Fluid Mechanics (for Rajasthan Technical University) Dr. Kamlesh Purohit Professor Dept. of Mechanical Engineering, JNV University, Jodhpur Dr. S.P. Harsha Assistant Professor Dept. of Mechanical & Industrial Engineering, IIT, Roorkee Dr. R.K. Purohit Retd. Associate Professor Dept. of Mechanical Engineering, JNV University, Jodhpur

Published by: Scientific Publishers (India) 5-A, New Pali Road, P.O. Box 91, Jodhpur – 342 001 (India) E-mail: info@scientificpub.com Website: www.scientificpub.com © Authors, 2013 All rights reserved. No part of this publication or the information contained herein may be reproduced, adapted, abridged, translated, stored in a retrieval system, computer system, photographic or other systems or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the authors/editors and the publishers. Disclaimer: Whereas every effort has been made to avoid errors and omissions, this publication is being sold on the understanding that neither the author (or authors of chapters in edited volumes) nor the publishers nor the printers would be liable in any manner to any person either for an error or for an omission in this publication, or for any action to be taken on the basis of this work. Any inadvertent discrepancy noted may be brought to the attention of the publishers, for rectifying it in future editions, if published. ISBN: 978-81-7233-816-9 eISBN: 978-93-86237-73-6 Lasertype set: Rajesh Ojha Printed in India

4ME3: FLUID MECHANICS

Syllabus UNIT -1 — Basic Definitions and Fluid Properties; Definition of Fluid,

Incompressible and compressible fluids, Fluid as a continuum, Mass, Density, specific weight, relative density, specific volume, Bulk modulus, velocity of sound Ideal fluid Viscosity. Newtonian and Non Newtonian fluid, Kinematic viscosity, Effect of temperature and pressure on viscosity, surface tension capillarity, vapour pressure and cavitation.

UNIT -2 — Kinematics and conservation of Mass : Flow classifications. Fluid velocity and acceleration, streamlines and the stream function. Pathlines and streak lines. Deformation of a fluid element, verticity and circulation. Irrotational and Rotational flow. Flownet, Laplace equation. Conser-vation of mass and the continuity equation for three dimensions. Fluid Momentum: The Momentum theorem Applications of the momentum theorem Equation of motion, Euler’s equation of motion Integration of Euler’s equation of motion. Bernoulli’s equation. Applications of Bernoulli’s Pitot tube, Equation of motion for Viscous fluid, Navier Stoke’s equation.

UNIT -3 — Orifice discharging free, Jet, vena contracts, co-efficient of contraction, velocity and discharge, coefficient of resistance. Orifices and mouthpieces Nozzles and weires. Flow Through Pipes: Reynold’s experiment Darcy’s Weisback equation. Loss of head due to sudden enlargements, contraction, entrance, exit obstruction, bend, pipe fittings. Total and Hydraulic grandient lines, Flow through pipe line. Pipes in series, parallel Transmission of power through pipes.

UNIT -4 — Laminar Flow: Simple solution of Navier Stokes equations. Hagen – Poiseuille flow. Plans Poiseuille flow and coutte flow. Turbulent Flow; Variation of friction factor with Reynold’s number. The Prandt Mixing length hypothesis applied to pipe flow, velocity distribution in smooth pipes, sough pipes. The Universal pipe friction laws, Colebrook. White formula. Dimensional Analysis: Buckingham variables, Model Similitude, Force ratio, Reynolds, Froude’s Mach, Weber and Euler numbers and their applications. Undistorted model distorted model scale effect.

UNIT -5 — The Boundary Layer: Description of the boundary layer. Boundary Layer thickness boundary layer separation and control. The Prandtl boundary layer equation. Solution for cominar boundary layer. The momentum equation for the boundary layer. The flat plate in uniform free stream with no pressures gradients. Approximate momentum analysis laminar boundary Aerofoils Theory. Flow round a body; Drag skin friction drag, pressure drag, combined skin friction & pressure drag (Profile drag) wave drag, lift induced drag. Flow past sphere & Cylinder.

Preface

It gives me immense pleasure in placing this book `Fluid Mechanics for RTU' in the hands of B.Tech IV Semester students of Mechanical Engineering of Rajasthan Technical University (RTU).

The subject matter has been evolved from first principles and the treatment is then extended to considerable depth so that this single book covers the complete course of IV Semester of RTU.

The importance of illustrative examples cannot be under estimated, as they help in understanding the concepts of the subject clearly. Besides, they also follow the style of question set in RTU.

We are thankful to colleagues, friends and students who encouraged us to write this book.

We express our appreciation and gratefulness to our publisher Tanay Sharma for his most cooperative, painstaking attitude and untiring effort in bringing out this book in a very short period.

We have used several references and we are grateful to those learned authors.

Jodhpur Dr. Kamlesh Purohit Dr. S.P. Harsha

Dr. R.K. Purohit

Contents

1. BASIC CONCEPT RELATING TO FLUIDS 1

1.1 Introduction 1 1.2 Definition of a Fluid 1 1.3 Fluid as a Continuum 1 1.4 Incompressible and Compressible Flow 2 1.5 Basic Definitions 2 1.5.1 Mass 2 1.5.2 Density 2 1.5.3 Specific Volume 2 1.5.4 Specific Weight 3 1.5.5 Relative density 3 1.6 Viscosity 3 1.6.1 Units of Viscosity 4 1.6.2 Dimensional Formula of Viscosity 5 1.6.3 Kinematic Viscosity 6 1.6.4 Units of Kinematic viscosity 6 1.6.5 Dimensional formula of kinematic viscosity 7 1.6.6 Newtonian and non Newtonian fluids 7 1.6.7 Effects of temperature and pressure on viscosity 8 1.6.8 Ideal Fluid 10 1.7 Compressibility and Elasticity of Fluids 11 1.8 Surface Tension 11 1.9 Capillarity 12 1.10 Pressure inside a Water Droplet, Soap and Bubble 15 Illustrative Examples 16

vi Fluid Mechanics

EXERCISES A. Theory 34 B. Unsolved problems 35

2 STATIC PRESSURE AND ITS MEASUREMENT 38

2.1 Introduction 38 2.2 Pressure at a Point 38 2.3 Pascal's Law 38 2.4 Euler's Differential Equations 40 2.4.1 The basic equation of hydrostatics 41 2.4.2 Pressure head 42 2.4.3 The hydrostatic paradox 43 2.5 Atmospheric Pressure 44 2.5.1 Fortin barometer 44 2.6 Application of the Basic Equation of Fluid Statics 45 2.7 Manometers 46 2.7.1 Piezometer 47 2.7.2 Simple manometers 47 2.7.3 Differential Manometers 49 2.8 Measurement of Small Pressure Difference 49 2.8.1 Inclined gauge 50 2.8.2 Micromanometers 51 2.9 Pressure Variation in a Compressible Fluid 52 2.9.1 Variation under Isothermal conditions 52 2.9.2 Variation under adiabatic conditions 53 2.9.3 Variation of pressure and density with altitude for a constant

temperature gradient 55

2.9.4 Variation of temperature and pressure in the atmosphere 55 ILLUSTRATIVE EXAMPLES 57 EXERCISES A. Theory 78 B. Unsolved Problems 80

3 FLUID STATICS 87

3.1 Introduction 87 3.2 Total Pressure 87 3.3 Centre of Pressure 87

Contents vii

3.4 Total Pressure and Centre of Pressure on A Vertical Plane Surface 87 3.5 Total Pressure and Centre of Pressure for Inclined Surface 90 3.6 Pressure Diagrams 92 3.7 Pressure on Curved Surfaces 94 3.7.1 General case of pressure on curved surfaces 94 3.7.2 Pressure on cylindrical surfaces 95 Illustrative Examples 96 EXERCISES A. Theory 126 B. Unsolved Problems 127

4 BUOYANCY AND FLOATATION 133

4.1. Introduction 133 4.2. Buoyancy 133 4.3. Centre of Buoyancy 134 4.4. Equilibrium of Floating Bodies 134 4.5. Metacentre 134 4.6. Metacentric Height 135 4.7. Stability of Floating Bodies-Metacentre and Metacentric Height 135 4.8 Experimental Method of Determination of Metacentric Height 136 4.9 Analytical Method for Metacentric Height 137 4.10 The Period of Roll of a Vessel 139 Illustrative Examples 139 Exercises A. Theory 150 B. Unsolved problems 151

5 KINEMATICS OF FLUID 153

5.1 Introduction 153 5.2 Description of Fluid Motion 153 5.3 Fluid Flow Classifications 154 5.3.1 Steady flow and unsteady flow 154 5.3.2 Uniform flow and non uniform flow 154 5.3.3 One, two and three dimensional flow 155 5.3.4 Laminar flow and turbulent flow 156 5.3.5 Rotational flow and Irrotational flow 156

viii Fluid Mechanics

5.4 Flow Lines 157 5.4.1 Streamline 157 5.4.2 Stream tube 158 5.4.3 Path line 158 5.4.4 Streak line 159 5.5 Velocity and Acceleration 160 5.5.1 Convective acceleration and Local acceleration 161 5.5.2 Tangential acceleration and normal acceleration 162 5.6 Principle of Continuity-Conservation of Mass Flow 163 5.6.1 One dimensional continuity equation 164 5.6.2 Continuity equation for three dimensional flow using Cartesian

Coordinates 165

5.7 Deformation of a Fluid Element 166 5.8 Circulation and Vorticity 169 5.8.1 Circulation for the rectangular element 171 5.8.2 Circulation for the circle 171 5.8.3 Vorticity 172 5.9 Stream Function and Velocity Potential 172 5.9.1 The stream function 172 5.9.2 Velocity Potential 175 5.10 Flow Net 178 5.10.1 Methods of drawing flow nets 179 5.10.2 Uses of flow net 182 5.10.3 Limitations of flow net 182 ILLUSTRATIVE EXAMPLES 182 EXERCISES A. Theory 194 B. Unsolved problems 196

6 DYNAMICS OF FLUID FLOW 199

6.1 Introduction 199 6.2 Euler's Equation of Motion 200 6.2.1 Bernoull's equation-Integration of Euler's equation along a

streamline for steady flow 201

6.2.2 Limitations on Bernoulli's equations 202 6.2.3 Modification to Bernoulli's equation 203

Contents ix

6.2.4 The physical significance of Bernoulli's equation 203 6.3 Applications of Bernoulli's Equation 204 6.3.1 Venturimeter 204 6.3.1.1 Venturimeter analysis 206 6.3.1.2 Vertical/Inclined Venturimeter 210 6.3.1.3 Use of differential manometer in venturimeter 212 6.3.2 Orifice meter 213 6.3.3 Flow nozzle or Nozzle meter 216 6.3.4. Flow tubes 218 6.3.5 Pressure recovery 218 6.4 Velocity Measurements 218 6.4.1 Static, stangnation and dynamic pressures 219 6.4.2 Pitot tube 220 6.4.3 Pitot- static tube 221 6.5 Momentum Equation 222 6.5.1 Impulse-Momentum equation 223 6.5.2 Momentum equation for two-and there dimensional flow along

a stream line 224

6.5.3 Momentum correction factor 225 6.5.4 Application of the momentum equation 226 6.5.4.1 Forces on a pipe bend 226 6.5.4.2 Force due to the diflection of a jet by a curved vane 227 6.5.4.3 Force at a nozzle 228 6.5.4.4 Reaction of a jet 228 6.6 Navier-Stokes Equations 229 6.2.1 Navier-Stokes equation in vector form and meaning of each

term 232

6.2.2 Navier-Stokes in cylindrical polar coordinates 233 Illustrative Examples 234 EXERCISES A. Theory 267 B. Unsolved Problems 269

7 FLOW THROUGH ORIFICES AND MOUTH PIECES 273

7.1 Introduction 273 7.2 Sharp Edged Orifice Discharging Free 273

x Fluid Mechanics

7.3 Hydraulic Coefficients 275 7.3.1 Coefficient of contraction (Cc) 275 7.3.2 Coefficient of velocity (Cv) 276 7.3.3 Coefficient of discharge (Cd) 276 7.3.4 Coefficient of resistance (Cr) 277 7.4 Experimental Determination of Hydraulic Coefficients for an Orifice 277 7.4.1 Experimental determination of coefficient of contraction 277

7.4.2 Determination of coefficient of velocity Cv 278 7.4.2.1. Jet distance measurement method (Trajectory method) 278 7.4.3 Determination of coefficient of discharge 280 7.5 Submerged Orifice 280 7.6 Partially Submerged Orifice 281 7.7 Sharp Edged Large Vertical Orifice with Rectangular Shape 282 7.8 Mouthpieces or Tubes 283 7.8.1 External cylindrical mouthpiece running full 284 7.8.2 Flow through convergent divergent mouthpiece 287 7.8.3 Borda's or Re entrant mouthpiece 289 7.8.3.1 Borda's mouthpiece running free 289 7.8.3.2 Borda's mouthpiece running full 290 7.9 Flow through an Orifice or a Mouthpiece under variable heads 292 7.9.1 General procedure for calculating time of emptying a tank

through an orifice /mouthpiece at its bottom 292

7.9.2 Time of emptying cylindrical tank 293 7.9.3 Determine the constant head under a head falling 294 7.10 Time of Emptying (or Filling) A Tank with Inflow 295 7.11 Flow of Liquid from One Vessel to Another 296 Illustrative Examples 297 EXERCISES A. Theory 308 B. Unsolved problems 310

8 FLOW OVER NOTCHES AND WEIRS 312

8.1 Introduction 312 8.2 Rectangular Weirs 312 8.2.1 Flow over rectangular weir 313 8.2.2 Flow over rectangular weir with velocity of approach 314

Contents xi

8.2.3 Empirical formulae for discharge over rectangular weir 315 8.3 Flow over a Triangular Weir (V-Notch) 317 8.4 Flow over a Trapezoidal Weir or Notch 319 8.4.1 Cippoletti weir 320 8.5 Long Based Weirs 322 8.5.1 Broad crested weir 322 8.5.2 Round nosed weirs 324 8.5.3 Crump weirs 324 8.6 Submerged Weirs 325 8.7 Ventilation of Weir 326 Illustrative Examples 327 EXERCISES A. Theory 340 B. Unsolved problems 341

9 FLOW THROUGH PIPES 344

9.1 Introduction 344 9.2 Reynolds Experiments 344 9.2.1 Reynolds number and its Significance 346 9.2.2 Laminar and turbulent flow 346 9.2.3 Critical Reynolds number 348 9.3 Fluid Friction 348 9.4 Head Lost Due to Friction in Pipes-Darcy's Weisbach Equation 349 9.4.1 Proof of the Darcy's Weisbach equation 349 9.4.2 Chezy's formula 351 9.4.3 Manning's formula 351 9.4.4 Hazen William's formula 351 9.5 Minor Losses 352 9.5.1 Head loss due to sudden enlargement 352 9.5.2 Head loss due to sudden contraction 354 9.5.3 Head loss at entrance to pipe 356 9.5.4 Exit loss 356 9.5.5 Head loss due to obstruction 357 9.5.6 Head loss due to bends, valves, Non symmetrical sections, etc. 358 9.6 Total Energy Line and Hydraulic Gradient 359 9.6.1 Total energy line (TEL) 359

xii Fluid Mechanics

9.6.2 Hydraulic gradient line (HGL) 359 9.7 Flow Through Pipe Lines 360 9.8 Flow through Pipes Connected in Series 362 9.9 Method of Equivalent Lengths 364 9.10 Flow through Pipes Connected in Parallel 365 9.11 Power Transmission through Pipes 366 9.11.1 Maximum power transmission efficiency 368 9.12 Flow through Nozzle at the End of a Pipe 368 9.12.1 Efficiency of Power transmission through nozzle 370 9.12.2 Condition for maximum power transmission through nozzle 370 9.12.3 Diameter of nozzle for maximum transmission of power

through nozzle 371

9.13 Water Hammer 372 9.14 Water hammer analysis 372 9.14.1 Rigid Water Column theory 373 9.14.2 Elastic pipe theory 375 Illustrative examples 377 EXERCISES A. Theory 402 B. Unsolved problems 405

10 LAMINAR VISCOUS FLOW 410

10.1 Introduction 410 10.2. Hagen-Poiseuille Flow 410 10.3. Plane Poiseuille Flow 416 10.4 Coutte Flow 420 Illustrative Examples 424 EXERCISES A. Theory 433 B. Unsolved Problems 435

11 TURBULENT FLOW THROUGH PIPES 437

11.1 Introduction 437 11.2 Turbulent Shear Stress 439 11.3 Boussines Eddy Viscosity 440 11.4. Prandtl's Mixing Length Theory 440 11.5 Shear Velocity or Friction Velocity 441

Contents xiii

11.6 Prandtl's Universal Velocity Distribution Equation for TurbulentPipe Flow

443

11.7 Hydrodynamically Smooth and Rough Boundaries 445 11.8 Velocity Distribution for Turbulent Flow in Smooth and Rough

Pipes-Prandtl Karman Velocity Distribution Equation 447

11.8.1 Velocity distribution in smooth pipes 448 11.8.2 Velocity distribution in rough pipes 450 11.9 Velocity Distribution for Turbulent Flow in Terms of Average

Velocity 451

11.9.1 Turbulent flow in smooth pipes 452 11.9.2 Turbulent flow in rough pipes 452 11.9.3 Difference between point velocity and average velocity for

smooth and rough pipes 452

11.10 Turbulent Pipe Coefficient 454 11.11 The Chronological Development of Turbulent Pipe Flow Theories 454 11.11.1 Smooth pipes and Blasius equation 455 11.11.2 Stanton and Pannell 456 11.11.3 Nikuradse experimental results using artificially rough pipes 456 11.11.4 The smooth and rough laws of Prandtl and von Karman 458 11.10.5 The Colebrook-White transition formula 461 11.11.6 Moody diagram for commercial pipes 464 11.11.7 Hydraulic Research station charts (HRS) Acketes 467 11.11.8 Barr explicit formula 468 11.11.9 Murdock formula 468 11.11.10 Swamee and Jain's explicit equation 469 11.11.11 S.E. Haaland's formula 469 11.12 Non Circular Pipes 469 11.13 Roughness of Pipes with Age (Old Pipes) 470 Illustrative Examples 470 Exercises A. Theory 486 B. Unsolved Problems

490

12 DIMENSIONAL ANALYSIS AND SIMLITUDE 492

12.1 Introduction 492 12.2 Dimensions, Dimensional Homogeneity and Units 493

xiv Fluid Mechanics

12.3 Dimensional Analysis 495 12.3.1 The Buckingham π - theorem 496 12.3.2 Selection of repeating variables 496 12.3.3 Determining the π groups 497 12.3.4 Some additional comments about dimensional analysis 498 12.3.5 Uniqueness of π terms 499 12.3.6 Limitations of dimensional analysis selection of variables -

superfluous and omitted variables 499

12.4 Modeling and Similitude 500 12.4.1 Geometric Similarity 501 12.4.2 Kinematic similarity 502 12.4.3 Dynamic similarity 503 12.4.4 Standard dimensionless numbers 503 12.4.4.1 Reynold's number (Re) 504 12.4.4.2 Froude's number ( RF ) 504

12.4.4.3 Mach's number (M) 505 12.4.4.4 Euler's number (Eu) 505 12.4.4.5 Weber's number (Wb) 506 12.5 Model Laws 506 12.5.1 Reynold's model law 507 12.5.2 Froude's model law 508 12.5.3 Mach model law 510 12.5.4 Euler's model law 510 12.5.5 Weber's model low 511 12.6 Undistorted and Distorted Models 511 12.7 Scale Effect 513 12.8 Comments on Model Testing 513 Illustrative Examples 514 EXERCISES A. Theory 537 B. Unsolved problems 539

13 BOUNDARY LAYER THEORY 544

13.1 Introduction 544 13.2 Description of the Boundary Layer 545 13.3 Boundary Layer Thicknesses 548

Contents xv

13.3.1 Boundary layer thicknesses δ 548 13.3.2 Boundary layer displacement thickness 548 13.3.3 Momentum thickness θ 549 13.3.4 Energy thickness δ** 550 13.4 Local Skin Friction and Average Skin Friction Drag Coefficient 551

13.4.1 Local skin friction drag coefficient Cf 551

13.4.2 Average skin friction drag coefficient CD 551 13.5 The Prandtl Boundary Layer Equations 551 13.6 Blasius Solution 552 13.7 Momentum Integral Boundary Layer Equation for Flat Plate or VON

Karman Integral Equation 557

13.7.1 Momentum integral equation for zero pressure gradient 561 13.8 Momentum Integral Method for Laminar Flow over a Flat Plate 562 13.9 Turbulent Boundary Layer 566 13.10 Combined Laminar and Turbulent Boundary Layers 569 13.11 Coefficient of Drag for Turbulent Boundary Layer for Rough Plate 571 13.12 Flow with a Pressure Gradient 572 13.13 Separation of Boundary Layer Flow 573 13.13.1 Examples of separation of boundary layer flow 576 13.13.2 Separation control 578 Illustrative Examples 580 EXERCISES A. Theory 602 B. Unsolved Problems 606

14 FLOW OVER IMMERSED BODIES 607

14.1. Introduction 607 14.2 Forces on Immersed Bodies-Drag and Lift 608 14.3 Drag on Immersed Bodies 611 14.4 Types of Drag 611 14.4.1 Skin friction drag 611 14.4.2 Pressure drag 612 14.4.3 Profile drag 613 14.4.4 Deformation drag 613 14.4.5 Wave drag 613 14.4.6 Induced drag 614

xvi Fluid Mechanics

14.5 Streamlined and Bluff Bodies 614 14.6 Drag on Sphere 615 14.6.1 Dynamics of sports ball 619 14.7 Drag on Cylinder 620 14.7.1 von Karman vortex street 623 14.7.2 Summary of flow regimes in flow past a circular cylinder 625 14.8 Lift 626 14.8.1 Magus effect and the circulation theory of lift

(Kutta-Joukowski theorem) 626

14.9 Lift of an Aerofoil 627 14.10 Aerofoil Terminology 628 14.11 Inviscid Flow Past a Two Dimensional Aerofoil 629 14.12 Real (Viscous) Fluid Past A Two Dimensional Aerofoil 631 14.13 Aerofoil Characteristics 632 Illustrative Examples 634 EXERCISES A. Theory 644 B. Unsolved Problems 645