.23) can be transformed to the form

P ¼ ’ P1;P2ð Þ (5.24)

where P ¼ 2DPd=lru2 ¼ l is the friction factor, P1 ¼ m=rudef ¼ Re�1;

with Re ¼ rudef =m, which is the Reynolds number, and P2 ¼ k=def ¼ k is the

relative roughness of pipe wall.

The dimensional analysis does not allow finding an exact form of the depen-

dence of the drag friction factor on the dimensionless groups- the Reynolds number

and relative roughness of the wall. However, it is possible to evaluate some

particular trends of the friction factor when Re and k are changing. Indeed. At

large values of the Reynolds number corresponding to fully developed turbulence,

the inverse Reynolds number P1 is much less than the value of the relative


roughnessP2. In this case the effect of viscosity is negligible, and the friction factor

is determined mostly by the relative roughness l � ’ðkÞ. In the second

corresponding to relatively small Reynolds number (laminar flows), the inverse

Reynolds numberP1 can be much large than the relative roughnessP2. Therefore,

in the latter case the roughness effects are weak. It is emphasized that at any finite

values of Re the influence of wall roughness on the friction factor of a pipe exists.

Only at Re ! 0, (5.24) reduces to the form l ¼ ’ðReÞ, whereas within the range ofmoderate Reynolds numbers the friction factor depends on both dimensionless

groups, namely l ¼ lðRe; kÞ. The latter form practically encompasses the whole

bulk of the available experimental data (Moody 1948; Schlichting 1979).

5.3 Flows in Irregular Pipes and Ducts

At applying the Pi-theorem to flows in irregular (non-circular) pipes or channels, it

is convenient to use an effective size def L½ � expressed through the cross-section

geometry. For example, the so-called hydraulic diameter dh defined as 4S P= , where

S is the cross-sectional area and P the wetted perimeter is used as an effective size

of irregular pipes and channels. In this case the form of the functional equation for

the pressure drop in irregular pipes and ducts remains the same as the regular ones

(with circular cross-section). However, as the parameter d in (5.19) the effective

size of an irregular pipe def is to be used. In particular, the Reynolds number Re is

defined based on def .As an example, we mention that the effective size of pipes with the elliptic cross-

section is defined as follows

def ¼ 2ffiffiffi2

p b

ð1þ e2Þ1=2(5.25)

where e ¼ b=a; and a and b are the large and small semi-axes of the ellipse.

The detailed data on pressure drop in the irregular pipes and ducts is available in

monographs of Shah and London (1978), Ward-Smith (1980), Ma and Peterson

(1997) and White (2008). Some results that display the effect of pipe shape on

resistance to laminar flows in the irregular pipes are presented in Table 5.1.

Table 5.1 Characteristics of irregular pipes

Shape of pipe Caracteristic size Constant in (5.22) Reynolds number

Circular def ¼ d 64 udn

Elliptic def ¼ 2ffiffiffi2

p ab

ða2þb2Þ1=264 udef

n

Equilateral triangular def ¼ h 160 uhn

Square def ¼ H 128 f ðxÞ= u2Hn

5.3 Flows in Irregular Pipes and Ducts 111

In this Table in addition to the previously introduced notation, h is a side of an

equilateral triangle, H is a side of a square, and f ðxÞ is a tabulated function of the

height to width ratio [ f ðxÞ changes from 2.253 to 5.333 when x varies from 1 to1(Loistyanskii, 1966)]. Note that the factor c in (5.22) for the friction factor for

laminar pipe flows is equal to 128, 160, and 64 for pipes with rectangular, equilat-

eral triangular and elliptic cross-sections, respectively (Table 5.1).

The friction factor for laminar flows in concentric and eccentric annuli can be

also presented in the form of (5.22). In these cases the factor c depends on the ratioof the effective diameters of the external and internal pipes, as well as on the value

of their eccentricity (see the Problems at the end of this Chapter).

5.4 Microchannel Flows

A detailed data on flows in microchannels with circular, rectangular, triangular and

trapezoidal cross-sections with hydraulic diameters in the range from 10�5m to

10�3m is available in a number of monographs (Incropera 1999; Celata 2000;

Kakas et al. 2005; Yarin et al. 2009) and review articles (Gad-el-Hak 1999;

Garimella and Sobhan 2003; Hetsroni et al. 2005a, b). In spite of the existence of

the numerous experimental and theoretical studies devoted to hydrodynamics and

heat transfer in laminar flows in microchannels, there is no consensus, in particular,

between the data on the friction factor of rough pipes and ducts, the effect of energy

dissipation, thermal conductivity of the wall and fluid, etc. Such situation

complicates understanding of the physical phenomena occurring in flows in

microchannels and sometimes leads to questionable ‘discoveries’ of specific

‘micro-effects’ (Duncan and Peterson, 1994; Ho and Tai 1988; Plam 2000;

Hezwig 2002; Hezwig andHausner 2003; Gad-el-Hak 2003). The shaky foundations

of a number of such results were revealed by comparison of the experimental data

with predictions of the conventional theory based on the Navier–Stokes equations.

In the framework of this theory, solutions corresponding to developed laminar flow

in straight microcannels involve a number of assumptions on flow conditions. They

are as follows: (1) the flow is driven by a static pressure drop in the fluid, (2) the

flow is stationary and fully developed, i.e. strictly axial, (3) the flow is laminar, (4)

the Knudsen number is small enough so that the fluid can be considered a contin-

uum, (5) there is no slip at the wall, (6) the fluid is incompressible Newtonian with

constant viscosity, (7) there is no heat transfer to (from) the ambient medium, (8)

the energy dissipation is negligible, (9) there is no fluid-wall interaction (except

purely viscous, so the electric, dispersive London and van der Waals forces are

neglected), (10) the micro-channel walls are smooth. Under these conditions

problems on flows in microchannels reduces to integrating (5.7) with no-slip

conditions over the wetted perimeter of the channel. The corresponding solutions

lead to the following results: the Poiseuille number Po is a constant, which is

determined by the micro-channel shape. The discrepancy between this result and


the experimental data on microchannel resistance to flow are interpreted sometimes

as a manifestation of some new effects inherent to microchannel flows. The

puzzling situation was considered in Hetsroni et al. (2005a) where the influence

of different factors that affect hydrodynamics in microchannels (e.g. the change of

the physical properties of fluid, the energy dissipation, etc.) was evaluated. In

addition, the conformity of the actual experimental conditions with those assumed

in the theoretical description of micro-channel flows was considered. It was shown

that only in certain cases the experimental conditions were consistent with the

theoretical assumptions. The experimental results corresponding to these cases

agree fairly well with the theory. In particular, in single-phase fluid flow in smooth

micro-channels of hydraulic diameter in the range from 15 mm to 4010 mm the

Poiseuille number is independent on the Reynolds number and equals 64. In single-

phase gas flows in micro-channels of hydraulic diameter in the range from 10.1 mmto 4010 mm and the Knudsen number 0:001 � Kn � 0:38; the friction factor agreesfairly well with the theoretical predictions for fully developed laminar flows (Yarin

et al. 2009). Therefore, it was concluded that the experiments on flows in smooth

micro-channels show no difference with macro-scale flow and no new physical

effects should be invoked. From the point of view of the dimensional that means

that the approach which is used for study of friction factors in macroscopic channels

can be equally used for micro-scale channels. In both cases the problem consists in

choosing as the governing parameters of fluid viscosity, mean velocity and macro-

or micro-channels diameter followed by transformation of the functional equation

for pressure drop to dimensionless form using the Pi-theorem. It is emphasized that

this statement refers to the application of the Pi-theorem to flows in smooth micro-

channels only. A number of experiments show that the Poiseuille number in rough

micro-tubes exceeds significantly the one in smooth micro-tubes (Qu et al. 2000;

Pfund et al. 2000; Li et al. 2003; Bahrami et al. 2006; Wang and Wang 2007; Li

et al. 2007). The latter shows that the set of the governing parameters responsible

for the friction factor of rough micro-channels has to include roughness of the wall.

5.5 Non-Newtonian Flows

Non-Newtonian fluids are those for which the rheological constitutive equation

differs from the Newton-Stokes one. In particular, it means that such fluids can

possess a yield stress or/and in simple shear flow the dependence of shear stress on

shear rate becomes nonlinear (i.e. the shear viscosity of such fluids is not constant at

given pressure and temperature as Fig. 5.4 shows).

The rheological constitutive equations of non-Newtonian fluids are more complex

than that of Newtonian ones. They can contain a number of parameters that account

for such specific features of non-Newtonian fluids as consistency index, the exponent,

yields stress which should be exceeded before flow starts, and viscoelastic relaxation

time (Wilkinson and Chen 1960; Astarita and Marrucci 1974; Bird et al. 1977).

5.5 Non-Newtonian Flows 113

As an example that demonstrate the application of the Pi-theorem in hydrody-

namics of non-Newtonian fluids, we consider laminar flow in a straight cylindrical

pipe of Bingham fluid possessing yield stress. Our consideration will be restricted to

the analysis of fully developed flows in the two cases corresponding to: (1) a given

volumetric flow rate (i.e. a given mean velocity in the pipe), and (2) a given

pressure drop along the pipe. In the first case the problem consists in finding the

friction factor, whereas in the second one in finding the volumetric flow rate. In

both cases we use the Pi-theorem to eatablish dependences of the unknown

characteristics on flow and geometrical parameters.

The shear behavior of Bingham fluids is given by the following equation

(Wilkinson and Chen 1960)

t� t0 ¼ m0 _g; t>t0 (5.26)

where t is the shear stress, t0 is the yield stress, m0 is the viscosity, and _g is the rateof shear.

We assume that flow of Bingham fluid in a pipe is dominated by shear near the

walls and thus consider (5.26) to be an adequate representative of the overall

(tensorial) rheological constitutive equation of the fluid. It is seen that (5.26)

contains two characteristic parameters: t0 and m0 which have the following

dimensions

t0½ � ¼ L�1MT�2; m½ � ¼ L�1MT�1 (5.27)

The pressure drop in flows of Bingham fluids in straight cylindrical pipes should

depend on the mean velocity u and pipe diameter d in addition to the rheological

parameters. Then, the functional equation for the pressure drop DP=l reads

Fig. 5.4 Flow curves for

time-independent non-

Newtonian fluids. 1-Bingham

plastic, 2-pseudoplastic fluid,

3-Newtonian viscous fluid,

4 dilatant fluid


DPl

¼ f ðm0; t0; u; dÞ (5.28)

Three governing parameters in (5.28) have independent dimensions, so that the

difference n� k ¼ 1: Then, in accordance with the Pi-theorem (5.28) reduces to the

following one

P� ¼ ’ðPÞ (5.29)

where P� ¼ DP=lð Þ=ma10 ua2da3 ; and P ¼ t0=ma01

0 ua02da

03 .

Bearing in mind that DP=l½ � ¼ L�2MT�2 and u½ � ¼ LT�1, we find that

the exponents ai and a0i are as follows: a1 ¼ 1; a2 ¼ 1; a3 ¼ �2; a

01 ¼ 1; a

02 ¼ 1;

and a03 ¼ �1: In accordance with that, (5.29) takes the form

f ¼ c

Re’�ðPÞ (5.30)

where f ¼ 2 DP=lð Þd=ru2 is the friction factor, c�’� ¼ 2’, c is dimensionless

constant equels 64 for flow in round pipe, and P ¼ t0d=m0u is the dimensionless

yield stress parameter called the Bingham number.

The dependences of the friction factor on the Reynolds number in (5.30) shows

that friction factor f is a function of two dimensionless groups, namely, the

Reynolds number Re and the Bingham number P. This dependence is shown

in Fig. 5.5 in log-log coordinates for different values of the Bingham number P.

For the lowest line P ¼ 0, which corresponds to Newtonian fluid. The kink on the

lowest curve corresponds to laminar-turbulent transition.

In the second case when the pressure drop DP=l is given, the volumetric flow rate

depends on the parameters m0; t0 and r0 (r0 is the radius of pipe)

Π

104

5.103

Rough

Smooth

Frict

ion

fact

or

0.01 0.1 1.0 10 100 .104

0.1

1.0

10

100

1000

Fig. 5.5 Friction factor

versus Reynolds number for

Bingham plastic fluids in

laminar regime

5.5 Non-Newtonian Flows 115

Q ¼ f ðDPl; m0; t0; r0Þ (5.31)

Three governing parameters in (5.31) have independent dimensions. Then,

according to the Pi-theorem, (5.31) transforms to the following one

P ¼ ’ðP1Þ (5.32)

where P ¼ Q= DP l=ð Þa1ma2o ra30�

;P1 ¼ t0=fðDP=lÞa01ma

02

0 ra03

0 g::.Bearing inmind the dimensions ofQ; r0; mo; DP=l and t0, we find the exponents

ai and a0i: They are as follows: a1 ¼ 1; a2 ¼ �1; a3 ¼ 4; a

01 ¼ 1; a

02 ¼ 0; a

03 ¼ 1:


Q ¼ DPl

r4om0

’tolroDP

� �(5.33)

The analytical solution of this problem leads to Buckingham’s equation

(Wilkinson and Chen 1960)

Q ¼ p8

DPl

r40m0

1� 4

3

2lt0r0DP

� �þ 1

3

2lt0r0DP

� �4( )

(5.34)

5.6 Flows in Curved Pipes

Flows in curved toroidal pipes demonstrate a number of characteristic features,

which result from the action of the centrifugal forces. The centrifugal-force–

induced pressure gradient generates the secondary flow in the pipe cross-section.

In the general case flows in curved pipes depend on the inertia, centrifugal and

viscous friction forces, which, in its turn, are determined by the pipe geometry, as

well as the physical properties of fluid. At a given pressure gradient the local

velocity in a curved pipe depends on the fluid density and viscosity, radius of the

pipe and its curvature, as well as the coordinates of a given point. The mean flow

characteristics, in particular, the pressure drop DP=l, are determined by the physical

properties of fluid, pipe geometry and mean velocity w0

DPl

¼ f ðr; m; r0;R;w0Þ (5.35)

where r0 and R are the radius of the pipe cross-section and the radius of curvature.

Equation (5.35) contains five governing parameters, whereas (5.15) corresponding

to flows in straight pipes only three of them: m; r0; and u0: The additional parameters


r and R are included in the set of the governing parameters in (5.35) in order to

account for the effect of the centrifugal acceleration of fluid in a curved pipe and

actual pipe geometry. The governing parameters in (5.35) have the following

dimensions

r½ � ¼ L�3M; m½ � ¼ L�1MT�1; r0½ � ¼ L; R½ � ¼ L; ½w0� ¼ LT�1 (5.36)

It is seen that in the present case the difference n� k ¼ 2, and (5.35) reduces to

the following form

P ¼ f ðP1;P2Þ (5.37)

where P ¼ DP=lð Þ=ra1wa20 r

a30 ; P1 ¼ R=ra

01w

a02

0 ra03

0 ; and P2 ¼ m=ra001 w

a002

0 ra003

0 .

In addition, the exponents ai; a0i and a

00i are found from the principle of dimen-

sional homogeneity as a1 ¼ 1; a2 ¼ 2; a3 ¼ �1; a01 ¼ 0; a

02 ¼ 0; a

03 ¼ 1; a

001 ¼ 1;

a002 ¼ 1; a

003 ¼ 1 Then, (5.37) takes the form

l ¼ ’ðd;ReÞ (5.38)

where l ¼ 2 DP=lð Þ 2r0ð Þ=rw20 is the friction factor, Re ¼ w0r0=n is the Reynolds

number, and d ¼ r0=R is the ratio of the cross-sectional radius of pipe to its radius

of curvature.

The above result is obviously very weak. Indeed, in writing the functional

equation (5.35), we implicitly assumed that flows in curved pipes are determined

by the radius of curvature R. Under this assumption, (5.35) is identical to the

functional equation for flows in straight irregular ducts, in particular, to the one

for rectangular channels where the friction factor depends on two dimensionless

groups: (1) the Reynolds number, and (2) the ratio of channel width to its depth. In

contrast with the flow in a straight pipe, when studying flows in curved pipes it is

necessary to account for a number of different factors that affect such flows through

the interaction of the inertial, centrifugal and viscous forces. Accordingly, the

functional equation for the pressure drop in a curved pipe as a dependence of

DP l=ð Þ on the acting forces reads

DPl

� �¼ f ðfi; fc; ff Þ (5.39)

where fi; fc and ff are the specific inertial, centrifugal and viscous forces,

respectively, with fj ¼ L�2MT�2 and subscript j ¼ i; c; f .

All the governing parameters in (5.39) have independent dimensions. Then,

according to the Pi-theorem, this equation takes the form

DPl

� �¼ cf a1i f a2c f a3f (5.40)

5.6 Flows in Curved Pipes 117

or

l ¼ c1fa1�i f a2c f a3f (5.41)

where l ¼ 2 DP=lð Þf�1i is the friction factor, ci ¼ 2c; and a1� ¼ a1 � 1.

Since the dimension of the friction factor l½ � ¼ 1; the dimension of the right

hand side of (5.41) has to be equal 1. There is a number of different combinations

of fi; fc and ff that are dimensionless. However, only one of them, namely,

f1=2i f

1=2c =ff has a clear physical meaning. It can be interpreted as the ratio of the

inertial forces to the viscous ones, i.e. as the natural analog of the Reynolds number

for flows in curved pipes.

Then, we can present (5.41) in the following form

l ¼ c1f1=2i f

1=2c

ff

!n

(5.42)

with n being a dimensionless constant.

Assuming that all the velocity components are proportional to the mean velocity

w0, which is defined as the ratio of the flow rate to the cross-sectional area of a

curved pipe, we can estimate fi; fc and ff as follows

fi � rw20

r0; fc � rw2

0

R; ff � m

w0

r20(5.43)

Substituting (5.43) into (5.42), one finds the following equation for the friction

factor

l ¼ ’ðkÞ (5.44)

where k ¼ d1=2Re is the Dean number, and ’ðkÞ is the function of the Dean number.

Thus, we arrive at a very impotent result: the friction factor of a slightly curved

pipe is determined by a single dimensionless group-the Dean number. Naturally,

this result can be obtained directly by transforming the Navier–Stokes equations to

the dimensionless form that contain two dimensionless groups, d and Re (Berger

et al. 1983). A number of dimensionless groups that characterize flows in curved

pipes can be reduced to a single one in the particular case of a flow in a slightly

curved pipe, i.e. d<<1 (Dean 1927, 1928). When the order of magnitude of the

inertial, centrifugal and viscous forces is the same, the system of dimensionless

equations for the fully developed flows in slightly curved pipes takes the form.

@u

@rþ u

rþ 1

r

@v

@a¼ 0 (5.45)


u@u

@rþ v

r

@u

@a� v2

r� w2 cos a ¼ � @P1

@r� 2

k

1

r

@

@a@v

@rþ v

r� 1

r

@u

@r

� �(5.46)

u@v

@rþ v

rþ uv

rþ w2 sin a ¼ � 1

r

@P1

@aþ 2

k

@

@r

@v

@rþ v

r� @u

@a

� �(5.47)

u@w

@rþ v

r

@w

@a¼ � @P0

@zþ 2

k

@

@rþ 1

r

� �@w

@rþ 1

r2@2w

@a2

� �(5.48)

u ¼ u0w0; v= ¼ v0 w0= ;w ¼ w

0w0= are the velocity components in the toroidal

coordinate system r0; a; y (cf. Fig. 5.6), P ¼ P rw2

0

�is the pressure

P ¼ PðzÞ þ Pðr; z; aÞ½ �, r ¼ r0=r0; s ¼ s

0=r0 ¼ Ry=r0 ¼ d1=2z, r

0denotes the dis-

tance from the center of circular pipe cross-section in its plane, a is the angle

between the radius vector r0and the plane of symmetry, y is the angular distance of

the cross-section from the pipe entrance.

Therefore, according to (5.45–5.48) the velocity components and pressure will

depend on the sole dimensionless group- the Dean number. As a result, all the

integral characteristics of flow, in particular, the axial pressure drop are also

functions of the Dean number. The friction factor of a curved pipe lc can be

expressed at small values of the Dean number k as (White 1929)

l ¼ 1þ 0:00306K

576

� �2

þ 0:0110K

576

� �4

þ ::: (5.49)

where l ¼ lc=ls; with lc and ls being the friction factors of the flow in curved and

the corresponding straight pipe, respectively, and K ¼ 2dRe2.At large values of the Dean number the asymptotic expressions for the ration of

the friction factors l are given by (Adler 1934; Ito 1959; Barua 1963; Mori and

Nakayama 1965)

l � 0:1k1=2 (5.50)

θ

S¢ = Rq

P(r′, a, q)α

v′w′

u′

R

r00

r′

z

Fig. 5.6 Pipe cross-section

and the toroidal coordinate

system

5.6 Flows in Curved Pipes 119

and (Van Dyke 1978)

l ¼ 0:4713k1=2 (5.51)

The correlation (5.49) agrees fairly well with the measurements of White (1929),

Adler (1934) and Ito (1959) at relatively small values of the Dean number ðk<102Þand small values of d ðd<10�3Þ. At a fixed d, an increase in the Dean number (i.e.

an increase in Re) results in a significant disagreement of the theoretical predictions

with experiment. The asymptotic formula of Van Dyke (1978) corresponding to

very large Dean numbers (k ! 1) agrees well with the experimental data at d<4 �10�3 and k<102: In all cases the disagreement of the theoretical and experimental

results stems from the change in the flow structure as the Reynolds number

increases.

In flows in helical pipes (Fig. 5.7) the set of the governing parameters is

supplemented by an additional dimensional parameter, the geometric torsion t½ � ¼L�1: Accordingly, the flow in helical pipes is determined by two dimensionless

groups (Germano 1989)

K ¼ 2dRe2; T ¼ t=r�Re

(5.52)

where r� ¼ R= R2 þ l2ð Þ is the modified curvature, and t ¼ l= R2 þ l2ð Þ is the

modified torsion.

5.7 Unsteady Flows in Straight Pipes

Consider axisymmetric flows of incompressible viscous fluids in straight cylindri-

cal pipes driven by given pressure gradient DP=l ¼ const, which is imposed

instantaneously on fluid at rest at t ¼ 0 (the startup flow). The reduced form of

the Navier–Stokes equations, which corresponds to this flow reads (Loitsyanskii

1966)

@u

@t� n

@2u

@r2þ 1

r

@u

@r

� �¼ 1

rDPl

(5.53)

The boundary and initial conditions corresponding in this case are

u ¼ 0 at r ¼ r0; u ¼ 0 at t ¼ 0 (5.54)

Equation (5.53) subjected to the conditions (5.54) shows that the velocity at any

point of pipe cross-section depends on six dimensional parameters. Four of the are

the given constants, namely, DP=l; r; m and r0, whereas the two others are


variables r and t (the radial coordinate in the cross-section and time). Accordingly,

the mean flow characteristics, in particular, the volumetric flow rate, depend on five

dimensional parameters (without r). Therefore, we can write the following func-

tional equations for the volumetric rate Q½ � ¼ L3T�1 and local velocity u½ � ¼ LT�1

Q ¼ f ðDPl; m; r0; r; tÞ (5.55)

u ¼ f1ðDPl; m; r0; r; r; tÞ (5.56)

The governing parameters in (5.55) and (5.56) have the following dimensions

DPl

� �¼ L�2MT�2; m½ � ¼ L�1MT�1; r0½ � ¼ L; r½ � ¼ L�3M; t½ � ¼ T (5.57)

Fig. 5.7 Helical pipe with the modified curvature R= R2 þ l2ð Þ and torsion t ¼ l= R2 þ l2ð Þ

5.7 Unsteady Flows in Straight Pipes 121

First we consider (5.55). It contains five governing parameters, three of them

have independent dimensions. Therefore, the difference n� k ¼ 2: In this case

(5.55) reduces to the following dimensionless equation

P ¼ ’ðP1;P2Þ (5.58)

with the dimensionless groups being given by the following expressions:

P ¼ Q= DP=lð Þa1ma2ra30a

; P1 ¼ r= DP=lð Þa01ma

02r

a02

o

� �; P2 ¼ t= DP=lð Þa

001 ma

002 r

000

h i:

Bearing in mind the dimensions of the volumetric flow rate and the governing

parameters, we find the values of the exponents ai; a0i and a

00i as

a1 ¼ 1; a2 ¼ �1; a3 ¼ 4; a01 ¼ �1; a

02 ¼ 2; a

03 ¼ �3; a

001 ¼ �1; a

002

¼ 1; a003 ¼ �1 (5.59)

Then, the expressions for the dimensionless groups P; P1 and P2 become

P ¼ QmDP=lð Þr40

; P1 ¼ r DP=lð Þr30m2

¼ c1; P2 ¼ t DP=lð Þr0m

¼ c1Fo (5.60)

For given conditions (the physical properties of fluid, the pipe radius and the

pressure gradient DP=lÞ the dimensionless group P1 ¼ c1 is a constant ( c1½ � ¼ 1)

whereas the dimensionless group P2 ¼ c1Fo where Fo ¼ tn=r20 is kindred to the

Fourier number, with nbeing kinematic viscosity.


Q ¼ DPL

� �r40m’ðFoÞ (5.61)

In order to compare the expression (5.61) with the known analytical solution

corresponding to this case, recast (5.58) as follows

Q ¼ DPl

� �r40m

1� cðFoÞ½ � (5.62)

where cðFoÞ ! 0as Fo ! 1, which means the asymptotic approach to the

Poiseuille law, as expected on the physical grounds.

The exact analytical solution of the problem is available and yields (Loitsyanskii

1966)

Q ¼ p8

DPl

� �r40m

1� 32X1k¼1

expð�l2kFoÞl4k

" #(5.63)


where lk are the roots of the equation J0ðlkÞ ¼ 0; with J0 being the Bessel functionof the zero order.

It is seen that the structure of (5.62) resembles that of (5.63). In both cases the

factors on the right hand side of these equations correspond to the fully developed

laminar flow in a straight pipe, whereas the factor in the parentheses approaches one

at large Fo as expected when transient effects fade.

Applying the Pi-theorem to (5.56), we arrive at

P ¼ ’ðP1;P2;P3Þ (5.64)

whereP ¼ u= DP=lð Þr40=m

; P1 ¼ r=r0; P2 ¼ c1; with c1½ � ¼ 1 being a constant.

Then, we obtain the following expression for the velocity profile u

u ¼ DPl

� �r20m’ðr;FoÞ (5.65)

where r ¼ r=r0:For comparison, the analytical solution for u is (Loitsyanskii 1966)

u ¼ DPl

� �r204m

1� ðrÞ2� �

� 8X1k¼1

expð�lkFoÞ J0ðlkrÞJ1ðlkÞ

( )(5.66)

where J1 is the Bessel function of the first order.

Problems

P.5.1. Determine the dependence of the friction factor of the concentric and

eccentric annuli with fully developed stationary flows on the governing parameters.

A concentric annulus (Fig. 5.8) is characterized by two geometric parameters,

namely: (1) the internal diameter d1 and (2) the external diameter d2. Then, thefunctional equation for the pressure gradient becomes

DPl

¼ f ðm; u�; d1; d2Þ (P.5.1)

where u� is the mean velocity.

Three of the four governing parameters in (P.5.1) have independent dimensions.

Then, according to the Pi-theorem, (P.5.1) reduces to

P ¼ ’ðP1Þ (P.5.2)

where P ¼ DP=lð Þ=ma1ua2� da32 ; and P1 ¼ d1=ma01u

a02� d

a03

2 :

Problems 123

Taking into account the dimensions of the parameters involved, DP l=½ � ¼L�2MT�2; m½ � ¼ L�1MT�1; u�½ � ¼ LT�1; d1½ � ¼ L; and d2½ � ¼ L, we find the

values of the exponents ai and a0i as a1 ¼ 1; a2 ¼ �1; a3 ¼ �2; a

01 ¼ 0; a

02 ¼ 0;

and a03 ¼ 1: Then, (P.5.2) takes the following form

l ¼ 1

Re’

d1d2

� �(P.5.3)

where l ¼ 2 DP=lð Þd2=ru2�; and Re ¼ u�d2=n:It is emphasized that the equivalent diameter de defined as

de ¼ 4S

P¼ d2 � d1 (P.5.4)

where S is the cross-sectional area, and P the wetted perimeter, can be used as one

of the governing parameters for flows in concentric annuli.

In the latter case the form of the dependence of the friction factor on the two

dimensionless groups involved retains the same form as (P.5.3), albeit the Reynolds

number Re should be replaced by Ree based on the equivalent diameter de:The analytical solution of this problem can be found by integrating the Navier–

Stokes equations, which in the present case reduce to

d2d1

d2d1

e

a

b

Fig. 5.8 Concentric (a) andeccentric (b) annuli


dP

dx¼ m

r

d

drrdu

dr

� �(P.5.5)

subjected to the no-slip boundary conditions

u ¼ 0 at r ¼ r1; u ¼ 0 at r ¼ r2 (P.5.6)

The solution leads to the following expression for the friction factor (Ward-

Smith 1980)

l ¼ 64

Ree’

d1d2

� �(P.5.7)

where

’ d1=d2ð Þ ¼ 1� d1=d2½ �2 ln d1=d2ð Þ= 1� d1=d2ð Þ2 þ 1þ d1=d2ð Þ2h i

ln d1=d2ð Þn o

.

In the case of a fully developed stationary flow in a straight eccentric annuls

(Fig. 5.8 b) the functional equation for the dependence of the pressure gradient on

flow and geometric parameters reads

DPl

¼ f1ðm; u�; d1; de; eÞ (P.5.8)

where e is eccentricity (Fig. 5.8b).

Applying the Pi-theorem to (P.5.8) we arrive at the following expression for the

friction factor

l ¼ 1

Ree’1

r1r2;e

re

� �(P.5.9)

where r1 and r2 are the radii involved, and re is the equivalent radius equal to

r2 � r1:The analytical solution describing flows in eccentric annuli is readily available

for comparison (Ward-Smith 1980)

l ¼ 64

Ree’

r1r2;e

re

� �(P.5.10)

In the limit r1=r2 ! 1 the function ’ on the right hand side of (P.5.10) takes the

form ’1 ¼ 2=3ð Þ 1þ 3=2ð Þ e=reð Þ½ �f g2.P.5.2. Determine the velocity profile in the cross-section of a cylindrical pipe

with fully developed laminar flow of Newtonian fluid and a given pressure gradient.

Also, determine the relation between the volumetric flow rate and fluid viscosity,

pressure drop and pipe radius.

Problems 125

At a given pressure gradient DP=l, the functional equation which determines the

velocity profile in the cross-section of a cylindrical pipe reads

u ¼ f m;DPl; r; r0

� �(P.5.11)

where u is the longitudinal velocity component corresponding to the radial coordi-

nate r, r0 is the pipe radius.Applying the Pi-theorem to (P.5.11), we arrive at the following dimensionless

equation

P ¼ ’ðP1Þ (P.5.12)

where P ¼ u ma1 DP l=ð Þa2ra30� �

; and P1 ¼ r=fma01 DP=l:ð Þa02r

a03

0 g:.Taking into account the dimensions of u½ � ¼ LT�1; m½ � ¼ L�1MT�1; DP l=½ � ¼

L�2MT�2; r½ � ¼ L and r0½ � ¼ L, we arrive at the equations� a1 � 2a2 þ a3 � 1 ¼ 0; � a

01 � 2a

02 þ a

03 � 1 ¼ 0

a1 þ a2 ¼ 0; a01 þ a

02 ¼ 0 (P.5.13)

� a1 � 2a2 þ 1 ¼ 0; � a01 � 2a

02 ¼ 0

From (P.5.13) it follows that

a1 ¼ �1; a2 ¼ 1; a3 ¼ 2; a01 ¼ 0; a

02 ¼ 0; a

03 ¼ 1 (P.5.14)

Thus, (P.5.12) takes the following form

u

DPr20=lm� � ¼ ’

r

r0

� �(P.5.15)

For the axial (maximum) velocity u ¼ um at r ¼ 0, (P.5.11) yields the followingfunctional equation

um ¼ f m;DPl; r0

� �(P.5.16)

Since all the governing parameters in (P.5.16) possess independent dimensions,

(P.5.16) takes the form

um ¼ cDPl

� �r20m

(P.5.17)



Comparing (P.5.15) with (P.5.17), we obtain

u

um¼ ’�

r

r0

� �(P.5.18)

where ’� r=r0ð Þ ¼ c’ r=r0ð Þ:Calculating the volumetric flow rate using the expression for u

Q ¼ 2pðr00

urdr (P.5.19)

we arrive at the following formula

Q ¼ 2pcDPl

� �r40m

ð10

’ðrÞrdr (P.5.20)

where r ¼ r=r0:The comparison of (P.5.20) with the exact analytical solution of the present

problem shows that cÐ10

’ðrÞrdr ¼ 1=16:

It is emphasized that the form of the dependence Q on the DP=L; m and r can bedirectly revealed by applying the Pi-theorem to the functional equation

Q ¼ fDPl; m; r0

� �(P.5.21)

Bearing in mind the dimensions of the volumetric flow rate, pressure drop, fluid

viscosity and pipe radius, we arrive at the following expression

Q ¼ cra10DPl

� �a2

ma3 (P.5.22)

where the factor c is a constant and a1 ¼ 4; a2 ¼ 1; and a3 ¼ �1.

Thus, for the volumetric flow rate we have the expression

Q ¼ cr40DPl

� �1

m(P.5.23)

The exact analytical solution of the present problem yields a numerical value for

the constant c: It is equal to p=8:P.5.3. Determine the dependence of the friction factor on the Reynolds number

for fully developed flows of viscous incompressible fluid in rough cylindrical pipes.

Let the characteristic sizes of the pipe and its roughness are d and ks; the

mean velocity of the fluid u; and fluid density and viscosity r and m; respectively.Then the functional equation for the pressure gradient is

Problems 127

DPl

¼ f d; k; r; m; uð Þ (P.5.24)

The set of the governing parameters in (P.5.24) includes three parameters that

have independent dimensions. Then, in accordance with the Pi-theorem, (P.5.24)

can be transformed to the following dimensionless form

l ¼ ’1

1

Re; k

� �(P.5.25)

where l ¼ 2 DP=lð Þd=ru2 is the friction factor, Re ¼ ud=n is the Reynolds number,

and k ¼ k=d is the relative roughness.

In the case of fully developed laminar flows the number of the governing

parameters reduces to four (d; k; m; and u), since density of the fluid does not

affect such flows. Then, (P.5.25) takes the following form

DPl

¼ f d; k; m; uð Þ (P.5.26)

Applying the Pi-theorem to (P.5.26), we obtain

l ¼ 1

Re’2 k� �

(P.5.27)

In the fully developed (in average) turbulent flows the effect of molecular

viscosity is negligible, and m can be excluded from the set of governing parameters.

Accordingly, the functional equation for the pressure gradient reads

DPl

¼ f ðd; k; r; uÞ (P.5.28)

The dimensionless form of (P.5.28) becomes

l ¼ ’3ðkÞ (P.5.29)

In connection with the above results, it is necessary to add the following

remarks. An explicit form of the dependences ’iðkÞ cannot be revealed in the

framework of the dimensional analysis. To determine the dependences ’iðkÞ, it isnecessary to recall some additional physical considerations or the experimental

data. The data of Nikuradse (1930) and Schiller (1923) show that the friction factor

of the conventional macroscopic rough pipes does not depend on k in fully

developed laminar flows and corresponds to the one following from the Poiseuille

law. Thus, the function ’2ðkÞ in (P.5.27) can be assumed to be constant equal to 64.

In the fully developed (in average) turbulent flow (at high values of Re) the friction

factor does not depend on Re and is fully determined by relative roughness.


Therefore, the dependence of l on Re for rough conventional pipes can be selected

separately for three characteristic flow regimes corresponding to laminar

ðRe<Recr1Þ, transitional ðRecr1<Re<Recr2Þ and turbulent ðRe>Recr2Þ flows, for

which (P.5.25), (P.5.27) and (P.5.28) are valid, respectively. For micro-channels

ð10 � d � 103mmÞ the recent measurements show that the friction factor

corresponding to fully developed laminar flows depends significantly on the value

of relative roughness (Yarin et al. 2009).

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Springer, Berlin


Chapter 6

Jet Flows


The subject of the present chapter is the hydrodynamics of laminar submerged jets

in the light of the dimensional analysis. Submerged jets are discussed in detail in

special monographs devoted to the jet theory (Pai 1954; Abramovich 1963; Vulis

and Kashkarov 1965), boundary layer theory (Schlichting 1979), as well as the

theory of turbulent flows (Townsend 1956; Hinze 1959).

Submerged jets belong to the vast class of laminar and turbulent shear flows.

They originate from issuing of viscous fluid into an infinite or semi-infinite space

filled with the same fluid. Jet flows are encountered in numerous engineering

applications such as steam and gas turbines, ejectors, jet engines, industrial

furnaces, pneumatic and ventilation systems, as well as in the phenomena typical

for the environmental flows. A wide variety of flow configurations is characteristic

of jet flows. There are, for example, mixing layers due to the interaction of two

parallel streams moving with different velocities, or jets of viscous fluid issued

from a nozzle or a slit into fluid which is moving or at rest. Wakes behind solid

bodies moving in fluid or plumes due to the action of buoyancy forces also belong

to the class of jet flows. Three main types of jet flows depending on conditions of

their formation can be distinguished: (1) submerged jets, (2) wakes, (3) plumes and

thermals (Fig. 6.1). The existence of convective motion that is predominantly

directed along the flow axis, as well as an intensive transversal mass and momen-

tum transfer due to strong shear in cross-section is typical for all kinds of jet flows.

Submerged jets can be classified by a number of characteristic features: the flow

regime (laminar, turbulent), geometry (plane, axisymmetric), mutual direction of

the jet spreading and motion of the surrounding fluid (co- and counter-flows, a jet in

crossflow), the interaction with the ambient fluid and solid walls (without any

contact with walls, or the wall and impinging jets). A detailed sketch of a planar

submerged jet of an incompressible fluid is shown in Fig. 6.2. It forms as a result of

mixing of viscous fluid issued from a slit into the same fluid as rest. The horizontal

lines a� a subdivide the domains of jet issued from the nozzle and the ambient



131

fluid at the initial cross-section x ¼ 0 where the jet originates. These are the lines

of a discontinuity of the tangential velocity component. In viscous fluid such

a discontinuity cannot be sustained. As a result of the interaction of the fluid in

the jet with the ambient fluid, a thin mixing layer with a steep continuous velocity

profile forms in the vicinity of the lines a� a. The viscous stresses developed at thejet periphery lead to the expansion of the shear layers downstream. The inner

boundaries dinðxÞ of the shear layers reach the jet axis at some distance xin from

the nozzle exit encasing the core with a plug velocity profile which originated in the

slit. Outside the core domain the velocity profile in the jet cross-section decreases

from its value in the core and asymptotically approaches to zero at the outer

boundaries of the shear layers. In the cross-section x ¼ xin; the shear layers

merge. At x>xin smooth velocity profiles with characteristic maximum at the axis

are observed.

The size and geometry of the slit or a nozzle, as well as the velocity distribution

at its exit affect the flow field in submerged jets. The influence of the issuing

Fig. 6.1 Jet flows.

(a) Submerged jet: 1-constant

velocity core, 2-the far field

of the jet. (b) Wake behind

a solid body 1. (c) Buoyantplume rising from a hot body 1

132 6 Jet Flows

conditions is significant only in a relatively short section of the jet adjacent to the

slit or the nozzle. The length of this section is about 20–30 times the slit or nozzle

cross-sectional size for plane and axisymmetric jets, whereas in three-dimensional

(non-axisymmetric) jets it is significantly larger (Trentacoste and Sforzat 1967;

Krothapalli et al. 1981; Ho and Gutmark 1987; Hussain and Hussain 1989).

Therefore, at large x the flow field of submerged jets practically does not depend

on the details of the issuing conditions and is fully determined by the integral

characteristics of the jets. The latter express conservation of the momentum and

energy fluxes along the jets. For a submerged jet issuing from a thin tube the total

momentum flux is (Landau and Lifshitz 1987)

Jx ¼þPrr cos yds (6.1)

whereQ

rr ¼ Pþ rvr2 � 2m@vr=@r is the component of the tensor of the momen-

tum flux in the r-direction, dS ¼ r2 sin ydyd’ is the surface element, r and P are the

density and pressure, vr is the r-component of the velocity, m is the viscosity;

r; y and ’ are the spherical coordinate frame with the center located at the jet axis

at the tube exit.

The total energy flux in the jet can be expressed as (Vulis and Kashkarov 1965)

Q ¼þ

rvrðh� h1Þ � k@T

@r

� �ds (6.2)

Fig. 6.2 Sketch of

a submerged jet. The inner

and outer boundaries of the

boundary layer are denoted

as din and dout, respectively


where h and T are the enthalpy and temperature, respectively, k is the thermal

conductivity, subscript1 corresponds to the undisturbed fluid far away from the jet

axis.

The submerged jets represent themselves boundary layers. Therefore, the bound-

ary layer theory can be used to study hydrodynamics of submerged jets. For laminar

planar submerged jets issuing from a slit, or a non-swirling (about the jet axis)

axisymmetric jets issuing from a pipe, the boundary layers equations have the

following form

ru@u

@xþ rv

@u

@y¼ m

yj@

@yyj@u

@y

� �(6.3)

ru@h

@xþ rv

@h

@y¼ k

yj@

@yyj@T

@y

� �(6.4)

Here (6.3) expresses the longitudinal momentum balance, (6.4)-the

corresponding energy balance; u and v are the longitudinal and transversal velocitycomponents, j ¼ 0 or 1 for the plane and axisymmetric jets, respectively; pressure

drop is omitted in (6.3), since pressure in the jets is practically constant over the

entire flow field. In (6.4) viscous dissipation is omitted, since it is negligible at

sufficiently small flow velocities.

The corresponding continuity equation reads

@ruyj

@xþ @rvyj

@y¼ 0 (6.5)

Using (6.5), one can transform the momentum and energy balance Eqs. 6.3 and

6.4 to the following form

@ru2yj

@xþ @ruvyj

@y¼ m

@

@yyj@u

@y

� �(6.6)

@ruDhyj

@xþ @rvDhyj

@y¼ k

@

@yyj@T

@y

� �(6.7)

where Dh ¼ h� h1.

The boundary conditions for planar and axisymmetric submerged jets read

y ¼ 0:@u

@y¼ @T

@y¼ @h

@y¼ 0; v ¼ 0 (6.8)

y ! 1 : u ! 0; T ! T1; Dh ! 0 (6.9)

134 6 Jet Flows

Integration of (6.6) and (6.7) across the jet cross-section yields the following

invariants

Jx ¼ð10

ru2yjdy ¼ const: (6.10)

Qx ¼ð10

ruDhyj ¼ const: (6.11)

It is emphasized that (6.10) expresses the fact that the total momentum flux Jxdoes not vary along the jet. Similarly, (6.11) expresses the fact that the total

excessive enthalpy flux Qx does not vary along the jet. Note that the expressions

for Jx and Qx for submerged turbulent jets are similar to those in (6.10) and (6.11).

However, in that case r; u; T and h imply their average values. In addition, it

should be assumed thatÐ10

u02 þ P� �

yjdy ¼ 0; where u0is the velocity fluctuation.

Swirling motion about the jet axis results in a non-uniform pressure distribution

across it. In this case the system of Eqs. 6.3 and 6.4 is supplemented by the

following two equations describing the balance between the centrifugal force due

to swirling and the corresponding pressure gradient across the jet and the azimuthal

momentum balance allowing for finding the swirling velocity component w

jw2

y¼ 1

r@P

@y(6.12)

ru@w

@xþ rv

@w

@yþ r

vw

y¼ m

1

yj@

@yyj@u

@y

� �� w

y2

� (6.13)

In addition, the longitudinal momentum balance of Eq. 6.6 is now modified by

the inclusion the longitudinal pressure gradient � @P=@x on its right-hand side.

Then, the following set of the integral invariants for the axisymmetric swirling jet is

obtained via integration of the modified momentum balance equation, as well as

(6.7), (6.12) and (6.13)

Jx ¼ð10

Pþ ru2 �

ydy (6.14)

Mx ¼ð10

ruwy2dy (6.15)


Qx ¼ð10

ruDhydy (6.16)

where Mx is the total moment-of-momentum flux.

The integral invariants for different types of submerged jets are presented in

Table 6.1.

The integral invariants (6.1) and (6.10) express conservation of the momentum

flux which is brought by the fluid issued from the jet origin. The relations (6.1) and

(6.10) are not identical. The first one accounts for the full momentum flux around

the momentum source, i.e. represent itself the momentum flux through a closed

surface S (Fig. 6.3a), whereas the second one-only the momentum flux through a

cross-section AA normal to the jet axis (Fig. 6.3b). The invariant (6.1) and the

closed surface are used to solve the problem of the submerged jet in the framework

of the full Navier–Stokes equations, whereas the invariant (6.10) and the cross-

sections of the type AA are used to solve the problem in the framework of the

boundary layer theory. It is emphasized that in the framework of the boundary layer

theory the integral invariant (6.10) does not account for the flow ‘history’

(Konsovinous 1978). (Schneider 1985) Nevertheless, the relation (6.10) is a fairly

good approximation of the relation (6.1) and is widely used in the theory of jets.

6.2 The Far Field of Submerged Jets

The understanding that the far field of a jet flow does not depend on the issuing

conditions and the related simplifications plays a pivotal role in the boundary layer

theory of submerged jets. It allows one to decrease the number of factors which

should be accounted for and to develop an asymptotic model of the flow. Namely,

when a jet is assumed to be issued from a pointwise source, it is natural to assume

that the far field of velocity in the jet is determined by physical properties of fluid

and total momentum flux (but not by flow details at the nozzle/pipe exit). Therefore,

we can write the functional equation for the longitudinal velocity component u as

u ¼ fuðr; m; Jx; x; yÞ (6.17)

The functional equation (6.17) contains the set of parameters which determine

the velocity at any point of the far field of a submerged jet. Any change of the flow

conditions (for example the appearance of a co- or counter flow of the surrounding

fluid) requires an expansion of the set of the determining parameters on the right

hand side of Eq. (6.17). In particular, when r and m are constant, the fluid properties

are characterized by a single parameter, namely, the kinematic viscosity n ¼ m=r,as it follows from the boundary layer equations. In this case (6.17) reduces to

136 6 Jet Flows

Tab

le6.1

Jetinvariants

Jetgeometry

Jet’sschem

e

Integralinvariant

Dynam

icproblem

Thermal

problem

Planejet

J x¼Ð1 0

ru2dy

Qx¼Ð1 0

ruc p

T�T1

ðÞdy

Axisymmetricjet

J x¼

2pÐ1 0

ru2ydy

Qx¼

2pÐ1 0

ruc p

T�T1

ðÞdy

Radialjet

J x¼

2px

Ðþ1 �1ru

2dy

Qx¼

2px

Ðþ1 �1ru

c pT�T1

ðÞdy

Walljet

J x¼Ð1 0

ru2Ðy 0

rudy

�� d

yQ

x Pr¼

1¼ð1 0

ruc p

T�T1

ðÞðy 0

rudy

0 @1 A dy

Qx a

d¼ð1 0

ruc p

T�T1

ðÞdy

6.2 The Far Field of Submerged Jets 137

u ¼ fuðn; Ix; x; yÞ (6.18)

where Ix ¼ Jx=r is the kinematic momentum flux.

Similarly we write the functional equation for the jet thickness as

d ¼ fdðr; m; Jx; xÞ (6.19)

or

d ¼ fdðn; Ix; xÞ (6.20)

Since at y ¼ 0; u ¼ um, with um being the axial maximal velocity in a jet cross-

section, we obtain from (6.17) and (6.18) the following equations for the axial

velocity in the planar and axisymmetric (round) jets

um ¼ fuðr; m; Jx; xÞ (6.21)

or

um ¼ fuðn; Ix; xÞ (6.22)

In a cross-section of an axisymmetric jet it is possible to select two characteristic

parameters: the jet (boundary layer) thickness and the axial velocity. These

parameters can be taken as the governing ones. Then, we can assume that velocity

at any point of the jet depends on umðxÞ; d and the distance from the jet axis to the

point under consideration y: Then, (6.18) is replaced with the following one

u ¼ fumðum; d; yÞ (6.23)

Fig. 6.3 Sketch of a jet flow.

(a) Flow in a jet issued from a

thin tube (the Navier–Stokes

approximation). (b) Jet from a

pointwise momentum source

(the boundary layer

approximation)

138 6 Jet Flows

6.3 The Dimensionless Groups of Jet Flows

The dimensions of the governing parameters can be expressed in the framework of

any system of fundamental units. For the dimensional analysis of jet flows it is

convenient to use the modified LMT system that includes three different length

scales Lx, Ly; and Lz, respectively for the x, y and z directions. In this case the

dimensions of the characteristic parameters are

u½ � ¼ LxT�1; v½ � ¼ LyT

�1; x½ � ¼ Lx; y½ � ¼ Ly; r½ � ¼ L�1x L�1

y L�1z M;

m½ � ¼ L�1x LyL

�1z MT�1; n½ � ¼ L2yT

�1; d½ � ¼ Ly (6.24)

In addition, the dimension of the momentum flux is

Jx½ � ¼ Le1x Le2y L

e3z M

e4Te5 (6.25)

where ei 6¼ 0 (i ¼ 1, 2, 3, 4, 5) are constants which depend on the jet configuration

(Table 6.2).

Applying the Pi-theorem to (6.17)–(6.23), we can transform them to the follow-

ing canonical form

P ¼ ’jðP1;P2:::Pn�kÞ (6.26)

where n� k is the number of dimensionless groupsP1; P2:::; whereas n and k arethe number of the governing parameters and the parameters with independent

dimensions, subscript j ¼ u; d refers to the velocity or thickness of the jet,

respectively.

Among the numerous cases corresponding to different values of n� k, the most

important ones are the following two: (i) n� k ¼ 1; and n� k ¼ 0: The former

corresponds to a self-similar flow in which dimensionless velocity depends on a

single dimensionless variable, whereas the latter – to a constant dimensionless

velocity or jet thickness. In the latter case u and d can be expressed in the form of

an explicit function of the governing parameters. As can be seen from (6.24) and

(6.25), the parameters r; m; Jx and x on the right hand side in Eqs. (6.17), (6.19)

and (6.21) have independent dimensions. Accordingly, we obtain the following

equations for the dimensionless velocity and jet thickness

Table 6.2 Exponents eifor different jetsJet geometry e1 e2 e3 e4 e5Plane 1 0 �1 1 �2

Axisymmetric 2 0 �1 1 �2

Radial 2 0 �1 1 �2

Wall 1 0 �2 2 �3

6.3 The Dimensionless Groups of Jet Flows 139

Pu ¼ ’ðP1Þ (6.27)

Pum ¼ c1 (6.28)

Pd ¼ c2 (6.29)

where

Pu ¼ u

ra1ma2Ja3x xa4; P1 ¼ y

ra01ma

02J

a03

x xa04

(6.30)

Pu;m ¼ umra1ma2Ja3x xa4

(6.31)

Pd ¼ d

ra01ma

02J

a03

x xa04

(6.32)

and c1, c2 are constants.The unknown exponents ai and a

0i (i ¼ 1; 2; 3; 4) are found from the expressions

(6.30)–(6.32) accounting for the dimensions of the parameters on the left and right

hand sides of these there. As a result, we arrive at the following sets of algebraic

equations

XLx

: �a1 � a2 þ e1a3 þ a4 ¼ 1

XLy

: �a1 þ a2 þ e2a3 ¼ 0 (6.33)

XLz

: �a1 � a2 þ e3a3 ¼ 0

XM

: a1 þ a2 þ e4a3 ¼ 0

XT

: �a2 þ e5a3 ¼ �1

and

XLx

: �a01 � a

02 þ e1a

03 þ a

04 ¼ 0

140 6 Jet Flows

XLy

: �a01 þ a

02 þ e2a

03 ¼ 1 (6.34)

XLz

: �a01 � a

02 þ e3a

03 ¼ 0

XM

: a01 þ a

02 þ e4a

03 ¼ 0

XT

: �a02 þ e5a

03 ¼ 0

where the symbolsPLx

;PLY

;PLz

;PM;PT

refer to summation of the exponents of

lengths and mass time scales, respectively.

Bearing in mind (6.30)–(6.32), we present (6.27)–(6.29) as follows

u ¼ ’uð�Þ (6.35)

um ¼ c1ra1ma2Ja3x xa4 (6.36)

d ¼ c2ra01ma

02J

a03

x xa04 (6.37)

where u ¼ u=um; � ¼ y=d; and ai and a0i depend on ei which is different for

different jet geometry.

6.4 Plane Laminar Submerged Jet

Consider the application of the Pi-theorem to study characteristics of plane laminar

submerged jets (Fig. 6.4). We deal with only the far-field region of the jet, as

explained before, and select the following governing parameters: r; m; Jx; x and ywhich have dimensions listed in (6.24) and (6.25). Four of the five governing

parameters have independent dimensions, which yields (6.34)–(6.37) that deter-

mine velocity distribution in jet cross-section, the velocity variation along the jet

axis, and the boundary layer thickness. Taking into account the values of the

exponents ei ði ¼ 1; 2; 3; 4Þ in (6.25) (see Table 6.2), we find from (6.33) and

(6.34) the values of the exponents ai and a0i

a1 ¼ � 1

3; a2 ¼ � 1

3; a3 ¼ 2

3; a4 ¼ � 1

3; a

01 ¼ � 1

3; a

02 ¼

2

3; a

03 ¼ � 1

3; a

04

¼ 2

3(6.38)

6.4 Plane Laminar Submerged Jet 141

In accordance with (6.36) and (6.37), we obtain

um ¼ c1J2xr2n

� �1=3

x�1=3 (6.39)

d ¼ c2rn2

Jx

� �1=3

x2=3 (6.40)

Using (6.35), (6.39) and (6.40), it is possible to estimate several important

characteristics of plane laminar submerged jets, in particular, the local Reynolds

number. The latter is defined by the average velocity, the boundary layer thickness

and the kinematic viscosity

Red ¼ <u>dn

(6.41)

where <u> ¼ 1d

Ðd=2�d=2

udy is the average velocity in jet cross-section.

Taking into account (6.35), (6.39) and (6.40), we arrive at the following expres-

sion for the local Reynolds number

Red ¼ c3Jxx

rn2

� �1=3 ð1=2�1=2

’ð�Þd� (6.42)

where c3 ¼ c1c2:

Fig. 6.4 Plane laminar jet

142 6 Jet Flows

Equation 6.42 shows that the local Reynolds number in plane laminar submerged

jet increases downstream as x1=3: At sufficiently large values of x the local Reynoldsnumber exceeds the critical value corresponding to laminar-turbulent transition

(Fig. 6.5).

A higher total momentum flux or a lower fluid viscosity increase the local

Reynolds number Red and the laminar-turbulent transition cross-section appro-

aches the jet origin. The observations show that the laminar sections of plane

submerged jets exist up to Re0 ’ 30 (Re0 is based on the outflow jet velocity

and the slit width, Andrade 1939).

6.5 Laminar Wake of a Blunt Solid Body

As the second example application of the Pi-theorem to jet flows consider plane

laminar wake behind a blunt solid body in a uniform stream of viscous fluid

(Fig. 6.1b). Similarly to flows in submerged jets, flows in laminar wakes can be

characterized by an integral invariant which accounts for the body characteristics.

In order to find this invariant we use the boundary layer equation

ru@u

@xþ rv

@u

@y¼ m

@2u

@y2(6.43)

Fig. 6.5 Laminar-turbulent transition (in cross-section A-A) in a submerged jet

6.5 Laminar Wake of a Blunt Solid Body 143

Consider the relative longitudinal velocity component

u1 ¼ u1 � u (6.44)

where u1 is the free stream velocity; the body is considered to be at rest, while the

stream impinges on it with velocity u1.

At a large distance from the body u1 becomes sufficiently small. Then (6.43) can

be linearized and takes the following form

ru1@u1@x

¼ m@2u1@y2

(6.45)

Integrating (6.45) from y ¼ �1 to y ¼ þ1 across the wake and accounting for

the fact that @u1=@y ! 0 at y ! �1, we obtain

Jx ¼ð1

�1ru1dy ¼ const: (6.46)

Below it will be shown that um depends on the physical properties of fluid,

velocity of the free stream, as well as on the integral invariant Jx

u1 ¼ f ðr; m; Jx; u1x; yÞ (6.47)

where the dimensions of r; m; Jx; u1; x; and y in the LxLyLzMT system of units are

L�1x L�1

y L�1z M; L�1

x LyL�1z MT�1; L�1

z MT�1; LxT�1; Lx and Ly; respectively.

We also write the functional equations for u1m and dw

u1m ¼ fwðr; m; Jx; u1; xÞ (6.48)

dw ¼ fdwðr; m; Jx; u1; xÞ (6.49)

The functional equation for u1 can be rewritten as follows

u1 ¼ f ðu1m; dw; yÞ (6.50)

Equation 6.50 for u1 expresses the physically realistic assumption that velocity

at any point of a wake cross-section is determined by three parameters: velocity at

the wake axis u1m; the boundary layer (wake) thickness dw and the distance from

the wake axis to the point under consideration y:Applying the Pi-theorem to (6.48), we obtain

Pm ¼ ’mðP1Þ (6.51)

144 6 Jet Flows

where Pm ¼ u1m=ra1ma2ua31xa4 ; P1 ¼ Jx=ra01ma

02u

a031xa

04 ; and a1 ¼ a2 ¼ a4 ¼ 0;

a3 ¼ 1; a1 ¼ 1; a01 ¼ a

02 ¼ a

03 ¼ a

04 ¼ 1=2:

Therefore, (6.51) takes the form

u1:mu1

¼ ’m

Jx

rmu1xð Þ1=2u1

( )(6.52)

The application of the Pi-theorem to (6.50), yields

u1 ¼ ’wð�Þ (6.53)

where u1 ¼ u1=u1m; and � ¼ y=dw:In order to determine the dependence of Jx on the drag force acting on a blunt

solid body, we use the overall momentum and mass balances for the rectangular

control volume ABCD encompassing the body (Fig. 6.1b). They read

ru22h�W �ðþh

�h

ru2dy ¼ 0 (6.54)

ru12h�ðþh

�h

rudy ¼ 0 (6.55)

where W is the drag force acting on the solid body, and 2h is height of the contour

ABCD.

From (6.54) and (6.55) it follows that W equals to

W ¼ðþh

�h

ruðu1 � uÞdy (6.56)

Taking into account that in the limit h ! 1, we can rewrite (6.56) as

W ¼ð1

�1rðu1 � uÞu1dy � u1

ð1�1

ru1dy (6.57)

The comparison of (6.46), (6.56) and (6.57), reveals that

Jx ¼ W

u1(6.58)

6.5 Laminar Wake of a Blunt Solid Body 145


u1mu1

¼ ’m

W

rmu1xð Þ1=2( )

(6.59)

The wake thickness can be found directly from (6.49) or (6.53) and (6.57). From

(6.53) and (6.47) it follows that

dw ¼ W

ru1u1mÐ1

�1’wð�Þd�

(6.60)

6.6 Wall Jets over Plane and Curved Surfaces

Laminar wall jets represent themselves a “compound” boundary layers, which are

similar to the free boundary layer of submerged jets on the one side, and to the near-

wall (Blasius) boundary layers on the other side (close to the wall). The presence of

the wall friction results in variation of the total momentum flux downstream the

wall jets. Akatnov (1953) and Glauert (1956) showed that there is an integral

invariant Jx ¼Ð10

ru2Ðy0

rudy� �

dy which remains constant along the laminar wall

jets over a plane wall.

As in the previous cases of jet flows considered in the present chapter, we begin

our consideration with formulation of the functional equations for the maximal

longitudinal velocity and the boundary layer thickness in the jet. These equations

are identical to (6.19) and (6.21). Since all the governing parameters in the

functional equation for the maximal longitudinal velocity and the boundary thick-

ness have independent dimensions, we obtain

um ¼ cura1ma2Ja3x xa4 (6.61)

d ¼ cdra01ma

02J

a03

x xa04 (6.62)

where cu and cd are constants, and the exponents ai and a0i are given by

a1 ¼ � 1

2; a2 ¼ � 1

2; a3 ¼ 1

2; a4 ¼ � 1

2; a

01 ¼ � 1

4; a

02 ¼

3

4; a

03 ¼ � 1

4; a

04

¼ 3

4(6.63)

Then the expressions for um and dd read

146 6 Jet Flows

um ¼ cuJxrm

� �1=2

x�1=2 (6.64)

dd ¼ cdm3

rJx

� �1=4

x3=4 (6.65)

It is emphasized that (6.39) and (6.40), as well as (6.64) and (6.65) coincide with

the expressions obtained as the exact analytical solutions of the boundary layer

equations (Table 6.3) (Vulis and Kashkarov 1965).

� ¼ 1

2F1ln

Fþ ffiffiffiffiffiffiffiffiffiffiFF1

p þ F1

ð ffiffiffiffiffiffiffiF1

p � ffiffiffiF

p Þ2þ

ffiffiffi3

p

F1ðarctan 2

ffiffiffiF

p þ ffiffiffiffiffiffiffiF1

pffiffiffiffiffiffiffiffiffi3F1

p � arctan1ffiffiffi3

p Þ,

� ¼ Byxb, F1 ¼ 1:7818, the dimensionless value J1 ¼Ð10

FF0d�:

The approach outlined above can be also used for the analysis of wall jets over

curved surfaces. This type of jets was studied theoretically by Wygnanski and

Champagne (1968). They found the integral invariant of the problem and the self-

similar solutions for both concave and convex surfaces in the case when the local

radius of curvature varies as x3=4(with x being reckoned along the surface). Below

we consider wall jets over curved surfaces in the framework of the dimensional

analysis.

In the case when the ratio of the characteristic jet thickness to the local radius of

curvature R is sufficiently small and curvature variations occur in such a way that

dR=dx � 1; the set of the governing boundary layer equations has the following form

ru@u

@xþ 1þ y

R

� �v@u

@yþ r

uv

R¼ � @P

@Xþ m 1þ y

R

� � @2u@y2

þ mR

@u

@r(6.66)

ru2

R¼ @P

@y(6.67)

Table 6.3 Characteristic constants for laminar and turbulent jetsa

Jet geometry

Laminar jet Turbulent jet

’ð�ÞAi Bi ai bi Ai Bi ai bi

Plane jet 12

ffiffi½p 3� 3J2x4r2n

12

ffiffi½p 3� Jx6rn2

� 13

� 23

ffiffiffiffiffiffiffiffiffi3Jx

8rk1=2

q1

2k1=2� 1

2� 1 ’Pð�Þ

Axisym-metric jet 3Jx8prn

ffiffiffiffiffiffiffiffiffi3Jx

8prn2

q � 1 � 1ffiffiffiffiffiffiffi3Jx8prk

q1

k1=2� 1 � 1 ’Að�Þ

Radial jet 14

ffiffi½p 3� 9J2x2p2r2n

12

ffiffi½p 3� 3J2x4prn2

� 1 � 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

3Jx16prk1=2

q1ffiffiffiffi2k

p � 1 � 1 ’Rð�ÞWall jet

ffiffiffiffiffiffiffiffiffiffiJx

4r2nJ1

q12

ffiffiffi4

p 3J2x J�11

4r2n2� 1

2� 3

4� � � � ’Wð�Þ

aA; B; a and b are the coefficients in the expressions for um and ’ :

um ¼ Axa; ’Pð�Þ ¼ ð1� tanh2�Þ; ’A ¼ ð1þ 18�Þ�2; ’Rð�Þ ¼ ð1� tanh2�Þ,

’Wð�Þ ¼ ðF132F

12 � F2Þ, the function F is determined by the transcendental equation

6.6 Wall Jets over Plane and Curved Surfaces 147

@ru@x

þ @

@y1þ y

R

� �rv

n o¼ 0 (6.68)

where x and y are the coordinates along and normal to the surface, u and v are thevelocity components in the x and y directions. When the local radius of curvature

R>0, the wall is convex outwards, whereas R<0 the wall is concave.

Equations (6.66)–(6.68) are subjected to the following boundary conditions at

the wall (y ¼ 0) and far from it

y ¼ 0; u ¼ v ¼ 0; y ¼ 1; u ¼ 0 (6.69)

Integrating (6.66)–(6.68) yields the invariant

Jx ¼ð10

ru 1þ y

R

� � ð10

ru2dy� 1

R

ð10

ru2ydy

8<:

9=;dy ¼ const (6.70)

where Jx½ � ¼ LxL�2z M2T�3:

At R ! 1, the invariant (6.70) takes the form of the Akatnov’s invariant

mentioned above. It is emphasized that there are two identical forms of the

integral invariant of the wall jet near a straight wall: (i) Akatnov’s (1953) invariant-

Jx ¼Ð10

ru2Ð1y

rudy

!dy; and Glauert’s (1956) invariant-Jx ¼

Ð10

ruÐ1y

ru2dy

!dy:

It is easy to show that they are identical.

From the physical point of view, we note that the velocity and the jet thickness in

the far field of the wall jet over a curved surface should depend on the fluid

properties, the integral invariant of the jet, as well as on the radius of curvature of

the wall

u ¼ fuðr; m; Jx;R; x; yÞ (6.71)

um ¼ fumðr; m; Jx;R; xÞ (6.72)

d ¼ fdðr; m; Jx;R; xÞ (6.73)

where u is the longitudinal velocity, um its local maximal value, and d is the jet

thickness.

Equation 6.71 can be presented in the form

u ¼ fuðum; d;R; yÞ (6.74)

Since two of the four governing parameters in (6.74) have independent

dimensions, it reduces to

148 6 Jet Flows

u ¼ ’uð�; rÞ (6.75)

where u ¼ u=um; � ¼ y=d; and r ¼ R=d:Applying the Pi-theorem to (6.72), we arrive at the following expression

um ¼ Jxrm

� �1=21ffiffiffix

p ’um

R

m3x3=rJxð Þ� 1=2

(6.76)

The self-similar solution of the present problem exists when R ¼ cx3=4; where cis a constant (Wygnanski and Champagne 1968). Substitution this expression in

(6.76) yields

um ¼ Jxrm

� �1=21ffiffiffix

p ’um

c

m3=rJxð Þ1=4( )

(6.77)

6.7 Buoyant Jets (Plumes)

The buoyant jets belong to a special class of jet flows developing due to the

influence of buoyancy forces on fluid motion (Fig. 6.1c). They are typical for

a number of phenomena in nature and engineering: mass, momentum and heat

transfer in the atmosphere and oceans, pollutant dispersal, etc. (Turner 1969; 1986;

Jaluria 1980). The buoyant jets were a subject of a number of theoretical and

experimental investigations of complex flows developing under the influence of

the inertial, viscous and buoyancy forces (Zel’dovich 1937; Batchelor 1954;

Morton et al. 1956).

Below we consider only a few simplest examples of buoyant jets that illustrate

the application of the Pi-theorem to buoyancy-driven jet flows. Following

Zel’dovich (1937), we consider laminar vertical buoyant jets (plumes) over

a pointwise horizontal wire or a pointwise sphere treated as sources of heat. In

the framework of the Boussinesq approximation, the boundary layer equations

describing flows in laminar buoyant jet, which are generated by such sources read

u@u

@xþ v

@u

@y¼ n

yk@

@yyk@u

@y

� �þ bg# (6.78)

@u

@xþ 1

yk@vyk

@y¼ 0 (6.79)

u@#

@xþ v

@#

@y¼ a

yk@

@yyk@#

@y

� �(6.80)

6.7 Buoyant Jets (Plumes) 149

where n and a; are the kinematic viscosity and thermal diffusivity, respectively, b is

the thermal expansion coefficient, u and v are the longitudinal and transverse

components of velocity in the vertical and horizontal directions x and y; respec-

tively (cf. Fig. 6.1c), # ¼ T � T1;with T1 being the ambient temperature, g is the

gravity acceleration, k ¼ 0 or 1 for the plane or axisymmetric problems,

respectively.

The solutions of (6.78)–(6.80) are subjected to the following boundary

conditions

y ¼ 0@u

@y¼ 0; v ¼ 0;

@#

@y¼ 0; y ! 1 u ! 0; v ! 0 (6.81)

Rewriting (6.80) in the divergent form

@u#

@xþ 1

yk@v#yk

@y¼ a

yk@

@yyk@#

@y

� �(6.82)

and integrating it in y from the wake center y ¼ 0 to its edge y¼1, we arrive at the

following integral invariant expressing conservation of the excessive convective

heat flux along the plume

Q ¼ð10

u#ykdy ¼ const: (6.83)

Equations 6.78–6.80 with the boundary conditions (6.81) and the integral invari-

ant (6.83) show that velocity and temperature in buoyant laminar vertical jets

depend on six dimensional parameters, namely,

gb½ � ¼ Ly�1T�2; n½ � ¼ L2T�1; a½ � ¼ L2T�1; Q½ � ¼ L2þkyT�1; x½ � ¼ L; y½ �¼ L (6.84)

Therefore, the functional equations for the velocity and temperature fields can be

presented by the following functions

u ¼ f1ðgb; n; a;Q; x; yÞ (6.85)

# ¼ f2ðgb; n; a;Q; x; yÞ (6.86)

In order to reduce the number of the governing parameters it is convenient to

introduce new variables

½ev � ¼ v

n1=2

h i¼ T�1=2; ½y�� ¼ y

n1=2

h i¼ T1=2 (6.87)

150 6 Jet Flows

Then (6.78)–(6.80) take the form

u@u

@xþ ev @u

@ey ¼ 1ey k

@

@ey ey @u@ey

� �þ bg# (6.88)

@u

@xþ 1ey k

@evyk@ey ¼ 0 (6.89)

u@#

@xþ ev @#

@ey ¼ 1

Pr

1ey k

@

@ey ey k @#

@ey� �

(6.90)

The boundary and integral conditions in new variables read

ey ¼ 0@u

@ey ¼ 0; ev ¼ 0;@#

@ey ¼ 0 (6.91)

ey ! 1 u ! 0; # ! 0 (6.92)

Q�1 ¼

ð10

u#ey kdey (6.93)

where Q�1

� ¼ LyTðk�1Þ 2= .

Then, the velocity and temperature fields are given by the following functional

equations (where it is assumed that the Prandtl number equals one)

u ¼ f1ðbg;Q�1; x; ey Þ (6.94)

# ¼ f2ðbg;Q�1; x; ey Þ (6.95)

The axial velocity in the buoyant jet corresponds to ey ¼ 0, which reduces (6.94)

to the following equation

um ¼ f1�ðbg;Q�1; xÞ (6.96)

All the governing parameters in (6.96) have independent dimensions. Therefore,

in accordance with the Pi-theorem, (6.96) takes the form

um ¼ cðbgÞa1ðQ�1Þa2ðxÞa3 (6.97)

where c is a constant and the exponents ai are determined using the principle of the

dimensional homogeneity as


a1 ¼ a2 ¼ 2

5� k; a3 ¼ 1� k

5� k(6.98)

Accordingly, (6.97) reads

um ¼ c bgQ�1

�2=ð5�kÞxð1�kÞ=ð5�kÞ

n o(6.99)

Since Q�1 ¼ Q=nðkþ1Þ=2; (6.99) takes the form

um ¼ bgQð Þ2=ð5�kÞn�ðkþ1Þ=ð5�kÞxð1�kÞ=ð5�kÞn o

(6.100)

Taking in (6.100) k ¼ 0 or 1, we arrive at the following expressions for the axial

velocity in plane and axisymmetric buoyant laminar jets (plumes)

um;plane ¼ c bgQð Þ2=5n�1=5x1=5n o

(6.101)

um;axis: ¼ c bgQð Þ1=2n�1=2n o

(6.102)

According to the Pi-theorem (6.94) corresponds to the case of n�k ¼ 1. There-

fore, it can be reduced to the following dimensionless form

P ¼ ’ðP1Þ (6.103)

where P ¼ u= bgð Þa1ðQ�1Þa2ðxÞa3 ; P ¼ y

�= bgð Þa

01ðQ�

1Þa02ðxÞa

03 ; and a1 ¼ a2 ¼

2=ð5� kÞ; a3 ¼ ð1� kÞ=ð5� kÞ; a01 ¼ a

02 ¼ �1=ð5� kÞ; a0

3 ¼ 2=ð5� kÞ.For k ¼ 0 or 1, (6.103) yields the longitudinal velocity profiles in plane and

axisymmetrical buoyant jets as

uplane ¼ bgQð Þ2=5n�1=5x1=5’plane y bgQð Þ1=5=n3=5x2=5n o

(6.104)

uaxis: ¼ bgQð Þ1=2n�1=2’axis: y bgQð Þ1=4=n3=4x1=2n o

(6.105)

Similarly the temperature distributions in buoyant laminar jets are given by

yplane ¼ Q4=5ðbgÞ�1=5n�2=5x�3=5’�plane yðbgQÞ1=5=x2=5y3=5

n o(6.106)

yaxis: ¼ Qn�1x�1’�axis: yðbgQÞ1=4=x1=2n3=4n o

(6.107)

The expressions (6.104)–(6.107) determine only the form of the self-similar

solutions for the velocity and temperature distributions in laminar buoyant jets.

152 6 Jet Flows

They can be also used to establish the exact analytical solution of this problem. In

particular, substitution of the expressions (6.104)–(6.107) into (6.78)–(6.80) allows

transformation of the system of the governing partial differential equation into

the system of ordinary differential equation for the unknown functions ’plane;’axisymmetric; ’�

plane; and ’�axisymmetric: It is emphasized that that the analytical

solution of this problem is found without any particular hypothesis about the

mechanism of mixing of hot fluid with the surrounding fluid in the buoyant jets.

It is interesting to note that the Reynolds numbers in plane and axisymmetric

laminar buoyant jets increase with x

Replane ¼ bgQx3=n3 (6.108)

Reaxisymmetric ¼ bgQx2=n3 (6.109)

That means that there is some critical distance xcr from the source where

transition of laminar flow into the turbulent one inevitably occur (Zel’dovich 1937).

The developed approach also allows the analysis of the velocity and temperature

fields in turbulent buoyancy jets. In this case the system of the governing equations

is written for the average flow characteristics and should be closed using a semi-

empirical model of turbulence, in particular, the Prandtl mixing length model.

Applying the Pi-theorem it is not possible to obtain the following expressions for

the longitudinal velocity and temperature distributions in plane and axisymmetric

turbulent jets. In particular, for plane turbulent jets

uT ¼ bgQx

� �1=3

’planeðy x= Þ; yT ¼ Q2=3ðbgÞ�1=3x�5=3’planeðy xÞ= (6.110)

and for the axisymmetric jets

uT ¼ bgQ x=ð Þ1=3’axis:ðy x= Þ; yT ¼ Q2=3 bgð Þ�1=3x�5=2’axis: y x=ð Þ (6.111)

Note that the expressions (6.110) and (6.111), as any expressions derived using

semi-empirical models of turbulence incorporate some constants involved in

these models.

Studies of turbulent buoyant jets widely use models based on the so-called

entrainment hypothesis which assumes a certain relation of the mean inflow

velocity across the edge of a turbulent jet with some characteristic velocity in

a given cross-section (Batchelor 1954; Morton et al. 1956). In the framework

of such models the following system of the “entrainment” equations can be

formulated

d

dzb2uum � ¼ 2abuum (6.112)


d

dz

1

2b2uu

2u

� �¼ l2b2ugym (6.113)

d

dz

l2b2uumgym1þ l2

� �¼ �b2uumN

2ðzÞ (6.114)

where a is the entrainment constant, l ¼ by=bu is the ratio of the widths of the

temperature and velocity profiles (l is assumed to be constant-l � 1:2Þ; NðzÞ ¼�g=r1ð Þ dr0=dzð Þ; with r0 and r1 being the density at a given height z and the

reference density in the environment. The change of the characteristic scales of

the problem with height z can be easily found by considering the dimensions of

the parameters involved. As a result, the velocity at the axis of an axisymmetric jet

um is determined as

um ¼ cM1=2z�1 (6.115)

where rM ¼ 2pÐ10

ru2��z¼0

rdr is the momentum flux at z ¼ 0; and c is a constant.

A number of instructive examples of the application of the entrainment model to

flows in buoyant jets of different types can be found in the surveys of Turner (1969)

and List (1982), as well as in the original works of Morton (1957, 1959), Turner

(1966, 1986), Papanicolaou and List (1982), Baines et al. (1990), and Bloomfield

and Kerr (2000).

Problems

P.6.1. Determine the velocity distribution along the axis of plane laminar

submerged jet.

The governing parameters in the present case are: n½ � ¼ L2yT�1; Ix½ � ¼

L2xLyT�2; and x½ � ¼ Lx: Therefore, the functional equation for the axial velocity

reads

um ¼ fmðn; Ix; xÞ (P.6.1)

Since the parameters n; Ix; and x have independent dimensions, (P.4.1) reduces

to the following one

um ¼ c1na1 Ia2x xa3 (P.6.2)

where c1 is constant.

154 6 Jet Flows

Taking into account the dimensions of um; n, Ix and x, we find the values of the

exponents ai: a1 ¼ �1=3; a2 ¼ 2=3; a3 ¼ �1=3. Substitution of the values of a1;a2 and a3 in (P.6.2) yields

um ¼ c13

ffiffiffiffiffiI2xnx

r(P.6.3)

P.6.2. Show that the local Reynolds number Red ¼ <u>d=n in the axisymmetric

laminar submerged jet is independent of x.The functional equations for u; um and d have the following form

u ¼ fuðr; m; Jx; x; yÞ (P.6.4)

um ¼ fu:mðr; m; Jx; xÞ (P.6.5)

d ¼ fdðr; m; Jx; xÞ (P.6.6)

where the governing parameters have the dimensions

r½ � ¼ L�1x L�1

y L�1z M; m½ � ¼ L�1

x LyL�1z MT�1; Jx½ � ¼ LxLyL

�1z MT�2; x½ �

¼ Lx; y½ � ¼ Ly: (P.6.7)

Equation (P.6.4) can be presented in the form

u ¼ fuðum; d; yÞ (P.6.8)

Applying the Pi-theorem to (P.6.8), we reduce it to the dimensionless form

u ¼ ’uð�Þ (P.6.9)

where u ¼ u=um, and � ¼ y=d:Bearing in mind that r; m; Jx and x have independent dimensions, we present

(P.6.5) and (P.6.6) as follows

um ¼ c1ra1ma2Ja3x xa4 (P.6.10)

d ¼ c2ra01ma

02J

a03

x xa04 (P.6.11)

where c1 and c2 are constants, and the exponents ai and a0i are equal to a1 ¼ 0;

a2 ¼ �1; a3 ¼ 1; a4 ¼ �1; a01 ¼ �1=2; a

02 ¼ 1; a

03 ¼ �1=2, and a

04 ¼ 1: Then,

we obtain

Problems 155

um ¼ c1Jxrnx

(P.6.12)

d ¼ c2rnx

rJxð Þ1=2(P.6.13)

Taking into account that <u> ¼ umÐþ1=2

�1=2

’ð�Þ�d�; we find

Red ¼ AJxrn2

� �1=2

¼ const (P.6.14)

where A ¼ c1c2Ð1=2

�1=2

’ð�Þ�d�:

P.6.3 Describe the velocity variation along the axis of a radial laminar

submerged jet (Table 6.1).

The axial velocity of radial jets depends on four governing parameters: ½r� ¼L�1x L�1

y L�1z M; ½m� ¼ L�1

x LyL�1z MT�1; ½Jx� ¼ L�2

y L�1z MT�2 and ½x� ¼ Lx that have

independent dimensions. According to the Pi-theorem,

um ¼ cra1ma2Ja3x xa4 (P.6.15)

where c is a constant, and the exponents ai have the following values: a1 ¼ �1=3;a2 ¼ �1=3; a3 ¼ 2=3; and a4 ¼ �1:

As a result, we obtain the following expression for the axial velocity in radial

laminar submerged jets

um ¼ c3

ffiffiffiffiffiffiffiJ2xr2n

sx�1 (P.6.16)

References

Abramovich GN (1963) The theory of turbulent jets. MIT Press, Boston

Akatnov NI (1953) Development of two-dimensional laminar incompressible jet near a rigid wall.

Proc Leningrad Polytec Inst 5:24–31

Andrade EN (1939) The velocity distribution in liquid-into-liquid jet. The plane jet. Proc Phys Soc

London 51:748–793

Baines WD, Turner JS, Campbell IH (1990) Turbulent fountains in an open chamber. J Fluid Mech

212:557–592

Batchelor GK (1954) Heat convection and buoyancy effects in fluid. Quart J Roy Meteor Soc

80:339–358

Bloomfield LJ, Kerr RC (2000) A theoretical model of a turbulent fountain. J Fluid Mech

424:197–216

156 6 Jet Flows

Glauert MB (1956) The wall jet. J Fluid Mech 1:625–643

Hinze JO (1959) Turbulence. An introduction to its mechanism and theory. McGraw Hill Book

Company, New York

Ho CM, Gutmark E (1987) Vortex induction and mass entrainment in a small-aspect-ratio elliptic

jet. J Fluid Mech 179:383–405

Hussain F, Hussain HS (1989) Elliptic jets. Part 1. Characteristics of unexcited and excited jets.

J Fluid Mech 208:259–320

Jaluria Y (1980) Natural convective heat and mass transfer. Pergamon, Oxford

Konsovinous NS (1978) A note on the conservation of the axial momentum of turbulent jet. J Fluid

Mech 87:55–63

Korthapalli A, Baganoff D, Karamcheti K (1981) On the mixing of rectangular jet. J Fluid Mech

107:201–220

Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, London

List EJ (1982) Turbulent jets and plumes. Annu Rev Fluid Mech 14:189–212

Morton BR (1957) Buoyant plumes in moist atmosphere. J Fluid Mech 2:127–144

Morton BR (1959) Forced plumes. J Fluid Mech 5:151–163

Morton BR, Taylor GI, Turner JS (1956) Turbulent gravitational convection from maintained and

instantaneous sources. Proc Roy Soc London A 234:1–23

Pai SI (1954) Fluid dynamics of jets. D. Van Nastrand Company, New York

Papanicolaou PN, List EJ (1982) Investigations of round vertical turbulent buoyant jets. J Fluid

Mech 195:341–391

Schlichting H (1979) Boundary layer theory, 8th edn. Springer, Berlin

Schneider W (1985) Decay of momentum flux in submerged jets. J Fluid Mech 154:91–110

Townsend AA (1956) The Structure of turbulent shear flow. Cambridge University Press,

Cambridge

Trentacoste N, Sforzat P (1967) Further experimental results for three-dimensional free jets.

AIAA J 5:885–891

Turner JS (1966) Jets and plumes with negative or reversing buoyancy. J Fluid Mech 26:729–792

Turner JS (1969) Buoyant plumes and thermals. Annu Rev Fluid Mech 1:29–44

Turner JS (1986) Turbulent entrainment: the development of the entrainment assumption and its

application to geophysical flows. J Fluid Mech 173:431–471

Vulis LA, Kashkarov VP (1965) The theory of viscous fluid jets. Nauka, Moscow (in Russian)

Wygnanski I, Champagne FH (1968) The laminar wall jet over curved surface. J Fluid Mech

31:459–465

Zel’dovich YaB (1937) Limiting laws of free-rising convective flows. J Exp Theoret Phys

7:1463–1465. The English translation in: Ya.B. Zel’dovich: Selected Works of Ya.B.

Zel’dovich, vol. 1. Chemical Physics and Hydrodynamics. Limiting laws of freely rising

convective currents (Princeton Univ. Press, Princeton, 1992).

References 157

Chapter 7

Heat and Mass Transfer


This Chapter deals with processes of heat and mass transfer in solid, liquid and

gaseous media. They have important implications for understanding various natural

phenomena as well as technological processes. A vast literature is devoted to heat

and mass transfer in motionless and moving media. Tens of thousands of journal

publications contain detailed information on modern methods of investigation of

heat and mass transfer problems (e.g. mathematical modeling, measuring thermohy-

drodynamical quantities, etc.). The results of numerous theoretical and experimental

researches in this field are generalized in a number of well-known monographs on

fluid dynamics (Landau and Lifshitz 1987; Schlichting 1979; Levich 1962), heat and

mass transfer (Kutateladze 1963; Spalding 1963; Kays 1975;White 1988), as well as

in reference books (Rohsenow et al. 1998; Kaviany 1994).

Heat and mass transfer in various media occur under the interaction of different

factors, which involve the effects of physical properties of the matter, its equation

of state and regime of motion, contact with the surrounding fluid or solid surfaces,

etc. In the general case the relevant physical processes are described by a system of

coupled non-linear partial differential equations that include the continuity,

momentum, energy and species conservation equations. This system of equations

should be supplemented by the equation of state and correlations determining the

dependences of physical properties of the matter on temperature and species

concentrations. Solving such complicated system of equations entails great

difficulties. Therefore, studing heat and mass transfer processes, as a rule, employs

several simplifying assumptions that lead to an approximate solution of the problem

at hand. A qualitative analysis of such complex phenomena as heat and mass

transfer in continous media is signicantly facilitated by applying the dimensional

analysis, in particular, the approach based on the Pi-theorem. The results discuss in

the present chapter demonstrate the applications of the Pi-theorem to conductive

heat transfer in media with constant and temperature-dependent thermal diffusivity,

convective heat transfer under the conditions of forced, natural and mixed



159

convection, as well as heat and mass transfer in laminar boundary layers and

laminar pipe and jet flows. Using the Pi-theorem for studing heat and mass transfer

under the conditions of phase change is also discussed in this Chapter.

7.2 Conductive Heat and Mass Transfer

7.2.1 Temperature Field Induced by Plane InstantaneousThermal Source

Consider the evolution of the temperature field due to a plane instantaneous thermal

source of strength Q releasing heat at t ¼ 0 and x ¼ 0 in an infinite medium with

constant (temperature-independent) physical properties (Fig. 7.1).

The source can be presented as

q ¼ QdðxÞdðtÞ (7.1)

where Q is a constant, and dð�Þ is the delta-function.The medium is at rest, heat is transferred by conduction only, and the thermal

balance equation that describe the one-dimensional excessive (relative to the initial

one) temperature field Tðx; tÞ at t>0 reads

@T

@x¼ a

@2T

@x2(7.2)

where a is the thermal diffusivity.

Integrating (7.2) by x from �1 to þ1 and accounting for the boundary

conditions at infinity @T=@xj�1 ¼ @T=@xj1 ¼ 0; we find that Q ¼ Ðþ1

�1Tdx

¼ const, i.e. Q is the problem invariant. At the initial time moment t ¼ 0 the

excessive temperature equals zero at any x>0. That means that the previous integral

is indeed equal to the heat source strength, while at t¼0 the temperature field is

Fig. 7.1 Temperature

distribution corresponding to

a plane instantaneous thermal

source acting at t ¼ 0 at

x ¼ 0 in an infinite medium

with constant properties. The

snapshots shown correspond

to ti>0

160 7 Heat and Mass Transfer

given as Tðx; 0Þ ¼ QdðxÞ: Then, the functional equation for the temperature

field reads

T ¼¼ f ða;Q; t; xÞ (7.3)

Analyzing the dimensions of the governing parameters ( a½ � ¼ L2T�1; Q½ � ¼ Ly;t½ � ¼ T; x½ � ¼ L, with y being the temperature scale of temperature and applying the

Pi-theorem, we arrive at the following equation

T ¼ Qffiffiffiffiffiax

p ’xffiffiffiffiat

p� �

(7.4)

where ’ ¼ ’ð�Þ; and � ¼ x=ffiffiffiffiat

p:

7.2.2 Temperature Field Induced by a Pointwise InstantaneousThermal Source

The approach of the previous sub-section can be also applied to the evolution of the

excessive temperature field triggered by a pointwise thermal source of strength Q,which acted at t¼0 and r¼0 in an infinite medium with constant thermal diffusivity

q ¼ QdðrÞdðtÞ (7.5)

where Q is a constant, and r is the radial coordinate in the spherical coordinate

system centered at the heat source.

The thermal balance equation that describe the evolution of the temperature field

at t>0 reads1

@T

@r¼ a

1

r2@

@rr2@T

@r

� �(7.6)

Integrating (7.6) by r from r ¼ 0 to r ¼ 1 and accounting for the boundary

conditions @T=@rjr¼0 ¼ @T=@rjr¼1 ¼ 0, yields the following invariant

ð10

Tr2dr ¼ Q ¼ const: (7.7)

where ½Q� ¼ L3y:

1 Equation 7.6 accounts for the spherical symmetry of the temperature field.

7.2 Conductive Heat and Mass Transfer 161

Then, the excessive temperature field satisfies the following functional equation

T ¼ f ða;Q; t; rÞ (7.8)

Taking into account the dimensions of the governing parameters in (7.8) and

using the Pi-theorem, we arrive at the dimensionless equation

P ¼ ’ðP1Þ (7.9)

where P ¼ T=aa1Qa2 ta3 ; and P1 ¼ r=aa01Qa

02 ta

03 .

Determining the exponent as a1 ¼ �3=2; a2 ¼ 1; a3 ¼ �3=2; a01 ¼ 1=2; a

02 ¼

0 and a03 ¼ 1=2, we arrive at the following expression for the temperature field

T ¼ Q

atð Þ3=2’

r

ðatÞ1=2 !

(7.10)

7.2.3 Evolution of Temperature Field in Mediumwith Temperature-Dependent Thermal Diffusivity(The Zel’dovich-Kompaneyets Problem)

The present sub-section is devoted to the evolution of temperature field in response

to an instantaneous plane energy source in medium which thermal diffusivity

depending on temperature. Very strong heat release in a substance is accompanied

by temperature rise of the order of tens or even hundred of thousands degrees. In

such cases the energy transport occurs mainly by radiation. Under these conditions

the radiant thermal diffusivity coefficient depends on temperature and can be

expressed as (Zel’dovich and Kompaneyets 1970; Zel’dovich and Raizer 2002)

w ¼ aTn (7.11)

where a and n are given constants, in particular, a is dimensional, a½ � ¼ L2T�1y�n

and n is dimensionless, n½ � ¼ 1:According to (7.11), the radiant thermal diffusivity coefficient w approaches to

zero at T ! 0: At high temperature in a heated zone Th � T1 ðTh and T1 are the

temperatures in the heated zone and the surrounding medium, respectively) it is

possible to assume that ambient temperature equals zero, i.e. T1 ¼ 0: In this case

heat can not be transferred instantaneously to large distances from the thermal

source. It spreads over substance with finite speed, so that there exists some

boundary that separate the heated zone from the cooled undisturbed one. In this

case head spreads in the form of a thermal wave as is shown in Fig. 7.2.


Let at t¼0 in plane x ¼ 0 thermal energy of E (say, Joule) is released per 1m2 of

surface. The evolution of the temperature field at t>0 is described by the thermal

balance equation

@T

@t¼ @

@xw@T

@x

� �(7.12)


x ! 1; T ! 0; x ¼ 0;@T

@x¼ 0 (7.13)

where the dependence of the radiant thermal diffusivity coefficient on temperature

is given by (7.11).

Integrating (7.12) in x from �1 to 1, we obtain the invariant of the present

problem

Q ¼ð1

�1Tdx (7.14)

where ½Q� ¼ ½E=rcP� ¼ Ly, where r and cP are density and the specific heat at

constant pressure of the matter, respectively.

From (7.11), (7.12) and (7.14) it follows that there are the following governing

parameters of the problem: two constants a and Q and two variables x and t

T ¼ f ða;Q; x; tÞ (7.15)

It is seen that three of the four governing parameters have independent

dimensions. Then, in accordance with the Pi-theorem (7.15) reduces to the follow-

ing dimensionless form

P ¼ ’ðP1Þ (7.16)

where P ¼ T=aa1Qa2 ta3 ; and P1 ¼ x=aa01Qa

02 ta

03 .

T

0 x

t1t2t3

Fig. 7.2 Temperature

distribution in response to a

plane instantaneous thermal

source at t ¼ 0 at x ¼ 0 in

medium with temperature-

dependent thermal diffusivity

7.2 Conductive Heat and Mass Transfer 163

Taking into account the dimensions of the parameters involved, we arrive at the

system of the six algebraic equations for the exponents ai and a0i:

2a1 þ a2 ¼ 0; 2a01 þ a

02 ¼ 1

� a1 þ a3 ¼ 0 ;�a01 þ a

03 ¼ 0 (7.17)

� na1 þ a2 ¼ 1; na01 þ a

03 ¼ 0

From (7.17) it follows that

a1 ¼ � 1

nþ 2; a2 ¼ 2

nþ 2; a3 ¼ � 1

nþ 2; a

01

¼ 1

nþ 2; a

02 ¼

n

nþ 2; a

03 ¼

1

nþ 2

(7.18)


T ¼ Q2

at

� �1=ðnþ2Þ’

x

aQntð Þ1=ðnþ2Þ

( )(7.19)

Substituting the expression (7.19) into (7.12) yields the following ODE for the

unknown function ’

ðnþ 2Þ d

dx’n d’

dx

� �þ x

d’

dxþ ’ ¼ 0 (7.20)

The boundary conditions for (7.20) are

’ðxÞ ¼ 0 at x ! 1;d’ðxÞdx

¼ 0 at x ¼ 0 (7.21)

where x ¼ x

aQntð Þ1=ðnþ2Þ :

The solution of (7.20) and (7.21) is (Zel’dovich and Raizer 2002)

’ðxÞ ¼ x20n

2ðnþ 2Þ� �

1� xx0

� �2" #1=n

(7.22)

at x<x0; and

’ðxÞ ¼ 0 (7.23)


at x ¼ 0, where x0 is a constant which is found from the energy invariant (7.14)

x0 ¼ðnþ 2Þ1þn

21�n

npn=2Gn 1=2þ 1=nð Þ

Gn 1=nð Þ

( )1=ðnþ2Þ(7.24)

In (7.24) Gð�Þis the gamma function. The position and velocity of the thermal

wave front are given by the following expressions

xf ¼ x0 aQntð Þ1=ðnþ2Þ(7.25)

vf ¼ x01

nþ 2

aQn

tnþ1

� �1=ðnþ2Þ(7.26)

where xf ðtÞ and vf ðtÞ are the current coordinate and velocity of the thermal wave front.

7.3 Heat and Mass Transfer Under Conditions of Forced

Convection

7.3.1 Heat Transfer from a Hot Body Immersed in Fluid Flow

The first attempt of theoretical investigation of this problem by applying the

dimensional analysis dates back to Lord Rayleigh (1915). He employed the Pi-

theorem for studying heat transfer from a hot body moving in an incompressible

fluid. Rayleigh assumed that five dimensional parameters, namely, (1) the charac-

teristic size of a body, say, a spherical particle, d; (2) its velocity relative to the

surrounding medium v; (3) the temperature difference between the body and the

undisturbed fluid far away from it DT; as well as (4) the fluid heat capacity c and (5)thermal conductivity k determine the rate of heat transfer h�

h� ¼ f ðd; v;DT; c; kÞ (7.27)

where DT ¼ Tw � T1; Tw and T1 are the body and undisturbed fluid temperature,

respectively.

The unknown rate of heat transfer h� and the governing parameters of the

problem d; v; DT; c and k have the following dimensions

½h�� ¼ JT�1; d½ � ¼ L; v½ � ¼ LT�1; DT½ � ¼ y; c½ � ¼ JL�3y�1; k½ �¼ JL�1T�1y�1 (7.28)

where J and y are the independent units of heat and temperature.

7.3 Heat and Mass Transfer Under Conditions of Forced Convection 165

The set of the five governing parameters contains four parameters with indepen-

dent dimension, so that n� k ¼ 1: Choosing as the parameters with the indepen-

dent dimensions d; v; DT; and k, we write in accordance with the Pi-theorem the

dimensionless form of (7.27) as

P ¼ ’ðP1Þ (7.29)

where P ¼ h�=da1va2DTa3ka4 ; and P1 ¼ c=da01va

02DTa

03ka

04 :

Using the principle of dimensional homogeneity,we find values of the exponents aiand a

0i : a1 ¼ 1; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a

01 ¼ �1; a

02 ¼ �1; a

03 ¼ 0; and a

04 ¼ 1:


h�

dkDT¼ ’

dvc

k

� �(7.30)

Defining heat flux as q ¼ h=S; with S being the surface of the body, we rewrite

(7.30) as

Nu ¼ ’ðPeÞ (7.31)

where Nu ¼ qd=kDT; and Pe ¼ dvc=k are the Nusselt and Peclet numbers,

respectively.

Equation 7.31 shows that the Nusselt number depends on a single dimensionless

group Pe. At a fixed Peclet number the rate of heat transfer h is directly proportionalto the temperature difference DT and the characteristic size of the body d, whereasthe heat flux q is inversely proportional to d:

The results of Rayleigh demonstrated the efficiency of application of the dimen-

sional analysis to problems related to convective heat transfer, which attracted a

significant attention to this approach which was thoroughly discussed in the fol-

lowing references: Riabouchinsky (1915), Brigman (1922) and Sedov (1993).

In particular, Riabouchinsky made a remark about the choice of the system of

units of in description of convective heat transfer phenomena. The system of units

that Rayleigh used includes three mechanical units (length L; mass M and time TÞand two independent thermal units for quantity of heat (understood as thermal

energy) J and temperature y: It is emphasized that it is possible to express the

dimension of temperature, and accordingly the other thermal quantities, by means

of the basic mechanical units LMT: Indeed, temperature can be related with the

average kinetic energy of molecules with the dimension ML2T�2½ � in the LMTsystem of units. According to the First Law of thermodynamics, the dimension of

heat in the LMT system of units ss J ML2T�2½ �: Then the dimensions of the

governing parameters in (7.27) are as follows

d½ � ¼ L; v½ � ¼ LT�1; y½ � ¼ ML2T�2; c½ � ¼ L�3; k½ � ¼ L�1T�1 (7.32)


Three parameters of the five in (7.32) have independent dimensions, so that n�k ¼ 2: In this case the dimensionless form of (7.27) reads

P ¼ ’ðP1;P2Þ (7.33)

where P ¼ h�=kDTd; P1 ¼ dvc=k; and P2 ¼ cd3:Thus, instead of (7.29) that determined the dimensionless rate of heat transfer as

a function of a single dimensionless group P1; we arrive at (7.33) where the

unknown quantity P is a function of two dimensionless groups (P1 and P2) that

makes its much less valuable. Essentially one sees that under the same conditions

(7.29) and (7.33) determine different dependences of the dimensionless heat trans-

fer rate P on the governing parameters. In particular, according to (7.29) the

dimensionless heat transfer rate P does not depend on the heat capacity c at

vc ¼ const, whereas (7.33) shows the existence of such dependence.

The seeming contradiction related to using two different systems of units in the

above-mentioned problem deserves the following comments:

1. By choicing governing parameters, one, in fact, makes some assumptions on the

structure of substance involved. In the frame of the continuum approach and

thermodynamics thermal phenomena are described in the macroscopic approxima-

tion fully the molecular structure of substance. Accoordingly, in this case

Rayleigh’s approach is correct, while Riabouchinsky’s counter-example is illegal.

2. The expression of dimensions of thermal quantities via mechanical units is based

on the First Law of thermodynamics that postulate the equivalence of all kinds of

energy, in particular, the thermal and mechanical ones. Therefore, in the general

case the set of governing parameters that determine the heat transfer rate should be

supplemented by two constants characterizing the relation of thermal energy to

mechanical one. These are the mechanical equivalent of heat j½ � ¼ ML2T�2J�1

and Boltzmann’s constant kB½ � ¼ ML2T�2y�1. Accordingly, Rayleigh’s set of the

governing parameters read

d; v; DT; c; k; j; kB (7.34)

Then the apppplication of the Pi-theorem leads to the following expression for

the heat transfer rate (Sedov 1993)

h

kDTd¼ ’ðdvc

k;jcd3

kBÞ (7.35)

When the effect of transformation of mechanical energy into heat (dissipation) is

negligible, (7.35) reduces to (7.30).

3. Many additional questions were raised by Rayleigh’s analysis (Brigman 1922). In

particular, concernswere driven by the fact that density and viscosity aremissing in

the set of governing parameters. The lack of these quantities in the set of the

governing parameters, seemingly diminishes the value of the result of the dimen-

sional analysis in this case.


Consider Rayleigh’s problem with the account for the fluid density and viscosity.

Then the functional equation for the heat flux q reads

q ¼ f ðd; v;DT; k; m; cP; rÞ (7.36)

where ½d� ¼ L; ½v� ¼ LT�1; r½ � ¼ L�3M; m½ � ¼ L�1MT�1; cP½ � ¼ JMy�1; and

q½ � ¼ JT�1L�2; k½ � ¼ LMT�3y�1; ½DT� ¼ yThe set of the governing parameters in (7.36) contains seven parameters, five of

them with independent dimensions, so that n� k ¼ 2: Take as the parameters with

independent dimensions d; DT; k; m and r and using the Pi-theorem transform

(7.36) to the following dimensionless form

P ¼ ’ðP1;P2Þ (7.37)

where P ¼ q=da1DTa2ka3ma4ra5 ; P1 ¼ v=da01DTa

02ka

03ma

04ra

05 ; and P2 ¼ cP=d

a001

DTa002 ka

003 ma

004 ra

005 :


exponents ai; a0i and a

00i as

a1 ¼ �1; a2 ¼ 1; a3 ¼ 1; a4 ¼ 0; a5 ¼ 0

a01 ¼ �1; a

02 ¼ 0; a

03 ¼ 0; a

04 ¼ 1; a

05 ¼ �1 (7.38)

a001 ¼ 0; a

002 ¼ 0; a

003 ¼ 1; a

004 ¼ �1; a

005 ¼ 0


Nu ¼ ’ðRe; PrÞ (7.39)

where Nu ¼ qd=kDT; Re ¼ vdr=m; and Pr ¼ cpm=k are the Nusselt, Reynolds

and Prandtl numbers, respectively.

In the particular case corresponding to creeping flows with small Reynolds

numbers (Re<1) the inertial effects are negligible. Then it is possible to omit

density from the set of the governing parameters and write (7.36) as follows

q ¼ f ðd; v;DT; k; m; cPÞ (7.40)

The dimensionless form of (7.40) is

Nu ¼ ’ðPrÞ (7.41)

where Pr is the Prandtl number.

The explicit form of (7.31) and (7.37) is determined either experimentally or by

solving the Navier-Stokes and energy equations. For flows of incompressible fluids


with constant physical properties the momentum and continuity equations can be

uncoupled and integrated independently of the energy equation. In this case the heat

transfer problem reduces to solving the energy equation with a known velocity

field. For creeping flows the expressions for the Nusselt number are typically

presented in the form of the series of the Peclet number. For example, Acrivos

and Taylor (1962) obtained the following expression for the Nusselt number for a

spherical particle at Re � 1 and Pr � 1 using the method of matched asymptotic

expansions

Nu ¼ 2þ Peþ Pe2 ln Peþ 0:829Pe2 þ 1

2Pe3 lnPe (7.42)

For calculating heat transfer from spherical bodies in a wide range of the

Reynolds and Prandtl numbers a number of empirical correlations have been

proposed (Soo 1990). In particular, the correlation valid in the range

1<Re<7� 104, 0.6<Pr<400 reads

Nu ¼ 2þ 0:459Re0:55 Pr0:33

(7.43)

It is seen that the form of the dependences NuðPeÞ and NuðRr; PrÞobtained in theframework of the dimensional analysis is identical to the form of correlations (7.42)

and (7.43) resulting from the analytical solution of the problem and experimental

data.

7.3.2 The Effect of Particle Rotation

The effect of rotation of a spherical particle on heat transfer is due to the influence

of the centrifugal force on fluid around the particle (Kreith 1968). The particle

rotation about an axis through its two poles promotes secondary currents that are

directed toward the poles and outward from the equatorial region. The superposi-

tion of the secondary flows on the main flow (driven by the inertial, buoyancy and

pressure forces) determines the hydrodynamic structure of the overall flow and its

evolution, as well as the general characteristics of heat and mass transfer.

In the general case the intensity of heat transfer from a spinning spherical

particle depends on nine dimensional parameters accounting for the effect of the

inertial, buoyancy and centrifugal forces: r; m; d; u1; k; cP; DT; ðbgÞ; and o.Accordingly, the functional equation for the heat flux from the particle reads

q ¼ f ðr; m; d; u1; k; cP;DT; bg;oÞ (7.44)

where ½b� ¼ y�1 is the thermal expansion coefficient, ½g� ¼ LT�2 is the gravity

acceleration, and ½o� ¼ T�1 is the angular speed of rotation.


Five governing parameters of the nine in (7.44) have independent dimensions so

that the difference n� k ¼ 4: Then, according to the Pi-theorem, (7.44) reduces to

the following form

P ¼ ’ðP1;P2;P3;P4Þ (7.45)

where P ¼ q=ra1ma2da3ka4DTa5 ; P1 ¼ u=ra01ma

02da

03ka

04DTa

05 ,P2 ¼ cP=ra

001

ma002 da

003 ka

004DTa

005 ; P3 ¼ bg=ra

0001 ma

0002 da

0003 ka

0004 DTa

0005 ; and P4 ¼ o=ra

1V1 ma

1V2

da1V3 ka

1V4 DTa1V

5 .

Taking into account the dimension of the heat flux q and those of the parameters

governing it, we find the values of the exponents ai; a0i; a

00i ; a

000i and a1Vi : They are

equal to

a1 ¼ 0; a2 ¼ 0; a3 ¼ �1; a4 ¼ 1; a5 ¼ 1

a01 ¼ �1; a

02 ¼ þ1; a

03 ¼ �1; a

04 ¼ 0; a

05 ¼ 0

a001 ¼ 0; a

002 ¼ �1; a

003 ¼ 0; a

004 ¼ 1; a

005 ¼ 0 (7.46)

a0001 ¼ �2; a

0002 ¼ 2; a

0003 ¼ �3; a

0004 ¼ 0; a

0005 ¼ �1

a1V1 ¼ �1; a1V2 ¼ 1; a1V3 ¼ �2; a1V4 ¼ 0; a1V5 ¼ 0

Bearing in mind (7.46), we rewrite (7.45) as follows

Nu ¼ ’ðRe; Pr;Gr;ReoÞ (7.47)

where Nu¼hd=k;Re¼u1d=n;Pr¼n=a;Gr¼bgðTw�T1Þd3=n2;and Reo¼od2=nare the Nusselt, Reynolds, Prandtl, Grashof and particle rotational Reynolds num-

bers, respectively, h¼q=DT is the heat transfer coefficient, and a¼k=rcP is the

thermal diffusivity.

The approaches based on the boundary layer theory are used for theoretical

description of heat transfer of a spinning particle (Dorfman 1967; Banks 1965;

Chao and Greif 1974; Lee et al. 1978). Under the conditions of forced convection

due to particle rotation the heat transfer coefficient depends on the rotational

Reynolds and Prandtl numbers, as is anticipated from (7.47). The dependence on

Reo may be expressed as (Kreith 1968)

Nu ¼ 0:43Re0:5o Pr0:4 (7.48)

for Reo<5 105; and

Nu ¼ 0:066Reo0:67Pr0:4 (7.49)

for Reo>5 105.


The comparison of a number of theoretical predictions with experimental data

on heat transfer of a spherical particle under the conditions of forced convection due

to its rotation shows that all the dependences NuðReo; PrÞ proposed have the form

Nu ¼ ARe0:5Pr0:4; with only the values of the factor A being different (Hussaini and

Sastry 1976). The empirical correlations for the average Nusselt number of a

spherical particle rotating in air at rest or in air stream were suggested by Eastop

(1973). In the former case the correlation reads

Nu ¼ 0:353Re0:5o (7.50)

whereas in the latter one (at Reo=Re>0:54) it has the following form

Nu ¼ 0:288Re0:6 1þ 0:167Reo

Re� 0:54

� �� (7.51)

7.3.3 The Effect of the Free Stream Turbulence

The free stream turbulence tends to enhance heat transfer from a particle to the

surrounding fluid due to the eddies penetrating from the external flow into the

particle boundary layer. The disturbances of the near-wall flow facilitate transition

to turbulence and shift the boundary layer separation point downstream over the

particle surface. The intensity of heat transfer from a heated spherical particle

immersed into turbulent flow depends on the physical properties of fluid: its density

r½ � ¼ L�3M; viscosity m½ � ¼ L�1MT�1; thermal conductivity k½ � ¼ JT�1y�1L�1;and specific heat cP½ � ¼ Jy�1M�1. It also depends on particle diameter d½ � ¼ L and

its velocity v1½ � ¼ LT�1; the temperature difference between the particle and

surrounding fluid DT½ � ¼ y; the characteristic velocity of turbulent fluctuationsev0h i

¼ LT�1 (ev0 ¼ffiffiffiffiffiv02

pis the root mean square of the turbulence velocity

fluctuations) and the integral scale of turbulence l½ � ¼ L: Accordingly, the func-

tional equation for the heat flux q½ � ¼ JT�1L�2 reads

q ¼ f ðr; m; k; cP; v1;DT; d; ev0 ; lÞ (7.52)

Equation 7.52 contains nine dimensional parameters, five of them having inde-

pendent dimensions. Then, according with the Pi-theorem, the dimensionless form

of (7.52) appears to be

P ¼ ’ðP1;P2;P3;P4Þ (7.53)


where P ¼ q=ra1ma2ka3DTa4da5 ; P1 ¼ cP=ra01ma

02ka

03DTa

04da

05 ; P2 ¼ v1=ra

001 ma

002 ka

003

DTa004 da

005 ; P3 ¼ ev0=ra

0001 ma

0002 ka

0003 DTa

0004 da

0005 ; and P4 ¼ l=ra

1V1 ma

1V2 ka

1V3 DTa1V

4 da1V5 and the

exponents ai; a0i; a

00i ; a

000i and a1Vi are equal to

a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a5 ¼ �1;

a01 ¼ 0; a

02 ¼ �1; a

03 ¼ 1; a

04 ¼ 0; a

05 ¼ 0;

a001 ¼ �1; a

002 ¼ 1; a

003 ¼ 0; a

004 ¼ 0; a

005 ¼ �1; (7.54)

a0001 ¼ �1; a

0002 ¼ 1; a

0003 ¼ 0; a

0004 ¼ 0; a

0005 ¼ �1;

a1V1 ¼ 1; a1V2 ¼ 0; a1V3 ¼ 0; a1V4 ¼ 0; a1V5 ¼ 1;

Taking into account the values of the exponents ai; a0i; a

00i ; a

000i and a1Vi , it is

possible to present (7.52) in the following form

Nu ¼ ’ðPr;Re;ReT ; lÞ (7.55)

where Nu ¼ qd=kDT; Pr ¼ cPm=k; Re ¼ v1dr=m; and ReT ¼ ev0=v1

v1dr=m

¼ TuRe (Tu ¼ ev0v1

is the turbulence intensity) are the Nusselt, Prandtl, Reynolds

and turbulent Reynolds number, respectively, l ¼ l=d is the dimensionless turbu-

lence scale.

Equation 7.55 shows that the contribution of turbulence to heat transfer is

determined by two dimensionless parameters accounting for the turbulence inten-

sity and the size of turbulent eddies. Depending on values of these parameters,

different conditions of heat transfer may be realized. When l 1; the particle

experiences a time-dependent flow, whereas when l<<1, the flow becomes

quasi-steady. In the latter case the heat transfer coefficient depends on the Reynolds

and Prandtl numbers, as well as on the turbulence intensity Tu (Kestin 1966)

Nu ¼ ’ðPr;Re; TuÞ (7.56)

In particular, for a spherical particle immersed in the turbulent flow with

2� 103<Re<6:5� 104 and 1<Tu<17%, (7.55) takes the form (Lavender and Pei

1967)

Nu ¼ 2þ f ðReTÞRe0:5 (7.57)

where f ðReTÞ ¼ 0:629Re0:035T for ReT<103 and f ðReTÞ ¼ 0:145Re0:25T for Re>103:The difference in the dependences f ðReTÞcorresponding to small and large values

of ReT are related to the peculiarities of flow about a particle at low and high

turbulence intensity. The influence of the free stream turbulence on the particle

boundary layer is weak enough at low Tu: When turbulence disturbances reach


some critical level, laminar-turbulent transition in the particle boundary layer occurs.

It is accompanied by significant changes in the flow structure, reducing dramatically

the particle drag.

7.3.4 The Effect of Energy Dissipation

The internal friction of viscous fluid near the surface of a solid body moving in it is

accompanied by the mechanical energy dissipation. The latter leads to heating the

body, which can result in a significant temperature increase of the body temperature

Tw above the temperature of the undisturbed fluid T1: In flows of viscous incom-

pressible fluid heating of solid bodies depends on physical properties of fluid,

relative velocity of the body to that of fluid and its characteristic size. These

considerations allow us to present the functional equation for equilibrium tempera-

ture difference DT ¼ Tw � T1 as follows

DT ¼ f ðp; m; k; cP; u; d�Þ (7.58)

where p; m and k are the viscosity, thermal conductivity, cp is the specific heat of

fluid, u is the relative velocity of the body, and d� its characteristic size.Since the set of the five governing parameters of the problem contains three

parameters possessing independent dimensions, we can present (7.58) in the fol-

lowing form

P ¼ ’ðP1;P2Þ (7.59)

where P ¼ DT=pa1ma2ca3P ua4 ; P1 ¼ k=pa

01ma

02c

a03

P ua04 ; and P2 ¼ d�=pa

001 ma

002 c

a003

P ua004 .


00i using the principle of

dimensional homogeneity, we find a1 ¼ 0; a2 ¼ 0; a3 ¼ �1; a4 ¼ 2;

a01 ¼ 0; a

02 ¼ 1; a

03 ¼ 1; a

04 ¼ 0; a

001 ¼ �1; a

002 ¼ 1; a

003 ¼ 0 and a

004 ¼ �1: Then,

we can transform (7.59) to the following form

DT ¼ u2

cP’ðPr;ReÞ (7.60)

The expression (7.60) shows that the temperature difference due to viscous

dissipation is determined by the relative velocity of the body, as well as values of

the Reynolds and Prandtl numbers. The evaluations show that in the two limiting

cases corresponding to small and large Reynolds numbers heating of a body moving

in fluid can be expressed as (Landau and Lifshitz 1987)

Tw � T1 ¼ c Pru2

cP(7.61)


Tw � T1 ¼ u2

cP’�ðPrÞ (7.62)

respectively. In (7.61) c is a constant that depend on the body shape; ’�ðPrÞ is afunction of the Prandtl number.

Essentially, that in both cases corresponding to small and large Reynolds

number body heating due to viscous dissipation is directly proportional to the

square of the relative velocity.

7.3.5 The Effect of Velocity Gradient

Consider heat flux from a spherical particle immersed in a uniform flow of hot

incompressible fluid with linear velocity distribution. Assume that the total heat

flux is a result of superposition of two components: one – due to the rectilinear

translational motion relative to the surrounding fluid, and the other one – due to

shear. In this case it is possible to write the equation for the total heat flux in the

form

qt ¼ qc þ qsh (7.63)

where qt; qc and qsh are the total heat flux, heat flux due to rectilinear translational

motion, and heat flux due to shear, respectively, [qt] ¼ [qc] ¼ [qsh] ¼ JT�1L�2.

The functional equations for each component of the total heat flux read

qc ¼ fcðr; m; d; u; k; cP;DTÞ (7.64)

qsh ¼ fshðr; m; d; g; k; cP;DTÞ (7.65)

where g T�1½ � is the shear rate, and u is the velocity of the particle, and d is the

diameter of a spherical particle.

Equations 7.64 and 7.65 contain seven governing parameters, five of which

having independent dimensions. According to the Pi-theorem, (7.64) and (7.65)

can be presented as

qc ¼ ra1ma2da3ka4DTa5’u

ra01ma

02da

03ka

03DTa0

5

;cP

ra01ma

02da

03ka

04DTa0

5

!(7.66)

qsh ¼ ra1�ma2�da3�ka4�DTa5�’sh

g

ra01�ma

02�da

03�ka

04�DTa0

5�;

cP

ra001�ma

002�da

003�ka

004�DTa00

5�

!

(7.67)


Divide the left and right hand sides of (7.63) by the product ra1ma2da3ka4DTa5 . As

a results, we arrive at the following dimensionless equation (accounting for the

equality ai ¼ ai�Þ

P ¼ Pc þPsh (7.68)

where P ¼ qt=ra1ma2da3ka4DTa5 ; Pc ¼ qc=ra1ma2da3ka4DTa5 ; Psh ¼ qsh=ra1ma2

da3ka4DTa5 ;Pc ¼ ’cðP1;P2Þ, Psh ¼ ’shðP1�;P2�Þ, P1 ¼ u=ra01ma

02da

03ka

04DTa

05 ,

P2 ¼ cP=ra001 ma

002 da

003 ka

004DTa

005 , P1:� ¼ g=ra

01�ma

02�da

03�ka

04�DTa

05� , and P2� ¼ cp=ra

001�

ma002�da

003�ka

004�DTa

005� .

Taking into account the dimensions of qt; qc; qsh; u; g and cP; as well as the

fact that the governing parameters r; m; d; k and DTpossess independent

dimensions, we find the values of the exponents ai; a0i; a

00i ; ai�; a

0i�; and a

00i� as

a1 ¼ 0; a2 ¼ 0; a3 ¼ �1; a4 ¼ 1; a5 ¼ 1

a01 ¼ �1; a

02 ¼ 1; a

03 ¼ �1; a

04 ¼ 0; a

05 ¼ 0

a001 ¼ 0; a

002 ¼ �1; a

003 ¼ 0; a

004 ¼ 1; a

005 ¼ 0 (7.69)

a01� ¼ 1; a

02� ¼ 1; a

03� ¼ �2; a

04� ¼ 0; a

05� ¼ 0

a001� ¼ 0; a

002� ¼ �1; a

003� ¼ 0; a

004� ¼ 1; a

005� ¼ 0

According to (7.69), (7.68) takes the form

Nut ¼ Nuc þ Nush (7.70)

Where Nut ¼ qtd=kDT; Nuc ¼ qcd=kDT; Nush ¼ qsh=kDT; P1 ¼ Re ¼ udr=m;P2 ¼ Pr ¼ cPm=k;P1� ¼ d2grcP=k and P2� ¼ Pr ¼ cPm=k.

At low Reynolds number the Nusselt number approaches the value of 2. In this

case (7.70) reads

Nut ¼ 2þ ’shðPe1�; PrÞ (7.71)

The analytical solution of the problem yields the following expression for the

Nusselt number which is valid for low Reynolds and Peclet numbers (Frankel and

Acrivos 1968)

Nu ¼ 2þ 0:9104

2pð Þ1=2Pe (7.72)

where Pe ¼ r2grcP=k and r is the particle radius.


7.3.6 Mass Transfer to Solid Particles and Drops Immersed inFluid Flow

From the physical point of view the diffusion flux to a reactive spherical particle

immersed in an infinite reactant flow is determined by the physical properties of

liquid (its density, viscosity and diffusivity), particle size, reactant concentration in

the undisturbed flow, fluid velocity, as well as the kinetics of heterogeneous

chemical reaction at the particle surface. When the rate of the surface reaction

exceeds the rate of reactant diffusion in the carrier fluid, the concentration of the

reactant at the particle surface equals zero. In this case the influence of the kinetic

characteristics of the heterogeneous chemical reaction at the particle surface on the

mass transfer rate in the fluid bulk is negligible and these characteristics should be

excluded from the set of the governing parameters. At small values of the Reynolds

number corresponding to the Stokes creeping flow, the fluid density does not affect

the flow field and can be safely excluded from the governing parameters. Assume

that the carrier phase is liquid. Taking into account the fact that typically in liquids

the admixture (e.g., reactant) diffusivities are much less than viscosity (D<<nÞ, it ispossible to omit n from the set of the governing parameters. Indeed, under this

condition the mass transfer results mostly from diffusion rather than convection.

Thus, the functional equation for the diffusion flux of a reactant in a liquid carrier to

a reactive particle takes the form

qm ¼ ’ðD; r0; u1; c1Þ (7.73)

where r0 is the radius of particle, u1 is the undisturbed flow velocity, and c1 is the

reactant concentration in the liquid carrier far from the particle.

The dimensions of the governing parameters are

D½ � ¼ L2T�1; r0½ � ¼ L; u1½ � ¼ LT�1; c1½ � ¼ L�3M (7.74)

Three governing parameters from the set (7.74) have independent dimensions, so

that the difference n� k ¼ 1: Then, accoring to the Pi-theorem, (7.73) transform as

P ¼ ’ðP1Þ (7.75)

where P ¼ qmr0=Dc1 ¼ Sh is the Sherwood number, and P1 ¼ u1r0=D ¼ Ped isthe Peclet number.

Small droplets immersed in liquid flow stay spherical and effectively

undeformable due to the action of the interfacial tension. However, the mass

transfer to the surface of a small reactive droplet of viscosity m2 immersed into a

reactant liquid solution solution of viscosity m1 can differ significantly from the

mass transfer to a solid reactive particle. This stems from the different hydrody-

namic conditions at the liquid-liquid and liquid-solid interfaces. The existence of a

non-zero interfacial tangential velocity at the liquid-liquid interface determined by


viscosities of two liquids determines the interfacial velocity as characteristic scale

of the problem. Then the functional equation for the diffusion flux qm takes the form

qm ¼ f ðD; r0; v�; c1Þ (7.76)

where v� is the absolute value of the interfacial velocity at the drop equator.

In this case the dimensionless form of (7.76) reads

P ¼ ’ðeP1Þ (7.77)

where e ¼ c m2=m1ð Þ.The explicit forms of the dependences (7.75) and (7.77) can be found only

experimentally or via theoretical solutions of the corresponding problem (Levich

1962). When the rate of the heterogeneous reaction at the particle surface is much

larger than the diffusion rate, the problem reduces to the integration of the species

balance equation. In the framework of the diffusion boundary layer, and for the

creeping flow velocity distribution it reads

vr@c

@rþ v’

r

@c

@y¼ D

@2c

@r2þ 2

r

@c

@r

� �(7.78)


c ! c1 at y ! 1; c ¼ 0 at r ¼ r0 (7.79)

where r; ’; and y are the spherical coordinates with the origin at the particle

center.

The Levich solution results in the following dependence of the Sherwood

number on the Peclet number

Sh ¼ 7:85Pe1=3 (7.80)

for a solid particle, and in the dependence

Sh ¼ 2ffiffiffiffiffiffi6p

p Pe1=2m1

m1 þ m2

� �1=2

(7.81)

for a liquid droplet, where Sh ¼ <qm>r0=Dc1 is the Shrwood number, <qm> ¼I=r20 is the average diffusion flux, I ¼

Rqmds is the total diffusion flux at the particle

or droplet surface.

The comparision of the results of the dimensional analysis given by (7.75) and

(7.77) with the analytical solution of the problem corresponding to (7.80) and (7.81)

reveals the benefits of applying the dimensional anaysis to study mass transfer to

solid particles and droplets in flows of liquid reactant solutions. It is seen that results


of the dimensional analysis correctly represent the qualitative character of the

dependence of the Sherwood number on the Peclet number, but do not allow finding

the exact form of this dependence.

The knowledge of an exact form of the dependence ShðPeÞ is important for

understanding of the laws of mass transfer to solid particles and droplets. As can be

seen from (7.80) and (7.81), two different scaling laws in the dependence on the

Peclet number exist, with the exponents being one third and one half for particles

and droplets, respectively. That shows that ratio of the mass transfer intensity to a

droplet to that of a comparable solid particle Shd Shp�

depends weekly on the Peclet

number and strongly on the viscosities of the droplet and surroundind medium.

7.4 Heat and Mass Transfer in Channel and Pipe Flows

7.4.1 Couette Flow

Consider heat transfer in fully laminar flow of incompressible fluid between two

parallel closely located infinite plates having different temperature. Let the lower

plate is motionless, whereas the upper one moves with a constant velocity U.

Assuming that temperatures of the lower and upper plates being equal to T0 and

T1, respectively (with T1>T0). In the case when T0 and T1 are constant, the fluid

temperature changes in y-th direction only (coordinate y is normal to the plates).

The set of the governing parameters that determine heat flux from the upper wall to

the fluid q, as well as the fluid temperature # ¼ T � T0 reads

k½ � ¼ LMT�3y�1; m½ � ¼ L�1MT�1; H½ � ¼ L; U½ � ¼ LT�1; #�½ � ¼ y; y½ � ¼ L(7.82)

where H is the gap between the plates, and #� ¼ T1 � T0.Accordingly, the functional equations for q and # have the following form

q ¼ fqðk; m;H;U; #�Þ (7.83)

# ¼ f#ðk;m;H;U; #�; yÞ (7.84)

Four governing parameters listed in (7.83) and (7.84) possess independent

dimension, so that the difference n� k for (7.83) and (7.84) equals 1 and 2,

respectively. Then the dimensionless form of (7.83) and (7.84) is

Pq ¼ ’qðP1qÞ (7.85)

P# ¼ ’#ðP1#;P2qÞ (7.86)


wherePq ¼ q=ka1Ha2Ua3#a4� ; P1q ¼ m=ka01Ha

02

Ua03#

a04� ; P# ¼ #=ka1�Ha2�Ua3�#a4�� ; P1# ¼ m=ka

01�Ha

02�Ua

03�#a

04� and

P2# ¼ y=ka001�Ha

002�Ua

003�#

a004�� .

Bearing in mind the dimensions of heat flux, temperature and the governing

parameters, we find the values of the exponents ai; a0i; ai�; a

0i and a

00i�as

a1 ¼ 1; a2 ¼ �1; a3 ¼ 0; a4 ¼ 1

a01 ¼ 1; a

02 ¼ 0; a

03 ¼ �2; a

04 ¼ 1

a1� ¼ 0; a2� ¼ 0; a3� ¼ 0; a4� ¼ 1 (7.87)

a01� ¼ 1; a

02� ¼ 0; a

03� ¼ �2; a

04� ¼ 1

a001� ¼ 0; a

002� ¼ 1; a

003� ¼ 0; a

004� ¼ 0

Using (7.87), (7.85) and (7.86) take the following form

Nu ¼ ’qðPrEcÞ (7.88)

# ¼ ’#ð�; PrEcÞ (7.89)

where Nu ¼ qH=kðT1 � T0Þ; Pr ¼ mcP=k; and Ec ¼ U2=cPðT1 � T0Þ are the

Nusselt, Prandtl and Eckert numbers, respectively, # ¼ #=#�; and � ¼ y=H:The analytical solution of the problem on heat transfer in Couette flow reads

(Bayley et al. 1972)

# ¼ � 1þ 1

2PrEcð1� �Þ

� �(7.90)

The heat flux from the top plate to the fluid is expressed using the analytic

solution as

q ¼ k@T

@y

��y¼0

¼ kTs � T0

H1� 1

2PrEc

� �(7.91)

The comparison of the results of the exact analytical solution of the problem

with its analysis based on the Pi-theorem indicates a certain insufficiency of the

latter approach. In particular, the dimensional analysis allows finding the dimen-

sionless groups only, but cannot reveal a number of the important peculiarities of

the process, for example, the reverse of the heat flux at high values of the product

PrEcwhen q changes sign.

7.4 Heat and Mass Transfer in Channel and Pipe Flows 179

7.4.2 The Entrance Region of a Pipe

The evaluation of the thickness of the thermal boundary layer within the entrance

region of heated pipe is tackled in the present subsection. Considering flows of

incompressible fluids it is possible to assume that the local thickness of the thermal

boundary layer dTðxÞ depends on five dimensional parameters: the density r,thermal conductivity k and specific heat of the fluid cP; its velocity in the undis-

turbed core u and the cross-section coordinate x:2 Accordingly, the functional

equation for the thickness of the thermal boundary layer is written as

dTðxÞ ¼ fTðr; k; cP; u; x; Þ (7.92)

In the system of units LxLyLzMTJy the dimensions of the governing parameters

and the thickness of the thermal boundary layer are expressed as

r L�1x L�1

y Lz�1Mh i

; k L�1x LyL

�1z JT�1y�1

�; cP½ � ¼ JM�1y�1; u½ �

¼ LxT�1; x½ � ¼ L; d½ � ¼ Ly dT Ly

�(7.93)

It is seen that all the governing parameters have independent dimensions.

Therefore, in accordance with the Pi-theorem, (7.92) takes the form

dTðxÞ ¼ cTra1ka2ca3p ua4xa5 (7.94)

where cT is a constant.

Accounting for the fact that dT½ � ¼ Ly, we find the values of the exponents ai.They are: a1 ¼ �1=2; a2 ¼ 1=2; a3 ¼ �1=2; a4 ¼ �1=2 and a5 ¼ 1=2. Then


dTðxÞ ¼ cTkx

rucP

� �1=2

(7.95)

or

dTðxÞ ¼ cTx Pr�1=2

Re�1=2d (7.96)

where Pr ¼ n=a and Re ¼ ud=n are the Prandtl and Reynolds numbers with n and abeing the kinematic viscosity and thermal diffusivity, respectively.

Taking into account that thickness of the thermal boundary layer in the cross-

section that correspond to the end of entrance region equals d=2, we find the thermal

entrance lenth lenT as

lenTd

¼ c�TRed Pr (7.97)

where c�T ¼ ð2cTÞ�2.


Equations (5.12) and (7.95) show that the dynamic and thermal entrance lengths

of a pipe are determined by its diameter, the physical properties of fluid and flow

velocity. Therefore, it is possible to use len and lenT as some generalized parameters

in studies of flow resistance and heat transfer in the entrance section of pipes.

Introducing the dimensional parameters as dP dx=ð Þ ru2ð Þ� � ¼ L�1 and

qs kDTð Þ=½ � ¼ L�1 (where qs is the heat flux at the pipe wall), we write the func-

tional equations for these parameters as follows

dP=dxð Þru2

¼ fPðlen; xÞ (7.98)

qskDT

¼ fqðlenT ; xÞ (7.99)

In both expressions the difference between the number of dimensional

parameters n and the number of parameters having independent dimensions k is

equal to 1. Then, according to the Pi-theorem, (7.98) and (7.99) are deduced to

dP=dxð Þlenru2

¼ ’P

x

len

� �(7.100)

qslen:TDT

¼ ’qðx

len:TÞ (7.101)

Substituting (4.14) and (6.65) into (7.100) and (7.101), we arrive at the following

dimensionless expressions

l ¼ 1

Red’PðXÞ (7.102)

Nu ¼ 1

Pr Red’qðXTÞ (7.103)

where l ¼ dP=dxð Þdc�½ �=ru2 and Nu ¼ qslenT=kDT are the friction factor and the

Nusselt number, as well as X ¼ x=dc�Red and XT ¼ X=dc�T Pr Red.Comparing (7.95) with (5.12), we find the relation between the thicknesses of the

thermal and dynamical boundary layers as

dTd

Pr�1=2

(7.104)

7.4.3 Fully Developed Flow

Consider heat transfer in laminar pipe flow of incompressible fluid. Let pipe

diameter be denoted as d. We will use the LMTJy system of units. Assume that

physical properties of fluid are constant and flow is fully developed dynamically

7.4 Heat and Mass Transfer in Channel and Pipe Flows 181

and thermally. Let heat flux at the wall qs ¼ hðtw � tmÞ is constant along the tube

with h½ � ¼ JT�1L�2y�1 being the heat transfer coefficient, and tw and tm being the

wall and mean (bulk) fluid temperature, respectively.

It is plausible to assume that heat flux at the pipe wall depends on the

physical properties of fluid ( m½ � ¼ L�1MT�1; k½ � ¼ JT�1L�1y�1; cP½ � ¼ JM�1y�1;

r½ � ¼ L�3MÞ; the mean velocity of flow u½ � ¼ LT�1; the difference between the

wall and bulk temperature Dt ¼ tw � tm½ � ¼ y and the diameter d½ � ¼ L

qs ¼ fqðr; m; k; cP; u;Dt; dÞ (7.105)

Equation 7.105 contains seven governing parameters, five of them with inde-

pendent dimensions. Applying the Pi-theorem, we transform (7.105) to the follow-

ing dimensionless form

P ¼ ’qðP1;P2Þ (7.106)

where P ¼ qs=ra1ma2ka3da4Dta5 ; P1 ¼ u=ra01ma

02ka

03da

04Dta

05 and P2 ¼ cP=ra

001

ma002 ka

003 da

004Dta

005 :

The exponents ai; a0i and a

00i are equal to

a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ �1; a5 ¼ 1

a01 ¼ �1; a

02 ¼ 1; a

03 ¼ 0; a

04 ¼ �1; a

05 ¼ 0 (7.107)

a001 ¼ 0; a

02 ¼ �1; a

003 ¼ 1; a

004 ¼ 0; a

005 ¼ 0

Then, we obtain

Nu ¼ ’dðRe; PrÞ (7.108)

where Nu ¼ qsd=kDt; Re ¼ udr=m and Pr ¼ cPm=k are the Nusselt number based,

which represents itself in this case the dimensionless heat flux at the pipe wall, the

Reynolds and Prandtl numbers, respectively.

In the particular case of creeping flows corresponding to very small Reynolds

numbers (Re ! 0Þ it is possible to omit u and m from the set of the governing

parameters and write the functional equation for the heat flux in the form

qs ¼ fqðr; k; cP; d;DtÞ (7.109)

All the governing parameters in (7.109) have independent dimensions. Accord-

ingly, this equation should take the following form

qs ¼ cra1ka2ca3P da4Dta5 (7.110)



Taking into account the dimension qS½ � ¼ JT�1L�2, we determine values of the

exponents ai as a1 ¼ 0; a2 ¼ 1; a3 ¼ 0; a4 ¼ �1 and a5 ¼ 1. Accordingly, the

Nusselt number is expressed as

Nu ¼ qsd

kDt¼ c ¼ const: (7.111)

The numerical value of the constant c in (7.111) that follows from the analytical

solution of this problem is 4.364 (Kays and Crawford 1980).

7.5 Thermal Characteristics of Laminar Jets

Consider thermal characteristics of plane submerged laminar jet when a heated (or

cooled) fluid is issuing from a narrow slit in the same cooler (or warmer) fluid,

which is at rest far from the jet origin. We assume that the difference between the

fluid temperature in the jet and the surrounding medium is sufficiently small, which

allows us to consider the fluid density and viscosity being constant. In this case the

flow and temperature fields of fluid in the jet are determined by two parameters: (1)

the kinematic viscosity n, and (2) thermal diffusivity a.Non-isothermal jets, in fact, form two boundary layers, the dynamic and thermal

one. In the general case their thicknesses d and dT are not equal. The relation between dand dT depends on the relative intensity of the momentum and heat transfer which is

characterized by the Prandtl number Pr ¼ n=a:Only when Pr ¼ 1; d ¼ dT ; whereaswhen Pr<1 and Pr>1; dT>d and dT<d; respectively (Fig. 7.3).

There are two underlying physical mechanisms of heat transfer in submerged

jets: (1) convection, and (2) thermal diffusion. Therefore, it is natural to assume that

the temperature field depends on the flow velocity characterizing the convective

Fig. 7.3 Plane jet. Thicknesses of the dynamic and thermal boundary layers at different values of

the Prandtl number

7.5 Thermal Characteristics of Laminar Jets 183

component of the heat transfer, as well as the thermal diffusivity characterizing the

conductive component. Then, the functional equation for the temperature field

reads

DT ¼ f ðu; a; qx; x; yÞ (7.112)

where DT ¼ T � T1½ � ¼ y; T1 is the temperature of the undisturbed fluid, a½ � ¼L2yT

�1 is the thermal diffusivity, qx ¼ Qx rcPð Þ=½ � ¼ LxLyT�1y where the axial

convective enthalpy flux Qx ¼Ð10

ruDhdy� �

¼ L�1z T�1J with h½ � ¼ JM�1 being

the enthalpy, in the LxLyLzMTJy system of units.

For the axial excess (maximal) temperature DTm, and the thermal boundary

thickness dT the following functional equations are valid

DTm ¼ fTðum; a; qx; xÞ (7.113)

dT ¼ fd:Tðum; a; qx; xÞ (7.114)

Since the governing parameters um(the axial maximal velocity in jet cross-

section), a, qx and x have independent dimensions, (7.113) and (7.114) take the form

DT ¼ c1Tua1m a

a2qa3x xa4 (7.115)

dT ¼ c2Tua�1

m aa�2q

a�3x x

a�4 (7.116)

where c1T and c2T are constants.

Then the exponents ai and a�i are found as

a1 ¼ � 1

2; a2 ¼ � 1

2; a3 ¼ 1; a4 ¼ � 1

2; a�1

¼ � 1

2; a�2 ¼

1

2; a3 ¼ 0; a�4 ¼

1

2

(7.117)

Therefore, (7.115) and (7.116) take the form

DTm ¼ c�1TQx

rcP

1

Jxn=rð Þ1=6Pr1=2

x�1=3 (7.118)

dT ¼ c�2T1

J2x=r2n� �1=3 Pr

�1=2

x2=3 (7.119)

where c�1T ¼ c1T=c1 and c�2T ¼ c2T=c1 with c1 being a constant from (5.28).


Comparing (6.40) and (7.119), we estimate the ratio of the thicknesses of the

dynamic and thermal boundary layers

ddT

ffiffiffiffiffiPr

p(7.120)

Using the Pi-theorem, we transform (7.112) to the following form

DTDTm

¼ ’T

y

dT

� �(7.121)

Bearing in mind (7.120), we present (7.121) in the form

DTDTm

¼ ’T �Pr1=2

(7.122)

where � ¼ y=d.The temperature distribution in cross-sections of plane laminar jet is shown in

Fig. 7.4. It is seen that profiles DTð�Þ=DTm are wider than profiles uð�Þ=um at

Pr<1; and narrower at Pr>1:

Fig. 7.4 Velocity and temperature distributions in cross-sections of plane laminar submerged jet.

1- Velocity and temperature distributions for Pr ¼ 1; 2-Temperature distribution for Pr < 1; 3-

Temperature distribution for Pr > 1

7.5 Thermal Characteristics of Laminar Jets 185

7.6 Heat and Mass Transfer in Natural Convection

7.6.1 Heat Transfer from a Spherical Particle Under theConditions of Natural Convection

Consider heat transfer from a warm spherical particle immersed into still fluid. The

temperature difference between the particle and surrounding fluid is the reason of

natural convection, which arises due to the existence of a non-uniform density field in

fluid with temperature distribution and thermal expansion. Under the conditions of

natural convection it is plausible to assume that heat flux from a warm particle is

determined by the physical properties of fluid- its density r; thermal conductivity k;viscosity m; specific heat cp; the particle and fluid temperature difference Dt ¼ tp �t1; aswell as the product of gravity acceleration g and thermal expansion coefficientb:

Accordingly, the functional equation for the heat flux reads

qt ¼ f ðr; m; k; cP;Dt; gb; r0Þ (7.123)

In the system of units LMTJy the dimensions of the governing parameters are

r½ � ¼ L�3M; m½ � ¼ L�1MT�1; k½ � ¼ JT�1L�1y�1; cP½ � ¼ JM�1y�1 Dt½ �¼ y; gb½ � ¼ y�1LT�2; r0½ � ¼ L (7.124)

while qt½ � ¼ JL�2T�1.

Five of the seven governing parameters in (7.123) possess independent

dimensions. Then, according to the Pi-theorem the dimensionless form of (7.123) is

P ¼ ’ðP1;P2Þ (7.125)

where P ¼ qt=ra1ma2ka3Dta4ra50 ; P1 ¼ cp=ra

01ma

02ka

03Dta

04r

a05

0 and P2¼gb=ra001 ma

002 ka

003

Dta004 r

a005

0 :

Bearing in mind the dimension of the heat flux qt JT�1L�2½ �, we find values of the


00i

a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a5 ¼ �1

a01 ¼ 0; a

02 ¼ �1; a

03 ¼ 1; a

04 ¼ 0; a

05 ¼ 0 (7.126)

a001 ¼ �2; a

002 ¼ 2; a

003 ¼ 0; a

004 ¼ �1; a

005 ¼ �3


Nu ¼ ’ Pr;Grð Þ (7.127)

where Nu ¼ qtr0=kDt; Pr ¼ cpm=k and Gr ¼ r2gbDtr30=m2 are the Nusselt, Prandtl

and Grashof numbers, respectively.


The explicit form of the dependence of the Nusselt number on Pr and Gr for awarm spherical particle found experimentally has the form (Raithby and Hollands

1998)

Nu ¼ 0:878ARa1=4 (7.128)

where A is a weak function of the Prandtl number, namely, 0.086<A<0.104 for Pr

varying from 2000 to 0.71 and Ra¼PrGr being the Rayleigh number.

At very small values of the Prandtl number (m ! 0Þ the functional equation for

the heat flux from a warm particle has the following form

qt ¼ f ðr; k; cp;Dt; gb; r0Þ (7.129)

Five governing parameters in (7.129) have independent dimensions. Then the

Pi-theorem allows transformation of (7.129) to the following dimensionless form

P ¼ ’ P1ð Þ (7.130)

where P ¼ qt=ra1ca2p ka3Dta4ra50 and P2 ¼ gb=ra

01c

a02

p ka03Dta

04r

a05

0 .


exponents ai and a0i as

a1 ¼ 0; a2 ¼ 0; a3 ¼ 1; a4 ¼ 1; a5 ¼ �1 (7.131)

a01 ¼ �2; a

02 ¼ �2; a

03 ¼ 2; a

04 ¼ �1; a

05 ¼ �3


Nu ¼ ðGrPr2 Þ (7.132)

7.6.2 Heat Transfer from Spinning Particle Under the Conditionof Mixed Convection

Consider the general case of heat transfer from a spherical spinning particle under

conditions ofmixed convectionwhen both natural and forced convection are essential.

We assume fluid to be incompressible and possess invariable physical properties. The

factors that determine the heat transfer intensity in the present case are: (1) the physical

properties of fluid-its density r, thermal conductivity k, viscosity m and specific heat

7.6 Heat and Mass Transfer in Natural Convection 187

cP; (2) particle linear and angular velocities relative to fluid u1 and o, respectively,particle diameter d, and temperature difference between the particle and fluid

Dt ¼ tP � t1. In the case when the effect of buoyancy force is significant, it is

necessary to supplement the system of the governing parameters with the product

bg(where b is the thermal coefficient and g gravity acceleration). Therefore, the

functional equation for the heat flux from the particle surface reads

q ¼ f ðr; k; m; cP; u1;o; d; bg;DtÞ (7.133)

In (7.133) the set of the nine governing parameters contains five parameters with

independent dimensions. Then, according to the Pi-theorem, the number of dimen-

sionless groups that determine the dimensionless heat transfer coefficient (the

Nusselt number) equals four. Choosing as parameters with independent dimensions

r; k; m; d and Dt, we arrive at the following equation

P ¼ ’ðP1;P2;P3;P4Þ (7.134)

where

P ¼ q=ra1ka2ma3da4Dta5 ; P1 ¼ u1=ra01ka

02ma

03da

04Dta

05 ; P2 ¼ o=ra

001 ka

002ma

003 da

004Dta

005 ;

P3 ¼ cP=ra0001 ka

0002 ma

0003 da

0004 Dta

0005 ; and P4 ¼ bg=ra

1V1 ka

1V2 ma

1V3 da

1V4 Dta

1V5 :

The values of the exponents ai; a0i; a

00i ; a

000i and a1Vi are equal to

a ¼ 0; a2 ¼ 1; a3 ¼ 0; a4 ¼ �1; a5 ¼ 1

a01 ¼ �1; a

02 ¼ 0; a

03 ¼ 1; a

04 ¼ �1; a

05 ¼ 0

a001 ¼ �1; a

002 ¼ 0; a

003 ¼ 1; a

004 ¼ 2; a

005 ¼ 0 (7.135)

a0001 ¼ 0; a

0002 ¼ 1; a

0003 ¼ �1; a

0004 ¼ 0; a

0005 ¼ 0

a1V1 ¼ �2; a1V2 ¼ 0; a1V3 ¼ 2; a1V4 ¼ �3; a1V5 ¼ �1


Nu ¼ ’ðRe;Reo; Pr;GrÞ (7.136)

where Nu ¼ qd=kDt; Re ¼ u1dr=m; Reo ¼ od2r=m; Pr ¼ mcP=k; and

Gr ¼ bgDtd3r2=m2 are the Nusselt, Reynolds, rotational Reynolds, Prandtl and

Grashof numbers, respectively.

In the particular case corresponding to heat transfer in motionless fluid at an

invariable Prandtl number, the heat transfer coefficient is determined by two


dimensionless groups, namely, the rotational Reynolds and Grashof numbers. The

measurements show that buoyancy effects are dominant at Reo<103 when the heat

transfer intensity is determined by natural convection (Tieng and Yan 1993). In the

range 103<Reo<9 103 a sharp change in the flow structure is observed. It

manifests itself in formation of a turbulent zone and jet eruptions which affect the

heat transfer process. At Reo>9 103 the forced convection is dominant. In this

case the contribution of the jet eruptions to the heat transfer is essential. For

calculations of the average Nusselt number for mixed convection, Tieng and Yan

(1993) suggested the following empirical relation which is valid at Pr ¼ 0:71 :

Nu3 ¼ Nu

3

n þ Nu3

f (7.137)

where Nu; Nun and Nuf are the overall Nusselt number and the average Nusselt

numbers for the natural and forced convection, respectively which are given by

Nun ¼ 2þ 0:392Gr0:31 (7.138)

at 1<Gr<105 and

Nuf ¼ 2þ 0:175Re0:583o (7.139)

at Gr ¼ 0:10<Reo<104 and Pr ¼ 0:71:

7.6.3 Mass Transfer from a Spherical Particle Under theConditions of Natural and Mixed Convection

Heterogeneous reactions taking place at the liquid-solid interface are the reason of

formation of non-uniform concentration fields in liquid solutions. That leads to

changing density of liquid solution and arising of buoyancy forces which drive

natural convection. When the dependence of liquid density on concentration of

reactant in solution is weak, the following approximation for solution density is

valid (Levich 1962)

rðcÞ rðc1Þ þ @r@c

� ��c¼c1

ðc� c1Þ (7.140)

where c and c1 are the local concentration and concentration far from the particle.

In this case the buoyancy force can be evaluation as follows

fb ¼ g rðc1Þ � rðcÞf g gðc1 � cÞ @r@c

� ��c¼c1

� agc1@r@c

� ��c¼c1

(7.141)

where g is the gravity acceleration and a is a known dimensionless constant (a<1Þ.


Under the conditions of natural convection, themass flux to a particle is determined

by the buoyancy force, concentration of the reagent in solution, reagent diffusivity and

particle size. Accordingly, the functional equation for the mass flux reads

qm ¼ f ðfb; c1;D; r0Þ (7.142)

Three governing parameters in (7.142) have independent dimensions. Therefore,

it follows from the Pi-theorem that (7.142) can be transformed to the following

dimensionless form

P ¼ ’ðP1Þ (7.143)

where P ¼ qmr=Dc1 and P1 ¼ fbr30=D

2c1 � agr30 @r=@cð Þjc¼c1=D2 are the

Sherwood and Grashof numbers, respectively.

The functional equation for the mass transfer to a particle under the conditions of

mixed convection reads

qm ¼ f ðfb; c1;D; r0; u1Þ (7.144)

where u1 is the velocity far from particle.

In this case the set of the five governing parameters contains three parameters

with independent dimensions. Therefore, the dimensionless form of (7.144) is

P ¼ ’ P1;P2ð Þ (7.145)

where P; P1 and P2 ¼ u1r0=D are the Sherwood, Grashof and Peclet numbers,

respectively.

7.6.4 Heat Transfer From a Vertical Heated Wall

Consider heat transfer from a vertical heated wall under the conditions of natural

convection. In the framework of the boundary layer approximation, the system of

the governing equations describing the flow and heat transfer near the vertical wall

reads

u@u

@xþ v

@u

@y¼ n

@2u

@y2þ gbðT � T1Þ (7.146)

@u

@xþ @v

@y¼ 0 (7.147)

u@T

@xþ v

@T

@T¼ a

@2T

@y2(7.148)


where n and a are the viscosity and thermal diffusivity, b ¼ 1=T1 is the thermal

expansion coefficient, g gravity acceleration, x and y are the coordinate axes directedalong the wall and normal to it and subscript1 denotes the ambient conditions.

The boundary conditions for (7.146)–(7.148) are as follows

y ¼ 0; u ¼ v ¼ 0 T ¼ Tw; y ! 1; u ! 0 T ! T1 (7.149)

where Tw is the wall temperature.

It is convenient to introduce the excess of temperature DT ¼ T � T1 and rewrite

(7.146) and (7.148) in the following form

u@u

@xþ v

@u

@y¼ n

@2u

@y2þ g�y (7.150)

u@y@x

þ v@y@y

¼ @2y@y2

(7.151)

where y ¼ T � T1ð Þ= Tw � T1ð Þ and g� ¼ g Tw � T1ð Þ=T1.


y ¼ 0 u ¼ v ¼ 0; y ! 1 u ! 1 y ! 0 (7.152)

Equations 7.147, 7.150 and 7.151 and conditions (7.152) contain three dimen-

sional constants ðn; a; and g�Þand two variables ðx and yÞ that determine the veloc-

ity and temperature fields. Therefore, one can assume that the following functional

equations are valid

u ¼ fuðx�; y; n; g�Þ (7.153)

y ¼ fyðx�; y; a; g�Þ (7.154)

where x� ¼ ex with e being a dimensionless constant (its numerical value is

determined from the consideration of dimensions and the condition of the existence

of a self-similar solution), x is the longitudinal coordinate.The dimensions of velocity u, temperature #, as well as of the governing

parameters x�; y; n; a; and g� (in the system of units LxLyLzMTy) are

u½ � ¼ LxT�1; y½ � ¼ 1; x�½ � ¼ Lx; y½ � ¼ Ly; n½ � ¼ L2T�1; a½ � ¼ L2yT

�1; g�½ �¼ LxT

�2 (7.155)

The set of the governing parameters in 7.153 and 7.154 contains three

parameters with independent dimensions, so that the difference n� k ¼ 1. There-

fore, the dimensionless forms of these equations read


Pu ¼ ’ðP1uÞ (7.156)

Py ¼ #ðP1yÞ (7.157)

where Pu ¼ u=xa1� na2ga3� ; P1u ¼ y=x

a01� na

02g

a03� , Py ¼ y=xa1�� aa2�ga3�� and

P1y ¼ y=xa01�� aa

02�g

a03�� .

Taking into account the dimensions of u; #; x�; y; n and a, we find the values ofthe exponents ai and a

0i

a1 ¼ 1

2; a2 ¼ 0; a3 ¼ 1

2; a

01 ¼

1

4; a

02 ¼

1

2; a

03 ¼ � 1

4(7.158)

a1� ¼ a2� ¼ a3� ¼ 0; a01� ¼

1

4; a

02� ¼

1

2; a

03� ¼ � 1

4

Then we obtain

Pu ¼ u

x�g�ð Þ1=2; P1u ¼ yg

1=4�

x1=4� n1=2

; Py ¼ #; P1y ¼ yg1=4�

x1=4� a1=2

(7.159)

According to the expressions (7.159), (7.156) and (7.157) take the the following

form

u ¼ ðx�g�Þ1=2’ð�Þ (7.160)

y ¼ #ð�ffiffiffiffiffiPr

pÞ (7.161)

where Pr ¼ n=a is the Prandtl number, and � ¼ yg1=4� =x

1=4� n1=2:

Bearing in mind that the stream function is defined as c ¼ Ð udy and

v ¼ �@c=@x, we find

c ¼ x1=4� g1=4� n1=2’ð�Þ (7.162)

v ¼ � e4x1=4� g1=4� n1=2 3’ð�Þ � �’

0 ð�Þn o

(7.163)

Substitution of the expressions for u; v and y into (7.150) and (7.151) yields the

following system of ODEs for the unknown functions ’ð�Þ and #ð� ffiffiffiffiffiPr

p Þ(Pohlhausen, 1921)

’000� þ 3’’�

00 � 2’02� þ # ¼ 0 (7.164)

#00� þ 3 Pr’#

0 ¼ 0 (7.165)


where subscript � corresponds to differentiation by �: Note, that imposing the

conditions that the equations for the velocity and temperature do not incorporate

an arbitrary constant e, it should be taken as e ¼ 4.

The boundary conditions for (7.164) and (7.165) are

� ¼ 0; ’ ¼ ’0 ¼ 0 # ¼ 1; � ! 1; ’

0� ¼ 0 # ! 0 (7.166)

The local heat flux qðxÞ is expressed as

qðxÞ ¼ �k@T

@y

� �y¼0

¼ �kcx�1=4 @#

@�

� ��¼0

ðTw � T1Þ (7.167)

where k is thermal conductivity, c ¼ g�=4n2ð Þ1=4: The value of @#=@�ð Þ�¼0depends

on the Prandtl number value. It can be found only by solving (7.164) and (7.165).

For example, @#=@�ð Þ�¼0 ¼ 0:508 for Pr ¼ 0:773(Schlichting 1979).

The total heat flux from a plate of length l to the surrounding fluid is

Q ¼ b

ðl0

qðxÞdx ¼ 0:5084

3bl3=4ckðTw � T1Þ (7.168)

where b is the plate width. The average Nusselt number based on Q is expressed as

Nu ¼ 0:478 Grð Þ1=4 (7.169)

where Gr ¼ gl3ðTw � T1Þ=n2T1 is the Grashof number.

7.6.5 Mass Transfer to a Vertical Reactive Plate Under theConditions of Natural Convection

Consider mass transfer to a vertical reactive plate in contact with a liquid reactant

solution. Assume that at the surface of this plate (but not in the solution bulk) takes

place a chemical reaction the rate of which is much large then the rate of reactant

diffusion. In this case the reactant concentration will be equal zero at the wall. As a

result, a not-uniform reactant concentration field arises near the plate. Since the

density of solution depends on reactant concentration, a non-uniform density field

arises as well. The latter results in natural convection that determines the intensity

of mass transfer to the reactive wall.

In the framework of the boundary layer approximation the momentum and

species balance equations of the problem have the form (Levich 1962)


u@u

@xþ v

@u

@y¼ n

@2u

@y2þ gc�c� (7.170)

u@c�@x

þ v@c�@y

¼ D@2c�@y2

(7.171)

where gc� ¼ ðgc1=rÞ @r=@cð Þc¼c1 ; c� ¼ ðc1 � cÞ=c1; g is gravity acceleration

due, r ¼ rðcÞ; D is the reactant diffusivity coefficient, and subscript 1corresponds to reactant concentration in the undisturbed solution.


u ¼ v ¼ 0 c� ¼ 1 at y ¼ 0; u ¼ v ¼ 0 c� ! 1 at y ! 1 (7.172)

Equations 7.170 and 7.171 and conditions (7.172) formally coincide with

(7.150) and (7.151) and conditions (7.152). Accordingly, the expressions for the

components of velocity and reactant concentration can be recast as follows

u ¼ x�gc�ð Þ1=2’ð�Þ (7.173)

v ¼ �e1

4x�1=4� gc

1=4� D1=2 3’ð�Þ � ’

0 ð�Þn o

(7.174)

c� ¼ #cð�ffiffiffiffiffiffiSm

pÞ (7.175)

where Sc ¼ n=D is the Schmidt number, � ¼ g�=4n2ð Þ1=4y=x1=4.Substitution of the expressions (7.173) and (7.175) into (7.170) and (7.171) leads

(at e ¼ 4Þ to the system of ODEs for the functions ’ð�Þ and #ð� ffiffiffiffiffiSc

p Þ

’000 þ 3’’

00 � 2’02 þ #c ¼ 0 (7.176)

#00c þ 3Sc#

0c ¼ 0 (7.177)

where primes ðÞ0 correspond to differentiation by the dimensionless variable �.The mass flux of reactant at the wall is found by calculating the derivative

@c�=@yð Þy¼0 ¼ @c�=@�ð Þ�¼0@�=@y and using Fick’s law. The expressions that

determine the local j and total J mass flux at the vertical wall of height h and

width b reads (Levich 1962)

j ¼ 0:7Sc1=4Dgc14n2r

@r@c

� �c¼c1

( )1=4c1x1=4

(7.178)

J ¼ 0:9Sc1=4gc14n2r

@r@c

� �c¼c1

( )1=4

bh3=4c1D (7.179)


7.7 Heat Transfer From a Flat Plate in a Uniform Stream of

Viscous, High Speed Gas

Consider laminar boundary layer over a wall subjected to parallel uniform stream of

viscous, high velocity, perfect gas. It is easy to see that pressure along the wall is

constant in this particular case. Then, the system of equations that correspond to this

problem reads (Loitsyanskii 1966)

ru@u

@xþ rv

@u

@y¼ @

@ym@u

@y

� �(7.180)

@ru@x

þ @rv@y

¼ 0 (7.181)

ru@h

@xþ rv

@h

@y¼ k

mcP

@

@ym@h

@y

� �þ m

@u

@y

� �2

(7.182)

P ¼ g� 1

grh; m ¼ m0f

h

h1

� �(7.183)

where r; u; v; P and h are the density, longitudinal and lateral velocity

components, pressure and enthalpy, respectively; m; k and cp are the viscosity,

thermal conductivity and specific heat of fluid, respectively; g ¼ cP=cV is the

adiabatic index (the ratio of specific heats at constant pressure or constant vol-

ume).The boundary conditions for (7.180)–(7.182) are

y ¼ 0 : u ¼ 0; v ¼ 0 h ¼ hw; y ! 1 : u ! u1; h ! h1; P ! P1 (7.184)

for the plate with a given constant temperature, and

y ¼ 0 : u ¼ 0; v ¼ 0;@h

@y¼ 0; y ! 1 : u ! u1; h ! 1; P ! 1 (7.185)

for thermally insolated wall (subscripts w and 1 refer to the wall and undisturbed

flow, respectively).

The density of the ambient gas (far away from the wall) determines (at a given of

the enthalpy, h1Þ the value of pressure, which follows from the first (7.183). Taking

into account this circumstance, it is possible to write the following functional

equations for the longitudinal velocity component and enthalpy

u ¼ fuðx; y; r1; u1; h1; hw;P1; m1; k1; cP1Þ (7.186)

h ¼ hhðx; y; r1; u1; h1; hw;P1; m1; k1; cPÞ (7.187)

7.7 Heat Transfer From a Flat Plate in a Uniform Stream 195

It shows that u and h are the function of ten dimensional parameters: x½ � ¼ L;

y½ � ¼ L; r1½ � ¼ L�3M; u1½ � ¼ LT�1; P½ � ¼ L�1MT�2; h1½ � ¼ JM�1;

hw½ � ¼ JM�1; k1½ � ¼ JT�1L�1y�1; cP½ � ¼ JM�1y�1 and m½ � ¼ L�1MT�1.

Taking into account that (7.182) includes the ratio k=mcP½ � ¼ 1 as a single

complex, it is possible to diminish the number of the governing parameters in

(7.186) and (7.187) and present them in the following form

u ¼ fuðx; y; r1; u1; h1; hw;P1; m1;k

mcPÞ (7.188)

h ¼ fhðx; y; r1; u1; h1; hw;P1; m1;k

mcPÞ (7.189)

Equations 7.188 and 7.189 contain nine governing parameters, four of which

have independent dimensions. Then, according with the Pi-theorem, (7.188) and

(7.189) reduce to the two following dimensionless equations

Pu ¼ fuðP1;P2;P3;P4;P5Þ (7.190)

Ph ¼ fhðP1;P2;P3;P4;P5Þ (7.191)

where Pu and Ph are the dimensionless velocity and enthalpy, respectively; the

dimensionless groups Pi (i¼1,. . .5) are given by the following expressions:

P1 ¼ y=xa01ra

021u

a031h

a041; P2 ¼ hw=x

a001 ra

0021u

a0031h

a0041; P3 ¼ P1=xa

0001 ra

00021 u

a00031 h

a00041 ; P4 ¼

m1=xa1V1 r

a1V21 u

a1V31 h

a1V41 and P5 ¼ k1 m1cP1=ð Þ=xaV1 raV21u

aV31h

aV41

Bearing in mind that Pu ¼ u=xa1ra21ua31ha41 and Ph ¼ h=xa1�ra2�1 ua3�1 ha4�1 , we can

rewrite (7.190) and (7.191) in the following form

u

u1¼ ’u

y

x;hwh1

;P

r1u21;

m1xr1u1

;k1

m1cP

� �(7.192)

h

h1¼ ’h

y

x;hwh1

;P1

r1u21;

m1xr1u1

;k1

m1cP

� �(7.193)

It is seen that the dimensionless velocity and enthalpy are determined by the

functions of two dimensionless variables y=x and m1=xr1u1and three constants.

That shows that (7.180)–(7.182) cannot be reduce ODEs. In order to solve this

problem it is necessary to use a special transformation (7.180)–(7.182). First of all,

we transform (7.180)–(7.182), with the boundary conditions (7.183) and the

correlations (7.183) to the dimensionless form normalizing the parameters by

their values in the undisturbed flow (u ¼ u=u1; h ¼ h=h1, etc.) and the variables

x and y by some length scaleL (x ¼ x=L and y ¼ y=LÞ. Omiting for brevity bars


over the dimensionless parameters and assumed that viscosity is a linear function of

temperature, we rewrite (7.180)–(7.183) as follows

ru@u

@xþ rv

@u

@y¼ @

@ym@u

@y

� �(7.194)

@ru@x

þ @rv@y

¼ 0 (7.195)

ru@h

@xþ rv

@h

@y¼ ðg� 1ÞM2

1m@u

@y

� �2

þ 1

Pr

@

@ym@u

@y

� �(7.196)

r ¼ 1

h; m ¼ hn (7.197)

Planar compressible problems described by (7.194)–(7.197) are further

simplified by employing the Dorodnitsyn (1942), Illingworth (1949), Stewartson

(1949) transformation (Loitsyanskii 1966; Schlichting 1979)

x ¼ x; � ¼ðy0

rdy (7.198)

The transformation reduces the compressible (7.194)–(7.197) to the icompressible

boundary layer equations, which take the following dimensionless form

u@u

@xþ ev @u

@�¼ @2u

@�2(7.199)

@u

@xþ @ev@y

¼ 0 (7.200)

u@h

@xþ ev @h

@�¼ 1

Pr

@2h

@h2þ ðg� 1ÞM2

1@u

@�

� �2

(7.201)


� ¼ 0 : u ¼ 0; h ¼ hw; � ! 1 : u ! 1; h ! 1 (7.202)

for plates with a constant temperature given, and

� ¼ 0; u ¼ 0;@h

@�¼ 0; � ! 1 : u ! 1; h ! 1 (7.203)

7.7 Heat Transfer From a Flat Plate in a Uniform Stream 197

In (7.199)–(7.201)

ev ¼ u@�

@xþ rv (7.204)

Equations (7.199) and (7.201) and the boundary conditions (7.202) and (7.203)

show that the dynamic problem formulated in new variables and coordinates

becomes autonomous and the dimensionless velocity u depends on two dimension-

less variables x and � as

u ¼ f ð�; xÞ (7.205)

Any combination of these variables is also dimensionless. Similarly to flows of

incompressible fluid, one can assume that u is self-similar in a sense that f in (7.205)is a function of one dimensionless variable

w ¼ a�

xa(7.206)

where a and a are some constants.

Introducing function ’wðwÞ instead of f in such a way that u is expressed as

u ¼ a’0wðwÞ (7.207)

we find the expressions for the stream function c and a transformed lateral velocity

component ev asc ¼

ðudy ¼ xa’ðwÞ (7.208)

ev ¼ � @c@x

¼ axa�1ða’0wx� ’Þ (7.209)

where prime denotes derivatives in w.Substitution of the expressions for u and ev into (7.194) yields

a

x2a’

000w þ a

x’

00w’ ¼ 0 (7.210)

Requiring that (7.210) become an ODE in w, while variable x cancels, one finds

that a ¼ a ¼ 1=2. Then (7.210) takes the form

’000w þ ’

00w’ ¼ 0 (7.211)


The boundary conditions for (7.211) are

w ¼ 0; ’ ¼ 0 ’0w ¼ 0; w ! 1; ’

0 ! 2 (7.212)

Correspondingly, the energy equation 7.203 reduces to the following ODE

h00w þ Pr’h

0w þ

Pr

4ðg� 1ÞM2

1’00w ¼ 0 (7.213)


w ¼ 0 : h ¼ hw; w ! 1 : h ! 1 (7.214)

for a constant temperature wall, and

w ¼ 0 :@h

@w¼ 0; w ! 1 : h ! 1 (7.215)

for the thermally-insulated one.

The problem (7.211) and (7.212) formally coincides with the problem on the

incompressible boundary layer near a plane wall (the Blasius problem). Accord-

ingly, the drag coefficient at m ¼ hn and n ¼ 1 is expressed as

cf ¼ 1

2

1ffiffiffiffiffiffiffiffiRex

p ’00wð0Þ (7.216)

where Rex ¼ u1x=n1 and ’00wð0Þ ¼ 1:328.

The solution of (7.213) leads to the following dependence for the the average

Nusselt number for a plate of length L

Nu ¼ Tw � TsTw � T1

f ðPrÞffiffiffiffiffiffiffiffiffiRe1

p(7.217)

where Ts is the stagnation temperature, Re1 ¼ uL=n.

7.8 Heat Transfer Related to Phase Change

7.8.1 Heat Transfer Due to Condensation of Saturated Vaporon a Vertical Wall

Consider the heat transfer process during condensation of saturated vapor on a cold

vertical wall. As a result of condensation, a thin liquid film forms at the wall

surface. This film flows downward due to gravity. Latent heat that is released due

7.8 Heat Transfer Related to Phase Change 199

to vapor condensation is transferred from film to the wall by conduction and

convection. In a thin liquid film flowing with low velocity (Re 1Þthe convectivecomponent of the heat flux qconv is negligible in comparison with the conductive

one qcond, which is supported by the following estimates

qcond � kL@T

@y

� �y¼0

� kLdðTs � TwÞ (7.218)

qconv � rucPdðTs � TwÞ (7.219)

In Eqs. (7.218) and (7.219) d is the liquid film thickness; rL; kL and cPL are thedensity, thermal conductivity and specific heat at constant pressure of the liquid, Tsand Tw are the vapor saturated corresponding to a given pressure and the wall

temperatures, respectively.

The estimations (7.218) and (7.219) show that

qconvqcond

Re Pr 1 at Re 1 and Pr 1 (7.220)

The assumptions adopted here were first used by Nusselt (1916) in a simplified

theory of heat transfer during film condensation of saturated vapor on a cold

vertical wall. Below we consider this problem using the Pi-theorem. First of all,

consider the parameters which can affect the heat transfer from a saturated vapor to

a cold vertical wall. The set of the governing parameters should inevitably include

the liquid density rL and viscosity mL as well as gravity acceleration g: It is

necessary to account for the latent heat of condensation Dhfv;which is the origin

of heat flux due to vapor condensation q. The set of the governing parameters

should also include the vapor density rV (in fact, the difference rL � rVÞ: How-ever, since rL>>rV ; it is possible to omit rV from the set of the governing

parameters. Accordingly, we can write the functional equation for the liquid film

thickness at the wall as follows

d ¼ f ðr; m;Dhfv; g; q; xÞ (7.221)

where x is the longitudinal coordinate reckoned down the wall.

In order to decrease the number of the governing parameters, it is naturally to

assume that the parameters g and mL group with the liquid density rL as the

specific weight grL and kinematic viscosity mL=rL ¼ nL: Then (7.221) transforms

into

d ¼ f ðgrL; nL;Dhfv; q; xÞ (7.222)

In the framework of the boundary layer approximation one can introduce three

different scales of length Lx, Ly and Lz for the longitudinal and two lateral directions


x, y and z, and use the system of units LxLyLzMTJ: Then the dimensions of the

governing parameters in (7.222) are expressed as

rLg½ � ¼ L�1y L�1

x MT�2; nL½ � ¼ L2yM; Dhfv � ¼ JM�1; q½ �

¼ JT�1L�1x L�1

z ; x½ � ¼ L (7.223)

It is seen that all the governing parameters have independent dimensions, so that


d ¼ cðrLgÞa1ðnLÞa2ðDhfvÞa3ðqÞa4ðxÞa5 (7.224)

where c is a constant.Taking into account the dimension of d½ � ¼ Ly, we find the values of the

exponents ai as a1 ¼ �1=3; a2 ¼ 1=3; a3 ¼ �1=3; a4 ¼ 1=3 and a5 ¼ 1=3.Then (7.224) takes the form

d ¼ cnLqx

ðrLgÞDhfv

� �1=3(7.225)

Assuming that q ¼ kL @T=@yð Þy¼0 � kL Ts � Twð Þ=d, we obtain

d ¼ c1kLmLðTs � TwÞ

r2LgDhfvx

� �1=4(7.226)

where c1 ¼ c3/4

Under the assumption that the temperature distribution within the liquid film is

linear, the local heat transfer coefficient is found as

a ¼ kLd¼ c2

r2LgDhfvk3L

mLðTs � TwÞ1

x

� �1=4(7.227)

where c2 ¼ c�11 is a constant.

The comparison of the predictions of Nusselt’s theory with the experimental

data on film condensation shows that there is a significant deviation of the

theoretical results from the experimental results (about 25%). The latter stems

from significant oversimplification of the complex phenomenon in Nusselt’s

theory which does not account for a number of factors that affect the heat transfer

intensity. These factors are: wave formation on the film surface, temperature

dependence of the physical properties of fluid, etc. (Baehr and Stephan 1998).

A detailed analysis of steady laminar film condensation (condensation of stagnant

and flowing vapor, the self-similar solution in the cases of free and

forced convection, etc.) can be found in the monographs by Fujii (1991) and

Stephan (1992).


7.8.2 Freezing of a Pure Liquid (The Stefan Problem)

Let the half-space 0bxb1 be filled with a pure liquid (say, water) at rest with an

initial temperature #21>0(# ¼ T � Tf ; where Tf is the freezing temperature)

(Fig. 7.5). On the left side of the water-filled domain a refrigerator is put at the initial

timemoment t¼0. Its temperature #10 is kept constant below the freezing temperature

#f : #10<#f ¼ 0: Because of the conductive heat transfer to the refrigerator, liquid

temperature decreases in the negative x direction.As a result of liquid cooling, near the

contact with the refrigerator forms an ice layer and its thickness increases in time. The

interface that separates the ice and liquid domains propagates into liquid (in the

positive x direction). Themain aim is in determining the rate of growth of the ice layer.

From the physical point of view it is plausible to assume that the ice layer

location of the freezing interface x is determined by the physical properties of liquid

and its ice (cP1; cP2; k1 and k2;with subscript 1 being used for ice and subscript

2-for liquid), latent heat of solidification qs; the initial temperature of liquid far

away from the refrigerator #20, the refrigerator temperature #10, densities of the ice

and liquid r1 and r2, as well as time t

x ¼ f ðcP1; cP2; k1; k2; #10; #20; r1; r2; qs; tÞ (7.228)

The dimensions of the governing parameters are as follows

cP½ � ¼ JM�1y�1; k½ � ¼ JT�1L�1y�1; #i½ � ¼ y; qs½ � ¼ JM�1; t½ � ¼ T (7.229)

Equations 7.228 and 7.229 show that the present problem contains ten dimen-

sional parameters, five of them possessing independent dimensions. Choosing as

the parameters with the independent dimensions r2; cP2; k2; #21 and t, we

trabsform (7.228) to the following form

P ¼ ’ðP1;P2;P3;P4Þ (7.230)

where the dimensionless groups are expressed as

P ¼ x

ra2cbP2k

g2#

e20t

o; Pi ¼ wi

rai2 cbiP2k

gi2 #

ei20t

oi

; (7.231)

J1 = J2 = Jf = 0

uf

0

Water J2, +>JfJ1.– <Jf ice

Fig. 7.5 Freezing of pure liquid


with wi being one of the following five parameters r1; cP1; k1; #10 and qs, andi ¼ 1; 2; 3; 4 and 5, respectively.

Determining the values of the exponents a; b; g; e; o; and ai; bi; gi; ei,oi, we

arrive at the following expression

x ¼ a2tð Þ1=2’ðr12; cP12; k12; #12; StÞ (7.232)

where a2 is the thermal diffusivity of liquid, r12 ¼ r1=r2; cP12 ¼ cP1=cp2;k12 ¼ k1=k2; #12 ¼ #10=#20, and St ¼ qs=cP2#20 being the Stanton number.

The speed of the freezing front is

Vs ¼ 1

2

a2t

1=2’ðr12; cP12; k12; #12; StÞ (7.233)

Equations 7.232 and 7.233 show that the freezing front propagates as t1=2,whereas its speed decreases as t�1=2.

In the framework of the dimensional analysis it is impossible to establish the

exact form of the dependence ’ðr12; cP12; k12; #12; StÞ and calculate the exact

values of x and Vs. In order to find the exact form of the solution, the corresponding

equations should be solved exactly. The system of the governing equations of the

Stefan problem reads

@#1

@t¼ a1

@2#1

@t2(7.234)

for the ice domain �1<x<x, and

@#2

@t¼ a2

@2#2

@x2(7.235)

for the liquid domain x<x<1.

The corresponding boundary conditions are

x ¼ x; # ¼ #1 ¼ #2 ¼ #f ¼ 0 (7.236)

x ¼ �1; #1 ¼ #10; x ¼ 1; #2 ¼ #20 (7.237)

In addition, the system of Eqs (7.234) and (7.235) is subjected to the following

boundary condition expressing the thermal balance at the freezing front

k1@#1

@x

��x¼x�

� k2@#2

@x

��x¼xþ

¼ qsrdxdt

(7.238)

This additional condition allows finding the coordinate of the freezing front x asa function of time. It is emphasized that the problem allows determining x only up


to an additive constant, which corresponds to the initiation of freezing from any

cross-section x (in particular, from x ¼ �1). Equation 7.234 and 7.235 with the

boundary conditions (7.236) and (7.237) admit solutions in the form of the follow-

ing functional equations for the temperature fields in the ice and liquid domains

#1 ¼ f1ða1; #10; x; tÞ (7.239)

#2 ¼ f2ða2; #20; x; tÞ (7.240)

Since the governing parameters in (7.239) as well as in Eq (7.240) contains three

parameters with independent dimensions, these equations can be transformed to the

following self-similar forms

#1 ¼ ’1ð�1Þ (7.241)

#2 ¼ ’2ð�2Þ (7.242)

where #1 ¼ #1=#10 and #2 ¼ #2=#20 with �1 ¼ x=ffiffiffiffiffiffia1t

pand �2 ¼ x=

ffiffiffiffiffiffia2t

p,

respectively.

The conditions (7.236) and (7.237) take the form

x ¼ x; ’1ð�1xÞ ¼ ’2ð�2xÞ ¼ 0 (7.243)

x ¼ �1; ’1ð�1Þ ¼ 1; x ¼ 1; ’2ð1Þ ¼ 1 (7.244)

where �1x ¼ x=ffiffiffiffiffiffia1t

pand �2x ¼ x=

ffiffiffiffiffiffia2t

p.

Accordingly, (7.238) shows that the only possible law of propagation of the

freezing front (up to an additive constant) is given by

x ¼ bffiffit

p(7.245)

where b is a constant (an eigenvalue of the problem) which is also detrmined by

(7.238).

In particular, substituting (7.241)–(7.245) into (7.234)–(7.238), one finds the

self-similar solutions as

’1ð�1Þ ¼ A1 þ B1erf ð�1=2Þ (7.246)

’2ð�2Þ ¼ A2 þ B2efrð�2=2Þ (7.247)

A1 ¼ 1; B1 ¼ 1

erf b=2ffiffiffiffiffia1

p� � ; A2 ¼erf b=2

ffiffiffiffiffia2

p� �1� erf b=2

ffiffiffiffiffia2

p� � ; B2

¼ 1

1� erf b=2ffiffiffiffiffia2

p� � (7.248)


and the following equation determining b

k1 exp �b=4a21� �

a1erf b=2a1ð Þ þ k2 exp �b=4a22� �

a2 1� erf b=2a2ð Þ½ � ¼ �qsr2bffiffiffip

p2

(7.249)

Problems

P.7.1. Show that the evolution of the thermal field driven by radiative heat flux in a

uniform medium after an instantaneous pointwise energy release by a powerful

source corresponds to a self-similar solution.

The equation that determines temperature field driven by radiative heat flux

reads

@T

@t¼ 1

r2@

@rwr2

@T

@r

� �(P.7.1)

where w ¼ aTn is the radiant thermal diffusivity coefficient, and it is assumed that

the initial energy release happens at r¼0.Integrating (P.7.1) from r ¼ 0 to r ¼ 1 yields the folloving integral invariant

4pð10

Tr2dr ¼ Q (P.7.2)

Equations P.7.1 and P.7.2 show that the problem under consideration contains four

governing parameters: two dimensional constants, a and Q, and two independent

variables r and t. Accordingly, the functional equation for the temperature field reads

T ¼ f ða;Q; r; tÞ (P.7.3)

The dimensions of the governing parameters are

a L2T�1y�n �

; Q yL3 �

; r L½ �; t T½ � (P.7.4)

Since three of the four governing parameters have independent dimensions,

(P.7.3) can be reduced to the following dimensionless equation

P ¼ ’ðP1Þ (P.7.5)

where P ¼ T=aa1Qa2 ta3 and P1 ¼ r=aa01Qa

02 ta

03 :

Problems 205

Bearing in mind the dimensions of T; a; Q; r and t, we arrive at the followingset of algebraical equations determining the exponents ai and a

0i

2a1 þ 3a2 ¼ 0; 2a01 þ 3a

02 ¼ 1

� a1 þ a3 ¼ 0;�a01 þ a

03 ¼ 0 (P.7.6)

� a1nþ a2 ¼ 1;�a01nþ a

02 ¼ 0

Equations P.7.6 yield the following values

a1 ¼ � 1

nþ 2=3; a2 ¼ 2

3

1

nþ 2=3; a3 ¼ � 1

nþ 2=3;

a01 ¼

n

3nþ 2; a

02 ¼

1

3nþ 2; a

03 ¼

1

3nþ 2

(P.7.7)

Then (P.7.5) takes the following form

T ¼ Q2

ðatÞ3 !1=ð3nþ2Þ

’r

ðaQntÞ1=ð3nþ2Þ

( )(P.7.8)

Substitution of the expression (P.7.8) into (P.7.1) leads to the following ODE

determining the function ’ðxÞ

ð’nþ1Þ00x þ2

xð’nþ1Þ0x þ

1

3nþ 2x’

0x þ

3

3nþ 2’ ¼ 0 (P.7.9)

where x ¼ r=ðaQntÞ1=ð3nþ2Þ, and subscript x denotes differentiation by x.

P.7.2. Determine mass flux to a pipe wall in the case of fully developed laminar

flow of liquid reactant solution. The rate of the heterogeneous reaction at the wall is

assumed to be infinite, and accordingly, the reactant concentration at the wall

equals zero.

A sketch of a fully developed (hydrodynamically) laminar flow is shown in

Fig. 5.1. In this case the flow field does not change with x (along the pipe). On the

contrary, the reactant concentration field, and in particular, the thickness of the

diffusion boundary layer near the pipe wall increases in flow direction. The reactant

concentration changes in a thin boundary layer since the kinematic viscosity is

typically much larger than the diffusion coefficient, n � D. Accordingly, the flowstructure and the reactant concentration distribution can be characterized by the

maximum velocity u0 at the pipe axis and the reactant concentration c0 in the core

of the flow. Thus, as the governing parameters that determine the diffusion flux of

the reactant toward the wall one can choose u0; c0, pipe radius R, the diffusion

coefficient D; as well as the longitudinal coordinate x as the parameter responsible


for the growth of the diffusion boundary layer thickness in flow direction. Then the

functional equation for the reactant diffusion flux reads

qm ¼ f ðu0; c0;R;D; xÞ (P.7.10)

The dimensions of the diffusion flux and the governing parameters are

qm½ � ¼ ML�2T�1; u0½ � ¼ LT�1; c0½ � ¼ L�3M; R½ � ¼ L; D½ � ¼ L2T�1; x½ �¼ L (P.7.11)

Three of the five governing parameters in (P.7.10) have independent dimensions.

Therefore, according to the Pi-theorem, (P.7.10) can be transformed to the follow-

ing form

P ¼ ’ðP1;P2Þ (P.7.12)

where P ¼ qm=ca10 D

a2Ra3 ; P1 ¼ x=ca01

0 Da02Ra

03 and P2 ¼ u0=c

a001

0 Da002Ra

003 :


00i , we find a1 ¼ 1; a2 ¼ 1;

a3 ¼ �1; a01 ¼ 0; a

02 ¼ 0; a

03 ¼ 1; a

001 ¼ 0; a

002 ¼ 1 and a

003 ¼ �1. Then, we obtain

Sh ¼ ’ðx;PedÞ (P.7.13)

where Sh ¼ qmR=coD and Ped ¼ u0R=D are the Sherwood and the diffusional

Peclet numbers, and x ¼ x=R:Equation P.7.13 shows that the dimensionless mass transfer coefficient is deter-

mined by two dimensionless groups x and Ped. In the framework of the dimensional

analysis it is impossible to specify this dependence further more.. However, this

problem can be solved in the framework of the dimensional analysis if the diffusion

equation is simplified in the boundary layer approximation, which allows another

set of the governing parameters to be chosen. For this aim consider the rigorous

mathematical formulation the problem (but not its exact solution). The reactant

diffusion equation in the thin diffusion layer near the pipe wall reads (Levich 1962)

2u0R

y@c

@x¼ D

@2c

@y2(P.7.14)

c ! c0 at y ! 1; c ¼ 0 at y ¼ 0 (P.7.15)

where y ¼ R� r; and r is the radial coordinate in the cylindrical system reckoned

from the pipe axis. of coordinates.

Introducing the new variable y ¼ y u0=DRð Þ1=3, transform (P.7.14) to the follow-

ing form

@c

@x¼ 1

2

1

y

@2c

@y2(P.7.16)

Problems 207

Equation P.7.16 with the boundary conditions (P.7.15) show that concentration cderends on three dimensional parameters

c ¼ f ðc0; x; yÞ (P.7.17)

Two governing parameters in (P.7.17) have independent dimensions, so that this

equation can be transformed to the following dimensionless form

P ¼ ’ðP1Þ (P.7.18)

where P ¼ c=ca10 xa2 and P1 ¼ y=c

a01

0 xa02 :

The corresponding values of the exponents ai and a0i are found as:

a1 ¼ 1; a2 ¼ 0; a01 ¼ 0 and a

02 ¼ 1=3. Then, (P.7.18) takes the form

c ¼ c0’ð�Þ (P.7.19)

where � ¼ y u0=DRxð Þ1=3.The reactant flux to the pipe wall is

qm ¼ D@c

@y

� �y¼0

¼ @c

@�

� ��¼0

Du0DR

1=3 1

x1=3c0 (P.7.20)

or

Sh ¼ @c

@�

� ��¼0

Ped1=3x�1=3 (P.7.21)

Using the expression (P.7.20), we find the cumulative mass flux of reactant to the

wall in a pipe section of length x as

I ¼ 2pRðqmdx ¼ Ac0DR

u0x2

DR

� �1=3

(P.7.22)

where A ¼ � 2p=3ð Þ @c=@�ð Þ�¼0

The exact analytical solution of the problem reads (Levich 1962)

I ¼ 2:01pc0DRu0x

2

DR

� �1=3

(P.7.23)


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(eds) Handbook of heat transfer, 3rd edn. McGraw-Hill, New York

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phenomena. Dover, New York


Chapter 8

Turbulence


The turbulence represents itself a very complicated hydrodynamic phenomenon

characterized by irregular unsteady fluid motion. It emerges in liquid and gas flows

at sufficiently high Reynolds numbers when laminar flow regime becomes unstable

and strongly perturbed. This process is accompanied by arising turbulent eddies of

different sizes which are, in their turn, sources of velocity disturbances at each point

of the flow field. The amplitudes and frequencies of such disturbances depend

on the Reynolds number value. The scales of these disturbances decrease as

Re increases, whereas their frequency is proportional to the Reynolds number.

An exceptional complexity of turbulence impedes the theoretical analysis of this

phenomenon. In this situation a number of important results (mostly qualitative)

may be obtained using the methods of the similarity theory and dimensional analysis

(Kolmogorov, 1941a, b; Obukhov 1941). Following these works we address briefly

some problems related to the uniform isotropic turbulence, as the examples

illustrating applications of the dimensional considerations to study turbulent flows.

The actual velocity vector v at any point of developed turbulent flows can be

presented as a sum of the mean velocity v (obtained by averaging the actual velocityover a long time interval) and fluctuation velocity v

0. The existence of turbulent

fluctuation velocity and other fluctuating hydrodynamic and thermal characteristics

significantly affect the flow structure, as well as the intensity of processes of

momentum, heat and mass transfer.

Turbulent flows can be represented schematically as a conglomerate of turbulent

eddies of different sizes that generate fluctuations of velocity, temperature, etc. The

influence of large and small eddies on flow field is essentially different. The large

eddies are the bearers of kinetic energy of turbulent flow, which is transferred

though a cascade of smaller and smaller eddies to the smallest ones. On the other

hand, the smallest eddies are subjected to significant viscous forces which dissipate

the transferred kinetic energy into heat. The energy transfer from large to small

eddies occurs practically without dissipation, so that the energy flux from large



211

eddies equals energy dissipation in the smallest ones. That allows the assumption

that in developed turbulent flows a continuous energy transfer from large to small

eddies takes place. Such peculiarity of turbulent flows has a principal meaning for

the evaluation of a number of important characteristics of developed turbulent

flows. Indeed, using the concept of the energy transfer from large to the smallest

eddies, it is possible to evaluate the dissipation of kinetic energy by analyzing

solely the large scale motions.

Taking into account the above-mentioned peculiarities of the energy transfer, it

is possible to neglect the influence of viscosity on flow characteristics within the

domain of large eddies. In this case the energy dissipation is determined by only

three parameters, namely, the fluid density r, the characteristic size of large eddies lwhich is of the same order as the external flow scale, and the mean velocity change

Du over the length l. Accordingly, the functional equation for the energy dissipationper unit time E takes the form

E ¼ f ðr;Du; lÞ (8.1)


according to the Pi-theorem, (8.1) can be written as

E ¼ cra1Dua2 la3 (8.2)


Taking into account the dimension E½ � ¼ L�1MT�3, we find the values of the

exponents ai as a1 ¼ 1; a2 ¼ 3 and a3 ¼ �1, which allows transformation of (8.2)

to the following form

e ¼ cDu3

l(8.3)

where e ¼ E=r is the energy dissipation per unit time and unit mass.

In saddition, the functional equation for the turbulent (eddy) viscosity associated

with the turbulent Reynolds stresses can be also expressed through the same set of

the governing parameters

mT ¼ f ðr;Du; lÞ (8.4)

Using the Pi-theorem, we arrive at the following expression

nT ¼ c1lDu (8.5)

where nT ¼ mT=r is the turbulent kinematic viscosity and c1is a dimensionless

constant.

212 8 Turbulence

Then the relation between the turbulent and physical (molecular) viscosities can

be expressed as

nTn¼ c1Re (8.6)

where Re ¼ lDu=n.Equation (8.6) shows that the ratio nT=n increases with the Reynolds number.

Consider now some local properties of turbulence that characterize its features at

the scales of the order l much smaller than the characteristic scale of large eddies lbut much larger than the scale of the smallest dissipative eddies l0. It is possible toassume that turbulence is isotropic in flow regions far away from solid surfaces.

Using this assumption, we evaluate first the change of velocity of turbulent motion

vlover a distance of the order of l. Choosing as the governing parameters the

energy dissipation e and length l, we present the functional equation for vl as

follows

vl ¼ f ðe; lÞ (8.7)

Bearing in mind the dimensions e½ � ¼ L2T�3; l½ � ¼ L and nl½ � ¼ LT�1, we arrive

at the scaling law

vl � ðelÞ1=3 (8.8)

It expresses the Kolmogorov-Obukhov law: the change of velocity over a small

distance l is scales as l1=3. The local Reynolds number for the smallest dissipative

eddies corresponds to the borderline where molecular viscosity begins to play role.

Therefore, this Reynolds number should be of the order of one. On the other hand, it

is determined by velocity vl corresponding to l ¼ l0, the scale of the smallest

eddies responsible for energy dissipation l0, and fluid viscosity n. Accordingly, wearrive at the following estimate of l0

l0 � l

Re3=4(8.9)

In the framework of the dimensional analysis it is possible to evaluate velocity

change over the smallest dissipative eddy scale l0 [which is given by (8.8) and

(8.9)], the frequency of velocity pulsations, as well as the turbulent pulsations of

temperature at l>> l0. In order to estimate the temperature fluctuations in a non-

uniformly heated fluid over a distance l, consider the dependence for the tempera-

ture difference Tl (temperature fluctuations) following Obukhov (1949). For this

aim, we use the expression for the energy dissipation due to thermal conductivity


ET � krTð Þ2T

� kTl l=ð ÞT

(8.10)

where ET is the energy dissipation due to thermal conductivity of fluid

ð ET½ � ¼ L�1MT�3Þ, k is the termal conductivity ( k½ � ¼ LMT�3y�1), Tis the actual

temperature.

Accordingly, the rate of dissipative changing of temperature may be estimated as

follows

E0T � wt

Tl l=ð Þ2T

(8.11)

where E0T is the rate of dissipative changing temperature ( E

0T

� � ¼ yT�1), and wt is theturbulent thermal diffusivity ( wt½ � ¼ L2T�1).

Assuming that E0T ¼ ’ T= , where ’ is a certain factor that determine the local

features of turbulence in a non-uniformly heated fluid, we can postulate the

following relation

wtTl l=ð Þ2T

¼ ’

T(8.12)

At small intensity of temperature fluctuations (Tl T= <<1) it is possible to

replace the actual temperature T by the mean temperature of fluid and assume that

’ ¼ wtTll

� �2

(8.13)

Thus, the factor ’ depends on three dimensional parameters, namely, (1) turbu-

lent thermal diffusivity wt, (2) temperature fluctuations Tl, and (3) the distance l

’ ¼ f ðwt; Tl; lÞ (8.14)

Since all the governing parameters in the functional (8.14) have independent

dimensions, according with the Pi-theorem, it takes the following form

’ ¼ wa1t Ta2l la3 (8.15)


Bearing in mind the dimension of ’ ( ’½ �¼y2T�1), we find using the principle

of dimensional homogeneity that the values of the exponents aiare:a1¼1;a2¼2;anda3¼�1. Then (8.15) reads

214 8 Turbulence

’ ¼ cwtTll

� �2

(8.16)

Accounting that wt � nt � lvl, and vl ¼ elð Þ1 3= , we arrive at the following

estimate for the temperature fluctuations

T2l � ’e�1=3l2=3 (8.17)

It is seen that temperature fluctuations like the velocity fluctuations are propor-

tional to l1=3 on the scale l>>l0.Below we consider in detail the applications of the Pi-theorem to a number of

important types of turbulent flows, in particular, turbulent near-wall flows, flows in

smooth and rough pipes and channels, as well as various kinds of turbulent jets.

8.2 Decay of Isotropic Turbulence

The behavior of isotropic turbulence is described by the von Karman and Howarth

(1938) [see also Pope (2000)]

@bdd@t

¼ 2n1

r4@

@r

�r4@bdd@r

�þ 1

r4@

@rðr4bdd�dÞ (8.18)

where bdd and bddd are the components of the second and third correlation tensor,

respectively.

At very the smallest-scale dissipative eddies the corresponding small velocity

fluctuations are dominated by the viscous effects and the second term on the right-

hand side in (8.18) can be omitted. Then the von Karman and Howarth equation

takes the form

@bdd@t

¼ 2n1

r4@

@r

�r4@bdd@r

�(8.19)

To find the solution of (8.19), it is necessary to impose the initial distribution of

the second correlation bddðr; tÞ, i.e. bddðr; 0Þ. Equation (8.19) admits the invariant

(the Loitsyanskii invariant L0) which can be determined through the initial condi-

tion bddðr; 0Þ (Loitsyanskii 1939)

L0 ¼ð10

r4bdddr (8.20)

8.2 Decay of Isotropic Turbulence 215

The value of the invariant does not change in time and is valid for all time

moments t > 0.

Consider the problem (8.19) and (8.20) assuming that the value ofL0 is non-zero

and finite, 0<L0<1, as the most plausible assumption (Landau and Lifshitz

1987). Introducing the new variables bdd ¼ bdd=L0 and t� ¼ 2t, we transform

(8.19) and (8.20) to the following form

@bdd@t�

¼ n1

r4@

@r

�r4@bdd@r

�(8.21)

ð10

r4bdddr ¼ 1 (8.22)

From (8.21) and (8.22) it follows that bdd depends on three governing parameters

bdd ¼ f ðt�; n; rÞ (8.23)

These parameters have the following dimensions t�½ � ¼ T; n½ � ¼ L2T�1 and r½ � ¼ L,whereas the dimension of the unknown quantity is bdd

� �¼ L�5. It is seen that two of

the three governing parameters have independent dimensions. Then, in accordance

with the Pi-theorem, the dimensionless form of (8.23) reads

P ¼ ’ðP1Þ (8.24)

where P ¼ bdd=ta1� n

a2 and P1 ¼ r=ta01� na

02 .

Using the principle of the dimensional homogeneity, we find the value of the

exponents ai and a0i as a1 ¼ a2 ¼ �5=2 and a

01 ¼ a

02 ¼ 1=2. Then, (8.24)

transforms to the following expression for bdd

bdd ¼ 1

nt�ð Þ52’

rffiffiffiffiffiffint�

p� �

(8.25)

To find an exact expression for the function ’, one should express the derivativesin (8.21) using (8.25)

@bdd@t�

¼ � 1

2

1

nt�ð Þ5=2t�5’þ x

d’

dx

� �(8.26)

n1

r4@

@rr4@bdd@r

� �¼ 1

nt�ð Þ5=2t�4

xd’

dxþ d2’

dx2

� �(8.27)

216 8 Turbulence

where ’ ¼ ’ðxÞ and x ¼ r= nt�ð Þ1=2.Substituting (8.26) and (8.27) into (8.21), we arrive at the following ODE for the

function ’ ¼ ’ðxÞ

d2’

dx2þ 4

xþ x2

� �d’

dxþ 2’ ¼ 0 (8.28)

The Loitsyanskii invariant takes the form

ð10

x4’ðxÞdx ¼ 1 (8.29)

The solution of (8.28) and (8.29) and a detailed analysis of the problem on decay

of isotropic turbulence can be found in the monographs of Sedov (1993) and

Barenblatt (1996).

8.3 Turbulent Near-Wall Flows

8.3.1 Plane-Parallel Flows

Turbulent flows over smooth and rough walls were a subject of a large number of

experimental and theoretical investigations. The most important results of these

works are combined in several well-known research monographs by Hinze (1975),

Rotta (1962), Schlichting (1979) and Monin and Yaglom (1965-Part 1, 1967-Part 2),

as well as in numerous text- and reference books on hydrodynamics. In the present

sub-section we focus our attention on the application of the Pi-theorem to reveal

some fundamental features of turbulent near-wall flows, while addressing an

interested reader to find the discussion of the other aspects of such flows in the

above-mentioned monographs.

We begin with the general form of the distribution of mean velocity in plane-

parallel flows over a plate. Let vbe the mean velocity vector in a plane-parallel flow

with the components u ¼ uðyÞ; v ¼ 0 and w ¼ 0 in the x, y and z directions,

respectively. It is seen that flow characteristics, in particular, velocity u, in such

flows depend only on the transversal coordinate y normal to the wall. It is plausible

to assume that local velocity uðyÞ is determined by the following four parameters:

the fluid density r½ � ¼ L�3M and viscosity m½ � ¼ L�1MT�1, the shear stress at the

wall t�½ � ¼ L�1MT�2 and the distance from the wall y½ � ¼ L

u ¼ f ðr; m; t�; yÞ (8.30)

8.3 Turbulent Near-Wall Flows 217

Three of the four governing parameters in (8.30) have independent dimensions,

so that according to the Pi-theorem, this equation transforms to the following

dimensionless form

P ¼ ’ P1ð Þ (8.31)

where P ¼ u=ra1ma2ta3� and P1 ¼ y=ra01ma

02t

a03� .

The values of the exponents ai and a0i are found as a1 ¼ �1=2 , a2 ¼ 0 ,

a3 ¼ 1=2; a01 ¼ �1=2; a

02 ¼ 1 and a

03 ¼ �1=2, which allows us to present (8.31)

as follows

u

u�¼ ’

yu�n

� �(8.32)

or

uþ ¼ ’ðyþÞ (8.33)

In (8.33) uþ ¼ u=u� and yþ ¼ yu�=n are the dimensionless velocity and distance

from the wall, respectively, u� ¼ffiffiffiffiffiffiffiffiffiffit�=r

pis the friction velocity.

Equations (8.32) or (8.33) expresses the Prandtl law-of-the wall (Prandtl,

1925b). The exact form of the function ’ yþð Þcan be determined in two limiting

cases corresponding to small or large values of yþ. At small yþ (in the viscous

sublayer) velocity is so low rgat it should not dependend on density, since the

inertia effect is negligible. Then the functional equation for the velocity in the

viscous sublayer reads

u ¼ f ðm; t�; yÞ (8.34)

Since all the governing parameters in (8.34) have independent dimensions, it

takes the form

u ¼ cma1ta2� ya3 (8.35)

where c is a constant.Determining the values of the exponents ai as a1 ¼ �1; a2 ¼ 1 and a3 ¼ 1, we

obtain

u

u�¼ c

yu�n

(8.36)

Equation (8.36) shows that fluid velocity increases linearly in y within viscous

sublayer. This conclusion agrees fairly well with the experimental data in the

0< yþ < 5.

218 8 Turbulence

Consider now the flow far enough from the wall. At large distances from the

wall, turbulent transfer plays the dominant role. That allows one to omit molecular

viscosity in the set of the governing parameters, and assume that flow

characteristics are determined by three parameters: r; t� and y. It is easy to see

that in this case it is impossible to assume m ¼ 0 in (8.30) and write the functional

equation for velocity in the form u ¼ f ðr; t�; yÞ. That would result in the unrealisticoutcome that velocity does not depend on y. Indeed, since all the governing

parameters in this equation have independent dimensions, it takes the following

form u¼ cra1ta2� ya3 , where c is a dimensionless constant, and a1 ¼�1=2;a2¼ 1=2

and a3 ¼ 0. At the same time, this set of governing parameters determines the

velocity gradient far from the wall, so that functional equation for the problem

can be written in the form

du

dy¼ f ðr; t�; yÞ (8.37)

It is emphasized that (8.37) implies a dependence of velocity gradient on r; t�and y due to the fact that at large Reynolds numbers, when the influence of

molecular viscosity is negligible, the value of the velocity gradient at each point

must be determined by only two parameters r; t� and the distance y (Landau and

Lifshitz 1979).

All governing parameters in (8.37) have independent dimensions. In this case the

Pi-theorem determines the following form of (8.37)

du

dy¼ w�r

a1ya2ta3� (8.38)

where w� ¼ x�1 is a constant. Also, a1 ¼ �1=2; a2 ¼ �1 and a3 ¼ 1=2.Then (8.38) takes the form

du

dy¼ u�

wy(8.39)

Integrating (8.39) yields

u ¼ u�wðln yþ cÞ (8.40)

The constant c in (8.40) is found from the matching condition at the outer

boundary layer of the viscous sublayer y0 � n=u�. There the velocity is close to

the friction velocity: u � u�. The latter results in the logarithmic velocity profile

u ¼ u�w

lnyu�n

(8.41)


The accuracy of the law-of-the wall (8.41) can be improved by including an

empirical constant in addition to the logarithmic term. Then, the comparison with

the experimental data shows that the von Karman constant w ¼ 0.4 and the updated

equation (with the additional constant) becomes u ¼ u� 2:5 ln yu�=nð Þ þ 5:1½ � ¼2:5u� ln yu�=0:13nð Þ as suggested by Coles (1955).

8.3.2 Pipe Flows

Consider fully developed turbulent flows in straight pipes with circular cross-

section of radius R. The local velocity in such flows is determined by the distance

from wall y ¼ R� r (r is the corresponding distance from the pipe axis), fluid

density r and viscosity m and friction velocity u� (Monin and Yaglom, 1971)

u ¼ f ðR; y; r;m; u�Þ (8.42)

The transformation of (8.42) to thee dimensionless form by with the help of the

Pi-theorem yields

P ¼ ’ P1;P2ð Þ (8.43)

where P ¼ u=ua1� Ra2ra3 ;P1 ¼ y=u

a01� Ra

02ra

03 and P2 ¼ m=u

a001� Ra

002 ra

003 .

Allying the principle of the dimensional homogeneity and find the values of the


00i as a1 ¼ 1; a2 ¼ 0; a3 ¼ 0; a

01 ¼ 0; a

02 ¼ 1; a

03 ¼ 0;

a001 ¼ 1; a

002 ¼ 1 and a

003 ¼ 1, we transform (8.43) to the form of a dependence of

the dimensionless velocity u=u� on two dimensionless groups y=R and u�R=n

u

u�¼ ’

y

R;u�Rn

� �(8.44)

There are two important cases when the function ’ y=R; u�R=nð Þ can be reducedto a function of one dimensionless variable. The first of them corresponds to small

values of the ratio y=R � 1, at which the dependence of the local velocity on radius

of the pipe becomes unimportant. In this case the functional equation for local

velocity u reduces to

u ¼ f ðy; r; m; u�Þ (8.45)

Transforming (8.45) to dimensionless form by using the Pi-theorem, we arrive at

u

u�¼ ’

u�yn

� �(8.46)

220 8 Turbulence

The second limiting case corresponds to flow in the so-called ‘turbulent core’,

i.e. the region surrounding the pipe axis. In this region the turbulent shear stress is

larger than the viscous one. Correspondingly, the functional equation for the

velocity gradient du=dy should be written as

du

dy¼ f ðy;R; u�Þ (8.47)

Applying the Pi-theorem to (8.47), we reduces it to the form

du

dy¼ u�

R’

y

R

� �(8.48)

Assuming that ’ y=Rð Þ ¼ �Rdc y=Rð Þ=dy with cð0Þ ¼ 0, and integrating (8.48)

from y to R, we arrive at the following equation for the defect of velocity that was

found by von Karman (1930)

u0 � u

u�¼ c

y

R

� �(8.49)

where u0 is the velocity at the boundary of the turbulent core.

Equation (8.49) expresses the law of the defect of velocity. It is valid within the

turbulent core of pipe flows where �1 <�< 1, with � ¼ y=R and �1 and 1 being thedimensionless coordinates of the boundaries of the turbulent core.

8.3.3 Turbulent Boundary Layer

The flow in the turbulent boundary layer over a flat plate is significantly different

from plan-parallel and pipe flows. The differences stem from the different

conditions at the outer boundary of the boundary layer and a pipe axis, as well as

from the dependence flow characteristics on coordinates. In particular, in plane-

parallel and pipe flows velocity depends only on the normal-to-wall coordinate y,whereas in turbulent boundary layers velocity depends on the normal y and longi-

tudinal x coordinates. Accordingly, in turbulent boundary layers the friction veloc-

ity is not constant but changes with x.Assume that the defect of velocity within the outer part of the turbulent boundary

layer u1 � u depends on local parameters of the flow corresponding to a fixed x,e.g. on the thickness of the boundary layer dðxÞ, the friction velocity u�, as well ason the velocity of the undisturbed fluid u1, and distance from the wall y. Then we

can write the following functional equation


ðu1 � uÞ ¼ f ðu1; d; y; u�Þ (8.50)

Applying the Pi-theorem, we transform (8.50) to the following dimensionless

form

P ¼ ’ P1;P2ð Þ (8.51)

where P ¼ u1 � uð Þ=u�; P1 ¼ y=d and P2 ¼ u1=u�.The experimental data on the velocity distribution in turbulent boundary layers

show that the dependence of the dimensionless group P on P1 is very weak

(Clauser 1956). That allows simplification of (8.51) to the form similar to the one

for pipe flows [(8.49)]

u1 � u

u�¼ ’

y

d

� �(8.52)

On the other hand, the flow in a thin viscous sublayer adjacent to the wall is

described by (8.36).

8.4 Friction in Pipes and Ducts

8.4.1 Friction in Smooth Pipes

The application of the Pi-theorem to study friction in smooth pipes of different

geometry in the case of laminar flow of incompressible fluid was considered in

Chap. 4. In the present subsection we briefly address this problem in relation to

friction in fully developed turbulent flows in smooth pipes. It is quite plausible to

assume that friction factor, and thus pressure gradient dP dx=½ � ¼ L�2MT�2, in such

flows is determined by four dimensional parameters: (1) the mean velocity

u½ � ¼ LT�1, pipe diameter d½ � ¼ L, as well as fluid density r� ¼ L�3M½ and viscos-

ity m½ � ¼ L�1MT�1

dP

dx¼ f ðu; d; r; mÞ (8.53)

It is emphasized that there is a principal distinction between the functional

equations for friction or pressure gradient in pipes with fully developed turbulent

and laminar flows. In the case of turbulent pipe flows, (8.53) contains fluid density

as one of the governing parameters. The latter shows that the inertial forces play an

important role in fully developed turbulent flows in pipes. In contrast, in steady-

state, fully developed laminar flow, the inertial forces are immaterial and the fluid

density is absent from the corresponding (8.53).

222 8 Turbulence

In (8.53) three of the four governing parameters have independent dimensions,

which allows transformation of (8.53) to the following dimensionless form

P ¼ ’ P1ð Þ (8.54)

where P ¼ dP=dxð Þ=da1ra2ma3 and P1 ¼ u=da01ra

02ma

03 .

Taking into account the dimensions of the pressure gradient dP=dx and the

governing parameters with independent dimensions, we find the values of the

exponents ai and a0i as a1¼�3; a2¼�1; a3¼�2; a

01¼�1; a

02¼�1 and a

03 ¼ 1.

Then, (8.54) takes the form

l ¼ ’ðReÞRe2

¼ cðReÞ (8.55)

where l ¼ 2 dP=dxð Þd=ru2 is the friction factor and Re ¼ ud=n:The friction factor l can be described by different empirical and semi-empirical

correlations for the function cðReÞproposed for smooth pipes, for example by the

Blasius correlation

cðReÞ ¼ 0:3164Re�1=4 (8.56)

(valid for Re < 105) or the Prandtl universal friction law for smooth pipes

1ffiffiffil

p ¼ 2:01 lg Reffiffiffil

p� �� 0:8 (8.57)

which is valid up to Re ¼ 3.4� 106 (Schlichting 1979).

8.4.2 Friction in Rough Pipes

Numerous experimental investigations performed during the last century show that

roughness significantly affects friction, and thus the pressure gradient, in turbulent

flows in rough pipes. The functional equation for the pressure gradient dP=dx in thiscase can be written as1

1 Strictly speaking, (8.58) should be written as: dP=dx¼ f ðu;def ;m;r;ks;a1;a2 � � �aiÞ; where a1;a2 �� ai are the parameters characterizing the shape and distribution of rough elements on the surface.

8.4 Friction in Pipes and Ducts 223

dP

dx¼ f ðu; def ; m; r; kÞ (8.58)

with def and k being the effective diameter and characteristic roughness amplitude

(corresponding to sand grains glued to the inner wall) of such pipes (the choice of

the characteristic scales of rough pipes is discussed in Herwig et al. 2008).

Transforming (8.58) to the dimensionless form with the help of the Pi-theorem,

we arrive at the following equation for the friction factor l

l ¼ ’ðRe; kÞ (8.59)

where l ¼ 2 dP=dxð Þd=ru2 and the relative sand roughness is k ¼ k=def .Equation (8.59) shows that the friction factor of rough pipes depends on two

dimensionless parameters: the Reynolds number based on the effective diameter of

a pipe and the relative sand roughness ks. Typically, three different regimes of

turbulent flows in rough pipes are distinguished: (1) the hydraulically smooth

region corresponds to u�k=n< 5, (2) the transition region-to 5< u�k=n< 70; and

(3) the completely rough region with u�k=n>>70. At small enough values of the

dimensionless ratio u�k=n, rough pipes are practically hydraulically smooth since

their friction factor depends only on the Reynolds number [as for example, in the

Blasius correlation (8.56)]. At large values of ks, the riction factor of rough pipes

depends practically only on the relative roughness. In such flows the quadratic

resistance law in which l depends only on ksrather than on Re (i.e. l is proportionalto ru2) is valid.

8.5 Turbulent Jets

8.5.1 Eddy Viscosity and Thermal Conductivity

In the aero- and hydromechanics of submerged turbulent jets the main physical

feature to be accounted for is the fact that the intensity of molecular momentum,

heat and mass transfer in these flows is negligibly small than the intensity of

turbulent transfer due to fluctuation motion of fluid associated with eddies. The

latter allows one to simplify the system of the turbulent Reynolds equations

describing turbulent jets of incompressible fluid

u@u

@xþ v

@u

@y¼ � 1

rdP

dxþ @

@yyj nþ nTð Þ @u

@y

(8.60)

@uyj

@xþ @vyj

@y¼ 0 (8.61)

224 8 Turbulence

where u, v and P are the longitudinal and lateral velocity components and pressure,

respectively (with bars denoting the averaged parameters), n and nT are the molec-

ular and eddy kinematic viscosities of fluid and subscripts j ¼ 0 and 1 corresponds

to the plane and axisymmetric jets, respectively.

The physically plausible assumption that nT >> n allows us to omit n in (8.60)

and reduce the transfer term in this equation to a simpler form

@

@yyj nþ nTð Þ @u

@y

¼ @

@yyjnT

@u

@y

(8.62)

As usual with the turbulent Reynolds equations, the system of (8.60–8.61)

should be supplemented by a semi-empirical correlation that determine the depen-

dence of the eddy viscosity on the characteristics of the mean velocity field. A

number of such correlations were suggested by Prandtl (1925a, 1942), von Karman

(1930) and Taylor (1932) in the framework of the semi-empirical theories of

turbulence.

There are two different approaches to express of the eddy viscosity: (1) differ-

ential approach, and (2) the integral one. The former implies that turbulent transfer

in shear flow (such as submerged jets) is determined by local features of the mean

velocity field in the vicinity of a considered point. In particular, as the governing

parameters that determine turbulent transfer one can take the local velocity gradient

of the mean flow du=dy, the fluid density r, as well as some characteristic length lthat account for the displacement of fluid elements in the lateral y-direction and is

termed the mixing length. Accordingly, the functional equation for the eddy

viscosity in one-dimensional shear flow with zero pressure gradient

dP=dx ¼ 0(including the submerged jet flows) reads

mT ¼ f1 r; l;du

dy

� �(8.63)

where mT is the turbulent eddy viscosity.

Since all the governing parameters in (8.63) have independent dimensions,

namely r½ � ¼ L�3M, l½ � ¼ L and du dy=½ � ¼ T�1, it reduces to

mT ¼ c1ra1 la2du

dy

� �a3

(8.64)

where c1 is a constant.Taking into account the dimension of the eddy viscosity mT½ � ¼ L�1MT�1, we

find the values of the exponents ai as a1 ¼ 1, a2 ¼ 2 and a3 ¼ 1, and obtain the

following relations for mT and nT

mT ¼ c1rl2du

dy(8.65)

8.5 Turbulent Jets 225

nT ¼ c1l2 du

dy(8.66)

In the case when we take into account both the first and second derivatives, the

functional equation for the eddy viscosity reads

mT ¼ f2 r; l;du

dy;d2u

dy2

� �(8.67)

Equation (8.67) contains four governing parameters, three of them have inde-

pendent dimensions. Then (8.67) reduces to

P ¼ ’ðP1Þ (8.68)

whereP¼ mT= ra1 du=dyð Þa2 d2u=dy2ð Þa3� �andP1 ¼ l= ra

01 du=dyð Þa

02 d2u=dy2ð Þa

03

� �.

Finding the values of the exponents ai and a0i as a1 ¼ 1, a2 ¼ 3, a3 ¼ �2; a

01 ¼ 0

a02 ¼ 1 and a

03 ¼ �1, we arrive at the following correlation

mT ¼ rdu=dyð Þ3d2u=dy2ð Þ2 ’

l

du=dyð Þ= d2u=dy2ð Þ

(8.69)

The expression for the eddy viscosity can be significantly simplified by using the

von Karman similarity hypothesis (von Karman 1930) which implies that the

mixing length l is determined by the local characteristics of the mean velocity

field in the vicinity of a point of consideration, in particular, by the values of the

derivatives du=dy and d2u=dy2as

l ¼ cl

du

dy;d2u

dy2

� �(8.70)

Bearing in mind (8.70), we can exclude l from (8.67) and transform this equation

to the following form

mT ¼ f2 r;du

dy;d2u

dy2

� �(8.71)


in accordance with the Pi-theorem, we obtain

mT ¼ c2ra1du

dy

� �a2 d2u

dy2

� �a3

(8.72)

226 8 Turbulence

where c2 is a constant.Taking into account the dimensions of mT , r, du=dy, and d2u=dy2, we find the

values of the exponents ai as a1 ¼ 1, a2 ¼ 3 and a3 ¼ �2. After that, (8.72) takes

the following form

mT ¼ c2rdu=dyð Þ3d2u=dy2ð Þ2 (8.73)

The integral approach to the eddy viscosity implies that turbulent transfer in

shear flows is determined by the integral flow characteristics (in particluar, for the

far field of a turbulent submerged jet by its total momentum flux Jx), as well as fluiddensity and local coordinates x and y

mT ¼ f r; Jx; x; yð Þ (8.74)

where the governing parameters have the following dimensions

r½ � ¼ L�3M; Jx½ � ¼ LjMT�2; x½ � ¼ L; y½ � ¼ L (8.75)

Since in the framework of the Pi-theorem the difference n� k ¼ 1, (8.74) takes

the form

P ¼ ’1ðP1Þ (8.76)

where P ¼ mT=ra1Ja2x xa3 and P1 ¼ y=ra

01J

a02

x xa03 .

For the axisymmetric jets (j ¼ 1) the exponents ai and a0i are found as a1 ¼ 1=2,

a2 ¼ 1=2, a3 ¼ 0; a01 ¼ 0, a

02 ¼ 0 and a

03 ¼ 1 which yields

mT ¼ rJxð Þ1=2’ y

x

� �(8.77)

Within the far field of turbulent axisymmetric jets, the distribution of the

longitudinal velocity component u can be presented as

u

um¼ c

y

d

� �(8.78)

where um is the axial velocity and d is the half-width of the jet.

Indeed, since the velocity profile in the far field of the jet is determined by

three dimensional parameters, namely the velocity at the jet axis um, the jet thicknessd, and a coordinate y of a point under consideration, the functional equation

u ¼ f ðum; d; yÞ is valid. It is easy to see that two of the three governing parameters

in this equation possess independent dimensions. Therefore, applying the Pi-theorem

results in (8.78).


Beating in mind (8.78), and the integral invariant of the axisymmetric jets

Jx ¼Ðy0

ru2ydy ¼ const, we obtain

ffiffiffiffiJxr

s¼ umd

ð10

cð�Þ�d�8<:

9=;

1=2

(8.79)

where � ¼ y=d.Substituting (8.79) into (8.77), we obtain the following expression for the

kinematic eddy viscosity

nT ¼ ðumdÞð10

cð�Þ�d�8<:

9=;

1=2

’y

x

� �(8.80)

In the particular case of ’ y=dð Þ being a week function ofy x= , it is possible to

assume that ’1 is a constant. Then, (8.80) becomes identical Prandtl’s relation for

the eddy viscosity in submerged turbulent jets (Prandtl 1942)

nT ¼ const� ðumdÞ (8.81)

According to (8.81), the kinematic eddy viscosity is constant in the axisymmet-

ric turbulent jets, whereas in plane turbulent jets nT ¼ nTðxÞ (Vilis and Kashkarov

1965). Corrsin and Uberoi (1950), Antonia et al. (1975), Chevray and Tutu (1978),

Chua and Antonia (1990) showed that in reality the eddy viscosity changes across

the axisymmetric turbulent jets. It appears to be constant in the inner part of the jet

flow and decreases rapidly at toward the external boundary of the mixing layer at

the jet edge.

The thermal analog of nT is the eddy thermal diffusivity aT : The experimental

data show that there is a certain difference between the values of nT and aT , so that

the turbulent Prandtl number PrT ¼ nT=aT is not equal to one (Table 8.1, Mayer and

Divoky 1966).

The experimental data also show that nT and aT , as well as the turbulent Prandtlnumber change over the cross-section of submerged turbulent jets (Fig. 8.1). The

mean values of PrT in the far field of submerged jets are about 0.75–0.8 (Vulis and

Kashkarov, 1965; Chua and Antonia, 1990). Note, that the values of the turbulent

Prandtl and Schmidt numbers practically do not depend on the physical properties of

fluid, i.e. on the molecular values of the Prandtl and Schmidt numbers. [The Schmidt

number represents itself the ration of the kinematic viscosity to diffusivity and in

mass transfer processes plays a similar role to that of the Prandtl number in the heat

transfer processes.] The experimental studies of the velocity, temperature or concen-

tration fields in turbulent jets of different fluids (mercury: Pr � 10�2; oil: Pr � 103)

228 8 Turbulence

and aqueous solt solutions (Sc � 103) reveal that the turbulent Prandtl and Schmidt

numbers are always about 0.75–0.8 (Sakipov 1961; Sakipov and Temirbaev 1962;

Forstall and Gaylord 1955). These data confirm that turbulent transfer of momentum,

heat and species is dominant in submerged turbulent jets. A comprehensive analysis

of a number of semi-empirical correlations for the kinematic viscosity and thermal

diffusivity can be found in the monograph by Hinze (1975)

8.5.2 Plane and Axisymmetric Turbulent Jets

Consider velocity distribution in the far field of a plane or axisymmetric turbulent

jet. The governing parameters in this case are selected as: (1) the fluid density r,(2) the total momentum flux Jxwhich is constant along the jet, (3) longitudinal and

Table 8.1 Turbulent Prandtl number in turbulent jet flows

Flow field Type of flow PrT Authors

Planar Wake of a heated cylinder 0.54 Fage and Falkner

Planar Heated jet 0.54 Reichardt

Planar Heated jet 0.42–0.59 Van der Hegge Zijnen

Axisymmetric Heated jet with tracers 0.74 Van der Hegge Zijnen

Axisymmetric Hydrogen jet 0.72 Keary and Weller

Axisymmetric Heated jet 0.7 Corrsin

Axisymmetric Heated jet 0.71 Forstall

Axisymmetric Submerged water jet 0.72–0.83 Forstall and Gaylo-rd

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5

Pr T

η

Fig. 8.1 Turbulent Prandtl number in the axisymmetric turbulent jet. ■-Corrsin and Uberoi

(1950), ~-Chevray and Tutu (1978), ●-Chua and Antonia (1990) (measurements 90 X- probe),� ¼ y=y1=2 is the dimensionless lateral coordinate, y1=2is the half-velocity coordinate where

u ¼ um=2, u and umare the local longitudinal velocity and the axial velocity, respectively.

Reprinted from Chua and Antonia (1990) with permission


lateral coordinates x and y determining the location of a point of consideration. The

eddy viscosity mT , of course, also affects the flow field, however, it is fully

determined by Jx; x and y and thus, is not an independent parameter. Therefore,

the functional equation for the longitudinal mean velocity u reads

u ¼ f r; Jx; x; yð Þ (8.82)

Here and hereinafter bars above mean flow characteristics are omitted for

brevity.

The dimensions of the governing parameters in (8.82) are

r½ � ¼ L�3M; Jx ¼ MT�2; x½ � ¼ L; y½ � ¼ L (8.83)

It is seen that three governing parameters possess independent dimensions, so

that the difference n� k ¼ 1. Then (8.82) reduces to the following form

P ¼ ’1ðP1Þ (8.84)

where P ¼ u=ra1Ja2x xa3 and P1 ¼ y=ra01J

a02

x xa03 .

Taking into account the dimensions of the parameters involved in the

expressions for P and P1 and applying the principle of the dimensional homoge-

neity, we find the values of the exponents ai and a0i as a1 ¼ �1=2, a2 ¼ 1=2,

a3 ¼ �1=2; a01 ¼ 0, a

02 ¼ 0, and a

03 ¼ 1. Accordingly, we obtain

u ¼ffiffiffiffiJxr

s’1

y

x

� �x�1=2 (8.85)

The functional equation for the axial velocity um and the jet thickness d are

um ¼ fu r; Jx; xð Þ (8.86)

d ¼ fd r; Jx; xð Þ (8.87)

Applying the Pi-theorem, we transform (8.86) and (8.87) to the following form

um ¼ c1rb1Jb2x xb3 (8.88)

d ¼ c2rg1Jg2x xg3 (8.89)

where c1 and c2 are constants, and the exponents bi and gi are equal to: b1 ¼ �1=2,b2 ¼ 1=2, b3 ¼ �1=2, g1 ¼ 0, g2 ¼ 0, and g3 ¼ 1.

As a result, we arrive at the following equations

230 8 Turbulence

um ¼ c1

ffiffiffiffiJxr

sx�1=2 (8.90)

d ¼ c2x (8.91)

Bearing in mind (8.89) and (8.90), we can rewrite (8.85) as follows

u

um¼ ’1ð�Þ (8.92)

where � ¼ y=d.The axial velocity um and the jet thickness di are defined by the profile of the i-th

characteristic being used as the characteristic scale, which allows us to present the

profiles of different flow characteristics in the form similar to that of (8.92). Indeed,

the functional equation for any of the flow characteristic Ni reads

Ni ¼ fi um; y; dið Þ (8.93)

where Ni ¼ ui;ffiffiffiffiffiffiu02

p;

ffiffiffiffiffiv02

p; u0v0 . . . which represent any of the mean velocity

components, velocity pulsations, the shear stress, etc.

Applying the Pi-theorem, we arrive at

Pi ¼ ’iðP�Þ (8.94)

where Pi ¼ Ni=uaim , P� ¼ y=di ¼ �, etc., as well as a1 ¼ a2 ¼ a3 ¼ 1 and a4 ¼ 2

for i ¼ 1,. . .4.Accordingly, we arrive at the following equations

u

um¼ ’1ð�Þ;


pum

¼ ’2ð�Þ;ffiffiffiffiffiv02

pum

¼ ’3ð�Þ;ffiffiffiffiffiffiffiu0v0

pu2m

¼ ’4ð�Þ (8.95)

Choosing as the characteristic scales of temperature and species concentrations

their values at the jet flow axis, as well as the thicknesses of the thermal and

diffusion layers dT and dc, respectively, we express the functional equations for

DT and DCj as follows

DT ¼ fT DTm; dT ; yð Þ (8.96)

DCj ¼ fC DCj; dc; y �

(8.97)

where DT ¼ T � T1, DTm ¼ Tm � T1, DCj ¼ Cj � Cj1, and T1 and Cj1 are the

temperature and concentration of the j� th species in in the undisturbed fluid

outside the jet.


The application of the Pi-theorem to (8.96) and (8.97) leads to

DTDTm

¼ ’Tð�TÞ (8.98)

DCDCm

¼ ’Cð�CÞ (8.99)

where �T ¼ y=dT and �C ¼ y=dC.Similar functional equations for fluctuations of temperature and species

concentrations read

ffiffiffiffiffiffiT 02

pDTm

¼ ’Tð�TÞ (8.100)

ffiffiffiffiffiffiffiC02

pDCm

¼ ’Cð�CÞ (8.101)

The assumption that the axial velocity (temperature or species concentrations)

and the thickness of the dynamic (thermal or diffusional) layer can serve as the

characteristic scales for submerjed jets is based only on the symmetry of these flows

and does not imply whether they are plane or axisymmetric. Therefore,

(8.95–8.101) are valid as for both plane and axisymmetric jets. These equations

can be used for generalizing of the experimental data for the mean and pulsation

velocity, temperature and concentration distributions in the far field of any axisym-

metric jets. The corresponding generalized experimental results for the velocity,

temperature and concentration in plane and axisymmetric turbulent jets are

presented in Figs. 8.2–8.6. It is seen that experimental data obtained at different

jet cross-sections collapse at single curves corresponding to the self-similar behav-

ior uncovered in the present subsection.

8.5.3 Inhomogeneous Turbulent Jets

The inhomogeneous turbulent jets emerge when turbulent mixing of gas streams of

different densities is encountered. The existence of non-uniform density field

significantly affects the aerodynamics of inhomogeneous jets, in particular, the

decay of the mean velocity, velocity pulsations, and concentration along the jet

axis, as well as distributions of these parameters in jet cross-sections, the jet

ejection features, etc. (Abramovich,1974; Panchapakesan and Lumley 1993). The

flow field in turbulent inhomogeneous jets is described by the continuity, momen-

tum and species balance equations for turbulent motion of the inhomogeneous

mixtures of variable density (Shin et al. 1982). Consider a plane or axisymmetric

232 8 Turbulence

turbulent jet of species 1 injected into space submerged by a mixture of species 1

and 2. After a number of simplifications of the mean momentum balance and

species balance equations, the integration of these equations across the jet yields

ð10

ru2yjdy ¼ Jx (8.102)

0

0.2

0.4

0.6

0.8

1

1.2a

b

0 0.5 1 1.5 2 2.5

12

y

y

u u m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.5 1 1.5 2 2.5 3

12

yy

u'2

u m

Fig. 8.2 Distribution of the mean (a) and (b) velocity pulsations in the cross- sections of turbulentjet. (a)x=d: ♦�118, ■�108, ~�103, ��76, *�65; (b) x=d: ♦�143, ■�129, ~�118, ��106,

*�95. Reprinted from Gutmark and Wygnanski (1976) with permission


ð10

ruDc1yjdy ¼ Gx (8.103)

where r ¼ r1�1 � r2

�1ð Þc1 þ r2�1½ ��1

is the mean local density of the gas mixture,

c1 is the mean local concentration of the injected gas, r1 and r2 are the mean densiy

of the injected (1) and ambient (2) species, Dc1 ¼ c1 � c11 is the excess concen-

tration of the injected gas, subscript 1 corresponds to the ambient conditions far

away from the jet axis, j ¼ 0 or 1 for plane and axisymmetric jets, respectively.

As the governing parameters determining the velocity and concentration fields in

the far field of submerged inhomogeneous turbulent jets it is natural to choose the

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.5 1 1.5 2

12

yy

u2 m

u′ v

′

Fig. 8.3 The distribution of the turbulent shear stress in cross-sections of planar turbulent jet. x=d:♦�143, ■�129, ~�118, ��106, +�95. Reprinted from Gutmark and Wygnanski (1976) with

permission

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

12

yy

ΔT ΔTm

Fig. 8.4 Radial profiles of the mean temperature DT=DTm ¼ T � T1 �

= Tm � T1 �

in the cross-

sections of an axisymmetric turbulent jet x=d: ◊�7.0, □�10, ~�15, ��20.21, +�25.47.

Reprinted from Lockwood and Moneib (1980) with permission

234 8 Turbulence

local density of the gas mixture r½ � ¼ L�3M, the total momentum flux along the jet

Jx ¼ LjMT�2, the total mass flux of the injected gas Gx½ � ¼ Lj�1MT�1; as well as thecoordinates x½ � ¼ L and y½ � ¼ L. (Note that Jxand Gxdo not change along the jet and,

thus are considered as given invariant constants). Accordingly, we can state the

following functional equations for the thickness of a turbulent inhomogeneous jet d,as well as for the axial velocity umand concentration Dc1min it

d ¼ f1 r; Jx;Gx; xð Þ (8.104)

um ¼ f2 r; Jx;Gx; xð Þ (8.105)

Dcm1 ¼ f3 r; Jx;Gx; xð Þ (8.106)

Applying the Pi-theorem to (8.104–8.106), we arrive at the following dimen-

sionless equations

Pk ¼ ’kðP�Þ (8.107)

where k ¼ 1; 2; 3; P1 ¼ d=ra1Ja2x Ga3x , P2 ¼ um=ra

01J

a02

x Ga03

x , P3 ¼ Dcm=ra001 J

a002

x Ga003

x

and P� ¼ x=ra�1J

a�2

x Ga�3x .

Bearing in mind the dimensions of d, um, and Dcm, and applying the principle ofthe dimensional homogeneity, we find the values of the exponents involved in

(8.107) as: a1 ¼ �1=2; a2 ¼ �1=2, a3 ¼ 1; a01 ¼ 0, a

02 ¼ 1, a

03 ¼ �1; a

001 ¼ �1=2,

a002 ¼ �1=2, a

003 ¼ 1; a�1 ¼ �1=2, a�2 ¼ �1=2, a�3 ¼ 1. Accordingly, we obtain the

following equations for d; um, and Dcm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3

1

yy2

e T e T.m

Fig. 8.5 Radial distributions of the root-mean-square temperature fluctuations in the cross-

sections of an axisymmetric turbulent jet x=d: ◊�7.0, □�10.0, D� 15:0, ��20.21, +�25.47,~�30.75, ♦�36.66; eT-rms of temperature fluctuations. Reprinted from Lockwood and Moneib

(1980) with permission


d

G2x=rJx

�1=2 ¼ ’1

x

G2x=rJx

�1=2( )

(8.108)

umJx=Gxð Þ ¼ ’2 x=

G2x

rJx

� �1=2( )

(8.109)

Dcm1 ¼ ’3 x=G2

x

rJx

� �1=2( )

(8.110)

Taking into account that the invariants Jx and Gx are constants, we find that

Jx ¼ r1u210d

jþ1, Gx ¼ r1u10Dc10djþ1, and Dc10 ¼ c10 ¼ 1 for issuing of pure

injected gas into space filled by another gas, where u10 and c10 are the initial

velocity and concentration of the injected gas and d is the nozzle diameter. Then,

we obtain

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25η

k

0

0.05

0.1

0.15

0.2

0.25

0.3b

a

0 0.05 0.1 0.15 0.2 0.25η

cC(c

, h)

k

cC¢ rm

s(c,

h)

Fig. 8.6 Concentration distribution in cross-sections of turbulent axisymmetric jet. (a) mean

concentration, (b)concentration pulsations. x d= : □�20, ◊�40, D�60, ��80; Reprinted from

Doweling and Dimotakis (1990) with permission

236 8 Turbulence

JxGx

¼ u10;G2

x

rJx

� �1=2

¼ffiffiffiffiffir1r

rd jþ1ð Þ=2 (8.111)

The gas mixture density in the far field of the jet is of close to the ambient one,

and it is possible to assume that r r2. Then (8.108–8.110) take the form

d ¼ ’1 xffiffiffiffio

p �(8.112)

um ¼ ’2 xffiffiffiffio

p �(8.113)

Dcm1 ¼ ’3 xffiffiffiffio

p �(8.114)

where d ¼ d=dffiffiffiffio

p, um ¼ um=u10, Dcm1 ¼ Dcm1 Dc10 ¼ Dcm1= , ðDcm1 ¼ cm1 when

c11 ¼ 0Þ, x ¼ x=d and o ¼ r2=r1.Thus, the variation of the longitudinal velocity and concentration of the issued

species along the axis of an inhomogeneous turbulent gas jet is determined by a

single dimensionless group xffiffiffiffio

p. The experimental data on the dependence Dcm1 �

xffiffiffiffio

pð Þ are presented in Fig. 8.7. The figure also includes the experimental data on

the axial decay of the excess of enthalpy in high temperature turbulent jets which

is expected to be indentical to the decay of the excess concentration. It is seen

that all the experimental data corresponding to the different inhomogeneous

ð0:27 � o � 8:2Þand high temperature ðo> 15Þ turbulent jets groip around a

single curve cm xffiffiffiffio

pð Þ. (Abramovich et al. 1974)

Choosing as the characteristic scales of velocity, concentration and the

corresponding fluctuations their values at the jet axis, we present distribution of

u,ffiffiffiffiffiffiu02

p,

ffiffiffiffiffiv02

p, and c1 and

ffiffiffiffiffiffiffic102

pas follows

Fig. 8.7 Turbulent mixing, axsisymmetric gas jets; the dependence cm1ðxffiffiffiffio

p Þ, with xffiffiffiffio

p:

□�0.27, ◊�1.3, D�8.2, ��1.5, ~�21, ♦�26


Ni ¼ f1 um;y; d �

(8.115)

Mk ¼ fk cm1; y; dð Þ (8.116)

where Ni ¼ u,ffiffiffiffiffiffiu02

por

ffiffiffiffiffiv02

pand i ¼ 1,2 and 3; also, Mk ¼ c1 or

ffiffiffiffiffiffiffic102

pand k ¼ 4

and 5.

Applying the Pi-theorem to (8.115) and (8.116), we arrive at the following

equations

u

um¼ f1

y

d

� �;


pum

¼ f2y

d

� �;

ffiffiffiffiffiv02

pum

¼ f3y

d

� �;c1cm1

¼ ’4

y

d

� �;ffiffiffiffiffiffiffi

c102pcm1

¼ ’5

y

d

� �(8.117)

Using a similar approach, we can find the following dimensionless equations for

the turbulent correlations u0v0, v0c0

, etc. in jet cross-sections

u0v0

u2m¼ f6

y

d

� �;v0c1

0

umcm1¼ f7

y

d

� �(8.118)

Equations (8.117) and (8.118) show that the profiles of the dimensional

characteristics corresponding to different cross-sections of the inhomogeneous jet

can be presented in the parametric planes Ni�, andMi� (where � ¼ y=dÞ in the formof universal dependences Nið�Þ and Mið�Þ (cf. Figs. 8.8–8.10)

8.5.4 Co-flowing Jets

Co-flowing turbulent jet is formed when a turbulent jet is issued into a uniform fluid

flow in an infinite domain. Detailed experimental data for the mean and

characteristics and turbulent pulsations in co-flowing turbulent jets were obtained

byMaczynski, 1962; Bradbury and Riley 1967; Antonia and Bigler 1973; Everitt and

Robins 1978; Abramovich et al. 1984; Nickels and Perry 1996. Some general

considerations of the features of co-flowing turbulent jets can be found in the

monograph by Townsend (1956) who analyzed the conditions corresponding to

self-similar flows in such jets. He also dealt with the effect of viscosity on flow

characteristics, e.g. the independence of the decay of the mean and pulsation

velocities along the jet axis on the Reynolds number, etc. For the additional details,

a reader can consult the above-mentioned works, while we restrict our discussion to

the applications of the Pi-theorem to study the aerodynamics of co-flowing turbulent

jets, and especially to generalize the experimental data for the far field in these flows.

238 8 Turbulence

A sketch of co-flowing turbulent jet with a uniform velocity profile at the orifice

exit x ¼ 0 is depicted in Fig. 8.11. At some distance from the jet orifice, the

initialvelocity profile transforms into a smooth profile with a characteristic maxi-

mum at the jet axis, which gradually decreases toward the outer edge of the jet

where the longitudinal velocity u matches the free stream velocity u1. In the

framework of the boundary layer theory the flow in co-flowing turbulent jets is

described by the momentum balance and continuity equations subjected to the

boundary conditions that account for the existence of the outer flow

y ¼ 0;@u

@y¼ 0; v ¼ 0 y ! 1 u ! u1 (8.119)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3yx

0

0.2

0.4

0.6

0.8

1

1.2b

a

0 0.05 0.1 0.15 0.2 0.25 0.3

u u mc 1 c m

1

yx

Fig. 8.8 The mean velocity (a) and mean helium concentration (b) distributions in the axisym-

metric turbulent jet. x d= : □�90, ◊�100, D�110, ��120. Reprinted from Panchapakesan and

Lumley (1993) with permission


where subscript 1 refers to the undisturbed co-flow.

The integration of the momentum balance equation across the jet accounting for

the boundary conditions (8.119) yields the following integral invariant of co-

flowing jets

Jx ¼ð10

ruðu� u1Þyjdy (8.120)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3

12

yy

12

y

y

c′1

c m1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4 a

b

0 0.5 1 1.5 2 2.5

u′ u m

Fig. 8.9 The distribution of the intensities of fluctuations, a – the root – mean-square of velocity

(u0=um) and b – concentration (c10=cm1)in cross-sections of the axisymmetric turbulent jet x=d:

□�50, ■�60, D� 70, ��80, ~�90, ♦�100, +� 110. Reprinted from Panchapakesan and

Lumley (1993) with permission

240 8 Turbulence

which is the momentum flux along the jet and where j ¼ 0 or 1correspond to plane

or axisymmetric jets, respectively.

Far from the orifice where the jet ‘forgets’ the initial conditions, the longitudinal

velocity is determined by fluid density r½ � ¼ L�3M, the total momentum flux

Jx½ � ¼ LjMT�2, the free stream velocity u1½ � ¼ LT�1 and the longitudinal and

lateral coordinates x½ � ¼ L and y½ � ¼ L

u ¼ f1 r; Jx; u1; x; yð Þ (8.121)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.05 0.1 0.15 0.2 0.25 0.3yx

c m1u

m

c 1′v

′

0

0.05

0.1

0.15

0.2

0.25

0.3a

b

0 0.5 1 1.5 2 2.5

u′v′

u m2

12

yy

Fig. 8.10 Variation of the Reynolds stress (u0v0=u2m)�a and radial scalar flux (c1 0v0=cm1um)

�b

across the axisymmetric turbulent jet. x=d: D� 50, □�60, ◊�70, ��90, ■�100, ~�110.

Reprinted from Panchapakesan and Lumley (1993) with permission


Beating in mind the dimensions of the governing parameters we, find that in the

present case the difference n� k ¼ 2 and, accordingly, (8.121) reduces to the

dimensionless equation

P ¼ ’1 P1;P2ð Þ (8.122)

where P ¼ u=u1, P1 ¼ x= Jx=rxjþ1ð Þ1=2 and P2 ¼ y= Jx=rxjþ1ð Þ1=2.Thus, the dimensionless velocity in co-flowing jets is a function of two dimen-

sionless groups. That means that the present problem has no self-similar solution.

Note that excluding u1 from the boundary conditions (8.119) by introducing the

excess of velocity u� u1 does not allow one to decrease the number of the

governing parameters and thereby obtain a self-similar solution, since such trans-

formation introduces u1 into the momentum balance equation.

Within the far field of co-flowing turbulent jets, velocity change across the jet is

small enough compared to the free stream velocity. This allows one to recover an

approximate self-similarity of co-flowing turbulent jets (Townsend 1956). Indeed,

the functional equations for the mean flow characteristics of plane co-flowing

turbulent jets can be written now as

U ¼ f2 r; Jx; x; yð Þ (8.123)

Um ¼ f3 r; Jxxð Þ (8.124)

d ¼ f4 r; Jx; xð Þ (8.125)

where U ¼ u� u1 is the excess of velocity, subscript m refers to the jet axis and ddenotes the jet thickness.

Applying the Pi-theorem to (8.123–8.125), we arrive at the following equations

u� u1ffiffiffiffiffiffiffiffiffiffiffiffiJx=rx

p ¼ ’2

y

x

� �(8.126)

Fig. 8.11 Sketch of co-

flowing turbulent jet with a

step-wise initial velocity

distribution

242 8 Turbulence

um � u1ffiffiffiffiffiffiffiffiffiffiffiffiJx=rx

p ¼ C1 (8.127)

d ¼ C2x (8.128)

where C1 ¼ ’2ð0Þ ¼ const.Equations (8.127) and (8.128) show that the excess of axial velocity in the far

field of plane co-folwing turbulent jets is inversely proportional to x1=2, whereas thejet thickness is directly proportional to x. These results agree fairly well with the

experimental data of Bradbury and Riley (1967) and Everitt and Robins (1978) for

plane co-flowing turbulent jets (cf. Figs. 8.12 and 8.13).

It is seen that d � x (with x ¼ x=y) and Dum � x�1=2 at x> 40 (d ¼ d=y,

x ¼ x=y, Dum ¼ Dum=u1, Dum ¼ um � u1, y ¼ Ðþ1

�1u=u1 u=u1 � 1ð Þdy. It is

emphasized that all the data corresponding to different values of the co-flow

parameter m(the ratio of the exit to the free stream velocity) collapse near single

curves dðxÞ and DumðxÞ in the parametric planes d� x and Dum � x.Using (8.126–8.128), we express the velocity distribution across the co-flowing

jet as

u� u1um � u1

¼ c3’2

y

d

� �(8.129)

where c3 ¼ c�11 .

0

0.5

1

1.5

2

2.5

0 20 40 60 80 100 120xq

,u ∞

qdΔu

m

Fig. 8.12 Variation of the axial velocity and thickness of co-flowing jet along the flow axis.

Reprinted from Bradbury and Riley (1967) with permission


The experimental data on the velocity distribution in different cross-sections of

co-flowing plane turbulent jet are presented in Fig. 8.14. The data related to the

distributions of turbulent pulsations across co-flowing turbulent ‘strong’ and ‘weak’

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100 120 140x

q

d q

Fig. 8.13 Variation of the thickness of a plane turbulent co-flowing jet along its axis. y is the

momentum thickness of the jet . u=u1: D�2.60, ◊�3.03, □�3.24, ��3.29, ~�3.78, ■�6.72,

+�17.08. Reprinted from Everitt and Robins (1978) with permission

0

0.2

0.4

0.6

0.8

1

1.2

-3 -2 -1 0 1 2 3

12

yy

U–U

1

U0

Fig. 8.14 Velocity distribution in cross-sections of plane turbulent jet issuing into a parallel

moving air stream. U0=U1: ◊�6.64, □�1.93, D�0,41, ��20.2, ~� Reprinted from Bradbury

and Riley (1967) with permission

244 8 Turbulence

jets (i.e. the jet flows where the excess of center-line velocity is much larger or

much smaller than the free stream velocity) are shown in Fig. 8.15 It is seen that in

both cases the experimental data can be generalized in the form of universal

dependences in the parametric planes Ni � � where Ni ¼ Du=Dum and u02= Duð Þ2.

8.5.5 Turbulent Jets in Crossflow

The aerodynamics of turbulent jets in crossflow was a subject of a number of

experimental works. They contain detailed data on the jet trajectory, decay of

centerline velocity, distributions of the mean and pulsation characteristics, as well

as the vorticity field in such jets (Chassaing et al. 1974; Moussa et al. 1977;

Andreopoulos and Rodi 1985; Fric and Roshko 1994; Kelso et al. 1996; Smith

and Mungal 1998). In addition, a number of the empirical and semi-empirical

correlations for predicting the trajectory of turbulent jets in crossflow and variation

of the flow characteristics along the curved jet axis was proposed (Keffer and

Baines 1963; Abramovich 1963). A self-consistent analysis of the aerodynamics

of turbulent jets in crossflow based on the similarity theory was recently proposed

by Hasselbrink and Mungal (2001). Following the ideas of the latter work, we

outline below the application of the dimensional analysis to study characteristics of

turbulent jets in crossflow.

Our aim is to describe the flow field in an incompressible turbulent jet in

crossflow with a known trajectory ycðxcÞfor a given blowing ratio

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.5 1 1.5 2 2.5 3

12

yy

u'2

U02

Fig. 8.15 Distribution of the velocity fluctuations in cross-sections of plane turbulent jets.

D-strong jet, □-weak jet, ▪▪▪-wake. Reprinted from Everitt and Robins (1978) with permission


ðrju2j r1v21� Þ1 2=

(Fig. 8.16). We will proceed from the assumption that the local

values of any of the flow characteristics Ni ¼ r; u; v � � � are fully determined by the

axial value of a given characteristic in the same cross-section Nic, the jet thickness

there d, the position along the centerline xc, and coordinates x; y of a point under

consideration. It is emphasized that the velocity vector v in the general case has

three components u; v and w, but following Hasselbrink and Mungal (2001), we

consider the flow to be planar and thus, are dealing with only u and v. Then we canwrite the functional equation for Ni as follows

Ni ¼ f Nic; x; y; xc; dð Þ (8.130)

where Ni ¼ r; u; v.The dimensions of Nic are LaMbTg, whereas the dimensions of the other

governing parameters in (8.130) are all L½ �. Therefore, in the case under consider-

ation the total number of the governing parameters is five, whereas the number of

the governing parameters with independent dimensions equals two, so that

n� k ¼ 3. Then, according to the Pi-theorem, (8.130) can be transformed to the

following dimensionless equation

Ni ¼ Nic’ �; g; xcð Þ (8.131)

where �, g and xc are the coordinates x,y and xc normalized by d.The density r and velocity component distributions can be expressed as follows

r ¼ rcðxcÞf1 �; g; xcð Þ (8.132)

u ¼ ucðxcÞf2 �; g; xcð Þ (8.133)

v1 � vð�; g; xcÞ½ � ¼ v1 � vcðxcÞ½ �f3 �; g; xcð Þ (8.134)

Fig. 8.16 Sketch of a jet in

crossflow. The origin of the

coordinates x andyis at thecenter of the nozzle issuing

the jet. Adenotes a jet cross-section with a center at

x ¼ xc and y ¼ yc. Subscriptsj;1and c correspond to the

nozzle, the undisturbed

crossflow and the centerline

of the jet, respectively.

Reprinted from Hasselbrink

and Mungal (2001) with

permission

246 8 Turbulence

where the functions f1, f2, and f3 are assumed to be universal within each of the

characteristic flow domains of jets in crossflow: (1) potential core, (2) the near-field

region, and (3) the far-field region.

In order to determine the value of the flow characteristics at the jet axis, it is

necessary to employ some additional relations that follow from the mass, momen-

tum and momentum excess (based on Dv ¼ v1 � vc) balance equations. In partic-

ular, they have the form

m I1rcucd2 (8.135)

J I2rcu2cd

2 (8.136)

y I3rcd2 v1 � vcð Þuc (8.137)

for the near-field of the jet, and

m I4rcv1d2 (8.138)

J I5rcv1ucd2 (8.139)

y I6rcv1 v1 � vcð Þd2 (8.140)

for the far-field of the jet where m ¼ mj þ m1 is the mass flux,Jand y are the

momentum and momentum excess fluxes, respectively, through a cross-section at a

given distance from the nozzle. In addition, I1 ¼ÐA

f1f2d�dg, I2 ¼ÐA

f1f22 d�dg

I3 ¼ÐA

f1f2f3d�dg, I4 ¼ÐA

f1d�dg, I5 ¼ÐA

f1f2d�dg and I6 ¼ÐA

f1f3d�dg, where A

denotes jet cross-section.

Neglecting the effect of the molecular viscosity, we assume that the functional

equations for the axial velocity and turbulent jet thickness in the near field region of

the jet are

uc ¼ c1 J; r1; xð Þ (8.141)

d ¼ c2 J; r1; xð Þ (8.142)

Taking into account the dimensions of J, r1 and x, we find

uc ¼ c1J

r1

� �1=2

x�1 (8.143)

d ¼ c2x (8.144)

where c1 and c2 are constants.


The dependence of vmon x found using (8.140) is

ðv1 � vcÞ � y

r1=21 J1=2x�1 (8.145)

For the far-field region of the jet flow we find the following expressions for uc; dand vc

uc � J

r1v1x�2 (8.146)

d � x (8.147)

ðv1 � vcÞ � J

r1v1x2 (8.148)

The relations (8.143–8.145) and (8.146–8.148) allow us to evaluate the change

in the velocity and thickness withing the near and far regions of of the jet. It is seen

that the centerline velocity fades according to the different laws at small and large

distances from the nozzle. In particular, at small distance from the nozzle the

centerline velocity fades as x�1, wereas at large x it fades as x�2.In both regions

the jet thickness is proportional to x.

8.5.6 Turbulent Wall Jets

Consider a turbulent wall jet outflowing from a two-dimensional slot in contact with

a wall into a fluid at rest. The velocity profile in a cross-section of the wall jet is

sketched in Fig. 8.17. It results from the interection of the jet with the solid surface,

Fig. 8.17 Sketch of turbulent

wall jet

248 8 Turbulence

as well as with the surrounding fluid. In the far-field region of turbulent wall jets the

profile of the longitudinal mean velocity component has a maximum located at

some distance y ¼ ym from the wall which depends on x. On both sides from the

maximum at y ¼ ym velocity gradually decreases to zero at y ¼ 0 (at the wall, due

to the no-slip condition) and y ! 1 (in the surrounding fluid at rest).

The measurements show that turbulent wall jets have a very complicated struc-

ture because of the interplay of the molecular viscousity and the inertial effects

(Launder and Rodi 1981; 1983). The influence of various factors on flow

characteristics is different in different domains of turbulent wall jets. Within the

near-wall region the molecular viscous effects are dominant, whereas at large

distance from the wall fluid inertia plays an important role. Such structure of

turbulent wall jets allows one to select two characteristic domains: (1) the inner

one close to the wall (with dominant viscous effects), and (2) the outer one in which

the inertial effects are dominant. In the two domains the velocity distribution is

affected by different parameters responsible for specific flow features in the inner

and outer parts of turbulent wall jets. In the framework of such model the charac-

teristic velocity and length can be chosen as the friction velocity u� ¼ tw=rð Þ1=2andlength d� ¼ n=u� in the inner domain, and the maximum longitudinal velocity umand the jet thickness d (e.g. corresponding to the velocity value u ¼ um=2Þ for theouter one.

At the Reynolds number being infinite, there exists a self-similar solution of the

problem (George et al. 2000; Bergstrom and Tachie 2001), and the functional

equation for the longitudinal velocity in the inner and outer domains can be written as

uin ¼ fin u�; y; vð Þ (8.149)

uout ¼ fout um; d; yð Þ (8.150)

Applying the Pi-theorem to (8.149) and (8.150), we transform them to the


Pin ¼ ’inðP1inÞ (8.151)

Pout ¼ ’outðP1outÞ (8.152)

where the dimensionless groups read

Pin ¼ u

u�; P1in ¼ yu�

n; Pout ¼ uout

um; P1out ¼ y

d(8.153)

Equations (8.151) and (8.152) show that the dimensionless velocity in the inner

and outer domains is a function of a single dimensionless group. The latter makes it

possible to collapse the experimental data for turbulent wall jets as a single curve

Uþ ¼ u=u� versus yþ ¼ yu�=n for the inner layer, and another single curve


u=u� versus y=d for the outer one .The experimental data for the mean velocity

profiles in the outer (y y1 2=

�) and inner (yþ) coordinates, and the variation of the

half-width with the streamwise distance in turbulent plane jet are shown in

Figs. 8.18, 8.19 and 8.20.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

u u m

12

yy

Fig. 8.18 The mean velocity profiles in the wall jet near a smooth wall.◊�Re0¼12500; x h= ¼50;

□�Re0¼12500; x h¼80= ;■�Re0¼9100; x h= ¼50; ��Re0¼6100; x h= ¼30; ~�Re0¼10000

(data by Karlsson et al., 1993). Reprinted from Tachie et al. (2004) with permission

1.00E+07

1.00E+08

1.00E+09

1.00E+10

1.00E+08 1.00E+09 1.00E+10 1.00E+11

xI0

n2

n2

y 1/2

I 0

Fig. 8.19 Variation of the half-width of turbulent plane wall jet in the streamwise direction. Re0:

�1240, ◊�11900, ~�10300, □�9100, ��6100, ♦�12500, ■�5900. Reprinted from Tachie

et al. (2004) with permission

250 8 Turbulence

In order to determine the character of variation of um and d along the turbulent

wall jet some addition considerations of the governing parameters that determine

flow in the outer domain should be involved. Narasimha et al. suggested that the set

of parameters should also include the momentum flux at the jet origin divided by

density I0½ � ¼ L3T�2 and molecular kinematic viscosity n½ � ¼ L2T�1 (Narasimha

et al. 1973). Accordingly, one can present the functional equations for um and d as

follows

um ¼ f1 n; I0; xð Þ (8.154)

d ¼ f2 n; I0; xð Þ (8.155)

Then, applying the Pi-theorem, we arrive at

umnI0

¼ ’1

xI0n2

� �(8.156)

dI0n2

¼ ’2

xI0n2

� �(8.157)

1.00E+00

1.00E+01

1.00E+02

1.00E+00 1.00E+01 1.00E+02 1.00E+03y+

U+

Fig. 8.20 The mean velocity profile in the inner coordinates in a turbulent plane wall jet. Re:

◊�12500, D�9100, □�6100. Reprinted from Tachie et al. (2004) with permission


8.5.7 Impinging Turbulent Jet

A sketch of the turbulent jet impinging normally onto an infinite solid wall is shown

in Fig. 8.21.

The impinging jets can be subdivided into three characteristic regions. In the first

one (1 in Fig. 8.21) free turbulent mixing of the jet with the surrounding medium is

dominant. In region 2 direct interaction of the jet with the solid wall is accompanied

by an abrupt deceleration of the jet and its deflection in the direction normal to its

initial path. In region 3 the jet-wall interaction resembles that one for a walljet. The

centerline velocity um gradually changes from its initial value at the nozzle exit u0to zero at the stagnation point at the wall surface. The distribution of the centerline

velocity corresponding to different distances from the nozzle exit down to the solid

surface is shown in Fig. 8.22 where u0 and d0 are the jet velocity at the nozzle axis

and the nozzle diameter, respectively. It is seen that the jet-wall interaction affects

significantly the longitudinal velocity umðxÞ.In order to gain an insight into the peculiarities of the velocity distribution we

apply -theorem. In this case the governing parameters determining the centerline

P the (for a ½Jx� ¼ MT�2 the total momentum flux ½r� ¼ L�3M, velocity are: the

fluid density and the ½h� ¼ L plane jet), the distance between the solid surface and

the nozzle exit. Accordingly, the functional equation for the centerline velocity

reads ½x� ¼ L coordinate

um ¼ f r; Jx; h; xð Þ (8.158)

Fig. 8.21 Sketch of the impinging turbulent jet

252 8 Turbulence

Since three of the four governing parameters in (8.158) have independent

dimensions, this equation reduces to the following dimensionless equation

P ¼ ’ðP1Þ (8.159)

where P ¼ um=ra1Ja2x ha3 and P1 ¼ x=ra01J

a02

x ha03 .

Taking into account the dimensions of um, r, Jx, h and x, we find the values of theexponents ai and a

0i as a1 ¼ �1=2, a2 ¼ 1=2, a3 ¼ �1=2; a

01 ¼ a

02 ¼ 0, and a

03 ¼ 1.


umu0

h

d0

� �1=2

¼ ’x

h

� �(8.160)

In (8.160) it is accounted for the fact that Jx � ru20d0.Equation (8.160) shows that dimensionless centerline velocity eu ¼ um=u0ð Þ

h=d0ð Þ1=2 is a universal function of a single dimensionless variable x� ¼ x=h. The

experimental data by Gutmark et al. (1978) on the centerline velocity distribution in

plane impinging turbulent jet are presented in Fig. 8.23. They correspond to different

valuesof the ratio h=d0 and the Reynolds number. Figure 8.23 shows that all experi-

mental data collapse around a single curve u� ðx�Þ according to the results of the

dimensional analysis. It is worth noting that near the wall there is a narrow layer

where the centerline velocity is proportional to the distance from the wall. This linear

relation corresponds to the inviscid stagnation flow. A significant deflection of the

dimensionless velocity u�from the one corresponding to a free jet takes place in the

region 0< x� � 0:2.

Fig. 8.22 Distribution of the centerline velocity at several cross-sections. Curve 1 corresponds

to a free jet, h ¼ h1 ¼ 1. Curves 2–4 correspond to different distances from the nozzle:

h2 > h3 > h4


Problems

P.8.1. Establish the functional dependence of the dimensionless centerline velocity

in the axisymmetric impinging turbulent jet on the dimensionless distance from the

wall.

The governing parameters which determine the centerline velocity in such a jet

are: r½ � ¼ L�3M, Jx½ � ¼ LMT�2, h½ � ¼ L and x½ � ¼ L. Accordingly, the functional

equation for the centerline velocity reads

u ¼ f r; Jxh; xð Þ (P.8.1)

Applying the Pi-theorem to (P.8.1), we obtain

P ¼ ’ðP1Þ (P.8.2)

where P ¼ um=ra1Ja2x ha3 and P1 ¼ x=ra01J

a02

x ha03 .

Taking into account the dimensions of um; r, Jx, h and x, we find the values of

the exponents ai and a0i as a1 ¼ �1=2, a2 ¼ 1=2, a

01 ¼ 0,a

02 ¼ 0, a

03 ¼ 1. Then we

obtain from (P.8.2)

umu0

� �h

d0

� �¼ ’

x

h

� �(P.8.3)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2x

h

u mh

1 2

u 0d 0

Fig. 8.23 The distribution of the normalized mean longitudinal velocity component

um=u0 h=d0ð Þ1=2along the jet center line. D�Re¼3 �104; h d¼100= , □�Re¼4:3 �104; h d= ¼40,

◊�Re¼5:6 �103; h d= ¼31,��Re¼5:6 �103; h d= ¼43:6, +�Re¼5:6�103; h d= ¼67:5. Reprintedfrom Gutmark et al. (1978) with permission

254 8 Turbulence

Note that in (P.8.3) accounts for the fact that Jx � ru20d20, where d0 is the

diameter of the nozzle.

P.8.2. Consider cavity formation at the free surface of a liquid pull by an

impinging axisymmetric turbulent gas jet. (1) Determine the dimensionless groups

of the flow, and (2) present the experimental data for the cavity depth and diameter

in an appropriate dimensionless form.

The flow under consideration is depicted in Fig. 8.24. An axisymmetric turbulent

gas jet is issued from a nozzle with the exit diameter dj. The jet is directed normally

to the unperturbed free surface of the pool. The distance between the nozzle exit and

the unperturbed surface in the liquid pool is h. The impinging jet causes a depres-

sion of the liquid surface. A sufficiently strong jet creates a visible cavity at the free

surface. Gas penetration into liquid is accompanied by the jet deceleration and

formation of the annular reverse gas flow. The liquid surface is deformed due to the

action of of the dynamic pressure and friction from the gas side as well as the cavity

shape is affected by liquid surface tension. In some cases the cavity surface is

unstable and gas bubble entrainment can take place there, the phenomenon which is

disregarded here. Thus, the cavity formation depends on several competing factors:

the physical properties of the gas and liquid phases, the initial jet diameter and

velocity of the jet, etc.

u0

h

Fig. 8.24 A cavity formed at

the surface of a liquid pool by

an impinging axisymmetric

gas jet

Problems 255

Assuming that the gas velocity distribution at the nozzle exit is uniform, we list

the governing parameters of the flow

rG L�3M� �

; rL L�3M� �

; mG L�1MT�1� �

; mL L�1MT�1� �

; (P.8.4)

s MT�2½ �, g LT�2½ �, uj LT�1½ �, dj L½ �, h L½ �where r and m denote density and viscosity, respectively, s is the surface tension,

g is the gravity acceleration, ujis the jet velocity at the nozzle exit, and subscripts

G and L refer to the gas and liquid, respectively.

Three of the nine governing parameters in (P.8.4) possess independent

dimensions. According to the Pi-theorem, the number of the dimensions groups

that determine the dimensionless characteristics of the cavity equals to six. These

are the following

rGL; h; ReG; ReL; Fr; We (P.8.5)

where rGL ¼ rG=rL, H ¼ h=dj. Also, ReG ¼ ujdj=nG, ReL ¼ ujdj=nL, Fr ¼ u2j =gdjand We ¼ djrGu

2j =s are the two Reynolds numbers, the Froude and Weber num-

bers, respectively, with n being the kinematic viscosity.

In the particular case where the Reynolds and Weber numbers are sufficiently

large and the viscous and surface tension effects are negligible, the number of the

dimensionless groups can be reduced significantly. At a large distance between

the nozzle exit and the unperturbed liquid surface (h >> djÞ the characteristics ofthe gas jet are mostly determined by its total momentum flux Jx ¼ rGu

2j d

2j p=4,

whereas the effect of the gas pressure at the cavity surface (which is determined by

the weight of the liquid displaced from the cavity) can be related to the specific

weight of the liquid g ¼ rLg L�2MT�2½ �. Then, the number of the governing

parameters reduces to three, namely, Jx, g and h, so that the functional equation

for the cavity depth hc becomes

hc ¼ f1 Jx; g; hð Þ (P.8.6)

In (P.8.6) the number of the governing parameters with independent dimensions

equals two. Accordingly, (P.8.6) reduces to the following dimensionless equation

P ¼ ’ðP1Þ (P.8.7)

where P ¼ hc=Ja1x ga2 and P1 ¼ h=J

a01

x ga02 .

Taking into account the dimensions of hc, h, g and Jx, we find that a1 ¼ a01 ¼ 1=2

and a2 ¼ a02 ¼ �1=3. Then (P.8.7) takes the form

hc

Jx=gð Þ1=3¼ ’

h

Jx=gð Þ1=3( )

(P.8.8)

256 8 Turbulence

or

hch¼ Jx

g

� �1=31

h’1

h

Jx=gð Þ1=3( )

¼ F1

Jxgh3

� �(P.8.9)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.5

1

1.5

2

2.5

3

3.5a

b

0 0.01 0.02 0.03 0.04 0.05Ix[lb]

d c[in

]h c

[in]

Jx[lb]

Fig. 8.25 Variation of the cavity diameter (a) and depth (b) with the distance between the nozzleand the unperturbed free surface (h) and the momentum flux of the jet. h in½ �: ~�2.0, +�3.0,

��4.0, D�5.0, □�5.0, ◊�7.0. Reprinted from Cheslak et al. (1969) with permission

Problems 257

Since the cavity diameter dc is related with its depth, it is possible to state the

functional equation for dc as follows

dc ¼ f2 Jx; g; hcð Þ (P.8.10)

Applying the Pi-theorem to (P.8.10), we arrive at

dchc

¼ F2

Jxgh3c

� �(P.8.11)

The experimental data on the cavity diameter and depth are presented in

Figs. 8.25a and 8.25b.

It is seen that an increase in h is accompanied by the growth of the cavity

diameter and a decrease in its depth. An increase in the total momentum flux of the

jet leads to an increase of the cavity depth and diameter. Equations (P.8.9) and

(P.8.11) show that the dimensionless cavity depth hc=h and diameter dc=hc are

functions of the dimensionless group Jx=gh3 and Jx=gh3c , respectively. Accordingly,one can expect that all the data points corresponding to different experimental

conditions should collapse at single curves hc=hð Þ Jx=gh3ð Þ and dc=hcð Þ Jx=gh3c �

in

the parametric planes hc=h versus Jx=gh3 and dc=hc versus Jx=gh3c . This result,

indeed, agrees with the data by Banks and Chandrasekhara (1963), as well as

with several other measurements.

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Theory of turbulent jets. Nauka, Moscow (in Russian)

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turbulent jet. J Fluid Mech 72:455–480

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flowing air stream. J Fluid Mech 61:805–822

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velocity gas jet through a liquid surface. J Fluid Mech 15:13–34

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versity Press, Cambridge

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layers on smooth and rough surfaces. Phys Fluids 13:3277–3284

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airstream. J Fluid Mech 27:381–394

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cross-stream. J Fluid Mech 62:41–64

258 8 Turbulence

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jets. J Fluid Mech 36:55–63

Chevray R, Tutu NK (1978) Intermittency and preferential transport of heat in a round jet. J Fluid

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Mech 306:111–144

Kolmogorov AN (1941a) Local structure of turbulence in incompressible viscous fluid for very

large Reynolds numbers. DAN SSSR 30(4):299–303, in Russian

Kolmogorov AN (1941b) Disperse energy at local isotropic turbulence. DAN SSSR 32(1):19–21

Landau LD, Lifshitz EM (1979) Fluid mechanics, 2nd edn. Pergamon, London

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helium. Part 2. Helium jet. J Fluid Mech 246:225–247

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Prandtl L (1942) Bemerkungen zur Theorie der freien Turbulenz. ZAMM 22:241–243

Prandtl L (1925b) Bericht uber Untersuchungen zur ausgebildeten Turbulenz. ZAMM 5:136–139

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momentum and heat in free turbulent jet of mercury. Izv AN Kaz, SSR, 22

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260 8 Turbulence

Chapter 9

Combustion Processes


Combustion presents itself complicated physicochemical process which proceeds

due to progressively self-accelerating exothermal chemical oxidation reactions

sustained by an intensive heat release. A strong dependence of the chemical

reaction rate on temperature according to the Arrhenius law determines a very

high sensitivity of combustion processes to small disturbances of the governing

parameters. It also determines an almost abrupt transition of reactive systems from

a low temperature state to a high temperature state which is associated with

ignition. The existence of a critical state corresponding to ignition, as well as the

ability of combustion oxidation reactions to sustain a self-propagating flame front

over reactive media represent themselves main features of combustion process.

Combustion of continuous gaseous media is described by the system of

equations including the Navier–Stokes, continuity, energy and species balance

equations

@r@t

þ rðv � rÞv ¼ �rPþrðmrvÞ (9.1)

@r@t

þr � ðrvÞ ¼ 0 (9.2)

r@h

@tþ rðv � rÞh ¼ rðkrTÞ þ qW (9.3)

r@cj@t

þ rðv � rÞcj ¼ rðrDrcjÞ �Wj (9.4)

where r, v, P, T, h and cj are the density, velocity vector, pressure, temperature,

enthalpy, and species concentrations, respectively, q is the heat release of the

combustion oxidation reaction (it is assumed that the whole complicated chemical



261

process can be reduced to a single equivalent reaction),W is the rate of the chemical

reaction, Wj is the rate of conversion of j� th species (positive for reagents, which

are consumed and negative for the reaction products which are produced), m; k andD are the viscosity, thermal conductivity and diffusivity, respectively. It is

emphasized that for simplicity all the transport coefficients and physical properties

of the parameters are assumed to be identical for all species involved.

The system of (9.1–9.4) should be supplemented by an equation of state of the

gas, a microkinetic law for the rate of chemical reaction and the correlations

determining the dependence of the physical properties on temperature. Solving

the highly nonlinear (9.1), (9.3) and (9.4) is extremely difficult. Therefore, the

analytical models of the combustion processes, as a rule, involve various

simplifications and approximations. Consider briefly some of them. First, we

discuss simplifications related to the chemical reaction rate W. For this aim, we

assume that: (1) the reactive mixture represents itself perfect gases, (2) the thermal

and species diffusivities are equal, and thermal conductivity and specific heat cP areinvariable, so that the enthalpy is expressed as h ¼ cPT, (3) a simple equivalent

single-step chemical reaction with the rate which can be factorized as a product of

two functions depending solely either on temperature or concentration

Wðc; TÞ ¼ ’ðcÞcðTÞ (9.5)

Moreover, we assume that there is only one limiting species in the system (fuel)

and, as a result, only one species (fuel) balance equation is to be considered. Its

concentration is denoted as c.Then, the energy and fuel balance equations read

@h

@tþ ðv � rÞh ¼ ar2hþ qWðc; TÞ (9.6)

@c

@tþ ðv � rÞc ¼ Dr2c�Wðc; TÞ (9.7)

where a and D are the thermal and mass diffusivity coefficients.

Assume that a ¼ D, i.e. the Lewis number Le ¼ a=D ¼ 1. Then, eliminating the

source terms from (9.6) and (9.7), we obtain the equation for the total (physical and

chemical) enthalpy

@H

@tþ ðv � rÞH ¼ ar2H (9.8)

where H ¼ hþ qc:For an isolated system @H=@nð ÞS ¼ 0; with S being the system envelop and n is

the normal to it. In this case (9.8) is integrated as

H ¼ const (9.9)

262 9 Combustion Processes

The constant in (9.9) is found from the condition that the maximum temperature

Tm corresponds to complete fuel consumption, i.e. to c ¼ 0: Accordingly, that theconstant is equal to cPTm, and thus the current fuel concentration is related to the

current temperature as

c ¼ cpðTm � TÞq

(9.10)

Therefore, the function ’ðcÞin (9.5) can be presented as

’ðcÞ ¼ ’cPðTm � TÞ

q

� �¼ ’�ðTÞ (9.11)

Then the reaction rate becomes

Wðc; TÞ ¼ ’�ðTÞcðTÞ ¼ WðTÞ (9.12)

Assuming the nth order reaction depending on temperature via the Arrhenius

law, i.e.Wðc; TÞ ¼ zcn exp �E=RTð Þ, we arrive at the following explicit expressionfor WðTÞ

WðTÞ ¼ zðTm � TÞn exp � E

RT

� �(9.13)

where z is pre-exponent, E is the activation energy and R is the universal gas

constant.

It is emphasized that (9.12) and (9.13) are valid only when the Lewis number

Le ¼ 1. In the general case when Le 6¼ 1; the distribution of the total enthalpy

within a reactive medium is not given by a constant according to (9.9) but is

more complicated. In combustion of gaseous mixtures, which is a particular case

of homogeneous combustion), H has extrema in the vicinity of the flame front. It

was shown in Zel’dovich et al. (1985) that this is the result of the energy redistri-

bution due to different rates of heat and mass transfer at Le 6¼ 1:An additional simplification that is widely use in the theory of thermal explosion

and ignition is related to the Frank-Kamenetskii transformation of the exponent in

the Arrhenius law. Following Frank-Kamenetskii (1969), we present the ratio

E=RTas

E

RT¼ E

RðT� þ DTÞ ¼E

RT�

1

1þ DT=T�ð Þ �E

RT�� E

RT2�DT (9.14)

where as the temperature T� is close to the temperature at which the chemical

reaction proceeds: to the initial temperature T0 in the problem of self-ignition, or to

the maximum temperature Tmin the problem of combustion wave propagation;

D ¼ T � T�:


According to (9.14), the exponent in the Arrhenius law takes the form

e�ERT � e�

ERT�ey (9.15)

where y ¼ E=RT2�

� �ðT � T�Þ.The expression (9.15) is an accurate approximation of the Arrhenius law at

temperatures near to T�, albeit without linearization. The latter feature is important,

since such phenomena as ignition, extinction or thermal explosions are basically

nonlinear and to be able to address them, nonlinear (even though simplified)

nature of the problem should be preserved (Frank-Kamenetskii 1969; Zel’dovich

et al. 1985).

An additional significant simplification of the problems related to combustion of

non-premixed gases can be achieved by means of the Schvab–Zel’dovich trans-

formation (Schvab 1948; Zel’dovich 1948). It allows removal of the non-linear

term WðTÞ from the all but one species balance equations and transforms them to

the form identical to the form for inert gases. In order to illustrate this approach, we

consider application of the Schvab–Zel’dovich transformation to the species

equations in the case of reaction between a single fuels and a single oxidizer

r@ca@t

þ rðv � rÞca ¼ rDr2ca �Wa (9.16)

r@cb@t

þ rðv � rÞcb ¼ rDr2cb �Wb (9.17)

where subscripts aand b refer to fuel and oxidizer, respectively.

Taking into account that the reaction ratesWa andWb are related by the stoichio-

metric relation dictated by the corresponding chemical reaction

Wa ¼ Wb

O(9.18)

with O being the stoichiometric coefficient, we transform the system of equations

(9.16) and (9.17) in the following way. We divide (9.17) by O and subtract (9.17)

from (9.16). As a result, we arrive at the following equation

r@b@t

þ rðv � rÞb ¼ rDr2b (9.19)

where b ¼ ca � cb=O.Equation (9.19) has the same form as the species balance equation the inert flows

without any chemical reactions. This equation admits solutions which differ up to a

constant, i.e. b0 ¼ bþ const: The latter allows one to reduce the boundary

conditions for axisymmetric reactive flows (submerged torches) to the form


y ¼ 0@b0

@y¼ 0; y ! 1 b0 ! 0 (9.20)

where is the radial coordinate.

9.2 Thermal Explosion

Thermal explosions corresponds to unstable states of reactive system at which any

initial temperature distribution leads to a disruption of the thermal equilibrium

resulting from a runaway rate of the exothermal chemical reaction with heat release

higher than the rate of heat removal to the environment. The latter leads to

progressive temperature growth and an abrupt transition from the initial low

temperature state to the final stable high temperature state.

In order to illustrate the nature of thermal explosions, we consider the thermal

regime of combustion of an ideally stirred reactor (for example, a jet-stirred reactor,

JSR) which represents itself a closed volume filled with a homogeneous reactive

mixture of fuel and oxidizer. The wall temperature of JSR is assumed to be fixed and

equal to a low temperature T0 imposed by the surroundingmedium. The temperature

field inside JSR is assumed to be uniform, which corresponds to an infinite rate of

mixing. Changes in the reactant concentrations in time are assumed to be negligible

during the extremely short period of time preceding thermal explosion. Then, the

specific rate of heat release by chemical reaction QI and the rate of heat removal to

the environment QII (both divided by the reactor volume V) read

QI ¼ qW ¼ const� exp � E

RT

� �(9.21)

QII ¼ hS

VðT � T0Þ (9.22)

where q is heat of reaction,W ¼ zceacsb exp �E=RTð Þ is the rate of chemical reaction

(without simplification by means of the Frank-Kamenetskii transformation), z is thepre-exponent, ca and cb are reactant concentrations (fuel and oxidizer, respec-

tively), e and s are constant (they denote the reaction orders in fuel and oxidizer,

respectively), h is the heat transfer coefficient at the outer wall of the reactor, S andVare the surface area and volume of JSR.

The curves of heat release QI and heat removal QII are plotted versus tempera-

ture T in Fig. 9.1. In the general case there are three intersection points which

correspond to the low (1), intermediate (2) and high temperature (3) states. It is easy

to see that intermediate state (2) is unstable whereas the states (1) and (3) are stable.

Indeed, any perturbation increasing the mixture temperature relative to that

corresponding to point 2 (T > T2Þis accompanied by an excess of the heat release

9.2 Thermal Explosion 265

rate over the intensity of the heat removal rate. As a result, temperature T should

keep increasing and thus, the intermediate point 2 is unstable. Also, if due to

a perturbation the temperature decreases (T < T2), the heat removal rate exceeds

the heat release rate. As a result, temperature T keeps decreasing and once more it is

seen that point 2 is unstable. On the other hand, similar arguments show that points

1 and 3 are stable. Among the possible mutual locations of curves QIðTÞ and

QIIðTÞthere are two particular ones where they are tangent to each other. These

two cases signify the two critical states: (1) the transition from the low- to high-

temperature state, which corresponds to the mixture ignition-point I in Fig. 9.1, and

(2) the transition from the high- to low-temperature state, which corresponds the

mixture extinction-point E in Fig. 9.1. Therefore, the thermal explosion

corresponds to the ignition condition, at which a low-temperature stationary state

of a reactive system becomes impossible. This is an outline of the non-stationary

theory of thermal explosion elaborated by Semenov (1935). On the other hand, the

stationary theory of thermal explosion developed by Frank-Kamenetskii (1969)

treats thermal explosion as the situation at which no stationary solution of the

thermal balance equation can be found. A detailed exposition of both theories of

thermal explosion can be found in the monograph by Zel’dovich et al. (1985);

see also Frank-Kamenetskii (1969) and Vulis (1961). Referring the interested

readers to these monographs, we restrict our consideration to the applications of

the Pi-theorem to the stationary problem of thermal explosion.

Consider stationary temperature distribution in a symmetric reactor with charac-

teristic size r0 and wall temperature T0:Assuming as before that changes in reactant

concentrations are negligible, we determine the governing parameters of the

1

I

QII

2

3

QIE

To T

Q

0

Fig. 9.1 The curves

corresponding to heat release

QI, and heat removal QII . The

lower, intermediate and upper

intersection points 1, 2 and 3,

respectively, correspond to

the stable lower temperature

state, intermediate (unstable)

state, and the stable high

temperature state


problem. From the physical point of view the local temperature inside a reactor

depends on the thermal conductivity of gas mixture k½ � ¼ JL�1T�1y�1, kinetic

factors in the Arrhenius law z exp �E=RTð Þ; namely on z½ � ¼ T�1; E½ � ¼Jmol�1 and R½ � ¼ Jy�1mol�1; the heat of reaction q½ � ¼ JL�3; the reactor size

r0½ � ¼ L; the wall temperature T0½ � ¼ y and the coordinate r L½ � of the point under

consideration

T ¼ f ðr0; r; T0; k; z;E;R; qÞ (9.23)

The temperature distribution defined by (9.23) satisfies the following conditions

T ¼ T0 at r ¼ r0;dT

dr¼ 0 at r ¼ 0 (9.24)

Equation (9.23) and the boundary conditions (9.24) contain eight dimensional

parameters including five parameters with independent dimensions. Then,

according to the Pi-theorem, (9.23) reduces to the following dimensionless equation

b ¼ ’ðx; g; b0Þ (9.25)

where b ¼ RT=E; b0 ¼ RT0=E; g ¼ qzRr20=kE and x ¼ r=r0.Equation (9.25) shows that using the Pi-theorem, it was possible to decrease the

number of the governing parameters from eight to three. However, even in this case

the study of the critical state which corresponds to thermal explosion is highly

complicated. In this situation, similarly to the analytical solutions of the combustion

theory discussed above, it is useful to employ a physically-based simplification,

namely the Frank-Kamenetskii transformation. Since the ignition (or thermal

explosion) process occurs at temperatures close to the wall temperature T0, simi-

larly to (9.15) we have

ze�ERT � ze

� ERT0e

E

RT20

ðT�T0Þ(9.26)

Then, (9.23) can be replaced by the following equation

DT ¼ f ðr; r0; k;ez; T0� ; qÞ (9.27)

where ez ¼ z exp �E=RT0ð Þ; T0� ¼ RT2

0=E and DT ¼ T � T0:Applying the Pi-theorem to the simplified (9.27), we arrive at the following

simpler dimensionless equation

# ¼ cðx; dÞ (9.28)

where # ¼ EðT � T0Þ=RT20 and d ¼ qEzr20=kRT

20

� �exp �E=RTð Þ is the Frank-

Kamenetskii parameter, which is the sole dimensionless constant on the right-

hand side in (9.28).

9.2 Thermal Explosion 267

The critical value of dcorresponding to the ignition (or thermal explosion) can be

found only by solving the thermal balance equation

r2x# ¼ �d expð#Þ (9.29)

subjected to the boundary conditions

# ¼ 0 at x ¼ 1;d#

dx¼ 0 at x ¼ 0 (9.30)

The corresponding critical condition found from (9.29) and (9.30) when the

stationary solution becomes impossible reads

d ¼ dcr ¼ const (9.31)

where dcr ¼ 0:88 for plane reactors and 0.33 for the spherical ones.

9.3 Combustion Waves

The present section is devoted to a simple estimate of the speed of combustion wave

that propagate in homogeneous infinite reactive media. The analysis is based on the

approach of the thermal theory of combustion that imply that combustion wave

propagation is a result of heat transfer from a high temperature reaction zone to

a relatively cold fresh mixture of fuel and oxidizer due to thermal conductivity. This

theory was developed by Zel’dovich (cf. Zel’dovich et al. 1985) and Frank-

Kamenetskii (1969). The thermal theory of combustion accounts for the main

features of the process, namely, the sharp dependence of the chemical reaction

rate on temperature, the intensive heat release within a thin reaction front, as well as

for the heat and mass transfer due to molecular thermal conductivity and molecular

diffusion. According to this theory, the mechanism of combustion wave propaga-

tion in homogeneous mixtures is the following. An instantaneous heating of a thin

layer of a preliminarily cold reactive mixture by an external source triggers

chemical reaction within the heated layer (Fig. 9.2). The heat released by the

exothermal chemical reaction, in its turn, leads to a further heating of the mixture

in this layer and its ignition under certain conditions. Heat transfer from the high

temperature zone to the cold mixture ensures heating and ignition of the neighbor-

ing layers, i.e. propagation of a self-sustained chemical reaction zone (the flame

front, or combustion wave) over reactive medium. At the transient stage of the

combustion wave propagation, the process develops under the conditions of a

continuous variation of the temperature and concentration fields. Also, the speed

of the combustion wave varies until its value will not approach the one

corresponding to the stationary regime of combustion.


In the framework of the thermal theory there are two main factors that determine

the speed of combustion wave in homogeneous reactive mixtures: (1) the exother-

mal chemical reaction accompanied by an intense heat release, and (2) the heat

transfer from the high-temperature reaction zone to the cold fresh mixture by

thermal conductivity. Under these conditions the governing parameters of the

process are as follows: the mixture density r½ � ¼ L�3M; thermal conductivity k½ � ¼LMT�3y�1; specific heat cP½ � ¼ L2T�2y�1, and the characteristic time of chemical

reaction tm½ � ¼ T determined by the maximal temperature. In addition, it is

assumed here that the Lewis number Le ¼ 1. Then, the thermal diffusivity a ¼ k/rcP ¼ D, and thus, the diffusion coefficient D should not be included separately in

the set of the governing parameters. Accordingly, the functional equation for the

speed of combustion wave uf� ¼ LT�1 has the form

uf ¼ f ðr; k; cP; tmÞ (9.32)

All the governing parameters in (9.32) have independent dimensions. Then,

according to the Pi-theorem, (9.32) transforms to

uf ¼ cra1ca2p ka3ta4 (9.33)


Using the principle of the dimensional homogeneity and accounting for the

dimensions of uf ; r; cP; k and tm, we arrive at the following system of equations

for the exponents ai

�3a1 þ 2a2 þ a3 � 1 ¼ 0

a1 þ a2 ¼ 0

�2a1 � 3a3 þ a4 þ 1 ¼ 0

a2 þ a3 ¼ 0

(9.34)

Fig. 9.2 The structure of a

combustion wave at a certain

moment of time. The wave is

propagating from right to left.I-heating zone, II-reaction

zone, III-high temperature

zone

9.3 Combustion Waves 269

Equations (9.34) yield

a1 ¼ � 1

2; a2 ¼ � 1

2; a3 ¼ 1

2; a4 ¼ � 1

2(9.35)

and thus

uf ¼ c

ffiffiffiffiffiatm

r(9.36)

The value of the dimensional constant c depends on the other dimensionless

groups of the problem. They are E=RT0 and E=RTm (Frank-Kamenetskii 1969). On

the other hand, when the Frank-Kamenetskii transformation (9.15) is employed to

simplify the Arrhenius law, the constant c depends a single dimensionless group

ym ¼ EðTm � T0Þ=RT2m combining the two previously mentioned groups.

Using (9.36) it is possible to estimate the effect of pressure on the speed of

combustion wave propagation. Assuming that c is a weak function of the dimension-

less groups E=RT0 and E=RTm, we see that uf �ffiffiffiffiffiffiffiffiffiffia=tm

p: The thermal diffusivity of

gases is inversely proportional to pressure, a � P�1. Based on the chemical kinetics

data, one can also expect that t�1m � rn exp �E=RTmð Þ � Pn exp �E=RTmð Þ; were

n is the reaction order. As a result, we obtain the flame speed as

uf � Pn�12 exp � E

2RTm

� �(9.37)

It is seen that the flame speed uf does not depend on pressure in the case of a firstorder reaction. On the other hand, in the case of a second order reaction the flame

speed uf is proportional toP1=2:

As was noted before, a detailed form of the dependence of the combustion wave

speed on the physicochemical and kinetic parameters can be found by solving the

energy and diffusion equations. At Le ¼ 1when the profiles of fuel concentration

and the normalized temperature are similar to each other, the problem reduces to the

integration of the energy equation. In the frame of reference associated with the

moving combustion front (flame) the equation reads

ruf cPdT

dx¼ d

dxkdT

dx

� �þ qWðTÞ (9.38)

In an infinite premixed mixture of fuel and oxidizer the boundary conditions for

(9.38) have the form

x ¼ �1; T ¼ T0; x ¼ þ1; T ¼ Tm (9.39)


It is easy to see that if T(x) is a solution of the problem (9.38) and (9.39), then

T(x þ c) (with c being an arbitrary constant) is also a solution. The latter means that

the constant c is undetermined in principle, and one of the boundary conditions

(9.39) becomes redundant. However, (9.38) contains a still unknown flame speed uf,which shows that a seemingly redundant boundary condition should be used to

find uf, i.e. the flame speed uf represents itself an eigenvalue of the problem (9.38)

and (9.39). A comprehensive discussion of the analytical solutions for the speed

of combustion waves in homogeneous mixtures can be found in the following

monographs and surveys: Frank-Kamenetskii (1969), Williams (1985), Zel’dovich

et al. (1985), Merzhanov and Khaikin (1992). Numerical solutions of this problem

were discussed in Spalding (1953), Zel’dovich et al. (1985) and Merzhanov

et al. (1969).

9.4 Combustion of Non-premixed Gases

Consider combustion of non-premixed gases in an adiabatic cylindrical chamber

(Fig. 9.3). The gaseous reactants are supplied through a core tube of cross-sectional

radius r1 (fuel) and an annular gap of thickness r2 � r1 (oxidizer). It is assumed

that the velocity distribution at any cross-section of the combustion chamber

(burner) is uniform, i.e. fuel and oxidizer are issued with the same speed and the

effect of viscous friction at the wall is negligible. The mass flux does not change

downstream in the chamber. In addition, it is assumed that the diffusion transfer in

radial direction is much large than in the longitudinal one. Regarding the rate of

chemical reaction at the flame front, it is assumed that it is infinite, and therefore,

concentrations of fuel and oxidizer at the flame front are zero. The assumptions

made follow those in the seminal work of Burke and Schumann (1928), as well as

the detailed analysis of the corresponding problem is covered in the monographs

by Vulis (1961), Williams (1985), Zel’dovich (1948) and Zel’dovich et al. (1985).

Below we discuss briefly the formulation of this problem and concentrate of the

application of the dimensional analysis to in this particular case.

r

x

r2r1Fuel

Oxidizer

Oxidizer

2

1Fig. 9.3 Sketch of a non-

premixed gas burner

9.4 Combustion of Non-premixed Gases 271

The above assumptions and the Schvab-Zel’dovich transformation allow us to

reduce the species balance equations to the following single equation we write the

governing equation in the form

ru@b@x

¼ rD1

r

@

@rr@b@r

� �(9.40)

[cf. (9.19)] where b ¼ ca � cb=O; ca and cb are the concentration of fuel and

oxidizer, respectively, O is the stoichiometric oxidizer-to-fuel mass ratio, ru is

the mass flow rate and rD the product of density and diffusion coefficient; both ruand rD are constant in the present case.

The boundary conditions for (9.40) read

x ¼ 0 : 0 r r1 b ¼ ca0; r1 < r < r2 b ¼ � cb0O

x > 0 : r ¼ 0@b@r

¼ 0; r ¼ r2@b@r

¼ 0

(9.41)

The conditions at x ¼ 0 determine the uniform distribution of fuel and oxidizer

at the burner inlet; the boundary conditions at x> 0 determine the flow symmetry

and correspond to the absence of the chemical reaction at the wall.

The solution of (9.40) with the boundary conditions (9.41) is (Zel’dovich

et al. 1985)

bðr; xÞ ¼ b�X1i¼1

CIJ0ðr’i=r2Þ expð�SI � xÞ (9.42)

The following notation is used in (9.42): b ¼ ca0 � ca0Oþ cb0ð Þ=O½ � r1=r2ð Þ2;Si ¼ rD=ruð Þð’2

i =r22Þ, Ci ¼ 2 cb0=Oð Þð1þ wÞ r1=r2ð ÞJ1ðr1’i r2Þ= = ’i J0ð’iÞ½ �2

n o;

w ¼ ca0O=cb0ð Þ; J0 ð Þ and J1 ð Þ are the Bessel functions of the first kind of zero

and first orders, and ’i are the roots of the equation J1ð’Þ ¼ 0: Concentrations ca0and cb0 correspond to fuel and gas at the burner entrance at x ¼ 0:

The corresponding approximate expression for the flame length xf ¼ lf is

(Zel’dovich et al. 1985)

lf ¼ ur22D’2

1

ln2ð1þ wÞ r1=r2ð Þ2J1 r1’1 r2=ð Þ

’1 J0ð’1Þ½ �2 w� ð1þ wÞ r1=r2ð Þ2h i (9.43)

where ’1 ¼ 3:83 and J0ð’1Þ ¼ �0:4:The expression (9.43) corresponds to combustion at the excess of oxidizer and

the whole fuel is consumed at a finite length xfwhich corresponds to the tip of curve1 in Fig. 9.3. In the opposite case when combustion proceeds at the lack of oxidizer,

the term J0ð’1Þ½ �2 in the denominator of (9.43) should be replaced by J0ð’1Þ½ �.


Then, (9.43) describes the distance at which all oxidizer will be fully consumed,

which corresponds to the right-hand side end of curve 2 in Fig. 9.3.

Consider the Burke-Schumann problem in the framework of the dimensional

analysis. The problem formulation reveals that at Le ¼ 1 when the thermal and

mass diffusivities are equal to each other, the field of the compound concentration

b in coaxial burner is determined by nine parameters

b ¼ f ðu;D; r; x; r1; r2; ca0; cb0;OÞ (9.44)

These governing parameters have the following dimensions

u½ � ¼ LxT�1; D½ � ¼ L2yT

�1; r½ � ¼ Ly; x½ � ¼ Lx; r1½ � ¼ Ly; r2½ � ¼ Ly;

ca0½ � ¼ 1; cb0½ � ¼ 1; O½ � ¼ 1(9.45)

Among of the six dimensional parameters in (9.45), three parameters have

independent dimensions. Therefore, it is possible to form three dimensionless

groups

r1r2;

r

r2;

xD

ur22(9.46)

Then (9.44) reduces to the following dimensionless form

b ¼ ’r

r2;xD

ur22;r1r2; ca0; cb0;O

� �(9.47)

As it was mentioned above, the concentrations of reactants at the combustion

front at xf ¼ xf ðrf Þ are equal to zero, so that b ¼ 0 there. Then, (9.47) yields

’rfr2

xfD

ur22;r1r2; ca0; cb0;O

� �¼ 0 (9.48)

Solving (9.48) relative to the dimensionless group xfD=ur22, we obtain

xfD

ur22¼ c

rfr2;r1r2; ca0; cb0;O

� �(9.49)

Equation (9.49) determines geometry of the diffusion flame of non-premixed

reagents. It contains four constants r1=r2; ca0; cb0 and O that account for the burner

geometry, as well as the characteristics of reactive system. The dependence

xfD=ur22 ¼ c rf =r2

� �found from the exact solution (9.43) is shown in Fig. 9.4. As

discussed, the shape of the diffusion flame depends on a relation between w ¼ca0O=cb0ð Þ and r1=r2ð Þ= 1� r1=r2ð Þ2

h iwhich determines where the system has

9.4 Combustion of Non-premixed Gases 273

either an excess of oxidizer or fuel. In the first case the flame tip is located at the

flow axis, whereas in the second one at the wall of the burner. Assuming in (9.49)

rf ¼ 0, we obtain the following expression for the diffusion flame length xf ¼ lf

lf ¼ Pe� cr1r2; ca0; cb0;O

� �(9.50)

where Pe ¼ ur2=D is the Peclet number and lf ¼ l=r2.Equation (9.50) shows that with r1 r2= ; ca0; cb0 and O being constant, the flame

length lf ¼ lf r2= � Pe i.e. lf � ur22 D= . The volumetric flow rate of the gaseous

phase is Gv � ur2e for the planar flame, and Gv � ur22 for the axisymmetric flame

(e ¼ 1is the unit of length). Then, we arrive at the conclusion that lf ;planar � Gvr2 D=and lf ;axisymm � Gv D= , i.e. the length of the axisymmetric flame does not depend on

the radius of combustion chamber, whereas the length of a planar flame is directly

proportional to r2 whenGv ¼ const. Note, that in the planar case the difference r2 �h is equal to the semi-height of the channel (Vulis 1961).

9.5 Diffusion Flame in the Mixing Layer of Parallel Streams

of Gaseous Fuel and Oxidizer

The flow and flame structure under consideration are sketched in Fig. 9.5. Two

uniform streams of gaseous non-premixed reactants moving over both sides of a

semi-infinite plate which ends at x ¼ 0 come in contact to each other. The fuel

1

2

0.1

0.50

xf D

ur22

rfr2

Fig. 9.4 Configurations of

the diffusion flame of non-

premixed gases. 1: The case

of the excess of oxidizer. 2:

The case of the excess of fuel


stream is supplied at y < 0, whereas the oxidizer-at y > 0. The ignition takes place

at the line x ¼ 0; y ¼ 0: When the rate of chemical reaction is large enough,

conversion of reactants into combustion products occurs within a thin reaction

zone that can be considered practically infinitesimally thin and viewed as the

flame front (Zel’dovich et al. 1985); cf. Fig. 9.5. Then, domain I in Fig. 9.5 is filled

with the oxidizer and fully converted combustion products, whereas domain II is

filled with fuel and combustion products. Chemical reaction takes place neither in

domain I nor in domain II but solely at the flame front. On the other hand, pure

mixing takes place in domains I and II.

In the framework of this model, and assuming the low Mach number (M << 1)

and h ¼ cpT(cP ¼ const), the velocity, temperature and fuel and oxidizer concen-

tration fields are determined by the following equations

ru@u

@xþ rv

@u

@y¼ @

@ym@u

@y

� �(9.51)

@ru@x

þ @rv@y

¼ 0 (9.52)

rucP@T

@xþ rvcP

@T

@y¼ @

@yk@T

@y

� �(9.53)

Fig. 9.5 Diffusion flame in the mixing layer of parallel streams of gaseous and oxidizer

9.5 Diffusion Flame in the Mixing Layer of Parallel Streams 275

ru@cj@x

þ rv@cj@y

¼ @

@yrDj

@cj@y

� �(9.54)

where subscript j corresponds to the j� th species (j ¼ a for fuel, and j ¼ b for

oxidizer).

The system of (9.51–9.54) is supplemented by the equation of state (9.55)

accounting for the fact that pressure is constant in the mixing layer, as well as the

dependences of the physical parameters on temperature (9.56)

rT ¼ const (9.55)

mðTÞ; kðTÞ; rDjðTÞ (9.56)

The boundary conditions at x > 0 for (9.51–9.54) read

y ! þ1; u ! uþ1; T ! Tþ1; ca ! caþ1 (9.57)

y ! �1; u ! u�1; T ! T�1; cb ! cb�1

At the flame front y ¼ yf ðxÞ the reactant concentrations are zero, since the

reaction rate is practically infinite, whereas the diffusion fluxes of fuel and oxidizer

are in stoichiometric ratio

T ¼ Tf ; ca ¼ cb ¼ 0 (9.58)

� Db @cb=@nð ÞfDa @ca=@nð Þf

¼ O (9.59)

where Tf is the flame temperature which is equal to the adiabatic temperature of

combustion of non-premixed fuel and oxidizer, O is the stoichiometric coefficient

and @=@n is the derivative along the normal to the flame front. It is emphasized that

the boundary condition (9.59) allows one to determine the location of the flame front.

The problem we are dealing with in the present section represents itself a

compressible flow. In such cases the Dorodnitsyn–Illingworth–Stewartson trans-

formation discussed previously in Sect. 7.7 of Chap. 7 allows one to reduce

compressible problems to the corresponding incompressible ones. In particular,

introducing new the variables x ¼ x and � ¼ Ry0

rdy and assuming that the

dependences mðTÞ; kðTÞ and rDðTÞ are linear, it is possible to reduce (9.51–9.54)

to the form identical to the incompressible equations corresponding to the same

flow geometry but with r ¼ const: The dimensional analysis of these system of

equations shows that there exists the self-similar solution of the dynamic, thermal

and species balance equations in the following form

u ¼ F0ð’Þ; DT ¼ yð’Þ; cj ¼ #ð’Þ (9.60)


with u ¼ 2u= uþ1 þ u�1ð Þ; DTa ¼ T � Tþ1� �

= Tf � Tþ1� �

; DTb ¼ T � T�1ð Þ=

Tf � T�1� �

; ’ ¼ �xg; � ¼ RY

0

rdy; � ¼ x; y ¼ y=l�ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiReð1þ mÞ½ �=2p

; x ¼x=l�; ra ¼ ra=raþ1; rb ¼ rb=rb�1; m ¼ u�1=uþ1; l� is an arbitrary

length scale, x and � are the Dorodnitsy n variables, and g is a constant.The functions Fj

0 ð’Þ; yjð’Þ and #jð’Þ are determined by the following ODEs

Fj000 þ 1

2Fj

00Fj ¼ 0 (9.61)

yj00 þ Pr

2yj

0Fj ¼ 0 (9.62)

#j00 þ Sc

2#j

0Fj ¼ 0 (9.63)

[subscript j ¼ a or b], with the boundary conditions

Fa0 ¼ 2

1þ m; ya ¼ 0; #a ¼ 1 at ’ ! þ1

Fb0 ¼ m

1þ m; yb ¼ 0; #b ¼ 1 at ’ ! �1

ya;b ¼ 1; #a;b ¼ 0 at ’ ¼ ’f

(9.64)

The integration of (9.61–9.63) with the boundary conditions (9.64) leads to the

following expressions for the velocity, temperature and concentration distributions

in the mixing layer

u

uþ1¼ 1

21þ mð Þ þ 1� mð Þerf ð’Þf g (9.65)

ya ¼ 1� erf ð’ ffiffiffiffiffiPr

p Þ1� erf ð’f

ffiffiffiffiffiPr

p Þ ; #a ¼ 1� 1� erf ð’ ffiffiffiffiffiSc

p Þ1� erf ð’f

ffiffiffiffiffiSc

p Þ (9.66)

for ’f < ’ <þ1; and

yb ¼ 1þ erf ð’ ffiffiffiffiffiPr

p Þ1þ erf ð’f

ffiffiffiffiffiPr

p Þ ; #b ¼ 1� 1þ erf ð’ ffiffiffiffiffiSc

p Þ1þ erf ð’f

ffiffiffiffiffiffiffiScÞp (9.67)

for �1 < ’ < ’f :To find the location of the flame front, we take into account the following

evaluations valid for the boundary layer in which the flame front slope relative to

the x-axis is small enough, so that cosðn; xÞ � 0 and cosðn; yÞ � 1: For example, at


O ¼ 15; Pr ¼ 1 and Rex ¼ 100� 1000; cosðn; yÞ � 0:99� 0:998. Under these

conditions @=@n ¼ @=@x � cosðn; xÞ þ @=@u � cosðn; yÞ � d=dy. Calculating the

derivatives @ca=@nð Þf and @cb=@nð Þf by using the expressions (9.66) and (9.67)

and substituting them into (9.59), we arrive at the following expression for the

coordinate of the flame front ’f

erf ð’f

ffiffiffiffiffiSc

pÞ ¼ 1� e

1þ e(9.68)

where e ¼ caþ1=cb�1ð Þ Da=Dbð ÞO�1:The results presented above are related to the aerodynamics of non-premixed

diffusion flames at small speed of fluid and oxidizer. Now we consider some

features of the non-premixed diffusion flames in high velocity flows where the

energy dissipation significantly affects the flame characteristics, such as, for

example, the flame front temperature. We will consider diffusion combustion of

non-premixed gases in the boundary layer formed when a high speed uniform semi-

infinite gaseous fuel flow comes in contact with a semi-space filled with gaseous

oxidizer at rest. Mixing of the gaseous fuel with gaseous oxidizer begins at cross-

section x ¼ 0: The ignition of the reactive mixture occurs by an external source

located at point x ¼ 0; y ¼ 0: As a result of the ignition, the reactive mixture in the

boundary layer forms a thin reaction zone that can be presented as an infinitely thin

flame front. The system of the governing equations describing velocity, enthalpy

and species concentration distribution in diffusion combustion of non-premixed

gases in high speed flows takes the following form (in the boundary layer approxi-

mation after the Dorodnitsyn–Illingworth–Stewartson transformation has been

applied; cf. Sect. 7.7, Chap. 7)

u@u

@xþ ev @v

@�¼ 1

Re

@2u

@�2(9.69)

u@h

@xþ ev @h

@�¼ 1

Pr

@2h

@�2þ g� 1ð ÞM2

þ1@u

@�

� �2

(9.70)

u@cj

@xþ ev @cj

@�¼ 1

Sc

@cf

@�2(9.71)

@u

@xþ @ev@�

¼ 0 (9.72)

where u and v are component of the velocity, cj is the concentration

u ¼ u=uþ1; v ¼ v=uþ1ð Þ ffiffiffiffiffiPr

p; ev ¼ rvþ u@�=@x; h ¼ h=hþ1; h ¼ cPT;

ðcP ¼ constÞ r ¼ r=rþ1; x ¼ x=l�; � ¼ �=l�; x and � are the Dorodnitsy

n variables, Re and Mþ1 are the Reynolds and Mach numbers, respectively.


The boundary conditions corresponding to the case of gaseous fuel issuing into

the oxidizer at rest (u�1 ¼ 0) are posed at the both edges of the mixing layer and

the flame front. They are as following

u ¼ 1; h ¼ 1; ca ¼ 1;@u

@�¼ 0 at � ! þ1

u ¼ 0; h ¼ h�1; cb ¼ 1 at � ¼ �1ca ¼ 0; cb ¼ 0 at � ¼ �f

(9.73)

The conditions (9.73) should be also added to the boundary conditions (9.59) to

determine the location of the flame front. Also, the boundary condition describing

the thermal balance at the flame front is needed. The latter is necessary to determine

the combustion temperature, since in high velocity flow it depends not only on the

heat of reaction but also on the heating due to the energy dissipation. The boundary

condition which expresses the thermal balance at the flame front reads

qrf Daf@ca@n

� �f

þ kf@T

@n

� �f

¼ kf@T

@n

� �f

(9.74)

where q is the heat reaction.

The dimensional analysis shows that there exists a self-similar solution of

(9.69–9.72) subjected to all above-mentioned boundary conditions. The self-similar

solution allows us to reduce the system of the partial differential equations of

the problem to the corresponding system of ODEs for the functions F ’ð Þ), yð’Þand #ð’Þ

Fj000 þ 2FjFj

00 ¼ 0 (9.75)

yj00 þ 2 PrFjyj

0 þ Pr g� 1ð ÞM2þ1 Fj

00� �2 ¼ 0 (9.76)

#j00 þ 2ScFj#j

0 ¼ 0 (9.77)

The boundary conditions for (9.75–9.77) read

F0a ¼ 1; ya ¼ 1; #a ¼ 1;F

00a ¼ 0 at c ! þ1

Fb0 ¼ 0; yb ¼ h�1; #b ¼ 1 at c ! �1y ¼ yf ; #a ¼ #b ¼ 0 at c ¼ cf

(9.78)

where F0j ¼ uj uþ1= ; yj ¼ hj hþ1= ; #j ¼ cj cj�1

;c ¼ � 2

ffiffiffix

p ; x and � are the

Dorodnitsyn variables, and Mþ1 is the Mach number of the undisturbed flow.

The boundary conditions (9.78) should be supplemented by the balance relations

(9.59) and (9.74) that determine the position of the flame front, as well as its


temperature. The first of the latter is determined (as in the flow with small velocity)

by (9.68). However, in high velocity flows the temperature of the flame front

depends not only on the physicochemical properties of the reactants but also on

the velocity of the undisturbed flow. The transformation of (9.74) leads to the

following relation for the flame temperature

TfTþ1

¼ 1þ qcaþ1cPTþ1

1

1þ eþ g� 1

2M2

þ1e

ð1þ eÞ2 (9.79)

where g ¼ cp cv= is the ratio of the specific heats at constant pressure and volume,

respectively.

The solution of (9.75–9.77) with the boundary conditions (9.58) that determine

the velocity, enthalpy and reactant and combustion products concentration fields, as

well as the configuration of the flame front and its temperature was found by Vulis

et al. (1968). A similar approach can be also used to study combustion of liquid fuel

in a stream of gaseous oxidizer which blows over its surface, for example, the

combustion of large oil spots (Yarin and Sukhov 1987). Solution of the latter

problem is very similar to the one described above, albeit it involves additional

thermal and mass balance conditions, which are required for calculation of the

temperature and vapor concentration at the free surface.

9.6 Gas Torches

Gas torches represent themselves submerged jets in which the intensive exothermal

chemical oxidation reaction (combustion) proceeds. The conversion of the initial

reactants into combustion products occurs in such jets within a thin high tempera-

ture zone that is identified with the flame front. The thickness of this zone can be

estimated by the dimension consideration. Since the combustion process is deter-

mined by the two general factors, (1) kinetics of chemical reactions and (2)

diffusion, we can assume that the thickness of the reaction zone depends on the

rate constant Z½ � ¼ T�1 of the chemical reaction and diffusivity D½ � ¼ L2T�1

d ¼ f ðZ;DÞ (9.80)

where Z ¼ Z0 exp �E=RTð Þ is the Arrhenius factor, k0 is the pre-exponential, E and

R are the activation energy and the universal gas constant, respectively.

According to the Pi-theorem, because k and D have independent dimensions,


d ¼ c � Za1Da2 (9.81)



Taking into account the dimensions of d; Z and D and applying the principle of

dimensional homogeneity, we find the values of the exponents ai as a1 ¼ �1=2;a2 ¼ 1=2 which transforms (9.81) as follows

d ¼ c

ffiffiffiffiD

Z

r(9.82)

Equation (9.82) shows that the characteristic size of the reaction zone in torches

of non-premixed gases (the diffusion flame) is the order of the flame front thickness

in homogeneous mixtures. Indeed,

d �ffiffiffiffiD

Z

r�

ffiffiffiaZ

r� uf

Z(9.83)

where a is the thermal diffusivity, uf �ffiffiffiffiffiffiaZ

pis the speed of combustion wave in

homogeneous mixtures.

The estimate (9.83) reflects the physical similarity of the processes that occur in

combustion of homogeneous reactive mixtures and in reaction zones of torches of

non-premixed gases. Assuming that the characteristic size of the mixing zone in a

gas torch is l (for submerged torches l is on the order of the boundary layer

thickness), we arrive at the following estimate of the relative thickness of the

reaction zone in diffusion flames

d �ffiffiffiffiffiffiffiD

l2Z

r¼

ffiffiffiffiffitktD

rexp

E

2RT

� �(9.84)

where d ¼ d=l; and tk ¼ Z�10 and tD � l2=D are the characteristic kinetic and

diffusion time, respectively.

It is emphasized that the estimate (9.84) is valid not only in laminar torches but

also in turbulent ones. In the latter case (9.84) implies not the molecular diffusion

and thermal conductivity D and k but rather their turbulent analogs. Equation (9.84)shows that the relative thickness of reaction zone d depends essentially on the

values of the kinetic and diffusion times. If the rate of chemical reaction is large

enough so that tk << tD; the relative thickness of the reaction zone is small

enough, since as tk tD ! 0= ; the value of d ! 0:Combustion temperature also affects significantly the thickness of the reaction

zone. An increase in T (due to combustion of high caloric fuels) is accompanied by

decreasing d: That allows one to assume that the chemical reaction of combustion

proceeds within a very thin (in the limiting case of an extremely small ratio tk tD= ,

in an infinitesimally thin) flame front at a temperature close to the maximum one.

Outside of flame front only the inert transfer of mass, momentum and energy take

plays in submerged laminar and turbulent torches. The latter makes it possible to

apply the method of the theory of the gas jets in studying gas torches (Abramovich

1963; Vulis et al. 1968; Vulis and Yarin 1978).

9.6 Gas Torches 281

One can distinguish two main types of gas torches: (1) torches of premixed

homogeneous mixtures, and (2) torches of non-premixed gaseous fuel and oxidizer.

In the first type of torches, a premixed reactive mixture is supplied directly into the

flame front, whereas in the second one a separate supply of reactants into the

reaction zone occurs. Accordingly, in homogeneous torches combustion process

is determined by the rate of chemical reaction, whereas in torches of non-premixed

gases it is determined predominantly by the mixing rates and to some extent by the

reaction rate. Moreover, in the non-premixed torches, as a rule, the mixing is the

limiting process. Therefore, torches of non-premixed gases are longer and less

intense than the homogeneous ones.

In homogeneous torches combustion is fully completed within the entrance

section of the jet, at a distance of about five nozzle calibers. In this case the

flame front is located near the boundary of the potential core of the jet. Under

such conditions the geometry of a homogeneous torch is determined by the velocity

distribution at the nozzle exit, as well as the speed of combustion wave in homo-

geneous mixture. Thus, the functional equation for the length of homogeneous

torch lf is

lf ¼ f ðu0; uf ; dÞ (9.85)

where u0 ¼ u0ðyÞ is the velocity of the reactive mixture at the nozzle exit, with ybeing transversal coordinate. Equation (9.85) also incorporates the speed of com-

bustion wave uf which represents the physicochemical characteristics of the reac-

tive mixture and can be calculated using the well-known methods of the combustion

theory (Zel’dovich et al. 1985; Williams 1985), and the nozzle diameter d.According to the Pi-theorem, (9.85) reduces to the following dimensionless form

lfd¼ ’

u0uf

� �(9.86)

It is seen that the relative lengths of homogeneous torches depend only on the

ratio of the issue velocity to the speed of combustion wave. At a known uf , thelength and shape of homogeneous torches are found by integrating the following

equation

dx

dy¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu0uf

� �2

� 1

s(9.87)

Integration of (9.87) in the case of a parabolic velocity profile at the nozzle exit

results in the following expression (Khitrin 1957)

lf � xf ¼ N sin’ cos’D’� 1� k2

k2Fð’;KÞ þ 1þ k

k2Eð’; kÞ

� �(9.88)


where sin2’ ¼ yf ðu� 1Þ; k2 ¼ u� 1ð Þ= uþ 1ð Þ; N ¼ ðu� 1Þ ffiffiffiffiffiffiffiffiffiffiffiuþ 1

p; D’ ¼

1� k2sin2’� �1=2

; u ¼ u0=uf ; y ¼ yf =r0; x ¼ xf =r0; lf ¼ lf =r0; r0 ¼ d=2 is the

nozzle radius, Fð’; kÞ and Eð’; kÞ are the elliptic integral the first and second

kind, respectively; cf. Fig. 9.6.

When the velocity distribution at the nozzle exit is uniform, i.e. u0 ¼ const, andthe outflow velocity of reactive mixture significantly exceeds the velocity of

combustion wave, the length of homogeneous torch is given by

lfd¼ u0

uf(9.89)

This relation shows that the length of laminar homogeneous torches is propor-

tional to the outflow velocity. This result agrees well with the experiments on

combustion in laminar torches of premixed gases, whereas the experimental data

for the turbulent torches show that their length weakly depends on u0, since ufT �un0; with ufTbeing the speed of turbulent flame and n � 0:7� 0:8:

Consider now vertical torches of non-premixed gases. First, we reveal the effect

of flow parameters on the length of such torches. For this aim, we begin with the

simplest example, namely a torch which forms when gaseous fuel is issued from a

cylindrical nozzle into a gaseous medium containing oxidizer, which is at rest far

x

y

u0

uf

q

Fig. 9.6 Sketch of a

homogeneous mixture torch

9.6 Gas Torches 283

away in the transverse direction from the nozzle. Assuming that the velocity of the

fuel jet is large enough, we neglect the any possible effect of buoyancy forces,

which might arise from the disparity of fuel and oxidizer densities. We also neglect

the effect of density change and assume that r is constant. At fixed initial

concentrations of fuel in the gas jet and oxidizer in the ambient medium, we state

the functional equation for the torch length lf as follows

lf ¼ f ðu0;D; dÞ (9.90)

Bearing in mind that gas torches features stem from those of submerged jets, we

define the dimensions of the governing parameters in the system of units LxLyMT

u0½ � ¼ LxT�1; D½ � ¼ L2yT

�1; d½ � ¼ Ly (9.91)

Since all the governing parameters possess independent dimensions, (9.91)

reduces to

lf ¼ cua10 Da2da3 (9.92)

where c is a dimensionless constant that depend on concentrations of the fuel in

the gas jet cg and oxygen in the ambientcox, as well as on stoichiometric fuel-to-

oxygen mass ratio O. The latter follows directly from the functional equation

for the torch length written with the account of the dimensional ( u0½ � ¼ LxT�1;

D½ � ¼ L2yT�1; d½ � ¼ Ly) and dimensionless ( O½ � ¼ 1; cg

� ¼ 1; cox½ � ¼ 1) charac-

teristics. Since the dimension of lf� ¼ Lx can be constructed as a combination

of the dimensions of the parameters u0;D and d, such an equation acquires the

form of lf ¼ ua10 Da2da3 f ðO; cg; coxÞ, i.e. the form of (9.92) where the constant

c ¼ f ðO; cg; coxÞ.Taking into account that the dimension of the torch length lf

� ¼ Lx and

applying the principle of the dimensional homogeneity, we find the values of the

exponents ai as a1 ¼ 1; a2 ¼ �1 and a3 ¼ 2: Accordingly, we obtain that in the

case of n ¼ D

lf ¼ cRe (9.93)

where lf ¼ lf d= and Re ¼ u0d=n is the Reynolds number.

It is seen that the length of laminar axisymmetric torches of non-premixed gases

(with n ¼ const) is proportional to the Reynolds number that is defined by the

outflow velocity and the nozzle diameter, whereas the length of turbulent torches

(with nT � u0d) lf does not depend on Re as follows from (9.93). (Hottel,

Hawthorne, 1949) The experimental data shown in Fig. 9.7 agree with these

predictions.

The length of non-premixed gases torches is large enough to assume lf >> 1:Then, the characteristics of non-premixed diffusion flames practically do not

depend on the outflow conditions at the nozzle exit and are determined by the


integral parameters of the flow: the total momentum flux and mass fluxes of species.

In order to determine these characteristics, we use the continuity, momentum and

species balance equations in the boundary layer form

@ruyk

@xþ @rvyk

@y¼ 0 (9.94)

ru@u

@xþ rv

@u

@y¼ 1

yk@

@yykt� �

(9.95)

ru@b@x

þ rv@b@y

¼ 1

yk@

@yykg� �

(9.96)

where t and g are the shear stress and the compound species flux, respectively, the

compound concentration b ¼ ca � cb þ 1; with c1 and c2 being the fuel and oxygenconcentration, respectively, ca ¼ caO=cb1; cb ¼ cb=cb1: Also, O is the stoichio-

metric fuel-to-oxygen mass ratio, subscript1 refers to the ambient conditions, k ¼0 and 1 for the plane and axisymmetric torches, respectively. It is emphasized that

(9.96) for the compound concentration b was obtained from the species balance

equations by using the Schvab–Zel’dovich method, which allows us to exclude the

source terms associated with the rate of chemical reaction.


IIIIII

Re0

f

Fig. 9.7 The dependence lf ðReÞof the length of non-premixed torches on the Reynolds number.

I-laminar torches, II-transitional torches and III-turbulent torches

9.6 Gas Torches 285

u ! 0 b ! 0 at y ! 1;@u

@y¼ @b

@y¼ 0 at y ¼ 0 (9.97)

for submerged torches, and

u ¼ 0 b ¼ 0 at y ! 1; u ¼ 0@b@y

¼ 0 at y ¼ 0 (9.98)

for the wall torches issued from a slit parallel to the adjacent wall (in the case k ¼ 0

only).

By integrating this system of equations with the boundary conditions (9.97), we

arrive (at r ¼ const) the following integral invariants for the free torches

ð10

u2ykdy ¼ eJx (9.99)

ð10

ubykdy ¼ eGx (9.100)

where eJx and eGx are constant determined by the conditions at the nozzle exit and,

thus, are given. It is emphasized that the invariant (9.100) stems from the fact that

the compound concentration b is described by (9.96) which does not involve any

term related to the rate of chemical reaction.

Assuming that the longitudinal convective species fluxes within the far field of

non-premixed torches are determined by the invariants eJx and eGx, we formulate the

functional equation for (ubÞm as follows

ðubÞm ¼ f ðeJx; eGxÞ (9.101)

where subscript m refers to the torch axis.

Taking into account that the dimensions of eJxh i¼ L2xL

1þky T�2 andeGx

h i¼ LxL

1þky T�1 are independent, we arrive at the simpler form of (9.101)

umbm ¼ ceJxfGx

(9.102)

where c is a constant.The dimensional analysis of the flow in the far field of submerged jets shows

(Chap. 6) that the problem under consideration based on (9.95) and (9.96) has the

following self-similar solutions

u

um¼ ’ð�Þ; b

bm¼ cð�Þ; um ¼ Axa (9.103)


where � ¼ y=d; d is the thickness of the boundary layer of the jet, and A and a are

constants, the functions ’ð�Þ and cð�Þ are determined by a system of the ordinary

differential equations that can be found transforming (9.94–9.96) using the Pi-

theorem.

The substitution of the expression (9.103) for the axial velocity um into (9.102)

yields

x�a ¼ bmAc1fGmfJm (9.104)

where c1 ¼ c�1 is the constant.

Taking in (9.104) x ¼ lf and accordingly bm ¼ 1 which corresponds to the

complete combustion at the flame tip, we obtain the following expression for the

length of non-premixed gas torch

l�af ¼ Ac1

fGxeJx (9.105)

Using the known expressions for the coefficients A and a (see Chap. 6) and

accounting for the fact that fGx ¼ pku0b0lkþ1 (where l0 is the nozzle characteristic

size and b0 is the given compound concentration value at the nozzle exit), we arrive

at the following relations for the dimensionless length of torches of non-premixed

gases of different types (Table 9.1). It is seen from Table 9.1 that the length of

laminar non-premixed torches depends on the issue velocity (through the Reynolds

number Re) and parameter e ¼ 1þ ðca0=cb0ÞO accounting for the reactant

concentrations and the stoichiometric fuel-to-oxygen mass ratio. On the other

hand, the length of turbulent torches depends only on the ratio of reactant

concentrations and the stoichiometric fuel-to-oxygen mass ratio. The constants

Ciin the expressions for lf in Table 9.1 depend on the type of the jet flow (free

or wall flow), its geometry (plane or axisymmetric), as well as on the flow

regime (laminar or turbulent). It is emphasized that the dependence of lf on the

reactant concentrations is different for different types of non-premixed torches.

For example, in plane laminar torches lf � e3, whereas in the axisymmetric ones

lf � e. That results from the different mixing intensity in various types of

submerged torches.

Table 9.1 Length of

diffusion torches of different

geometry

Type of flow lfPlane laminar torch C1Ree3

Plane laminar wall-torch over an adiabatic plate C2Ree4

Axisymmetric laminar torch C3ReePlane turbulent torch C4e2

Axisymmetric turbulent torch C5e

9.6 Gas Torches 287

9.7 Immersed Flames

Consider an immersed diffusion flame formed by a jet of gaseous oxidizer issuing

into liquid reagent (fuel); cf. Fig. 9.8. As a result of the gas–liquid interaction,

namely its breakup into bubbles and atomization of liquid in the form of liquid

droplets, a two-phase jet-like flow forms in the liquid medium.

The general characteristics of combustion process in this situation, such as the

intensity of heat release, completeness of combustion, etc., are determined to a

considerable extent by the volumetric content of the gaseous phase mv: Dependingon the value of mv, the immersed jet flow acquires a form of either bubbly or gas-

droplet jets (Abramovich et al. 1984).

At a high initial temperature of reactants (or due to the presence of an external

igniter, chemical reaction between fuel and oxidizer begins. In consequence of

heating and vaporization of liquid reagent due to the chemical reaction between fuel

vapor and the oxidizer supplied as the gas jet, recognizable domains filled with

vapor–oxidizer mixture containing droplets of reactive liquid (fuel) or bubbles

filled by the oxidizer/fuel vapor mixture are formed in the liquid. A high tempera-

ture zone-a diffusion flame is formed in the two-phase mixture. In the case when

combustion products are gaseous, the diffusion flame is located in an open cavity

(Fig. 9.8a). On the other hand, when combustion products represent an immiscible

liquid, the diffusion flame is located in a closed gas cavity (Fig. 9.8b). Its size

depends not only on the physicochemical properties of reactants and the intensity of

their mixing but also on the boiling temperature of combustion products that

determine the position of the condensation zone. Sukhov and Yarin (1981, 1983)

developed a theory of laminar immersed flames in the case when combustion

products are gaseous. In this case, assuming that the rate of chemical reaction is

infinite, the thermal conductivity and diffusion coefficients are constant and the

effect of buoyancy force is negligible, the problem reduces to the following system

of equations

f1

f1

2

2

x

y

x

y0 0

a b

Fig. 9.8 Sketch of the immersed flame. 1: Gaseous phase (a mixture of vapor of reactive liquid,

gaseous oxidizer and combustion products). 2: Reactive liquid


ui@ui@x

þ vi@ui@y

¼ niyk@

@yyk@ui@y

� �(9.106)

@

@xðuiykÞ þ @

@yðviykÞ ¼ 0 (9.107)

ui@Ti@x

þ vi@Ti

@y¼ aiyk

@

@yyk@Ti@y

� �(9.108)

ui@cj@x

þ vi@cj@y

¼ Djyk @

@yyk@cj@y

� �(9.109)

Xj

cj ¼ 1 (9.110)


y ¼ 0@ui@y

¼ 0; vi ¼ 0;@ca@y

¼ @cb@y

¼ 0; ca ¼ 0;@T1@y

¼ 0

y ¼ yf ðxÞ ca ¼ cb ¼ 0; cc ¼ 1; T ¼ Tf

y ¼ y�ðxÞ u1 ¼ u2 ¼ u�;@u1@y

¼ r21n21@u2@y

; cb ¼ 0; T1 ¼ T2 ¼ T�

y ! 1 u2 ¼ 0; T2 ! T21

(9.111)

where x and y are longitudinal and transverse coordinates, u and v are the longitu-dinal and transverse velocity components, T and cj are the temperature and concen-

tration, n; a and Dj are the kinematic viscosity, thermal and species diffusivities,

respectively, k ¼ 0 or 1 for the plane and axisymmetric flows, subscripts f and �correspond to the combustion front and the liquid–gas interface, subscript 1 and

2 refer to the gaseous and liquid phases, and j ¼ a; b and c correspond to the fuel,

oxidizer and combustion products, respectively, r21 ¼ r2 r1 and n21 ¼ n2 n1:==The conditions (9.111) should be also supplemented with the Clausius–

Clapeyron equation determining the equilibrium concentration of vapor at the

liquid–gas interface, as well as by the mass and thermal balances at the flame and

the interface. The Clausius–Clapeyron equation reads

ca� ¼ w exp � qeRaT�

� �(9.112)

where ca� is the vapor concentration at the interface, w is the pre-exponential factor,

qe is the heat of evaporation.

9.7 Immersed Flames 289

In the framework of the boundary layer theory, the slope of the flame front to the

flow axis is sufficiently small, so that @=@nð Þf � @=@yð Þf ; with n being the normal

to the flame front. Then assuming the Lewis number Le ¼ 1; we obtain the balanceconditions at the flame front and the liquid–gas interface in the following form

@cb@y

� �f

þ O@ca@y

� �f

¼ 0 (9.113)

@T

@y

� �af

� @T

@y

� �bf

þ q

rcP

@ca@y

� �f

¼ 0 (9.114)

v1cc� þ D@cc@y

� ��¼ 0 (9.115)

k1@T1@y

� �� k2

@T2@y

� ��þ r1D

@ca@y

� ��¼ 0 (9.116)

r2v2� � r1v1�ca� þ r1D@ca@y

� ��¼ 0 (9.117)

where k and cP are the thermal conductivity and specific heat, respectively, and q isthe heat of reaction.

The conditions (9.111) should be supplemented with the integral invariants that

are needed to obtain a non-trivial solution. Using the Schvab–Zel’dovich variable

and integrating (9.106) and (9.109) across the immersed jet, we arrive at the

following invariants

ðy�0

u21ykdyþ r21

ð1y�

u22ykdy ¼ Ix ¼ const (9.118)

ðy�0

u1bykdy ¼ Gx ¼ const (9.119)

where b ¼ 1þ Dc; Dc ¼ cb O= � ca and r21 ¼ r2 r1= .

Introducing the dimensionless variables ui ¼ ui=u0 and y ¼ y=y0, we transform(9.118) and (9.119) as follows

ðy�0

u2i ykdyþ r21

ð1y�

u22ykdy ¼ 1 (9.120)


Zy�0

u1bykdy ¼ 1 (9.121)

where u0 and y0 are the scales of velocity and length defined as u0 ¼ Ix Gx=ð Þ andy0 ¼ G2

x Ix=� �1= kþ1ð Þ

.

The totality of the boundary conditions (9.111), balance equations (9.113–9.117)

and the integral relations (9.120) and (9.121) fully determines the problem on the

immersed diffusion torch. It allows finding the profiles of all characteristic para-

meters, values of temperature at the flame front, the temperature and concentrations

at the interface surface, as well as the lengths and configurations of the flame front

and gaseous cavity (Yarin and Sukhov 1987). However, instead of describing the

theoretical solution of (9.106–9.110), following the approach of the present book, we

focus our attention at the calculation of the length and configuration of the immersed

diffusion flame using the dimensional analysis. For this aim, we use the approach

developed inVulis et al. (1968) andVulis andYarin (1978) for the investigation of the

aerodynamics of diffusion flames. It consists in the calculation of the axial velocity

and concentration and then determining the flow parameters corresponding to the

flame front. In the framework of the model of an infinitely thin flame front

(corresponding to the assumption on an infinitely large rate of chemical reaction),

the length of the immersed flame is found from the following conditions: ca ¼ cb ¼ 0

at x ¼ lf , with subscript f corresponding to the tip of the torch.In order to determine the velocity in the immersed flame, we use the approach

developed in Chap. 6 for the free laminar jets. We write the following functional

equations for the local velocity u; the axial velocity um, and the thickness of

gaseous cavity y�

ui ¼ fiðum; y; y�Þ (9.122)

um ¼ fmðIx; n1; xÞ (9.123)

y� ¼ f�ðIx; n1; xÞ (9.124)

where subscripts i ¼ 1 and 2 correspond to the gaseous and liquid phases,

respectively.

Note that the kinematic viscosity of liquid n2; as well as densities of both phasesr1 and r2 are not included in the sets of the governing parameters in (9.122–9.124)

because they are accounted for in the expression for the kinematic momentum Ix:To transform (9.122–9.124) to the dimensionless form, we use the LxLyLzMT

system of units. The dimensions of the parameters involved in (9.122–9.124) in this

system of units are as follows

ui½ � ¼ LxT�1; um½ � ¼ LxT

�1; Ix½ � ¼ L2xLyT�2; n1½ � ¼ L2yT

�1 y½ � ¼ Ly; x½ � ¼ Lx

(9.125)


It is seen that two governing parameters in (9.122) have independent

dimensions, whereas the dimensions of the governing parameters in

(9.123–9.124) are independent. Then, according to the Pi-theorem, (9.122–9.124)

take the form

ui ¼ umci ’ð Þ (9.126)

um ¼ Aixe (9.127)

y� ¼ Bxg (9.128)

where Ai ¼ ciu�10 I2x d

kþ10 =nkþ1

1

� �1= 3�kð Þ; B ¼ c� n2d20=Ix

� �1= 3�kð Þ; e ¼ � k þ 1ð Þ=

3� kð Þ, g ¼ 2= 3� kð Þ; ui ¼ u=u0; um ¼ um=u0; y� ¼ y=y0; ’ ¼ y=y�; ci and c�are constants.

Equations (9.126–9.128) show that the system of PDEs determining the velocity

distribution in the immersed laminar torch can be reduced to a system of ODEs.

Therefore, the velocity can be expressed as a function cð’Þ of a single variable

cið’Þ ¼F

0ið’Þ’k

(9.129)

where the function Fð’Þ should to satisfy the following conditions

’ ¼ 0F

01

’k

� �¼ 0;

F01

’k¼ 1; F1 ¼ 0

’ ¼ 1 F01 ¼ n21F

02;

F01

’k

� �0

¼ r21n21F

02

’k

� �0

’ ! 1 F02

’k! 0;

F02

’k

� �0

! 0

(9.130)

Here primes denote differentiation with respect to ’; and n21 ¼ n2 n1=ð Þ.Determine the concentration distribution. Since the physically realistic self-

similar solution of (9.109) is absent under the condition ca� ¼ const, we use the

integral method of calculation distribution of b: Approximate the actual concentra-

tion profile by the series

b ¼X1n¼0

anðxÞy (9.131)

where the coefficients anare found from the conditions at the flow axis and the

interface surface


y ¼ 0 db dy= ¼ 0; b ¼ bm; y ¼ y�ðxÞ b ¼ b� (9.132)

Taking into account three terms of the series (9.131), we arrive at the expression

b ¼ bmð1� ’2Þ þ b�’2 (9.133)

Using (9.121) and (9.133), we find the dependence bmðxÞ as

bm ¼ x�kþ13�kðA1B

kþ1Þ�1 � b�I1n o

F1ð1Þ � I1½ ��1(9.134)

where I1 ¼Ð10

’2F01ð’Þd’.

Bearing in mind that at the flame front bf ¼ 1, and that at the tip of the diffusion

flame bm ¼ 1, we arrive at the following equations for the shape and length of the

immersed torch

yf ¼ Bx2= 3�kð Þ bm � 1

bm � b�

� �1=2

(9.135)

lf ¼ ðA1Bkþ1Þ�1 � b�I1

n o 1�kð Þ= 1þkð ÞF1ð1Þ � I1ð1� b�Þ½ � k�3ð Þ= kþ1ð Þ

(9.136)

where lf is the length of the immersed flame, and subscript f corresponds to the

flame front.

The correlations (9.135) and (9.136) are qualitative, since they contain factors

A and B that incorporate the unknown constants ci and c�, as well as the vapor

concentration at the interface surface b�: The latter can be determined from the

Clapeyron–Clausius equation for a given temperature at the interface T�: The actualvalues of the factors Aand B are found from (9.106) and (9.107) after substituting

into these equations the expressions (9.127–9.129)

Ai ¼ ðI2 þ r21n221I3Þ�2= 3�kð Þ Re

6� 5k

� � 1þkð Þ= 3�kð Þ;A2 ¼ n21A1

B ¼ ðI2 þ r21n221I3Þ1= 3�kð Þ Re1

6� 5k

� ��2= 3�kð Þ(9.137)

where I2 ¼R10

F01

� �2=’kd’; I3 ¼

R11

F02

� �2=’kd’; and Re1 ¼ u0y0=n1.

Accordingly, the expression (9.136) takes the form

lf ¼ Re1

6� 5kðI2 þ r21n

221I3Þ 1�kð Þ= 1þkð Þ F1ð1Þ � I1ð1� b�Þ½ �ðk�3Þ kþ1Þ=

(9.138)


The effect of various parameters on the characteristics of the immersed flames is

clearly visible through the dependence of its length on the reactants temperatures,

the issue velocity, composition of gaseous oxidizer, etc. In particular, an increase

in the reactants temperatures is accompanied by shortening of the immersed

torch length. This results from an increase in vapor concentration the interface

(a decrease in b�Þ and an increase of the term I3ð1� b�Þ in (9.138). An opposite

effect takes place at increasing the latent heat of evaporation: an increase in the

value of qe leads to a significant growth of the torch length lf :As with the other types of diffusion flames, the characteristics of the immersed

flames depend on the flow geometry. For example, the length of the plane (k ¼ 0)

and axisymmetric (k ¼ 1) flame is inversely proportional, respectively, to the third

and first powers of the factor F1ð1Þ � I3ð1� b�Þ½ � which accounts for the effect of

vapor concentration at the interface surface. It is emphasized that in both cases the

length of the immersed flames is directly proportional to Re: A possible presence of

an inert admixture in the gas jet also affects the flame characteristics (Sukhov and

Yarin 1983, 1987). An increase in the content of an inert admixture in the gaseous

oxidizer jet is accompanied by a significant decrease in the length of the axisym-

metric and plane flames (cf. Fig. 9.9).

Problems

P.9.1. Evaluate the burning time of a liquid fuel droplet at its combustion in

a stagnant atmosphere that contains gaseous oxidizer.

The process of droplet burning involves a number of simultaneously happening

physical processes such as liquid vaporization, mixing of gaseous reagents and

combustion of the resulting vapor–oxidizer mixture. These processes are also

accompanied by heat and mass transfer, as well as by the diminishment of the

droplet size and surface and the corresponding displacement of the reaction zone.

Accordingly, a theoretical description of droplet combustion implies solving the

coupled non-steady equations governing the mass, momentum, energy and species

transfer in the liquid and gaseous phases (Yarin and Hetsroni 2004).The non-linear

terms accounting for the heat release and species consumption involved in these

equations make the theoretical analysis of the problem extremely difficult.

Fig. 9.9 The effect of an

inert admixture of the length

of plane (k ¼ 0) and

axisymmetric (k ¼ 1)

immersion flames


Therefore, as a rule, droplet burning is studied using models based on a number of

simplifying assumptions: (1) the chemical reaction rate is infinite, (2) the droplet

temperature is uniform, (3) the effects of buoyancy and radiant heat transfer are

negligible, (4) the Lewis number equals one, (5) the physical properties of the liquid

and gaseous phases are constant.

The assumption (1) allows one to consider the reaction zone as an infinitesimally

thin flame front which separates the flow field into two domains: the inner one (near

the droplet surface) filled with a mixture of the fuel vapor and combustion products,

and the outer one filled with a mixture of the oxidizer and combustion products.

Also, in this case the vapor and oxidizer concentrations at the flame front are equal

to zero. The other assumptions make it possible to use the lumped capacitance heat

transfer model for the droplet, as well as to consider a spherically-symmetric flame

(for the burning in a stagnant atmosphere). The flame temperature equals the

adiabatic combustion temperature in this case. Moreover, the droplet surface tem-

perature changes only slightly during the combustion process and can be taken as

a constant equal to the boiling temperature of liquid fuel.

Based on the above-mentioned simplifications, assume that during the combus-

tion process the droplet diameter d depends on: (1) densities of liquid fuel and

gaseous oxidizer, (2) species diffusivity (assumed being identical for all the

components), (3) total enthalpy of fuel, (4) latent heat of evaporation, (5) droplet

initial diameter, and (6) time

d ¼ f ðr1; r2;D; qt; qe; d0; tÞ (P.9.1)

Here r1 and r2 are the density of gaseous and liquid phases, respectively, D is

the diffusivity, qt ¼ c01q=Oþ cPðT1 � TsÞis the total enthalpy of the fuel with qbeing the heat of reaction, c01 the oxidizer concentration in the surrounding

medium, O the stoichiometric oxidizer-to-fuel mass ratio, qe the latent heat of

evaporation, cP the specific heat of gaseous phase, T1 and Ts being the ambient

and saturated temperature, respectively; d0 is the initial droplet diameter, and t istime.

The dimensions of the governing parameters involved are

½r1� ¼ L�3M; ½r2� ¼ L�3M; ½D� ¼ L2T�1; ½qt� ¼ JM�1;

½qe� ¼ JM�1; ½d0� ¼ L; ½t� ¼ T (P.9.2)

Five of the seven governing parameters possess independent dimensions. Then,

according to the Pi-theorem, the number of dimensionless groups of the present

problem is equal two, and (P.9.1) reduces to the following dimensionless equation

P ¼ ’ðP1;P2Þ (P.9.3)

where P ¼ d=d0; P1 ¼ r1=r2ð Þ Dt=d20� �

and P2 ¼ qt=qe ¼ B (B is the Spalding

transfer number.

Problems 295

Droplet is burnt completely at the moment t ¼ tb when d ¼ 0. At that moment

(P.9.3) yields ’ðP1b;P2Þ ¼ 0; where P1b ¼ r1=r2ð Þ Dtb=d20

� �: Solving the latter

equation for P1b, we obtain the following expression for the droplet burning time

tb ¼ r2r1

d20DcðBÞ (P.9.4)

It is emphasized that the analytical solution of the problem yields the following

expression for tb

tb ¼ r2r1

d208D

lnð1þ BÞ½ ��1(P.9.5)

References

Abramovich GN (1963) The theory of turbulent jets. MTI Press, Cambridge

Abramovich GN, Girshovich TA, Krasheninnikov SY, Sekundov AN, Smirnova IP (1984) Theory

of turbulent jets. Nauka, Moscow (in Russian)

Burke SP, Schumann TE (1928) Diffusion flames. Ind Eng Chem 20:998–1004

Frank-Kamenetskii DA (1969) Diffusion and heat transfer in chemical kinetics, 2nd edn. Plenum

Press, New York

Hottel HC, Hawthorne WR (1949) Diffusion in laminar flame jets. In: Proceeding of third

symposium on combustion and flame and explosion phenomena, Williams & Wilkins,

Baltimore, pp 254–266

Khitrin LN (1957) The physics of combustion. Moscow University, Moscow (in Russian)

Merzhanov AG, Khaikin BI (1992) Theory of combustion waves in homogeneous media. AN

SSSR, Chernogolovka (in Russian)

Merzhanov AG, Khaikin BI, Shkadinskii KG (1969) The establishment of a steady-state regime of

flame propagation after gas ignition by an overheated surface. Prikl Mech Tech Phys 5:42–48

Schvab BA (1948). A relation between temperature and velocity fields in gaseous flame. In: The

investigation of the process of fossil fuel combustion, Gosenergoizdat, Moscow-Leningrad, pp

231–248 (in Russian)

Semenov NN (1935) Chemical kinetics and chain reactions. Oxford University Press, Oxford

Spalding DB (1953) Theoretical aspects of flame stabilization: an approximate graphical method

for the flame speed of mixed gases. Aircraft Eng 25:264–276

Sukhov GS, Yarin LP (1981) Combustion of a jet of immiscible fluids. Combust. Explos. Shock.

Waves 17:146–151

Sukhov GS, Yarin LP (1983) Calculating the characteristics of immersion burning. Combust.

Explos. Shock waves 19:155–158

Vulis LA (1961) Thermal regime of combustion. McGraw-Hill, New York

Vulis LA, Yarin LP (1978) Aerodynamics of a torch. Energia, Leningrad (in Russian)

Vulis LA, Ershin SA, Yarin LP (1968) Foundations of the theory of gas torches. Energia,

Leningrad (in Russian)

Williams FA (1985) Combustion theory, 2nd edn. Benjamin-Cummings, Menlo Park

Yarin LP, Hetsroni G (2004) Combustion of two-phase reactive media. Springer, Berlin

Yarin LP, Sukhov GS (1987) Foundations of combustion theory of two-phase media.

Energoatomizdat, Leningrad (in Russian)

Zel’dovich YB (1948) Toward a theory of non-premixed gas combustion. J Tech Phys 19:199–210

Zel’dovich YB, Barenblatt GI, Librovich VB, Makhviladze GM (1985) Mathematical theory of

combustion and explosion. Plenum Press, New York


Author Index

A

Abrahamsson, H., 259

Abramovich, G.N., 131, 156, 232, 237, 238,

245, 258, 281, 288, 296

Acrivos, A., 169, 175, 209

Adamson, T.C., 102

Adler, M., 119, 120, 129

Adrian, R.J., 102

Akatnov, N.I., 146, 148, 156

Alhama, F., 7, 38

Andrade, E.N., 143, 156

Andreopoulos, J., 245, 258

Antonia, R.A., 228, 229, 238, 258, 259

Anton, T.R., 80. 101

Armstrong, R.C., 129

Astarita, G., 113, 129

B

Baehr, H.D., 39, 69, 201, 209

Bagananoff, D., 157

Bahrami, M., 113, 129

Baines, W.D., 154, 156, 245, 259

Balachander, R., 260

Banks, R.B., 258

Banks, W.H.H., 170, 209

Barenblatt, G.I., 7, 23, 37, 217, 258, 296

Barua, S.N., 119, 129

Basset, A.B., 20, 101

Batchelor, G.K., 54, 69, 71, 101, 149, 153, 156

Bayazitoglu, Y., 139

Bayley, F.J., 179, 209

Bearman, P.W., 76, 81, 101

Berger, S.A., 118, 129

Bergstorm, D.J., 249, 258, 260

Berlemont, A., 80, 101

Bernulli, D., 10

Bigler, R.W., 238, 258

Bird, R.B., 113, 129

Blackman, D.R., 4, 37

Blasius, H., 47, 69

Bloonfield, L.J., 154, 156

Boothroyt, R.G., 84, 101

Boussinesq, J., 80, 101

Bradbury, L.I.S., 238, 243, 244, 258

Brenner, H., 71, 102

Bridgmen, P.W., 11, 23, 37, 87, 101, 166,

167, 209

Britter, R.E., 63, 69

Buchanan, H.J., 75, 102

Buckingham, E., 23, 37

Burke, S.P., 271, 296

C

Campbelle, I.H., 156

Carpenter, L.H., 81, 102

Celata, G.P., 112, 129

Champagne, F.H., 147, 149, 157

Chandrasekhara, D.V., 258

Chang, E.J., 80, 101

Chao, B.T., 170. 209

Chassaing, P., 245, 258

Chen, A.M.L., 113, 114, 116, 130

Chen, C.S., 105, 129

Cheslak, F.R., 257, 259

Chevray, R., 228, 229, 259

Cho, Y.I., 209

Chua, L.P., 226, 229, 259

Claria, A., 258

Clauser, F.H., 222, 259

Clift, R., 71, 78, 101

Coles, D., 220, 259

Corrsin, S., 228, 229, 259


DOI 10.1007/978-3-642-19565-5, # Springer-Verlag Berlin Heidelberg 2012

297

Crawford, M.E., 39, 69, 183, 209

Culham, J., 129

D

Dandy, D.S., 80, 101

Dean, W.R., 118, 129

Derjagin, B.M., 96, 101

Desjanquers, P., 101

De Witt, K.J., 209

Desjonquers, P., 101

Didden, N., 63.69

Dimotakis, P.E., 236, 259

Divoky, D., 228, 259

Dodson, D.S., 102

Dorfman, L.A., 170, 209

Dorodnizin, A.A., 197, 209

Douglas, J.F., 7, 38, 45, 69

Doweling, D.R., 236, 259

Dowirie, M.J., 101

Du, D.X., 130

Dunkan, A.B., 112, 129

Dwyer, H.A., 80, 82, 101, 102

E

Eastor, T.D., 171, 209

Emery, A.E., 105, 129

Eriksson, J., 259

Ershin, S.A., 296

Eskinazi, S., 259

Everitt, K.M., 238, 243–245, 259

F

Fargie, D., 105, 129

Fendell, F.E., 82, 102

Flakner, V.M., 68, 69

Forstall, W., 229, 259

Frankel, N.A., 175, 209

Frank-Kamenetskii, D.A., 263, 264, 266, 268,

270, 271, 296

Fric, T.F., 245, 259

Friedman, M., 105, 129

Fujii, T., 201, 209

G

Gad-el-Hak, M., 112, 129

Garimella, S., 112, 129

Gaylord, E.M., 229, 259

George, J., 249, 258

George, W.K., 249, 259

Germano, M., 120, 129

Gilis, J., 129

Girshovich, T.A., 258, 296

Glauert, M.B., 146, 148, 157

Gloss, D., 130, 259

Couesbet, G., 101

Grace, J.R., 101

Graham, J.M.R., 76, 81, 101, 102

Greif, R., 170, 209

Gua, Z.Y., 130

Gutmark, E., 133, 157, 233, 234, 253, 254, 259

H

Hadamard, J.S., 85, 102

Hagen, G., 103, 129

Hamel, G., 43, 69

Hamilton, W.S., 80, 102

Happel, J., 71, 102

Hartnett, J.P., 209

Hassager, O., 129

Hasselbrink, E.F., 245, 246, 259

Hausner, O., 112, 130

Hawkis, G.A., 30, 38

Hawthorne, W.R., 284, 296

Herwig, H., 110, 112, 129, 130, 224, 259

Hetsroni, G., 71, 102, 112, 113, 130, 294, 296

He, Y.-L., 130

Hinze, J.O., 28, 38, 131, 157, 217, 229, 259

Ho, C.-M., 112, 130, 133, 157

Hollands, K.G.T., 187, 209

Hottel, H.C., 284, 296

Hoult, D.P., 60, 69

Howarth, L., 215, 259

Huntley, H.E., 7, 38, 45, 69, 88, 102

Huppert, H.E., 60, 61, 63, 69

Hussain, F., 133, 157

Hussain, H.S., 133, 157

Hussaini, M.Y., 171, 209

I

Illigworth, C.R., 197, 209

Incorpera. F.P., 112, 130

Ipsen. D.C., 4, 38

Ito. H., 119, 120, 130

J

Jaluria, Y., 149, 157

Jeng, D.R., 209

Jones, J.B., 30, 38

Jonicka, J., 260

298 Author Index

K

Kakas, S., 112, 130

Kandlicar, S.G., 110, 130

Karamcheti, K., 157

Karanfilian, S.K., 80, 102

Karlsson, R.I., 250, 259

Karman, Th., 55, 69, 215, 221, 225, 226, 259

Karthpalli, A., 133, 157

Kashkarov, V.P., 54, 70, 131, 133, 147, 157,

228, 260

Kassoy, D.R., 82, 102

Kaviany, M., 159, 209

Kays, W.M., 39, 69, 159, 183, 209

Keffer, J.F., 245, 259

Kelso, R.M., 245, 259

Kenlegan, G.H., 81, 102

Kerr, R.C., 154, 156

Kestin, J., 9, 30, 38, 172, 209

Khaikin, B.I., 271, 296

Khitrin, L.N., 282, 296

Kolmogorov, A.N., 211, 259

Kompaneyets, A.S., 162, 210

Konsovinous, N.S., 136, 157

Kotas, T.J., 80, 102

Krashenninikov, S.Yu., 258, 296

Kreith, F., 169, 170, 209

Kuta teladse, S.S., 18, 38, 158, 209

L

Lamb, H., 83, 90, 102

Landau, L.D., 37, 38, 52, 53, 54, 69, 71, 94,

95, 98, 102, 103, 130, 133, 157, 159,

173, 209, 216, 219, 259

Launder, B.F., 249, 259

Lavender, W.J., 172, 209

Lawerence, C.J., 102

Lee, M.H., 170, 209

Levich, V.G., 29, 38, 65, 69. 94, 95, 102,

159, 177, 189, 193, 194, 207–209

Levi, S.M., 96, 101

Librovich, V.B., 296

Li, D., 130

Lifshitz, E.M., 37, 38, 54, 69, 71, 98, 102,

103, 130, 133, 157, 159, 173, 209,

216, 219, 259

Lim, T.T., 259

Liron, N., 129

List, E.J., 154, 157

Li, Z., 113, 130

Li, Z.X., 113, 139

Lockwood, F.C., 234, 235, 259

Lofdahl, L., 259

Loitsyanskii, L.G., 22, 28, 38, 42, 43, 69,

75, 101–103, 105, 112, 120. 122,

123, 130, 195,,197, 209, 269

London, A.L., 111, 130

Lumley, J.L., 232, 239–241, 260

Lykov, A.M., 18, 38

M

Maczynski, J.F.J., 238, 259

Madrid, C.N., 7, 38

Ma, H.B., 111, 130

Makhviladze, G.M., 296

Mala, G.M., 130

Marrucci, G., 113, 129

Martin, B.W., 105, 129

Maxey, H.R., 80, 101, 102

Maxworthy, T., 63, 69

Mayer, E., 228, 259

Mc Laughlin, J.B., 80, 102

Mei, R., 80, 102

Merzhanov, A.G., 271, 296

Messiter, L.F., 102

Mikhailov Yu,A., 18, 38

Maneib, H.A., 234, 235, 259

Monin, A.S., 217, 220, 260

Moody, L.F., 111, 130

Mori, Y., 119, 130

Morton, B.R., 149, 153, 154, 157

Mosyak, A., 130

Moussa, Z.M., 245, 259

Mungal, M.G., 245, 246, 259, 260

N

Nakayama, W., 119, 130

Narasimha, R., 251, 260

Narasyan, K.Y., 260

Nicholles, J.A., 259

Nickels, T.B., 238, 260

Nikuradse, J., 109, 129, 130

Nusselt, W., 200, 209

O

Obasaju, E.D., 101

Obukhov, A.M., 211, 213, 260

Odar, F., 80, 102

Oseen, C.W., 80, 102

Owen, J.N., 209

Author Index 299

P

Pai, S.I., 131, 157

Panchapakesan, N.R., 232, 239–241, 260

Papanicolaou, P.N., 154, 157

Parthasarthy, S.R., 260

Pei, D.C.T., 172, 209

Perry, A.E., 238, 259, 260

Persson, J., 259

Peterson, G.P., 111, 112, 129, 130

Pfund, D., 113, 130

Plam, B., 112, 130

Pogrebnyak, E., 130

Pohlhausen, E., 63, 70, 192

Poiseuille, J., 103, 130

Pope, S.P., 215, 260

Prabhu, A., 258

Prandtl, L., 218, 225, 228. 260

Q

Qu, W., 113, 130

R

Raithby, C.A., 187, 209

Raizer, G.P., 162, 164, 209

Ramaswamy, G.S. 4, 38

Rao, V.V.L., 4, 38

Rayleigh, L., 165, 209

Raynolds, O., 35, 38

Rector, D., 130

Riabouchinsky, D., 166, 209

Riley, J.J., 80, 102, 238, 243, 244, 258

Robins, A.G., 238, 243–245, 259

Rodi, W., 245, 249, 258, 259

Rohsenow, W.M., 159, 269

Rosenhead, L., 43, 70

Roshko, A., 245, 259

Rotta, J.C., 217, 245, 260

Rybczynskii, W., 85, 102

S

Saffman, P.S., 79, 93, 102

Sakipov, Z.B., 229, 260

Sananes, F., 258

Sasty, M.S., 171, 209

Schiller, L., 104, 129, 130

Schlichting, H., 39, 52, 58, 64, 67, 70, 75,

102–104, 107, 110, 111, 131, 157, 159,

193, 197, 209, 217, 223, 260

Schneider, W., 136, 157

Schumann, T.E., 271, 296

Schvab, B.A., 264, 296

Sedov, L.I., 6, 10, 15, 23, 38, 43, 54, 55, 70, 71,

83, 87, 102, 166, 209, 217

Sekundov, A.N., 258, 296

Semenov, N.N., 266, 296

Sforzat, P., 133, 157

Shah, R.K., 111, 130

Sherman, F.S., 55, 70

Shekarriz, A., 130

Shih, C.C., 75, 102, 232

Shin, T.-H., 232, 260

Shkadinskii, K.G., 296

Sichel, M., 259

Simpson, J.E., 60, 70

Skan, S.W., 68, 69

Smirnova, I.P., 258, 296

Smith, S.H., 245, 260

Sobhan, C., 112, 129

Soo, S.L., 71, 102, 169, 209

Spolding, D.B., 159, 209, 271, 296

Sprankle, M.L., 102

Spurk, J.H., 23, 38

Stephan, K., 39, 69, 201, 209

Stephenson, S.E., 258

Stewardson, K., 197, 209

Stokes, G.C., 44, 70, 87, 102

Sukhov, G.S., 280, 288, 291, 294, 296

T

Tabol, L., 129

Tachie, M.F., 249–251, 258, 260

Tai, Y.-C., 112, 130

Tang, G.-H., 130

Tao, W.-Q., 130

Taylor, G.I., 157, 225, 260

Taylor, T.D., 169, 209

Temirbaev, D.Z., 229, 260

Tieng, S.M., 189, 210

Towendsend, A.A., 131, 157, 238, 242, 260

Trentacoste, N., 133, 157

Trischka, J.W., 259

Turner, J.S., 149, 154, 156, 157

Turner, A.B., 209

Tutu, N.K., 228, 229, 259

U

Uberoi, M.S., 228, 229, 259

300 Author Index

V

Vasiliev, L.L., 130

Vulis, L.A., 54, 70, 131, 133, 147, 157,

228, 260, 266, 271, 274, 280, 281,

291, 296

Van Dyke, M., 120, 129

W

Wang, H.L., 113, 130

Wang, Y., 113, 130

Ward-Smith, A.C., 103, 111, 125, 130

Weast, R.C., 38

Weber, M.E., 101

Weiss, D.A., 58, 59, 70

Wenterodt, T., 130, 259

White, C.M., 119, 120, 130

White, F.M., 111, 130, 159, 210

Wilkinson, W.L., 113, 114, 116, 130

Williams, F.A., 271, 282, 296

Williams, W., 6, 38

Wolfshtein, M., 259

Wosnik, M., 259

Wygnanscki, I., 147, 149, 157, 233, 234, 259

Y

Yaglom, A.M., 217, 220, 260

Yalin, M.S., 90, 92, 93, 102

Yan, A.S., 189, 210

Yao, L.-S., 129

Yarin, A.L., 58, 59, 70

Yarin, L.P., 71, 102, 110, 112, 113, 129, 130,

280, 281, 288, 291, 294, 296

Yenter, Y., 130

Yavanovich, M.M., 129

Z

Zel’dovich Ya, B., 149, 153, 157, 162, 164,

209, 263, 264, 266, 268, 271, 272, 275,

282, 296

Author Index 301

Subject Index

A

Acceleration effect, 80–81

Activation energy, 263, 280

Applying the II-theorem to transform PDE into

ODE, 63

Archimedes number, 19

Arrhenius law, 261, 263–264, 267, 270

B

Bessel function, 123, 272

Biot number, 19

Bond number, 19

Brinkman number, 14, 19

Buoyant jet, 149–154

C

Capillary number, 19, 95

Capillary waves in liquid lamella after a weak

drop impact onto a thin liquid film,

58–60

Clausius–Clapeyron equation, 289, 293

Co-flowing turbulent jets, 238–239, 242–244

Combustion of non-premixed gases, 264,

271–274, 278

Combustion waves, 268–271

Continuity equation, 7, 13, 31–34, 49, 52, 56,

67, 68, 71, 77, 85, 104, 134, 168, 239

Convective heat and mass transfer, 2, 160, 166

Couette flow, 178–179

Critical conditions, 268

D

Damkohler number, 17, 19, 21

Darcy number, 19

Dean number, 19, 118–120

Deborah number, 19

Delta function, 160

Diffusion boundary layer over a flat reactive

plate, 65–67

Diffusion flame, 273–281, 284, 288, 291,

293, 294

Dimensional and dimensionless parameters,

3–7

Dimensionless groups, 1, 3, 9, 13, 18–22, 26,

29, 40, 45, 48, 51, 53, 73, 81, 84, 87,

90–91, 110, 111, 115, 117, 118, 120,

122, 124, 139–141, 167, 179, 188, 189,

196, 202, 207, 220, 242, 249, 255, 256,

270, 273, 295

Dorodnitsyn–Illingworth–Stewardson

transformation, 197, 276, 278

Drag

of a body partially dipped in liquid, 87

of a deformable particle, 84–86

on a flat plate, 73–76

force, 1, 4, 7, 23–28, 71–101, 145

of an irregular particle, 82–84

on a solid particle, 76–82

of a spherical particle at low, moderate and

high Reynolds number, 1, 76–79

E

Eckert number, 19, 179

Eddy viscosity and thermal conductivity,

224–229

Effect

of energy dissipation, 112

of the free-stream turbulence, 1, 81,

171–173

of particle acceleration, 80


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Effect (cont.)of particle-fluid temperature difference,

1, 82

of particle rotation, 1, 79, 169–171

of velocity gradient, 174–175

Ekman number, 19

Enthalpy, 5, 16, 17, 134, 135, 184, 195, 196,

237, 261–263, 278, 280, 295

Entrance

flow regime, 106–108

region of pipe, 106, 107, 180–181

Euler number, 14, 17, 19, 22

F

Flow

in curved pipes, 116–120

in irregular pipes, 111–112

over a plane wall which has instantaneously

started moving from rest, 44–47

in straight rough pipes, 1

Fourier number, 122

Frank-Kamenetskii

approximation, 263, 264

parameter, 266, 267

Freezing of a pure liquid, 202–205

Froude number, 14, 17, 73, 86, 87, 256

Fully developed flow

in rough pipes, 109–111

in smooth pipes, 109

G

Gas torches, 2, 280–287

Grashof number, 19, 170, 186, 188–190, 193

H

Hadamard–Rybczinskii formula, 85

Heat

of reaction, 265, 267, 279, 290, 295

release, 19, 21, 66, 162, 261, 265, 266, 268,

269, 288, 294

Heat transfer

accompanying condensation of saturated

vapor on a vertical wall, 199–201

in channel and pipe flows, 2, 178–183

under the conditions of phase change, 160

from a flat plate in a uniform stream of

viscous high-speed perfect gas,

195–199

in forced convection, 165–178

from a hot particle immersed in fluid flow,

2, 165–169

in mixed convection, 2, 187–188

in natural convection, 186–194

from a spherical particle, 186–187

from a spinning particle, 170, 187–188

from a vertical hot wall, 190–193

I

Ideally stirred reactor, 265

Immersed flame, 288–294

Impinging turbulent jet, 252–254

Inhomogeneous turbulent jet, 232–238

J

Jacob number, 19

Jet flow, 43, 51, 131–156, 160, 225, 228, 229,

231, 245, 248, 287, 288

K

Knudsen number, 19, 112, 113

Kutateladze number, 19

L

Laminar

boundary layer over a flat plate, 14, 47–51

flow near a rotating disk, 55–58

flows in channels and pipes, 103–129

jets issuing from a thin pipe, 134

submerged jet, 51–54, 131, 141–143,

154–156, 185

wake of a solid body, 143–146

Lewis number, 18, 19, 262, 263, 269, 290, 295

M

Mach number, 19, 275, 278, 279

Mass

diffusivity coefficient, 262

flux, 23, 190, 194, 206, 208, 235, 247, 271,

285

transfer from a spherical particle in natural

and mixed convection, 189–190

transfer in forced convection, 165–178

transfer to a vertical reactive plate in natural

convection, 193–194

transfer to solid particles and drops

immersed in fluid flow, 176–178

304 Subject Index

Micro-channel flows, 112–113

Microkinetic law, 262

Mixing length, 153, 225, 226

N

Navier–Stokes equations, 7, 13, 27, 28, 31, 32,

34, 39, 52, 56, 67, 71, 77, 96, 104, 105,

112, 118, 120, 124, 136, 168, 261

Newton’s law, 28, 79

Nondimensionalization of the governing

equations, 1, 16

Non-Newtonian fluid flows, 1, 18, 113–116

Nusselt number, 15, 20, 166, 168–172, 175,

179, 181–183, 186–189, 193, 199

O

Oscillatory motion, 75–76

P

Peclet number, 14, 17, 20, 21, 166, 168, 169,

175–178, 190, 207, 274

Plane jet, 137, 147, 183, 250, 252

Prandtl number, 18, 20, 65, 82, 151, 168–170,

172–174, 179, 180, 182, 183, 186–188,

192, 193, 225, 228–229

Pre-exponential, 280, 289

Propagation of viscous-gravity currents over a

solid horizontal surface, 60–63

R

Rate of conversion, 262

Rayleigh number, 20, 187

Reynolds number, 1, 2, 14, 17, 18, 20–22, 25,

27, 28, 34, 72–84, 86, 88, 89, 91, 96, 97,

103, 105, 107, 110–111, 113, 115, 117,

118, 120, 124, 128, 142, 143, 153, 155,

168–170, 172–176, 180, 182, 188, 189,

211, 213, 219

Richardson number, 20

Rossby number, 20

S

Schmidt number, 18, 20, 30, 67, 194, 228, 229

Schvab–Zel’dovich transformation, 264, 272

Sedimentation, 1, 90–93

Self-similar solution, 39, 41–43, 58, 63, 67,

149, 152, 191, 201, 204, 205, 242, 249,

276, 279, 286

Semenov number,

Shear stress, 33, 36, 45, 50, 74, 77, 78, 88, 113,

114, 217, 221, 231, 234, 285

Sherwood number, 20, 176–178, 190, 207

Similarity, 1, 3, 11, 21–22, 211, 226, 245, 281

Single-one-step chemical reaction, 262

Spalding transfer number, 296

Stoichiometric coefficient, 264, 276

Stokes equations, 97

Strouhal number, 14, 17, 20, 75

T

Taylor number, 20

Temperature field, 150, 151, 153, 160–165,

183, 184, 191, 204, 205, 265

induced by a plane instantaneous thermal

source, 160–161

induced by a pointwise instantaneous

thermal source, 161–162

Terminal velocity of small heavy spherical

particle in viscous liquid, 87–90

Thermal

boundary layer over a flat plate, 63–65

characteristics of laminar jets, 183–185

diffusivity coefficient, 162, 163, 205, 262

explosion, 2, 263–268

particle in viscous liquid, 87–90

Thin liquid film on a plane withdrawn from a

pool filled with viscous liquid, 93–96

Total enthalpy, 263, 295

Total momentum flux, 52, 54, 133, 135, 136,

143, 146, 227, 229, 235, 241, 252, 256,

258, 285

Transfer in turbulent jet, 224

Turbulent

jet, 2, 135, 147, 153, 215, 224–254

wall jet, 248–251

Two-phase flow, 19, 71, 93

U

Unsteady flows in straight pipes, 120–123

V

Van-der Waals equation, 30–31

W

Wall jet over plane and curved surfaces,

146–149

Weber number, 20, 84, 100, 256

Subject Index 305

Experimental Fluid Mechanics

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