OPEN CHANNEL TURBULENT BOUlYDARY LAYERS

AND WALL JE:TS ON ROUGH SURFACES

A Thesis Submitted to the College of

Graduate Studies and Research

in Partial Fuifiliment of the Requirements

for the Degree of Doctor of Philosophy

in the Department of Mechanical Engineering

University of Saskatchewan

Saskatoon

BY

Mark Francis Tachie

November 2000

Q Copyright Mark Francis Tachie, 2000. Al1 rights reserved.

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PERMISSION TO USE

- The author grants petmission to the University of Saskatchewan Libraries to make

this thesis available for inspection. Copying of this thesis, in whole or in part, for

scholarly purposes may be granted by my supervisors (Professors Ram Balachandar and

Donaid J. Bergstrom), the Head of the Department of Mechanical Engineering, or the

Dean of the College of Engineering. It is understood that any copying or publication or

use of this thesis or part theteof for financiai gain shail not be ailowed without my

written permission. It is also uaderstood that due recognition to me and the University of

Saskatchewan must be granted in any scholarly use which may be made of any material

in this thesis.

Request for permission to copy or to make other use of materiai in this thesis in

whole or in part should be addresseci to:

Head of Deparement of Mechanicd Engineering

University of Saskatchewan

57 Campus Mve

Saskatoon, Saskatchewan

SM SA9

For many industrial and environmental flows, the momentum and convective heat

transfer rates at the d a c e are determined by the turbulence structure in the near-wall

region. Although many flows of practicai interest occur on rough surfaces, our ability to

predict rough wall turbulent flows Iags fh behind the correspondhg technology for

smooth surfaces. This provides reasonable grounds for additionaI refined rough wall

measurements wiîh the expectation of improving our physical understanding of

practically relevant turbulent flows.

This thesis reports a comprehensive experimental investigation of wall roughness

effects on the characteristics of a turbulent boundary layer and wall jet. The

measurements are obtained for a hydraulicdy smooth as well as three geomeûically

different rough surfaces using a LDA system. For the smooth wali measurements, data

are obtained in the viscous sublayer which then d o w the wall shear stress to be

accurately detennined, Data presented include the streamwise and wall-normal

components of the mean veIocity and theh fluctuations, Reynolds shear stress as well as

distributions of turbulence kinetic energy budgets and d g Iength. An insighrful

presentation of the results requires that the correct scaling laws must be used. in the case

of the turbulent boundary layer, the appropriateness of the classical log law proposed by

Millikan (1938) to mode1 the overlap region of the mean flow as well as the recent

power laws proposed by BarenbIatt (1993) and George and Castilio (1997) is aiso

examined. The present results are Uiterpreted within the wntext of the 'wall similarity

hypothesis', which States that, outside the roughness sublayer, turbulent motions are

independent of wall mughness at sufficiently high ReynoIds numbers.

The boundary layer results show that, irrespective of the specific d a c e

conditions, the power law proposed by George and CastilIo (1997) has miportant

advantages over the log law both in modeling the mean velocity profiles as well as

predicting the wall shear stress. The results also show that the characteristics of the

turbulence structure and the transport tenns depend on the specific geomeûy of the

roughness elements, which suggests that rough wall turbulence models must expiicitly

account for the specific geometry of the roughness eIements in order to predict the

mixing characteristics accurately. This promises to provide significant challenges to

rough wall turbulence models. in the case of turbulent wall jet, it is observed that surface

roughness increases the inner layer thickness but the jet half-width does not show any

important sensitivity to d a c e roughness. The fact that the spread rate is not altered by

d a c e roughness suggests that a wall jet is a complex flow in which the mechanisrns of

near-wall damping are not the same as in a simple boundary layer. The spread rates for

the jet half-widîh are higher than the values obtained in eariier measurements. This may

be attriiuted to the high background ttrrbuience levels in the present flow. It is also

observed that the streamwise evolution of the mean flow is nearly independent of initial

conditions when scaled using the exit kinematic mumentum.

1 would like to express my pmfound appreciation trr my supervisors, Professors

Ram Balachmdar and Donald J. Bergstrom. Thank you for your guidance, advice,

thoughtfûl insights and so generously taking pur t h e s to discuss varied aspects of this

research. My sincere tfianlcs to you and your families for your love and support for my

famil y.

My sincere thanks go to my advisory cornmittee members: Professors E. M.

Barber, R. W. Besant, J. il. Bugg and D. Sumner for their usehi suggestions. 1 would

also Iike to thank my Extemai Examiner Dr- V. C. Patel for his very useful comments

and suggestions. Special thanks to Professors G. J. Schoenau and S. YanacopoIous, for

their motivation throughout my graduate program. Technical assistance provided by

Dave Deutscher and Ed Ailen is gratefdiy acknowledged. My appreciation aiso goeç to

the secretanes in the Departmats of Mechanicd and Civil Engineering for their varied

help and support.

It is my pleasure to acknowledge the encouragement and supports received fiom

my wife, Mom and Dad and ail rdatives who heIped in many ways to make this program -

a success. Special thanks also go to ML Anthony Yadaar, Richard Abada, Nathan Kotey,

Franklin Krampa-MorIu, Shyamsundar Ramachandran, Deighen Blakely, Mrs. Theodora

Kokumaù, Dr. ikechukwuka Oguocha, Dr. and Mrs. Amoako, Dr. and M. Adzanu, Dr.

and Dr. Mpofb, Mr. and Dr. Gana, Abena Owum-Boakye, JJustina Anaab, Opeyemi and

Busola Fagbemiro and Dr. and Ms. Kumaran for the interest show in this program and

aiso for their supports for my family.

My thanks aiso go to Professor R 1. Karlsson of the RoyaI institute of

TechnoIogy, Sweden, and his CO-workers for making their wail jet data and ment

unpublished work on 'Similarity Theory for Plane Turbulent Wall Jet' avdable to me.

Financiai supports f?om the Natural Sciences and Engineering Research Council of

Canada (NSERC), the University of Saskatchewan Graduate Scholarship and the donor

of Clarence Forsberg Metnorial Scholarship are acknowledged and sincereky

appreciated.

DEDICATION

This thesis is dedicated tu my wre (Akofo) und son (Sedem) for their love andpatience

throughour this program; and to Mr. L M. K Adlgndeh for his interest in my

educan'on.

TABLE OF CONTENTS

PERMISSION TO USE

ACKNOWLEDGEMENT

DEDICATION

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

NOMENCLATURE

1. rNTRODUCTTON

1.1 GeneraI Remarks

1.1.1 Turbulence

1.1.2 Equations of Motion and Notations

1.1.3 Motivation for Near-wall Turbulence Research

1.1.4 Canonid and Cornplex Near-wall Flows

1.1.5 Research Methodologfes

1.1.6 Measuring Devicw

1.2 Reynolds Nimiber Effects and Scaling Issues

1.3 Surface Roughness and the Wall Similarity Hypothesis

1 -3.1 Dennitions and T d n o l o g y

1.3.2 Some Characteristics of Surface Roughness

1 2.3 Types o f Surface Roughness

1.3.4 The Wall SUniIarity Hypothesis

1.4 Turbulent Wali Jets

1.4.1 Definition and NomencIature

1.4.2 Applications of Wall Jets

1 -5 Some Characteristics of Open Channel Flows

xii

xvii

1.6 Summary

1 -7 Objectives and Scope

I .7.1 Objectives

L.7.2 Scope

2. LITERATURE REVIEW

2.1 Theoreticai Analysis

2.1. I Turbulent Boundary Layers

2.1. L .1 Scaling Law for the Inner Layer

2.1. L -2 Scaling Law for the Outer Layer

2.1.1.3 Scaling Laws for the Overlap Region

2-1.13.1 The Logarithmic Law

2. I -1.32 The Power Laws

2.1- 1 -4 Determination of Shear Sûess

2.1.1.5 Scaling the Turbulence Quantities

2.1.2 Turbulent Wall Jets

2.1.2.1 Scaiing the Transverse Profiles

2.1 2.2 Skùi Friction Conelation

2.1.2.3 Streamwise Development

2.2 Previous Studies

2.2.1 Turbulent Boundary Layers

2.3.1-1 Reynolds Number Effects

2.2.1.2 Surface Roughness Effects

2.2.1.3 Effects of Elevated Freestream TurbuIence

2.2.1.4 Turbulent Boundary Layer in Open Channel Flows

22.2 Turbulent Wall Jets

2.3 State of Knowtedge and Refïnement in objectives

3. INSTRUMENTATION AND EXPERIMENTAL DETAILS

3.1 The Laser DoppIer Anemometry

3.2 Errors in LDA Measuranents

3.3 ExperimentaI Set-Up

3.3.1 The Opai Channel Hume

3.3.2 The Watl let Facility

3.3.3 Description of Surface Roughness

3.4 Instrumentation and Measutement Procedure

3.5 Experimentd Details

3.5.1 Boundary Layers

3 S. 1.1 Series A: Reynolds Number Effects

3.5.1 2 Series 8: 1 -D Smooth and Rough WaU Experiments

3.5.1.3 Series C: 2-D Smooth and Rough WaIl Experiments

3.5.2 Wall Jet Experiments

3.5.2.1 Series D: 1 -D Smooth and Rough Wall Experiments

3.5.2.2 Series E: 2-D Smooth and Rougb WaU Experiments

3.6: Uncertainty Estllnates

4. SURFACE ROUGHNESS AND LOW REYNOLDS NUMBER EFFECTS

ON THE STREAMWBE VELOCITY COMPONENT 79

4.1 Determination of Wall Shear Stress 79

4.2 Mean Veiocity Distriions 84

4.2. t Mean ProfIIes in Outer Coordinates 84

4 6 2 Mean Velocity M i Profiles 86

42.3 Mean Profifes in Inner Coordinates 90

4.2.4 Cornparison Between Log Law and Power Laws 93

4.3 Turbuience intensity 100

4.4 SkewnessandFlatnessFacbts 1 OB

4.5 Triple Correlation 1 I l

4.6 Summary 114

5. EFFECTS OF SURFACE ROUCI'HNESS ON TURBULENCE

STRUCTURE

5.1 Detennmation of Friction velocity

Mean VeIocity Profiles

5.2.1 Outer Coordinates

5.2.2 Inner Coordinates

Turbulence htensity and ReynoIds Stresses

Shear Stress Correlation Coefficient

Stress Anisotropy Tensor

Skewness and Flatness Factors

Triple Correlation

Energy Budgets

Mixing Length and Eddy Viscosity

5.10 Summary 153

6.CHARACTERISTICS OF ATURBULENT WALL JET ON SMOOTH AND

ROUGH SURFACES 155

6.1 Flow Qualification 155

6.1.1 Exit Profiles t 55

6.1.2 Effects of Reverse Flow and Flow Development 158

6.1.3 Test for Two-Dimensiondity 163

6.2 Streamwise Evolution of the Mean Flow 166

6.2.1 Velocity Decay 166

6.2.2 ûrowth of Inner Layer and Jet HaIf-width 169

6.3 Transverse Velocity Profiles 172

6.3.1 Mean Velocity Protiles 172

6.3.1.1 Determination of Friction Veiocity 172

6.3.1 2 Mean Profiles in Outer Coordinates 175

6.3.1 3 Mean Profles in b e r Coordinates 178

6.3.2 Reynolds Shear Stress and Turbuience Intensities 181

6.3.3 Triple Correlation 188

6.3.4 Energy Budgets 188

6.3.5 Mixing Length and Eddy Viswsity 195

6.4 Summary 198

6. SUMMARY, CONCLUSIONS, CONTRIBUTIONS AND FUTURE WORK 200

7.1 Summary 200

7.1.1 Turbulent Boundary Layer 200

7.1 -2 TurbuIent Wail Jets 203

7 2 Conclusions 204

73.1 Turbulent Boundary Layer 204

7.22 Turbulent Wall Jets 205

7.3 Contributions 207

7.4 Recommendations for Future Work 207

APPENDIX A: AN OVERVTEW OF A LDA 22 1

APPENDIX B: ERRORS IN LDA MEASUREMENTS 225

APPENDIX C: UNCERTAINTY ANALYSIS 236

LIST OF TABLES

Table 2.1 : Summary of log law constants for turbulent w d jets 39

TabIe 2.2: Summary of some earlier wall jet studies 54

Tabte 2.3: Summary of sbte of knowledge regarding roughness effects on

turbuIence structure 59

Table 3.1: Optical and operating parameters of the LDA 67

Table 3.2: Summary of test conditions for Series A 7 1

Table 33: Summary of test conditions for Series B 72

Table 3.4: Summary of test conditions for Series C 74

TabIe3.5: Summaryoftestconditions forseries D 76

Table 3.6: Summary of test conditions for Series E 77

Table 4.1 : Summary of fiction velocity and deviation of measured U kom log iaw profiIe at y' = F 80

Table 4.2: Summary of skin fiction velocity and wake parameter for Series B 83

Table 4.3: Summary of power law constants and fiiction velocity 94

Table 5.1 : Test conditions and boundary Iayer parameters 118

Table 6- 1 : Sumary of exit conditions 157

Table C. 1 : Typicd uncertainty estimates for Test C-SMH 237

Table C.2: Typicd uncertainty estimates 23 8

LIST OF FIGURES

Fig. 1.1 Definition sketch of roughness elememts

Fig. 1.2 Schematic of a turbulent waI1 jet

Fig. 3.1 A schematic of open channel flume

Fig. 3.2 A schematic of wall jet facility

Fig. 3.3: Pictures showing sections of (a) perforateci pIate (PF) and (b) wire mesh (WM) rough surfaces

Fig. 4. la Determination of Ut for smooth wail data: Linear and poiynomial fits to near-wail data

Fig. 4.1 b Determination of ïi and U, for smooth and rough wall data

Fig. 4.2 Distriiutions of mean velocity in outer coordinates (a) smooth data at various Ree (b) smooth and rough wall data

Fig. 4.3 Distn'bution of mean velocity defect (a) inner coordinates (b) outer coordinates

Fig. 4.4a Mean defect profiles for smooth waii

Fig. 4.4b Velocity defect profiles for smooth and rough surfaces

Fig. 4.5 Velocity disüiiution in inner variables: (a) smooth wall data at various RQ (b) cornparison between smooth and rough waU data

Fig. 4.6 Log law and power law profiles on smooth surface

Fig. 4.7 Log law and power law profiles for sand grain data

Fig. 4.8 Log law and pwer Iaw profiles for wire mesh

Fig. 4.9 Distributions of turbulence intensity (a) present and previous canonicai boundixy layer flows (b) present and previous studies at high f b s t m m turbulence

Fig. 4. IO Turbdence intensity profiles at various Ra (a) inner variables (b) outer coordinates

Fig. 4.1 1 Variation of turbuience intensity profles on smooth and rough wdls (a) inner variables (b) outer coordinates 1 07

Fig. 4.12a Distributions of skewness factors at various RQ 110

Fig. 4.12b Distniutions of £labas factors at various RQ 110

Fig. 4.13 Variation of eu3> with Reynolds number (a) inna scaling (b) mixed scaiing derived h m AP 112

Fig. 5.1 Mean veiocity in outer coordinates (a) streamwise; (b) vertical

Fig. 5.2 Mean profiies in inner cocdhwes (a) lines represent log law pronles (ô) lines: solid @olynomjal fits); dash (composite profiles) 122

Fig. 5.3 Disîriiution of mean velocity gradient 124

Fig. 5.4 (a) Streamwise turbulence intensity in outer coordinates 126 (b) Streamwise Reynolds stress in b e r coordinates 126

Fig. 5.5 (a) VerticaI turbulence intensity in outer variables (b) Verticai ReynoIds stress in inner variables

Fig. 5.6 Reynolds shear stress on smooîh and rough srafaces (a) outer scaling (b) inner scaiing 130

Fig. 5.7 Distniution of shear correlation 133

Fig. 5.8 Distriiution of stress anisottopy teasor (a) streamwise; (b) vertid; (c) shear

Fig. 5.9 Distn'butions of skewness and flatness factors (a), (c) i ~ e r coordinates; (b) , (d) outer cootdinates

Fig. 5.1 0 Distniutioos of tripie correlation (normalized by uI'U,) 141

Fig. 5.1 1 (a) Triple wmlaîion (Cs> + <ut>) (b), (c), (d) elernents of e ~ ~ e f g y budget

Fig. 5.12 (a), (b) Distniutions of eddy viscosity on smooth and rough d c e s (c), (d) Distn'buîions of mixing length on smooth aad rough sirrfaces 150

Fig. 6. Ia: Mean velocity profiles at jet exit 156

Fig. 6-1 b: stresmwisc turbuiertce QuctPation at jet exit 156

Fig. 62: Mean velocity profiles at various locations for Test E-SM1 (a) streamwise (b) verticai component 160

Fig. 6.3: Turbulence intensity and Reynolds shear stress at aownstream locations (a) streamwise (b) vertical turbulence intensity (c) Reynolds shear stress

Fig. 6.4: Streamwise momentum flux at various downstream locations (a) mean velocity data (b) contribution tiom turbulence fluctuation

Fig. 6.5: Vm-ation of maximum velocity with streamwise distance (a) conventional (b) kinematic momentum scaling

Fig. 6.6: b e r layer thickness on smooth and mugh surfaces

Fig. 6.7: Variation of jet half-width with streamwise distance (a) conventional (b) kinematic momentum scaiing

Fig. 6.8: Variation of maximum velocity with hdf-width

Fig. 6.9a: Linear and polynomial fits to near-wall data on a smooth surface

Fig. 6.9a: Variation of skin fiction coefficient with Reynolds number

Fig. 6.10: Mean velocity profiles in outer wordinates (a) smooth wail (b) cornparison between smooth and rough wall

Fig. 6.1 1 : Mean velocity profiles in inner wordinates (a) smooth surface (b) rough surface

Fig. 6.12: Distriiutions of streamwise turbulence intensity (a) cornparison between present and previous data in outer variables (b) present and previous data m inner coordinates (c), (d) present smooth and rough waii data in outer coordinates

Fig. 6.13: Vertical turbulence fluctuations (a) comparison to previous data (b) cornparison between present smooth and rough wall data

Fig. 6.14: Dismlutions of Reynolds shear sîress (a) present and previous data (b) comparison between present smooth and rough wall data

Fig. 6.15: Distriiutions of triple correlation

Fig. 6.16: Mean and turbulence &ta and corresponding cuve fits (a), (b) data for Test E-SM2 (x/b = 30) (c), (d) Test E-SG2 (xlb = 50)

Fig. 6.17: Distributions of (<d>+<u%WJm for srnoth and rough wall 192

Fig. 6.1 8: Turbdeuce kinetic energy budgets (a) production (6) advection and turbuknce dilfùsion 193

Fig. 6.19: Cornparison between present and previous wdl jet and free jet 196

Fig. 6.20: Distriiutions of (a) rnixing length for turbulent wdl jet and k jet (b) eddy viscosity for turbulent wall jet and fiee jet 197

Fig. B. 1 : Data processing using various sarnpling schemes (a), (b) mean profles (c), (d) turbulence fluctuations 228

Fig. B.2: Data processing using various sampling schemes using various sampling schemes (a) Reynolds shear stress (b) stress correlation coefficient (c), (d) skewness 229

Fig. B.3: Mean and turbulence statistics in turbulent buundary layer at various angles of tilt (a) rnean profiles (b) turbulence intensity

(c) skewness (d) flatness factor 233

Fig. B.4: Mean and turbulence statistics in turbulent wall jet at various angles of tiit (a) mean profiles (b) îurùulence intensity and shear sû-ess 235

NOMENCLATURE

ACRONYMS

AIP: DNS: LD A:

LES:

NOTAR:

PIV:

SNR:

STOL:

STOVL:

VISA:

VIT A:

Asymptotic invariance Principle Direct Numerical Simulation Laser Doppler Anemometer

Large Eddy Simulation

No Tai1 Rotor

Particle Image Velocimetry

Signal-to-noise-ratio

Short Takeo ff and Landing

Short Takeoff and Vertical Landing

ENGLISEI SYMBOLS

A:

b:

bij:

B :

C:

Cr:

Ci:

Co:

d:

fi:

G: F:

h:

H:

J:

k:

power law constant

slot height (mm)

stress anisotropy tensor

additive constant in log law (aiso a power law constant)

power law constant

skin fiction coefficient

power Iaw constant

power Law constant

pipe diameter (mm)

dimensionless functionai relationship for the inner Iayer

dimensionIess fixnctional relationship for the outer layer

fiatness factor

depth of flow (mm)

bomdary layer shape factor

jet exit momentum flux (kg/s2)

average roughness height (mm)

k :

KI:

K2:

L :

M:

N:

P:

R:

Re:

Ra:

Rej:

Rek:

turbulent kinetic energy (m7/s2)

maximum velocity decay constant

maximum veIocity decay constant

mixing length (mm)

jet exit rnomentum flux (kg/&

number of sarnples

pressure (hJ/rn7)

pipe radius (mm), aiso as correlation coefficient in uncertainty analysis

Reynolds number

Reynolds nurnber based on jet exit (bulk) velocity and slot height

Reynolds number based on jet exit (maximum) velocity and slot heigbt

Roughness Reynolds nurnber based on average roughness height and fiction

vekocity

Rh: Reynolds number based on inner layer thickness and maximum velocity

Reynolds number based on momenhm thickness

Reynolds nimiber based on depth of flow and fiiction velocity

skewness factor

streamwise turbulence intensity ( d s )

streamwise component of mean velocity (mls)

skin fiction velocity ( d s )

vertical turbulence intensity ( d s )

vertical component of mean velocity ( d s )

spanwise turbulence intensity (mls)

spanwise component of mean velocity ( d s ) -

streamwise distance (m)

vertical or waii-normal distance (m)

y l ~ : jet haIf-width (mm)

yo: virmal origin (mm)

y,: ümer layer thickness of a w d jet (mm)

z: spanwise distance (m)

GREEK SYMBOLS

a: power law exponent

9: power law exponent

6: boundary layer tbickness (mm)

6': boundary layer dispiacement thickness (mm)

fi ReynoIds number based on boundary tayer thickness and friction velocity

6 stress anisotropy ten~oi

AB': roughness function

power Iaw exponent

Von Kman constant

absolute viscosity (Ns/m2)

kinematic viscosity (m%)

eddy viscosity (m'lç)

Coles wake parameter

boundary layer momentum thickness [mm)

density (kg/m3)

m r in bearn-crossing angie (percent)

wall shear stress @/rn2)

SUBSCRIPTS

j: jet exit

max: maximum

u: strearnwise component

v: wall-normai/verticaI component

SUPERSCRIPTS

+: normaiizaîion by viscous tmits

CHAPTER 1

INTRODUCTION

The study presented in this thesis pertains to experimental investigations of complex

near-wall turbulent flows. Specificaily, the stmctwe of both a turbulent boundary layer

and wall jet created in an open channel flow on smooth and rough surfaces is examined.

From the perspective of near-wdl turbulence research, a turbulent wail jet is a near-wall

flow that is a degree more complex than a canonical turbulent boundary Iayer. An

insightful interpretation of the wall jet data requires that the mcture of the turbulent

boundary layer must be examined first, In open channel flows, when experiments are

required to be canied out at low Froude numbers, i.e. in the sub-critical range. there is a

limitation of working at low velocities and. therefore, low values of Reynolds number

based on momentum thickness. In this case, low Reynolds number effects on the

turbulence structure must be examined. Furthmore, systematic investigation of surface

roughness and low Reynolds number effects on the turbulence structure requires that the

correct scaling laws be used to anaiyze the resuits.

in this chapter, some features of turbulence with specific reference to its

compIexity and practical importance are introduced. The basic equations of motion and

motivations for near-wail turbulence research are presented. An ovewiew of Iow

Reynolds number effects and scaiing issues in near-waü turbulent flows is dso

presented. With regard to d a c e roughness, some characteristic features, definitions and

terrninology are introduced. Some nomenclature and applications of a turbulent wail jet

are also presented. In order to faditate discussion and cornparison to a zero-pressure

gradient turbuIent boundary Iayer and fuIIy developed duct flow in subsequent chapters,

some characteristics of open charnel flows .xe discussed. The overall objectives and

scope of the present study are aiso outLined.

1.1 GENERAL REMARKS

1.1.1 Turbulence

Fluid flow turbulence is a phenornenon encountered in many scientific and technologicai

disciplines: in industrial and environmentai flows, combustion, aerodynamics,

meteorology, hydrodynamics and uceanography. lt presents some of the most difficult

probIems both in the fundamental understanding of its physics and in applications, some

of which are still unresolved in spite of extensive research for well over a century.

Indeed, turbulence has been characterized as the last, great, unsolved problem of

classicai physics by several physicists including the Iate Nobel Prize winner Richard

Feynman (Zhou and SpeziaIe, 1998). Horace Lamb (I916), one of the leading

hydrodynamicists of the last century, after d i s d g ai l the branches of hydrodynmics

known to him, fïnaily had to deaI with htrbdence and remarked. "It remains to cal1

attention to the chiefoutstanding dificuly of our abject" (see Bradshaw, 1990).

There are extremely different points of view on turbulence when it is viewed as a

fluid flow phenornenon, a11 of which have in common its complexity. In bis book

'Turbulence in Fluids', Marcel Lesieur (1987) gave the following notion

The jlows one cali's 'turbulent' mgv possess faiev dtrerenr dvnamics, m q be

three-dimensional or sometimes quasi-two-dimensional, may exhibit well

organked stnrctures or otheTwise. A common propeq which is required of them

Ls that t h q should be able to miic ~ansported quantiries much more rapidlv than if

only molecular dijüsion processes were involved.

Turbulence is characterized by its richness in scales, randomness and enhanceci

mixing property. Fwm a practical point of view, its ability to enhance mWng is certainiy

the most important. An engineer, for instance, is mainly concerneci with the knowledge

of turbulent drag a d o r heat transfer coefficients. in spite of its complex features as weil

as ûustration and challenges that are usuaiIy encountered, turbulence continues to

receive considerabte research attention as evidenced in the amount of human and

material resources dedicated to studying it in recent years.

1.13 Equations of Motion and Notrition

In the present study, Cartesian coordinates are adopted: (x, y, z) are used to denote

streamwise, vertical or waii-normal, and spanwise directions, respectively. The

components of mean velocity and turbulence fluctuations in these directions are denoted

by (U, V, W) and (u, v, w), respectively. In Cartesian tensor notation, the meaa and

fiuctuating values in the positive xi direction are denoted by Ui and ui, respectively. Here,

and in subsequent chapters, i = 1, 2, 3 denote the streamwise, vertical and spanwise

direction, respectively. Furthmore, the suffix summation conventions usuaiIy adopted

when disçJssing Cartesian tensor are implied. The superscript '+' is used to represent

quantities in wall units. For example, U' = U/U, u' = u/ü, and y- = y U ~ V , where Ur is

the fiction velocity (to be defined later) and v is the kinematic viscosity.

On the basis of the continuum fluid assunption (e.g. Townsend, 1976), the

dynamics of turbulence is adequately descnied by the continuity and Navier-Stokes

equations. Using tensor notation, the continuity and the Reynolds-averaged Navier-

Stokes equations for an incompressible fluid are, respectiveIy, given by

au. aui l a p a au, -+ul-=--- ax, p axi +-1 ax, --(uiuj)) ax, at

where P is the thennodynamic pressure, p is the fluid density, v is the kinematic

viscosity, and <yu? denotes turbulent or Reynolds stresses.

1.13 Motivation for Neru-Waii Turbulence Research

Many fluid tlow processes encountered in engineering, enviromentai, and industnustnal

appIications are turbulent and are direcîiy influenceri by a soiid boundary. Most of the

flow dynamics take pIace in the near-wall region. in simple wall-bounded flows, for

example, the peak values of the Reynolds shear stress and production of turbulence

kinetic energy occur in the near-wd region. Furthemore, haî, mass and momentum

tr&m rates at the surface are controlled by the turbulent flow structure in the near-wall

region. In near-wail turbulence, the wall shear stress is one of the most important

parameters of interest. An accurate determination of the wall shear stresq is important

because, as wilI be seen in Chapter 2, most of the scaling laws involve the fiction

velocity or wail shear stress. The wail shear stress is also of primary importance fiom a

practical perspective since it is directly related to the drag force. Progress in near-wall

turbulence research could lead to a better understanding of the turbulence structure and

skin tnction characteristics so that momentum, heat, and mass transfer rates at the waiI

can be accuately predicted, Consequently, near-wall turbulence has attracted

considerable research attention and continues to be of current research interest.

1.1.4 Canonical and Complex Near-waii Flows

In the present study, a turbulent boundary layer is dehed as a near-wail flow in which

the following two regions cari be identified: 1) a v q thin layer in the immediate

neighborhood of the wall in which the velocity gradient normal to the wail (au/ay) is

very large and viscosity exerts an essentiai infiuence, and 2) an outer region where

viscous effects are unimportant- The terni canonical flow will refer to a zero pressure

gradient turbulent boundary layer (at low freestream turbulence) or M y developed flow

in ducts with smooth d a c e s . Because of their simplicity, both in physics and geometry,

canonical flows are the most wideIy researched flows.

AIthough resuIts obtained h m these studies have improved our physical

understanding of the near-waii turbulence stnicture, many practical flows are much more

complex. For example? in flow over an airfoi1 or turbine blade, rapid changes in pressure

gradients and surface curvature may occur. Blowing or suction may aiso be applied at -

some point dong the airfoil as a fom of boundary layer control. In industriai

applications, the mass transfer of interest may relate to corrosion of the wall materiai, in

which case corrosion is responsiile for generating surface roughness. For environmental

flows involving dispersion of waste products in rivers and streams, the strearnbed is

aiways rough, These extra strains, c.g. pressure gradient, surface roughness and

ma tu re , may introduce additiond physics which complicates the hubulence stnicture.

For the purpose of accurate prediction of practically relevant near-wail turbulent flows,

additionai research on relatively more compiex flows are necessary.

1.1.5 Research Methodologies

Near-wall turbulence has been extensively investigated using experimentai techniques,

and theoretical and numerical analyses, i.e. turbulence modeling and direct numericd

simulation (DNS). Experirnentd investigation, using both single and multi-point

measurements as well as flow visualization, is the approach most widely used in

turbuience research.

Turbulence modeling, of varying complexity, has aIso played a significant d e in

turbulence research. The low-ReynoIds number two-equation models have been widely

used in predicting a variety of tubdent flows encountered in practice. Eddy-viscosity

models have regaineci popularity as cornponents of two-Iayer turbulence models (Rodi et

al., 1993). For engineering flows dominated by complex flow physics, the Reynolds

stress transport equation becomes a viable alternative because the various terms in the

governing equations are treated exactiy.

The arriva1 of high performance cornputing resources in recent years has opened

the possïbility of direct numerical sïmuIation (DNS) of turbulent flows by solution of the

three-dimensionai Navier Stokes equations. The DNS has been very successfil in

calculating relatively low Reynolds number turbulent flows with simple geometry. The

DNS has ais0 the capability to adequately resolve the diffusive sublayer and accurate

statistics obtained fiom DNS have provided insight into the physics of turbulence. In

some sense, DNS can be regarded as a cornpanion (or numerical) experiment to the

actuaI physicai experiment because it can generate much cornprehensive information on

the turbulence structure of the flow field, Even though DNS is not yet a technique for

compIex engineering flow computations, the data obtained fiorn DNS caiculation,

together with physical experllnental data, have enhanced the development of physicdy

correct turbulence models.

1.1.6 Measuring Devices

In the area of experimentai investigation of turbulent flows, conventional instruments

such as thermal anexnometers and Pitot-tubes are most extensively employed. However,

the suitability of Pitot-tubes and conventional thennai anetnometers for velocity

meaSuTernents in wons of high turbulence Ievel has been repeatedy questioned.

Furthemore, the spatial resolution of Pitot-tube and thermal probes is usually poor and it

is difficult to make masurement in the viscous sublayer especiaiiy if cross-wires are

empIoyed.

With the arn'val of optical devices such a$ particle image velocimetry (PM and

Iaser DoppIer anmorneter (LDA), and recent advances in signal processing technology,

it is now possii te to obtain sophisticated measuements, which provide remarkable

insight into the near-wail turbulence structure- Since optical anemometers are non-

intrusive. they exert minimal interférence on the flow field. The LDA is dso suitab le for

near-wall measurements and in regions of high Iocal turbulence intensity. in contrast to

Pitot-tube and thermal probes, the LDA exhibits a linear veIocity-fiequency relationship

and constant instrument sensitivity. in spite of these attractive features, care is required

in the proper use of the LDA.

12 REYNOLDS NUMBER EFFECTS AND SCALING ISSUES

Reynolds numbers encountered in practice are d l y very hi@. Due to hardware and

equipment limitations, Reynolds numbers of the flows invesfigated experimentally or

nmmicaIly are orders of magnitude lower than those encountered in practice. It is

thetefore important to know whether r d t s obtained fiom relatively low Reynolds

number experiments or numericd calculations cm be extrapolateci to the higher

ReynoIds numbers encountered in engineering and environmental appIications.

The concept of Reynolds number similarity has been wideIy used in fluid

dynamics research. When simiiarity assuraptions are apptied to na-wall turbulent

flows, they imply that individual turbulence statistics obtained ftom different facilities

and at different Reynolds numbers will coiiapse ont0 a single curve when they are made

dimensionless using the proper velocity and length scales. Implicit in the above

assumption is that the proper scaiing laws need be identified in a systematic

investigation of Reynolds number effects. in near-wall turbulence research, two possible

velocity scaies are the fiction velocity U, (= [r,Jpll", where r, and p are the wall shear

stress and fluid density, respectively) and fteestream velocity U,. The ratio U&, which

is directiy related to the skin fnction coefficient, is known to be Reynolds number

dependent. It should be pointed out that an inaccurate estimate of the friction velocity

may mask any systematic examination of Reynolds number effects. Although high

Reynolds number experiments will be useful in examining what the (upper) limits are,

low Reynolds number experiments are required for sorting differences in scaiing iaws

because of the more rapid variation of Ut and U, in this regime (George and Castille,

1997). Another motivation for Iow Reynolds number experiments is that the viscous

sublayer is relatively thick so that a more accurate estimate of the friction velocity can be

made using data obtained in the viscous sublayer. In this case, the influence of

measurement uncertainties is kept minimaI.

13 SURFACE ROUGHNESS AND THE WALL SlMILARITY

HYPOTHESIS

13.1 Deunitions and Terminology

Before discussing roughness and its eff- on the turbulence structure, some tenns and

notations that are fiequently encotmtered in this and subsequent chapters are defbed.

Figure 1.1 is a schematic of a rough d a c e and aiso defines some geometrical features

of the roughness elements. in this figure, k denotes a representative average roughness

height; y, is the wall-normal distance measured fiom the top plane of the roughness

eIements; and y, is the virtual orïgin and represents the distance between y, and the

location at which the mean velocity goes to zero (Le. U = O). Therefore. for a rough

surface, the effective wall normal distance y = y, + y,. 0th related tenninology for y,

includes fluid-dynamic height origin (e-g. Raupach et ai., 1991) or error-in-origin (Perry

et d., 1969; Bandyopadhyay, 1987). Perry et ai. (1969) suggested that y. is a measure of

the interaction between the meau flow and roughness. For a given roughness height, the

virtuai ongin must satisQ the foiiowing constraint: O < y, < k. The exact value of y,,

however, depends on the roughness elements and other geometric factors. For sand grain

roughness, for example, the data compiled by Nezu and Nakagawa (1 993) suggest y. =

0.15 - 0.3k. Krogstad et al. (1992) reportai a value of y, = 0.25k for their wue mesh

roughness.

1 Ground plane

Figure 1.1 : Definition sketch of roughness elements

The term roughness sublayer is the counterpart of the viscous sublayer in a smooth

wall turbulent boundary layer. It refers to the entire layer which is dynamically

influenced by length s d e s associateci with roughness elements. Typically, it extends

fiom the wail to 2 - 5 roughness heights, i.e. y 5 5k (Raupach et ai., 1991). The

2

roughness Rqmolds number is dehed as Rq, = kU&. Physically, it represents the ratio -.

of a typicai roughness length s a l e to the viscous iength scale. \

13.2 Some Characteristics of Surface Roughness

Following the classicai sand roughness experiments by Nikuradse (1933), it is generally

accepted that for R% > 5, the d a c e is considered as being hydrauiicdly rough. It is

important to recognize that a flow that is hydrauiicdly smooth in one sense rnay be fdIy

rough fiom other perspectives, For example, in the case of high Prandtl and Schmidt

number fluids, the diffusive sublayer is extremely thin so that it may lie entirely within

the momentum viscous sublayer. While such a surface rnay be considered hydraulically

smooth fiom the point of view of momentum transport, the roughness elernents may

likely protnide beyond the thmal diftiisive sublayer.

Turbulent flow over rough d a c e s is a complex phenornenon. In the vicinity of

the roughness elernents, the flow is spatiaily heterogeneous, may be three-dimensional

and no longer parallel to the ground plane (see Figure 1.1). In this case, spatial (rather

than Mie) averaging is desirable aithough ihis is difficdt to wry out in physical

experiments. The passage of eddy motion over the roughness elements causes locd

eruptions which may increase the vertical masdmomentum interchange (Gatski, 1985).

Details of the turbulent structure in this region are aiso controlled by the specific

geometry of roughness elements. The locd turbulence intensity close to the roughness

elements is usuaiIy hi& in which case standard techniques such as cross-wùe

anemometty d e r fiom substantiai measurernent mors that are often difficult to

diagnose and correct. These complicating features may explain the relatively slow

progress in our physical understanding of rough wall turbulent boundary layen in

cornparison to the smooth wall counterpart. Furtherrnore, rough wail turbulence research

has not benefited much h m DNS because of the additional complexity introduced by

the geometry of roughness elements as weii as intricate physics of the flow.

AIthough the average height is very ofien used as the characteristic length scaie for

roughness elements. other length scaies and geometric factors such as aspect ratio,

roughness element dimensions and element separation may considerably influence the

dynamics of fiow over rough surfaces. For example. the mugh waIl data compiled by

Bandyopadhyay (1987) showed that the upper criticai value of Rq, above which the flow

regime is fùIly rough decreases as the span-to-heigth aspect ratio (Vk) increases. in the

case of evenly distriiuted sand grains, the upper critical value is about 55 to 70 while it

is approximately 12 - 15 for an aspect ratio O/k) greater than 12. Since most of the

mughness parameters depend on the specific geometry of roughness elernents, it is

relativeIy difficult to develop a unifling theory for rough wail turbulent flows.

133 Types of Surface Roaghness

in considering the effects of surface roughness, ofien the specific characteristics of the

roughness elements have been given minimal attention. Two main types of roughness

have been identified in the literanire. FoUowing the temiinology of Perry et ai. (1969),

these are referred to as k-type or d-type roughness. if the roughness function (Le. the

parallel shifi between smooth wail and rough wail velocity profiles on a semi-

logarithrnic plot) depends on Rq, it is texmed k-type roughness. Experiments have

shown that the k-type scaiing is not obeyed by transverse grooved surfaces when the

cavities are narrow or on a smooth surface wiih a series of depressions. This type of

roughness, as opposed to the sand-grain or k-type roughness, scales with outer variables,

i.e. the boundary layer thickness, 6, or the pipe diameter, d, and is known as d-type.

Depending on the ratio of the groove spacing to height, such a surface rougimess may

d u c e the non-dimensionai Reynolds stress near the surface. Of particdar importance to

waiI turbulence control are smail-scde longitudinal grooves (or nilets) which can

produce [ocal shear reduction of up to 50 percent and net drag reduction of the order of

1 0 percent (Walsh, 1982; Walsh and Lindemann, 1 984).

13.4 The Waii Similarity Eypothesis

In his book 'The Structure of Turbulent Shear Flows', Townsend (1976) States

While geomenically similar flows are expected to be ajvramical!~ similar if their

R q m l d ~ numbers are the same, their structures are also v q nearly similarfor

al/ Rqnolds numbers which are large enough to allow @l&) ncrbulentflow.

This is one of the several statements of the ReynoIds number sirnilarity impIied in

Section 1.2. Perry and Abell (1977) extended the above notion to rough wall boundary

Iayers. A more general statement of the similarity concept, which is referred to as the

''waii smiilarity hypothesis" was given by Raupach et al. (1991) as foIIows

Outside the roughness (or viscous) mblayer, the turbulent motions in a bounday

layer ut high Rqvnolds number are independent of the wall roughness and the

viscosity exceptfor the role of the wall in setting the velociy scale U, the height y

and the boundary layer thickness 6.

The above notion suggests that the effects of surface roughness are confined to the

irnmediate vicinity of the roughness elements so that the turbulence structure over a

significant portion of the boundary layer should be unchanged in spite of substantiai

aiterations to the surface characteristics of the waI1. This has important implications for

rough wall turbulence models as will be discussed in subsequent chapters.

1.4 TURBULENT WALL JETS

1.4.1 Definiton and Nomenclature

A turbulent wall jet is a shear flow directed dong a wall, where by virtue of the initiaily

supplied momentum, at any downstream station, the streamwise velocity over some

region within the flow exceeds that in the externai stream (Launder and Rodi, 198 1). A

wail jet also constitutes part of more complex flows as in the case of the flow regime

downstrearn of an impinging jet.

A definition sketch that also serves to define some of the flow nomenclature is

shown in Figure 1.2. In this figure, x and y denote distances in the streamwise and

vertical directions, respectively; U and V are the streamwise and verticai components of

the mean velocity; Uj is the jet exit velocity; b is the slot height; U, is the local

maximum velocity, y,,, and yin, respectively, denote the verticai Iocations where U, and

OSU, occur. In the present study, y, and y1.p will be referred to as the inner thiclcness

and the jet half-width, respectively,

Figure 1.2: Schematic of a turbulent wall jet

The flow field is traditionally divided into two regions: an inner layer which

extends fiom the wall to the point of maximum velocity (i.e. y 5 y,), and an outer region

which stretches fiom the point of maximum velocity to the outer edge of the flow

(Le. y > y,). in this context, a turbulent watl jet may be thought of as a composite flow

made up of two interacting shear Iayers: an inner region, which possesses many of the

characteristics of a turbulent bomdary layer, and an outer region, which, though

influenced by the solid wail, is stnicturaily simiiar to a fke plane jet. The interaction

between the srnail-scale dominated inner layer and the large-scaie dominated outer layer

creates a complex structure that is characterized by uitense mixing. This region is still

poorly understood and poses the greatest challenge to numerical models.

1.42 Applications of WaU Jets

The turbulent wail jet has received considerable research attention, prompted mainly by

its important and diverse technoIogicai applications, e.g. in boundary Iayer control and

film cooling technology. Among the various types of control techniques, boundary layer

control is probably the oldest and most economically important (Gad-el-Hak, 1996). The

turbulent wall jet has been suggested as the most prefened and straightfomard fiow

separation controI technique applied to military fighters and STOL transports (Gad-el-

Hak and BushneU, 1991). By using a wall jet to alter the locations and strength of

vomces formed at a i r d whgtips, wingibody junctions as well as around slender

bodies such as missiles and aircraft fuselages, the amount of lifi and drag are effectively

modified. A tangentid jet blowing over the upper surface of a rounded trailing-edge

airfoil is also employed to set an effective Kutta condition by fixing the location of

separation. This technique is also used to stabilize a trapped vortex and has been

employed to achieve increased super-rnanoeuverability of helicoptem and controllability

of a i r d flying at low angles of attack. Thus wall jet flows may be found on airpiane

wings (e.g., F-104 Starfighter, A-6 Crusader) and more recently on helicopter tail booms

(NOTAR).

investigation of a wdl jet in a cross ffow has been stimulated primarily by

problems of interaction between lifting jets and crosswinds used by VSTOL aircraft.

Aerodynamic interaction between a hovering a i r d and the ground environment has

also been recognized to be a dominant factor in the successful development of VSTOL

technology. In order to detennine the impact an aircraft has on the surrounding

environment while hoverhg in ground effect, adequate knowledge of important design

parameters such as d a c e temperature and ptessure at impingement, acoustic noise and

veIocity decay of the g m d plane waIi jet is necessary. Suflicient understaading of the

velocity characteristics is also necessary to avoid unnecessary aerodynarnic loading on

ground persoanel, buildings, and other aircrafi. -

In order to irnprove the thexmai performance of modem gas turbines used in either

aircraft engines or power production systems, specifications for turbine inlet temperature

continue to increase while cooling airflow is kept minimal. The specific power of a gas

turbine also depends on turbine inlet temperature. in spite of noticeable progress made in

turbine blade metallwgy, a reasonable lifetime of turbine biades can be ensured only if

an efficient surface-cooling mechanism is employed. Film cooling has been suggested

(MacMullin et al., 1989; Lakehai et ai., 1998) as one of the most efficient cooling

rnethods for such devices. It is claimed that this method is more efficient than internai

convection cooling because of the relativeiy Iow heat-transfer characteristics of air. The

wall jet has been identified as one of the most efficient film cooling devices in gas

turbine applications.

Other widespread appIications of waII jet. for heat and mass tramfer modification

can be found in the automobile defioster and deflectors used in conditioned air-

circulation systems (Launder and Rodi, 1983). The design and position of such

defiectors become especialiy crucial in large-scale onesf-a-kind applications such as

found in a concert auditorium. Turbulent walI jets are also of particdar interest in

agricdture to irnprove air circulation in poultry houses (Blackwell et al., 1990), and in

mziny other indutrial applications to effect enhanced drying, leaching of soiids and

toughening of glass.

Investigation of wall jets, aside h m practical applications, has alsri drawn

considerable fundamental interest in the past because it has the characteristics of both a

boundary layer and a free jet The wall jet has, therefore, k e n identified as a prototypical

flow for investigating the physics of complex near-wall turbulence as weIl as improving

our physical understanding of the interaction between a boundary layer and fiee shear

flow. Furthexmore, the influence of elevated fieestrem turbulence on fluid dynamics

and convective heat transfer has recently been recognized as a major factor in turbine

blade design (MacMullin et al., 1989). in this respect, a wall jet bas also been used in

such research efforts to simulate the high free-strearn turbulence encountered in

turbomachinery.

1.5 SOME CEARACTERISTICS OF OPEN CHANNEL nows

The structure of most wail-bounded flows is considered to be similar. However, there are

some specific and important differences arnong these flows. Since the present study

pettains to rneasurements in an open channe1 flow and the results will be compared CO

other near-walI turbulent flows, some of the important and unique characteristics of open

channel flows are summarized below.

1. in an open channel bomdary layer, the maximum streamwise mean velocity may

occur beIow the tIee surface (Tominago et al., 1989; Nezu and Nakagawa, 1993; Shi

et al., 1999). This unique feature is refend to as 'vebcity dip', and is attnibuted to

secondary flows (Neni and Nakagawa, 1993).

2. Near the fke d a c e of an open chanael flow, the background turbulence levei is

substantially higher than fkestmm turbulence intensities reported in typical wind

tunnel experiments.

3. Similar to zero pressure gradient turbulent boudary layers (e-g. Gad-el-Hak and

Bandyopadhyay, 1994), the existing open channel flow litetature indicates that the

outer wake parameter 0 shows a Reynolds number dependence. The LDA data of

Nezu and Rodi (1986) suggested an asymptotic value of ïi = 0.2. This value is

considerably lower than the asymptotic value of ïi = 0.62 reported by Coles (1 987).

4. In open channel flows, the vertical motions are tesrtained in the interfacial or free

d a c e region by the damping efféct of the fke surface (e.g. Komori et al., 1993;

Borne et al., 1995). The DNS results of Komori et al. (1993) indicate that the

turbulent kinetic energy of the vertical motion is re-distriiuted to the spanwise and

streamwise motions through the pressure fluctuation, This causes an increased stress

anisotropy in the vicinity of the fie sudice of an open channel flow in cornparison

to outer edge of canonicai turbuient boundary layers.

1.6 SiJMMARY

Near-wall turbulence is a cornplex fluid flow phenmenon. The skin fiiction behavior is

of both practical and theoretical interest, For flow over rough surfaces, the presence of

vortical structures, which are present in the roughness-element wakes, furîher

compiicates the turbulence structure especiaüy close to the roughness elements. The hi&

local turbulence intensity close to the mughness elements saggests that conventional

thmal anemometers may be making important measurement mors close to the w d

where most of the flow dynamics occur. -

AIthough the physics of canonical near-wail flows is relatively well understood,

mady due to refined measurements and direct numericd simulations (DNS). our

physical understanding of practicdly relevant (ix. complex turbulent flows) is deficient.

Low Reynolds number effects and scaling issues remain important research questions.

AIthough the wall similarity hypothesis would be very attractive for turbulence

modeling, since it suggests that the turbulence stnicture over srnooth and rough surfaces

is essentiaily the same, its validity needs czitical verification. This provides reasonable

grounds for a systematic experimentd investigation of surface roughness and its effect

on the near-wail turbulence structure. Furthemore, by treating roughness as a

modification of the inner layer, or the outer region of a nnbulent wall jet as a

modification of the outer layer of a turbulent boundary layer, an improved understanding

of the interaction of the inner and outer layers may be reaiized. Therefore, investigation

of a turbulent boundary layer and a wall jet over smooth and rough surfaces, apart h m

practical motivations, would also promote a better understanding of complex nnbulent

flow.

1.7 OBJECTIVES AND SCOPE

1.7.1 Objectives

ïhe purpose of this research is to examine the structure of tucbuient boundaxy layers and

w d jets on smooth and different types of rough surfàces with an overall objective of

irnproving our physical understanding of the na-wail turbulence structure- The

objectives are:

1. To examine low Reynolds number effects and scaling laws for turbulent boundary

layers. The understanding obtained &om these resuIts is then used to accomplish the

principal objective which is stated next.

2. To examine the interaction between the inner and outer layers of a turbulent

boundary layer and wall jet on smooth and rough surfaces.

1.7.2 Scope

The scaiing Iaws for the mean vebcity and its higher order moments. as well as the

reIevant experimental and numencal literature on smoorh and rough wail turbulent

boundary layers and wall jets are reviewed in Chapter 2. In the light of our m e n t

understanding of these flows, M e r refinements are made to the objectives stated in

Section 1.7.1. An overview of the LDA system, description of experimental fadities and

surface roughness as weIl as instnmientation and experimentd details are given in

Chapter 3. In Chapter 4, one-component smwth and mugh wall velocity measurements

in turbulent boundary layers are reported. The sets of data presented in this chapter are

used to examine low Reynolds number effécts, scaihg issues and effects of the specific

roughness geometry on the turbulence structure. The effects of surface roughness on

higher order turbulence statistics such as Reynolds stresses, mple correlation as weII as

the energy budget, mixing length and eddy viscosity distn'butions are discussed in

Chapter 5. The understanding obthed regarding the turbuience structure on smwth and

rough wall turbulent boundary layers is used as the bais for interpreting the relatively

more complex wall jet data in Chapter 6. A summary, the major conc1usions and

contriiutions from the present research are given in Chapter 7.

CHAPTER 2

Scahg laws for the mean velocity and turbulence statistics are reviewed in this chapter.

In the case of the overlap region of the mean turbulent boundary layer, the scaling laws

proposed by classical theones as weU as recent power laws fomulated by BarenbIatt

(1993) and George and Castillo (1997) are considered, The techniques used to determine

the wall shear stress are discussed. Both conventionai and recent scaiing laws proposed

to describe the streamwise evolution of turbulent wail jets are aiso discussed. Finaily, the

recent and relevant experimental and numericai studies on turbulent boundary Iayers and

wail jets are reviewed.

2.1 THEORETICAL ANALYSIS

2.1.1 Turbulent Boundary Layers

Scaling laws derived fiom theoretical analysis have played a significant role in

interpethg near-wail experimemtd data It is generaily accepted that the dynamics of a

turbulent flow is describeci by the Navier-Stokes equation, i.e. Eqn. (1.2). For near-wall

turbulent flows, the two-Iayer concept forms the basis of mterpreting events and a h

constnicting mathematical models. According to the two-layer concept, the flow

stmcture consists of two distinct regions: 1) an inner layer, i.e. viscous sublayer and

b&er region, where viscous effects dominate; and 2) an outer region where inertid

effkcts dominate. At sd5cÏentIy hi& Reynolds numbers, dassical theories (e.g.

asymptotic expansion (MiIlikan, 1938) and mixiag iength (Prandtl, I932)), suggest an

overlap region between the inner and outer Iayers, On îhe other hand, more ment

analyses propose power laws to descriie the overlap region. In the foilowuig subsections,

s c a h g laws for the b e r and outer layers as well as the overlap region are reviewed.

Some of the availabIe techniques used to determine the wall shear stress are also

discussed.

2.1.1.1 Scahg Law for the lnner Loyer

In the classical approach, dimensional analysis of the dynamicd equations and bomdary

conditions leads to a scaihg Iaw for the mean velocity profiIe. In the irnmediate vicinity

of a solid boundary, the flow dynamics is assumed to depend on the distance fiom the

wall (y), the wall shear stress (T,) and the fluîd properties, Le. kinematic viscosity (v)

and density (p). From dimensional considerations, the following dimensionless

fimctional reIationship is obtained for the mean velocity

U+ =fi[y', 5'1 (2. la)

where U' = UN, y* = yü4v and UT = [~dp]'" is the fiction velocity. The parameter 6+

(= 6U&) is a ReynoIds number basai on the boundary layer thickness (8) and the

fiction velocity, and indicates the ratio of the outer to the inner Iength s d e s . If the

dimensionless functional reiationship 6 is independent of Reynolds number, i-e.

v'=f;-m (2- 1 b)

it i m p k complete sllnilarity exists in the inner region. Eqn. (3Ib) is commody referred

to as the universai law of the wd.

The velocity distribution in the near-wall region (i.e. the viscous sublayer and

lower part of the buffer region) wilI be of considerable interest in determining the skin

fnction and dso for constnicting a composite velocity profile. Using a Taylor series

expansion togetfier with the continuity equation and no-slip condition at the wdl, the

mean velocity can be expressed by the following relation

v = y' + c 4 T + c5f5 + HOT (2.2)

where the coefficients Q and CS may Vary slightly with Reynolds number and HOT z

higtier order tenns. Recent LDA measurements (Eriksson et al., 1998) suggested cd =

-0.0003 f 0.000 1 while George and Castillo (1997) proposed a value of cs = 13.5 x 10".

For a mugh surface, the characteristic length scales rnay also include the average

roughness height, k, and any additional length scales needed to completely characterize

the roughness. If the viscous length (v/Ut) and the average roughness height (k) are

chosen as the only relevant length scales in order to preserve the generality of flow over

both smooth and rougti surfaces (Raupach et al., 1991), one can define a roughness

Reynolds number, Rq, (= kU&).

2.1.1.2 Scriüng Law for the Outer Layer

In the outer region, the wall acts to retard the locd velocity in a way that is independent

of viscosity (v), but dependent on the distance h m the wall (y), the boundary layer

thickness (6) and an outer veiocity scale U,. in the case of a fdly developed duct aow,

the length scde is given by the radius (R). It is important to note that, in contrast to the

inner layer, no equident theory has been proposed by the classicd theories for the outer

layer. On the basis of experimental evidence and the need to attain sùnilarity in the outer

region, the mean velocity profile for this region is ofteri presented in a defect form.

Acwrding to classical tlieories, the vetocity =ale for both turbuIent boundary Iayers and

duct flows is the fiction velocity, Le. U, = U, The ment theory derived for a canonical

zero-pressure gradient turbulent boundary Iayer by George and Castillo (1997) showed

that the proper outer velocity scale is the freestream veIocity, i.e. U, = U,. The velocity

distribution in the outer region is given by

where fo expresses the dimensiodes

(2.3a)

functional relationship. If £, is independent of

Reynolds number, complete sirnilarity exists in the outer region, Le.

2.1.1.3 Scaüng Laws for the Overlap Region

At a sufficientIy hi& Reynolds number (0, classical theories suggest an overlap region

between the inner and outer layers where both layers interad. in this region, the inner

length scale (vWt or k) is presumably too small to controi the dynamics of the flow, and

the outer length scde () is presumably too large to be effective (Tennekes and Lumiey,

1972). If this occursl the dynamics of the flow is independent of aiI Iength scales except

the distance h m the wail (y).

The scdhg Iaw for the mean velocity in the overlap region has been of

considerabie interest to the flGd dynamics community because it leads directly to a skin

f?iction relation. En the overlap region, the scaling law for the mean velocity is obtaHied

by matchhg the b e r and outer scaling laws. The specific form of the scaling law in this

region depends on the additionai assumptions made in the course of the matching

process. The classical theories (Millikan, 1938; Clauser, 1954; Panton, 1990) propose a

log law for both duct flows and turbulent boundary Iayers. The ment pipe flow analysis

(Barenblatt, 1993) and zero-pressure gradient theory proposed by George and Castillo

(1997) indicaie that the overiap region is described by a power law. Long aad Chen

(1981) and George and Castillo (1997) showed that although the mean velocity in the

overlap region is logarithmic in the case of pipe flows, the scding law for turbulent

boundary layers is entirely different.

For a turbulent boundary layer, Long and Chen (198 1) remarked that it is strange

that the overlap region between the viscous inner and outer layen whkh is characterized

by inertia does not depend on both inertia and viscosity, but oniy on inertia. They

suggested that this might be a wnsequence of irnpropedy matchhg two layers whicti do

not overlap. They also showed that irrespective of Reynolds number, there exists a

'mesoiayer' which intBides between the hum and outer layers and preveats the overlap

of the cIassical theory. in spite of these recent deveIopments, the iog Iaw continues to be

the more prefaable scaling law used in the anaIysis of both turbulent boundary layers

and duct flows. Sreenivasan (1989) argued that dthough the power law used by

engineers to descriie the mean velocity profile has beea discredited by scientists ever

since Millikan (1938) derived the log law fiom asymptotic arguments, the basis for the

power law is a prion as sound as that for the logarithmic Iaw. George and Castillo

(1997) pointed out that it is very difficult to distinguish a logarithic law fiom a weak

power iaw using experünentai data alone since one can be expanded in terms of the

other.

in spite of specific differences among researchers, it appears that a power law is

more suitable for low Reynolds number flows (e.g. Djenidi et al., 1997; George and

Castillo, 1997; Zagarola et al., 1997). The boundary layer analysis of George and Castillo

(1997) showed that the overlap region consists of a mesoiayer (30 < y- < 300) and an

inertial subIayer (y- > 300). It was argued that the logarithmic portion of a boundary

layer (i.e., 50 < y* < 150) is just a portion of the mesolayer. Based on ernpirical evidence,

Zagarola et ai. (1997) proposeci that for pipe flows, the mean velocity consists of two

distinct regions, a power law region for 50 I y- S 500 or O. LR- (the upper limit king

dependent on Reynolds number), and a log law region for 500 S y 5 O.IR*. Recent

refined measurements and DNS r d t s at Iow Reynolds numbers showed that the overlap

region graduaüy disappears as the Reynolds number decreases. It foIIows that at low

Reynolds numbers, a log law region may not appear. This h a important implications for

low Reynolds number flows (especialIy on a rough surface as will be shown in this

study) because without a weii-defined log law region the usefùlness of the Clauser pbt

technique to determine the skin fnction is severely dimimdimimshed.

2.1.13.1 The Logarithmic Law

According to classical theones (Millikan, 1938; Clauser, 1954), the inner and outer

layers can be matched in the M t of infinite ReynoIds number, i.e. assuming complete

simiIarity, to obtain the following log law for smooth-wall turbulent flows

In Eqn. (2.4) the log Iaw constants (i.e. the von Karman constant, K and the additive

constant, B) are assumed to be universa1 and independent of Reynolds number. The

exact values differ slighdy fiom one researcher to the other; in the present study, the

following values are adopted: K = 0.41 and B = 5.0.

For a mu& wall boundary layer, the mean velocity profile may be written in the

following form

where AE3' is the roughness function which represents the @araIIel) shift between

smooth-wall and rough-wail velocity protiles on a semi-logarithmic plot. The specific

value of AB' depends on the roughuess Reynolds number as well as the roughness

geometry. As mentioned earli- the Iog law is the most widely used scaling Iaw for both

turbulent duct flow and boundary layers, and as mch is the formulation presented in most

undergraduate fiuid mechanics texts. It also forms the basis of the Clauser chart

technique used to determine the wail shear stress.

2.1.132 Power Laws

Over thc past decade, power laws have received increasing attention as an alternative

formulation for the mean velocity profile in boundary layer flows. Various types of

power Iaw formulations have emerged in recent years depending on the specific

assumptions made. in the present study, the formuIations proposed by Barenblatt (1993)

and George and Castillo (1 997) are considered.

Barenbrait (1993) [BPI

The power law by Barenblatt (1993) was specifically formulateci for pipe flows. He

explained the theoretical basis of both the log law and power law, and offered an

argument in favor of a power law to describe the rnean velocity. His formulation is based

on an incomplete similarity assumption for the overlap region, which implies that the

flow in this region is Reynolds number dependent. The power Iaw proposed by

Bareublatt (1993) is of îhe form

u+ = C(y3" (2.6)

where C and a are constants that Vary slowly with Reynolds number. The power law

constants are given by the following asymptotic expansions

On the basis of the pipe flow experiments of Nikuradse (1933), Barenblatt and

Prostokishin (1993), hereafter denoted as [BPI, proposed the following values: a1 = 1.5,

cl = 11 fi and el = 2.5. where only the 6rst term and the fïrst two t- are retained for

a and C, respectively. More recently, Zagarola et al, (1992) used their super-pipe data,

which covecs the range of 3 I x 103 I Re (= 2RUIv) I 35 x 106, to recdibrate the power

law constants. At Iowa Reynolds numbers, their values of C are significantly lower than

the values of [BPI while their values of a are higher than the values of [BPI.

George and C d i o f.991) [Gq

George and Castillo (19971, hereafter denoted by [GC], used what they tenned the

Asymptotic Invariance Principle (AiP) to formdly derive a different power law for the

overlap region of the canonical zero-pressure gradient boundary layer. They assumed

complete similarity in the inner and outer layers in the limit of infinite Reynolds

nurnbers. According to their theory, the appropriate velocity scales for the inner and

outer layers are Ur and U, respectively. It should be mentioned that unIike the classical

theory, which assumed the outer velocity scale Uo to be identicai to the fiction velocity

(Le. U, = Ur), George and Castitlo (1997) derived U, = U, h m sirnilarity

consideratioas. Since the ratio of inner and outer velocity scales (Le. UJJ,) is Reynolds

number dependent, it follows that the overlap region must admit Reynolds number

dependence except in the limit of infînite Reynolds number. Using a na-asymptotic

analysis in the overlap region, they showed that the mean velocity is descriied by a

power law at large Reynolds numbers. In inner and outer coordinates, respectively, their

form of the power law bewmes

rr'= ci&+

The coefficients Ci and Co as well as the exponent y are dependent on the Reynolds

number 8. In the above relations, the parameter a (or a 3 represents a stiifi in the ongin

for meamring y, associateci with the growth of the mesoiayer region (30 I y I 300).

They pointed out that the asymptotic approach of y to a mail value makes it possible to

approximately recover the log Iaw relation of the classical theones. In this case, the

additive constant in the classical log law is identicai to Ci, which may Vary from 7 to 10.

Note, however, that these values are substantiaiiy higher than the typical value of B - 5-0, but fa11 within the range of 4 I B I 12 reported in some Iow Reynolds number

experiments. As noted by [GC], neither the near-wall profile (Eqn. (2.2)) nor the overlap

profile (Eqn. (2.9)) is vaiid at y" = 15. They proposed the following composite velocity

profile to descnie the mean velocity in the viscous sublayer, the buffer region and the

overlap region

where d = -16 and d is a damping parameter chosen as d = 8 x IO" to fix the transition

fiom the viscous wall region to the overlap region at y" = 15.

2.1.1.4 Determination of Sheu Stress

in any near-wall turbulence research, one of the most important parameter to determine

is the waii shear stress, and hence the Ection velocity. An accurate detemination of the

waii shear stress is important h m pradcd point of view and also in view of its

devance in scaling the mean velocity as well as turbulence quantities. Following

George and Castillo (1997), one may physically view the wall shear stress as measuring

the forcing of the inner tiow by the outer, or alternatively, as measuring the retarding

effect of the huer flow on the outer. This would suggest a strong interaction between the

wall shear stress and the outer flow structure so that consideration rnust be given to the

specific flow structure in the outer layer in an accurate determination of the fiction

velocity.

The rnethods used to detemine the walI shear stress include direct measurement

(e.g. with a floating element gauge), performing a momenturn balance, extrapolatïng the

Reynolds shear stress to the wall, or by fitting the mean velocity to a standard profile. If

the Reynolds number is high enough for a welldefined overlap region to exist, the waiI

shear stress is commoniy determined by fitting the Iogarithmic profile (Eqn. (2.4)) to the

mean velocity data. This approach is known as the Clamer plot technique. The use of the

Clauser plot technique is well established for turbulent flow over a smooth surface at

low-fieestream turbulence intensity, and has also been assumed to be valid in high

fieestrem turbulence flows (Hancock and Bradshaw, 1983; Thole and Bogard, 1996).

Although the Clamer pIot technique (Eqn 2.5) fias been used in some earlier rough wall

boundary layers, some studies demoastrated that a Ciauser technique for mugh waII

boundary layers may not be reliabfe. Perry et al. (1969) remarked that due to two

additiond roughness variables (i.e. the roughness shifl., AB' and the virtual ongin, y,),

the Clauser pIot technique for tinding the waIl shear stress would be inaccurate.

As mentioned earlier, the overlap region is negligily small at low Reynolds

numbers, especiaily on a rough surface. tn this case a more - reliable estimate of the wail

shear stress cm be made by fitting to the mean velocity data in both the overlap and

outer regions. For a turbulent boundary layer developing over a rough surface? the

complete velocity profile is given by

where il is Coles' wake parameter and w is a imiversd fimction of y/& Eqn. (2.12)

indicates that description of a measured velocity profile on a rough wail requires the

determination of four parameters, narnely: U , AB*, il and y,,. A reduction in the number

of parameters tr, be fitted is obtained by choosing to work with the defect form of the

velocity profile given by Eqn. (2.3). By subtracting U- h m its value U; at the edge of

the boundary Iayer, the toughness parameter AB' is eliminated and Eqn. (2.12) becomes

which indicates that the velocity deficit in the outer region is strongly dependent on the

magnitude of the wake parameter iï. The wake-parameter is generaiiy regarded as

dependent on streamwise location. Coles (1956) initiaily proproposeci that for a smooth-waü

zero pressure gradient turbulent boundary layer, il would be 0.55 at hi& Reynolds

nimibers, but later (1987) gave an asymptotic value of 0.62. The recent smooth waU

experirnents by Osaka et al. (1998) exhibiteci a Reynolds number dependence for Il.

However, an asymptotic vaiue of 0.62 was obsewed at sufficientiy hi@ Reg, where Reg

is the ReynoIds number based on bomdary layer momentum ttiickness 8. For sub-criticai

smooth-walI open channe1 fiows, Nezu and Rodi (1986) aIso reportai a Reynolds

number dependence, but indicated that the wake parameter rernains nearly constant at ï i

- 0.2 at suficientiy hi& Reynolds nurnbers. Xinyu et al. (1995) made LDA

rneasurements in super-critical open channel flows at varying bed sIopes and obtained a

valueof ïi = 0.3.

A commoniy used form of Eqn. (2.13) for the velocity distribution in zero pressure

gradient boundary layers on a rough wall is Hama's (1 954) formulation. For srna11 values

of y/& Eqn. (2.13) is dominated by the logarithrnic term and is therefore written as

For Iarger values of y/& the wake contriiution dominates and Hama proposed the

following fûnction

In both cases, the dispiacement thickness 6*, is used as îhe reference Iength s d e . Eqns.

(2.14) and (2.15) wrmect smoothIy at y/6'~: = 0.045 or y16 = 0.15. Bandyopadhyay

(1987) suggests that the Hama profile codd be fitted to obtain a diable estimate of Ut

irrespective of the &e. He also argued that since the Clauser technique matches the

profile in the logarithrnic region, which is thin, thae are only a few data points to work

with. in contrast, the profile matching using Hama's formulation covers virtually the

entire region.

It has, however, been observeci (e.g. Bandyopadhyay, 1987; Perry et al., 1987;

Krogstad et al., 1992) that the value of friction velocity U, obtained h m the Hama

formuIation (Eqns. (2.14) and (2.15)) is consistendy higher than that obtained from

either a mornentum balance or by extrapoIating the Reynolds stress to the wall.

Bradshaw (1987) suggested that this may be due to the strength of the wake, as impIied

by Eqn. (2,15), being too small. With recent evidence of the dependence of ri on Reg,

mughness and (high) turbulence levels, the usefiilness of a defect law such as that of

Hama which fixes the value of ïI may be limiteci for the present expainenta1 conditions.

As will be shown subsequentIy, incorrect wake slrength may contaminate an esbmate of

the skin fiction coefficient, and hence the roughness shift, in rough wall flows.

As an aiternative to Hama's formulation, Krogstad et ai. (1992) employed a

correiation that does not impiicitiy &c il but raîher dlows its value to be optimized.

They used the formulation proposeci by Fidey et ai. (1966), and later used by both

Grandie (1976) and Hancock and Bradshaw (1983), namely

Eqa (2.16) is the simplest polynomiaI s a t i w g the two boundary conditions (correct

slope and fùnction values) both near the w d and the boundary layer edge. Krogstad et

al. (1 992) combined Eqns. (2.13) and (2.16) to obtain

which is a more sophisticated expression for the mean velocity profile which can be

fitted to the experimental data to obtain the optimized values of U,, ïi and y,. Of special

importance is the expiicit detemination of the wake strength n, and the expectation of a

more accurate estimate of the fiction veIocity, UT.

For a smooth wall turbulent boundary layer, if a sufficient number of data points is

obtained in the linear viscous sublayer, i.e. the near-wall region where U' = y*, a more

accurate estimate of the wall shear stress can be obtained using the relation

Recent LDA measurements indicate that the linear profile (Le. Cf = y3 is strictly valid

only for y* I 4. This requirement is t w stringeut to be met in many physical

experiments, especiespeciaiiy if Pitot-tube and hot-wire probes are used. On the other hand, by

fitting a polynomial (Eqn. (2.2)) to the veiocity data in the near-wail region, the useful

extent of the viscous region in detemiining the wail shear stress can be increased. Durst

et al. (1998) used a fi* order polyuomial to descriie their na-wall mean velocity

profiles. in the present study, Eqn. (22) tnmcated at the £ifth order, i.e.

u'=Y++coy"+csy+S

is adopted in detemiining the wall shear stress for the smooth wall data.

The power laws proposed by Barenblatt (1993) and George and Castillo (1997)

were also used to derive skin fiction relations. The skin fnction relation obtained from

the formulation pmposed by Barenblatt (1993) was shown (see Djenidi et al., 1997) to be

of the following form

For the power law derived by George and Castillo (19971, the skin fiiction relation was

aiso shown to be a power iaw and is of the form

Djenidi et ai. (1997) applied the theory of Barenblatt (1993) to Iow Reynolds number

smooth-wall boundary layers. They simply defined Re = U&v and were able to obtain

skin fiction velocities that agreed with the wnesponding vaiues measured by a Preston

tube to within f 1.5 percent.

2.1.1.5 Scaling the Turbulence Quantities

Although the classicai theories proposed U: as the appropriate scaie for the Reynolds

shear stress, the scaiing laws for the other turbulence quantities are, in general, Iess

obvious. Moa of the earlier boundary laya analyses adopted U: for nonnaiizing the

nomai Reynolds stress components and U: for the various terms in the energy budget

(see for example, Krogstad and Antonia, 1999). Accordmg to the analysis of George and

Castillo (1997), the proper velocity seale for the normal Reynolds stresses is U: while

the shear stresses were shown to scale on u:. Their analysis aiso showed that the àpIe

comlations, stress production as weU as dissipation scale on the mixed velocity scaie,

2.12 Turbulent WaU Jets

2.1.2.1 Scaiing the Transverse Profiies

The scaling law for the mean velocity in the overlap region of a turbulent wall jet has

drawn considerable conmversy and continues to be of current research interest. Many

wall jet investigators assumed a similarity between the inner regions of a wall jet and a

turbulent boundary layec so that the overlap regioo is also describeci by the classical log

law. However, there is a considerable inconsistency in the log law constants (Le. the von

Karman constant K and additive constant B) reportecl by various investigators. A

summary of the log law constants reportecl in some earlier studies is given in Table 2.1.

ï h e technique used to determine the wall shear stress is also given.

I I 1

Myers et ai. (1963) 1 Clauser plot 1 0.41 1 4.9

Author(s) 1 Technique

W a and Eslcinazi (1 964) / Reston tube 10.48 (11.4

K 1 B

Pai and Whitelaw (1 969) 1 Razor blade 1 0.52 1 9.0

Karlsson et al. (1993) 1 Veiocity gradient at the wail 1 0.41 1 5.0

Alcaraz et ai. (1977)

Wygnanski et ai. (1 992)

t

Abrahamsson et ai. (1994) 1 Velocity gradient at the wail 1 0.41 1 5.0

Table 2.1 : Summary of Iog law constants for turbulent wall jets

Velocity gradient, momentum

0.56

0.4 1

8.0

5.5

It is clear from Table 2.1 that while some investigators (e.g. Karisson et al., 1993;

Abrahamsson et ai., 1994) observeci log law constants identical to the values used in

bomdary layer analysis, others (e.g. Kruka and Eskinazi, 1964, Pai and Whitelaw, 1969)

did not. It should be noted that the data reported by Wygnanski et al. (1992)

demonstrated a universality of îhe slope (K) but the additive constant (B) showed

Reynolds nurnber dependence. MacMuIlin et ai. (1989) (not shown) reported a tendency

of the slope of the log region to decrease with increasing turbulence intensity and

downstream distance.

A nurnber of reasons have been proposed to explain the discrepancies reported in

the literature regarding the log law constants. Laundm and Rodi (1983) attributed the

disparity in the log law constants to possible m r s in meamring the wall shear stress

and hence the tiiction velocity Ut. hother possible reason is attempting to fit the log

law over too wide a portion of the inner region. It is important to note that non-existence

of a well-denned log law region with universal log law constants has important

implications for both experimentatists and numericai anaiysts. For example, use of the

Clauser plot technique to detemine the wail shear stress as well as the conventionai

"waii funetion" used to tesolve the near-wail region in near-wail flow computations

cannot be employed in waII jet research.

Some waii jet researchets afgued that the inner region of a wall jet has a character

quite different h m that of a turbulent bomdary layer. Hammond (1982) indicated that

there is no weil-dehed log iaw regioa for the w d jet. He presented an analysis of the

complete velocity and proposed a composite velocity profile to describe the mean

velocity profile ftom the wall up to the edge of the flow. George et al. (2000) extended

their AIP anaiysis (origmiüy applied to a turbulent boundary layer) to derive similarity

theory for nubulent wall jets. Their analysis showed that the inner region of a turbulent

wall jet is identical to that of a turbulent bounrlrlry layer. More specifically it was shown

that Eqns. (2.2), (2.9) and (2.10) can be used to describe the inner region. The scaiing

laws for the turbulence quantities for turbulent wall jets were also shown to be identical

to those they obtained for a turbulent boundary layer.

2.1 2.2 Skin Friction Correlation

A problem plaguing wall jet analysis stems fiom the difficulties of measuring the wall

shear stress. A sumrnary of the skin fiction relations is given in the review article of

Launder and Rodi (1981). The skin fiction correlations reportai show considerable

scatter. Some of the inconsistencies reported in the literature are attnbuted to the lack of

twodimensionality, to the thinness of the inner layer and to poor experimentai

techniques (Wygnanski et al., 1992). Many devices commonly usai to determine the

wall shear stress, and hence the skin fiction, in turbulent boundary layers rely on the

validity of the log law. However, as demonstrated in Table 2.1, the imiversality of the

log law constants has been repeatedly questioned. Because the skin fiction coefficient is

such a rninor contributor to the wali jet growth rate, aîtempts (e-g. Schwan and Cos-

1961) to estimate the skin fiction h m a mornentum balance ofhm give highly

implausibIe results. Furthexmore, the shear stress fails off so rapidly with distance that it

is usually not possiile to determine the wall shear stress by extrapolating the Reynolds

shear stress to the wall (Launder and Rodi, 198 1).

Bradshaw and Gee (1962) reported skin friction measurements using a Preston

tube and proposed the following skin friction correlation:

Cf = 0.0315Re - 0.182 rn (2.22)

where Re, is the local Reynolds number based on Um and y,. Among the skin fiction

correlations available pnor to the review of Launder and Rodi (1981), Eqn. (2.22)

a p p m to be the most satisfactory correlation for a wall jet in stagnant surroundings in

the range 3 x lo3 c R h < 4 x 10'. Hammond (1982) derived the following 'optimum'

skin fiction formula for the plane wall jet

Cf = 0.06675Rem 4.258 (2.23)

Eriksson et al. (1998) also used their LDA data to develop the following skin fiction

relation

Cf = 0.0179Rern 4.113 (2.24)

George et al. (2000) showed that the skin fiction Iaw for a tlirbulmt wail jet is also a

power law and is identicai to that derived for turbulent boundary layers (i.e. Eqn. (22 1)).

2.1.23 Streamwise Development

The strearnwise evolution of the flow has traditionaily been s d e d using the dot height

(b) and the exit velocity (p). Accordmg to Launder and Rodi (198 l), the growth rate of

the jet haff-width and the decay of maximum velocity are, respectively, given by the

following reisttions

where KI and K2 are presurned to be constants. Whenever the above scaling laws are

used both the veIocity decay and spread rates showed Reynolds number and facility

dependence. Recent measurements reported growth and decay rates that showed

important sensitivity to Reynolds nurnber as weil as to the types of measuring devices

used. For example, the veiocity data reported by Wygnanski et ai. (1992) showed distinct

Reynolds number dependence. Abrahamsson et al. (1994) reported a spread rate that

varied fiom 0.075 to 0.081, dependiig on the exit Reynolds number, while Schneider

and Goldstein (1994) reported values in the range 0.074 - 0.082, depending on the

measuring devices used.

Narasimha et al. (1973) suggested îhat scaiing of the relevant distances by the

characteristic dimension of the nozzle and exit velocity rnight be enonmus. Instead, they

proposed d i n g the streamwise w01ution of the flow by the momentun flux (I = u$)

and kinematic viscosity (v) of the fluid. The more ment paramehic anaIysis by

Wygnanski et al. (1992) and the wall jet similarity theory proposed by George et al.

(2000) support Narasimha's suggestion. Following Narasimha's suggestion, the

maximuin velocity decay and spread of the wake half-width (yin) are shown (e.g.

Wygnanski et ai., 1992; George et al, 2000) to be power laws of the form

where A, B, a and p are constants that may depend slightly on initial conditions. From

similarity considerations, George et ai. (2000) showed that the local maximum veiocity

(U,) and the jet half-width (yii2) are also related by a power law as follows

w h ~ e C and y are constants that may depend on initiai conditions. Using the LDA

measurements of Eriksson et al. (1998), îhey recomrnended the foIlowing values: C =

1.85 and y= -0.528.

2.2 PREMOUS STUDiES

23.1 Turbulent Boundary Layers

2.2.1.1 Reynolds Number Efiects

ReynoIds number effects have been the focus of a number of previous near-wall

turbulent flow studies (Spalart, 1988; Antonia et al., 1990, Durst et d., 1998). Purtell et

ai. (1981) investigated Reyno1d.s number effects in a zero pressure gradient turbulent

bouadary layer. The Reynolds numbers examinai were in the range 450 < Ree < 5 100.

Th& resdts showed that the overlap region did not disappear even at the lowest Re0

examined. They observed that the outer wake parameter showed a distinct Reynolds

number dependence for Ree < 2000. In huer coordinates, distniutions - of the streamwise

turbulence intensity were similar for y* < 15 while a much greater degree of similarity

was noted when the boundary layer thickness was used as the normalizing length scale.

Wei and Wiarth (1989) made measurements in a fully developed channel over a wide

range of Reynolds nurnbers and concluded that the region of Reynolds number similarity

is limited to y+ I 10. Harder and Tiderman (1991) reported measurements in a fiilly

developed charme1 flow and observeci that for y+ 5 50, their profiles are independent of

Reynolds number. So et al. (1 996) investigated Reynolds number effects in zero pressure

gradient turbulent boundary layers (1410 I Ree S 15400) as well as fiilly developed

channel and pipe flows (180 5 Re, l 8760) using Reynolds stress models. nie results

show that Reynolds number effects are very distinct in the inner region, these effects are

less distinct for turbulent boundary Iayers in comparkon to pipe and channel flows. n ie

LDA measurements by Ching et ai. (1995) at 400 I Ree I 1320 showed that the effect of

ReynoIds number is felt d o m into the viscous sublayer.

One of the most refined sets of near-wail measurements was made by Durst et ai.

(1998). Their measurements were made in a M y developed - channel flow using a high

resolution LDA. The Reynolds mnnbers (based on bulk velocity and channe1 width)

varied h m 2500 to 9800. After applying ali knom corrections to their data, they

obsemed tint the streamwise turbulence intensity scaied on inner variables for y- I 50.

Furthemore, the peak value of the profiles was found to be 2.55, independent of

Reynolds number. Osaka et al, (1998) examined Reynolds number effects in turbulent

hundary layers. ïhey observed a reasonatik collapse of the mean velocity in the near-

wall region. ïhe u- profiles showed Reynoids number independence for y' I 20 and the

peak values were found to be insensitive to ReynoIds number. More recently,

Balachandar and Ramachandran (1999) reporteci LDA measurements in open channel at

180 < Ree < 480, thus extending the database to lower vaiues of Ra. They identified an

overlap region, albeit narrow, with K tbat is independent of Ree. Within the range of

Reynolds number considered, they obsmed the outer wake parameter to decrease with

increasing Ree.

In spite of some specific différences among findings of previous investigators,

Reynolds number effects in turbulent boundary layers are weak for Ree > 3000 (Antonia

et ai., 1990). An excellent review of Reynolds number effects in wall-bounded flows was

made by Gad-el-Hak and Bandyopadhyay (1994). They showed that even at the highest

Reynolds number flows available in the literature turbulence quantities scaled using

inner variables show Reynolds number effects.

The scatter among measurements has been attn'buted, in part, to resolution

p b l e m s associated with measuring techniques and inaccuracies in diagnostic

instruments. For example, in the case of hot-wire measurements, Johansson and

Alnredsson (1983) showed that îhe maximm vdue of the normaiized streamwise

turbulence intensity decreases h m u*= 2.9 for 1' = 2.5 to u = 2.1 for 1" = 100, where 1"

is length of the hot-wire probe in w d units. In a related study, Johansson and Aifiedsson

(1986) examineci the effects of Reynolds number and probe length on IL' distriiutions.

They found that in the near-wail region c 30), distributions of t showed a

dependence on probe length but were independent of Reynolds number. Gad-el-Hak and

Bandyopadhyay (1994) recommended that for diable meamment of turbulence

quantities, especiaiiy in the vicinity of the wall, probe lengths Iess than the viscous

sublayer thickness are required. Some of the scatter observed previously cm dso be

attriiuted to inaccurate values of Ur. Note that a UT value that is 5 percent too high will

pull the uC profile down and to the right with an overail enor in the distniution that is

higher &an 5 percent.

22.1.2 Surface Roughness Effects

Subsequent to the classicai sand grain pipe flow experiments of Nikuradze (1933), a

number of rough wall turbulent boundary layer measurements have been reporteci (Perry

et ai., 1969; Antonia and Luxton, 197 1; Bandyopadhyay, 1987; Peny et al., 1987; Hirota

et ai., 1993). A comprehensive review of both theoretical and experimental knowledge

of rough wall hirbulent bomdary laym was given by Raupach et al. (1991). Frnvya and

Fujita (1967) reporteci measurements on sand grain roughness and wire-screen with

different pitch-to-diameter ratio, t/d. in the case of the wire-screen data, they observed

that the effect of roughness increases to a maximum for 5 < t/d < 9 and decreases when

t/d> IO.

In most of the earlier rough waii investigations, minimal attention was given to the

specific f o m of the outer Iayer, Le. the outer wake component. Mils and Hang (1983)

remarked that extensive rough waii turbulent ùomdary layer experiments carrieci out at

Stanford University gave skin fiction coefficients that deviated fiom the Prandtl-

Schlichting (1934) formulation by as much as 25 percent. They attriiuted the disparity to - the neglect of the role of the wake component of the velocity profile in the Prandtl-

Schiichting formulation. A number of previous rough wail experiments were re-

evaluated by Tani (1987) and the values of n obtained fell in the range of 0.4 - 0.7. The

d-type roughness experiment of Osaka and Mochimchi (1988) at Ree = 5300 gave ll=

0.68. Recent boundary layer experiments by Krogstad et ai. (1992) on a rough d a c e

indicated il = 0.7, which is distinctiy different fiom the asymptotic value proposed by

Coles.

A nmber of rough wall boundary tayer caicuIations have been reported. Cebeci

and Chang (1978) and Krogstad (1991) used eddy viscosity and mixing Iength models,

respectively, to compute the mean velocity. Tarada (1990) and Zhang et ai. (1996)

employai different low Reynolds ntnnber k - ~ models and observed fair agreement

between caIculations and experiments. Patel(l998) used k-E and k-o models to calculate

both the mean velocity and Reynolds shear stress. Predictions of the mean flow,

especially the rougimess shi& were comparable to experimental data but the Reynolds

shear stress was in error over most of the boundary layer. -

Although the global effect of surface roughness on the mean flow is relatively well

tmderstood, considerable inconsistencies are reported regardhg roughness effets on

higher order moments. Grass (1971) reported rough wall measuranents ushg the

hydrogen-bubble technique at different values of the roughness ReynoIds nmber. He

observed that outside the roughness sublayer, v' is invariant of wall conditions. Wood

and Autonia (1975) concludecl fiom their investigation that the influence of surface

roughness is confinai to the wall region. Sabot et al. (1977) reported large differences in

the spanwise w' and vertical v+ components of the Reynolds stress aIthough the

streamwise component u was observed to be independent of wall conditions. Raupach

(1 98 1) made cross-wire measurements over cylindrical roughness elements arrangeci in

different patterns. It was found that outside the roughness sublayer, secondader

moments when normalized by Ut are universal and independent of surface roughness.

However, the third-order moments as well as production and turbuIence diffusion tenns

in the energy budget showed important sensitivity to the specific roughness

concentration.

Measurements over uniform spheres were reported by Ligrani and Moffat (1985).

The roughness Reynolds number considered varied h m transitionally rough to fully

rough re@es. Their results showed a lack of collapse in u' and v but the Reynolds

shear stress <u'v*> and the correlation coefficients were invariant with Rq, and

fkesbeam conditions. It was also observed that, for both transitionally and fully rough

regîmes, the d i f i o n terms are a l t d by surface roughness. Furthemore, turbulence

production caused by Reynolds shear stress increases with increasing roughness

Reynolds number. The d-type mugh wail experiment of Osaka and Mochizuki (1988)

aIso showed that significant differences exist between smooth and rough wall

measufements even at y = 0.66. Krogstad et al. (1992) compared measurements over a

smooth surface and wire screen roughness. Their d t s showed that u" is not sensitive

to the surface condition but the v' profile over the rough sinface is significantly higher

than observed for the smooth surface- The LDA measurements over smooth and d-type

roughness reported by Djenidi et ai. (1996) showed important differences between the

smooth and rough wall data at significant distances from the swFaces.

Mazouz et ai. (1994) investigated the turbulence structure over different types of

surface roughness with varying rougbness geometry. It was observed that the skewness

of the streamwise velocity fluctuation showed distinct dependence on the span-to-height

ratio of the roughness elements. However, skewness of the vertical velocity fluctuation

as welI as the streamw'se and vertical components of the flabiess factor did not show any

important sensitivity to roughness geometry. Mazouz et ai. (1998) compared

measurements ovw a smooth wail and a k-type roughness generated using square cross-

sectioned two-dimensionai bars. Their resuits revealed that u* profiles are independent of

surface roughness but vy and w* profiles over the entire channe1 are lower on the rough

wail than the corresponding smooth wdl data. Krogstad and Autonia (1999) made

measurements on wire mesh and lateral rods of equivalent rougtiness shift (AE33. niey

found that the disûibutions of Reynolds stresses depend on the specific form of d a c e

roughness. Using u-v quadrant andysis they also showed ttiat the near-wall diffusion is

bighiy dependent on the d a c e geometry. The mple products also showed distinct

roughness dependence.

The stress anisotropy tensor b, is an important turbulence parameter. Hem, bi =

<uiu2/2k - 8,j.U whm 2.k (= u2 + $ +d) is the hubulm~e kinetic ai erg^ and is the

Kronecker delta so that 6, = I if i = j and 6, = 0, othenirise. Although the shear stress

anisotropy (bld in turbulent duct fiows appears to be independent of surface roughness,

the streamwise (bI ,) and vertical (b2) stress tenson were found to be significantly higher

for a mugh surface than observed for a smooth surface (Sabot et ai., 1977; Mazouz et ai.,

1998). Compareci to the DNS resuits of Spalart (1988), the rough wdi boundary layer

measurements reported by Shafi and Antonia (1995) showed a reduction in the normal

stress anisotropy tensors. This is in contrast to the observations made in fidIy developed

channel flows.

2.2.13 Effects of Elevated Freestream Turbulence

As rnentioned in Chapter 1, the background turbulence levels near the free d a c e of an

open channel boundary layer are relatively higher than reported for canonical zero

pressure gradient turbulent boundary layers. in order to facilitate discussion and

cornparison with the existing Merature, some related boundaq layer experiments

conducteci at elevated fi.eestream turbulence intensity are briefIy reviewed. For smooth-

wail boundary layers at elevated turbulence intensity (Tu), Bradshaw (1978) argued that

the log law holds when there is local equilt'brium in the near wall region. Hanmck and

Bradshaw (1989) measured various tenns in the turbulence energy transport equation at

Tu I 6 percent and found the boundary layer to be in local equil1Ibnm. ThoIe and

Bogard (1996) extended the existing smooth-wall data to turbdence intensity values as

high as Tu = 20 percent Among otfier findings, they conii.rmed the vdidity of the log

law at high fkestmm turbulence and noted significant alterations of the outer region of

the bomdary layer. Based on the measrired velocity spectnmi, they found that at Tu = 20

percent, the fieestmm turbuience penetrates deep into the wall region. Experimental

evidence also suggests that the strength of the wake is strongly altered at high kstream

turbulence levels. Blair (1983) and Hancock and Bradshaw (1983. 1989) showed that as

the fieestream turbulence increases, the outer region of the boundary layer exhibits a

depressed wake region. At a turbulence Ievel of Tu = 5 percent, for example, the wake

was essentially nonexistent, in the tecent smooth-wall study of Thole and Bogard (1996),

an asymptotic value of ïi = -0.5 was observed.

2.2.1.4 Turbulent Boundary Layer in Open Channel Flows

There is a considerable amount of Iiterature on turbulent boundary layers in open

channel flow, see for example, Steffler et al. (I983), Nezu and Rodi (1986) and

Tominago et al. (1989). An exceflent review of the literature existing prior to 1993 is

given by Neni and Nakagawa (1993). More recent studies include the LDA

measurernents qorted by Xinyu et al. (1995), Baiachandar and Ramachandran (1999)

as well as LES and DNS results of Komori et al. (1993) and Borne et al. (1995) and Shi

et al. (1 999).

in open channel bomdary layers, the Moody chart (with pipe diameter replaced by

four times the hydrauiic diameter) has been recommended for the prediction of the skin

fiction (see for example ASCE Task Force, 1963). Other techniques widely used for

skh m o n measuremmts in open channel flows include UT = @ln, where g is the

acceleration due to gravÏty, h is the depth of flow and S denotes the channel slope.

Although the fiction velocity determineci using this relation is found in many previous

experiments to be in fair agreement with the values obtained using other techniques, it is

important to note that the former gives an average value rather than a locai one. As

rightly pointeci out by Nezu and Nakagawa (1993), the value of UT determined using the

channel slope may not be adequate for the evaluation of turbulence characteristics.

Simila. to other near-wail flows, distributions of the mean velocity are 0 t h

interpreted in the context of inner and outer scaling Iaws discussed in Section 2.1. Nezu

and Rodi (1986) and many other researchers indicated that the overiap region is well

describeci by a logarithmic law with universal constants identical to those used in

boundary layer analysis. The proper outer velocity and Iength scales are the maximum

velocity (U,) and depth of flow (h). In order to facilitate cornparison to earlier canonical

turbulent boundary layers, the bounA?iry Iayer thickness (Q, which is defined as the

vertical distance at which U = 0.99U, is adopted as the outer length scaie.

23.2 Turbulent Wail Jets

Some of the earliest measurements in a turbulent wall jet indude those of Forthmann

(1934) and Sigaila (1958) over smooth surfaces, and the rough wall rneasufements

reported by Rajaramam (1965) and Sakipov et ai. (1975). The extensive wall jet

literature existing prior to 198 1 was critically reviewed by Launder and Rodi (198 2,

1983). Some of these studies are summarized in Table 2.2. in Table 22, Rej is the

Reynolds number based on exit velocity (Uj) and dot height (b), x is streamwïse distance

relative to the exit, dyinld~ is the growtù rate of the jet-half-width, M(x) denotes the

local momentum flux and J (= utb) is the jet mommtum b d on exit conditions.

Bradshaw & Gee (1962) 6. L 339 - 0.07 1 0.43-0.47

1459

Schwartz & Cosart (196 1) 1 L3.5 - 42 [ 29 - 85 1 0.056 - 0.085 1 0.60 - 0.80 Myers et al. (1963) 7.1-56 12-190 0.077 0.65

I l I l

Tailland & Mathieu (2967) 1 11-25 1 33-200 1 0.075 0.89 I

Wygnanski et al. (1992) 1 3.7-19 1 0 - 140 1 1 1.0

Abraharnsson et ai. ( 1994) 10-20 O - 150 0.075 - 0.081 0.8 - 1 .O

Schneider & Goldstein (1 994) L 4 43 - 1 10 0.074 - 0.082 0.80

Eriksson et aI. ( 1 998) 10 0-200 0.078 0.85 - 1.0 Venas et ai. ( t 999) 15.2 125

Table 2.2: Summary of some earlier wail jet studies

Despite the Large body of Iiterature existing at that the, the review articles

revealed that accurate, consistent and comprehensive data sets were lacking and the

physics of the flow was still not weiI understmd. Some of the important observations

and conclusions drawn h m these rwiew articles are as foUows:

1. Almost a11 the measurements comLISIdered in these reviews were obtained using

Pitot-tubes and hot-wires with an obvious spatial-resolution Limitation, especially

when cross-wires are d. Acçurate and reliabie near-wdi data were scarce.

2. Many of the studies repoxted in the literature lack two-dimensiondity. This was

attniuted, in part, to inaccurate memement of the mean exit velocity.

3. The log law constants and skin friction correlation showed considerable scatter.

4. Scaling the streamwise evolution of the fiow with the slot height and exit

velocity showed considerable scarter. On the basis of the spread rates available at

that t he , a value of 0.073 fl.002 was recornmended.

5. Turbulence measurements were scarce. OnIy a few experimenters measured ai1

the Reynolds stress components. Furthemore, higher order statistics mch as

triple products and energy budgets were not sufficiently known.

A number of measurements have ken reported subsequent to the reviews of

Launder and Rodi. These studies attempt to address some of the important research

questions that were unanswered. Dakos et al. (1984) investigated a heated wail jet on

both plane and curved surfaces. Measurements reported include Reynolds stresses, heat

fluxes and triple velocity correlations. Wygnaaski et al. (1992) reported measurements

over a wide range of Reynolds numbers using hot-wire probes. The streamwise

turbulence intensity data showed significant Reynolds nurnber dependence. At a given

inlet Reynolds number, it was aiso observed that distniution of turbulence intensity

varied appreciably with streamwise distance for 60 < xh < 120, aIthough most studies

reported similarity in both mean and turbulence quantities for db > 20. Karlsson and co-

workers (e.g. Karlsson et al., 1993; Eriksson et ai., 1998) reported one of the most

comprehensive measufements using high spatial resolution LDA. They were able to

resolve the mean velocity d o m to y' = 1. The mean and ReynoIds stress data reported

by Eriksson et ai. (1998) showed similarity for xfb 2 40. It was &O shown chat the total

production of turbulence caused by the normal stresses is srnail compared to shear stress

production term. Fwtherrnore, the production of turbulence caused by shear stress

showed two peaks, one very close to the waii and a relatively higher one in the vicinity

of y = y1:2.

Schneider and Goldstein (1994) made measurements using LDA, Pitot-tube and

hot-wires. h contrast to measirrements obtained using the LDA, the mean velocity data

obtauied using hot-wires did not go to zero at large distances fiom the wall. The mean

data obtained using the Pitot-tube were unacceptably low in the outer region. This

appears to be characteristic of al1 Pitot-tube measurements reporteci in the Iiteran~. The

Reynolds stresses obtained using the LDA were significantiy higher than the data

obtained using cross-wires. In a related study, Venas et ai. (1999) compared their puIsed

hot-wire data to the LDA measuranents of Karlsson et ai. (1993) and Schneider and

Goldstein (1994) as weU as measurements obtained by Abrahamsson et ai (1994) in the

sarne faciity but with conventional hot-wires. in the case of the meaa veiocity

distribution in the outer region, they observai good agreement between LDA and the

pdsed-wire data but these profiles were quite different h m the profiles obtained using

conventionai thermal anernometry. The Reynolds stresses obtained using LDA and

pulsed hot-wire showed good agreement but were found to deviate significantly h m the

measurements obtained from the conventional hot-wire over most part of the fiow.

Accurate prediction of the turbulent wail jet has been a major challenge to

turbulence modelers. The unique characteristic of zero shear not coincident with the zero

mean velocity gradient suggests that gradient transport models may not be appropriate

for computation of turbulent wall jets. Launder and Rodi (1983) summarized some of the

earlier wail jet computations. Gerodimos and So (1997) assessed some of the exiçting

near wall two-equation (k-E and k-a) models for their abifty to replicate the mixing

behavior between the outer jet-2ike layer and b e r walf layers. Using the experimental

data of Karbson et ai. (1992) and Wygnanski et ai. (1992), they conchdeci that ail the

modeIs are capable of replicating the Reynolds number effects. However, prediction of

the near waII asymptotic behavior, spread rate and decay of maximum velocity was poor-

Yamamoto (1997) used a multiple-time-scaie Reynolds stress to cornpute the piane wdl

jet measurements of Irwin (1973). The mode1 gave a reasonable prediction for the mean

velocity distribution and the spread rate. The spanwise and vertical stresses were in good

agreement with measurements but süeamwise stress and shear messes were over

predicted. More recently, Vasic (1999) compareci the performance of two equation and

Reynolds stress models to the measurements reporteci by Karisson et al. (1992). It was

concluded that the Reynolds stress mode1 successfiilly predicted the velocity decay, but

results h m the two-equation models were in error. None of the models was able ta

predict the skin friction reasonably weU. Furthermore, the Reynolds stress modeIs gave a

supaior prediction of the Reynolds stress but prwlictions h m the two-equation models

were unacceptable over most region of the flow.

23 STATE OF KNOWLEDGE AND REFINEMENT IN OBJECTIVES

In the previous sections of this chapter, the scaiing laws for the mean and turbulence

quantities were reviewed. Some of the widely used skin fnction relations were

summarized. In view of the strong interaction between the inner and outer Iayer, it was

emphasized that skin fnction cotrelations which explicitly takes the specific stmcture of

the outer flow into consideration should be prefmed. The existing iiterature on the

turbulent wall jet as well as Reynolds number and surface roughness effects in turbulent

boundary layers was briefly reviewed.

Regarding Reynolds number effects, no definitive statement could be made as to

the extent to which it persists. Some of the inconsistencies can be explained by poor

spatiaI resolution and inaccurate skin fiction measurements. Although the fiction

veIocity is used exclusively to scale both the mean velocity and turbulence statistics,

most of the scaling laws proposeci by George and Castillo (1997) suggest otherwise. It

will be of interest to see how the flow structure varies with Reynolds number when the

recent theory is used to anaiyze the data.

A summary of the dimiutions of turbulence intensities and ReynoIds shear stress

outside the roughness sublayer in some studies is given in Table 2.3. It is clear h m this

table that, in spite of extensive research efforts, the present state of knowledge regarding

roughness effects on the turbulence structure is conüadictory.

[ Author(s) 1 Flow 1 u' 1 V* 1 u-vT 1 Grass (1971)

Ligrani & Moffat (1985)

Mazouz et al. (1998)

Table 2.3: Summary of state of knowledge regarding roughness effects on

turbulence structure (ZPG r zero-pressure gradient)

Krogstad & Antonia (1 999)

Most of the earlier mu& waII measurements were made using hot-wires which

Open channel flow

ZPG bouudary Iayer

Developed duct ff ow

may have been affecteci by the high locaI turbulence levels in the vicinity of roughness

elernents. In some of the previous experiments, ody rough waIl measurements are

ZPG boundary layer

conducted and the results compared with smooth wall data conducted at different Rq or

invariant

higher

lower

in different facilities. As is well known, turbulent flows are very sensitive to initial or

boundary conditions so that measurements obtained in different facilities or at different

higher

conditions may not be similar in al1 details. This may suggest that with exception of wall

invariant

invariant

lower

conditions, attempts shouid be made to match al1 other initial conditions as much as

possible so that def i te conclusions could be drawn with regards to surface roughness

invariant

invariant

higher

effects on the turbulence structure. Furthemore, the existing rough waIl literature

higher

indicates that accurate measurement of skin fiction, - especially at low Reynolds

numbers, stll poses a challenge to experimentaiists.

in the Iight of receut LDA and pulsed hot-wire measurements, it appears the scatter

in earlier turbulent wali jet data can be attniuted, at least in part, to the well-known

problems of Pitot-tube and conventional hot-wires close to the wall and in regions of

hi& local turbulence intensity. Aithough studies reported in recent years attempt to

address some of the open questions and issues raised by the review articIes by Launder

and Rodi (1981, 19831, some important and practically devant research questions

remain unanswerecl For example, higher order statistics such as triple cornfations

cernain unknown wWe information on energy budgets remains limited. Although most

practical flow systerns in which walI jets are found are hydrauiically rougfi,

measurements of turbulent wail jets on rough d a c e s are rather scarce. Perhaps with the

exception of the Pitot-tube rneasurements reported by Rajaratnam (1965) and Sakipov et

ai. (1975), the effects of h e roughness on h e hydrodynamic characteristics of

turbulent waII jets are not knowa.

On the bais of our current undersbnding on turbulent bouudary layers and wail jets, the

objectives of this study are re-stated as foliows:

To examine the appropriateness of the different scaiing laws proposed for the

mean velocity and turbuience statistics in a turbulent boundary tayer.

To detennine diable methods for the evaluation of skin fiction in near-wall

turbulent flows over smooth and rough surfâtes.

To uivestigate the effects of wall roughness on the mean and nnbulence statistics

m open chute1 turbulent boundary layets and waIl jets using different types of

mughness elements.

To provide benchmark data for rough wall turbuient bounday layers and wail

jets in open channel for the parpose ofdeveIoping pracîicai turbulence modeIs.

CHAPTER 3

INSTRUMENTATION AND EXPERIMENTAL DETAILS

h this chapter, an overview of the LDA systern is given. Some of the problems, which

may h o d u c e significant measurement m r s , are also reviewed in this chapter.

Descriptions of materials used to mate the surface roughness as well as the test

facilities. instrumentation, and measurement procedure are given. Experimental details

and sumrnaries of test conditions for both the boundary layer and wall jet experiments

are also presented.

3.1 THE LASER DOPPLER ANEMOMETER

Laser Doppler anemometry is the measurement of fluid velocity by detecting the

Doppler fiequency shifi of Iaser Iight that fias been scattered by srnail particles moving

with the fluid. A laser Doppler anemometer (LDA) system consists of a laser source, an

opticai arrangement, a photo-detector that couverts iight into electrical signais and a

signal procasor. The various components are discussed in Appenduc A.

3.2 ERRORS IN LüA MEASCTREMENTS

In spite of the non-intrusive characteristic of the LDA and iîs suitability for turbulent

flow measurements, its potentid to provide highiy accurate measurements is sometimes

not realized because of some inherent problems. Some of the well-known sources of

memernent mors include velocity bias, presence of multiple particles in the

measuring volume, gradient broadening, and mors due to noise and non-orthogonaiity - of beam mssing. These errors are discussed in Appendix B. The resu1t.s of preliminary

experiments conducteci to examine some of these effects are dso reporteci in Appendix

B. Steps taken to minimize or correct possible meamexnent errocs are also discussed.

With regard to velocity bias, experiments were conducted using three different

sampling schemes (Appendix B.1). The results showed that the maximum deviation

observed for each statistic is comparable to the correspondhg measurement

uncertainties. Based on experimentd investigations of Johnson and Barlow (1989) (see

Appendix B.2), it is i n f d that the streamwise component of mean velocity as well as

streamwise and vertical components of turbulence fluctuations are nearly independent of

the spanwise dimension of the probe volume. Howwer, the Reynolds shear stress may

be underestimated by as much as 12 percent.

Analytical treatments and experiments carried out by Durst et ai. (1995, 1998) and

Eriksson et al. (1999) are Summarized in Appendix B.3. Their results suggest that the

effects of gradient broadening on mean and turbulence qmtities are negligible for the

present system and experimentd conditions. They also showed that m r s due to noise

are negiigibie except in the immediate vicinity of the wail (Appendix B.4). u1 the present

smdy, steps are taken to minimue such mrs.

In order to obtain data very close to the waii, the fiber-optic probe is pitched

towards the wall at an angle $. Prelirninary experhents were wnducted to examine the

angle of tilt (B) on the mean velocities and higher order turbulence statistics (Appendix

B.5). It is concluded that for fi 2 9, except for the vertka1 turbulence fluctuations, the

turbulence statistics do not show any significant dependence on the angle of tilt.

3 3 EXPERIMENTAL SET-UP

33.1 The Open Channel Flume

The boundary layer experiments were wnducted in a rectangular cross-section open

channel flume. A schematic of the flume is shown in Figure 3.1. The flwne is 0.8 m

wide, 0.6 m deep, and 10 m long. The sidewalls of the flume were made of transparent

tempered giass to facilitate velocity measurements using a laser Doppler anmorneter. A

contraction and several stilling arrangements used to reduce any large-scale turbulence

in the flow preceded the straight section of the channel. The channel bottom was made

of brass and the slope was adjustable. For the present experiments, the channel bottom

was horizontal. The experiments were wnducted on a hydrauiically smooth and three

different types of surface mughness. The various rough surfaces are descnied in Section

3.3.3.

3 3 2 The Waii Jet Facility

Figure 3.2 shows an overview of the set-up for the wail jet experiments. The important

dimensions are also indicated. The wail jet test facility was screwed on to the bottom of

the open channel flume descnied in Section 3.3.1. The inlet of the nozzie was pIaced 3

m downstream of the channel contraction, The nozde has a contraction ratio of 9 to 1

and was designed foIiowing More1 (1975) in order to avoid flow sepration. Depending

u ?ontraction (a) Top view

reduction (Not drawn to scak)

10 rn B

9, - t - 3 j ~ - 4 ~ ~ m Ï n

rn

Fig. 3.1 : A schematic of the open channel fluxne

&

0.6 m

_t *To weighing

\ . . - -t il-

tank id^ for (b) Side view

ulence

t=6mm,b=IOmm. (Not dram to scaIe)

Fig. 3.2: A schematic of the wali jet f d t y

on the exit jet veiocity Uj, the water level downstream the exit varied fiom 350 to 400

mm above the floor of the test faciity. The ratio of dot thickness (t) to the slot height (b)

was th = 0.6 while the width (w) of the dot to the dot height was w h = 79. A weir

dowflstfcam kept the water leveE constant. The slot exit was preceded with sûaw packing

to reduce any large-de disturbance in the approaching flow, Tbe wail jet experiments

were conducted on a smooth wall and a rough surface created from sand grains as

descn'bed next.

3 3 3 Description of Surface Roughness

In order to examine the effect of d a c e roughness on near-wall flows, three

geornetricaiiy différent types of surface roughness were employed in addition to a

hydrâulically smooth surface:

Figure 3.3: Pichires showing sections of (a) perforated plate (PF) and (b) wire

mesh (WM) mugh surfaces

1 A 1 A-mm thick and 1.5 rn long sheet with circular perforations arrayed in a

hexagonal pattern. The perforation diameter was 2 2 mm 4 t h a 4.0-mm spacing

between centers. This configuration gives an openness ratio of approxmiately 43

percent. A picture of a section of the perforated plate (PF) is shown in Figure

3.3a.

2 A uniformiy and closely distniuted 1.2-mm nominal diameter sand graias (SG).

The sand grains were coated on to a 1.75-m long plywood sheet using double-

sided tape.

3 A 1.3-m long stainiess steel wire mesh (WM). The mesh was made of 0.6 mm

diameter wires with 7.0 mm centerline spacing, giving a ratio of centerline

spacing to wire diameter of about 12. A section of the wire mesh roughness is

shown in Figure 3.3b.

3.4 INSTRUMENTATION AND MEASUREMENT PROCEDURE

The velocity measurements were obtained using a single- and two-component fiber-

optic probe LDA system. nie LDA system is powered by a 300 mW Argon-Ion laser

(Dantec inc.). The optical elements include a 40 MHz Bragg ce11 to remove directional

ambiguity, a 1.96 beam expansion unit, a beam splitter, a color separator and a 500 mm

focusing lem. The laser beam is separated into green (A = 514.5 nm) and blue light (A =

488 nm). The two-component system uses a four-beam two-wlor configuration arranged

at right angles to each other. The measWng volume dimensions (based on the e-' light

intensity cut-off point) for the present configuration were 0.12 mm x 0.1 2 mm x 1.4 mm.

A photo-multiplier (PM) configured in backscatter mode is used to coI1ect scattered Iight

received Eom the measuring volume. The optical and operating parameters of the LDA

system are summarïzed in Table 3.1.

1 Wavelength of laser (nm) 1 514.5 (green), 488 @lue) 1 l

Diameter of laser beam (mm) 1 1.35 I

Focal length of the bdnsmitting lem (mm) 1 400 (Series B), 500 (dl other tests) 1 Beam separation (mm) 1 38 (Series B), 74.5 (dl others) 1

Measuring volume dimensions 1 0.19 x 0.19 x 4.09 mm' (Series B)

Number of f i g e s

Fnnge spacing (pm)

36

5.422 (Series B), 3.2842 (al1 others)

I

Bragg ce11 140 M H z

1 Beam expansion unit

Table 3.1 : Optical and operating parameters of the LDA

O. 12 x O. 12 x L .4 mm3 (dl other tests)

1.96 (for al1 tests except Series 3 )

In the present sets of experiments, no artificial seeding was used since there were

enough scattering particles (i.e. naturaily occming hydrosois) in the flow. The use of

naturaily and uniformiy occurring seeding is expected to minimize velocity bias towards

higher velocities (McLaughIin and Tidennan, 1973). Scattered Iight from the rneasuring

volume is digitally processeci with a 58N40 Flow Velocity AnaIyser (FVA) that is

hterfaced to a microcornputer using a 58G110 PC interface board. The measurement

process, data aquisition and data processing are controlled by type 46S51 FLOware,

which is a user-fiiendly professional software package developed by DANTEC. The

trigger of the LDA system is set in such a way that no signal is obtained when one of the

laser beams is blocked. The bandwidth parameters affecthg the arrivai and transit time

clock rates, the opticai shift and hardware fiIter values werp set to 'best choice vaIuesT

acwrding to the recommendations of DANTEC. With these settings, the influence of

noise on the rneasured data is expected to be minmiai. Particuiar attention was paid to

the validation parameters that affect whether data are converteci by FLOware and

whether invaiid data cm exist in the converted data me. In this regard, both velocity

channeis were enabled and the rejection levels were set in accordance with the present

optical parameters and the flow conditions following the suggestions made by

DANTEC.

The transmitting opticd elements were cleaned before the commencement of the

tests. Prior to the measurement of each set of data, the bottom wall as well as the

sidewdl of the test faciIity were cleaned. These were found necessary in order to

minimize extraneou Eght scattered fiom particles distributed through~ut the

illuminating beams. %or to each measurement series, data were acquired in a repetitive

mode, which means that acquisition is performed on line with data displayed on the

acquisition window. This mode is used to examine the quality of data, h g e count

mors, and signal-to-noise ratio (SNR) validation level m r . In the event of

unsatisfactory error levels or poor quaiity of data, the aecessary optical parameters and

validation levels are met. Karisson et ai. (1993) and Diirst et al. (1995) have pointed out

that even a mal1 misalignment of the fier-optics probe on the order of 1' in the x-y

plane could cause large m r s in vertical component of the mean velocity and its

fluctuation. With this in mînd, care was taken to minimize any possiile misalignment. in

most of the experiments descnibed below, the probe was pitched towards the bottom wall

but in a way that no significant measurement errors are introduced. On the basis of

preIiminary reSPLts, which are summarized in Appendix B.1, no correction for velocity

bias was applied.

3.5 EXPERIMENTAL DETAILS

3.5.1 Boundary Layer

Three sets of boundary layer experiments were conducted. The first set (Series A) was

conducted on a hydraulically smooth surface at five different Reynolds numbers. in

order to stay in the sub-criticai range (Le. Froude number less than unity), it was not

possible to attain Reynolds nurnbers based on rnornentum thickness higher than 3300. in

spite of this limitation, these measurements dlowed some scaling issues and Reynolds

number effects to be examined. The major objective of the second (Series B) and third

(Series C) sets of boundary layer experimmts was to examine the effects of surface

mughness using a single- and two-component LDA, respectively. As will be discussed

in Chapter 5, the configuration of ttie present two-component LDA system could not

perrnit measurements very close to the wall. in the boundary layer experiments

descnied beiow, the change in water surface elevation was less than 1 mm over a

streamwise distance of 600 mm implying a negligiile pressure gradient. No d a c e

waves were observecl at the fiee surfxe. Extensive prelllninary experiments showed that

the variations of the mean and turbulence quantities across the channel (i.e. spanwise

direction) are comparable :O the measurement uncertainties at the rniddle third of the

channel. In al1 the experiments reporteci in the next sections, the measurements were

acquired at the centerhe of the channeI.

35.1.1 Series A: Reynolds Nwnber Effects

As rnentioned above, the object of these measurements is to investigate Reynolds

number effects on the stmmwise cornpotlent of the mean velocity and its higher order

turbulence statistics. The measurements were made on a hydraulically smooth d a c e at

five différent fieestream velocities. The depth of flow was kept constant at h = 100 mm

and the channel aspect ratio (AR = Bk) was 8. To ensure a fiilly turbulent boundary

Iaym, a 1-mm diameter rod located 4.5 m downstream of the contraction and spanned

the width of the flurne was used to trip the flow. The rneasurements were obtained at 750

mm (ie. 750d) downstream of the trip. For rnost of the experiments in this series, the

LDA probe was tilted at P - 2" towards the bottom wail. The vdidated data rate varied

fiom 6 to 10 Hz close to the wall and approximately 50 Hz at distances remote corn the

wail. The maximum duration of data acquisition at each measuring location was set to

750 seconds while the maximum sample size was set to 10000. Typical sample size at a

rneasuring point varied from 5000 to 10000.

The test conditions are summarized in Table 3.2. In this table, U, denotes the local

maximum velocity, Tu is the turbulence intensity (u/U,) at the outer edge of the

bomdary layer (y = 6), Reg and Rq, are the Reynolds number based on the momentum

thickness 0 and depth of flow, respectively. Re, (= hua) is the Reynolds number based

on the fiction velocity and depth of flow, K is the Reynolds number based on the

bomdary Iayer thickness and the fiction velocity, H is the boundary layer shape factor,

Cf is the s h fiction coefficient, and 1; (= 1,Ua) is the vertical dimension of the probe

measuring volume in viscous units. The present values of 1; are adequate to resoIve

both the mean and turbuience statistics down to the wall (Gad-el Ha. and

Bandyopadhyay, 1994). According to the investigation of Johnson and Barlow (19891,

the spanwise dimensions of the measuring volume in walI uni& (1 1 < 1; < 39) are not

expected to cause any signifïcant effect on the mean and turbulence intensity. On the

basis of Re& the present fIows may be considered as low ReynoIds number turbulent

boundary layers. The present values of Rq, are comparabIe to most open channel flow

data avaiIable in the hterature. The range of Re, considered herein is, however,

considerably higher than those reported for fùily devetoped (closed) duct flows.

Table 3.2: Summary of test conditions for Series A.

Test

A-SM1

A-SM;!

A-SM3

A-SM4

A-SM5

3.5.12 Series B: 1-D Smooth and Rough Wall Expehents

In this set of experiments, measurements were obtained on a hydrauiicdy smooth and

three geometncaiiy different rough surfaces. The rough sucfaces consist of the

perforated pIate (PF), sand grain (SG), and wire mesh (WM) descnied in Section 3.3.3.

For each surface condition, measurements were made at three different veIocities and

depths of flow. To ensure a turùuient boundary layer, a trip was Iocated 3.5 m

doWIlStream of the contraction and sparmed the width of the flume. The trip was

composed of 3-mm (median diameter) pebbles giued to the bottom of the channel as a

40-mni Iong strip. The perforated plate (PF) and sand grain roughness (SG) were Iocated

at about 1.1 m downstrearn of the trip, whde the wire mesh screen was located 1 2 m

downstream of the trip.

U,

15.3

20.9

34.5

54.2

62.1

Tu(%)

2.0

2.4

2.1

2.0

3.2

Ree

750

1080

1450

2400

3250

Rq,

15,300

20,900

34,500

54,200

62,100

H

1.30

1.3 1

1-29

1.25

1.25

Cr

(X 1 O")

5.33

4.76

4.36

4.23

4.01

F

440

560

930

1450

1610

Re,

790

1020

1610

2410

2780

1;

1.0

1.3

2.0

3.0

3.5

For convenience, a reference axid position (x = 0) was located 1.3 m downstream

of the trip. For the smooth surface (SM), measurements were made at an axial station of

x = 0.50 m. For the wire mesh roughness (WM), measurements were made at axial

locations of x = 0.30 and 0.50 m for each test condition. The measurements on the

perforated plate and sand grain roughness were conducted at x = 0.10, 0.25 and 0.52 m,

for each test condition. The data reporteci in this study are measurements obtained at x =

0.50 or 0.52 m. A summary of the important test conditions for this set of measurements

is &en in Table 3.3.

Test

B-SM 1

B-SM 3

B-PF 1

B-PF 2

B-PF 3

B-SG 1

B-SG 2

B-SG 3

B-WM 1

B-WM 2

B-WM 3

Type of

surface

smooth

smooth

smooth

perforated

perforated

perforated

sand grain

Sand grain

sand grain

wire mesh

wire mesh

wire mesh

Depth, U, Tu(%) 6 0 H

Umm) W s ) (aty = 6) (mm) (mm)

100 0.737 3.1 46 3.56 1.29

80 0.331 3.3 48 4.17 1.33

Table 3.3: Summary of test conditions for Series B

3.5.13 Series C: 2-D Smooth and Rough Wdï Experiments

The measurements in Series C were obtained to examine the effects of surface toughness -

on the turbulence structure using a two-component LDA. The use of a two-component

LDA allowed the Reynolds shear stress, the mixing length and eddy viscosity as well as

the turbulent energy budget to be examined. Measurements were obtained on a smooth

surface (SM), sand grain (SG) and wire rnesh (WM) roughness. A trip composai of 3-

mm (median diameter) pebbles glued to the bottom of the channel as a 40-mm long strip

was located 5.0 m downstream of the contraction. The measurements were made at an

axial location of 1.0 m downstream of the trip. For the smooth wall rneaSuTements, the

probe was slightly pitched towards the bottom waii. Coincident data rates of 7 Hz very

near the wall and 30 to 50 Hz away h m the wall were typical depending on the local

veiocity. The maximum duration of data acquisition was set to 1500 seconds. Dependhg

on the local velocity and distance away from the wall, typical sample size at a measuring

point varied h m 10000 to 20000. Due to a hardware limitations, two-component

measurments could not be made close to the wall. In order to resolve the streamwise

velocity statistics d o m to the waI1, one-cornponent LDA rn-ents were dso made

for each test condition. The streamwise turùulence statistics obtained h m the two-

component measutements were compareci to the corresponding measurements made

using the one-component LDA. For each test condition, the two sets of data were found

to agree witbin measurement uncertainties.

A summary of some important flow parameters is given in Table 3.4. The probe

dimension in the waii-normal direction was in the range 1.8 1 1,' 1 3.5 which is high

enough to permit reliabIe meamrements of both meau velocity and turbulence intensity.

The spanwise dimensions in w d units are in the range 22 < 1; < 41. Based on the

results of Johnson and Barlow (1989), the present u' and vT profiles will be unaffecteci

but <u*vi may be underestimateci by as much as 12 percent. hasrnuch as the spanwise

dimensions in wall units are approximately constant for smooth and rough wall

experiments at similar ûeestream conditions (i.e. for SML and SGL; and also for SMH,

SGH and WMH), underestimation of <uTvT> is expected to be similar. Because the

purpose is to examine the effects of d a c e roughness on the txrbulence structure, no

attempts are made to conect the Reynolds shear stress for probe volume effects.

Table 3.4: Test conditions and boundary layer parameters for Series C

3.5.2 Waii Jet Erperiments

The wall jet experiments were conducted on a smooth walI (SM) and a sand grain (SG)

rough d a c e . It is impossiIe to mate a turbulent waI1 jet in a stfil surrounding in an

open channe1 flow. As wii be s h o w in Chapter 6, the outer edge of the present waiI jet

is characterized by high background turbulence levels and recirculating fiow.

Notwithstanding these effects, the d t s obtained h m the flows consldered here are

similar in many respects to existing data in the literature. in wall jet expaiments, it is

well known that inaccurate measurements of the exit velocity profile make a significant

contriiution to lack of conformity of the local momentum flux to two-dimensionality

(Launder and Rodi, 1981). In view of this and also due to the importance of the exit

momentum flux in scaling the streamwise evolution of the mean flow, measurements

were obtained at the jet exit (x/b = O) in the case of the smooth wall tests.

Two sets of wall jet measurements were made. The fkt set (Series D) pertains to

single-component measurements on a smooth wall and a sand grain rough surface,

Mea~u~ements at the exit as well as several locations downstream of the jet exit were

made. This dowed the effects of surface roughness on the streamwise component of the

mean velocity and its higher order moments to be made. This set of data is also used to

examine the streamwise evolution of the wall jet. in the second set of expaîments

(Series E), two-cornponent measurements on srnooth and sand grain rough surfaces are

reported. Measurernents obtained in Series E enabled turbulence statistics such as

Reynolds shear stress, triple correlations, and energy budgets to be analyzed.

3.5.2.1 Series D: l-D Smooth and Rough Waii Erperiments

In this set, measurements were obtained at the slot and several distances up to 100 dot

heights downstream of the exit. For the smooth w d measurements, the probe was tiited

about 2" towards the bottom waU. The turbdence intensity in the central region of the jet

exit varied h m 3 to 6 percent. The samphg rate varied h m 7 Hz in regions of low

velocity to 80 Hz in regions of high IocaI velocities. The maximum sampiing time and

maximum sampie size at a mea-g location were set to 1Oûû seconds and 10000,

respectively. Typical sample sizes varied h m 5000 to 10000. The boundary layer

thickness at the exit is approximately 2 to 3 mm. -

Depending on the nurnber of downstream locations traverseci, the duration of one

s a of experiments was in excess of 100 hours- For each test condition, a minimum of 10

measurements of the mass flow rate was obtained using an electronic weighng tank.

The standad deviation of the mean bulk velocity Ub cdculated h m the mass flow rate

varied fiom 2 to 5 percent. The higher values were typical for experiments that ran for a

longer period of time.

The important test parameters are summarized in TabIe 3.5. Here, U, is the

maximum velocity at the jet exit, Ub is the b& mean velocity determined h m rnass

flow rate measurements. Rej and R a are the Reynolds nuniber based on exit conditions

(Uj and b) and (Ub and b), respectively. The momentun thickness determined h m the

exit veIocity profiles is denoted as 8 and H is the boundary layer shape factor.

Tabk 3.5: Summary of test parameters for Series D

3.5.22 Series E: 2-D Smooth and Roagh Waii Experiments

in this set of experiments, single- and two-component velocity measurements were

obtained on smooth and rough surfaces. For the two-component meaçuremmts, the

probe was pitched about 3' towards the bottom wdl so that data couid be obtained doser

to the wall. Following the preliniinary results summarized in Appendix B.2, it is

concluded that no significant mors were caused in the measurements of U, V, u and

<UV>. The vertical turbulence fluctuation, Le. v, rnay be slightly contaminated, but as

noted in Appendix B 3 , any possrile m r may be comparable to the corresponding

measurement uncertainties in v, The sampling rate varieci h m 7 Hz in regions of low

velocity to 60 Hz in regions of high 1oca.i velocities. The maximum sampling t h e and

sampie size at each measuring location were set to 1500 seconds and 15000,

respectively. Depending on the local veIocities typical sample size varied h m 10000 to

l5OoO.

E-SM2 1.023 0.857 L0300 9000

E-SGI 1.304 1.117 13100 12000

Table 3.6: Summary of test parameters for Series E

A summary of some miportant test parmeters is given in Table 3.6. For Test E-SMl,

measufements were obtained at x/b = 0, IO, 30-40, 50,60,70, 80 and 100. For each of

Test E-SM2, E-SGI , E-SG2, rneasufements were obtained at x/b = 30 and 50.

The analyses and results of the above experirnents are reported in the next three

chapters. In Chapter 4, the boundary layer measurements obtained in Series A and B are

discussed while the r d t s for Series C are discussed in Chapter 5. The waIl jet data

obtained in Series D and Series E are discussed in Chapter 6.

3 -6 Uncertainty Estimates

In this section, statistical uncertainties, at the 95 percent confidence level, are presented

for the mean velocities and turbulence fluctuations. A more complete uncertainty

analysis is presented in Appendix C. For the boundary layer, measurement uncertainty in

the mean velocities (U and V) and the turbulence intensities (u, v) is l a s than t percent.

Close to the wall, the enor in u is estimated to be about 4 percent. The maximum

uncertainty in the Reynolds shear stress is about 12 percent. These uncertainty bounds

are similar to those obtained by Schwarz et aI. (1999). In the inner layer of the wall jet,

the estimates are similar to those outlined for the turbulent boundary layers. The

uncertainty in the outer Iayer is substantially higher due to the local turbulence intensity

as weU as reduction in sample size. Typicai estimates in the outer region are as follows:

+ 2.5 percent for the mean veIocities, f 5 - 10 percent for the turbulence intensity and

Reynolds shear stress.

SURFACE ROUGEiNESS AND LOW REYNOLDS NUMBEREFFECTS ON

THE STREAMWISE VELOCITY COMPONENT

In this chapter, smooth and rough wail measurements of the streamwise component of

the mean velocity and its higher order turbulence statistics are reportecl. The techniques

used to determine the wail shear stress for both the smooth and rough wail data are

discussed. Measuments obtained in Series A are used to compare the conventionai

scaling Iaws with the more recent theory proposed by George and Castillo (1997). This

aiso allows the effect of Reynolds number on the mean and turbulence quantities to be

examined. The effects of surface roughness on the mean velocity and turbulence

intensity are examined using the measuements obtained in Series B. Finally, the

appropriateness of the power laws proposed by Barenblatt (1993) and George and

CastiIlo (1997) to desmie the mean velocity profile on both smooth and rough &ces

is assessed- The waii shear stress values obtained h m these power laws are also

compared to those obtained h m other reliable and widely used techniques.

4.1 Determination of Waii Shear Stress

An accurate determination of the wall shear stress (or the fiction vel&ty) is important

because of its relevance in scahg the mean velocity and turbulence quanàties. For flow

over a smooth d a c e , a reliabie estimate of the wall shear stress can be obtained h m

the velocity gradient at the wail. Because of the thinness of the linear viscous sublayer,

sufficient data could not be obtained in this region for some of the smooth wall

experiments. Consequently, the velocity gradient at the wall, whenever possiile, as well

as fourth (v = f + c4y4) and fifth (UT = y + 4y'S + cs i5 ) order polynomiai fits to the

near-wail data are used to estimate the fnction velocity for the smooth wail data.

Table.4.1: Summax-y of fiction vetocity and deviation of measmeci U' from

log law profile at y' = F

The near wall data and the conesponding Iinear as weU as the fourth and f i f i

order polynomial fits to some of the smooth waI1 data sets (A-SM 1, A-SM2, A-SM3) are

shown in Figure 4.Ia The coefficient q = - 0 . 0 2 7 is in good agreement with the value

of -0.OOO3 (H.ûûû1) suggested by Eriksson et al. (1998); the value of ci = 1 3 . 4 ~ 1 0 ~ is

0.7 percent lower than that proposeci by George and CastiUo (1997). The slight

difference between the present value for cs and that recommended by George and

Castillo (1997) may be due to Reynolds number effects. Figure 4.la shows that the

experimental data agree fauly well with the linear fit for y' I 5, while agreement is good

20

O A-SM1 (Re, = 7 50) A A-SM2 (Re, = 1080) O A-SM3 (Re, = 1450)

15

U' ----U-=y- - 0 2 7 x [ 0 ~ ~ y ~ + i3.4x10'f5

1 O

5

O O 5 - 10 15

Y Fig. 4. la: Detemination of U- for smooth wail data:

Linear and polyno&ial fits to near-wail data

Fig. 4. Ib: Determination ofïI and Ut for smwth and mugh walI data

(Lmes mdicate nts to Eqns- 2.13 and 2.16)

for yT I 14 in che case of the fifth order polynomial. The values of Ut and the skin

fiction coefficients for measurements obtained in Series A are given in Table 4.1. The

values of Ut determinecl by fitting a fifth order polynomial to the near-wall data agree

with the corresponding value obtained fiom the velocity gradient (whenever possible) to

within + 1.5 percent. in Table 4.1, AUrn- denotes the deviation of the mean velocity at

y' = F'. The relevance of this parameter will be discussed in a later section.

For the rough wall data, the values of Ur and il were determined following the

optimization procedure outhed in Section 2.1.4. The optimization was carried out by

fitting Eqn. (2.13) to the experimentaI data, using Eqn. (2.16) for the wake bction.

Specificdy, the iterated values of Ur and n that gave the best fit to Eqn. (2.13) while

enswing a log-Iinear relation with K = 0.41 were sought. As mentioned earlier, the

present technique does not impiicitly fix the value o f the wake parameter n. Instead it

allowed the value of Iï to be optimized with the expectation of ensuring a diable

estimate of Ut to be made. Since the wall shear stress is influenceci by the outer layer, the

correlation adopted here is expected to yieid some important advantages over Hama's

formulation, as wiil be shown subsequently. Compared to the Clauser plot method,

which uses only data in the overlap region, the technique adopted here uses more data

points in the course of the profile matching since data in both the overIap and outer

regions are employed, For the smooth wall data in Series B, Ut and il were obtained

using the above optimimtion technique. The fiction velocity Ut was also detennined

independentiy fimm the velocity gradient at the waI1. This ailowed a cornparison between

UT obtained fiom the two different methods to be made. The difference between the U,

values as obtained fiom the veiocity gradient and optimization technique was less than 4

percent for al1 cases.

I I I 1 1 1

B-SGI 1 2620 1 2.95 1 4.50 1 0.25 1 2.8 1 35

Table 4.2: Summary of skin fiction veIocity and wake parameter for Series B

in Figure 4.1 b, fits of Eqn, (2.13) and (2.16) to some of the experimentai data

(Tests B-SM2, B-PFI, B-SG3 and B-WM2) are shown. Foliowing Press et ai. (1987),

the chi-square dimiution of the fitted m e and the experimentai data was computed in

the region 0.1 I y16 < 1 as a quantitative measure of the goodness-of-fit for the plots

shown in Figure 4Ib. It was noted that the fits gave a gwd representation of the

experimental data at 99.5 percent addence level.

The UT and ïi values obtained for the smooth and rough wall data are also

summarked in Table 42. The present values of ïi obtained for the smooth wail data are

similar to the value of 0.1 and 0.16 reporteci by Kirkgoz and Ardichoplu (1997) and

Nem and Rodi (1986), respectively, in open channel flow experiments at similar test

conditions. It is, however, important to note that the present smooth wall values of rI are

significantly lower than the suggested value of 0.55 for a typical smooth wail zero

pressure gradient bomdary layer- This difference is most likely due to both the fiee

d a c e effects present in open channel flow and the elevated turbulence levels. Similar

to the observations made by Krogstad et ai. (1992) for a zero pressure gradient boundary

layer, TabIe 4.2 shows a clear variation in the relative strength of the wake with the type

of roughness. Specifically, the smooth plate has the Iowest wake strengh (II = 0.1)

while the wire mesh has the highest strength (il = 0.48). This observation suggests that

the effects of surface roughness are not confùied to the wall region but affects the outer

flow more than imptied by the 'waü similarity hypothesis'.

4 2 MEAN VELOCITY DISTRIBUTIONS

4.2.1 Mean Profdes in Outer Coordinates

The mean velocity profiles for the smooth wall data (Series A) in outer coordinates are

shown in Figure 4.2a The characteristic 'blunt' profile typicai of a ttirbdent boundary

layer is clearly evident Figure 4.2a shows that the mean profile becomes more 'Mi' as

Ree mcreases. The velocity dip for this set of data is minimal

O A-SM 1 (Re, = 750) A A-SM2 (Re, = 1080)

L .O U A-SM3 (Re, = 1450) r A-SM4 Oie, = 2400)

fi O A-SM5 (Re, = 3250)

05

0.0 0.0 0.2 0.4 0.6 0.8 1 .O

u/U

Fig. 42. Disrnions of mean velocity in outer variables (a) Ssmooth data at various Re, (b) Smooth and mu& waii data

Figure 4.2b compares the smooth and rough wall data (Series B) in outer

coordinates. Here only the data up to y = 1.256 are shown while the inset shows the

profiles up to the free surface for some of the data sets. The rough wall data are las

'full' when compared to the smooth wail profile. It is also clear fiom this figure that the

wire mesh exhibits the highest deviation fiom the smooth profile whiIe the perforated

pIate shows the lest deviation. This suggests that even though the wire itself has the

smallest diameter (0.6 mm compared with 1.2 mm for the sand grain, and 1.4 mm for the

perforation depth), the mesh roughness exhibits the greatest resistance for the present set

of test conditions. Furîher evidence of this trend will be discussed in subsequent

sections. The inset shows that the mean data in the vicinity of the free surface are about

94 to 96 percent of the correspondhg local maximum value. Compared to measurements

obtained in Series A (see Figure 4.2a), the inset shows that the characteristic velocity dip

is more exireme for Series B.

4.2.2 Mean Velocity Defect Profiles

The velocity defect profiles for Series A are shown in Figure 4.3. in Figure 4.3a the

conventional velocity scale U, is used while the fieestmm velocity U, is used as the

scaling velocity in Figure 4.3b. In both figures, the boundary layer thiclaiess, 6 is used to

normalize the vertical distance. in Figure 4.3a, no systematic ReynoIds number effects

cm be observecl for y 2 0.026. On the other hand, Figure 4.3b which uses the scaling

proposed by George and Castillo (1997), shows a srnail but systematic decrease in the

mean velocity defect at similar y16 as Re0 increases.

Fig. 43: Distribution of mean velocity defect (a) inner coordinate; (b) outer coordinate

The distriiutions of the velocity defect for the smooth wall data sets (Series B) are

compared with the results of Thole and Bogard (1996) at both lower and higher

fieestrem nirbuIence values in Figure 4.4a The solid Iine is a fit to Hama's function

(Eqns. (2.14) and (2.15)) which is qresentative of a correlation which f ies the value of

il implicitly. It shodd be noted that the study of Thole and Bogard (1996) considered a

boundary layer with significant and sustained freestream turbulence, while our data

pertains to an open channel flow where the notion of 'Ykmeam turbulence" becornes

ambiguous. Even though our study considered a boundary layer in an open channel, the

present srnooth data fdl within the envelope of Thole and Bogard's boundary layer data

for a zero pressure gradient.

In Figure 4.4% the defect velocity profiles for ail data sets are consistently lower

than would be predicted by Hama's functions. in fact, if in the course of the optimization

technique, one Uisists on the smooth waI1 data (Test B-SM3) following the Hama fit by

fixing iï = 0.55, a Ur value of about 0.0165 m/s (corn@ with 0.0210 mis obtained

from Eqn. (2.13) and (2. t6)) wodd be pdicted. This would give a skin fiction

coefficient that is 40 percent Iowa than otherwise obtained. The fiction velocity

obtained fiom Hama's fonndation for the Test M M 3 data wouid also require a dope

of a' = 3.2 and an additive constant of C = 72 for the experimentd data to follow the

log law. Furthemore, the universality of = Y' would be invalidateci. It should be

recaiied that for the smooth case, the value of U, obtained h m the present oph'mi;ration

procedure ciosely matched that detefmined h m the slope of the velocity profile at the

wali. Following Bradshaw (I987), one may attriiirte the lower vahe of Ur obtained h m

20

2625 3.1 1380 3.3

15 1750 2.4 A Thole & Bogard 1140 1.0

Ue- - U- 4 Thole& Bogard 750

10

5

O 0.0 0 2 0.4 0.6 0.8 1 .O

Fig. 4.4a. Mean d e f ~ t profiles for smwth waH

O B-SM1 A B-PF1 d B-PR

B-SG1 V B-WM1

Fig. 4.4b. Velocity defect pronles for smooth and mu& daces, solid üne is a fit to Hama profile (ïi = 0.55)

the Hama function (and hence Cf) to the strengih of the wake, as implied by Eqn. (2.15),

being too large for the present experiments. This result leads - one to conclude that use of

a correlation such as Hama's which implicitiy fixes the wake strength at ïi = 0.55 is an

erroneous approach for an open channel boundary layer as well as other turbulent

boundary layers with a weak wake component (i.e. a low ïi value).

In Figure 4.4b the smooth and rough wail data in Series B are shown. The

strengths of the wake produced by the perforated plate and sand grain roughness are

nearly equal (II = 0.24 - 0.36), while the wire mesh has the strongest wake strength.

Consistent with the ïi values summarized in Table 4.2, the wire mesh profile which has

the highest ïi value foIIows the Hama formulation most closely. One may conclude fiom

Figure 4.4b that the veiocity profiles for the rough wail boundary layers being studied are

significantly different h m the smooth case in the outer part of the flow.

4.23 Mean Proiiies in Inner Coordinates

Figure 4.5a shows the distributions of the mean profiles in inner coordinates for the

smooth wail data (Series A). The log law constants adopted are K = 0.41 and B = 5.0.

The coiiapse of the profiles at various Ree is to be expected in the viscous sublayer as

well as in the overlap region, As Ree increases, the extent over which the experimental

data colIapse onto the logarithmic law increases. This is consistent with the trend with 6

shown in Table 4.1.

B-SM2 A B-PF1 4 B-PF3 0 B-SG1 I B-SG3 V B-WM1

increasing mugbness effecr

Fig. 45. VeIocity d i s tn ion in imier variables (a) Smooth wail data at various Re, (b) CornpanSon between snooth and mu& waIi data

The Reynolds number effects in the outer region cm be examined h m the

deviation AU-" between the mean data and the log Iaw profiIe at y- = F. The

+ .

maximum deviation, AU,, 1s related to the strength of the wake, n as follows: AU-*

= 2 n / ~ The values of AU,' are summarized in Table 4.1. Using this relation, the

corresponding values of the wake parameter are ri = O and O. 1 at Ree = 750 and 3250,

respectively. The evident reduction of the wake strength for dl the profiles may be

attri'buted to the reiatively higher turbdence Ievels in the outer channet flow as weil as

the IÏee d a c e effect.

The rough wall data are compared with the smooth waIl profiles in Figure 4.5b.

The effect of sinface rougbness is to shift the velocity profile dom and to the right

relative to the profile on a smooth wall. The SM in the velocity profile, and to a lesser

degree the shape of the profile, is strongly dependent on the type of roughness- The

roughness function (AE33 of each profne and îhe corresponding roughness Reynolds

nurnber (Rek = kU&) are also summarized in Table 4.2. For the tests conducted on the

perforated plate (Tests B-PFl, B-PF2 and B-PF3), no noticeable shift (AB-) was

observai suggesting that the mean velocity profile is essentiaily similar to the smooth

wail data except for the strength of the wake. A possiile explanation for this observation

is that the flow withia the cavities (i.e. îhe perforations) does not interact substantially

with the bulk flow.

Figure 4.5b also shows rhat men though the mesh diameter is only half as thiçk as

the average diameter of the sand gramS, it gave the highest roughness fimction. It shouid

be noted that for flow over mugh surfaces, the total drag consists of both viscous and

fom drag. The conûiiution of the form drag would depend on the onset of vortex

shedding proces, which in nmi depends on the W f i c geometry of the roughness

elements. Furthmore, Bandyopadhyay (1987) indicated that the criticai roughness

Reynolds number (Rd beyond which the flow regime becomes fully rough decreases

with increasing span-to-height (Mc) ratio of the mughness elements. The criticai values

of Rq, for sand grain (Vk = 1) and the present wire mesh (Vk = 12) are approximately 55

and [5, respectively (see Bandyopadhyay, 1987; Figure 26). This may explain the higher

roughness effect observed for the wire mesh compared to the sand grain data The

velocity profiles for the wire mesh roughness plotteci in Figure 4 3 show a kink and a

dramatic change in slope at y' = 15. These features are possibly an artifact of the

periodic vortices shed over the wire. Obviously, the mean velocities in this region

represent the time-averaged values of these periodic vorticd structures.

42.4 Cornparison Between Log Law and Power Laws

Consideration is now tumed to the appropriateness of using a power law to descriie the

mean velocity in the overlap region. The resuits for some of the tests (Tests A-SM& C-

SMH, C-SGL, C-SGH, C-WMH, B-WMI) are shown and discussed. For the power law

formulation of Barenblatt (1993), the initial values of the constants a and C were

computed using Eqn. (2.7) and (2.8). However, an improved fit to the experimentd data

was obtained by sIightly modifLing both a and C. The modified values, which were dso

used to compute LIo, are Summanzed in Table 43- For C-Sm, no modification was

necessary for a but the modified value of C was found to be 0.8 percent iower than the

initial estimate- For A-SM2, on the other hand, the ciifference between the modified

values of a and C, and those detemiined h m Eqn. (2.7) and (2.8) was 3.1 and 4.9

percent, respectively. In view of the sensitivity of the skin fiiction velocity to the values

of the power law constants, these modest ciifferences between the modified and initial

values for a and C are significant.

In the case of the formuiation proposed by George and Castillo (1997), the value of

Co needs to be prescn'bed. It was observed that a more accurate estimate of Ua is

obtained by setting Co = 1.00. AIthough Co depends on Reynolds number, the range of

Reynolds numbers considered here is too narrow for any variation in Co to be important.

Therefore, a value of Co = 1 .O0 is assumed for both the smooth and rough-wall analysis

reported h m . The constants wed in fitcing the power law formulation of George and

Castillo ( 1997) are a h summarized in Table 4.3.

Table 4.3: Summary of power law constants for [BPI and [GC] and friction velocity

Test

A-SM2

C-SMH

C-SGL

C-SGH

C-WMH

B-WM1

a

0.166

0.152

0.184

0.1'75

Y

0.133

0.139

0.180

0.195

0.261

0260

C

7.5

8-14

5.9

5.9

Ci

8.8

8.7

6.0

5.2

3.0

2.76

UU

1.03

2-17

2.21

3.27

AU,

(%)

1.0

2.7

22.0

44.1

Ue

1.03

2.20

1.69

2.65

2.82

3.94

AU,

(%)

1.0

1.3

5.0

2.9

2.8

4.6

Figures 4.6a and 4.6b show the velocity distributions for A-SM2 and C-SMH,

respectively, as weli as the corresponding fits to the power law derived by Barenblatt

(1993) and the composite profile proposed by George and Castillo (1997). The

logarithmic law (with K = 0.41 and B = 5.0) is also shown for cornparison. The power

law of Barenblatt (1993) and the log law adequately represent the mean profile in the

overlap region. With the exception of a 'kink' observed at y' = 16, the composite profile

of George and Castillo (1997) closely matches the velocity h m the wall up to y- = c. As mentioned earlier, the 'kink' at y' = 16 may be attniuted to the fact that neither Eqn.

(3.2) nor Eqn. (2.9) is valid in the neighborhood of y' = 15.

As will be discussed later, the success of the power law of Barenblatt (1993) in

describing the velocity profile close to y' = is partly due to the negligiile wake

component observed for the present smooth wall data. It should be noted that in the

lower part of the overlap region, Le. 30 < y+ < 120 and 40 < y c 300 in Figure 4.6a and

4.6b, respectively, the power and log law profiles are aimost indistinguishable fiom each

other. The values of the fiction veIocity UO derived h m the constants used in fitting

Barenblatt (1993) and George ami Castillo (1997) are reported in Table 43. These values

are in excellent agreement with the values determineci from the velocity gradient at the

wd, the maximum deviation being less than 3 percent.

The measurements obtained on the sand grain mughness are shown m Figure 4.7.

Also shown are the corresponding log Iaw and power law profiles. Aiîhough a' (Eqm.

2.9 and 2.1 1) rnay depend on the roughness Reynolds number, and even the specific

(a) A-SM2

--

O Measurements - Log law ---. Composite profile [GCJ (Eqn. 2.1 1) - - - - Power law [BPI (Eqn. 2.6)

O Measurements - Log law ---. Composite profile [Gq

Power Iaw [BPI

Fig. 4.6: Log law and power Iaw profles [BPI and [GC] on smooth surface

%

Fig. 4.7: Log law and power law profiles [BPI and [Gq for sand grain data

geometry of the roughness elements, a value of a+ = -16 was usai for aii the rough wall

data as weii. The formulation of George and Castillo (1997) fits the profiles reasonably

well up to y- = 0.9c. On the other hand, the power law of Barenblatt (1 993) and the log

law coilapse with the experimental data over a more Iimited range of y-. This can be

attributed to the relatively larger wake component observed for the sand grain data in

comparison to the srnooth wall data. The values of Uo obtained ffom the skin fiction

Iaws deriveci by Barenblatt (1993) and George and Castillo (1997), and the

corresponding deviations 6om the reference values given in Tables 4.2 and 5.1, are

reporteci in Table 4.3. The values obtained f?om George and Castillo (1997) are in very

good agreement with those reportai in Table 4.2 and S. 1, while the values obtained fiom

Barenblatt (1 993) are unacceptably high.

In Figure 4.8, some of the velocity distributions obtained on the wire mesh

roughness (C-WMH and B-WMI) are presented. It should be noted that the wire mesh

data showed wake components (il = 0.52 and 0.48 for C-WMH and B-WMl,

respectively) that are significantly larger than those observed for the smooth surfaces and

also, presumably, in pipe flows. fn view of the poor performance of Barenblatt (1993) for

the sand grain data, the power law formulation of Barenblatt (1993) was not applied to

the wire mesh data. Oniy the profiies derived h m George and Castille (1997) are shown

in the figure, The good agreement between the measurements and fitted profiles is

markable. For the purpose of comparison, the log law profile to C-WMH is aIso

shown. It shouid be noted that while the composite profile follows the data for C-WMH

up to y" = 1500, the range of applicability of the log law does not extend beyond

Fig. 4.8: Log Iaw and power law profiles [GCJ for wire mesh

y' = 160. For Test C-WMH, oniy 5 data points collapse on the log law. An immediate

implication of this observation is that the usehlness of the log law in low Reynolds

number flows over rough surfaces reduces as the roughness effect or the wake

component increases. In contrast, the power law proposed by George and Castillo (1997)

was able to follow the velocity profile over most of the extent of the boundary layer. As

shown in TabIe 4.3, the friction velocities predicted h m George and Castillo (1997)

formulation agree with the values obtained fiom the profile matchhg technique to within

+ 5.0 percent.

4.3 TURBlTLENCE INTENSITY

Figure 4.9a compares the distributions of u* obtained for Tests A-SM3. A-SM4 and B-

SM1 to the boundary layers of Ching et al. (1995) and Osaka et al. (1998), and the fully

devetoped channel data reported by Johansson and Alfredsson (1982) and Harder

and T idman (1 99 1 ). Al1 profiles (previoudpresent and channeüboundary layer)

colIapse reasonably well in the near-wall region (y* < 30). The peak values of the

profiles agree to within f 5 percent Beyond y* = 30, the data of A-SM4 anc! B-SM1

which have Ree values similar to the data of Osaka et a1 (1998), show consistently higher

values than the data of Osaka et ai. (1998). This can be amiuted, in part, to the

characteristic high background turbulence Ievels in the 6-ee d a c e region of open

channe1 flows.

It is evident h m Figure 4.9a that the values of u* at the fk surface of the open

chamei aperiments are comparable to the centerline value of 0.78 f 10 percent

4

O B-SM3

3 A Thoie & Bogard (Tu = 1%) A Thole & Bogard (Tu = 10%)

Hancock & Bradshaw (Tu = 3.5%) u O Hancock & Bradshaw (Tu = 4.6%)

2

1

O 0.0 0.4 0.8 1.2 1.6

y/&

Fig. 4.9: Distrïïutions of turbuience intensity (a) Ment aud prwious canonid ùomdary iayer flows (6) Preseuî and previous boundary layers at hi@ fiestmm turbulence

wmpiIed for duct flows by Durst et al. (1998). The present values are, however,

significantiy hiber than those obtained at the outer edge of canonical tubdent boundary

Iayers. It is also observed that the canonical boundary layer profile of Osaka et ai. (1 998)

fdls off more rapidly than the profile of A-SM4, perhaps, because of the lower

turbulence levels at the outer edge of their flow. The deviations between the open

channel data, i.e. Tests A-SM4 and B-SM1, on the one hand, and the M y developed

channel measurements of Johansson and Alfiedsson (1982) and Harder and Tiderman

(1991), may be due to the relatively higher values of Rq, in the open channel data.

Although the data for Tests A-SM4 and B-SM1 were obtained in the same facility,

some deviations are apparent between the two sets of data in the Mcinity of the free

surface. For example, Test B-SM1 shows a slight increase beyond y- = 2000. It shodd

be recalled h m Figure 4 2 that for measurements in Series B, the velocity dip is more

extreme as evidenced in the mean vaiue closest to the fiee surface being about 96

percent of the local maximum. For Series A, on the other hand, the veiocity dip is

minimal. It should also be pointed out that the region over which the u profile increases

in the outer region corresponds to the region beyond the velocity dip. in this region, both

-<UV> and aUlay are negative. As wiii be shown in Chapter 5, the magnitude of these

quantities inmeases as the velocity dip becomes more extreme. The increase observed in

u+ pmnles of Test B-SM1 compared to Test A-SM4 rnay be due to higher production of

turbulence kinetic energy close to the fiee surface for the fornier data

The turbulence intetlsity for the smooth case is also compared to other boundary

Iayer memements at eIevated freestream turbulence in order to assess the effect of the

turbulence intensity of the exterior flow. Although cornparisons are made to the

tioundary layer measurements of Thole and Bogard (1996) and Hancock and Bradshaw

(1983) at similar and different fieestmm turbulence intensities, the present case

considers an open channel flow wbere the Qow region outside the boundary Iayer is

somewhat different, both in ternis of mean flow structure and turbulence length scale.

The distn'butions of the turbulence intensity for the smooth wall data in Series B as well

as the data reported by Thole and Bogard (1996) and Hancock and Bradshaw (1983) are

shown in Figure 4.9b. Here, the friction velocity and the boundary layer thickness are

used as the normalizing scales. The agreement between the present smooth data and the

other shidies at comparable intensities appears reasonable. Specifically, the present data

and the data of Hancock and Bradshaw (1983) at Tu = 3.45 percent are similar in the

range y < 0.756. Figure 4.9b also suggests hat as the fieestream turbulence increases, the

intensity profile becornes more fiat, impiying that the outer turbulence is penetrating

more deeply into the boundary layer.

Figure 4.10a shows the uA profles for the present smooth wall data (Series A)

using inner variables. For these profiles, some scatter exists in the near-walI region. in

view of the difficulty in d e t e r d g the position y = O, an uncertainty of about f0.025

mm is possible* In waü n i t s , this uncertainty varies b r n y- = 0.3 for Test A-SM1 to y*

= 0.8 for A-SM4. If an uncertainty of f0.5 is ailowed in y', the collapse in the near-waII

region wi l i impmve significantly. Aiiowing for ihis mcertainty and ais0 uncertainties in

O A-SM 1 (Re, = 750) 1 A A-SM2 (Re, = 1080)

A-SM3 (Ree = 1450)

Fig. 4.10: Turbulence intensity profiles at various Re, (a) inner variabIes; (b) outer coordinates

u', Reynolds number sixniiarity may be claimesi for y' l30. Compared to the mean flow,

for which the profiles collapse up to y'- = 250, this implies that Reynolds number effects

in the turbulence intensity profiles "penetrate deeper" into the inner region than for the

mean profiles. The peak vdue for the present profles is (u3, = 2.73 W.05 and these

values occur in the range 13 < y' < 15, irrespective of Ree. These Iocations are in good

agreement with the vdues obseryed in open chamel LDA experiments and DNS rauits

(e.g. Kornori et al., 1993; Borne et al., 1995). From Figure 4.10a, systematic deviations

are apparent in the profles for y' 2 30.

in hi@ ReynoIds number near-wall fiows, the turbulence intensity is

approximately constant over the constant-stress region. Gad-el-Hak and Bandyopadhyay

(1994) remarked that u asymptotes to 2 in the constant-stress layer. The present profiles,

in good agreement with the trend reported by Purtell et ai. (1981) at similar Reynolds

numbers, indicate that a constant-u' region does not exist in low Reynolds number

turbulent bounhy layers. It is apparent h m Figure 4.10% however, that as Ree

increases the vdues of u systematicdIy increase towards the asymptotic value in the

region over which the constant-stress is presumed to uccur, At Reg = 750, for example

the value of u' in this region is about 1.5 while a value o f f = 1.82 is attained at RQ =

2400.

Figure 4.LOb shows the distn'butions of streamwise turbulence intensity in outer

variables. This is the scaiing law derived h m the recent boundary Iayer andysis of

George and Castilio (1997). In outer scaling, the profiles coIIapse W y weii in the near-

wall region (i.e. y/6 < 0.01). The collapse is aiso remarkable for yi6 > 0.2, in apparent

contradiction to the observations made in Figure 4.10a which uses inner scaling. As one

would expect (see for example George and Castille, 1997) the largest Reynolds number

effects are evident in the overlap region. Furthemore, there is a tendency for the location

at which the peak value of u occurs to move closer to the wall as Re0 increases. It is also

evident fiom Figure 4.1 Ob that as Ree inmeases, the peak value of u/U, reduces.

Figure 4.1 1 compares the turbulence htensity profles on smooth and rough

d a c e s . In Figure 4.1 la, the boundary layer thickness is used as the normaiizing length

scaie, whiIe in Figure 4.1 1 b the viscous length scaie is adopted. From Figure 4.1 1 b, the

location (y',) at which each data set attains its maximum value is confined to the range

10 < y ' < 15. It is also evident h m Figure 4.1 1 that the smooth wall and perforated

plate data exhibit dightly higtier peak values. The smooth wall data, however, faII more

rapidly and beyond y+ - 20, they become consistently lower than al1 the other

data sets up to y* - 300. in general, at similar outer turbulence levels, the turbulence

intensity profiles tend to be more flat as the roughness effect increases. It is important to

note that, in the constant-stress (or overlap) region, the mugh wall data indicate u = 2.3

which is distinctiy higher than the asymptotic value recommended for high Reynolds

number smooth waiI data. This is ctear evidence of the inauence of roughness extending

beyond the roughness sublayer.

O &SM1 A EPFI

BSGI v EWMI

Fig. 4.1 1, Variation of turbuience intensity on smooth and rough siirfaces (a) outer variabki (b) inner variables

4.4 SKEWNESS AND FLATNESS

investigation of stnictud information in wail-bounded flows within the context of near-

waIl turbulence production mechanism has received considerable attention (e-g. Kline et

al., 1967; Kim et ai., 1971). Low Reynolds number flow visualizations show that Iow-

speed fluid in the near-wall region occasionaiiy enrpts violently into the high-speed outer

region of the boundary layer. FoUowuig Kline et al, (1967) and Kim et ai. (1971), this

process is referred to as bursting. They also concludeci that essentially al1 the turbuIent

production occm durhg the bursting process. Corino and Bnidkey (1969) indicated that

the ejection phase of the bursting p c e s s is foiiowed by a large-scde motion of

upstream fluid that manates fiom the outer region and sweeps the wail region of the

previously ejected fluid. Subsequent studies (e-g. Raupach, 1981) used conditional

sampIing methodologies (e.g. VITA, VISA and quadrant andysis) to W e r our

understanding of the turbulent structure. However, the structural infonnation obtained

using these techniques couId be ambigous (Gad-eI-Hak and Baudyopadhyay, 1994).

- I I , ?

The skewness (Si = u, ' / utr ) aod flatness (Fi = u,' 14 ) fictors, (where ui

denotes the instantaneous turbulence fluctuation in the positive i-direction), give usetuI

quantitative information regarding the temporal distriiution of the veIocity fluctuafion

around its mean value. A non-zero skewness factor indicates the degree of temporal

asymmetry of the random fluctuation, e-g. acceleration versus deceleration or sweep

versus ejection. Since the skewness retains the sign infonnation, it contains vaiuable

statisticai information reIated to c o h m t structures and can be used to extract s t r u d

information without ambigu@ or subjectivity (Gad-el-Hak and Bandyopadhyay, 1994).

A flamess factor Iarger than 3 is generatly associated with a pe&y signal as for example

that produced by intermittent turûdent events. In this section, the skewness and flatness

factors are used to document some structural information regarding open channel

turbulent boundary layers.

Distriiutions of the skewness (Su) and flatness (Fu) of the streamwise component

of the turbulence fluctuations are show in Figures 4.12a and 4. t2b, respectively.

Sïmilar to the refined LDA meamements of Durst et aI. ( t 995) and DNS results of Kim

et al. (1986) in fMy developed duct flows, the Su increases in the linear viscous

sublayer. The peak value of 0.87 in the present flows is also in gmd agreement with the

value of 0.85 reported by Durst et al. (1995). The large values of Fu in the near-wdl

region (Figure 4.12b) are a manifestation of intemittent bursting events that take place

there. in the near-wall region, the in-rush phase of the bursting cycle bnngs in high-

velocity fluid fiom the outer layer. This results in large-amplitude positive u fluctuations

and high positive values of Su in the near-waIi region (Simpson et ai., 1981). Beyond

y" = 5, Su and Fu values decrease with incfeasing waU distance. The Su profiles change

sign in the buffer region (at Y+ 5: 12) and exhibit a near-wall clip (a local minimum) at

y& = 15 - 20. The location at which Su changes sign aiso corresponds to the position

where Fu attains its near-wd minimum value and aIso to the location where u-,,

occurs. Beyond this location, Su and Fu profiles increasc slightly and stay approximately

constant in the overiap region. The values of Su and Fu in this region are close to the

Guassian values of 0 and 3, respectively. Consistent with a wider region of overlap at

Fig. 4.12b: Distriaution of flamess hctor at various Re,

2 1 1

1 -

S .

0 -

O A-SM1 A A-SM2

A-SM3 -

-

-1 -

-2

- v A

1 L 1

1 10 100 Io00 *

Y Fig. 4.12a: Distriiution of skewness factor at various Re,

higher Ree, there is a tendency for the region of 'near-constant' Su and Fu to increase

with increasing Ree.

Although there is some scatter in Su and F, profiles in the vicinity of the fiee

surface, these values deviate significantly h m the Gaussian values. It is also observed

that the skewness profiles show a dip in the neighborhood of y = ij+ and increase as the

free surface is approached. This observation appears to be unique to an open channel

flow. In the outer edge of canonid turbulent boundary layers, the flow is intermittent

and the flatness factors are high (typicdy 6 to 7 as reported by Andreopoulos et ai.,

1984). On the other hand, the Fu data obtained in fdy developed duct flows using the

LDA and DNS (e.g. Kim et al., 1986; Niederschdte et al,, 1990; Durst et ai., 1995) show

near-constant values in the overIap and the core regions of the channel. Furthemore,

typical values of Fu in the overiap and core regions for fuUy developed flows varied fiom

3 to 4. The high vaiues of Fu (about 4 - 7) observeci close to the fiee d a c e in the

present study (Figwe 4.I2b) are presumably a signature of intermittent large-scale

negative u fluctuations which occm as a resdt of the large eddies driving the fluid fiom

the low velocity region.

4 5 TRIPLE CORRELATION

The gradient of the triple correlation <u3> conmiutes to the streamwise diffusive flux of

the streamwise kinetic energy (uz), The distn'butions of <us for Series A are shown in

Figure 4.13. b e r scaüng is used in Figure 4.13% while the mixed scaiing derived by

George and Castillo (1 997) is adopted in Figure 4.13b- The data for Test B-SM 1 are also

decreasing Re,

-050 1 I I

1 E-3 0.0 1 0.1 1 3

Y/&

Fig. 4.13: variation of eu3> wîth Reynolds number (a) Inner scaling (b) Mixeci scaling derived h m AIP

shown for comparison. In Figure 4.13% the profiles show a reasonable collapse in the

near-wall region (i.e. y < 250). Here, the peak values as well as the locations of the

near-wall maximum (y+ = 7) and minimum = 20) are nearly independent of Reynolds

number. In the outer region, however, the profiles obtained at lower Reynolds number

are closer to zero.

In Figure 4.13b, the distribution obtained frorn the re-scaled high Ree canonical

turbulent boundary layer data obtained by Krogstad and Antonia (1999), Le. [KA99], is

also shown. The trend shown by the present results is similar to that reporteci by

Krogstad and Antonia (1999). Each profile has a maximum value at a location that lies

within the viscous sublayer. kt the region that corresponds to the buffer region, the

profiles show a systematic Reynolds number dependence. More specifically, the absolute

values are higher at lower Reynolds numbers. The profiles also change less rapidly with

the wall-normal distance at lower Reynolds number in the inner region. The location at

which the near-wall minimum occurs is closer to the wall at higher Reynolds numbers.

Beyond this location, both the present and earlier measurements show a systematic

variation as the Reynolds nurnber becomes higher. Al1 the profiles are nearly flat in the

region corresponding to the overlap region. It is also observai that the high Reynolds

number profde of Krogstad and Antonia (1999) shows a more extended overlap region

in comparison to the open channel flow data.

4.6 SUMMARY

Measurements of the sireamwise wmponent of the mean velocity and higher order

turbulence statistics were obtained in mooth and rough wall open channel turbulent

bouudary layes. in analyzhg the data, the scaling Iaws derived from classical theones as

weii as the recent scaling laws proposed by BarenbIatt (1993) and George and Castiiio

(1997) were used. in order to assess the effect of the moderate turbulence intensity level

in the channel flow outside the boundary layer, the data were compared to bomdary

layer data in the literature at different k s m a m turbulence intensities. For the smooth

wail data, the fiction velociry was obtained from the velocity gradient at the walI or by

fitting fourth and f i f i order poIynomiaIs to the near-waiI data. in the case of the rough

walI data, a velocity deféct profile was fitted to each data set to determine the strength of

the wake and the skin îriction coefficient. h fitting the velocity defect law, a correlation

which did not fix the value of ïi mipiicitly, was found to yield a more consistent and

accurate estimate for the skin fiction coefficient than a formulation such as that of Hama

which fixes the vaiue of ïi.

The power laws proposed by Barenblatt (1993) and George and Castillo (1997)

were found to descnie the mean veIocity for the smooth wail data almost to the outer

edge. The values of the fiction veIocity obtained h m the skin fiction laws derived

fiom these power laws were in excellent agreement with the corresponding values

obtained fiom other diable and widdy accepted techniques. The power law derived by

Barenblatt (1993) was not suitable for modeling the mean velocity profiles over the

rough d a c e s considerai in the present study. Furthermore, the fiction velouties

obtained Eom the co~~esponding power-law skin fnction relation were found to be about

20 to 40 percent higher than the values obtained h m - other reliable techniques. in

contrast, the power law proposed by George and Castillo (1997) was found to do an

excellent job of descniing the mean veIocity over a significant extent of the boundary

Iayer. The values of the fiction velocity predicted h m their skin friction law were in

very good agreement (Iess than 5 percent variation) with the values obtained fiom a

velocity defect matching technique.

The mean defect profiles do not show any sensitivity to Reynolds number when

conventional inner scaling is used. in contrast, the present d t s show a stight but

systematic Reynolds number dependence when the scaiing derived fiom the AP, i.e. the

freestream velocity, is used. h inner coordinates, the turbulence intensity profiles show

important dependence on Reynolds number except for y' c 30. When outer coordinates

are used, the turbuIence intensity profiles show Reynolds number dependence only in the

overlap region. The skewness and flatness tictors appear to be independent of Reynolds

number but the triple correlation shows distinct Reynolds number dependence.

The effect of mughness on both the mean velocity and, to a lesser extent, the -

turbulence intensity, varied for the three different roughness elements. The vaiue of the

wake parameter, II, was aIso observeci to vary with roughness eIement. These

observations are at variance with the "waii smiilarity hypothesis' wbich suggests that the

effects of d a c e roughness should be contined to the roughness sublayer. Even though

the boundary layer in an open chmel flow is idaenced by the fke surface, many of the

flow characten'stics, in particuiar those that pertain to d a c e roughness, are similar to

those observed in a canonical zero p r m e gradient bomcky layer.

EFFECTS OF SURFACE ROUGHNESS ON TURBULENCE STRUCTURE

in this chapter, two-dimensional boundary layer measurements on smooth and rough

surfaces obtained in Series C are reported. The data presented include the mean velocity,

Reynolds stresses, triple correlation, approximate energy budgets as welI as mixing

Iength and eddy viscosity. The recent theory pmposed by George and Castillo (1997)

and conventional scaling laws are used to anaiyze the data Cornparisons to ather smooth

and rough wdl boundary layer measurements and DNS r d t s are made. The data and

discussion presented in this chapter provide insight into the effits of surface roughness

on the turbulence structure and its impIication for rough walI turbuknce models.

5.1 DETERMINATION OF FRICTION VELOCITY

For the smooth waii measurements, the fiction veiocity was determineci using the hear

= y') and near-wall polynomial fit (LT = f + Qy* + ~ 5 ~ ' ' ) discussed in Chapter 4

(Section 4.1). The coefficients obtaineed were cq = -0.28~10" and q = 13.6x1o6, which

are in gwd agreement wia the vaIues obtained in Chapter 4 and dso with the values

recommended by Eriksson et al, (1998) and Gmrge and Castillo (1997). The fiction

veiocities obtained h m the near-wall data are denoted by UT and are summarized in

Table 5.1. The pwer Iaw and skin Ection relation proposed by George and Castilio

(1997) were aIso used to determine the friction velocity. These values are denoted by UQ

and are also summarized in Table 5.l. irrespective of the specific wall conditions, the

value of Ua is in excellent agreement with the corresponding - value Ut.

in the case of the rough wali data, the optimization procedure outline in Chapter 4

(Eqns. (2.13) and (2.16)) as well as the skin friction relation proposed by George and

Castillo (1997) (Eqn. 2.20) were used to detemine the fiction velocity. These values

are denoted as Ut and Ur?, respectively, in Table 5.1. The vaIues of the wake parameter

il for both the smooth and rough waii data are aiso summarized in Table 5.1. Sirnila. to

the observation made in Chapter 4, the values obtained on the rough wail are

significantly higher than the smooth wall data The roughness Reynolds number Rq, for

each of the rough wall data are also summarized in Table 5.1.

( d s ) ( d s ) C-C-SMH 492 1900 223 2.22 0.10

Table 5.1: Test conditions and hundary layer parameters.

C-C-SML

C-C-SGH

C-C-SGL

C-C-WMH

5 2 MEAN VELOCITY PROFILES

52.1 Outer Coordinates

The distnIbutions of the streiimwise component of the mean velocity in outer coordinates

are shown in Figure 5.h At the oater edge, i.e. y > 6, each velocity profile

34.8

53.1

35.1

53.4

2050

2180

2180

2600

1.57

2.73

1.78

2.90

1.55

2.65

1.69

2.83

0.08

0.30

0.25

0.52

33

26

17

O C-SMH C-SML

A C-SGH A C-SGL

C-WMH

Fig. 5.1 Mean velocity in outer coordinates (a) stceamwise; (b) vertical (Lmes represeat best fits to expimental data)

119

shows a siight dip where the local maximum value (Ud occurs beIow the tÏee surface

and aU@y is negative in the vicinity of the fiee surface. As expected, the rough-wall

profiles me kss 'full' compared to the smooth surface. Near the k e surface, it is evident

that the magnitude of aU/ay is higher for the lower depths of flow. The dismiutions of

the mean velocity in the vertical direction (V) are shown in Figure 5.1b. The gradient

aV/ay is dso negative near the free surface. It follows fiom continuity consideration that

au/& > O (i.e. a slight acceleration) close to the fiee surface. The velocity dip and hence

the magnitude aV/ay are higher for the wire mesh data than observed for the smooth

wall.

For the purpose of subsequent analysis, gradients of both the mean and turbulence

quantities (e.g. aU/ay and aS/ay) are required. The procedure of estimahg derivatives

directly h m experimentai data point is sensitive to 'noise' and may yield erroneous

resuits. Alternatively, by generating good analytical or frmctional fits to the experimentd

data, more reliable derivatives can be obtained. This alternative procedure is adopted in

the present analysis. In developing the curve fits, y-steps of O.OOO56 and y*-steps of 0.5

(wall units) were used To obtain the gradients, the m e s w a e graphicalIy

differentiated and smwthed over 5 data points. The 5 data points over which the

smoothhg was done cortespond to 2.5 wall units, which is comparable to the waii-

nomal dimension of the probe volume in viscous anits (1.8 < 1,' c 3.5). The fits to the

mean vertical velocity profiles for C-SMH, C-SGH and WM are shown in Figure 5Jb.

An assesment of the goodness of fit for each m e was made by evaluahng the

coefficient of detcl ' tion (R2). Th, value of R~ caicuiated for eafh cuve was higher

than 0.99. An alternative assessrnent using a chi-squared distribution at a 99.5 percent

confidence level indicated that the curves are good representatives of the experimentai

data.

522 Inner Coordinates

The streamwise cornponent of the mean velocity in inner variables is shown in Figure

5.2a using a semi-log scaie. The correspondhg logarithmic profiles are shown as hes.

The smooth wall data show a gwd agreement with the logarithmic Iaw in the range 25 <

y' < 350. Compared to the srnooth waii data, the rough-wall profiles show the expected

downward-right shifi. However, the region over which experimental data and

logarithmic profiles overlap is Iimited for the rough surfaces. Consistent wilh the values

of ïi shown in Table 5.1, the wake components for the rough-wall data are much larger

than for the smooth wali data.

The data for C-SMH, C-SGH and C-WMH as weii as the corresponding composite

profiles, Le. Eqn. (2.1 l), (dashed lines) are shown in Figure 5.2b. For the composite

profiles, the value of a' (= -16) recommended by George and Castillo (1997) was

adopted. In the viscous sublayer and the buffer region of the smmth waii data, the

values of Q and cs used in fitting the fifi order polyaomid to the near-wall data are

adopted. As observed in Chapter 4, the composite profles match the veIocity data fiom

the w d up to y' .= $ (= 850) except for the 'kink' observed at y' - 15. Afthough the

composite profile desmies the overlap region reasonably well, it does not extend to the

fi.ee surface, For the purpose of subsequent anaiysis, polynomials were also fitted to

Fig. 52: Mean pro6ks in hum coordinates (a) Lines represent log law profiles (b) ha: soIid @oIynomial nts); dashed (composite profles)

.- , -

-

-

5 -

O C-SMH O C-SML A C-SGH A C-SGL O C-wh4H .

c-

Tests C-SMH, C-SGH and C-WMH. The polynomial fits are shown as solid lines in

Figure 5.2b. For each test, the polynomid fit is observed to descnie the experimentai

data fiom the wdl to the fkee surface better than the composite profile.

The mean velocity gradients Wlay- obtaùied h m the poiynomial lits for

C-SMH, C-SGH and C-WMH as well as fiom the composite profile for C-SMH are

shown in Figure 5.3. In the viscous sublayer (Le. y < 5) both profiles give aV/ay- = 1

for C-SMH. The data obtained fiom the composite profile show some discontinuity in

the region 10 S y 5 30. This may be mibuteci to the singularity in Eqn. (8) at y' = a (=

16) and 'kink' observed in the mean velocity profile in the neighborhood of y- = 15

(Figure 52b). The iW/ily- profiie obtained h m the polynomial fit, on the other han&

is reasonably srnooth across the enth depth of flow, With the exception of the disparity

observed between the two C-SMH profiles in the region 10 I y' l 30, the values of

aU+/ay' fiom both profiles compare favorably. Cornparison between aU'/ay- profiles

obtained h m the composite profila and the curve fits for the rough data is very good.

The values of dU'lay' h m the DNS data obtained in a channel flow = 180,

where hc = hU&) by Kim et al. (1986) were reportecl by Cenedese et ai. (1998). Their

profile is aiso shown in Figure 5.3 for the pilrpose of cornparison. For y& < 100, which

represents the viscous sublayer, the b d e r region and the overlap region of the DNS

data, the deviations between C-SMH and DNS profiles are within f 5 percent. The

present aU'iayA profles are neatIy independent of waiI conditions in the overlap region,

i.e. 30 < y+ < 200. For y+ > 200, howevet, the C-WMH profile is consistently higher

- c"-SMH: polpmial fit LE-4 C-SMH: composite profile

--- C-SGH: polynomial fit . S . - * C-WMH: polynomid fit

1 E-5 + DNS data of Kim et al. (1986)

Fig. 53: Distnhüions of mean velocity gradient

than the smooth wall data. This is consistent with the trend shown by ïi values

summarized in Table S. 1. In the subsequent analysis, values of ~ ~ / a y ' obtained &om

the polynomial fits are adopted.

53 TURBULENCE INTENSITY AND REYNOLDS STRESSES

The distniutions of the streamwise turbulence intensity and Reynolds stress are shown

in Figures 5.4a and 5.4b, respectively. in Figure 5.4% the profiles are normalized using

the fieestrem velocity (U,) which is the correct velocity scale according to the recent

theory proposed by George and Castillo (1997), while Figure 5.4b uses the conventional

inuer scaling (U,). Irrespective of the specific wall conditions, the profiles increase

slightly with the vertical distance in the ûee surface region (Figure 5.4b). As remarked

in Chapter 1, this is characteristic of open channel flows and has been observed in earlier

LDA measurements and DNS r d t s . This phenomenon is attriiuted to the suppression

of vertical turbuience fluctuations at the fiee surface and a concomitant energy re-

distniution h m the verticai component to the streamwise and spanwise component via

pressure-strain (Komori et ai., 1993).

The C-WMH profile shows a flat and broad 'hump' at y16 = 0.06 - 0.2 (or 65 < y&

< 250). According to Ligrani and Moffat (1985) this is a salient feature of a W y rough

surface and also represents a region where production of longitudinal turbuience energy

is important. They aiso specdated that the large 'hump' may be a result of important

ejection-sweep cycle ciifferences due to roughness, which according to the flow

visualization d t s of Grass (1971) are associated with the detailed mechanics of Iow

O - C - S m C-SML

8 A ---- C-SGH

-1

u - .-.-- C-WMH

6

4

2

O

Fip, SA: (a) Streamwise turbulence intensity m outer coordinates (b) Streamwise Reynolds stress in huer coordinates

momentum fluid entrainment at the bed suffice following the inrush phases. As the

roughness effect reduces, the 'hump' becomes less flat and less broad. Figure 5.4a

shows higher values for the rough surfaces than for the smooth wall. AIthough the

average physicai roughness height of C-WMH (Le. the wire diameter) is only about 50

percent of the vahe for C-SGH (Le. the nominai diameter of sand particles), C-WMH

shows significantly higher vaIues over most of the flow. in flow past bluff bodies (e-g.

Kiya and Matsumura, 1988), the instantaneous velocity is decomposed into a tirne-mean,

a phase-averaged or coherent component and an incoherent or random component. The

relatively higher values obtiiined for the wire mesh may be due to the contriiution of the

coherent motion associated to the vortices shed fkom the wire.

The trend observed in Figure 5.4 and subsequent results suggest that the extent to

which roughness influences the turbulence structure depends on the specific geometry of

the rougfiness elernents. The deviation of the C-WMH profile h m the smooth-wall data

(C-SMH) persists up to y = 0.86 which corresponds to about 50 roughness heights away

fiom the wail. This is at variance with the waiI similarity hypothesis, which implies that

any influence of waU roughness on the turbulent structure should be confineci to about 5

roughness heigfits. In inner coordinates (Le. Figure 5.4b), the peak values for the present

smooth walI data occurred at y' = 13. The srnooth wall data are considerably higher than

the rough-waii values in the near-wd region (y' c 30). For y+ 1 30, and aIIowing for

rneasurement uncertainties, the smooth w d and sand grain profiIes at similar freestrearn

conditions are comparable, Compareci to the smooth and sand grain data, the wire mesh

protile is corisistently higher up to y* = 1000. Curve fits to C-SMH, C-SGH and WMH

data are aiso included for subsequent analysis.

The distributions of the vertical component of turbulence intensity and Reynolds

stress are shown in Figure 5.5a and Figure 5 3 , respectively. Due to limitations in

spatial resolution, reiiable data could not be obtained in the very near-walI region (y/S <

0.02). DNS results of fiee surface flows indicate chat the vertical fluctuation decreases to

zero at the fiee surface. Furthmore, the decreast is rapid and occurs in a thin region

close to the fiee d a c e . Limitations in the present system could not allow this region of

interest to be captured. Figure 5.5a shows that the rough-wall data are significantly

higher than the smooth-wall profiles. The present smooth wall profiles are comparable to

earlier meamrements. in inner coordinates, Le. Figure 5 3 , it is also observai that the

rough-wail data are distinctiy higher than the smoath wall data over a significant part of

the boundary layer. Curve fits to the data are also shown in Figure 5 3 .

The distrïiutions of the ReynoIds shear stress are shown in Figures 5.6a and 5.6b

using outer and inner scaiing, respectively. It is of interest to note that the shear stress is

negative for ail w d conditions in the region where aUlay and aV/ay are negative, i.e.

y16 > 1.5. The peak values of -<n"v"> for the smooth wall data (Figure 5.6b) are about

0.6 to 0.7. Alîhough these vaiues are comparabIe to the LDA measurements reported by

Komori et al. (1993) and Xinyu et aI, (1995) at low Reynolds nurnbers, they are lower

than the asymptotic value of L (Le = UA, which is the value to be expected at

high Reynolds nimibers, Measurements of Reynolds shear stress reported in the

C-SML C-SGH C-SGL C-WMH

b ---- C-SGH A C-SGL 0 .----- c-WMH

Fig. 5 3: (a) Vertical turbulence intensity m outer variabIes (b) Verticai Reynolds stress in Imier variabIes

O C-SMH I 0 C-SML A C-SGH A C-SGL

- < U V > / ' O C-WMH

1 I

(b) - O C-SMH

C-SML ---- A C-SGH A C-SGL

-.--- O C-WMH

n

Fig. 5.6: Reynolds shear srress on smooth and rough sirrfaces (a) outer scahg (b) inner scaling

130

literatrne showed considerable ReynoIds number dependence. For example, the LDA

channel data reporteci by Wei and Wfllmarth (1989) showed peak values that varied

h m 0.6 to 0.9. in the twbulent boundary Iayer rneasurements reportecl by Ching et al.

(1995), the nonnalized Reynolds shear stress showed peak values that varied from 0.8 to

1.0. It should be remarked that the D A systems used by Wei and Willmarth (1989) and

Ching et al. (1995) have better spatiaI resdutions than the system used in the present

study. Johnson and Barlow (1989) recommended that a spanwise dimension of the probe

voturne less than 15 viscous units is required for accurate measurernents of the Reynolds

shear stress. Unfortunate[y, this requirernent couid not be met in this study. It is,

therefore, not dear whether the relatively low peak values observed for the present

smooth wd data are due to low Reynolds nurnber ef fm or Iimitations in spatial

resoIution,

It is of interest ta note that in the wall region, i.e. y" < 200, the data for C-SML

(12 = 2 1) are slightly higher than those for C-SMH (1: = 32). It should dso be pointed

out that the data rate for C-SML was lower than that obtained for C-SMH. The relativdy

higher peak observed for C-SML may be am'buted to the smdler vdue of 1; and Iower

data rate for C-SML in compatison to C-SMH. If the peak for C-SMH is increased by 12

percent to 'correct' for a possi'ble und erestimation of the measured shear stress (Johnson

and Barlow, 1989), the peak values would then be comparable to previons boundary

layer tlows at comparable ReynoIds numbers. No correction was applied to the data

piotted in Figure 5.6 since the vaiues of 1,' at similar h e a m conditions do not vary

much among the smooth and mu&-wd data so that any possible effects of 1,' on <w>

wiII be neady independent of wail conditions.

Irrespective of the scahg use& the shear stress shows important sensitivity to the

specific w d conditions. For example, in b e r sding, Figure 5.6b shows peak values

about 30 and 60 percent higher for C-SGH and C-WMH compared to C-SMH. The

distinction among the various profiles is observed over most of the boundary layer. For

the purpose of subsequent andysis, curve fits to C-SMH, C-SGH and C-WMH data are

also shown. The m e s descnie the experimentai data over the depth of ff ow reasonably

well except for C-WMH wtiere the curve is slightly higher than the experimental data

for y- > 1000.

5.4 SHEAR STRESS CORRELATION COEFFICIENT

The distriiutions of the correlation coefficient are shown in Figure 5.7. The data

obtained on the smooth d a c e are Iower than those obtained for the rough surfaces in

the inner region (y16 < 02) but independent of waii conditions for y18 > 0.2. The peak

values are in the range 0.35 k 0.02, which are lower than observed in high Reynolds

number boundary Iayer flows. As observai earlier, u and v' obtained for C-SMH are

similar to earlier measurements but the present peak values of <u-v+> are Iowa than

most of the existing boundary layer data. A 12 percent incrase to account for any

possible effect of the long spanwise dimension of the rneasltfement volume would

improve agreement between the profiles and typicd high Reynolds number carnuicd

boundary layer profîies. However, the profles shown in Figure 5.7 compare favorably to

the LDA data of Xinyu et ai. (1995) whose Reynolds shear stress profiles are similar in

magnitude to the present data and also to values infened h measurements and the

DNS &ta of Komori et al. (1993).

O C-SMH a C-SML A C-SGH A C-SGL

C-WMH

-

-

-

-

> U

4.2 ! t , . S . a v l

o. 1 1 fi

Fig. 5.7: Dhbution of shear correlation

5 5 STRESS ANISOTROPY TENSOR

In order to quanti@ the differences between the stress distribution on smooth and rough

surfaces, the stress anisompy t m o r bij was evaluated, where bij = <uiuj>/q - 1/36ij, q (=

11) = u2 + $ + 2, and 4j = 1 if i = j and O othmvise. The streamwise and vatical

components of the tensor are denoted, respectivdy, by bli and b2, while b33 and blz

denote the spanwise and the shear components, respectively. Since the spanwise stress

was not measured, these vaIues were approximated. Spanwise data for open channel

flows are scarce. The few measurements and DNS r d t s show considerable scatter.

Following earlier boundary layer d t s , the following approximation was used: w" =

K(u*~ + v+'), where commonly used value of K for high Reynolds number boundary

layers is 0.5 (see e.g., Antonia and Luxton, 1971; Cutler and Johnston, 1989). The DNS

results of Spalart (1988) at Ree = 1410 (see Rodi et al., L993) and the open channel flow

analysis of Neni and Nakagawa (1993) suggest K = 0.4. The more recent boundary Iayer

measurements of Skare and Krogstad (1994) aIso showed a preference for K = 0.4. in

the present analysis, K = 0.4 was adopted.

Figure 5.8a shows the dimiutions of bri for hth smooth and rough wall data,

while and bit are shown in Figure 5.8b and 5.8c, respectivety. The values of blr

obtained on the smooth surface show higher values for yi6 < 0.2. For each surface, bît

shows a trend that is opposite to bri. Figure 5.8a and 5.8b suggest that surface roughness

reduces the anisotropy close to the w d . Compared to the smooth wall data, Figure 5 . 8 ~

shows that the magnignitude of bkz is higher for the mugh wall data close to the waii. The

negative peak values of b Iz are mpAiveIy 0.10 and O. 1 1 for the smooth and rough wail

- O C-SMH ----- A C-SGH .*.*** n C-WMH

0.10

O C-SMH 0.05 A C-SGH

9 2

Fig. 5.8: Dismiution of stress anisotropy tensor

measurements. These values are about 5 to IO percent lower than the smooth and rough

waii measurements obtained by Antonia and Luxton (1 971) but approxhately 30 to 40

percent Iowa than the typical value of 0.14 for relatively hi& Reynolds numbers.

Shafi and Antonia (1 995) compared their rough wall boundary layer to the smooth

wall DNS results of Spalart (1988). in the inner layer, the trends observai for bI 1 and bn

are qualitatively similar to those noted in the present study. . [n the case of b12, they

found no important différences between the smooth and rough waI1 data. Mazouz et al.

(1998) made a similar comparison between smooth and rough wall measwements in a

channel. They aIso reviewed a number of prwious channe1 and pipe flow measurements

over smooth and rough surfaces. In contrast to the present observation and that of Shafi

and Antonia (1995), they concluded that the stress tenson bi and & in duct flows over

rough d a c e s are higher tban the wrresponding smwth wdl values although blz

profiles were not sigaificantly affmed by surfme roughness.

Figure 5.8d shows the distniutions of Abij = fii/ - bG3/b,{ where superscript s and

r stand for smooth and rough, respectively. For C-WMH, Ab1 1 decreases h m 17 percent

at y16 = 0.04 to 5 percent at y16 = 0.15. The corresponding values for AbE at these

locations are 20 and 7 percent, respectively. Compared to the smooth waii data, bit for

C-WMH indicates deviations as high as 60 percent at y/6 = 0-04 and about 10 percent at

y16 = 0.15. In each case the data for C-SGH are slightiy lower than the data for C-WMH.

5.6 SKEWNESS AND FLATNESS FACTORS

The distriiutions of the streamwise (Su) and vertical (Sv) skewness factors for the

srnooth and rough w d data are shown in Figure 5.9a and 5.9b using inner and outer

coordinates, respectively. The streamwise (Fu) and vertical (F,) components of the

flatness factors are plotted in Figure 5 .9~ and 5.9d usuig inner and outer coordinates,

respectively. For the smooth wall data, Su data are positive close to the waII (y* < 13,

and decrease with increasing wail distance. The hi& positive vaiues of Su in this region

are possibly due to the arrivai of high-speed fluid (i.e., acceleration-dominated events)

h m regions away h m the wail. The Su profile for C-SMH changes sign at y- = 12

which also corresponds to the point of maximum u' (Figure 5.4b) and location of

minimum Fu (Figure 5.9~) . in the overlap region, the smooth wall data are negative

suggesting that large negaîive signais occur much more frequently. Beyond the overlap

region, Su initiaily decrease to a local minimum (yt = 800) before increasing towards the

free surface. The large negative values of Su in the outer region are evidence of

deceleration-dominatd events.

Earlier rneasurenients and DNS d i s in smooth wall îurbulent boundq layers

and hlly developed duct flows (e.g. Kim et al., 1986; Durst et al., 1995; Grmther et ai.,

1998) show that Sv is positive in the viscous s u b l a ~ , shows a dip in the b d w region

and remains positive for y+ > 30, However, the data reporteci by Kreplin and Eckehann

(1979) did not show any region of negative S,, in the buffer region. Reliable data for Sv

could not be obtained for y* < 30 in the present measurements. W~th the exception of a

few data points, S, data are consistentiy positive ovu a signifiant portion of the

C-SGH A A

Fig. 5.9: Distniution of skewness and thmess factors (a), (c) innet coordinates; (b), (6) outer coordinates

boundary layer (y' < 900). For 120 < y < 600, Sv is almost the reverse of Su. The

present smooth wall data for Su and Sv are qualitativeIy similar to canonical turbulent

boundary layers and duct flows (e-g. Ki. et ai., 1986; AndreopouIos et ai., 1984, Durst

et al., 1995; Gunther et ai., 1998) in the inner layer. The present data are, however,

distinctly different fiom canonicai boundary Layer flows in the outer region which may

be due to fiee surface effects.

Consideration is now tumed to the effects of surface rougimess on the skewness

and flaîness factors. The trends show by the mugh walI data are qualitatively similar to

that observed for the smooth wdl data. AIthough v*l (Figure 5.5) shows sensitivity to

walI conditions, Sv appears to be independent of the waii condition. This observation is

sirnilar to that made by Mazouz et ai. (1994) over smooth and grooved surfaces.

However, S,, profiles show important sensitivity to surfàce roughness for y' < 700. In

contrast to the negative values obtained on the smooth surface (C-SMH), the data

obtained for C-SGH is approximately zero over most part of the overlap region. In the

case of C-WMH, S, is consistently positive for y- < 120. This may suggest that, in

contrast to the smooth wail data, the overlap region of the wire mesh data are dominated

by hi&-speed fluid. Raupach (1981) and Mazouz et ai. (1994) aiso reported higher

values of Su for rough d a c e s mmpared to smooth wall data in the inner region.

With the exception of the large positive vaiues observai in the viscous sublayer

for C-SMH, Fa for botfi smooth and mugh w a h are close to the Guassian value of 3.

Irrespective of the waii condition, the F v profiles are higher than the Guassian value over

the entire depth of ffow but the dam m the overlap region do not deviate much h m 3.

The large values of Fu and Su for C-SMH in the viscous sublayer are manifestation of

intermittent bursting events that take place there. The large values of Fu and F, and the

corresponding large negative values of Su and Sv in the outer region are signatures of

intermittent large-scale negative fluctuations which occur as a result of the large eddies

driving the fluid fiom the low velocity region. The relatively larger values of F,

compared to Fu close to the fiee surface may suggest that v signais are more intermittent

than u signais near the fiee surface. The u and v signais in the vicinity of the fiee surface

do not show any sensitivity to waii conditions.

5.7 TRIPLE CORRELATION

The gradients of the third-order turbuknce statistics are important because they are

associated with the transfer and redistriiution of turbulent kinetic energy. In the present

study, the following triple products are measured: <us, cg>, <us> and <&P.

Following the anaiysis of George and Castillo (1 997), the triple products are nonnaiized

by &+.L

The dimibution of eu3>, wbich is mciated with the ~aospai of <us by the

turbdent motion in the streamwise direction, is shown in Figure 5.10a. The distributions

of CU%> and cd>, which repramt turbulent wnsport of <II% and <A, respectively,

in the vatical direction are shown m Figures 5-lob and 5.10~. The turbulent work done

by the Reynolds shear stress in the vertical direction is represented by -=Su> and is

shown in Figure 5-lûd. The magnitude of eu3> is àgnificantly larger than the other

triple products. The trends shown by <us and c h , ir. Figures 5.lOa and 5-iOd, are

O c-SMH 4.0 1 A C-SGH

O c-WMH

(b) cul* O C-SMH A C-SGH O C-WMH

0.050 -

(dl < U V ~ O C-SMH A C-SGH O C-WMH

Fig 5. IO: Distributions cf *le correlation (normalized by uU2u3

qualitaiively similar. Both sets of data are positive only in the viciaity of the wall (y/6 <

0.M). They show two dips at y16 = 0.03 and 0.4. The trends shown by <u2v> and <&+,

i.e. Figure 5.10b and 5,10c, which make most of the contriiution to turbulent d i f i o n

in the energy budget are opposite to those observed for < u s and &P. The profiles of

CS> and CU%> are positive over most of the boundary layer. These profiles exhibit two

peaks, which are Iocated at y16 = 0.03 and 0.4, i.e. positions at which Cu3>? and <?LI>

showed their dips. The peak values for <u2v> are higher (about twice) than for cg>.

Except for possiile Reynolds nimiber effects, the present data are qualitatively similar in

the inner region to earlier canonical boundary layer rneasurements (e.g. Bandyopadhyay

and Watson, 1988; Krogstad and Antonia, 1999). The present profiles are, however,

distinctiy different trom the canonical boundary layer data both in shape and sign in the

outer layer, possibly due to free surface effects.

From Figures 5JOa and 5 .10~~ it is apparent tbat the roua wail data (C-SGH and

C-WMH) are considerably higher than the smooth wall data in the inner Iayer. The

effect of roughness on <u% is conhed to y16 = 0.1 but it @sts up to y/S = 0.5 in the

case of cg>. In the outer layer, the pronles are almost independent of spcific w d

conditions. in the case of CU%> and <II+, roughness effects appeat to be more

important in the intermediate region where the magnitude of the data obtained on the

rough d a c e s is higher than the smooth wail data.

For the purpose of modehg the turbulence diffusion term in the kinetic energy

equation, it is the derivaiives of the tripie products with respect to y Mther than the

afnial values thai are required- Figure 5.1 la shows a plot of the sum of cd> and CU%>.

These terms are the major contributors to the turbulence d i s o n tetm in the energy

budget. The Iocation of the outer (and targer) peak is closer to the wail as the roughness

effect inmeases. The correspondhg curves used to obtain their decivatives are also

stiown. The profiles for the rough surfaces are much higher than the smooth wall profile

in the inuer region. As will be shown subsequently (Figure 5.1 Ib), the non-zero d u e s

of a(<u'v+S>)/ày in the vicinity of the free silrface suggest a non-negiigible turbulence

diffusion near the fiee d a c e .

5.8 ENERGY BüDGETS

Consideration is now turned to the turbulent kinetic energy budget. For an

incompressible fluid, the exact transport quation for turbulent kinetic energy (k) is

given by (e.g. Hinze, 1975)

I n m IV v

where, 1 denotes total advection of turbulence kinetic energy; II denotes pressure and

turbulence diffusion; IiI represents turbulence production by the mean flow; IV denotes

viscous diffusion; and V represents viscous dissipation.

in the present analysis, the foliowing approximations are adopted:

1. The mean flow is two-dimensional so that W = O and aW/& = O. From

wntinuity consideration, it follows that ilU/& = dV/ily.

C-SMH O - C-SGH A ----- C - M 0 * * - - - -

dissipation t

-25 0.00 0.25 050 0.75 1 .O0

Y/&

ii. Measmement of au/& showed that this term is negligibly srnaII.

iii. According to the DNS results for canonicai turbulent boundary layers by S p i a r t

(1988) and open channel boundary layer by Komori et ai. (1993), the viscous

diffusion may be neglected since its importance is restricted to the viscous region

near the wail.

iv. Measurernents of the pressure diffusion term are scarce. B a d on the DNS

resuits of Komori et ai. (1993) in open channel flow, the pressure d i f i o n t m

is estimated to be small compared to the other tenns.

v. The shear stress <vw> is estimated to be negligibly small compared to <UV>.

On the basis of the above assumptions, the approximate two-dimensionai steady

state tramport equation for turbulent energy is given by

where 1 = advection of energy by the mean flow; II = production by shear and normal

stresses; iII =- turbulent diffusion; and IV = viscous dissipation. Some of the above terms

could not be direcîiy mesurai and are approximated as discussed mbsequentiy.

a. Advecîion

The advection of energy by the verticd component of the mean velocity, Le. Vak/ay,

was directly measured but UaW& wuid not be measured. According to the

measurements of Skare and Krogstad (1994), UaW& and Vau* are of the same order

of magnitude but oppsite in sign so îhat the total advection by the mean flow shouId be

smd. The channel data on smooth and rough mfbces reprted by H h t a et al. (1992)

showed that the total advection term is indeed negIigiile across the channel. The

boundary layer measmments on smooth and rough surfaces reporteci by Antonia and

Luxton (1 97 1) and Krogstad and Antonia (1999) also showed similar results.

b. Production

TurbuIence production caused by streamwise and v h c d wmponents of the normai

stresses, i.e. U%U& and v%v@y, were computed. Since the mean flow is assumed to

be two-dimensional and aW& = O, w%wlaz, which was not measured is assumed to

be negligibly smdl compared to u%u/& and SaV/ay. Subsequent to assumption (v)

above, the main contriiutor to turùulence production causai by shear stress is

-<w>aU/ay. This term was aiso rneasured directly.

c. Turbulence Diffusion

Turbulence diffision by the streamwise and spanwise components of turbulence

fluctuation, i.e. &uk>/ax and &w>laz were not measured. These tems are assumed to

be negligible compared to &vbldy. Following previous approaches (e.g. Bradshaw,

1967; Antonia and Luxton, 1971; Krugstad and Antonia, 1999), a<vk>lay is

approximated by 0 . 7 5 ~ ( d v ~ & ) l d y .

d Dissipation Rate

The dissipation term was not measured but is obtained tiom the net energy Unbalance, -

Stnctiy speaking, the imbalance comprises the dissipation and ail neglected terms (eg,

pressure and viscous diffusion). However, the neglected ternis are assumed to be mal1

compared to the dissipation rate.

Al1 the tenns considerd in the following discussions are obtained by fitting cums

to the experimental data. These cuve fits have been discussed in earliw sections.

Following the analysis by George and Castillo (1997), di the tenns in the energy budget

are normaiized by u,u,%.

Figure 5.1 1b shows the distributions of advection and turbulence diffusion terms

for both the smooth and rough surfaces. in the region 0.1 S y/6 I 0.35, the diffusion

tems on al1 d a c e s are positive (Le. gain) while the profiles are negative (loss) for 0.4

5 y/6 I 1. In the region 1 < y16 < 2, the profiles show minimai gains and beyond this

region, slight Iosses are observeci for each surface. Close to the fiee surface, the

diffiision terms are srnail but not negligible, e.g. compared to the production term

(Fi- 5.1 lc). As the roughness effect increases, the magnitude of tiirt,uience diffusion

also iacreases. Furihennore, the locations of the inÏlermost peaks and dips on the rough

surfitces are doser to the w d compared to the smooth waii profiles.

The advetion of kinetic energy by the vertical component of mean velocity, i.e.

Vbklay, is negative (ioss) for y/S c 2, These ternis are slightiy positive beyond y16 = 2

but are: srnaller than the magnitude of îhe corresponding difbion terms in this region-

Over most of the flow, the magnitude of the approximate advection term obtained on the

rough d a c e s is higher than the conespondhg smooth wall data. Distniutions of the

production terms, -<uv>aU/ily and d % U / a x are shown in Figure 5.11~. For the

purpose of clarity, cS>av/ay is not shown. hspective of wall condition, cuLh~/&

and <S>av/ay are neariy the same but opposite in sign so that their sum is negligily

smali comparai to îhe individual nomai production temis as welI as production caused

by the Reynolds shear s m s . Figure 5.1 1c shows that at y16 = 0.1, cu%u/ax is about

30 to 50 percent of the corresponding <w>aU/ay. Compared to the smooth wail data,

production caused by normal and shear saesses obtained on the rough wdls show

significady higher values over most part of the flow. In the inner layer, the higher

vaiues observed for the rough surfaces may be attriiuted to the higher stress observed

for the rough surfaces (Figures 5.4-5.6). in the outer layer where the stresses are nearly

independent of walI conditions, the higher production observed on the KOU& surfaces

may be attniuted to higher values of dU/ay for the rough surfaces (see Figure 5.3).

The energy irnbalance is estimated as the difference between production and

turbulence diaision. On buis of the BSSUrnptions made earlier, i.e. viscous diffusion and

pressure diffusion terms are negligiile, and aiso in view of earlier experimental evidence

that advection of turbulence by the mean flow is negiigibly d l over most part of the

flow, one would expect îhe mibalance to approximate the dissipation. Figure 5.1 1d

shows the distn'butions of turbulence production, diffision and dissipation (Le.

imbalance) for smooth (C-SMH) and wire mesh (C-WMH) data The data obtained for

C-SGH are not shown for the purpose of clarity. However, the pronles for C-SGH fall

between the corresponding profiles for the smooth (C-SMH) and wire mesh (C-WMH)

data The dissipation rate for the smooth surface is considerably lower than the

corresponding data obtained for the wire mesh over most region of the flow. For 0.1 I

y16 I 0.3, dissipation is nearly equal to production, irrespective of the specific wdl

condition. However, a baiance between production and dissipation rate deteriorates in

the region y16 > 0.4.

5.9 MIXING LENGTFI AND EDDY VISCOSITY

Although the methodology of modelling turbulent tlows via mixing length and eddy

viscosity does not incorporate the exact physicai pmcesses, it bas been successfu1 in

predicting simple shear fiows on smooth and rough d a c e s (e.g. Cebeci and Chang,

1978; Antonia et ai., 1991). Furtherrnore, one-equation eddy viscosity models have

recently regained popularity as components of two-laye; modeIs where an eddy viscosity

mode1 is used to resoIve the near-waii region and more general models such as k-E or

Reynolds-stress-equation modeIs are employed outside the wail region (Rodi et al.,

1993). Within this conte* the effects of strrface roughness on the distriiutions of the

mixing length and eddy viscosity are examined in this section.

The mixjng Iengîh and eddy viscosity depend on the relative magnitude of the

turbulent shear stress and the mean veIocity gradient. In order to facilitate discussion on

the mkhg length and eddy viscosity, the parameter R = -<uv>'/arr'/ht for C-SMH,

C-SGH and C-WMH is pIotted in Figure 5.12a. Since two-component measucements

+ DNS: ~ e ; = 670 R

0.4 - CSMH ---- c-SGH ..*--. c-WMH V Krognad & Antonia (smooth)

0.3 - + Krognad & Antonia (rough) O Antonja & Luxton (smooth)

U6 , O Antonia & Luxton (rough)

Fig. 5.12: (a), (b) Dim'tnbiions of edcty viscosity on srnooth and rough sdàces (c), (d) Distributioas of mixing length on smooth and rough surfaces

codd not be obtained very close to the wall, the values of R for y' < 30 may not be

reliable. In the inner region where aU'/ay' is almost independent of waU conditions, the

higher vaiues of R observed in Figure 5.12a for the rough surfaces are due to the

relatively higher shear stress obtained on the rough d a c e s (Figure 5.6). in the outer

region, aCT'/ay' is considerably iower for C-SMH (Figure 5.3) so that R is higber for

C-SMH than observed for C-SGH and C-WMH.

The parameter R aiso represents the ratio of eddy viscosity (vd to moIecuIar

viscosity (v), Le. R = vt&. According to Rodi et al. (1993), this quantity is an indicator

of the iduence of viscous effects on the flow and therefore if Reynolds number effects

are hportant. The data deduced eom Spaiart's (1988) DNS results at Re = 670 and

1410 (Rodi et al., 1993) are also shown in Figure 5.12a for cornparison. Close to the

wail the agreement among aü the data sets is good. in the overlap region, the preswt

profiles do not deviate significady h m the DNS data at Rei, = 1410. The present data

and DNS results show that the eddy viswsity in the outer region is much dependent on

aU'/ay'. The DNS resuits at Ree = 1410 as well as C-SGH and C-WMH, which have

relative higher ïi values and presumably higher vdues of au+/*+ in the outer region,

show Iowa values wmpared to their respective peaks which occur in îhe overlap region.

In conîrast, the DNS resuits at Ree = 670 and C-SMH, which have low ïi values and

presumably lowa values of aU'/dy+ show much higher values in the outer region,

Figure 5.12b shows dis t r i ions of the eddy viscosity in outer coordinates. It shouid be

recalled that avldy* and -<u+v+> are both negative in the k d a c e region. The rke in

the profiles for y16 > 1.2 in Figure 5.12b are caused by the negative vahes of the shear

stresses in this region.

The distriiutious of the mixhg length L = -~u&v+>'"lau/aY are shown in Figures

5-12c and 5.12d using inner and outer scaling, respectively. Rodi et aI. (1993) computed

the mixing length distniutions h m the DNS data of Spalart (1988). These data are also

shown in Figure 5.12~. For y' < 80, the agreement between the present &ta and the

DNS resuits is reasonable, In general, the data for C-SMH and the DNS resuIts show

similar trends. This similady may be due to the fact that for these flow conditions,

arr'/ay+ tends to zero much faster than <u'v">.

The smooth and rough wail boundary layer measurements obtained by Antonia

and Luxton (1971) and Krogstad and Antonia (1999) are compared to the present data in

Figure 5.12d. Close to the wall, ali measurements (both earlier and present) are

approximately descnaed by the relation L = 0.41~. The present and eariier

measucernents are in good agreement for y16 < 03. Althougti for C-SMH is

lower than the earlier smwth wali profiles, the C-SMH data are higher in the outer

region due their characteristic low values of aUldy in this region. The cu*v+> profile as

well as ïI for C-WMH are comparabIe to the data reported by Autonia and Luxton

(197i) and Krogstad and Antonia (1999). These features may exphin the good

agreement between C-WMH and the earlier &ta set up to y16 = 0.6. Due to the dip in

ve1ocîw close to the fiee d a c e , the data for C-WMH are distinctly different hm the

bomdary layer profûes.

5.10 SUMMARY

Two-component LDA meaSUTements of turbulent boimdary layers created in an open

channel on smooth and rough surfices a . reported in this chapter. The data presented iu this

chapter indicated some important distinctions between a boundary layer in open channel

flow and a canonicd zero-pressure gradient boundary layer in the outer region, Le. y16 > 1.

These ciifferences are likely due to free surf'ace effects as well as the characteristic hi&

background turbulence levels in open channel flows. However, there are strong similarities

between a turbulent boundary layer in open channel flow and a zero pressure counterpart in

the wall region which allow one to draw general conciusions regarding the effects of surface

roughness on na-wall turbulence stnicture.

The present measurements show that surface mughness effects are not confined to

the roughness sublayer as implied by the wall similarity hypothesis. With regard to the

mean flow, the effect of d a c e roughness penetrates deep into the outer edge of the

flow and substantially inmases the value of the wake parameter il over the

corresponding srnooth waii value. For the low Reynolds numbers considerai herein, the

relatively higher vaiues of ïi obsecved for the rough d a c e s result in a limited overlap

region which may render the use of Clauser plot techique of determining the skin

fiiction unreiiable.

The r d t s presented in this chapter indiate that the turbulence intensities and the

Reynolds stresses are sensitive to the specific waii roughness. For the streamwise

component, the data obtained on the smooth and sand grain roughness do not show

important differences. Although the physical diameter of the wire mesh is srnaller than

the sand grain diameter, the data obtained on the wire mesh are significantiy Iarger than

the values observed for the smooth surfaces. The vertical component of the normal

stresses and the shear stress show even greater sensitivity to wall condition.

Disiributions of stress anisotropy tensors also show dependence on surface roughness in

the inner layer. lt was found that roughness promotes a tendency toward isotropy close

to the wall. The triple products invoiving the streamwise and vertical components of

turbuient fluctuations were computed. When these statistics are normalized by the

scaling suggested by George and Casriiio (1997) îhey show sensitivity to surface

rougbness. The streamwise component of skewness factor shows higher values for the

rough surfaces than for the srnooth surface but the vertical component of skewness

factor as well as the flatness factors do not show any sensitivity to surface roughess.

The distributions of turbuient diffusion, production and dissipation showed a strong

dependence on the specific mgh elments. These observations suggest îhat rough waI1

turbulence models rnust expiicitly acwunt for the specific geometry of roughness elements

in order to accurately predict the tramport characteristics of the flow. This promises to

provide significant ctiallenges to turbulence models. The dimiutions of the eddy viscosity

and mixing Iength show that the ReynoIds shear stress and the velocity gradient dominate

these quantities in the inner and outer regions, respectively. Consequently, the d g lengt.

and eddy viscosity are higher for the rough surfaces m the inner region wMe the relatively

bwer values of velocity gradient for the smooth surfàce cause signifkaut increase in these

parameters in the outer region,

CHAPTER 6

CHARACTERISTICS OF A TURBULENT WALL JET ON SMOOTH GND

ROUGH SURFACES

Wall jet measurements over smooth and rough surfaces are reported in this chapter. The

data reported indude the mean velocities, Reynolds stresses, tripIe products and

distrt'butions of the energy budget as weU as mixing length and eddy viscosity.

Compareci to eatlier studies, the present flows are significantly modifiai by reverse flow

as well as high background turbulence levels close to the kee surface. in order to

faditate cornparison with previous works, issues regarding quality of flow and two-

dimensionaiity are discussed. With regards to the mamwise development of the mean

fiow, both conventional scaling and scaling laws proposed by Narasunha et ai. (1973)

are applied. Some of the scaling laws used for the boundary layer anaiysis in Chapters 4

and 5 are applied to the huer region of the waii jet. The inner and outer regions of the

wall jet are compared to the structure of a turbuient boundary layer and turbulent k e

plane jet, respectively.

6.1 n o w QUALIFICATION

6.1.1 Exit Profiies

Since the streamwise evolution of the flow may depend on initial conditions such as the

sIot momentuni, it is important to document the exit profiles. Figure 6.1 shows

distniutions of the mean and fluctuating stteamwise components of the velocity for

Fig. 6. la: Mean velocity profiles at jet exit

Fig6. lb: Stream* turbulence fluctuaiion at jet ex3

some of the smooth waii tests. The velocity data and vertical distance are nomalized by

the centerline mean velocity Uj and slot height b, respectively. The profiles become

more 'full' as the exit Reynolds number inmases. in contrast to many experiments (e-g.

Karlsson et al., 1993 [KEP]; Abrahamsson et al., 1994 [ml; Schneider and Goldstein,

1994 [SG]) where tophat veIocity profiles have been reported, the present mean profiles

are flat only over the centrai 30 to 40 percent of the dot. The streamwise turbulence

intensity is flat, to within k5 percent, over the middle 20 percent of the dot. The

centerline turbulence uitensity varies fiom 3 to 5 percent, which is an order of magnitude

higher than values reported in the literaîure. A doser examinaiion of the lower half of

the turbulence intensity indicates a Reynolds nmber effect, where the peak value

decreases with increasing exit Reynolds number. Furthmore, the peak value occurred

doser to the wall at a higher Reynolds number.

Table 6.1 : Summary o f exit conditions

The characteristics of the mean profiles at the slot are summarized in Table 6. 1. In

this table, Uj denotes the maximum velocity; Ub is the buik velocity obtained f?om mass - flow rate measurement using an electronic weighing tank; U, is the mean velocity

obtained by integratuig the exit profile; the slot momentum J = puj% and M. = ~ N ' A ~

is a finite ciifference approximation of the exit momentum flux. The ratio h = Uj/Uo in

the present experiments varieci fiom 1.17 to 1 .B. These values are higher than the value

of 1.10 reporteci by Schneider and Goldstein (1994) and other waIl jet studies but are

comparable to the values of 1.15 to 1.20 obtained by Durst et al. (1998) in their low

Reynolds nurnber channel flow experiments. The differences between the bulk velocity

determineci h m mass flow rate measurement (Ub) and the co~~esponding value obtained

by integrating the exit velocity profile (U,) were l e s than 3 percent. It is important to

note h m Table 6.1 that the value of the exit momentum flux J is about 30 to 40 percent

higher than the corresponding value M, determined by integraihg the exit velocity

profile. in view of possiile dependence of streamwise development on initial conditions

such as the exit velocity profile or source momentum (Launder and Rodi, 198 1; George

et ai., 2000 [GAEKLW]), the values of M, are preferred for scahg purposes in

subsequent anaiysis.

6.1.2 Effeets of Reverse Fiow and Fiow Development

Wd jet experiments conducted in srnail enclosures are often influenced by secondary

flows. The streamwise evolution of the flow may depend on secondary flows so that a

reliable calcdation of decay and growth rates will be obtained only if data in the region

of minimal secondary flow effects are coasidered. The present w d jets are created in an

open charme1 and are characterized by a significant reverse flow and high background

turbulence intensity, especially at distances remote h m the slot.

Distniutions of the mean velocity and Reynolds stresses at various downstream

locations were examined in order to identify the region over which modification of the

turbulence structure by the reverse 80w is minimal. Figures 62a and 62b, respectively,

show streamwise and vertical wmponents of the mean profiles for Test E-SM1 in the

region 10 I xh 2 100. The mean velocities are nonnaiized by U, while the vertical

distance is nonnalized by ~ I Q . The pronles are show up to y/yl:2 = 5 so that data close

to the fiee surface, where effects of retum flow are expected to be most extreme, could

be examined. Figure 6.2a shows that for xh 2 30, reverse flow is present. At x/b 5 60,

the magnitude of the reverse or return flow in the vicinity of y/yin = 4 is less than 5

percent of the local maximum velocity U,. With mcreasing downstream distance, the

influence of îhe retum flow becornes more extreme. At x/b = 80 and 100, the negative

velocities in the vicinity of y/yir, = 3 are 13 and 30 percent, respectively, of the local

maximum value. It should be pointeci out that data reported in most ofthe earlier snidies

terminate at y/yllz = 2. There is considerable scatter among the mean velocity

distniutions obtained by various researchers in the outer region of the flow (ylyIn 2

1.3). Note that if consideration is limiteci to data in the region ylyin I 1.5, it would be

concluded h m Figure 6 2 that no effects of 0ow reversai are ptesent. Figure 6.2a also

shows that the mean streamwise profiles at xh 2 30 wllapse reasonably weII. With

regard to the vertical mean vefocity profles, Figure 62b shows that there are no

systematic deviatiom among the profiles obtained at x/b 1 70 in the region y/ym I 1 .S.

Fig. 62: Mean velacity at various downstream locations for Test E-SM 1 (Re,, = 1 1500) (a) streamwise (b) vertid component

At xlb > 70, the innuence of the reverse flow becornes important. In contrast to the

streamwise component for which secondary fiow are ümited to > 1.5,

Figure 2b shows that V could be modifted up to the wali.

The distn%utions of the Reynolds siresses are shown in Figures 6.3 using outer

scaling. Figures 6.3a and 6.3b show that the streamwise and wall-normal components of

turbuience fluctuations coliapse reasonably weli in the region 30 I xh I 60. The effects

of secondary flow become severe beyond xh = 70. At these downstream distances, the

turbulence leveis in the outer region increase appreciably and penetrate almost down to

the wall. On the other hand, distriiutions of the shear stress (Figure 63c) do not show

any systematic trend in the region y/yIe < 0.8 for 30 -< Xlb -< 80. It is conduded h m

Figure 6 . 3 ~ that for x/b S 80, the e f f i of fiow reversai on the Reynolds shear stress are

limited to ylyin > 0.8.

There are a number of studies available in the litemtm for which secondary fhws

appear to be less severe than observed in the present study, yet the Reynolds stresses fail

to wlhpse in the region where the mean pro6Ies become self-pesaving. For example,

in the wind tunnel meamments reported by Wygmnski et al. (1992) m, the -

normaiized sîreamwise turbulence intensity incceases with downsi~am distance. Al1

îhree mmpouents of the normal stresses as well as shear stresses reported by Dakos et

al. (1984) in a CO-flowing pIane turbulent waii jet also hdicate a simiru trend- A semi-

bounded flow like a turbulent w d jet may not be strictiy seIf-preserving. Very o h , the

iack of seiGpreservation for a waii jet is attn'buted to the fact that the maximum velocity

x=100b

0.0 0.2 0.4 0.6 0.8

vNm

Fig. 63: Distriïutions of (a) Stfeamwise turbuIence imensity (6) Vertical turbulence intensity (c) ReynoIds shear stress

U, and the Reynolds stresses decay at different rates with downstream distance.

However, the data reported by invin (1973). [KEP] and [AJL] and many others

identified a region over which the turbulence stresses are nearly self-preserving. It seems

reasonable to speculate that the Iack of collapse observeci in the present flow at x/b 2 70

may be due to the influence of reverse flow in the outer region.

6 J Test for Two-dimensionaüty

For h e plane jets Kotsovinos (1976) suggested that the following constraints: wlb > 10

to 20 and x/w < 2, where w is the slot width, wouId ensure a satisfactory two-

dimensionaiity in the plane of symmetry. This criterion suggests that for a given dot

height and exit Reynolds number, a jet created in a large enclosure is more likely to

confonn to two-dimensionality for larger downstream distances han a corresponding

flow created in a smaller enclosure. For a turbulent wall jet, a number of criteria such as

uniformity in mean velocity data in the crosswise direction (e.g. Gartshore and

HawaIeshka, 1964; [WKH]) have been proposed to examine twodimensionality.

Launder and Rodi (1981) recornmended that conservation of two-dimensional

momenturn flux should be used as a more critical criterion. If the recommendation of

Kotsovinos (1976) is adopted, satisfactory two-dimensionality caa be claimed for ail the

measurement locations considered herein- A simiIar conclusion is atnved at if spanwise

variation in streamwise component of the mean velocity is adoptecl. in the present study

the recommendation made by Launder and Rodi (1 98 1) is adopted as the principal test in

exarnining the two-dimensionaüty of the flow.

The variation of local momentum flux with streamwise distance is shown in Figure

6.4. Figure 6.4a shows the momentum flux (Mu) due to the mean velocity alone while

Figure 6.4b shows the hctiond contribution of streamwise turbulence fluctuations

(Mu). For a given test condition, the maximum momentum flux occurred at xh = 30 - 50

rather than at the slot. The maximuin vaIues are approximately 20 to 30 percent higher

than the corresponding value @&,) obtained at the slot, A review of measwements

obtained in plane kee jet by Ramaprian and Chandrasekhara (1985) showed a sirnilar

trend. For the experiments reviewed in that paper, it was observed that the maximum

local momentum flux couId be as hi& as 55 percent higher than the exit value.

FoUowing Ramaprian and Chandrasekhara (1985), the increase in momentum flux cm

be attriiuted to the presence of negative pressure supporteci by the turbulence

fluctuations in the cross-stream vebcity component. It can be shown fiom the

momentum equation that even smaii pressure changes can account for significant

changes in momentum flux. For each set of data shown in Figure 6.4a, the local

momentum flux is normalized by the maximum value.

in contrast to a plane fke jet where the momentum integral is considered to be a

conserveci quantity, there is a continuous loss of streamwise momentum flux with

streamwise distance in the case of a turbulent walI jet. Although part of the losses is

oftea attri'buted to the presence of the solid d a c e , it is g e n d l y considered that a solid

wall conaibutes very little to the total momentum loss. Launder and Rodi (1981)

estimateci that about 8 percent of the exit momentum flux codd be atûiibuted to

fiictionai losses at x/b = 100. In the present andysis, losses due to waU fiictiorî are

estimated to be l e s than IO percent. This would suggest that satisfactory two-

Fig. 6.4: Streamwise momennmi fiux at various downstream iocations (a) data h m mean veiocity profiles ody (b) coniribuîion h m turbulence fluctuation

dimensionality may be realized if the foiiowing condition holds: Mu 2 0.9& or 75

percent of the local maximum momenhrm flux. If the above constraint is used,

satidactory two-dimensionality can be clairneci for x/b 5 100 (Figure 6.4a). For each

test, the axial locations at which the above condition is fulfilled correspond to the

position at which the mean velocity profles show satisfactory self-similarity. The

farthest downstream distance at which the local value of Mu is p a t e r than 90 percent of

the exit momennn flux reduces as the Reynolds number decreases.

The fractional conmiution of u increases b m a value of 0.5 percent at the dot to

as high as 25 percent at large downstream distances (Figure 6.4b). The large values far

downstream are due to the high turbulence levels at these locations. in the study of

[ML], which was conducted in a large enclosure to minimize secondary ff ow effects, the

strearnwise turbulence intensity contributes about 10 percent to the total local

momenhmi flux in the self-preserving region. It is clear h m Figure 6.4b that at x/b 5

70, MJ(Mu+MU) does not exceed 12.5 percent. For each test, the axial locations at

which u contri'butes l e s than 12.5 percent of the total momentum flux coincide with the

Iocaîions at which the u profiles retain the characteristic wall jet shape.

6.2 STREAMWBE EVOLUTION OF TBE MEAN now

6.2.1 Velocity Decay

The variation of normaiized maximum velocity Um with streamwise distance is shown in

Figure 6.5. in Figure 6Sa, Uj and b, are used as the appropriate velocity and Iength

scales. Here, b, = (b - 28), where 8 is the exit bomdary layer momentum thickaess.

Fig. 65: Variation of maximum velocity with streamwise distance (a) conventional (b) kmematic momentcm çcaling (Lines denote fits to Eqn. 2 2 ï )

l.oo&, I I L I r

.

0.75 - U ~ ~ N , ~ .

- O DSMI (Re,= 12100)

A A DSM2 (Re, = 8700)

O @ O D-SM3 (Re,,=W)

ii v E-SMlCR%=115Oo) - @ [KEpIIRt,=LoOoo)

0 D-SG I (Reb = 12000) A MG2(Re,= 10000)

@ i D-SG3 = 5900)

8 -

I

The choice of b,, rather than b, is expected to account for the shape of the exit mean

profiles (Ramaprian and Chandrasekhar% 1985). Irrespective of the wall conditions, the

data obtained at the lowest Reynolds number decay more rapidly. The data reported by

[WKHJ and [kn] aiso showed a similar Reynolds number dependence, The data of

m] is in fair agreement with the present smooth wail data. At simila. strearnwise

location, the rough waii data are slightly Iower than the corresponding smooth waii data.

The deviation of the rough wail data h m the smooth waii data is greatest in the region

of flow development (x/b I 20).

The variation of U, with streamwise distance for the smooth wall data using the

exit kinematic momentum M, (Le. Eqn. (2.27)), which is the proper scaling according to

Narasimha et al. (1973), is shown in Figure 6 3 . Only data in the region where the

mean veIocity profiles are nearly self-preserving and the effects of reverse flow minimal

are considered here. in contrast to the systematic Reynolds number effect obsmed in

Figure 6.54 the choice of kinematic momentum as the scaling parameter makes the

decay rate independent of Reynolds number. Also shown in Figure 6.5b are best 6ts of

Eqn. (2.27) to the present data and those of m. The values of the power law

exponent (a) for both sets of experiments are within 4 percent of eacb other but the

multiplicative constants (A) are significandy different These differences may possibly

be due to differwt initial conditions such as Reynolds nmber and source momentum.

The near constancy of a suggests universality of Eqn. (2.27). Figure 6 3 aiso supports a

power law variation of u,' with x in the self-preserving region.

6.2.2 Crowth of Inner Layer and Jet Haif-width

The variation of the inner Iayer thichess, y,, with strearnwise distance for smooih and

rough wali data is shown in Figure 6.6. An accurate detemination of y, is generally

difficult. in the neighborhood of y, y-steps for smooth and rough surfaces were 1.0 mm

and 1.5 mm, respectively. The uncertainty in y, is estimated to be less than 10 percent.

It is clear ffom Figure 6.6 that wall jet flow over a rough surface has a considerably

thicker inner layer. More specifically, the local value of y, is about 40 and 25 percent

higher for the rough waIl data at x/b = 30 and xh = 70, respectively,

The variation of the jet half-width yiiz with x for smooth and rough surfaces is

shown in Figure 6.7a using the conventional scaling law. in sharp contrast to the

sensitivity of inner thickness to surface condition, the spread rate for the jet half-width is

nearIy independent of the surface condition. Smiilar to some earlier studies, (e.g. [M,

WKH]), the spread rate shows a Reynolds number dependence. More specificdy, the

spread rate increases with decreasing Reynolds number. The present growth rates Vary

fiom 0.085 to 0.090 fiir the highest and lowest Reynolds nimibers, respectively. The

present growth rates are higher than the value of 0.0073 recommended by Launder and

Rodi (1 98 1) for plane wall jets but lower than those obtained in fiee plane jet studies. As

will be shown in a later section, the relatively higher spread rates obtained for the

present smdy may be due to the high turbulence levels and turbulence production in the .

outer region of the flow.

Figure 6.7b shows the plots of y12 with x for the smooth w d data using the

scaling mmmended by Narasmiha et ai. (19731, i.e. Eqn. (2.28). In contrast to the

Fig. 6.6: b e r layer thickness for smooth and rough surfaces

- presait B = 1.22, b = 0.89 m: B= L.445, f! =O.881

IEIO IEI 1

XMehr=

Fig. 6.7: Variation of half-width with streamwise distance (a) conventional (b) kinematic momentum scaimg (Lmes denote fits to Eqn. 228)

distinct Reynolds dependence observai in Figure 6.7% the use of kinematic momentum

to scaie the data renders the spread rate for the jet half-width nearly independent of - Reynolds number. The best fits of Eqn. (228) to the present data and those of [WKH]

are in good agreement with each 0 t h . The power law exponents are within 2 percent of

each other. Furthmore, the value of (3 is less than 1 indicating a non-linear spread rate

of yin in accordance of the ment similarity theory proposai by [GAEKLW].

A plot of U, versus yin for the smooth wail data is shown in Figure 6.8. The best

fit of Eqn. (2.29) to the present data set and a fit with the constants recommended by

[GAEKLW] are also shown. The values of C in both fits are identical and the difference

between the power Iaw exponents (y) is les than 1.5 percent. This slight difference may

be due to a dependence on initid conditions. The present value of y is Iess than -0.5 as

required by similarity consideration [GAEKLW].

63 TRANSVERSE PROFILES

63.1 Mean Velocity Distribulioas

63.1.1 Determination of Waii Shenr Stress

in view of the existing debate regarding the scaiing law for the overlap region of the

mean veiocity, an accurate and independent determination of wdl shear stress is critical.

The technique used to determine the fnction velocity for the smooth wail jet is M a r to

that used in the boundary Iayer andysk (Chapter 4), Le. using the veIocity gradient at

the wall or fitting a fifth order polynomiai to the near-wali data. The near-wali data and

conesponding linear and &lh order poIynomial fits are shown in Figure 6 . 9 ~ ~

Fig. 6.8: Variation of maximum velocity with jet half-width (Lines represent fits to Eqn. 2.29)

Y -

Fig. 6.9a: Linear and polynomial fits to near-wall data

O DSM3 ---- Bradshaw and Gee ( 1 %2)

Fig. 6.9b: Variation of skui fiction coefficient with Reynolds number

The similarity between these data and those reported in Figure 4.la implies that the

mean velocity profile for a turbulent boundary layer and wall jet is identical in the near- -

wali region.

Figure 6.9b shows the distribution of skin fiction coefficient Cr with Reynolds

nimiber Re,,, (= U,,ydv). The skin fiiction correlations of Bradshaw and Gee (1962),

Hammond (1982) and Eriksson et ai. (1998) are shown for comparison. The present Cr

values are in gwd agreement with the correlation proposed by Bradshaw and Gee

(1962) and Eriksson et al. (1998). However, the present sets of data at Re, < 3000 are in

better agreement with Bradshaw and Gee's correlation, while the data at Re, > 5000 are

better descnied by the correlation of Eriksson et ai (1998). Hammond's correlation is in

fair agreement with the present data for Rem > 7000 but would substantially over predict

the present values of Cf for Rem I 7000. For the range of Rem considered herein, the skin

fiction coefficients obtained for the rough surface are in the range 0.008 I Cf I 0.012.

These values are substantially higher than the corresponding values obtained on a

smooth surface at similar Re,.

63.1.2 Mean Profiles in Outer Coordinates

The mean pronies in the self-presenring region for some of the smooth wall experiments

are shown in Figure 6.10a in outer coordinates, i.e. U, and yin. The profiles coUapse

reasonably well. The inset shows a comparison between the present data and the hot-

wire data of [WKH] and [AL] as well as LDA data of w]. The agreement among the

present and previous &ta sets is excellent up to ylyin = 1.5. The agreement

Fig. 6. IO: Mean veiocity profües m outer coordinates (a) smooth wd (b) cornparison between smooth and rough waii data

between the present data and the LDA data of [KEP] is monable up to y/yin = 2.5.

However, the D A &ta sets show lower values at the outer edge of the flow compared

to the hot-wire data sets. This trend is consistent with previous comparisons made by

[SG] and Venas et al. (1999) and has been attrr'buted to instniment limitations although

secondary flow effects could not be wmpletely d e d out. As remarked earlier, the

effects of reverse 80w on the mean velocity profiles is not any more severe than in

previous studies where attempts were made to minimize secondary flow effects.

The mean velocity profiles in outer coordinates for the rough waiI experiments are

shown in Figure 6.1 Ob. A smooth wail profile is also shown for compan%on. The smooth

and rough waIl data coilapse reasonably weiI in the outer part (y/yI,t > 0.5) of the flow.

However, systematic and significant deviations are observed ciose to the wall. As

already shown in Figure 6.6, the locations at which the maximum mean velocity

ocmeci are farther rernoved h m the waIl in the case of the rough-waIl profiles than for

the smooth wail data. The data in the region y/yiZ 1 0.4 is shown as an inset so that the

near-walI region can be more cIoseIy examined. Similar to the observations made in

Chapters 4 and 5 for turbulent boundary Iayer, the smooth wall profiles are more full

than the corresponding rough waii profiles in the inner region. Furthemore, the rough

wall profiles become Iess fiil1 as the roughness effect increases. For a turbulent boundary

layer, it is claimed that d a c e roughness enhances entrainment of irrotational flow into

the inner layer owing to higher surface drag over a mugh surface (Krogstad et al., 1992).

It may be speculated that the relatively thicker inuer layer (y,) observed for a turbulent

walI jet over a rough d a c e (Figures 6.6 and 6.10b) is caused by the same mechanism.

63.13 Mean Proales in Inner Coordinates

Distniutions of the mean velocity in inner coordinates are shown in Figure 6.1 la for the

smooth waU data. A data set of W P ] is also shown for cornparison. The present data, in

excellent agreement with those of p l , show that a weIldefined log region does exist

although the extent of the overlap region is relativeiy shorter than observed for a

turbulent boundary layer (Chapta 4 and 5). Launder and Rodi (198 1) suggested that the

narrow overlap region identified above may be due to incursions of Iarge-scale eddies

h m the outer shear Iayer bearing a shear stress of opposite sign. The present data aiso

show that as the dot Reynokds number increases, the extent of the log law region

increases slightly.

AIthough Re,,, increases with increasing streamwise distance for a given Rej, there

is no noticeable increase in the log law region with increasing streamwise distance. This

observation should be contrasteci to a turbulent boundary Iayer for which the overlap

region increases indefinitely with increasing x (or 6'). As can be inferred f3om the

values of <uv>/WW2 (Figure 6.3~) and U:/LJ,' (Figure 6.9b), the wail shear stress

appears to decrease faster than the shear stress in the outer region. As a mnsequence, the

relative strength of the outer flow and its encroachment on the inner layer increase

progressively with downstream distance (Launder and Rodi, 198 1 ).

in accordance with the sunilarity theory proposeci by [ G A E W , the resuits

presented in Figures 6.9a and 6.11a suggest a striking similarity between the imer

region of a wail jet and that of the turbulent boundary. Since the overlap region of a waii

Fig. 6.1 I : Mean veiocity in inner coordinates (a) smooth srnface (b) rough surface

jet is well described by a Iogarithmic law with values of K and B that are universal and

identical to those used in boundary layer anaiysis, the use of the Clauser plot technique

to estimate the skin fiction is justified for a turbulent wali jet. From a practical point of

view, the limited extent of the log law region could lead to significant enor unless

caution is exercised so that the region over which data is wnsidered is not too wide. For

the same reason, turbulent walI jet modelers can make use of the 'wail function' that is

widely used in boundary layer computations.

Determination of friction velocity for a turbulent wail jet over a rough surface is a

challenging task. For Test D-SGI, an attempt was made to use a momentwn balance

fiom one-component measurements at x/b = 30, 33, 35, 40 and 64 to evaluate the wail

shear stress. The value of skin fiction obtained h m this method was considerably

lower than the correspondhg valws obtained from the nu-wall data for the smooth

wall data at similar Rk. This is not suprising since previous atternpts to detemine the

wall shear stress fiom rnomentum bahce were unsuccessfiil even for wail jets over

smooth surfaces (Schwarz and Cosart, 1961). The present srnooth wall data suggest a

welldefined overlap region with universal log Iaw constants (Figure 6.1 la).

Furthemore, Figure 6-10 shows that the inner region of the mugh wail data is larger

than for the smooth wall data, Therefore, the existence of a log Iaw region with universai

constants for the smooth wall and the evideuce of a relatively tbicker b e r layer for the

rough wdl data are used to justifj a Clauser chart technique for the rough wdl data. The

uncertaïnty in Ut detecmifled h m this approach could be hi& perhaps of the order of

10 percent.

The velocity profiles in inner coordinates for the rough waIl data are shown in

Figure 6-1 1b. As expected and has been observed in boundary layer studies, the rough

wall data show a downward-right shift with respect to the log law profile for a smooth

surface. At similar xfd, the roughness shifi increases with increasing Reynolds nurnber.

For a given slot Reynolds number, the roughness effect inmeases with increasing

downstrearn distance.

63.2 Reynolds Shear Stress and Turbulence Intensities

The Reynolds stresses are shown in Figures 6.12 and 6.13. Dismiutions of the

streamwise turbulence intensity obtained using one-component LDA are shown in

Figure 6.12. The use of the one-component system ailowed measurernents closer to the

wall than when ihe two-component LDA system is used. in tests for which both one-

and two-cornponent measurements were made, the profiles obtained using one-

component compared favorably to the streamwise component of the corresponding two-

component meamements. in al1 cases the deviation was Iess than 5 percent, which is

comparable ta the variation in Ub obtained h m m a s fiow rate measurements.

in Figure 6.12% the present measurements are wmpared to the hot-wire data of

[AL] and the LDA measurements reporteci by [KEP] and [SG]. The present data set is

in gmd agreement with the profle of W P ] for y/yln < 1.5. The outer peak values and

their corresponding locations are smiilar in both snidies. The data compiled by

Ramapian and Chandrasekhara (1985) f i a turbulent fke plane jet indicated peak

values that vary h m 0.22 to 0.31, which are substantially higher than observed in

Figure 6.12a In the outer part of the flow, however, the present profle shows

O D-SM1 x = Z O b

DSGl x = 2 O b QB 4 DSGl x - 3 0 b

Fig. 6-12: Distriiutions of streamwise tiirbdence mtensity (a) present and prwious smooth data in outer coordinates (b) present and previous smooth data m inner coordinates (c), (ci) present smwth and mu& data in outer coordniates

Fig. 6.13: Vertical turbulence fluctuations (a) comparison to previous data (b) comparison between smwth and rough data

significatltiy higher trnbufence levels. Compared to the present profiles, the data

obtained by [ A L ] are in good agreement in the inner region but are Iowa in the outer

region. The LDA data of [SG] show persistentiy higher levels in the inner region but

their peak vaiue is Iocated cioser to the wail.

In inner coordinates, Figure 6.12b shows a good agreement between the present

data and the profile obtained by [KEP]. The vdue of the inner peak and the

correspondhg location were found to be k7 = 3 and y' = 14 - 15, in good agreement

with the boundary Iayer data reported in Chapters 4 and 5. It is concluded fiom Figure

6.12b that distniution of streamwise turbuience intensity in the huer region is similar

for both a turbulent wall jet and boundary layer.

Figure 6.12~ compares the present smooth and rough wail &ta in outer

coordinates. The smooth and rough wall profiles are simiiar in the outer part of the flow.

The values of the outer peak and their wall-normal locations at similar axial Iocations

appear to be independent of sirrfkce conditions. in order to examine any possible effect

of surface rougbness on the inner region, the smooth and rough waii profiles are shown

in Figure 6.126 for y/ylr, < 0.25. It is observed that d a c e roughness increases the

turbulence intensity in cornparison to the smooth wail data

The waiI-norma.1 component of turùulence fluctuations is compared to the data

reporteci by [KEP] and [AJL] in Figure 6.13a using outer scaiing. In contrast to the

streamwise fluctuation and ReynoIds shear stress, the v d c a i turbulence fluctuation

decreases monotonically h m its maximum vaIue, i.e. at y/yln = 0.7 - 0.9, towards the

wall. The present data are consistently higher than the earlier measurements over the

entire depth of flow. The peak values for the present profiles are about 0.17 to 0.19,

compared to typical values of 0.1 8 to 0.24 reported for a turbulent free jet (Ramapnan

and Chandraselchara, 1985). It should be noted fiom [KEPl's profile that the turbulence

fluctuation goes to zero rapidly in a thin region close to the wall. The LDA system could

not allow data acquisition very close to the wall. ïhe data obtained closest to the wall in

the present tests are substantialiy higher than those obtained by [KEP]. This may suggest

that the high background turbulence levels observed in the outer Iayer penetrate deeper

into the wall region than indicated by the streamwise turbdence intensity.

A cornparison between the present smooth and rough data at x/b = 50 is shown in

Figure 6.13b. The relativeiy higher values observed for Test E-SG2 in the near-wa1I

region may be due to a severe encroachment of the high turbulence levels close to the

iÎee surface. With regard to possr%Ie roughness effects, it appears reasonable to compare

Test E-SM 1, E-SM2, and E-SG 1, al1 of which have simiiar turbulence levels in the outer

region. In contrast to the observation made for the streamwise turbulence intensity, these

profiles do not show any syaematic roughness effects. Since data could not be obtained

very close to the wail as was obtained for the streamwise component, it is likely that the

region over which roughness effect dominates was not captured. It should be recailed,

however, that for the boundary layer data reported in Chapter 5, waii roughness effects

are noticed at farther distances h m ttie wall in the case of the vertical component than

for the streamwise turbulence i n t d t y . It is important to note that there are two

cornpethg effects modifjing the structure of the flow; 1) wail roughness which m a e s

the flow away h m the wall, and 2) high tiirbdence levels whose influence is b m the

more energetic outer region and towards the ma.-wall region. The interface between the

inner and outer layers may dampen these effects to different extents. in connection with -

the data shown in Figwe 6.13a, it seems reasonable to speculate that the effects of high

turbuimce levels dominate down to the near-wall region. This may not aiways be the

case since the rougimess effect of the sand grain used in this study is minimal. In this

regard, it will be of interest to investigate interaction between elevated £teestrem

turbulence and surfaces with significantly greater roughness effects or lower fieestrem

turbulence than ansidered here.

Figure 6.14a shows a comparison between the present Reynolds shear stress

profiles and data obtained by [KEP], [Ml and [SG]. For ~ l y ~ ~ < 0.5, al1 the profiles,

both present and previous, collapse to within meaSuTement uncertainties. Important

differences are, howevm, observed among the different sets of data in the outer region.

The outer peak values for the pfesent profles are higher than those reportai by [KEP]

and [ml but are comparable to the values obtained by [SG]. For the data reporteci by

[ML], the outer peak is located at ylyln = 0.66, while the present data and the other

LDA data (Le. [KEP] and [SG]) indicate a peak at ylyin = 0.8 - 0.9. For a turbulent 6%

plane jet, the data compiled by Ramaprian and Chandrasekhara (1985) indicated peak

values that Vary from 0.02 to 0.026, which are higher than observed in the present

measurements. Compared to the smooth data, Figure 6.14b shows no important

çensitivity to wall conditions.

A

~ J Y , , . = &

0 E-SML=x=SOb A E-SM2 = x = SOb

1 - I E-SGI=x =50b A E-SG2 = x = 50b

O - 1 1

-0.01 0.00 0.0 1 0.02 0.03

Fig. 6.14: Dktriidons of Reynolds shear stress (a) present md previous smootb waii &ta (b) present mooth and mu& wall data

633 Triple Correlation

The distributions of the following triple correlation: <u3>, <u%-, & and <Su>

nomalized by um3 for Tests E-SM2 (xh = 30,50) and E-SG2 (xlb = 50) are shown in

Figure 6.1%. The trends observai here are qualitatively similar to the hot-wire data

reported by irwin (1973) and Dakos et aI. (1984)- in contrast to the observation made for

the boundary layer data in Chapter 5, the wall jet profiles do not show any systematic

dependence on surface conditions. in the outer region, the profiles pas through zero at

0.80 I ylyiiz 5 0.95 which is dose to the location at which the maximum Reynolds

stresses occurred (Figure 6.14) The locations of zero crossing in the memement

reported by Irwin (1973) and Dakos et ai. (1984) are 0.7 I ylylr, S 0.8 and 0.9 I y l y ~ I

0.97. Each profile shows an i ~ e r and outer peak.

6.3.4 Energy Budgets

in compuhg the energy budgets for the wail jets, the approximations and assumptions

made in Chapter 5 for the turbulent boundary layers are applied. Measurernents of

turbuience energy budgets for wdl jets are scarce. Furthmore, there are no DNS data

for a turbulent waU jet so that not ail the assumptions and approximations impiied in the

foIIowing discussion can be justified. These comments notwithstaadiing, the subsequent

discussion provides some insight into the turbulence energy budgets. As indicated in

Chapter 5, the use of experimentai data and their derivatives may give rise CO significant

emrs so that curve fits to data points and their derivatives are used here. in order to

show the quaiity of curve fits used in the subsequent analysis, experimentd data and

thW mrresponding fits for two sets of data are shown in Figure 6.16. Figures 6.16a and

l (c) -a=*

Fig. 6.15: Triple comIation (nonnalized by LI rn ')

0.4 1 . t . T . I .

(b) Test E-SM2 x = 30b

F i g 4.16: Mean and turbulence data and correspondmg curve fi& (a) mean (b) tubdence data for Test E-SM2 x = 30b (a) mean (b) turbuience data for Test E-SG2 x = 50b

6.16b, respectively, show the mean and Reynolds messes for Test E-SM2 (x = 30b)

while Figwes 6.16~ and 6.164 show the comsponding distniutions for Test E-SG2 (x =

50b). An assessrnent of goodness-of-fit using RI and chi-square analysis showed that the

curves descnie the experimental data reasonably well over most of the flow.

Figure 6.17 plots the nmi of < u b and <$>, which are the major conmbuting

correlations to turbulence dif i ion in the turbulence kinetic energy transport equation.

The corresponding best fit to each set of data is also shown. It is clear that energy is

transporteci away fiom the two peaks. The dip between the inner and outer peaks

occurred at y/yir, = 0.3 - 0.4. The differences beniveen the srnooth and rough wali data

are pmbably due to the higher turbulence leveIs observed for the rough data (Fig. 6. L4b,

for example) rather than wall roughness effects.

Distniutions of energy production causeci by the shear stress <uv>aU/+ and

vertical nomai stress &V/ay are shown in Figure 6.18a The profiles of <uv~aU/ay

reported by Eriksson et ai. (1999) m] is aIso shown for cornparison. in each case, two

peaks can be inferreci for the production caused by shear stress. The b e r peak is

retatively higher than the outer peak. In each case the outer peak occurred in the vicinity

of maximum shear stress, i.e. = 0.8 - 0.9. Over most of the flow, the verticai sîress

production is mail compared to that caused by shear stress. The production caused by

the strearnwise normal &ess tt2a~Iax nearly balances the vertical wntniution except

close to the waii so that the totaI production by normal stresses does not make any

important contri%ution to turbulence euergy production except close to the wd.

Fig. 6.17: Distriiutiom of ( < v h z v > ) l l l fa awoth and mu& w d data

advcttion diffusion E-SM2 x = 3ûû - - - E - S M 2 ~ = 5 0 b - - - - E-SG2~=50b ----• - -

1.

4-02 I 1 1 1

0.0 0.5 1 .O 1.5 2.0 2.5

y%?

Fig. 6.18: Turbulence 1Metic energy budget (a) production by normai and shear stress (b) advection and turbulence diaision

Production caused by shear stress changes sign (Le. becornes slightly negative) in

the near-walI region but the total production does not change sign owing to the

contribution fiom the normal stresses. Since the mean velocity profiles for the present

data and the data obtained by [KEP] collapse (Figure 6.10a) and the Reynolds shear

stresses are similar for y/yI.? < 0.5, the good agreement between the present values of

<uv>dUlay and the corresponding profile for [EKP] is not surprising. The higher values

in the outer region for the present sets of data are consistent with the trends observed for

the shear stress in Figure 6.14a. The peak values for the present profiles are about 0.13 -

0.15 wbich are about 10 percent higher than reported by PKP], but are comparable to

the 6ee pIane jet data reported by (Ramaprian and Chandrasekhara, 1985).

Distributions of turbulence diffusion and advection by the vertical component of

the mean velocity (Figure 6.18b) are quditatively sirnilar to those reported by Irwin

(1973). A significant amount of turbulence is diffused outward from the inner peak and

inward h m the outer peak. The present diffusion terms at y/yln = 0.1 - 0.2 are several

h e s higher than the data reported by Irwin (1973). This and the high values for ylylr, >

1.8 are most likely due to the scatter observed in Figure 6.17. Data in this range may not

be reliable.

The dissipation term was approximated by the net imbalance. The values chse to

the wail are impiausïble mainly due to the hi& values of the diffusion tenns discussed in

the m o u s paragraph. A cornparison among the various energy budgets for Test E-

SM2 ( x = 50b) and the wail jet data of Irwin (1973) as well as the plane fke jet

measurements reported by Bradbury (1965) is shown in Figure 6.19. It is striking to note

that with the exception of the advection term, the present prunles fa11 within the enveIop

of the corresponding wall jet and free jet data obtained by Irwin (i973) and Bradbury

(1965), respectively. This is consistent witb the spread rate abserveci in the present study

the values of which are intermediate to typical vaIues of 0.073 and 0.1 reported for wall

jet and fiee jets, respectively.

63.5 Mixing Length and Eddy Viscosiîy

Distributions of mixing length and eddy viscosity are shown for Test E-SM2 at x/b = 30

and 50 in Figure 6.20. It should be recaikd ihat the mean velocity is maximum in the

neigbborhood of ylytn = 0. LS - 0.17 which may explain the singularity observed in this

region. The near-wall data (inset) indicate a region of rapid increase that is foiiowed by a

region of near-constant disûiiution and then a rapid decrease towards zero. The W n g

length dimiution for a fiee jet calculated by Bradbury (1965) is aIso shown in Figure

620a. Beyond the location of maximum mean velocity, the present data and calculation

show a simiIar trend. For 0.5 I yfytr, < 2, the distriiutions are nearly constant and

similar.

Figure 6.20b shows the eddy viscosity Erom the present measurements as weli as

the hot-wire data of Bradbury (1965) and LDA rneasurements reported by Ramapian

and Chandrasekhara (1985). The wall jet and iÎee jet data are comparable in the outer

region.

Present Bradbury ENlrin Production - O Diffusion - V V Advection --- 0 Dissipation - - - - A A

Fig 6.19: C o m p a . between present data and previous plane waii and free jets

E-SM2 x = 30b E-SM2 x = 50b

4 Bradbury - a- - Ramaprian and Chandrasekhara

1.25

1 .O0 -

0.75-

W.? - 0.0-

Fig. 6.20: (a) Dismibutions of mullng length for wali jet and fkee jet (b) Distributions of eddy viscosïty for wall jet and fke jet

- .. \

1 \. 1

1 1

0.00-A . 1 I L 1

0.0 0.5 1 -0 1.5 2.0 2.5

I I I 1

-

0.03

0.02.

I 0.01 -

- E-SM2 x = 30b ---- E-SM? x = 50b

11 I I 0.00 r-) I 0.05 0.10 0.15 : ‘

; 9

0.00.

- E-SM2 x = 30b - ---. E-SM2 x = 50b

6.4 SUMMARY

Measurements of turbulent wall jets on smwth and rough surfaces in an open channe1

were obtained using a LDA. Aithough the present fiows are modifiai by reverse flow

and high background turbulence intensity, the twbulence structure is similar to

previous studies in which secondary tlow effects are minimal. In andyzing the

sîreamwise evolution of the flow, both conventionai and scahg laws proposed by

Narasimha et al. (1973), Wygnanski et d. (L992) and George et al. (2000) are used.

The results show that the inner layer is relatively thicker for the rough surface but

the jet hdf-width is independent of wall conditions. Similar to earIier findings,

application of the conventionai scaling law makes the velocity decay and spread rate of

the jet half-width ReynoIds number dependent. The spread rates observeci in this study

are considerably higher than reportai in eariier investigations. On the other hanci, the

decay and spread rates do not show any important dependence on Reynolds nmber

when kinematic momentun scaiing is adopted. The fact that the spread rate is not altered

by surface roughness supports the premise of a previous numerical study (Gu and

Bergstrom, 1994) that a waü jet is a cornplex flow in which the mechanisms of darnping

are not the same as in a simple turbulent bomdary layer. The present study aiso provides

support for a power law decay and growth rates for the mean velocity and jet half-width.

Furthemore, the power law constants found in the present study are in good agreement

with the values obtained in some eariier measurements and those recommended h m the

similarity theory proposed by George et ai. (2000).

The inner region of a turbulent wail jet is sirnilar to that of a turbulent boundary

Iayer. in contrast to some eariier arguments, a welldefiaed log law region with universai

log iaw constants was identifid. Therefore, the popular Clauser chart technique for skin

fiction meamrements and the use of wall fimctions to resolve the near wall region are

vaiid, at least in principle. The skin fiction coefficients over a rough surface are

significantly higher than obtained over a smooth wall. One other important effect of

surface roughness on the mean flow is that it displaces the location of maximum velocity

farther away fiom the waII. The streamwise turbulence intensity in the near-wail region

is found to be considerably higher for the mugh wail than for the srnooth surface. These

results, in contradiction to the wall similarity hypothesis, show that the effects of surface

roughness on the structure of a turbulent waii jet penetrate beyond the roughness

sublayer. However, the normal turbulence intensity and shear stress do not show any

important sensitivity to surface roughness. Since this is inconsistent with the findings

made for turbulent boundary Iayers (Chapters 4 and S), it is speculated that the hi@

turbulence levels nea. the fiee d x e may be making severe encroachment down to the

near-wall region,

Cornparison to eariier waU jet and fke jet investigations show some similarity

between the energy budgets. A tenn by term cornparison showed that almost al1 the

energy terms obtained in this study are intemediaie to corresponding waII jet and fke

jet data. This observation and the high values obtained for the Reynolds stress in the

outer region may explain the present spread rate king higher than the wall jet data

avaiiable in the iiterature and lower than corresponding values obtained in free jet

experiments.

CHAPTER 7

SUMMARY, CONCLUSIONS, CONTRIBUTIONS AND FUTURE WORK

In this chapter, a sumrnary and the major conclusions and contributions of the present

study are given. Some important implications of the present findings for na-wall

turbulence models are also discussed. Finaily, recommendations for future work are

ouîiined.

7.1 SUMMARY

A program of study was undertaken to provide further insight into near-wall turbulence

structure. Specificaily, the effects of wail roughness on a . open channe1 turbulent

boundary layer and a turbulent w d jet were investigated. The experiments were

conducted using a single and a two-component LDA system. The results were

interpreteà using the conventional scaIing laws and the recent theories proposed for

turbulent boundary Iayers and wail jets. The boundary layer and wail jet experiments

and results are summarized in the foliowing sub-sections.

7.1.1 Tnrbuient Boundary Loyers

The boundary layer measurements were obtained on a smooth surface, and surface

mughnesses created h m three ge~me~cal ly different roughness elernents (i.e. sand

grains, perforateci plate and wire mesh) so that the specific geometry of wail conditions

on the turbulence structure could be examineci. The effects of low Reynolds number on

the turbulence structure were dso examined. The data presented in this study include

mean velociy, turbulence intensities, Reynolds stresses and triple correlations, skewness

and flatness factors, as weI1 as distn'butions of approximate turbulence kinetic energy

budgets, rnixing length, and eddy viscosity. The rough walI data were interpreted within

the context of the wall similarity hypothesis, which suggests similarity between the

turbulence structures on both smooth and rough siIrfaces, except in the roughness

sublayer.

Most of the techniques available for the determination of the wall shear stress were

discussed. in the case of flow over a srnooth mrhce, it was shown that, the usefuf extent

of the viscous sublayer in detennining the wall shear stress could be increased without

sacrificing accuracy, by fitting a fifth order polynomial to the near-wall data. In the case

of the rough wall data, a vebcity defect profiIe was fitted to each data set to determine

the strength of the wake and the skin fiction coefficient. in fitting the velocity defect

law, a correlation which did not 6x the vaiue of n implicitly but aiiowed its value to be

optimized, was found to yield a more consistent and accurate estirnate for the skin

fiction coefficient than other fomiuIations which fix the value of n. The fiction laws

proposed by Barenblatt (1993) and George and Castilb (1997) were aiso applied to the

smooth and rough waü turbdent boundary layers.

In analyzing the boundaty laym data, both anventionai scaling laws and recent

theories were considered. SpeEifically, the dassical logarithmic Iaw and the power laws

fonndated by Barenblatt (1993) and George and Castilio (1997) were used to mode1 the

overlap region of the mean velocity profiles over smooth and rough surfaces. The

present study is the fïrst to extend these power laws to rough waiI turbulent boundary

layers. The vaiues of the fiction velocity obtained fiom the power laws proposed by

Barenblatt (1993) and George and Castillo (1997) were compared to those obtained fiom

the near-wall data and velocity profile matchhg technique. For the turbulence quantities,

the friction velocity and scaling laws proposed by George and Castillo (1997) were

applied.

Even though the boundary layer in an open channel flow is influenced by the fkee

surface, the results presented in this study showed that many of the flow characteristics,

in particular those that pertain to surface roughness, are similar to those observed in a

canonid zero pressure gradient boundary layer. With regard to Reynolds number

effects on the turbulence structure, the application of different scaiing laws gave

different conclusions. The use of inner scaiing laws showed that similarity for the

streamwise turbulence intensity is ümited to y' < 30. On the other hand, the scaiing law

proposeci by George and Castillo (1997), Le. outer scaling laws, suggested similarity in

the very near-waii region and the outer layer, but the overlap region showed Reynolds

number dependent. The present resuIts aIso uidicated chat the effect of d a c e roughness

on the turbulence structure is aot confinai to the roughness sublayer as impIied by the

waU similarity hypothesis. Enstead, surface roughness increases the turbulence

fluctuations, Reynolds stresses, triple correlations and the cornponents of the turbuIent

kinetic energy budgets. Zt was aiso observed that the extent to which the mean and

turbulence quantities are modified by d a c e roughness depends on the specific

geometry of the roughness elements. -

7.1.2 Turbulent Wall Jets

The wail jet measurements were obtained on a smooth d a c e and surface roughness

created fiom sand grains. The data presented in this study include mean veiocity,

turbulence intensities, Reynolds stresses and triple correlations as well as distributions of

approximate turbulence kinetic energy budgets, mixing length and eddy viscosity. The

streamwise evolution was analyzed using both conventional scales and scaling laws

proposed by Narasimha et aI. (1973), Wygnanski et al. (1992) and George et al, (2000).

For the srnooth wail data, the fiiction velocity was independentiy deterrnined fiom

velocity gradient at the wail or by fitting a fi& order polynomiai to the data in the

region y' < 15. This was criticai to an independent examination of whether a well-

defined log law region with universal constants exists or not. In the case of the rough

wail data, the fiition velocity was obtained using a Clauser plot technique.

Aithough the present waii jet measurements were somewhat modified by reverse

flow and high background turbulence intensity close to the fiee surface, the mean

velocity and turbulence quantities were found to compare favorably to other diable data

in the fiterature. The effects of surface roughness appear to modiQ the mean flow

significantiy as evidenced in higher skin fiction coefficients as well as a thicker inner

Iayer and more rapid velocity decay for Qows over a rough surface compared to a

smooth surface. Wth the exception of the streamwise turbuknce intensity in which case

the near-waii daîa were foimd to be wnsiderabIy higher for the rough wall, none of the

other turbuIence statistics showed any important sensitivity to surface roughness.

Comparison to earlier wail jet and fiee jet investigations showed some similarity

between the energy budgets. A tenn by t e m cornparison showed that, except for the

advection terni, al1 the energy tenns obtained in this study are within the envelop of

earlier wall jet and fiee jet data.

7 3 CONCLUSIONS

The major conclusions of the present study are swnmarized in the following subsections.

7.2.1 Turbuient Boundary Layers

1. The scaling Iaws proposed by George and Castillo (1997) are more suitable for

examinhg low Reynolds number effects.

2. The mean profiles showed a systematic Reynolds number effect as evidenced in

the systematic variation of the outer wake parameter with Reynolds number. in

inner coordinates, the present resuits indicate Reyuoids number similarity for the

turbulence intensity in the region y+ < 30. AppIication of the scaling law

proposed by George and Castillo (1997) suggests that Reynolds number effects

on the trcrbuience intensity are confineci to the overlap region.

3. The power iaws proposed by Barenblatt (1993) and George and Castilio ( 1997)

are excelIent alternatives to the logarithmic law profiles for a smooth wall data,

In the case of rough waii turbulent boundary layers, the formulation proposed by

George and C d o (1997) is more suitable than the bg Iaw and the power 1aw

formuiated by Barenblatt (1993), both in modehg the velocity data and

prediction of skin friction,

4. The effect of surface roughness on the mean velocity profiles extends to the outer

edge of the flow and inmases the value of the wake parameter ïI over the

corresponding smooth wail data The ïI values ais0 depend on the specific

geometry of the roughness elements.

5. The effects of surface roughness on the turbulence quantities such as the

Reynolds stresses, triple wrrelations, and the distributions of the energy budgets,

eddy viswsity and mixing length are not confmed to the roughness sublayer as

implied by the wail sirnilarity hypothesis. Instead, d a c e roughness modifies

the turbulence structure outside the mughness subalyer in a way that depends on

the W f i c geometry of the roughness elements.

6. The present resuits show that surface roughness substantiaily increases the peak

values of the wall-normal wmponent of the turbuIence fluctuations and the

Reynolds shear stress. Furthermore, the stress tensor and the turbulence difiùsion

term in the turbulence kinetic energy equation are significantly modined by the

specific geomeûy of the roughness elements. These tindings suggest that for

rough wail turbulence models to be able to predict the transport properties

accurately, they mut exphcitly accoitat for the specifîc geometry of the

roughness elemenîs.

7.2.2 Turbuient Waü Jets

1. The spread of the jet half-width and the decay of the maximum velocity for the

turbulent w d jets show distinct Reynolds number dependence when they are

scaled using the dot height and exit velocity. However, the spread and decay

rates are independent of Reynolds number when kinematic mornentum is used as

the appropriate scaling parameter. This observation supports the use of the

kinematic momentum for scaling the streamwise evolution of a turbulent wall jet.

2. nie spread rates for the jet haif-width obtained in the present experiments are

significantly higher than the values reported in previous measurements.

3. The skin fiction coefficient as weIl as inner thickness and maximum velocity

decay are larger for rough surfaces cornpanxi to a smooth d a c e . However, the

spread rate for the jet half-width is independent of wall roughness.

4. The present smooth w d data show a striking sirnilarity between a turbulent

boundary layer and a turbuient wall jet in the inner region. The mean velocity

profiles are weii modeled by a Iogarithmic law with universal constants that are

identical to values used in boimdary layer analysis. The distributions of mixing

length and eddy viscosity in the outer region are similar to classicai turbulent fiee

jet data in the literature.

5. The similarity between the uiner region of a turbulent boimdary layer and that of

a wall jet provides evidence for the appropriateness of 'wall functions' for

turbulent wail jet caidations.

6. With the exception of the streamwise wmponent of the turbulence fluctuations,

the statistics do not show any important sensitivity of d a c e roughness. This

observation suggests that the turbuIent turbulent waü jet is a complex flow in

which the mechanism of damping is not the same as in a relatively simpier

turbulent bomdary Iayer.

7.3 CONTRIBUTIONS

The present study provides additional insight into roughness effects on low Reynolds

number turbulent boundary layers. The major contriiutions of this study are summarized

as follows:

1. A compIete and comprehensive set of rough wall measurements in open channel

trrrbulent boundary layers.

2. The fïrst study to apply the power laws proposed Barenblatt (1993) and George

and Castillo (1997) to rough wall turbulent boundary layers.

3. The 6rst comprehensive study of turbulent waii jets on a rough surfka.

7.4 RECOMMENDATIONS FOR FUTURE WORK

On the basis of the above conclusions and ouf current understanding of near-waII

turbulent structure, the foliowùig ~e~0rnmendati0~1~ are relevant for fiitirre work

1. The usefulness of the power formulations to mode1 the mean velocity depends on

one's sbiiity to accurately determine the pwer law constants. In this regard,

additional theoretical anaiysis and renaed measurements at higher Reynolds

numbers and with larger roughness elemenîs are required to cali'brate the pwer

Iaw constants.

2, Application of very high spatial resolution LDA systems and multi-point devices

such as PIV wodd be us& to explore the turbulence sîructwe in the immdate

vicinity of the mughness element, and also the free surbce region of an open

Channel flow.

3. Investigation of a turbuient w d jet with varying roughness effects and

fkstrearn turbulence Uitensities will provide further insight into the interaction

between the inner and outer Iayers.

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AN OVERVIEW OF A LDA

hi this section, an overview of the various components of a LDA system is

presented. The pararneters and condition that deserve particular considerations for

accurate flow measurements are aIso indicated,

A.1 Laser Source

Laser light bas some attractive characteristics that make it very suitable for LDA

application. A laser beam is both monochromatic, i.e. its energy is concentrated in

extremely narrow bandwidth, and coherent wfiich means al1 emitted radiation has the

same phase, both in space and in t h e . A laser beam is also highiy coIlimated and has a

plane of minimal cross section referred to as the 'waist' of the beam. The light intensity

across the beam has a Gaussiaa distribution. The beam diameter is defined as the

distance between points wbere the iight intensity has dropped to e*2 of the maximum

value. In selecting a laser source for application, important consideration is given to

power requirements, cost and the Iight t'requency. The light fiequency is particuiarly

important because it affects the h g e spacing, the photo-detector quantum efficiency

and scattering h m partictes. The amount of power scattered defines the signal level and

the signai-to-noise ratio (SNR). The beam diameter is also an important parameter

because it influences the measuring voIume dimensions.

A.2 Transmitîing Optics and Meauring Volume

The transmitting optics consists of a beam splitter, a Bragg ceii and a convergent

lem or mirror. It may also contain filters, polarizers, h m path equalizers and a beam

expander- For a dual beam system, which is the arrangement used in the present study,

an intense, highly collimateci light beam h m the faser is divideci mto two coherent

beams of equal power by the beam spiitter. As is weU known, the Doppler fiequency is

not dependent on the sign of the direction of the veiocity. Therefore, a positive and a

negative velocity of the same magnitude wiil result in the same Doppler shifi. To

remove directionai ambiguity, the laser beams are fiequency shifted by Bragg celIs. The

two coherent beams are transmitted through a focussing lem and directed into the flow.

Ideaily, the two beams shouid intersect at the focal point of the Iens, Le. their respective

beam waists. At the point of intersection, the two beams give rise to an interference

pattern or h g e s . The spatial region h m which measurements are obtained is

essentially the intersection of these beams. This region is referred to as the measuring

volume. The m&ng volume is defmed by the Iocus of e-' intensity points and is

ellipsoidal in shape. The dimensions of the control volume depend on the wavelength A

of the laser beam and the optic parameters. The number of ûinges and the f i g e spacing

within the measuring volume depend on the opticai parameters and the size of the

measuring volume.

A 3 The RceeMng Optics and Photodetoctors

When a scattering particle passes through a measuring volume, it scatters or

reflects the incident iight. The scattered iight is coilected by a set of receiving lenses and

focused ont0 a photo-detector. The photo-detector utilizes die ' photoelectric effect ', i.e.

the absorption of photons and mission of elecmns to convert the opticai signai into an

electrical signal for processing. The photo-cmt is subject to several sources of noise:

shot noise which may be due to random fluctuations in the rate of collected photons and

background illumination, and electronic or thema1 noise which may be due to

amplification of m e n t within the photo-detector or in ememal amplifiers. Among the

various types of photo-detectors in use cucfentiy, photo-muhipliers (PM) yield the best

signai-to-noise ratio (SM) because of theh nearly noise-h intemal amplification

(Durst and Sender,L990). Acwrding to the analysis of Durst and Sender (1990), PM is

most suitable for the range of veiocities considerd in the present study. The present

systern uses a photo-multiplier (PM).

A.4 Signai Processing Systems

The selectiou of a signal pcessor for fluid flow measuremaits depends on the

type of signal generated, for exampie, high or Iow partick density and aiso on flow

information desired. The appearance of the photo-detector output signal depends on the

collected light intensity, on the number of particles crossing the rneasuring volume at

any one time and aiso on the scattering cbracteristics of the particies. At extremely low

particle density, the signai consists of a train of pulses corresponding to individual

collected particles. In this case, special techniques are required to recover the Doppler

hpency . On the other hand, if the particle density is extremely high so that many

particles are present in the measurhg volume at any tirne, the DoppIer signai is

continuous but its phase and amplitude would Vary randomiy. This randomness

intmduces an additional enor in the Doppler fiequency measurement, called 'ambiguity

noise'. If the particle density is sufficientiy large to provide quasi-continuous signai but

low enough for the measuring volume to contain at most a single scattering particle at

any t h e , the signai received by the photo-detector will consist of a series of 'bursts'

corresponding to particle crossing. Each bwst can be viewed as an amplitude-modulated

sinusoidal bct ion of fiequency fi. The amplitude modulation depends on the light

intensity variation within the measuring volume, while differences between bursts refiect

differences between particle sizes and crossing paths. A particle passing through the

meamhg volume will cross a certain number of h g e s per unit tirne. Using a suitable

signal processor, the signals are processed for the determination of the Doppler

frerluency. If the ftequency fD of f i g e crosshg is known, the velocity of the particle is

given by

where C is a calikation factor which depends on the optical parameters. Eqn (A.1)

shows a linear velocity-fiequency relationship.

AS Soding Scattering particles are the basic source of the Doppler signd. The particks may

typicdy be 0.1 to IO pn in diameter. According to Adrian (1983), the scattering

particles have more influence on the quality of the signai than any other component of

the system. For example, the signal süength can be increased by 10' to 10' by increaskg

the paj?icle diameter h m several tenths of a micron to severaI microns. Improvements

of these ordm of magnitude are difficult, expensive, or pertiaps, impossible to achieve

by increasing the laser power or otherwise improving the opticai systern.

The velocity measured by the LDA system is that of the scattering particle.

Therefore ody if the scattering particles faithfully follow any changes of the flow

velocity can one expect the rneasurernents to yield veiocity data that accurately represent

the flow velocity. If the scatterhg particles are too large or if their density is too high

then, as a r d t of inertia, they may not respond to velocity changes sufficiently rapidly.

The aerodynamic size of a scattering particle, which is a masure of its ability to

faithfulIy follow the flow, is one of the most important properties of an individual

scattering particle. The signal-to-noise-ratio (SNR) that it produces is aiso important. A

hi& S N R requires that the particle is an effective scatterer. The concentration and

uniformity of the particle population also play important roles. Ideally, particles that

have the same density as the Buid, large effective area in regard to scattering power,

very uniforni properties fiom one particle to the other, easily controlled concentration,

and Iow expense are desirable. For liquid flows, the velocities are usually maIl and the

primary limitation on particle size cornes fiom the settling velocity rather than the abiliq

to follow the flow. In water flows, naturally occirrring hydroso1s are convenient and

o h yield satisfactory results.

An ideaI system considers a single particle in the rneastrring volume at any one

tirne. In densely seeded flows, with the receiving optics configured in backscatter mode,

validation of Doppler signal fiom multiple particles within the m&g volume may

contaminate the accuracy of flow measurements. This is particularly true if the

measuring volume is long. If the particle concentration is low, the streamwise and

vertical velocity fluctuations would be independent of the spanwise extent of the LDA

measuring volume. Another positive side of relatively low &ta rate is that it is unlikely

to rneasure muitiple particles in one time window.

APPENDIX B

ERRORS IN LDA MEASUREMENTS

In this section, some of the common errors encountered in LDA measurements are

discussed. Procedures required to correct or minirnize these errors are also outlined.

Preliminary experiments conducted to verifj some of the m r s are also reported and

discussed.

B.l Velocity Bias A burst mode or individual realizatioa LDA operates on signals generated by

single particles passing through the measurement volume and produces measurement of

the velocity of the particle whle it is in the conmi volume. During periods of relatively

high velocity, more particles are measured per unit t h e than in periods of relatively low

velocity. ï h e arrival rate of the measurable particle is, in general, not statistically

independent of the flow velocity which brings hem to the measurernent volume

(Edward, 1987). If the flow statistics are calculated by simpiy suniming the velocities of

al1 the measured particles and dividing by the nimiber of particles, i.e. particle averaging,

the statistics may be seriously biased.

In turbulent flows, veIocity bias occurs when the particle measurement rate, f, is

correlateci to the magnitude of the insbntaneous velocity vector Ui at a point in the flow

field (McLaughlin and Tiderman, 1973). Many corrections and sampling strategies have

been proposed to eliminate velocity bias (e-g. McLaughlin and Tiderman, 1973; Bamett

and Bently, 1974; Buchave, 1975; Stevenson and Thompson, 1982). Attempts have also

been made by a number of researchers to experimentaIIy ver@ some of these analyticd

studies. The resuits obtained are inconclusive as to the magnitude and even the existence

of velocity bias. These experimental resuits nonHithstanding, many LDA users routinely

correct for velocity bias. It shouid, however, be noted that if bias does not occur, such

routine bias coc~ections couid Iead to signifiant enors.

Giel and Barnett (1978) experimentdy examined statistical bias in a confined

subsonic air jet. They compared velocity parameters obtained by averaging individual

realization laser velocimeter data to measurements obtained ushg pitot-tube and hot-

wire probes. It was concluded that no consistent bias exists. Stevenson et al. ( 1982)

reviewed most of the existing experimental studies. They also reported measurements in

a rearward facing step at various locations characterized by different levels of

turbulence. For Iow turbulence level (of the order of 1 percent), the results obtained

using ensemble average or particle averaging was found to be identical to time average

values. At higher turbulence intensity (25 and 35 percent), mean veIocity obtained using

particle averaging was significantly biased at low particle mival rates. Adams and Eaton

(1988) made measurements in a backward-facing step using D A , pulsed hot-wire and

thermal tuR They concluded that depending on the particle rate and signai-to-noise

ratio, the particle average might not be biased. It was remarked that in such a situation,

the use of bias-eliniination algorithm such as the I-D and 2-D McLaughlin-Tiderman

correction or residence-time weighting scheme would result in 'over-correction' of the

bis . in processing th& LDA data, Adams and Eaton (1988) used three different

sarnpling schemes including particle averaging. Their results show that the data obtained

using particle averaging comparai most favorably with those obtained using the thermal

tuft,

in view of the above discussions, preIiminary experiments were conducted to

examine if serious velocity bias exists. The data were analyzed using the following three

different sampluig aigorithms.

1. Unweighted or particle averaging (PA)

2. Residence-tirne weighting (RT)

3. inter-arrivai time weighting (IT).

The results obtained for one set of experiments are shom in Figure B,1 and B.2.

in order to quanti@ the differences among the thtee data sets, the standard deuiation was

computed at each measuremeat Iocation, Figure B-la shows the distn'butions of the

streamwise component of the mean velocity. As this figure and the subsequent plots

show, the individual profiles compared with each other reasonably well. It should also

be noted that in most cases, the data obtained using particle averaging (PA) lie between

those obtained using residence-the weighting (RT) and inter-arrivai time weighting

(17'). Compared to the ûeestream velocity (U,), the variation among the three sets of data

varies fiom 0.9 percent in the vicinity of the wail to 0.03 percent at the outer edge of the

flow. Figure B.lb shows the distn'butions of the vertical component of the mean

velocity, V. The maximum and minimum variations among the data sets are 0.026V-

close to the wall and 0.003V- in the outer region.

Figure B.lc and B.ld show plots of the streamwise (u) and vertical (v)

components of the turbulence intensity. Compared to the peak values, the standard

deviation among the three sets of data varies from 1.6 percent (close to the wall) to 0.2

percent (near the free surface) for u and h m 1.8 percent (close to the wall) to 0.4

percent (near the free suffice) for v.

Figure B.2a and B.2b show dirn%utions of the Reynolds shear stress and stress

correlation coefficient, respectively. Close to the wall, the Reynolds shear stress data

sets agree to within 4.8 percent of the peak value and 0-1 percent of the peak value in the

outer part of the flow. The maximum variation observeci in the correlation coefficient

was 4.1 percent of the peak value. The sirearnwise and verticaI skewness distn'butions

are shown in Figure B.2c and B.24 respectively. The standard deviation at each y-

location is compareci with the Gaussian value of 3. The deviations were generally less

than 5 percent and 10 percent, respectively for the streamwise and v d c a l components.

For each turbulence statistics discussed above, the maximum deviation among the three

sets of data is comparabIe to the correspondmg statistical uncertainty estimates shown in

Appendix C.

Fig. B 1 : Data processing using different sampiing schernes (a), (b) mean veIocity profdes (c), (d) turbulence fluctuations h m vaious samphg schemes

Fig. 6.2: Data processing using diffezent sampling schemes (a) Reynolds shear stress (b) shear stress correlation (c), (d) skewness factors

B.2 Multipb Particies in Mersuring Volume

in practice more than one particle may be present in the measuring volume so the

photo-detector u d l y receives light scattered h m particles distributed throughout the

illuminating beams. In denseIy seeded flows or in a long tneasuring volume, the

probabiIity of the presence of muitiple particles within the measuring volume is high.

When multiple particles are present in the measuring volume, Doppler signai on the U

and V channel may be validated simu~taneously but may not corne tiom the same

particle. This may cause the Reynolds shear stress to be tmderestimated.

Johnson and BarIow (1990) investigated the effect of spanwise dimension 1; (=

I,U&) on two-component LDA rneasltremeuts in a turbulent boundary layer at Ree =

1440. The spanwise dimensions considered were in the range 7 S 1,' 5 44 and the

sampling rate was set to 25 Hz. The measurements were compared to the DNS d t s of

Spalart at Ree = 1410. They concludeci that the streamwise component of mean velocity

as well as stremwise and vertical components of velocity fluctuations is nearly

independent of the spanwise dimension. it was observed that the Reynolds shear stress

decreases as 1; inmases. The strongest dependence was observed nearest to the walI (y'

< 10) where the values obtained using 1,- = 6.7 and 43.6 were found to be 30 and 50

percent, respectiveiy, lower than the DNS results of Spalart (1988). At y = 38 and 71,

the probe with 1; = 43.6 gave values that were 12 percent lower than obtained h m 1; =

6.7. They recommended that accurate rne~surement of the Reynolds shear stress requires

a spanwise extent of the measirring volume to be less than 15 viscous units.

B3 Gradient Broadeaing

Due to the finite site of the measrning volume, LDA data are not really point

rneasurements but Mtegrated in space over the measuring voIume. Finite volume size

may came large velocity gradients and rnay dso present difficuity in accuratety locating

the wall (y = O). lfthere exists a non-negligiik m m velocity gradient in the measuring

volume, the r d t i n g probability fuLlction wiIl be broadened and skewed. As a resuit,

the-averaged turbdence properties, especidy in the viçinity of the w d , will show a

dependence of the measuring volume size and require volume correction. ûurst et ai.

(1 995, t 998) and Eriksson et al. (1999) discussed the effects of finite measuring volume. -

The effects of finite measuring voIume on measured mean and turbulence

quantities were discussed by Durst et al. (1995, 1998) and Eriksson et al. (1999). They

developed the following correction foxmulas

where i is the i& velocity component; subscript n and O represent the quantity actuaily

measured and the corrected data, respectively; dm denotes the probe volume in the

vertical direction, and HOT = higher order terms. The above expressions show that

correction for the mean velocity depends on the second derivative of the mean velocity

while the correction for the turbulence intensity is proportional to the gradient of the

mean velocity. In view of the linearity of the instanmeous streamwise velocity in the

viscous sublayer, it folIows h m Eqn. (3.3) that the meamWise turbulence intensity

when normaiized by the local mean velocity has the foiiowing iîmiting behavior

Eriksson et ai. (1999) concIuded h m their wall jet data (dm- = 1. L) that outside y = 6,

gradient broadening can be neglected The near-walI measurements reported by Durst et

al. (1995) reveaied that the effect of measuring voIume on the streamwise turbulence

intensity is neg2igibIe for y 2 2.5.

B.4 Errors due to noise

Emneous contribution to LDA measurements may be due to electronic noIse

resuiting fiom signal processing equipment as well as light scattered fiom mail

impurities on the solid wall and test windows. Errors due to noise have been treated by

Durst et ai. (1995) and Eriksson et al. (1999). The near-wall data of Durst et al. (1995)

showed that the effect of electronic noise is negligiile for y' 2 2, Eriksson et al. (1999)

made a systematic investigation of system error in their walI jet measurements. They

found that the effect of system error on vertical component of turbulence fluctuations is

about 10 percent at y = 4, becoming larger as the wail is approached. The influence of

system noise on the Reynolds shear stress, however, was found to be negligible. Error

due to extraneou sources such as impurities on solid walls and test windows are

difficult to quanti@.

B.5 Nonsrthogonality

In LDA measurements, it is a common practice to tilt or pitch the fiber-optic probe

towards the wail in order to obtain velocity data closer to the wail (Karlsson et al., 1993.

Swain and Schultz, 1999). When the probe is pitched towards the wail at an angle PT the

measured vertical component will be tilted relative to the nomai of the wail. This may

contaminate the velocity data, especially if P is large. Karlsson et al. (1993) discussed

these and other effects and developed formulas to correct quantities actuaily measured. lt

was shown that for $ S 3.3', the vaiue of the vertical component of turbulence

fluctuation is artificially increased by 6 percent at y& = 3 but the influence of tilt can be

neglected for y' 2 6. The effects of p on the streamwise component of the mean and

turbulence intensity were found to be negligiile.

In the present study, two sets of preliminary experiments were conducted to study

the effects of p on the mean velocities and their higher order moments. One of the

experiments was conducted in a smooth-waii turbuIent boimdary Iayer using a single-

component LDA, The tilt angles were in the range 0' I p I 3'. The d t s for the mean

velocity and turbulence intensity in inner coordinates are shown in Figure B.3a and

B.3b, respectively. The data for skewness and ffatness are shown in Figure B3c and

Fig. B.3: Mean and tubdence statistics in a bounday Iaya at various angles of tilt (a) mean profles (b) W e n c e intensity (c) skewness (d) flamess factor

B.34 respectively. It is observai that deviations among profles obtained at different

angies of tilt are within measurement uncertainties. It is therefore concIuded that for B I 3", the streamwise component of mean velocity, turbulence intensity as well as skewness

and flainess factors are independent of f3.

Two-çomponent measurements in a turbulent wail jet were aiso made for 0" 5 B I 5'. The results for the mean velocities (U and V) and Reynolds stresses (u v and <UV>)

are shown in Figure B.4. With the exception of the verticai component of the velocity

fluctuations, deviation among the profiles are within measurement uncertainties. It is

therefore concIuded that no significant dependence on tilt for P 1 5".

Fig. B4: (a) Mean veIocity profles at various angies of ült (b)Turbulence intensity and shear stress at various angles of tiIt

üNCERTAINTY ANALYSIS

The uncertainty d y s i s presented below are based on the methodology outiined

Kline and McC1intock (1953) and Moffat (1988). A 95 percent confidence interval is

assumeci in the following analysis. The main source of error in LDA measurements is

the uncertainty in the beam spacing calculation or how accurately the processor cm

deduce the fiequency present in each burst. This, in tum, depends on an accurate

determination of the bearn-crossing angle vanta and Smith, 1973; Schwarz et al., 1999)

and the signal-to-noise ratio (Castro, 1986). in addition to the above considerations, the

uncertainty in statisticai quantities will also depend on both the sample size (N) and rms

Ievei.

In the present analysis, consideration is given to the following

1. Except in the immediate vicinity of a solid wall, a photomultiplier has very hi&

signai-to-noise-ratio (SNR).

2. The DoppIer signal is band-pass filtered to remove the pedestal (non-oscillating part)

fiom the signal. The sensitivity to noise is further reduced by the use of a three IeveI

detection scheme.

3. Siringent validation levels are used in course of data acquisition to reject spurious

data

4- The f i g e bias angle is expected to be minimal by the application of a frequency

shift of 40MHz (Durst et al., 1993).

A methodology for estimating uncertainty in LDA measurements was developed

by Yanta and Smith (1973) and Schwarz et al. (1999). They derived the following

relations for the aeamwise and vertical components of the mean velocity, respectively:

The correspondhg expressions for the streamwise and vertical components of

turbulence fluctuations and the Reynolds shear stress are, respectively, given by:

where co is the m r due to uncertainty in the determination of the beam-crossing angle,

N is the number of sarnples and R is the shear stress correlation coefficient.

Following Schwarz et ai. (1999), a value of 6, = 0.4 percent is adopted in the

present analysis. Typicai estimates of uncertainties for the mean and fluctuating

quantities are given in Table C.1 using the test conditions for Test C-SMH. The values

simimarized in this table are very similar to those reporteci by Schwarz et ai. (1999) in

their boundary layer LDA measurements.

--

Table C. 1 : Typical unceRainty estimates for Test C-SMH

Region

Near-wall (y- < 15)

Overlap (y' = 60)

h t e r (yt = 750)

In the i ~ e r region of the waii jet, the errer estimates are comparable to those

reporteci in Table C. 1. However, the u n c d t i e s in U, V and <UV> in the outer edge of

the wall jet are wnsiderably higher due to the high local turbulence levels and Iowa

U (%)

0.5

0.4

0.4

V(%)

0.4

0.4

u(%)

0.8

0.7

0.6

v(%)

0.6

0.5

<UV>(%)

12.0

2.5

data rates. TypicaI uocertainty esthnates in the outer region of the wai1 jet are as follows:

f 2.5 percent for U and V; I 5 to 10 percent for u, v, and <UV>.

+ 5 %

2 2.5 % for a smooth surface

k 5 - IO % for a rough surface

1 Triple pmducts 1 f 10 % 1 1

su, sV J f 10% I

Fu, L l t 1 5 %

Table C.2: Typical uncertainty estimates

It should be pointed out that the uncertainty estimates summarized in Table C. i do

not consider mors due to electronic noise, The signal-to-noise-ratio is expected to

decrease as the wall is approached because of the decrease of the velocity and aiso due

to extramous refiection h m the w d . This would in turn increase the uncertainty in the

turbulence statistics in the vicinity of the wall. Ching et ai. (1995) obtained repeated

measurements at a given y+ in the near-wail region. Their results showed that for y' <

15, the ucertainty in u and v are, mpdve ly , f 4 and 9 percent. Typical estimates for

other parameters at 95 percent coatidence ievel are Summanzed m Table C.2.