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Fully Developed Turbulent Smooth andRough Channel and Pipe Flows

Der Technischen Fakultat derUniversitat Erlangen-Nurnberg

zur Erlangung des Grades

DOKTOR-INGENIEUR

vorgelegt von

Osama Abdelsattar Bayoumy Saleh

Erlangen 2005

Als Dissertation genehmigt von derTechnischen Fakultat derUniversitat Erlangen-Nurnberg

Tag der Einreichung : 20.06-2005Tag der Promotion: 26.08-2005

Dekan: Prof. Dr.-Ing. A. WinnackerBerichterstatter: Prof. Dr. Dr. h.c. F. Durst

Prof. Dr. Ch. Egbers

To my mother, my wife, my children,the memory of my father,and to my Professors

ABSTRACT

In this thesis, fully developed, turbulent channel flows with smooth walls, withone rough and one smooth wall and with two rough walls, were studied, providingsome useful and extended information about these kinds of flows.

Fully developed, turbulent plane smooth channel flows were studied theo-retically and experimentally. A Reynolds number range up to 1.1 ×105 wascovered. The value of the von Karman constant, κ, was found to be 1/e forthis kind of flow, confirming the recent findings of Zanoun and Durst [2003].Similarity conditions between channel flows with smooth surfaces, which possessdifferent dimensions, were investigated theoretically. Experiments were carriedout to check the theoretical findings and very good agreement was obtained.

Another objective of the present work concerned fully developed, turbu-lent plane channel flows with one rough and one smooth wall. This kind offlow shows interesting properties that have not been studied systematically, inspite of the availability of suitable experimental and numerical means. Themaximum velocity lies off-centre, closer to the smooth wall, and its location is notidentical to that of zero shear stress of the flow. Furthermore, the mean velocitydistribution close to the rough wall can only be plotted in wall coordinates if thewall location for zero shear stress could be defined. Methods to do this are givenutilising a two-component LDA system that was employed to measure the meanvelocity profiles and the Reynolds stress profiles for exact determination of thecorresponding position of the zero shear stress. Using the available theoreticaland experimental facilities, the effective height of the channel was determined aswell as the reference point, which is the middle of the channel, for the velocitymeasurements.

Fully developed, turbulent plane channel flows with rough surfaces werealso studied in some depth. Previously, to calculate the shift between the smoothlog law lines and the rough ones, one needed information about the height ofthe employed roughness. Without roughness height information, theoreticalinvestigations were carried out to calculate the shift between the smooth log lawline and the rough ones. The procedure was confirmed through correspondingexperimental measurements. Furthermore, a new method to measure the staticpressure in the rough regime was successfully introduced using Pitot tubes.

Finally, some of the theoretical findings were confirmed by the data ofNikuradse, available in the literature, for smooth and the rough pipe flows.

ACKNOWLEDGEMENTS

Praise to Allah, The Cherisher and Sustainer of the Worlds; Most Gracious,Most Merciful; Master of the Day of Judgment.

I would like to express my deep appreciation to Prof. Dr. Dr. h.c. F.Durst for his irreplaceable encouragement, good humor and support to conductmy Ph.D. work at LSTM-Erlangen. I am grateful to him for his excellentmonitoring of the study through numerous discussions and useful advice.

I am also appreciative to Dipl.-Ing. Ozgur Ertunc, the head of our group,for valuable discussions and suggestions during the different phases of my work.

I wish also to express my sincere gratitude to Dr. E. Zanoun and Dr. A.Nassar for their continuous help and useful advice during the entire researchtime I spent at LSTM-Erlanegn.

I would like also to thank the members of the thesis committee, Prof. Dr.E. Schlucker, Prof. Dr. Ch. Egbers and Prof. Dr. H. Munstedt.

I appreciate the help received from the workshops, in particular from Mr.Pavlik, who built the water channel, Mr. Heubeck, who prepared the airchannels, Mr. Horst, for preparing the LDA system and Mr. Zech, for providingthe pressure transducers. Special thanks are due to all members of the secretarialoffice, Mrs. Paulus, Mrs. Wagner and Mrs. Grasser, and to the adminstrationoffice, in particular Dr. Mohr.

I am very grateful to Dr. Rauf, Dr. Merimeche, M.Sc. Matti, M.Sc.Haddad, M.Sc. Alhamamrah, Dipl.-Ing. Epple, M.Sc. Lemouedda, M.Sc. Unsal,Dipl.-Ing. Sahiti and M.Sc. Koksoy for helpful discussions and detailed assis-tance. Also, many thanks are due to Dr. Al-Salaymeh and Dr. Ray for their help.

Thanks are also due to my colleagues Dipl.-Ing. Lienhart, Dr.-Ing. Asoud,Dipl.-Ing. Abu-Sharkh, Dr.-Ing. Bakry, Dipl.-Ing. Nishi and Dr.-Ing. Avdic fortheir help.

The author gratefully acknowledges the financial support received fromthe Egyption government and also that from LSTM-Erlangen, UniversitatErlangen-Nurnberg. The test section installation received support through theUniversitatbund Erlangen-Nurnberg.

I must offer special thanks to my mother for graciously instilling in methe value of education and supporting me throughout my education. Finally,

v

thanks are due to my wife and my children for being behind me at all time.

NOMENCLATURE

a Overheat ratioA ConstantA/D Analog to digital converterB Constant/channel widthBSA Burst spectrum analyzerc, C, c

′

Constantscf Wall skin frictioncp Specific heat at constant pressured Pipe diameterdf Fringe distanceD, D1 Constantse Base of natural logarithmEcorr. Corrected output of HWAE, Emeas. Measured output of HWAf Focal length/frequencyfD Doppler frequencyg Gravitational accelerationh Half of the channel heightH Channel full heightk Roughness heightks Height of the sand roughnessk+

s Roughness Reynolds numberl Wire lengthL Length of channel test sectionP Pressure forcePatm Atmospheric pressurePst Static pressureP0 Stagnation pressurer Pipe radiusRa Wire resistance at ambient temperatureRw Hot-wire working resistance

vii

t TimeT TemperatureTa Air temperatureTref Reference temperatureTmeas Measured temperatureTwire Wire temperatureu, U Streamwise mean velocityUi Cartesian velocity componentsUmax Maximum velocityu

′

Streamwise velocity fluctuationsuτ Wall friction velocityv

′

Velocity fluctuations in normal directionxi Cartesian coordinatesx Streamwise distancey Normal wall distancez Spanwise distance

viii

Subscripts

r, R Roughs, S Smoothw Wall

Superscripts

∗ Nondimensional quantity+ In wall coordinates

Greek letters

κ von Karman constant/specific heat ratioλ Friction factorλw Thermal conductivityρ Fluid densityµ Dynamic viscosityν Kinematic viscosityθ Angleτ Mean wall shear stressδij Kronecker delta∆B The shift between the log law lines of the

smooth and the rough walls∆U+ Roughness functionγ Exponent of power lawε Distance between the wall and the position

of zero shear stress

Dimensionless Numbers

Nu Nusselt numberRe Reynolds numberReτ Friction Reynolds numberPr Prandtl number

Contents

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiACKNOWLEDEGMENTS . . . . . . . . . . . . . . . . . . . . . . . . ivNOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . viLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

1 INTRODUCTION 11.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aims of the Work and Layout of the Thesis . . . . . . . . . . . . 2

2 BACKGROUND AND LITERATURE SURVEY ON PLANECHANNEL AND PIPE FLOWS WITH SMOOTH ANDROUGH WALLS 52.1 Pioneer Investigations . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Recent Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 THEORETICAL ANALYSIS 183.1 Basic Equations for Fully Developed Channel Flows . . . . . . . . 183.2 Similarity of Smooth Channel Flows . . . . . . . . . . . . . . . . . 21

3.2.1 Relation Between the Wall Shear Stresses for the Small andthe Large Channel . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Similarity Procedure . . . . . . . . . . . . . . . . . . . . . 223.2.3 Second Condition of Similarity . . . . . . . . . . . . . . . . 24

3.3 Two-Dimensional Asymmetric Turbulent Channel Flows . . . . . 253.4 Theoretical Investigations of Rough Channel Flows . . . . . . . . 28

4 EXPERIMENTAL APPARATUS AND MEASURING TECH-NIQUES 324.1 Experimental Apparatus Used for Symmetric Channel Flow . . . 32

4.1.1 Smooth Channel Test Section . . . . . . . . . . . . . . . . 324.1.2 Fabrication and Determination of Roughness . . . . . . . . 354.1.3 Rough Channel Test Section . . . . . . . . . . . . . . . . . 35

4.2 Measuring Techniques Used in Symmetric Channel Flows . . . . . 354.2.1 Hot-Wire Anemometry (HWA) . . . . . . . . . . . . . . . 354.2.2 Pressure and Temperature for the Air-Flow Experiments . 39

CONTENTS ii

4.2.3 Wall Distance Apparatus . . . . . . . . . . . . . . . . . . . 404.2.4 New Method to Measure the Pressure Gradient in Fully

Developed Rough Wall Bounded Flows . . . . . . . . . . . 414.3 Channel Test Section with One Smooth and One Rough Wall . . 434.4 Measuring Techniques Used in Asymmetric Channel Flows . . . . 46

4.4.1 Laser-Doppler Anemometry (LDA) . . . . . . . . . . . . . 464.4.2 Determination of the Middle Point of the Channel . . . . . 50

5 RESULTS, ANALYSES AND DISCUSSION 525.1 Results of Smooth Channel Investigations . . . . . . . . . . . . . 52

5.1.1 Pressure Measurements and Wall Shear Stress . . . . . . . 525.1.2 Velocity Measurements in Smooth Channels . . . . . . . . 575.1.3 Confirmation of the Findings of Zanoun and Durst . . . . 585.1.4 Check of the Similarity Condition for Smooth Channels . . 65

5.2 Pressure Measurements and Friction Factrors in Asymmetric Flows 765.3 Velocity Measurements in Asymmetric Flows . . . . . . . . . . . . 775.4 Determination of the Effective Height of the Asymmetric Channel 815.5 Pressure Measurements and Wall Shear Stress in the Channel with

Rough Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.6 Velocity Measurements in the Channel with Rough Walls . . . . . 90

5.6.1 Checking the Value of κ in the Rough Case . . . . . . . . . 915.6.2 Checking the Validity of the Relation Between the Normal-

ized Wall Distance in the Rough and Smooth Cases . . . . 955.7 A Procedure to Utilize the Suggested Method to Recalculate the

Roughness Height . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 SUMMARY AND CONCLUSIONS 107

BIBLIOGRAPHY 109

FORTRAN PROGRAM 117

List of Figures

2.1 Universal velocity profile for turbulent flows through pipes, afterN. Scholz [1955]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Similarity of smooth channel flows. Corresponding points are shown. 253.2 Sketch of the vertical channel with one smooth and one rough surface. 273.3 Similarity of smooth and rough channel flows. . . . . . . . . . . . 30

4.1 Sketch of the channel flow test section with the temperature, pres-sure and velocity measuring equipment. . . . . . . . . . . . . . . . 33

4.2 Photographs showing the different stages of preparation of therough walls: (a) the original plate, (b) gluing the emery paperon the plate and (c) the plate to be used. . . . . . . . . . . . . . . 35

4.3 Sketch of hot-wire anemometry probe. . . . . . . . . . . . . . . . 364.4 Photograph of position calibration set-up. . . . . . . . . . . . . . 404.5 Sketch of the new method for measuring the pressure difference. . 424.6 Comparison between the pressure gradient measured using Pitot

tube and that using pressure taps. . . . . . . . . . . . . . . . . . . 424.7 Sketch of the water channel test section showing the measuring

equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.8 Photograph of the experiment showing the test facility and the

measuring techniques. . . . . . . . . . . . . . . . . . . . . . . . . 454.9 Fringe spacing of the LDA control volume. . . . . . . . . . . . . . 474.10 Doppler frequency to velocity transfer function for a frequency-

shifted LDA system. . . . . . . . . . . . . . . . . . . . . . . . . . 494.11 Sketch of the equipment used to determine the middle of the channel. 504.12 Relation between the position of the laser beam and the corre-

sponding voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Pressure gradient distribution along part of (a) the small channeland (b) the large channel at different Reynolds numbers. . . . . . 53

5.2 Comparison between the measured pressure gradient in the smalland large channels for different Remean. . . . . . . . . . . . . . . . 54

LIST OF FIGURES iv

5.3 Comparison between the wall shear stress in the small and largechannels, for different Remean. . . . . . . . . . . . . . . . . . . . . 55

5.4 Relation between the wall shear stress in the small and largesmooth channels at corresponding Reynolds numbers. . . . . . . . 56

5.5 Nikuradse’s data: comparison between the wall shear stress inpipes with different diameters. . . . . . . . . . . . . . . . . . . . . 57

5.6 Samples of the velocity profile for the small smooth channel. . . . 585.7 Samples of the velocity profile for the large smooth channel. . . . 595.8 Velocity distribution in the small channel over a wide range of

Reynolds numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 605.9 Velocity distribution in the large channel over a wide range of

Reynolds numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 615.10 Velocity gradient in the small channel over a range of Reynolds

numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.11 Velocity gradient in the large channel over a range of Reynolds

numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.12 Log-law diagnostic function, Ξ, for the law of the wall in a small

2-D plane channel flow. . . . . . . . . . . . . . . . . . . . . . . . . 645.13 Log-law diagnostic function, Ξ, for the law of the wall in a large

2-D plane channel flow. . . . . . . . . . . . . . . . . . . . . . . . . 655.14 Power-law diagnostic function, Γ, for the law of the wall in a small

2-D plane channel flow. . . . . . . . . . . . . . . . . . . . . . . . . 665.15 Power-law diagnostic function, Γ, for the law of the wall in a large

2-D plane channel flow. . . . . . . . . . . . . . . . . . . . . . . . . 675.16 ln(du+/dy+) − ln y+, at Reynolds number 15822. . . . . . . . . . 685.17 ln(du+/dy+) − ln y+, at Reynolds number 19264 . . . . . . . . . . 695.18 ln(du+/dy+) − ln y+ representation of the mean velocity gradient

over the range of Reynolds numbers in the small channel flow. . . 705.19 ln(du+/dy+) − ln y+ representation of the mean velocity gradient

over the range of Reynolds numbers in the large channel flow. . . 705.20 Similarity between the small and large channels at Re = 58000. . 715.21 Similarity between the small and large channels at Re = 68000. . 715.22 Similarity between the small and large channels at Re = 31000. . 725.23 Similarity between the small and large channels at Re = 37000. . 725.24 Similarity between the small and large channels at Re = 47000. . 735.25 Relation of Reτ y/H versus U+ for a wide range of Re. . . . . . . 735.26 Nikuradse’s data: similarity between the small and large pipes, Re

= 4000 and Re = 105000, respectively. . . . . . . . . . . . . . . . 745.27 Nikuradse’s data: similarity between the small and large pipes, Re

= 105000 and Re = 725000, respectively. . . . . . . . . . . . . . . 745.28 Nikuradse’s data: relation of Reτ versus U+ for a wide range of Re. 755.29 Pressure gradient distribution along the channel with rough surface

at different Reynolds numbers. . . . . . . . . . . . . . . . . . . . . 76

LIST OF FIGURES v

5.30 Estimated pressure gradient at different Reynolds numbers for thepresent flow facility. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.31 Velocity profiles at different Reynolds numbers. . . . . . . . . . . 785.32 The total shear stress versus the position, Re=44356. . . . . . . . 795.33 Position of zero total shear stress at different Reynolds numbers. . 795.34 Wall shear stress in both the smooth and rough sides at different

Reynolds numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 805.35 Uncorrected and corrected values of dU+/dy+ after the iteration,

Re = 44356. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.36 Wall shear stress for the smooth and rough cases. . . . . . . . . . 845.37 ∆H+ variance with Reynolds number. . . . . . . . . . . . . . . . 855.38 Pressure gradient determination along the rough channel presented

for different Reynolds numbers. . . . . . . . . . . . . . . . . . . . 865.39 Comparison between the measured pressure gradient for the

smooth and rough channel flows. . . . . . . . . . . . . . . . . . . 875.40 Comparison between the wall shear stresses for the smooth and

rough channel flows. . . . . . . . . . . . . . . . . . . . . . . . . . 885.41 Comparison between the calculated (Equation 5.24) and the mea-

sured Darcy friction factors for the rough channel. . . . . . . . . . 885.42 Nikuradse’s rough data: comparison between the calculated (Equa-

tion 5.24) and the measured Darcy friction factors. . . . . . . . . 895.43 Samples of the velocity profile for the channel with rough walls. . 905.44 Velocity distribution in the rough channel over a range of Reynolds

numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.45 Diagnostic function Ξ of the law of the wall in 2-D rough channel

flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.46 Diagnostic function Γ of the law of the wall in 2-D rough channel

flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.47 ln(du+/dy+) − ln y+, at Re=15021. . . . . . . . . . . . . . . . . . 945.48 ln(du+/dy+) − ln y+, at Re=60150. . . . . . . . . . . . . . . . . . 955.49 Experimental values of the constant D at different values of

√

τwr

τws

. 96

5.50 Comparison between the log-law in the case of a smooth channelat Re = 69149 and a rough channel at Re = 60122. . . . . . . . . 99




5.54 Comparison between the calculated and the measured values of ∆B.1015.55 The behavior of ∆B at different values of U+. . . . . . . . . . . . 102

LIST OF FIGURES vi

5.56 Nikuradse’s data: values of the constant D at different values of√

τwr

τws

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.57 Nikuradse’s data: comparison between the log-law in the case of asmooth pipe at Re = 9200 and a rough pipe at Re = 680000. . . . 104

5.58 Nikuradse’s data: comparison between the log-law in the case ofsmooth and rough pipes. . . . . . . . . . . . . . . . . . . . . . . . 104

5.59 Nikuradse’s data: cComparison between the log-law in the case ofa smooth pipe at Re = 725000 and a rough pipe at Re = 427000. 105

1 Sketch of the Fortran program. . . . . . . . . . . . . . . . . . . . 117

Chapter 1

INTRODUCTION

1.1 State of the Art

The research work summarized in this thesis is a continuation of the fluidmechanics studies carried out at the Institute of Fluid Mechanics of theFriedrich-Alexander University Erlangen-Nuremberg into fully developed, tur-bulent channel flows. Extensive studies were carried out and presented inrecent publications by Zanoun et al. [2003] and Durst et al. [2003]. Theinvestigated flows were those generated in channels with hydrodynamicallysmooth surfaces. In a recent publication, Durst et al. [2005] analyzed all theexisting experimental and numerical data of fully developed, turbulent, planechannel flows with respect to the general existence of the logarithmic velocityprofile for high Reynolds number flows. The logarithmic law was found to existover a range of 150 ≤ y+ ≤ 0.7 Reτ , where Reτ = h+, the channel half-heightnormalized with wall variables. A very rational method was applied to yield thevon Karman constant κ = 0.362, e.g. see Durst et al. [2003], which is in a veryclose agreement with the value published by Zanoun et al. [2003], κ = 0.37.The investigations confirmed that there are consistent sets of data for the meanvelocity of fully developed, turbulent, plane channel flows available that provideall practical mean flow information needed in engineering applications. However,this consistent information exists only for hydrodynamically smooth channelwalls, and similar consistent mean velocity information does not exist for channelflows with rough walls. The present investigations, therefore, were aimed atthe study of channel flows with smooth and rough walls. In this work, thesuggestions of Zanoun et al. [2003] and Durst et al. [2003], where they suggestedthat the logarithmic velocity profile exists over nearly 70% of the completeboundary layer adjacent to both walls starting from y+ = 100, were confirmed.

The von Karman constant, κ, in the logarithmic law was found out to beequivalent to 1/e by Zanoun et al. [2003]. In the author’s own investigations,

1.2 Aims of the Work and Layout of the Thesis 2

the findings of Zanoun et al. [2003] and Durst et al. [2005] were confirmedfor smooth channel walls. Furthermore, the similarity conditions for smoothchannels with different heights were deduced. However, the author’s main workwas concentrated on fully developed, turbulent channel flows with rough walls.It is for this reason that the present investigations were aiming for a consistentpresentation of the mean velocity profiles of all fully developed, turbulent, planechannel flows with rough walls, independent of the type of roughness employed .It is shown that κ = 1/e, found by Zanoun et al. [2003] and Durst et al. [2005],is a consistent value also for rough walls.

Starting with the smooth wall, it is shown that all the rough wall meanvelocity distributions can be expressed by a logarithmic law that is downwardshifted by ∆B with respect to the log-law of the smooth wall. ∆B is found todepend on the momentum loss of the flow to the channel walls. The relationshipexpressing ∆B as a function of the ratio (τw)rough to (τw)smooth of the samechannel is derived. To verify some of the results obtained through the theoreticalconsiderations, experimental testing and appropriate measuring equipment wereset up to study fully developed, turbulent, plane channel flows with smoothand/or rough walls. The entire test facility is described and the measurementscarried out are outlined. The resultant relationship between the normalized meanvelocity profile for smooth walls and the corresponding profile for rough wallsis explained. Both theoretical derivations and the experimental verificationspermit a good insight into the general properties of fully developed, turbulent,plane channel flows with smooth and rough walls.

Again, verification experiments were performed and results are presentedto prove the major results of theoretical considerations. Hence the present workcan be considered a closing study of the smooth channel flow work carried outat LSTM-Erlangen and as the start of rough wall channel flow investigations.

1.2 Aims of the Work and Layout of the Thesis

In wall-bounded flows, it is of some importance to consider the similarity con-ditions between different walls. Therefore, it was the aim of this work to studythe influence of the rough wall on the effective height, on the velocity profilesand also on the wall shear stress. There are some open questions regarding theeffective height of the channel under investigation according to the influence ofthe roughness, the position of the maximum velocity and the value of the wallshear stress. Also, an important question arose regarding the shift between thelog-law lines in the case of smooth and rough wall flow. Taking into accountthese considerations and also what was found in the literature survey about theposition of the maximum velocity and the total shear stress, the decision was


made to go more deeply into these kinds of flows, trying to answer some of theopen questions related to them. The aim of this work was, therefore, to establishdetailed measurements in smooth channels with different heights to deduce theexisting similarity condition between them. Also detailed measurements hadto be made on asymmetric channel flow, to estimate accurately the wall shearstress and to determine the effective height of the channel precisely. It is highlydesirable to carry out detailed velocity measurements and to determine the wallshear stress using a high-fidelity technique that measures both u

′

and v′

com-ponents at the same time. Furthermore, extensive and accurate measurementsin the case of channels with rough walls have to be conducted to estimate theshift between the logarithmic law lines independent of the height of the roughness.

Therefore, the present study was planned to address the following points:

• Checking the similarity condition between smooth channels of differentheights.

• Studying the effect of roughness on the velocity profile.

• Determining the value of the shift of the zero shear stress far from the centerof the channel.

• Determining the effective height of the channel.

• Estimating the shift between the log-law lines in the smooth and roughcases without any information about the roughness.

• Analysing the data of Nikuradse according to the new procedures.

In summary, the present study was aimed at investigating the similarity conditionbetween smooth channels with different heights, studying the effect of roughnesson the effective height and the wall shear stress and to deducing the shift betweenthe lines of the logarithmic law corresponding to the smooth and rough wallsindependent of the roughness. The main parts of the thesis are described belowwith more detail.

Chapter 2 gives a brief literature survey of fully developed channel flow inboth smooth and rough surfaces. It summarizes the work which dealt with suchkinds of flow, showing the types of roughness that exist and how they affect thedifferent flow parameters.

Chapter 3 introduces theoretical considerations and governing equationsfor the different kinds of flow as a basis for the experimental studies. It presentsalso the similarity conditions between smooth channels with different heights.The procedure for determining the effective height of the channel because of the


influence of roughness is introduced. Further, a new way to consider the shiftbetween the smooth and the rough log-law lines is discussed in details. Moreover,the new procedure followed to solve the suggested problems is presented.

Chapter 4 presents the basis of the laser-Doppler anemometry, two-componentlaser-Doppler nemometry and hot-wire anemometry as the measuring techniquesutilized to carry out the present investigations. The experimental test facilitiesand the measuring techniques are described. The chapter outlines also consid-erations on the equations applied to the present measurements and the dataprocessing.

Chapter 5 presents the results for the three main measurements: smooth,asymmetric and rough wall fully developed, turbulent, plane channel flows. Itshows the consistency of the data of Durst and Zanoun. Effects of roughnesson the velocity profile, on the maximum velocity and on the wall shear stressare also introduced. Moreover, it gives the results of the new procedures andhow they agree with literature data. Furthermore, analysis of the experimentaldata of Nikuradse on turbulent flow in smooth and in sand-grain coated pipes ispresented.

Chapter 6 summarizes the outcome of the present study and presents someconcluding remarks explaining the approach that needs to be chosen to studyflows of this kind in order to yield correctly the relative shear stresses betweenthe two walls. Reliable and new methods for determining the similarity betweenthe smooth channels, the effective height and the shift between the smooth andthe rough wall log-law lines are introduced. Finally, conclusions are drawn andsuggestions for further work are presented.

Chapter 2

BACKGROUND ANDLITERATURE SURVEY ONPLANE CHANNEL AND PIPEFLOWS WITH SMOOTH ANDROUGH WALLS

The main task of the present work was concerned rough wall flow investigations.For this reason, the literature survey will be concentrated on such kinds of flowsafter a brief introdution to smooth wall flows.

The study of wall-bounded turbulent flows is of a vital importance inpractical engineering applications such as ventilation, air conditioning systems,heat exchangers and nuclear reactor cooling systems. For instance, channel flowhas proven to be an extremely useful framework for the study of wall-boundedturbulence and for the basic understanding of internal shear flows. In addition,pipe flow is one of the most easily reproducible flows in the laboratory forfundamental research and for many industrial applications. Therefore, theseclassical types of flows have attracted the attention of a large number of fluidmechanics researchers, utilizing both experimental and numerical approaches.

A number of workers have made detailed turbulence measurements of fullydeveloped flow in a circular-sectioned pipe and in a plane symmetric channel(Laufer [1951, 1954]; Comte-Bellot [1963]; Coantic [1967]; Clark [1968]; VanThinh [1967]) with smooth walls.

Zanoun [2003] presented a good review concerning wall bounded flows ina smooth regime.

2.1 Pioneer Investigations 6

Turbulent flow in rough tubes and channels has been studied over the lasttwo centuries by many investigators. Flows of this nature can be found inengineering systems of significant technological interest such as turbine bladeinternal cooling, advanced gas-cooled nuclear reactors, heat exchangers andcooling of microelectronic devices. Other examples of great relevance havebeen pointed out by McEligot and McEligot [1994]. Turbulent flow over roughsurfaces has been a subject of increasing interest in the areas of fluid dynamicsand heat transfer in recent years. Because of the improvements in measuringtechniques, the subject of rough wall investigations has been treated extensivilyin recent years. Therefore, this literature survey will be classified into twosections, concerning old and recent investigations.

2.1 Pioneer Investigations

Darcy [1858] made comprehensive and very careful tests on 21 pipes of castiron, lead, wrought iron, asphalt-covered cast iron and glass. With the exceptionof glass, all pipes were 100 m long and 1.2 to 50 cm in diameter. He notedthat the discharge was dependent on the type of surface and on the diameterof the pipe and its inclination. According to his data, the resistance factor λfor a given relative roughness k/r varies only slightly with the Reynolds number(k is the average depth of roughness and r is the radius of the pipe; Reynoldsnumber Re= u d

ν, where u is the average velocity, d is the pipe diameter and

ν is the kinematic viscosity). The friction factor decreases with increasing theReynolds number and the rate of decrease becomes slower with greater relativeroughness. For certain roughnesses, his data indicated that the friction factorλ is independent of the Reynolds number. For a constant Reynolds number,λ increases markedly with increasing the relative roughness. Bazin [1902], afollower of Darcy, completed the latter’s work and derived from his own andDarcy’s test data an empirical equation in which the discharge is dependent onthe slope and diameter of the pipe.

Mises [1914] performed a very valuable work, treating all of the then-knowntest results from the viewpoint of similarity. He chiefly obtained, from theobservations of Darcy and Bazin with circular pipes, the following equation forthe friction factor λ in terms of the Reynolds number and the relative roughness:

λ = 0.0024 +

√

k

r+

0.3√Re

(2.1)

which is valid for high Reynolds numbers. On the other hand, for small values ofthe Reynolds number, he proposed the following form:

λ = (0.0024 +

√

k

r)(1 − 1000

Re) +

0.3√Re

√

1 − 1000

Re+

8

Re(2.2)


The term “relative roughness”, i.e. kr, was first used by Mises [1914],

where k is the absolute roughness.

Proof of similarity for flow through rough pipes was furnished by Stanton[1911], who studied pipes of two different diameters. Two intersecting threadshad been cut in the inner surfaces of the pipes. In order to obtain geometricallysimilar depths of roughness, he varied the pitch and depth of the threads indirect proportion to the diameter of the pipe. He compared for the same pipethe largest and smallest Reynolds numbers obtainable with his apparatus andthen the velocity distributions for various pipe diameters. Perfect agreementin the dimensionless velocity profiles was found for the first case, but a smalldiscrepancy appeared in the immediate vicinity of the walls for the second case.Stanton thereby proved the similarity of flow through rough tubes.

Schiller [1923] made further observations regarding the variation of thefriction factor λ with the Reynolds number and the surface type. His tests weremade with drawn brass pipes. He obtained rough surfaces in the same manneras Stanton by using threads of various depth and inclinations on the inside ofthe test pipes. The pipe diameters ranged from 8 to 21 mm. His observationsindicated that the critical Reynolds number is independent of the type of wallsurface. He further determined that for greatly roughened surfaces, the quadraticlaw of friction is effective as soon as turbulence sets in. In the case of less severelyroughened surfaces, he observed a slow increase in the friction factor with theReynolds number. However, Schiller was not able to determine whether thisincrease goes over into the quadratic law of friction for high Reynolds numbers,since the Goettingen test apparatus at that time was limited to Re ≈ 105. Hisresults also indicated that for a fixed value of Reynolds number, the frictionfactor λ increases with increasing surface roughness.

Hopf [1923] made a comprehensive review of a numerous earlier experi-mental results about roughness and found two types of roughness in relation tothe resistance equation for rough pipes and channels. The first kind of roughnesscauses a resistance which is proportional to the square of the velocity; this meansthat the coefficient of resistance is independent of the Reynolds number andcorresponds to relatively coarse and tightly spaced roughness elements such ascoarse sand grains glued on the surface, cement or rough cast iron. In such cases,the nature of the roughness can be expressed with the aid of a single roughnessparameter, the so-called relative roughness k/r, where k is the height of aprotrusion and r denotes the radius or the hyraulic radius of the cross-section.It is concluded that in this case the resistance coefficient depends on the relativeroughness only. Fromm [1923] and Fritsch [1928] carried out experiments on


pipes and channels of different hydraulic radii but of the same absolute roughnessto determine the relation between the dimensionless coefficient of resistance λand k. They found that for geometrically similar roughness, λ is proportional to(k

r)0.314. The second type of resistance equation occurs when a small number of

the protrusions is distributed over a relatively large area, such as those in woodenor commercial steel pipes. In such cases, the resistance coefficient depends onboth the Reynolds number and the relative roughness.

The variation of the velocity distribution with the type of wall surface isof great importance, in addition to the law of resistance. Observations on thisproblem were made by Darcy, Bazin and Stanton. Fritsch [1928] made suchobservations at the suggestion of von Karman, using the same type of apparatusas that of Hopf and Fromm. The channel had a length of 200.0 cm, a width of15.0 cm and a depth varying from 1.0 to 3.5 cm. A two-dimensional conditionof flow existed, therefore, along the short axis of symmetry. He investigated thevelocity distribution for smooth, corrugated (wavy), rough (floors, glass plateswith light corrugations), rough (ribbed glass) and toothed (termed saw-toothedby Fromm) wall surfaces. He found that for the same depth of channel, thevelocity distribution, except for a layer adjacent to the walls, is congruent for allof these types of surfaces if the loss of head is the same.

Tests on channels with extremely coarse roughness were made by Treer[1929], who observed both the resistance and the velocity distribution. Fromthese tests and from those of other investigators, he found that the velocitydistribution depends only on the shear stress, whether this is due to variationsin roughness or in the Reynolds number.

The numerous, and in part very painstaking, tests which were available atthat time covered many types of roughness, but all lay within small rangesof Reynolds number. A systematic, extensive and careful investigation on theeffects of Reynolds number and relative roughness on friction factor and velocitydistribution in pipe flow was performed by Nikuradse [1933]. He used circularpipes covered on the inside, as tightly as possible, with sand of a definite grainsize glued on the wall. By choosing pipes of varying diameters and by changingthe grain size, he was able to vary the relative roughness, ks/r, from about1/500 to 1/15, where ks is the height of the sand roughness. He obtainedexperimental data for six different degrees of relative roughness with Reynoldsnumbers ranging from 104 to 106. He concluded the following:

• For small Reynolds numbers, the resistance factor is the same for rough andsmooth pipes. The projections of roughening lie entirely within the laminarlayer for this range.

• An increase in the resistance factor was observed with increasing Reyn-


nolds number. The thickness of the laminar layer was of the same order ofmagnitude as that of the projections.

• For higher values of Reynolds number, the resistance factor was found tobe Re independent. All the projections of the roughening extend throughthe laminar layer and the resistance factor λ is expressed by the simpleequation

λ =1

(1.74 + 2 log rks

)2(2.3)

• The velocity distribution is given by the general expression

U+ = A + B logy

ks(2.4)

where B = 5.75 and A = 8.48.

Scholz [1955] plotted the velocity distribution in the completely rough regimeusing the following equation:

U+ = 5.75 logyuτ

ν+ D1 (2.5)

with D1 = 8.5 − 5.75 logksuτ

ν. The plot is shown in Figure 2.1. The diagram

consists of a family of parallel straight lines withksuτ

νplaying the part of a

parameter. The value of k+s =

ksuτ

ν= 5 corresponds to hydraulically smooth

walls, the range betweenksuτ

ν= 5 to 70 corresponds to the transition from the

hydraulically smooth to the completely rough regime and forksuτ

ν> 70 the flow

is completely rough; see Schlichting [1981].

2.2 Recent Investigations 10

Figure 2.1: Universal velocity profile for turbulent flows through pipes, after N.Scholz [1955].

2.2 Recent Investigations

Several experimental studies have been performed to provide more informationabout the pressure drop, velocity distribution and turbulent flow structure nearrough walls, e.g. by Molki et al. [1993], Stukel et al. [1984], Nourmohammadiet al. [1985], Bandyopadhyay [1987] and Yokosawa et al. [1989]. More researchwork in connection with heat transfer as a function of the ratio of roughnessheight to hydraulic diameter, spacing between roughness elements and Reynoldsnumber was done by Dipprey and Sabersky [1963], Han [1984] and Ichimiaya[1987]. In addition, Wassel and Mills [1979] and Youn et al. [1994] related thefirst two magnitudes, rib height and rib pitch, to an equivalent sand roughnessas studied by Nikuradse [1933].

Powe and Townes [1973] investigated the turbulence structure for fully de-veloped flow in rough pipes. The method used to determine the turbulencestructure involved examination of the fluctuating velocity spectra in all threecoordinate directions. An important conclusion of this work was that in the


central region of the pipe, the flow was relatively independent of the natureof the solid boundary. In contrast, the flow near the wall presents a markeddependence on the nature of the solid boundary.

Several different ways of representing the effects of rough surfaces on tur-bulent flows have been proposed previously. Webb et al. [1971], in theirexperimental study of tubes with rough walls, developed a friction factorcorrelation based on the law of the wall similarity. In more recent work, Koh[1992] presented an equation to represent the mean velocity distribution acrossthe inner layer of a turbulent boundary layer, and used this velocity profile toderive a friction factor correlation for fully developed turbulent pipe flow. Moreadvanced analysis was performed by Youn et al. [1994], who deduced a newapproach where the transport equation and the turbulence model are solved onlybeyond a specified position away from the wall, and all the boundary conditionsare specified at this location.

A classical numerical approach to turbulent flow over rough surfaces wasproposed by Cebeci and Chang [1978]. By using the algebraic eddy viscosity ofCebeci and Chang [1978], and incorporating a suggestion by Rotta (in Cebeci[1978]) to represent the influence of the rough surface on the flow near thewall, they dealt with the incompressible rough wall boundary layer flow. Theapplicability of the procedure was confirmed by the comparison of calculatedand measured values of the skin friction available in the literature.

Pimentel et al. [1999] presented a low computational cost semi-analyticalprocedure for the solution of incompressible fully developed turbulent flowthrough symmetric and asymmetric ducts with rough walls. Using a modifiedalgebraic turbulence model to represent the influence of rough surfaces, theysolved the flow field in ducts with different wall roughnesses. They presented thevelocity distribution and the friction factor in two different geometries, circularducts and parallel plates, and compared them with experimental and numericaldata available in the literature, and obtained satisfactory agreement.

Hanjalic and Launder [1968] demonstrated in an interim report that themean velocity distribution between the plates of the channel exhibited differentregimes. In the vicinity of the walls, the velocity profile displayed the universalvariation appropriate to each surface. That is, near the smooth wall, the profilewas in good accord with the logarithmic “law of the wall”

u+S =

1

κln y+

S + c (2.6)


and correspondingly, near the rough wall, the variation was well described by

u+R =

1

κln

y

k+ c

′

(2.7)

In the above equations, u+ is the local streamwise velocity, normalized by the

friction velocity (τw

ρ)

1

2 , and the subscripts S and R denote whether the smooth

or rough wall shear stress, respectively, is used in the friction velocity. In thelatter equation, k denotes the characteristic height of the roughness and c

′

is aconstant independent of Reynolds number.

A number of workers have observed, in the case of asymmetric flows, thatstationary values of the mean velocity gradient and zero shear stress werenon-coincident, i.e. the plane of zero shear stress lies substantially closer tothe smooth wall than the plane of maximum velocity. This was observed byKjellstrom and Hedberg [1968] and Lawn [1970] in flow through annuli, byTailland and Mathieu [1967] in wall jets and by Beguier [1965] in an asymmetricplane jet.

Hanjalic and Launder [1972] presented the results of a detailed experimen-tal examination of fully developed asymmetric flow between two parallel planes.The asymmetry was introduced by roughening one side of the channel, whreasthe other side was left smooth, and the ratio between the shear stresses onthe two surfaces was 4:1. They found that there is an appreciable separationbetween the planes of zero shear stress and the maximum mean velocity. Inmore recent work, Parthasaraty and Muste [1994] confirmed the non-coincidenceof the planes of maximum velocity and zero Reynolds stress.

Reports of velocity profile measurements at rough surfaces are numerous,but few of them were concerned with the fully rough regime at an artificialroughness in fully developed turbulent flow in a closed channel. Measurementsfor free surface flow over a bed composed of hemispheres were performed byBayazit [1976]. He found a strong decrease in the von Karman constant κ inthe case of rough wall flow, when the flow depth was reduced. Because of thedifferent boundary conditions compared with a closed duct, his results cannot begeneralized. Investigations of certain two-dimensional roughnesses reported byHanjalic and Launder [1968, 1972], Lawn [1970], Perry et al. [1969] and Aytekinand Berger [1977] yielded values of the von Karman constant κ which deviatedfrom the generally accepted values of κ = 2.5 or 2.39.

Meyer [1980] performed experiments on fully developed turbulent flow in arectangular channel with variable aspect ratio to determine the parameters ofthe velocity profiles over two-dimensional rectangular roughnesses. He measured


the pressure and velocity profiles in the fully rough flow regime in channelswith one and two rough walls. His results showed that the slopes of thenon-dimensional velocity profiles in the smooth and rough zones decrease withincreasing the relative roughness, height and drag of the rough wall, contrary tothe generally accepted assumption of a constant profile slope.

Liou and Kao [1988] presented laser-Doppler velocimetry measurements ofthe mean velocity and turbulent intensity in turbulent flow past a pair of ribs ina rectangular duct with aspect ratio 2. The Reynolds number, based on the ducthydraulic diameter, was 2.0× 103 − 7.6× 104. The experiments covered ribs withrib to duct height ratios of 0.13-0.33 and with rib width to height ratios of 1-10.The critical rib height above which and the critical Reynolds number belowwhich the flow patterns become asymmetric were determined. The degree ofturbulence enhancement by the asymmetric and symmetric flows was compared.

Yokosawa et al. [1989] measured fully developed turbulent flow along asquare duct, two opposite walls of which were roughened, with a hot-wireanemometer. Velocities and stresses are presented and compared with measure-ments taken in a square duct with four smooth walls. Symmetric results, withrespect to the axes of symmetry of the duct cross-section, were obtained in everymeasured quantity.

Fujita et al. [1990] conducted an experimental study on fully developedturbulent flows through rectangular ducts with one rough wall. The distributionsof the mean velocities and turbulent stresses over the whole cross-section in thefully developed region were measured by hot-wire anemometry.

Sugiyama et al. [1993] performed a numerical analysis on fully developedturbulent flows in a square duct with two roughened facing walls using analgebraic stress model. The calculated results were compared with the exper-imental data and other published numerical results. The calculated results ofthe distributions of normal stresses were in reasonably good agreement with theexperimental data.

Hosni et al. [1993] carried out experimental measurements of profiles ofmean velocity and distributions of boundary-layer thickness and skin frictioncoefficient from aerodynamically smooth, transitionally rough, and fully roughturbulent boundary-layer flows for four surfaces, three of which were roughand one smooth. The rough surfaces were composed of 1.27 mm diameterhemispheres spaced in staggered arrays 2, 4 and 10 base diameters apart onotherwise smooth walls. The incompressible turbulent boundary-layer rough-wallair flow data were compared with previously published results on another, similarrough surface. It was shown that fully rough mean velocity profiles collapse


together when scaled with the momentum thickness, as was reported previously.However, this similarity cannot be used to distinguish roughness flow regimes,since a similar degree of collapse is observed in the transitionally rough data.Observation of his new data showed that scaling on the momentum thicknessalone is not sufficient to produce similar velocity profiles for flows over surfacesof different roughness character. The skin friction coefficient data versus theratio of the momentum thickness to roughness height collapse within the datauncertainty, irrespective of roughness flow regime, with the data for eachrough surface collapsing to a different curve. Calculations were made using thepreviously published discrete element prediction method and were comparedwith data from the rough surfaces with well-defined roughness elements. It wasshown that the calculations were in good agreement with the experimental data.

Sugiyama et al. [1995] conducted a numerical analysis of turbulent struc-ture and heat transfer in a square duct with one rough wall. A numericalanalysis of heat transfer was carried out for fully developed turbulent flow in astraight square duct with one roughened wall by using an algebraic Reynoldsstress model and turbulent heat flux model. Calculated results were comparedwith experimental data available presented by Hirota et al. [1992].

Hirota et al. [1995] measured a fully developed turbulent flow in a rect-angular duct Results were presented for quantities such as coefficient of flowresistance, local wall shear stress and turbulent shear stresses. The turbulentshear stress that was normal to the rough wall becomes much larger than thatof the smooth duct, but that parallel to the rough wall showed almost the samedistribution as the smooth duct, qualitatively and quantitatively.

Turbulent flows over rough surfaces are often encountered in practice andare of special interest in engineering. For example, in the atmosphere, theunderlying surface is usually rough. Also, pipes and ducts cannot, in general,be regarded as hydraulically smooth, especially at high Reynolds numbers, andit is not viable, either economically or from a technical viewpoint, to produceaerodynamically smooth surfaces.

Leonardi et al. [2003] mentioned that because of experimental shortcom-ings, there is an incomplete understanding of the physics of flows over roughsurfaces. The classical scheme based on the results of Nikuradse [1933], Clauser[1954], Rotta [1962] and Perry et al. [1969] relies entirely on the effect of theroughness on the mean velocity distribution. The effect of the roughness is toshift the mean velocity profile, with respect to that on a smooth wall, by anincrement ∆U+, referred to as the roughness function, i.e.

U+ =1

κln y+ + C − ∆U+ (2.8)


where κ and C are constants. The roughness function ∆U+ depends on thedensity, height and nature of the roughness and does not depend on y.

Antonia and Krogstad [2001] showed that for different rough surfaces withnominally the same ∆U+, the Reynolds stress distributions across the layer var-ied significantly, especially those involving the wall-normal velocity fluctuation.

Tachie et al. [2000] conducted an experimental study to investigate the effects ofroughness on the structure of turbulent boundary layers in open channels. Thestudy was carried out using the laser-Doppler anemometer in shallow flows forthree different types of rough surfaces and also a hydraulically smooth surface.The flow Reynolds number based on the boundary layer momentum thicknessranged from 1400 to 4000. The boundary layer thickness was comparable to thedepth of the flow and the turbulence intensity in the channel flow varied from 2 to4%. The defect velocity profile was correlated using an approach which allowedthe skin friction to vary. Wall roughness also led to higher turbulence levels inthe outer region of the boundary layer. The profound effect of surface rough-ness on the outer region and the effect of channel turbulence on the main flowindicates a strong interaction, which must be accounted for in turbulence models.

Gawad [2000] calculated the turbulent flow through a channel with twoopposite rough walls. He performed a numerical investigation of fully developedturbulent flow in square channels with two opposite roughened walls. Themain objective of his study was to determine the effects of the roughnessheight-to-hydraulic diameter ratio and Reynolds number on friction and heattransfer coefficients. The range of the roughness height-to-hydraulic diameterratio extends from 0.047 to 0.12. The roughness effect was introduced viamodifying boundary conditions in the wall vicinity. His results compared wellwith the experimental data available in the literature.

The effects of surface roughness on the mean velocity and temperatureprofiles were well reviewed by Raupach et al. [1991]. Krogstad and Antonia[1999] investigated the effects of surface roughness on a turbulent boundary layerby comparing measurements over two rough walls with measurements from asmooth wall boundary layer. The two rough surfaces have very different surfacegeometries, although designed to produce the same roughness function, i.e. tohave nominally the same effect on the mean velocity profile. They observeddifferent turbulent transport characteristics for the rough surfaces. Substantialeffects on the stresses occur throughout the layer, showing that the roughnesseffects are not confined to the wall region.

Kussin and Sommerfeld [2002] carried out experimental studies on turbu-lence modification in horizontal channel flows with different wall roughnesses.


Detailed measurements in a developed horizontal channel flow (length 6 m,height 35 mm; the length is about 170 channel heights) were presented usinglaser-Doppler anemometry for the determination of air velocity. The conveyingvelocity could be varied between 10 and 25 m/s. For the first time, the degreeof wall roughness could be modified by exchanging the wall plates. The effect ofthese parameters on the mean and fluctuating velocity profiles in the developedflow were examined. It was shown that the wall roughness decreases the meanvelocity and enhances the fluctuating velocities owing to irregular wall bouncing.

The most important effect of roughness is the change of the mean velocityprofile near the wall, with consequent modification of the friction coefficient.Nikuradse expressed the velocity profile as

U+(y) =1

κln

y

ks

+ 8.5 − 1

κΠW (

y

δ) (2.9)

Over rough walls, there is a question of which origin to use for the distance y.The shift ∆y from some reference location is usually determined empirically tomaximize the quantity of the logarithmic fit in Equation (2.9), and it is typicallysome fraction of k. Raupach et al. [1991] thoroughly discussed this issue, whichwas important for interpreting the experimental results.

Finally, Jimenez [2004] reviewed the experimental evidence on turbulentflows over rough walls. Two parameters were important: the roughness Reynoldsnumber k+

s which measures the effect of roughness on the buffer layer, and theratio of the boundary layer thickness to the roughness height, which determineswhether a logarithmic layer survives. He concluded that the subject needsextensive and careful experimental work to discuss the conflicts among theprevious experimental results.

Therefore, the primary purpose of the present work was to extend the previousexperimental studies, with particular attention being given to the streamwisemean velocity distribution and the turbulence statistics in two-dimensional fullydeveloped turbulent rough plane-channel flows.

From the early work, it was concluded that the turbulent velocity profilesin the rough plane-channel flows should obey a logarithmic law of the form

U+ =1

κln

y

k+ B (2.10)

It is clear that this expression for the logarithmic law depends extensively on thedetermination of the roughness height, i.e. k, which was not easy to determineaccurately. This convinced the present author to look again at the mean veloc-ity profile and to renew the situation in wall-bounded fully developed turbulent


rough flows. Hence the present work was aimed at direct measurements of thetime-averaged velocity distributions in the channel with rough walls and relatingit to the corresponding velocity distributions in the channel with smooth walls in-dependent of the roughness height. For this reason, the value of the von Karmanconstant κ for the logarithmic law of the wall was derived from the followingequation:

ln(dU+/dy+) = f(ln y+) (2.11)

A wide range of Reynolds numbers was covered to satisfy the validity and theuniversality of the new procedure. The present investigations were carried outfor an air-driven plane-channel flow of aspect ratio (width/height) 12:1, coveringa range of Reynolds number up to Rem ≈ 1.1 × 105. Furthermore, informationon the effective height of the channel because of the influence of roughness is alsopresented. Also, the similarity between the smooth channels of different heightswas studied. Mean velocity measurements were carried out for rough channelflows using hot-wire anemometry. Accurate shear stress measurements using atechnique independent of the mean velocity allowed the resultant velocity profilesto be normalized with the appropriate shear velocity uτ and the correspondingviscous length lc = ν/uτ scales. The experimental results and their analysis yield-ing this result and some other interesting features of the mean flow investigationsare also reported.

Chapter 3

THEORETICAL ANALYSIS

3.1 Basic Equations for Fully Developed Chan-

nel Flows

Research on turbulent flows basically started with the discovery of Reynolds in1883 that pipe flows, depending on a dimensionless number, later named theReynolds number, consist of two different modes, either laminar or turbulent.Reynolds [1883] also found that the transition from the laminar to the turbulentmode sets in intermittently by “flashes” that occur in localized regions whenthe Reynolds number exceeds a so-called “critical” value. As the Reynoldsnumber increases, the frequency of these “flashes” increases until a state of fullydeveloped turbulence is obtained in the downstream direction of the so-calledcore region. All these fundamental properties of pipe flows were later foundto represent common features of wall-bounded flows and, hence, also occur innominally two-dimensional channel flows. This part of the present work wastherefore aimed at studying the similarity condition between smooth channels,the effective height of the channel because of the influence of roughness and theshift between the smooth and the rough log-law lines. This requires consideringthe fundamental equations of motion presented in general Cartesian form asfollows:

Conservation of mass (continuity equation):

∂ρ

∂t+

∂ρUi

∂xi= 0 (3.1)

where ρ is the fluid density, t the time and Ui are the Cartesian velocitycomponents.

Conservation of momentum:

ρ(

∂Uj

∂t+ Ui

∂Uj

∂xi

)

= − ∂P

∂xj− ∂τij

∂xi+ ρgj (3.2)

3.1 Basic Equations for Fully Developed Channel Flows 19

where P denotes the pressure, τij the stress tensor and gj the gravity.

For the Newtonian fluids, the momentum transport term τij expressedas

τij = −µ(

∂Ui

∂xj

+∂Uj

∂xi

)

+2

3µδij

∂Uk

∂xk

(3.3)

where µ is the molecular viscosity and δij is the Kronecker delta.

All velocity variables of Equation (3.2) in the general case of turbulentflows consist of a mean velocity component Uj and a fluctuating component u

′

j,e.g. the velocity can be given in the form

Uj = Uj + u′

j (3.4)

Replacing all variables in Equation (3.2) by their mean and fluctuating compo-nents, and taking the time average, Equation (3.2) gives the well-known Reynolds-averaged Navier-Stokes equation in the following standard form:

ρ[

∂Uj

∂t+

∂

∂xi(UiUj + u

′

iu′

j)]

= − ∂P

∂xj− ∂

∂xiτij + ρ gj, (3.5)

For the stationary turbulent flows, Equation (3.5) yields

∂

∂xi

[ρ u′

iu′

j + ρ UiUj + τij] = − ∂P

∂xj

+ ρ gj (3.6)

where−ρ u

′

iu′

j = the turbulent momentum transport tensor;

−ρ UiUj = convective mean momentum transport due to mean motion;τij = time-averaged molecular momentum transport;P = pressure force on unit volume;

ρ gi = gravitional force on unit volume.

Equation (3.1) then takes the form

∂U i

∂xi= 0 (3.7)

Substituting Equation (3.7) into Equation (3.6) leads to

ρ(

U i∂U j

∂xi

)

= − ∂P

∂xj

+∂

∂xi

(

µ∂U j

∂xi

− ρ u′

iu′

j

)

(3.8)

where ρ gj = 0 for horizontal flow.

3.1 Basic Equations for Fully Developed Channel Flows 20

For two-dimensional flow, the mean velocity and all gradients in the span-wise direction, z, are negligible. In addition, for fully developed flow allcomponents and gradients are not depend on the streamwise distance, x, exceptfor the streamwise pressure gradient, which is needed for driving the flow throughthe channel. As a result, Equation (3.6) reduces to the following:

x-momentum:

−µ∂2U

∂y2+

∂ρ uv

∂y= −∂P

∂x(3.9)

y-momentum:∂ρ uv

∂x= −∂P

∂y= 0 (3.10)

z-momentum:

0 = −∂P

∂z(3.11)

where u = U1, v = U2, w = U3, x = x1, y = x2 and z = x3.

Integrating Equation (3.9) with respect to y yields

−µ

ρ

∂U

∂y+ uv = −y

ρ

∂P

∂x+ constant (3.12)

The total shear stress must be zero at the centerline of the channel, i.e. aty = H/2, since U is maximum and the Reynolds shear stress, −ρuv, is zero. Asa result, Equation (3.12) reduces to

0 = −H

2ρ

∂P

∂x+ constant ⇒ −H

2

dP

dx= constant (3.13)

Moreover, at the wall, i.e. y = 0, the turbulent velocity fluctuations shouldsatisfy the no-slip condition. As a result, the Reynolds shear stress is zero, henceEquation (3.12) could be rewritten as

−µ

ρ

∂U

∂y= constant ⇒ ∂U

∂y= constant (3.14)

The constant in Equations (3.13) and (3.14) is the wall shear stress τw. Therefore,Equations (3.13) and (3.14) indicate that an exact balance between the wallshear stress and the net pressure force acting on the flow exist. Hence, fromEquations (3.13) and (3.14), one can conclude that the wall shear stress can bedetermined by pressure gradient measurements along the channel centerline and

3.2 Similarity of Smooth Channel Flows 21

using Equation (3.13), which indicates an integral momentum balance betweenthe wall shear stress and pressure force:

τw = −H

2

(

dP

dx

)

⇒ uτ =

√

τw

ρ(3.15)

Such measurements require that the assumption of two-dimensionality of themean flow velocity distribution exists.

3.2 Similarity of Smooth Channel Flows

3.2.1 Relation Between the Wall Shear Stresses for theSmall and the Large Channel

To have a relation between the wall shear stress in the case of both smoothchannels, i.e the large and the small, at the same Reynolds number, the followingprocedure was carried out:

It is well known in the field of fluid mechanics that the wall shear stresscan be computed from the following relation:

τw =cf

2ρ U

2(3.16)

where cf is the wall skin friction, ρ is the density of the fluid and U is the meanvelocity of the flow.

Introducing the last equation once for one smooth channel of height hand then for a smooth channel of height H, with H > h, leads to the following

For the small channel:τwh

=cf

2ρ U

2(3.17)

After multiplying the nominator and the denominator of the right-hand side byh2ν2 and also taking cf = 0.0624 × Re−0.25 (see Zanoun [2003]), τwh

takes theform

τwh= 0.0312 ρ ν2 Re

7/4h /h2 (3.18)

For the large channel, one can find also

τwH= 0.0312 ρ ν2 Re

7/4H /H2 (3.19)

Dividing Equation (3.18) by Equation (3.19) for constant density, ρ = constant,and constant volume, V = constant, leads to

τwh

τwH

=Re

7/4h /h2

Re7/4H /H2

=H2Re

7/4h

h2Re7/4H

(3.20)


which, at the same Reynolds number, Reh = ReH , leads to

τwh

τwH

=H2

h2(3.21)

3.2.2 Similarity Procedure

The conservation of momentum equation in the general Cartesian coordinatesreads

ρ

(

∂Uj

∂t+ Ui

∂Uj

∂xi

)

= − ∂P

∂xj− ∂τij

∂xi+ ρgj (3.22)

From Equation (3.22), one can deduce the momentum equation, expressed forthe total stress distribution(τ21)total, for fully developed channel flows, as follows:

− dP

dx1+

d(τ21)total

dx2+ ρg1 = 0 (3.23)

Hence the total shear stress distribution of the channel flow reads

(τ21)total =dP

dx1x2 + constant (3.24)

The total shear stress must be zero at the centerline of the channel, i.e. aty = H/2 = h, since U is maximum and the Reynolds shear stress, −ρ uv, is zero:

0 =dP

dx1h + C ⇒ C = − dP

dx1h (3.25)

Then Equation (3.24), after replacing the value of the constant C, takes the form

(τ21)total =dP

dx1

x2 −dP

dx1

h ⇒ (τ21)total = − dP

dx1

h(1 − x2

h) (3.26)

Introducing the wall shear stress, τw = − dP

dx1h, yields

(τ21)total = τw(1 − y

h) (3.27)

For (τ ∗21)total =

(τ21)total

τwand

y

h= y∗, one can obtain the normalized shear stress

distribution as follows:(τ ∗

21)total = 1 − y∗ (3.28)

Hence, for every channel flow one obtains a similar shear stress distribution.


The total shear stress is also given by (τ21)total = µdU1

dy− ρu2u1, from which and

Equation (3.27) one can write

µdU1

dy− u2u1 = τw(1 − y

h) (3.29)

and, hence, the momentum equation for the normalized velocity distributionyields, if inner variables are used for the normalization, the following equation:

dU+1

dy+= 1 − y+

h++ u2u1

+ (3.30)

where U+1 = U1/uτ , y+ =

yuτ

νand u2u1

+ =u2u1

+

u2τ

with uτ =

√

τw

ρhas been in-

troduced for the normalization. Introducing Reτ = h+, the final equation reads

dU+

dy+= 1 − y+

Reτ+ u2u1

+ (3.31)

The above considerations show that every channel flow that has the same(τ ∗

21)total distribution yields the same normalized mean velocity distributions ifReτ are identical for both considered flows.

It is usual to normalize the velocity profiles for channels of different di-mensions to yield the same law of the wall for all Re. For the same Reτ values,the normalized velocity profiles of channels with different dimensions give thesame line. Here the question is raised of what the values of the normalizedvelocities are at the same distance from the wall and where they lie on thelog-law profile. This section, in contrast to the previous investigations, willestablish how the same location, i.e. y = Y , transfers to the log-law profile if theReτ values for the small and large channels, of heights h and H, respectively, arethe same.

In the following steps, the condition of the similarity between the smoothchannels with heights h and H will be deduced.

In the case of a channel of height h, the law of the wall can be writtenas

u+ =1

κln y+ + B (3.32)

where u+ =u

uτ, y+ =

yuτ

νand uτ =

√

τwh

ρ,


and in the case of a channel with a larger height H, where H > h:

U+ =1

κln Y + + B (3.33)

where U+ =U

Uτ, Y + =

Y Uτ

νand Uτ =

√

τwH

ρ.

Equation (3.32) for the small channel can be transformed to the case ofthe large channel by multiplying the nominator and the denominator by Uτ asfollows:

u+ =1

κln

yuτ

ν

Uτ

Uτ+ B (3.34)

which leads to

u+ =1

κln Y + uτ

Uτ+ B (3.35)

At the same Reynolds number from Equation (3.21), one hasτwh

τwH

=H2

h2, which

in turn leads touτ

Uτ

=H

h. Introducing this into Equation (3.35) leads to

u+ =1

κln Y +H

h+ B =

1

κlnY + +

1

κln

H

h+ B (3.36)

Comparing Equation (3.36) with Equation (3.33) leads to the following importantrelation:

u+ = U+ +1

κln

H

h(3.37)

which means that at the same Reynolds number the value of the normalizingvelocity in the small channel exceeds the same value in the large channel by the

factor of1

κln

H

h, as shown in Figure 3.1.

3.2.3 Second Condition of Similarity

It is well known in the field of fluid mechanics that the normalized velocity profilefor the smooth surfaces can be given as follows:

U+ =1

κln y+ + B (3.38)

where κ is the von Karman constant, U+ = u/uτ , y+ = y uτ/ν and B is an ad-ditive constant. Multiplying the the nominator and the denominator of Equa-tion (3.38) by h, the half-width of the channel, leads to

U+s =

1

κln

y uτ h

ν h+ B =

1

κln

Reτ y

h+ B (3.39)

3.3 Two-Dimensional Asymmetric Turbulent Channel Flows 25

Figure 3.1: Similarity of smooth channel flows. Corresponding points are shown.

where Reτ = h uτ

νand s refers to the small channel.

For a channel of larger height, H, Equation (3.39) reads

U+b =

1

κln

Reτ Y

H+ B (3.40)

Subtracting Equation (3.39) from Equation (3.40) leads to

U+b − U+

s =1

κln

Reτ b

Reτ s

(3.41)

which means that the behavior of the normalized velocity in the case of a largechannel is the same as that of a small channel but stretched in the positive y+

direction by a value of1

κln

Reτ b

Reτ s

.

3.3 Two-Dimensional Asymmetric Turbulent

Channel Flows

Figure 3.2 shows a sketch of the asymmetric channel flow which will be treatedin this section. Starting from Equation (3.8), for two-dimensional flows, the


following forms can be given:

ρ(U∂U

∂x+V

∂U

∂y) = −∂P

∂x+

∂

∂x(µ

∂U

∂x+µ

∂U

∂y)+

∂

∂y(µ

∂U

∂x+µ

∂U

∂y)−ρ[

∂

∂x(u′u′+

∂

∂y[u′v′])]

(3.42)and

ρ(U∂V

∂x+V

∂V

∂y) = −∂P

∂y+

∂

∂x(µ

∂V

∂x+µ

∂V

∂y)+

∂

∂y(µ

∂V

∂x+µ

∂V

∂y)−ρ[

∂

∂x(u′v′+

∂

∂y[v′v′])]

(3.43)where U = U1, V = U2, u

′

= u′

1 and v′

= u′

2.

For fully developed turbulent flow, all streamwise gradients with the ex-ception of the streamwise pressure gradient are zero. As indicated in Figure 3.2,y = 0 refers to the smooth wall and y = H refers to the rough wall.

Formulating the continuity equation, one can have∂V

∂y= 0, hence

0 = −∂P

∂x+

∂

∂y(µ

∂U

∂y− ρ u′v′) (3.44)

or

0 = −∂P

∂y− ρ

∂

∂y(v′2) (3.45)

By integrating Equation (3.45), one can derive

P (x, y) = Π(x) + ρ v′2 (3.46)

where ρ v′2 = f(y) and Π(x) is external pressure, since the flow in the streamwisedirection is fully developed. Obviously, from the above equation one can concludethat the pressure over the cross-section of the channel depends on the streamwisecoordinate and on the spanwise coordinate. However, the pressure change in they direction is very small, and therefore can be neglected.

Equation (3.44) can be given in the following form, which is valid for two-dimensional asymmetric turbulent flows:

0 = −dΠ

dx+

d

dy(µ

dU

dy− ρ u′v′) (3.47)

Since

τyx = µdU

dy− ρ u′v′ (3.48)

integrating the above equation leads to

τyx = C +dΠ

dxy (3.49)


Figure 3.2: Sketch of the vertical channel with one smooth and one rough surface.

where C is an integration constant. The integration constant can be determinedfrom the condition that at the position at which y equals ε (see Figure 3.2), theshear stress becomes zero. Introducing this into Equation (3.49) leads to

C = −εdΠ

dx(3.50)

Hence Equation (3.49) now takes the form

τyx = (y − ε)dΠ

dx(3.51)

The value of the wall shear stress on the smooth plate can be determined byputting y = 0 in Equation (3.51), which leads to

τws = −εdΠ

dx(3.52)

The determination of the pressure gradient can be done by static pressure readingson the flat surface. One can find ε, the distance between the smooth wall and

3.4 Theoretical Investigations of Rough Channel Flows 28

the position of zero shear stress, by calculating the total shear stress using therelation

τtotal = −µ∂U

∂y+ ρ uv (3.53)

where τtotal is the total shear stress.

The wall shear stress on the rough surface can be derived by setting y = H inEquation (3.51), which leads to

τwr = (H − ε)dΠ

dx(3.54)

According to Equations (3.52) and (3.54), to calculate the wall shear stress inboth smooth and rough cases one has to know the pressure gradient, ε and theeffective height of the channel. This means that one has to measure the pressuregradient in the streamwise direction, the mean velocity profile and u

′

v′

profileand determine the point at which the total shear stress is zero and the distancebetween this point and the smooth wall is taken as the value of ε. The mostimportant parameter is the effective height of the channel, which will also betreated.

3.4 Theoretical Investigations of Rough Chan-

nel Flows

Regarding flows with rough walls, it is common practice to write the velocityprofile as

u+r =

1

κln

yr

k+ C (3.55)

where κ is the von Karman constant, k is the height of roughness element andC is an additive constant. This equation can be written in the following usefulform:

u+r =

1

κln

y+r

k++ C =

1

κln y+

r − 1

κln k+ + C (3.56)

where y+r =

yruτ r

νand k+ =

kuτ r

ν. Because

1

κln k+ is also constant, then the last

equation might take the form

u+r =

1

κln y+

r + D (3.57)

It is well known that the velocity profile in the case of smooth walls takes theform:

u+s =

1

κln y+

s + B (3.58)


where y+s =

ysuτ s

ν.

It is common practice that the velocity profile in the case of rough wallsis downwards with respect to that for smooth walls (see Figure 2.1). Taking thisinto account, one can write Equation (3.57) in the form

u+r =

1

κln y+

r + B − ∆B (3.59)

where ∆B is the downward shift of the log-law velocity profile due to roughnesswhich caused increase of the momentum loss to the walls. This also correspondsto a shift of the velocity profile in the positive y+ direction according to

u+s = u+

r ⇒ 1

κln y+

s + B =1

κln y+

r + B − ∆B (3.60)

and then one obtains

∆B =1

κln y+

r − 1

κln y+

s (3.61)

Taking into account that the following relationship holds:

y+r

y+s

=yruτ r

νysuτ s

ν

=yruτ r

ysuτ s

(3.62)

leads toy+

r

y+s

= Duτ r

uτ s

= D

√τwr√τws

⇒ y+r = D y+

s

√τwr√τws

(3.63)

where D is a constant that is the ratio between yr and ys at u+r = u+

s .

Replacing the value of y+r in Equation (3.61) leads to the following useful

relation:

∆B =1

κln

D√

τwr√τws

y+s

y+s

=1

2κln D

τwr

τws

(3.64)

This equation is useful because it gives the shift between the smooth and therough log-law lines without any information about the roughness. This meansthat at the same value of the normalizing velocity in the smooth and roughchannels, the shift can be determined only from measurements of the wall shearstress in both cases, as shown in Figure 3.3.

It is also of crucial importance to deduce the roughness height accurately.Using Equation (3.64) with the universal law of Nikuradse for the rough walls,one can find the roughness height as follows. Nikuradse defined the universallaw for the rough walls in the following way:

u+r =

1

κln

y

ks+ 8.5 (3.65)


Knowing ∆B from Equation (3.64), one can calculate u+r at different values of

y and then, using Equation (3.65), one can directly calculate the correspondingvalues of the roughness height, i.e. ks.

Figure 3.3: Similarity of smooth and rough channel flows.

The following parameters are required for checking the above theoreticalconsiderations

• detailed mean velocity measurements

• pressure gradient measurements

• the position of the zero shear stress

• the effective height of the channel under the influence of roughness

• the values of the constant D in Equation (3.64)

• the values of the shift between the log-law lines of the smooth and roughcases.


All the above parameters will be treated in the next chapter.

Chapter 4

EXPERIMENTALAPPARATUS ANDMEASURING TECHNIQUES

In this chapter, the experimental test facilities and the measuring techniqueswhich were employed in the present work are introduced. The experimental testfacilities include smooth channel test section, asymmetric channel test sectionand rough channel test section. The experimental techniques used include hot-wire anemometry, laser-Doppler anemometry and pressure transducers and Pitottubes.

Symmetric Channel Flow Investigations

4.1 Experimental Apparatus Used for Symmet-

ric Channel Flow

4.1.1 Smooth Channel Test Section

The experiments were carried out at LSTM-Erlangen using two smooth channelflow test sections, one of them as sketched in Figure 4.1. The surface roughnessof each channel test section was measured accurately as ±0.25 µm, which interms of wall units was less than 0.03 for the maximum Reynolds number.The channels were therefore considered to be hydrodynamically smooth at allReynolds numbers reported in this study. The geometric dimensions of thecross-sections of the channels showed a width B of 600 mm and a height H of 50mm for the small channel and a width B of 600 mm and a height H of 100 mm forthe large channel, providing channel aspect ratios of 12:1 and 6:1, respectively.These aspect ratios were considered sufficient, (see, e.g., Dean [1978]) to ensure

4.1 Experimental Apparatus Used for Symmetric Channel Flow 33

the required two-dimensionality of the investigated turbulent, plane-channelflows. The total lengths of the channels were both 6 m, corresponding to L/H

Centrifugal Blower

Honeycomb

Perforated Plates

Screens

Settling Champer

PC & DAQ Card Hot-Wire Anemometry

HWA Test Section

Scany Valve

Pressure Transducer

Temperature Sensor

Pressure Taps

Temperature Sensor

Oil-Film Test Section

95 H

115 H

50

600

Tripping Test Section Test Section Dimensions

Air Filter Blower Inlet

Blower Outlet

Figure 4.1: Sketch of the channel flow test section with the temperature, pressureand velocity measuring equipment.

ratios of 120 and 60, respectively. The channels were constructed of threesections, each with a length of 2 m, connected together. A two-cubic arc inletnozzle to the channel made out of hard wood was used between the plenumchamber and the channel test section to ensure a smooth inlet flow. The actualmeasuring location was taken at a distance from the channel inlet of X = 115H.This length was considered to be sufficient to ensure a fully developed turbulentchannel flow before reaching the measuring test section (see, e.g., Comte-Bellot[1963]), and was far enough away from the channel outlet to ensure no outletdisturbances to the flow. Hence, prior to carrying out the actual measurements,the conditions were set up correctly to ensure the two-dimensionality of the flowand its state of full development by carefully choosing the channel dimensionsand measuring locations. The actual air flow was provided by a centrifugalblower which had a maximum capacity of 10 m3/s and was powered by a 20 kWmotor. The blower outlet was connected to a well-designed settling chamber toensure the uniformity of the flow entering the channel inlet. After the outlet ofthe blower, in the downstream direction, the first essential flow control in theplenum chamber was located and consisted of two perforated plates with 52%solidity having square openings of 10× 10 mm. Perforated plates were separatedby 30 cm from the blower and from each other. The second passive flow-controldevice in the plenum chamber was a ‘honeycomb plate’ with a mesh size of 8mm diameter and a total length of the tubes of 160 mm. These passive flowcontrol devices inside the plenum chamber were located in such a way as toyield a well-controlled inlet flow to the actual channel test section. The entireflow was setup in accordance with suggestions made by Loehrke and Nagib [1972].

4.1 Experimental Apparatus Used for Symmetric Channel Flow 34

The flow rate for each investigated Reynolds number was controlled bychanging the speed of the radial blower blades by means of a frequency convertercontrol unit, providing impeller rotational speeds of approximately 100–2000rpm. This corresponded to a mean velocity range of the channel flow from 3to 50 m/s with a centerline turbulence level of less than 0.3% at the axis ofthe channel inlet cross-section. A special Venturi flow nozzle at the inlet ofthe settling chamber was used for measuring the mean volume flow rate andconsequently the mean flow velocity through the channel test section for eachinvestigated Reynolds number. In addition, the mean velocity was also obtainedby integrating the velocity profile at the measuring station for each Reynoldsnumber to ensure a good assessment of the mean flow velocity, U , as time flowarea averaged velocity for each Re case. Good agreement was achieved of about±1% and the mean flow velocity was then used to compute the mean-basedReynolds number of the flow:

Rem =U h

ν(4.1)

Reynolds numbers ranging up to Rem ≈ 1.1 × 105 were covered in these investi-gations.

Furthermore, the present channel test sections were prepared with pres-sure tappings for wall pressure measurements along the channel at differentlocations. Pressure tappings were installed along the test section’s top wall ofthe channel (the wide side of the cross-section). These were employed over a 5m length of the channel test section, where 12 pressure tappings were located toprovide the streamwise pressure gradient, dP/dx, distribution for each investi-gated flow. Three static pressure taps of 500 µm diameter at each x−locationwere carefully installed at each of the 12 pressure-measuring locations, one atthe centerline of the channel and two on both sides, 10 cm apart from the centerpoint for the pressure measurements in the streamwise direction. Care was takento ensure that the inner surface of the top side of the channel, where the holeswere drilled, was free from drilling problems (i.e. smoothness was insured aroundthe pressure tappings). All pressure measurement points were connected to ascanny valve to facilitate switching from one point to another and the correspond-ing static pressure was then measured and recorded for different air flow velocities.

Hence, as the above description shows, the test facility was designed andbuilt carefully to carry out measurements over the required range of Reynoldsnumbers.

4.2 Measuring Techniques Used in Symmetric Channel Flows 35

4.1.2 Fabrication and Determination of Roughness

Emery papers were glued on the upper and lower walls of the channel to preparethe rough surfaces. First, the plates were covered with polystyrene adhesive.Subsequently, emery paper of 200 cm length and 60 cm width was pressed onthe plate surfaces. Care was taken to prevent any folds. After approximately4-5 hours, the plates can be used. Figure 4.2 shows photographs of the differentstages of the preparation of the rough surface.

Figure 4.2: Photographs showing the different stages of preparation of the roughwalls: (a) the original plate, (b) gluing the emery paper on the plate and (c) theplate to be used.

4.1.3 Rough Channel Test Section

The rough channel test section was kept the same as the smooth small channeltest section after replacing the upper and lower smooth plates by rough plates.The dimensions of the cross-section of the rough channel were 600 × 47.50 mm,providing a channel aspect ratio of 12.6:1. This aspect ratio was consideredhigh enough, e.g. Dean [1978], to ensure the required two-dimensionality of theinvestigated turbulent, plane-channel flows. The total length of the channel setupwas 6 m, corresponding to an L/H ratio of 126.

4.2 Measuring Techniques Used in Symmetric

Channel Flows

4.2.1 Hot-Wire Anemometry (HWA)

Basic Principle

The primary measuring technique for the experimental investigations made forsymmetric smooth and rough channels, to obtain the time history of the fluidvelocity at a point in a flowing fluid, was thermal anemometry. Thermal anemom-etry is a widely employed method and a good history of this technique can be


found in a paper by Freymuth [1968]. The initial work on thermal anemometrygoes back to the nineteenth century, but the main period for research and develop-ment started in 1946 and is still continuing. The fundamental concept of thermalhot-wire anemometry is based on the convective heat transfer from a heated wireplaced in a fluid flow. It consists mainly of a minute metallic wire sensor whichis fixed to the ends of a slender holder (Figure 4.3), then placed the flow formeasurements. The wire is electrically heated (Joule effect) and simultaneously

Air Flow

Electric Current

Sensor (Thin wire) Wire support

(St. St. needles)

Figure 4.3: Sketch of hot-wire anemometry probe.

cooled by the convective heat transfer induced by the lower temperature incidentflow. The main feature that allows the hot wire to be used as a measurementdevice is that its electrical resistance depends on its temperature. Therefore,by direct or indirect measurements of the wire resistance variations, one caninfer the heat transfer taking place and then obtain the incident flow information.

The heat loss from the hot wire is affected by many factors, which can besummarized as follows:

• Fluid physical properties such as density (ρ), viscosity (µ), heat capacitance(cp) and heat thermal conductivity (λ).

• Flow parameters, e.g. velocity (Ui), temperature (T ) and pressure (P ).

• Properties of wall material such as thermal conductivity (λw) and dimen-sions (δ).

• Wire configurations (l, d) and overheat ratio (a).

The heat transfer from the wire is mainly dominated by forced convection and,to minimize the effect of the conductive end losses of heat to the prongs, the


wire should be as long as possible, i.e. l/d ≥ 200, the thermal conductivity of thewire should be low and the wire should have a high temperature coefficient ofresistance. However, at very low flow velocities, the natural convection from thehot-wire probe becomes significant but in most hot-wire applications the radiationheat transfer from the wire is very small and can be neglected.

Hot-Wire Calibration

Prior to performing velocity measurements, each hot-wire probe has to becalibrated in a well-defined flow field using an appropriate calibration technique.Such a calibration is usually carried out in the undisturbed flow region at aflow nozzle exit or outside a boundary layer, yielding an electrical signal whichdepends on the thermodynamic properties of the fluid, the overheating ratio ofthe wire and the calibration velocity. The resulting calibration function wassubsequently applied to measure local flow velocities in unknown flow fields,e.g. near walls in boundary layer, pipe and channel. For this purpose, theoverheating ratio for the wire temperature was chosen to be equal to that ofthe calibration. In this way, numerous flow fields can be investigated and usefulresults can be provided for laminar and turbulent flows. Other anemometerssuch as Pitot tubes and the laser-Doppler anemometer, are usually used forhot-wire calibrations.

A fourth-degree polynomial was chosen to fit the calibration data withinan accuracy of better than ±1%. To ensure that the original calibration curvewas maintained during one entire set of hot-wire measurements, the calibrationcurves were rechecked after each set of measurements covering the entire rangeof velocities experienced in the near-wall region for each investigated flow case.If the deviations of the calibration were more than ±1%, the entire set ofdata wwere rejected and the measurements for the corresponding Reτ wererepeated. The temperature of the air stream inside the wind tunnel was keptconstant within ±0.2 ◦C during the calibration procedure and also duringthe measurements so as to yield accurate hot-wire velocity results. Once thecalibration curve had been established and the least-square curve fitting equationhad been obtained, the flow field measurements were performed.

Hot-wire Anemometry

The velocity profile measurements reported here were carried out using aconstant-temperature anemometer. To adjust the system, the instructionsprovided in the manual were followed, both for the calibration of the systemand for its employment for smooth and rough channel flow investigations. Thehot-wire measurements of the local velocity were carried out with a normalprobe (DANTEC, Type 55P15), equipped with a wire of 5 µm diameter and an


active wire length of 1.25 mm, providing an aspect ratio of l/d=250. Hence thewire had a sufficiently large aspect ratio to suggest a negligible influence of theprongs on the actual velocity measurement. All calibrations and measurementswere performed with an 80% overheat ratio, a = (Rw − Ra)/Ra, where Rw is theoperational hot-wire resistance and Ra is the resistance of the cold wire, i.e. atambient air temperature.

Before each set of measurements, the hot-wire probe was calibrated againstvelocity measured with a Pitot tube. The Pitot tube was installed directly atthe centerline of the channel cross-section and its output was connected to aprecision pressure transducer for both stagnation, P0, and static, Pst, pressuremeasurements. In addition, the air temperature inside the tunnel was measuredat all times during measurements within an accuracy range of ±0.05 ◦C. Toobtain the velocity dynamic head of the Pitot tube, the mean static pressure wasmeasured 6D (where D is the diameter of the Pitot tube) in the downstreamdirection of the Pitot opening.

Along a streamline at the channel centerline, integrating the momentumequation results in the well-known Bernolli equation, which applies between apoint in the flow and stagnation point on the same streamline:

P0

ρ=

Pst

ρ+

U2

2(4.2)

where ρ is the air density. By rearranging the terms of the above equation, themean velocity U of the air flow at channel entrance was obtained:

U =

√

2 (P0 − Pst)

ρ(4.3)

Hence the time-averaged air velocity for calibration was simply calculated by mea-suring the pressure difference between the stagnation pressure P0 and the staticpressure Pst using a differential pressure transducer. The ambient conditionswere monitored before and during each test run using an electronic barometerand thermometer. All measuring equipment were connected to an A/D converterboard from National Instruments with 16-bit resolution and 8 input channels.In addition, a LABVIEW program was used for acquiring and processing all themeasured data.

Correction of HWA Output for Ambient Temperature Drift

The output signal amplitude of the hot wire is temperature dependent, hence acorrection for temperature drift is necessary if the temperature of the workingfluid cannot be kept constant during calibrations and measurements. Different


methods are available in the literature to deal with the problem of the air tem-perature effect on hot-wire output. Good treatments of the problem were givenby Bearman [1970], Bremhorst [1985] and Crowell et al. [1988]. A direct cali-bration of the hot-wire output E with flow velocity U at air temperature Ta fora given wire resistance Rw is commonly carried out. This is one of the mostaccurate methods of establishing the velocity and temperature sensitivity of ahot wire working in a constant-temperature mode. To establish the dependenceof the hot-wire output curves on ambient fluid temperature, velocity calibrationsare usually performed at different air temperatures by measuring the hot-wireoutput for different air velocities. As a result, the calibration data are usuallyrepresented in the form

E = f(U)Ta=C or E = f(Ta)U=C (4.4)

For small temperature changes, i.e. approximately ±5◦C, the changes in thePrandtl number, the ratio Twire/Tmeas. and heat conductivity can be neglectedand, from the correlation Nu = F(Re, Pr, Twire/Tmeas.), the heat transfer coef-ficient is unchanged for the same velocity and hot-wire diameter. Under theseconditions, Bearman [1971] introduced the following expression to correct hot-wire output for the temperature drift:

Ecorr.∼= Emeas.

[

1 − ε

2σ

]

(4.5)

where ε = (Tref. − Tmeas.)/Tref. and σ = 1 − (Twire/Tref.), Tref. is a referencetemperature, Tmeas. is the measurement temperature and Twire is the wire tem-perature. For more information about hot- wire anemometry, see Brunn [1995].

4.2.2 Pressure and Temperature for the Air-Flow Exper-

iments

To provide the basis for the present data analysis, pressure measurements werecarried out to obtain the wall shear stress, τw, in channel for each investigatedRem of the flow. The pressure transducer, of Valdyne differential type, wasused for measuring the pressure with an accuracy of ±0.25%. Through carefullydrilled small holes along the channel smooth wall, two-dimensional pressuregradient measurements were obtained. The distance before the first station forthe pressure measurements was 30 cm, which corresponded to an X/H ratio of30 from the inlet of the channel. The last station for wall pressure measurementswas far enough away from the channel outlet, i.e. X/H = 10 from the channelend, to ensure that there were no outlet disturbances to the investigated flow.The resultant mean static pressure measurements at various locations in thefully developed flow region were then used to evaluate the streamwise pressure

gradient,dP

dx, which in turn was employed to obtain the wall shear stress and


the wall friction velocity, uτ .

In addition to the pressure measurements and corresponding to the airstream temperature in the channel, the air density and kinematic viscosity werecalculated for the purpose of normalization using the following relations fordensity:

ρ =(Patm + Pst)

<T(4.6)

and Sutherland’s correlation for the kinematic viscosity

ν = 1.458 × 10−6 T 3/2

ρ(T + 110.4)(4.7)

where Patm is the atmospheric pressure and < = 279.1 J/kg K is constant for airunder the ideal gas law.

4.2.3 Wall Distance Apparatus

In the present study, it was very important, for the velocity measurements, toknow the exact distance of the hot wire from the wall. To determine this positionthe calibration procedure proposed by Bhatia et al. [1982] and Durst et al. [2001]was used. It dictates no-flow measurements of the hot-wire output E for differentwall distances y between the wire and the upper surface of a specially preparedcalibration block. The calibration block was made from the same material as theplate for the flow investigations. The distances were determined very preciselywithin ±5µm with a microscope as shown in Figure 4.4 . This distance was takenas the position of the hot wire from the wall. Zanoun [2003] gave more details ofthis method.

�

�

�

Microscope�

Calibration�block�

Hot-wire�holder�

Hot-wire�probe�

Figure 4.4: Photograph of position calibration set-up.


4.2.4 New Method to Measure the Pressure Gradient in

Fully Developed Rough Wall Bounded Flows

Pressure taps are normally used to measure the pressure gradient in the caseof smooth wall bounded flows. In the case of rough wall bounded flows, it isnot possible to use pressure taps because of the large disturbances near thewall according to the influence of rough wall. For this reason, a new method tomeasure the pressure gradient is needed. This new method is described as follows.

It is well known that using normal Pitot tubes, one measures the totalpressure, Ptotal = Pstatic + 1

2ρu2. In a fully developed 2-D channel flow, the

velocity of the flow after a particular height does not change in the flowdirection. Hence, if the total pressure at some point in the fully developed regionis measured using a Pitot tube,

P1total = P1static +1

2ρu2

1 (4.8)

and then the total pressure is also measured in the fully developed region, at adownstream point which has the same height of the first point,

P2total = P2static +1

2ρu2

2 (4.9)

The velocities at the two points are equal, i.e. u1 = u2, because they have thesame height. Subtracting Equation (4.8) from Equation (4.9) leads to

P1total − P2total = P1static − P2static +1

2ρu2

1 −1

2ρu2

2 = P1static − P2static (4.10)

It is clear from Equation (4.10) that the difference between the values of thetotal pressure gives directly the static pressure difference between the two points.

Knowing the distance between the two points, the pressure gradient be-tween them can be calculated. Figure 4.5 shows a sketch of this new methodshowing the positions of the Pitot tubes.

To ensure the validity of this method, it was checked first using smoothchannel flows. The pressure was measured once using pressure taps and thenusing Pitot tubes. The pressure gradients in both cases were calculated andcompared. Very good agreement was found between the two methods. Figure4.6 shows a comparison between the pressure gradient calculated from themeasuremnts of the pressure using pressure taps and using Pitot tubes.


Pressuretransducer 3



Pressure taps

Air flow

Pitot tube 2 Pitot tube 1

Figure 4.5: Sketch of the new method for measuring the pressure difference.

Figure 4.6: Comparison between the pressure gradient measured using Pitot tubeand that using pressure taps.

4.3 Channel Test Section with One Smooth and One Rough Wall 43

Asymmetric Flow Investigations

4.3 Channel Test Section with One Smooth and

One Rough Wall

Because questions arose regarding the effect of roughness in fully developedturbulent channel flows, such as what the effective height of the channel isbecause of the influence of the roughness, the effect of roughness on the velocityprofile and also the wall shear stress, measurements were carried out yieldingadditional information on turbulent channel flow with one smooth and onerough surface. For this purpose, a vertical water channel was employed at theInstitute of Fluid Mechanics (LSTM) of the Friedrich-Alexander-University ofErlangen-Nurnberg.

A water flow facility was set up which permitted mean velocities of up to5 m/s to be obtained. Data were obtained over the range 2700 ≤ Re ≤ 44400.The flow facility and the instrumentation used for the present investigations areshown in Figures 4.7 and 4.8.

Figure 4.7: Sketch of the water channel test section showing the measuring equip-ment.

The channel test section was made of a high-precision smooth aluminum.Asymmetry was introduced by roughening one of the planes while the other wasleft smooth. The geometric dimensions of the cross-section of the channel werewidth B = 200 mm and height H = 10 mm, i.e. providing a channel aspect ratioof 20 : 1. This aspect ratio was considered to be large enough (e.g. Dean [1978]


recommended 7:1) to ensure the required two-dimensionality of the investigatedturbulent, plane channel flows. To drive the flow at a constant, pre-chosenflow rate, a total head tank was installed with a water head of approximately6 m. The water was supplied from the discharge to the overflow tank by aradial pump. The adjustment valve was used to control the flow rate, whichwas measured by means of a Coriolis flow meter, Model IFT9703IC3N2Z. Thechannel test section of dimensions L × B × H = 1.0 × 0.20 × 0.01035 m waspreceded by a rectangular contraction chamber (0.15 × 0.20 m). Upstream ofthe contraction, an 80 mm long honeycomb with 6 mm diameter cells was placedto improve flow uniformity. Downstream of the honeycomb, grids of 1 mm meshsize were installed to reduce the free stream turbulence intensity. Two Plexiglasplates, spanning the whole length of the channel, were used as side-walls toallow optical access to the flow. The height of the channel was kept constant byusing precise metal pieces between the base plates of the channel. The velocitymeasurements were performed using a DANTEC two-component laser-Doppleranemometer. Addition of milk was used for seeding the flow.

Furthermore, similarly to the preparation of the smooth channel for thepressure measurements, the present asymmetric channel test section was pre-pared with pressure tappings along the channel at differnt locations on thesmooth side.


Figure 4.8: Photograph of the experiment showing the test facility and the mea-suring techniques.

4.4 Measuring Techniques Used in Asymmetric Channel Flows 46

4.4 Measuring Techniques Used in Asymmetric

Channel Flows

4.4.1 Laser-Doppler Anemometry (LDA)

Measurement Principles of LDA

LDA is a widely accepted tool for fluid dynamic investigations in gases and liquidsand has been used as such for more than three decades. It is a well-establishedtechnique that gives information about flow velocity. Its non-intrusive princi-ple and directional sensitivity make it very suitable for applications with revers-ing flow, chemically reacting or high-temperature media and rotating machinery,where physical sensors are difficult or impossible to use. It requires tracer parti-cles in the flow.

Features

LDA has the following features:

• no calibration required

• velocity range 0 to supersonic

• one, two or three velocity components simultaneously

• measurement distance from centimeters to meters

• flow reversals can be measured

• high spatial and temporal resolution

• instantaneous and time averaged

• the ability to measure in reversing flows.

Principles

The basic configuration of an LDA consists of :

• a continuous- wave laser

• transmitting optics, including a beam-splitter and a focusing lens

• receiving optics, comprising a focusing lens, an interference filter and aphotodetector

• a signal conditioner and a signal processor.


Advanced systems may include traverse systems and angular encoders.

A Bragg cell is often used as the beamsplitter. It is a glass crystal with avibrating piezo crystal attached. The vibration generates acoustic waves actinglike an optical grid. The output of the Bragg cell is two beams of equal intensitywith frequencies f0 and fshift. These are focused into optical fibers that bringthem to a probe. In the probe, the parallel exit beams from the fibers are focusedby a lens to intersect in the probe volume.

The probe volume

The probe volume is typically a few millimeters long. The light intensity ismodulated owing to interference between the laser beams. This produces parallelplanes of high light intensity, so-called fringes. The fringe distance df is definedby the wavelength of the laser light and the angle between the beams:

df =λ

2 sin( θ2)

(4.11)

This is shown in Figure 4.9. Each particle passage scatters light proportional tothe local light intensity. Flow velocity information comes from light scattered

Figure 4.9: Fringe spacing of the LDA control volume.

by tiny ‘seeding’ particles carried in the fluid as they move through the probevolume. The scattered light contains a Doppler shift, the Doppler frequency fD,which is proportional to the velocity component perpendicular to the bisector ofthe two laser beams, which corresponds to the x−axis shown in the probe volume.

The scattered light is collected by a receiver lens and focused on a photo-detector. An interference filter mounted before the photo-detector passes onlythe required wavelength to the photo-detector. This removes noise from ambientlight and from other wavelengths.


Signal processing

The following steps are involved in the operation of signal processing:

• The photo-detector converts the fluctuating light intensity to an electricalsignal, the Doppler burst, which is sinusoidal with a Gaussian envelope dueto the intensity profile of the laser beams.

• The Doppler bursts are filtered and amplified in the signal processor, whichdetermines fD for each particle, often by frequency analysis using the robustfast Fourier transform algorithm.

• The fringe spacing df provides information about the distance travelled bythe particle.

• The Doppler frequency fD provides information about the time t = 1fD

.

• Since velocity equals distance divided by time, the expression for velocitythus becomes velocity U = df × fD

Determination of the sign of the flow direction

The frequency shift obtained by the Bragg cell makes the fringe pattern move ata constant velocity. Particles which are not moving will generate a signal of theshift frequency fshift. The velocities Upos and Uneg will generate signal frequenciesfpos and fneg , respectively, as illustrated in Figure 4.10. LDA systems withoutfrequency shift cannot distinguish between positive and negative flow directionsor measure zero velocity, but LDA systems with frequency shift can distinguishthe flow direction and measure zero velocity.

Two- and three-component measurements

To measure two velocity components, two extra beams can be added to the opticsin a plane perpendicular to the first beams. All three velocity components canbe measured by two separate probes measuring two and one components, withall the beams intersecting in a common volume. Different wavelengths are usedto separate the measured components. Three photo-detectors with appropriateinterference filters are used to detect scattered light of the three wavelengths.

Modern LDA systems employ a compact transmitter unit comprising theBragg cell and color beam splitters to generate up to six beams: unshifted andfrequency-shifted beams of three different colors. These beams are passed to theprobes via optical fibers.


Figure 4.10: Doppler frequency to velocity transfer function for a frequency-shifted LDA system.

Seeding particles

Liquids often contain sufficient natural seeding, whereas gases must be seededin most cases. Ideally, the particles should be small enough to follow the flow,yet large enough to scatter sufficient light to obtain a good signal-to-noise ratioat the photo-detector output. Typically, the size range of particles is between 1and 10 µm. The particle material can be solid (powder) or liquid (droplets).

In the present study, a two-component laser- Doppler anemometer(DANTEC2D FlowLite system, which is the latest laser technology in new FlowLite LDAsystems) was used in order to make reliable measurements. The new FlowLite2D is the first turnkey fiber system to offer the possibility of measuring twovelocity components simultaneously. Two lasers are integrated in the compactoptical system and linked to a two-component LDA probe. The FlowLite 2D isequipped with two visible and frequency-stable lasers with total laser power of20 mW. The two-component LDA system consisted of a 10 mW He:Ne laser and30 mW Nd:YAG laser, a 5 m long fiber cable, a 60 mm diameter probe and alens with a focal length of f = 160 mm. The signals from photodetectors areprocessed by a two-channel Burst spectrum analyzer (BSA) which was controlledby a PC over an Ethernet network adapter. The system power was 30 mW


lasers, which gives a better laser power balance. Durst et al.’s book [1976]provides a thorough description of laser- Doppler anemometry.

4.4.2 Determination of the Middle Point of the Channel

It was essential for the distance between the measuring volume of LDA and areference location, in this case the smooth wall, to be determined accurately forthe proof of the analytical considerations. In oreder to achieve this, a small pin (ofdiameter 4 µm) was constructed, as a reference point, in the asymmetric channelunder the condition that its end is exactly at the center of the channel. Withthe aid of the laser system and other equipment such as a traverse mechanism,down mixer, filter and multimeter, one can exactly reach the end of the pin bymeasuring the scattered light intensity. Figure 4.11 shows the arrangement ofthis equipment to achieve this end. The following procedure was carried out todetermine the middle of the channel.

• Starting to adjust the laser beam using the traverse mechanism which ismoving in the three directions until one receives a signal in both directions,i.e. the direction of the pin and the perpendicular to it.

• Moving the laser beam perpendicular to the pin and recording the voltageof the signal at each position.

• Taking the point at which the voltage has the highest value as the middlepoint of the end of the pin, which is middle point of the channel.

To ensure the procedure is accurate one has to go a little bit on the pin (69 µm)and again take the same measurements. This gives the maximum voltage at thesame point.

Repeating the last step going deeply on the pin (109 µm) also gives thesame point as the maximum one. Figure 4.12 shows the resultant of thesemeasurements.

PhotoDiode

Multi-meterLaser

DownMixer

OutputPin

FilterV = 0

Figure 4.11: Sketch of the equipment used to determine the middle of the channel.


Figure 4.12: Relation between the position of the laser beam and the correspond-ing voltage.

Chapter 5

RESULTS, ANALYSES ANDDISCUSSION

5.1 Results of Smooth Channel Investigations

5.1.1 Pressure Measurements and Wall Shear Stress

A wide varitiey of measuring techniques are available to obtain the wall shearstress experimentally and therefore the skin friction data. Both Winter [1976]and Fernholz et al. [1996] presented good reviews of the various techniques tomeasure wall skin friction.

The local skin friction is an essential quantity for investigating wall-boundedshear flows. For instance, precise estimations of the parameters of the logarith-mic velocity profile require accurate and independent values for the wall shearstress from velocity measurements. In channel flows, mean pressure gradientmeasurements along the channel test section provide an accurate method toobtain the wall skin friction data. Therefore, pressure measurements were carriedout in the present study for each investigated channel flow Rem value to obtainthe wall shear stress τw. Through carefully drilled small holes along the channelwall, pressure gradient measurements were carried out. Three static pressuretaps across the channel were carefully installed at each measuring location inthe flow direction for mean pressure measurements. The mean static pressure ateach location in the fully developed flow region was then averaged over the threepressure holes and used to evaluate the streamwise pressure gradient, dPw/dx,which in turn was employed to obtain the wall shear stress. As a result, the wallshear stress (τw) and the corresponding wall friction velocity (uτ) were calculatedas follows:

τw = −h(

dPw

dx

)

, uτ =

√

τw

ρ(5.1)

5.1 Results of Smooth Channel Investigations 53

In addition to the mean pressure measurements, the air stream temperature inthe channel test section was measured to calculate the air density and kinematicviscosity using the following relations for the density:

ρ =(Patm + Pst)

<T(5.2)

and Sutherland’s correlation for the kinematic viscosity:

ν = 1.458 × 10−6 T 3/2

ρ(T + 110.4)(5.3)

where Patm is the atmospheric pressure and < = 279.1 J/kg K is the specificconstant for air employed in the ideal gas law.

Figure 5.1 presents examples of the mean static pressure measurements atdifferent locations along the smooth channel test sections starting at x/H = 20from the channel inlet and ending at x/H = 80. The figure clearly indicatesthat, because of the linear relationship between the pressure and the distance,that the flow field was fully developed for all cases under investigation, at leastas far as the pressure distribution in the flow direction was concerned.

Figure 5.1: Pressure gradient distribution along part of (a) the small channel and(b) the large channel at different Reynolds numbers.

In addition to the mean pressure gradient measurements, the bulk flowvelcoity for each set of measurements, i.e. for each Rem value, was obtainedby measuring the dynamic pressure through the inlet nozzle of the flow facilitydepicted in Figure 4.1, where a uniform and well-defined flow field existed.The pressure difference between the nozzle inlet and throat of the nozzle wasmeasured utilizing a high-precision pressure transducer. Moreover, the airtemperature inside the settling chamber was monitored at all times during


measurements within an accuracy range of ±0.05 ◦C.

Utilizing the above measurements, i.e. the mean pressure gradient mea-surements provided in Figure 5.2, in connection with the bulk velocity, thewall skin friction data were obtained independently of the velocity profilemeasurements. Figure 5.3 shows a comparison between the wall shear stress inthe case of a small smooth channel and a larger smooth channel.

Thereafter, the wall friction velocity uτ and the kinematic viscosity ν wereused for scaling all results to yield the normalized velocity distribution over thechannel half-width.

Figure 5.2: Comparison between the measured pressure gradient in the small andlarge channels for different Remean.

In order to ensure the validity of the pressure measurements, the followingprocedure was performed.

Zanoun [2003] in his study of the smooth channel introduced a semi-empiricalrelation betwen the Reynolds number and the skin friction factor:

cf = 0.0624 Re−0.25 (5.4)

where Re is based on half of the channel height.


Figure 5.3: Comparison between the wall shear stress in the small and largechannels, for different Remean.

Also, the wall shear stress and the skin friction factor are correlated asfolllows:

τw =cf

2ρ U

2(5.5)

Taking the last relationships into account, one can find a correlation between thewall shear stress in the case of two smooth channels of different heights as follows.

The wall shear stress in the case of a small channel of height h is givenby

τwh=

cf

2ρ u2 = 0.0312 Re−0.25

h ρ u2h2ν2

h2ν2= 0.0312ρ

ν2

h2Re

7/4h (5.6)

and for a large channel of height H

τwH=

cf

2ρ U

2= 0.0312 Re−0.25

H ρ U2 H2ν2

H2ν2= 0.0312ρ

ν2

H2Re

7/4H (5.7)

Dividing Equation (5.6) by Equation (5.7) leads to

τwh

τwH

=H2

h2

Re7/4h

Re7/4H

(5.8)

At the same Reynolds number, Equation (5.8) gives(

τwh

τwH

)

Reh=ReH

=H2

h2(5.9)


The last equation was used to check the present pressure measurements. Inthe present work, the ratio between the heights of the smooth channels was2:1, which means that the ratio between the wall shear stresses must be 1:4 atthe same Reynolds number. The relations between the wall shear stresses andthe Reynolds numbers, shown in Figure 5.3, were fitted using the least-squaresmethod and then the corresponding values at the same Renolds numbers werecompared, as shown in Figure 5.4. The average value of the ratio between themeasured wall shear stresses was 0.248 at the same Reynolds numbers, whichwas in a very good agreement with the theoretical value.

Figure 5.4: Relation between the wall shear stress in the small and large smoothchannels at corresponding Reynolds numbers.

The same check of pressure measurements was made for the data of Niku-radse, who measured the wall shear stress for smooth pipes with differentdiameters, i.e. 1, 2, 3 and 5 cm. The behavior of his measured wall shear stresseswith the Reynolds number is shown in Figure 5.5. According to the theoreticalcalculations, the ratio between the wall shear stresses at the same Reynoldsnumbers must be 1:4, 1:9 and 1:25, respectively. The relations between the wallshear stresses and the Reynolds numbers shown in Figure 5.5 were fitted usingthe least-squares method and then the corresponding values at the same Renolds


numbers were compared. Table 5.1 shows the comparison between the measuredand theoretical values of the ratio of the wall shear stress at the same Reynoldsnumber for Nikuradse’s data.

Figure 5.5: Nikuradse’s data: comparison between the wall shear stress in pipeswith different diameters.

It is clear that there is good agreement between the measured and theo-retical values.

5.1.2 Velocity Measurements in Smooth Channels

Velocity profile measurements in the smooth small channel of height 5 cm, andin the larger smooth channel of height 10 cm were carried out using a DANTEC55P15 constant-temperature anemometer. The maximum Reynolds number was68000 for the small channel and 110000 for the large channel. The resultantvelocity profiles are shown in Figures 5.6 and 5.7 for the small and large smoothchannels, respectively.


Case Calculated MeasuredFirst case 0.25 0.26

Second case 0.11 0.12Third case 0.040 0.048

Table 5.1: Comparison between the measured and calculated values of the ex-pected ratio of the wall shear stress at the same Re for Nikuradse’s data.

Figure 5.6: Samples of the velocity profile for the small smooth channel.

5.1.3 Confirmation of the Findings of Zanoun and Durst

Zanoun et al. [2003] and Durst et al. [2003] suggested that the logarithmicvelocity profile exists over approximately 70% of the complete boundary layersadjacent to both of the channel walls starting from y+ = 150. They also foundthat the von Karman constant κ in the log-law to be equivalent to approximatelly1/e. Initially, as mentioned in the Introduction, the findings of Zanoun et al.[2003] and Durst et al. [2003] about the logarithmic law and the von Karmanconstant will be checked for the present smooth channel flows.

Diagnostic Functions for the Law of the Wall

In wall-bounded flows, for sufficiently high Reynolds number, an overlap region,where the mean velocity profile may be represented by either a logarithmic or


Figure 5.7: Samples of the velocity profile for the large smooth channel.

a power law, exists between flow in the wall layer and flow in the outer region.In addition, it is common to express velocity data in wall units and, therefore,the characteristic velocities, uc = uτ and Uc = Uτ , and viscous lengths, lc = ν/uτ

and Lc = ν/Uτ , scales were used for normalization with both small units andlarge units. The wall friction velocities uτ and Uτ used for normalization werededuced from the pressure gradient measurements in the downstream part of thetest section where a fully developed flow existed, as explained earlier. The meanvelocity profiles were then normalized using the corresponding shear velocity toyield u+ = u/uτ for the small channel and U+ = U/Uτ for the large channel andthe wall distance with the viscous length to give y+ = y/lc for the small channeland Y + = Y/Lc for the large channel. A summary of the dimensionless meanvelocity distributions u+ = F (y+) and U+ = F (Y +) are presented in Figure 5.8for the small channel and in Figure 5.9 for the large channel.

The logarithmic representation of the mean velocity distribution in thislayer was derived originally by von Karman [1930] based on similarity hypothe-sis, by Prandtl [1932] on the basis of the mixing length theory and by Millikan[1938] using asymptotic analysis. Therefore, the normalized velocity profiles inFigures 5.8 and 5.9 were analyzed with respect to the question of whether theprofile in the inertial sublayer behaves in a logarithmic manner or not. Utilizing


Figure 5.8: Velocity distribution in the small channel over a wide range ofReynolds numbers.

the wall friction data, the proposed log-law takes the following well-known form:

u+ =1

κln y+ + B, U+ =

1

κln Y + + B (5.10)

for the small and large channels, repectively, where κ is the von Karman constantand B is an additive constant.

The present study also embraced the question of whether the velocity pro-file obeys a power law, as proposed in the earlier investigations of Millikan [1938]and more recently by Barenblatt [1993], Wosnik et al. [2001] and Oberlack[2001], in the form

u+ = Cy+γ, U+ = CY +γ

(5.11)

where C and γ are empirical constants, but are often Reynolds number dependent.

In addition, it is of crucial importance in data processing to obtain themean velocity gradient to evaluate the law of the wall. The normalized meanvelocity gradient was then obtained from the corresponding mean velocities, u+

and U+, differentiated with respect to the normalized wall distance, y+ and Y +,using the central difference scheme and also the three-point scheme proposed byGrußman and Roos [1994]:

du+

dy+=

y+i+1 − y+

i

y+i+1 − y+

i−1

u+i − u+

i−1

y+i − y+

i−1

+y+

i − y+i−1

y+i+1 − y+

i−1

u+i+1 − u+

i

y+i+1 − y+

i

(5.12)


Figure 5.9: Velocity distribution in the large channel over a wide range ofReynolds numbers.

The results for the mean velocity gradients, du+/dy+ = f(y+) anddU+/dY + = f(Y +), are presented in Figures 5.10 and 5.11 for the smalland large channel flows, respectively.

The resultant velocity distributions were then analyzed with respect tothe question of whether the velocity profile in the investigated channel flowsbehaves in a logarithmic or in a power-law manner. To answer this questionand to see more clearly the effect of the Reynolds number on the mean velocityprofile, the following diagnostic functions suggested by Osterlund et al. [1999,2000] and Wosnik et al. [2001] are introduced:

Ξ =[

y+dU+

dy+

]−1

, Γ =y+

U+

[

dU+

dy+

]

, (5.13)

which represent the normalized slopes of the mean velocity distributionsin either the logarithmic or the power region, respectively, and the behavior ofboth functions is shown in Figures 5.12 and 5.14 for the small channel and inFigures 5.13 and 5.15 for the large channel flows, respectively.


Figure 5.10: Velocity gradient in the small channel over a range of Reynoldsnumbers.

A constant behavior of Ξ for high enough Reynolds numbers leads to theexistence of a logarithmic layer, supporting Millikan’s [1938] argument that alogarithmic law is expected in a high Reynolds number turbulent channel flowwith a constant value of the von Karman constant. In the inertial sublayer, allΞ profiles in both Figures 5.12 and 5.13 showed a constant slope at y+ ≥ 150for the small channel and Y + ≥ 150 for the large channel, which means that thelogarithmic law is a good representation of the mean velocity measured in theoverlap region for the channel flow. As a result, the behavior of the Ξ functionindicates clearly that the normalized mean velocity profiles of two-dimensionalfully developed turbulent plane-channel flows are well described by a logarithmicvelocity distribution. On the other hand, a constant behavior of the power-lawdiagnostic function Γ indicates that the mean velocity profile should behavein a power-law form. However, the general trend of the power-law diagnosticfunction in Figures 5.14 and 5.15 is a monotonic decrease in the overlap regionwhen plotted versus wall distance. Therefore, the power law is far from useful todescribe the mean velocity profile in the overlap region, e.g. see Clauser [1954],who came to the conclusion that no universal values can be assigned to C and γ.

To proceed further to obtain the exact values of the law of the wall, theauthor adopted the new approach, first suggested by Zanoun and Durst [2003]rather than following that in the earlier investigations, e.g. Osterlund et al.


Figure 5.11: Velocity gradient in the large channel over a range of Reynoldsnumbers.

[1999, 2000]. The new approach mainly depends, as proposed by Zanoun[2003], on the natural logarithm of the mean velocity gradient versus the naturallogarithm of the normalized wall distance, and for this purpose selected samplesof the data are shown in Figures 5.16 and 5.17 for the small and the channelflows, respectively.

For instance, for Reτ = 835 in the small channel by fitting ln(du+/dy+)versus ln y+ using the least-squares curve fit over the wall interval fromy+ = 150 to y+ = 585, which represents about 50% of (H+/2), the meanvalue of the von Karman constant κ was found to be 0.368. Similarly, carryingout the evaluation of the logarithmic law of the wall over the wall intervalwhere the log-law fits the data well, the von Karman constant was estimatedover the range of the Reynolds numbers in both the small and large channel flows.

For the purpose of the current analysis, all the data shown in Figures5.10 and 5.11 were re-plotted in the form ln(dU+/dy+) versus ln y+ over the wallintervals 150 ≤ y+ ≤ 75% of (H+/2) for the small channel and 150 ≤ Y + ≤ 75%of (H+/2) for the large channel and the results are shown in Figures 5.18 and5.19, respectively. All higher Reynolds numbers, data for the small and largechannel results and corresponding to the different Reynolds numbers showedthe same pattern where the log-law fits the data within the experimental error.The region where a good fit of the data to the log-law was considered for the


Figure 5.12: Log-law diagnostic function, Ξ, for the law of the wall in a small 2-Dplane channel flow.

least-squares curve fit, yielding a reliable value of the von Karman constant.

From Equation (5.10), the logarithmic law of the wall can be re-writtenas follows:

U+ =1

κln y+ +B, ⇒ dU+

dy+=

1

κy+⇒ ln

(

dU+

dy+

)

= − ln y+ +ln(

1

κ

)

(5.14)

Using an optimized least-squares curve fit for the best fit of each individual caseover the new upper and lower limits resulted in a value of 1 ± 2.5% for theintercept ln (1/κ) (see Figures 5.16 and 5.17). As a result, the data in Figures5.16 and 5.17 suggest (within an accuracy of measurements of ±2.5%) that thereexists a functional relationship for the far distant region of the normalized meanvelocity distribution, given by

ln(

dU+

dy+

)

= − ln y+ + 1 ± 2.5% (5.15)

Moreover, the least-squares curve fit of the experimental results shown in Fig-ure 5.18 for small channel flow as one set of data considering the lower limity+ = 150 and the upper limit y+ ≤ 0.75h+ produces a value of the interceptln (1/κ) very close to 1.0, and for the large channel also. Hence, taking the


Figure 5.13: Log-law diagnostic function, Ξ, for the law of the wall in a large 2-Dplane channel flow.

intercept in Equation (5.15) as equal to 1.0 results in

ln(

dU+

dy+

)

= − ln y+ + 1 ⇒ ln(

y+dU+

dy+

)

= 1 ⇒(

y+dU+

dy+

)

= e (5.16)

κ =(

y+dU+

dy+

)−1

=1

e(5.17)

which is an interesting result deduced from the author’s experimental data con-firming the findings of Zanoun et al. [2003]. This readily suggests a logarithmicregion for the far field of the normalized mean velocity profile with a von Karmanconstant κ = 1/e. It is worth noting that this experimental finding has no an-alytical foundation; however, the present experimental result will provide strongsupport for a future rigorous theory to prove that the von Karman constant isequal to 1/e.

5.1.4 Check of the Similarity Condition for Smooth Chan-

nels

To check the validity of the similarity condition between the smooth channels,which was deduced in Chapter 3, the following steps were carried out:


Figure 5.14: Power-law diagnostic function, Γ, for the law of the wall in a small2-D plane channel flow.

• The measured velocity profiles of both the small and large smooth channelsat the same Reynolds number were selected.

• The normalized values of the velocity u+ and U+ for the small and largechannels, repectively were calculated using the estimated values of the fric-tion velocities uτ and Uτ .

• Equation (3.37) was used to calculate u+ from the measured values of U+

utilizing the heights of the smooth channels and the von Karman constant.

• The values of u+ resulted from Equation (3.37) and those calculated fromthe direct measurements were compared.

Figure 5.20 shows the similarity between the small and large channels at aReynolds number of 58000.

The above procedure was applied to different values of Reynolds numberand very good agreement was observed. Figures 5.21–5.24 show samples of thecases of similarity between the small and large channels at different Reynoldsnumbers. It is clear, according to these figures, that good agreement betweenthe values of u+ was found in both cases.


Figure 5.15: Power-law diagnostic function, Γ, for the law of the wall in a large2-D plane channel flow.

The second condition of similarity between smooth walls which was dis-cussed in Chapter 3 was applied for the present experimental data for thesmooth channels. A selected sample is shown in Figure 5.25. At the same y/H,the values of U+ for the small channel and large channels were calculated. Itwas found that U+

b = U+s + 1/κ ln Reτ b

Reτ s

, confirming the findings in Chapter 3.

Nikuradse’s data cover smooth pipes over a range of Reynolds number upto 3.2 × 106. The data of Nikuradse for smooth walls does not help in the lastsimilarity procedure, which was deduced at equal Reynolds numbers. For thisreason, a general similarity condition for smooth walls at any Reynolds numberwas investigated through the following procedure.

If the Reynolds number of the large channel related to the Reynolds num-ber of the small channel through the relation ReH = C Reh, where C is aconstant, then, introducing this into Equation (3.20) leads to

τwh

τwH

=H2

C7/4h2(5.18)

and introducing this also into Equation (3.36) leads to the following similarityrelation for smooth walls of different heights at any Reynolds number:

u+ = U+ +1

κln

H

C7/8h(5.19)


Figure 5.16: ln(du+/dy+) − ln y+, at Reynolds number 15822.

To check the validity of the similarity condition for the data of Nikuradseat different Reynolds numbers, the following steps were carried out:

• The measured velocity profiles of Nikuradse’s data at different Reynoldsnumbers were selected.

• The constant C in Equation (5.18) was calculated for each case.

• The normalized values of the velocity u+ and U+ for the small and largepipes were calculated using the given values of the friction velocities uτ andUτ .

• Equation (5.19) was used to calculate u+ from the measured values of U+

utilizing the diameters of the smooth pipes (instead of the heights of thechannels) and the von Karman constant.

• The values of u+ resulting from Equation (5.19) and those calculated fromthe direct measurements were compared.

Figure 5.26 shows the similarity between the small and large pipes at Reynoldsnumbers 4000 and 105000, respectively.


Figure 5.17: ln(du+/dy+) − ln y+, at Reynolds number 19264 .

The above procedure was applied to different values of Reynolds numberand different pipe dimeters and very good agreement was observed. Figure 5.27shows samples of the cases of similarity between the small and large pipes forthe measurements of Nikuradse.

It can be concluded that for smooth pipes or channels with different di-mensions, it is sufficient to measure only the velocity profiles for one of them overa wide range of Reynolds numbers. Utilizing the present similarity condition, onecan calculate the corresponding normalized velocity profiles for the other pipesor channels, at the corresponding Reynolds numbers, knowing their dimensionsand the von Karman constant.

The second condition of similarity between smooth walls which was dis-cussed in Chapter 3 was applied to the experimental data of Nikuradse forsmooth pipes. A selected sample is shown in Figure 5.28. At the same y/H, thevalues of U+ for the small and large pipes were calculated. It was found thatU+

b = U+s + 1/κ ln Reτ b

Reτ s

, confirming the findings in Chapter 3.


Figure 5.18: ln(du+/dy+) − ln y+ representation of the mean velocity gradientover the range of Reynolds numbers in the small channel flow.

Figure 5.19: ln(du+/dy+) − ln y+ representation of the mean velocity gradientover the range of Reynolds numbers in the large channel flow.


Figure 5.20: Similarity between the small and large channels at Re = 58000.




Figure 5.25: Relation of Reτ y/H versus U+ for a wide range of Re.


Figure 5.26: Nikuradse’s data: similarity between the small and large pipes, Re= 4000 and Re = 105000, respectively.

Figure 5.27: Nikuradse’s data: similarity between the small and large pipes, Re= 105000 and Re = 725000, respectively.


Figure 5.28: Nikuradse’s data: relation of Reτ versus U+ for a wide range of Re.

5.2 Pressure Measurements and Friction Factrors in Asymmetric Flows 76

5.2 Pressure Measurements and Friction Fac-

trors in Asymmetric Flows

Measurement of the local skin friction is an essential aspect of wall-boundedshear flows. The asymmetric water channel described in Chapter 4 was used toclarify the findings of this kind of flow mentioned in Chapter 3. The asymmetrywas introduced by roughening one of the surfaces of the channel while the otherwas left smooth.

To provide the basis for the asymmetric channel data analysis, pressuremeasurements were carried out for each investigated Rem of the flow to obtainthe wall shear stress τw according to Equations (3.52) and (3.54). Throughsmall holes carefully drilled along the channel smooth wall, pressure gradientmeasurements were obtained. The mean static pressure measurements atdifferent stations in the fully developed flow region were then used to evaluatethe streamwise pressure gradient, dP/dx, which in turn was employed to obtainthe wall shear stress and the wall friction velocity uτ . Figure 5.29 showsthe pressure gradient distribution along the asymmetric channel at differentReynolds numbers. The slopes of the linear relations between the distance from

Figure 5.29: Pressure gradient distribution along the channel with rough surfaceat different Reynolds numbers.

the inlet and the pressure at each Reynolds number give the estimated values

5.3 Velocity Measurements in Asymmetric Flows 77

of dP/dx. Figure 5.30 shows the estimated values of the pressure gradient atdifferent Reynolds numbers, which is important for calculating the wall shearstress in both smooth and rough surfaces.

Figure 5.30: Estimated pressure gradient at different Reynolds numbers for thepresent flow facility.

5.3 Velocity Measurements in Asymmetric

Flows

Using the DANTEC 2D LDA system, mentioned in Chapter 4, the velocityprofiles and also the two velocity components for different Reynolds numbersfor the asymmetric water channel were carried out. Figure 5.31 shows thedistribution of mean velocity between the rough-smooth parallel surfaces of thechannel for six values of Reynolds number, where the Reynolds number is basedon the mean velocity and half-height of the channel. The profiles are stronglyasymmetric and are appreciably dependent on Reynolds number. This Reynoldsnumber influence may be attributed to the fact that the ratio of the rough wall

shear stress to the smooth wall shear stress increases approximately as U1

5max; for

more details, see Hanjalic and Launder [1972]. It is clear from Figure 5.31 thatthe maximum velocity is not exactly at the center of the channel but is shiftedtowards the smooth wall.


Figure 5.31: Velocity profiles at different Reynolds numbers.

To determine the shift exactly, the position of the zero total shear stress,

which is given by τ = µdu

dy− ρu′v′ , was determined at each investigated

Reynolds number. A sample of the calculated shear stress for the Reynoldsnumber 44356 is shown in Figure 5.32. It is clear that the position of the zeroshear stress shifted towards the smooth wall more than the shift of the maximumvelocity, confirming the findings of Hanjalic and Launder [1972]. Figure 5.33shows the positions at which the total shear stress is equal to zero for differentReynolds numbers. The distance between the smooth wall and this position ofthe zero shear stress gives directly the value of ε at each Reynolds number whichis important for calculating the wall shear stress for the smooth and rough wallsin asymmetric flows.

Friction Factor Estimation

Now all the requirements of Equations (3.52) and (3.54) are available, i.e. thevalues of dP/dx and ε, then one can calculate the wall shear stress in both sides ofthe channel, i.e. the smooth and the rough sides. Figure 5.34 shows a comparisonbetween the estimated values of the wall shear stress for the smooth and roughwalls in the asymmetric channel at different Reynolds numbers.


Figure 5.32: The total shear stress versus the position, Re=44356.

Figure 5.33: Position of zero total shear stress at different Reynolds numbers.


Figure 5.34: Wall shear stress in both the smooth and rough sides at differentReynolds numbers.

5.4 Determination of the Effective Height of the Asymmetric Channel 81

5.4 Determination of the Effective Height of the

Asymmetric Channel

The effective height of the channel is the height of the channel under theinfluence of roughness. It is questionable and difficult to determine exactly theeffective roughness in the case of rough walls in channels and pipes. A numberof researchers have raised this difficulty and how it is important to determinethe correct origin or the reference point for the measurements to guarantee thespatial accuracy; see, e.g., Moore [1951], Clauser [1954], Perry et al. [1969] andKoh [1992]. In this work, a new way to determine the effective roughness wasdeveloped.

From the measurements of the velocity profiles, one can calculate the shiftfrom the center of the channel, i.e. ε, from which one can calculate the wallshear stress at the rough surface similar to the last procedure, i.e. τwr, which inturn leads to the friction velocity uτ .

It is generally accepted, according to Hanjalic and Launder [1968], thatthe normalized velocity profile close to a rough wall can be expressed as:

U+ =1

κ(ln

y

ks

) + C (5.20)

where U+ =u

uτand ks is the roughness height. Equation (5.20) can be easily

transformed to the following form:

U+ =1

κ(ln

y+

k+s

) + C ⇒ U+ =1

κ(ln y+) + C − 1

κ(ln k+

s ) (5.21)

since y+ =yuτ

νand k+

s =ksuτ

ν.

Taking the derivative of Equation 5.21 leads to the following useful form:

dU+

dy+=

1

κy+⇒ y+dU+

dy+=

1

κ(5.22)

The last equation is useful because it gives a simple and clear relation betweenU+ and y+ in the vicinity of the rough wall, ignoring the effect of the roughnessheight and also the additive constant.

To determine the effective height of the channel, the following iterationprocedure using a FORTRAN program was applied:

• Assuming an initial value of the height of the channel, i.e. H.


• Calculating the wall shear stress in the rough case, i.e τwr, using Equa-tion (3.54).

• Calulating the rough friction velocity, i.e. uτ , from the relation uτ = (τwr

ρ)

1

2 .

• Calculating U+ and y+ from the relations U+ =u

uτ

and y+ =yuτ

ν.

• Assuming the von Karman constant κ to be 1/e.

• Calculating y+dU+

dy+and

1

κ.

• Checking the relation in Equation (5.22).

• Calculating the rms value between y+dU+

dy+and

1

κy+for the data range

which the log-law fits the data well.

• Repeating the whole procedure using H + δH instead of. H

• Repeating the whole procedure using H − δH instead of. H

• Comparing the rms values in all cases and take the value of H at which therms value is the smallest one as the effective height of the channel.

A sketch of the Fortran program is shown in the Appendix. Now one candetermine exactly the value of the effective height of the channel, i.e. H, fromwhich one can calculate the wall shear stress at the rough surface, i.e. τwr usingEquation (3.54).

Repeating the iteration method for each Reynolds number leads to the correc-tions for the effective heights for each Reynolds number. From the estimatedvalues of the corrections, one can easy calculate the required effective height ofthe channel at each Reynolds number. Table 5.2 shows the different values ofthe corrections for the effective height and the corresponding effective heights ofthe asymmetric channel at different Reynolds numbers. It is clear from Table5.2 that the effective height corrections are inversly proportional to the Reynoldsnumbers but the effective height is directly proportional to the Reynolds number.

Figure 5.35 shows a sample of the uncorrected and corrected values of

y+dU+

dy+using the iteration method. Using the corrected values of the H, one

can calculate the estimated values of the wall shear stress corresponding to therough wall. The estimated values of the wall shear stress in both smooth andrough walls at different Reynolds numbers were calculated and are summarized


Re Correction (µ)m Effective height (µ)m44356 090 1026035418 130 1022026883 160 1019015458 180 1018009779 190 1016007500 210 10140

Table 5.2: Values of the corrections to the effective heights and the correspondingeffective heights at different Reynolds numbers.

in Table 5.3 and shown in Figure 5.36.

Re τws τwr

44356 0.4562 0.659835418 0.2973 0.405226883 0.1787 0.235515458 0.05841 0.075979779 0.01659 0.038927500 0.01659 0.02143

Table 5.3: Wall shear stress for the smooth and rough cases.

The normalized values of the corrections in the position, ∆H+, can be

calculated from the relation ∆H+ =∆Huτ

νat each Reynolds number. The

resultant values of ∆H+ versus the Reynolds numbers are shown in Figure 5.37.

As a result, one can conclude that as the Reynolds number increases, thelocation y wher u = 0 goes deep into the roughness, i.e. more projection of theroughness penetrates into the flow.


Figure 5.35: Uncorrected and corrected values of dU+/dy+ after the iteration,Re = 44356.

Figure 5.36: Wall shear stress for the smooth and rough cases.


Figure 5.37: ∆H+ variance with Reynolds number.

5.5 Pressure Measurements and Wall Shear Stress in the Channel with RoughWalls 86

5.5 Pressure Measurements and Wall Shear

Stress in the Channel with Rough Walls

The method for measuring the static pressure, which was discussed in Chapter 4,was used to determine the static pressure in the channel of two rough walls.Results of static pressure measurements in the streamwise direction are presentedin Figure 5.38 for the channel with rough walls.

Figure 5.38: Pressure gradient determination along the rough channel presentedfor different Reynolds numbers.

The mean static pressure measurements, Pst, in Figure 5.38 were used toevaluate the static pressure gradient, dP/dx, which in turn was employed toobtain the wall shear stress and the wall friction velocity, uτ , as follows:

τw = −h(

dP

dx

)

, uτ =

√

τw

ρ(5.23)

As a result, the wall skin friction data were obtained independently of thevelocity profile using the pressure gradient measurements provided in Figure5.39. Thereafter, the wall friction velocity uτ and the kinematic viscosity ν wereused for scaling to yield the normalized velocity distribution over the channelhalf-width. A comparison between the wall shear stress for the smooth andrough channel cases is shown in Figure 5.40.

In order to ensure the validity of the pressure measurements for the rough


Figure 5.39: Comparison between the measured pressure gradient for the smoothand rough channel flows.

channel flow, the following procedure was followed.

The theoretical relationship betwen the Reynolds number and the skinfriction is given by the following relation:

√λ = 5.66

uτ

U(5.24)

where λ is Darcy friction factor; see Nikuradse [1933] and Colebrook [1939]. Thewall shear stress calculated from the measured values of the pressure gradientand that calculated using the last equation are compared in Figure 5.41, showingvery good agreement between the corresponding values.

The same check was made for the rough pipe data of Nikuradse, whomeasured the wall shear stress for pipes with different diameters, i.e. 9.94, 4.94,4.87, 4.82, 2.474 and 2.434 cm, and also with different roughness heights. Asample of the comparison between the measured and caculated Darcy frictionfactors for Nikuradse’s measurements is shown in Figure 5.42, which shows goodagreement between the calculated and measured values.


Figure 5.40: Comparison between the wall shear stresses for the smooth andrough channel flows.

Figure 5.41: Comparison between the calculated (Equation 5.24) and the mea-sured Darcy friction factors for the rough channel.


Figure 5.42: Nikuradse’s rough data: comparison between the calculated (Equa-tion 5.24) and the measured Darcy friction factors.

5.6 Velocity Measurements in the Channel with Rough Walls 90

5.6 Velocity Measurements in the Channel with

Rough Walls

Velocity profile measurements for the channel with rough walls were also carriedout using a DANTEC 55P15 constant-temperature anemometer over the entirerange of Reynolds numbers up to 60000. The resultant velocity profiles for thechannel with rough walls are shown in Figure 5.43.

Figure 5.43: Samples of the velocity profile for the channel with rough walls.

In order to check the validity of the theoretical considerations for the rough wallswhich were discussed in Chapter 3, the following steps were followed:

• The value of κ in the rough case must be the same as its value in the smooth

case, i.e. κ =1

e.

• The validity of the relation between the normalized value of the distance inthe rough and the smooth case, i.e. Equation (3.63), should be checked.

• The new procedure, i.e. Equation (3.64), should be utilized to calculatethe shift ∆B between the log-law lines in the smooth and rough cases andcompare the calculated and experimental values.

• Applying the new procedure to the data of Nikuradse to caculate the valuesof ∆B and compare them with his experimental values.


5.6.1 Checking the Value of κ in the Rough Case

To ensure that the value of the von Karman constant κ in the rough case is thesame as its value in the smooth case, the diagnostic functions for the law of thewall, which were shown before for the smooth case, were treated also for therough case.

A summary of the dimensionless mean velocity distributions, U+ = F (y+), forthe rough and smooth channels are presented in Figure 5.44. The diagramconsists of a family of parallel straight lines with shifts directly proportionalto the Reynolds number. The normalized velocity profiles in Figure 5.44 were

Figure 5.44: Velocity distribution in the rough channel over a range of Reynoldsnumbers.

analyzed also with respect to the question of whether the profile in the inertialsublayer behaves in a logarithmic manner or not. Utilizing the wall friction data,the proposed log-law for the rough channel takes the following well-known form:

U+ =1

κln

y

k+ B, (5.25)

where κ is the von Karman constant, k is the roughness height and B is anadditive constant. The last equation can be easily transformed to the followinguseful form:

U+ =1

κln y+ − 1

κln k+ + B (5.26)


where k+ = kuτ

ν. The distributions of the dignostic functions, i.e. Ξ and Γ, for

the channel with rough walls are shown in Figures 5.45 and 5.46, respectively.

Figure 5.45: Diagnostic function Ξ of the law of the wall in 2-D rough channelflow.

As mentioned before, a constant value of Ξ, for high enough Reynolds numbers,suggested the existence of a logarithmic layer. In the investigated flow, allΞ profiles in Figure 5.45 showed a constant slope for y+ ≥ 150 for all roughchannel flows, which means that a logarithmic law is a good representation ofthe mean velocity measured in the overlap region for the rough channel flow. Asa result, the behavior of Ξ indicates clearly that the normalized mean velocityprofiles of two-dimensional fully developed turbulent rough plane-channel flowsare well described by a logarithmic velocity distribution. On the other hand,a constant behavior of the power-law diagnostic function Γ indicates that themean velocity profile should behave in a power-law form. However, the generaltrend of the power-law diagnostic function in Figure 5.46 is a monotonic decreasefor all rough channel flows when plotted versus the normalized wall distance.Therefore, the power law is far from useful to describe the mean velocity profilein rough wall channel flows.

To proceed further to obtain the characteristic values of the law of thewall, a selected sample of the data is shown in Figures 5.47 and 5.48 for roughchannel flows.


Figure 5.46: Diagnostic function Γ of the law of the wall in 2-D rough channelflow .

For instance, for Reτ = 860 in the rough channel, by fitting ln(du+/dy+)versus ln y+ using the least-squares curve fit over the wall interval from y+ = 150to 600, which represents about 55% of (H+/2), the mean value of the vonKarman constant κ, was found to be 0.366. Also for Reτ = 3423, by fittingln(du+/dy+) versus ln y+ using the least-squares curve fit over the wall intervalfrom y+ = 500 to 2567, which represents about 75% of (H+/2), the mean valueof the von Karman constant κ, was found to be 0.3685.

Similarly, carrying out the evaluation of the logarithmic law of the wallover the wall interval where the log-law fits the data well, the von Karmanconstant was estimated over a wide range of the Reynolds numbers in the roughchannel flows. The results of ln(dU+/dy+) versus ln y+ shown in Figures 5.47and 5.48 showed the same pattern, where the logarithmic law fits the data wellwithin the experimental error for different Reynolds numbers.

All the higher Reynolds number data, i.e. Reτ > 1500, for the channelwith rough walls and corresponding to the different Reynolds numbers showedthe same pattern, where the logarithmic law fits the data within the experi-mental error. The region where a good fit of the data to the logarithmic lawwas considered for the least-squares curve fit yielding a reliable value of the vonKarman constant. From Equation (3.59) the logarithmic law of the wall in the


Figure 5.47: ln(du+/dy+) − ln y+, at Re=15021.

rough channel can be re-written as follows:

U+ =1

κln y+ + B − ∆B, ⇒ dU+

dy+=

1

κy+⇒ ln

(

dU+

dy+

)

= − ln y+ + ln(

1

κ

)

(5.27)Using a least-squares curve fit for the best fit of each individual case over thenew upper and lower limits resulted in a value of 1 ± 2.5% for the interceptln (1/κ) (see Figures 5.47 and 5.48). As a result, the data in Figures 5.47 and5.48 suggest (within an accuracy of measurements of ±2.5%) that there exists afunctional relationship for the far distant region of the normalized mean velocitydistribution, given by

ln(

dU+

dy+

)

= − ln y+ + 1 ± 2.5% (5.28)

Moreover, the least-squares curve fit of the experimental results shown in Fig-ure 5.47 for the rough channel flow as one set of data considering the lower limity+ = 150 produces a value of the intercept ln (1/κ) very close to 1.0. Hence, bytaking the intercept in Equation (5.28) as equal to 1.0 results in

ln(

dU+

dy+

)

= − ln y+ + 1 ⇒ ln(

y+dU+

dy+

)

= 1 ⇒(

y+dU+

dy+

)

= e (5.29)

κ =(

y+dU+

dy+

)−1

=1

e(5.30)


Figure 5.48: ln(du+/dy+) − ln y+, at Re=60150.

which is an interesting result deduced from the author’s experimental data forthe channel with rough walls. This readily suggests a logarithmic region for thefar field of the normalized mean velocity profile with a von Karman constantof κ = 1/e. It is worth noting that this experimental finding has no analyticalfoundation; however, the present experimental result will provide strong supportfor a future rigorous theory to prove that the von Karman constant is equal to1/e for the rough channel flows.

5.6.2 Checking the Validity of the Relation Between the

Normalized Wall Distance in the Rough and SmoothCases

To check the validity of Equation (3.63), the present velocity measurementsin the channel with rough walls were analyzed and compared with the cor-responding velocity measurements in the channel with smooth walls at thesame values of u+. Table 5.4 shows samples of this comparison. It is clearfrom Table 5.4 that at the same value of u+, the normalized wall distancevalue in the rough case, y+

r , is related to normalized wall distance value inthe smooth case, y+

s

√

τwr

τws

through a constant value, D. The former procedure

was applied to a set of data. Samples of the calculated values of the constantD are summarized in Figure 5.49 and also shown in Tables 5.4, 5.5 and 5.6


and for different values of the wall shear stress. The whole procedure indicatesthat Equation (3.63) is valid, universal and can be used in relevant applications.

Figure 5.49: Experimental values of the constant D at different values of√

τwr

τws

.

A number of samples were selected to check the validity of the new rela-tion to calculate ∆B, i.e. Equation (3.64). Figure 5.50 shows the first sample,a comparison between the log-law values of the smooth and rough channels atReynolds numbers 69149 and 60122, respectively.

Initially, the requirements of Equation (3.64), i.e. κ, D, τws and τwr, haveto be prepared. According to the last investigations, the value of κ was taken as1/e. The value of the constant D was deduced from measured values of the walldistance in both cases. The values of the wall shear stress for both cases werededuced from the static pressure measurements. Introducing these values intoEquation (3.64) leads to a value of the shift ∆B of 2.31.

Second, the lines in Figure 5.50 were fitted using the least-squares methodto calculate the value of ∆B, which gave a value of 2.32. The difference betweenthe values is 5× 10−3 % which means that there is very good agreement betweenthe measured and the calculated values. Figures 5.51–5.53 show a comparisonbetween the log-law lines in the case of smooth and rough channels at differentReynolds numbers. The above procedure was applied for these cases and someother cases, and very good agreements were obtained. Figure 5.54 shows a


Type u+ y+ uτ y+r /y+

s

√

τwr

τws

D

Rough 18.7863 453.3488 1.70005 1.9259 3.3704 0.571Smooth 18.7450 235.3837 0.50439Rough 20.3035 793.3604 1.70005 1.9661 3.3704 0.583Smooth 20.2017 403.5149 0.50439Rough 21.2705 1133.372 1.70005 1.9826 3.3704 0.588Smooth 21.1431 571.6462 0.50439Rough 22.1827 1586.7208 1.70005 1.8874 3.3704 0.560Smooth 22.1854 840.6562 0.50439

Table 5.4: Experimental values of the constant D at√

τwr

τws

= 3.37.


s

√

τwr

τws

D



τwr

τws

= 1.78.

comparison between the calculated and measured values of the shift ∆B.It is clear that the new procedure is mainly dependent on the evaluationof U+. For this reason, it is important to check realiably the influence ofchoosing different values of U+. Some random values were taken for U+ and thecorresponding values of ∆B were calculated. There is no difference between thevalues of ∆B at the different values of u+. This means that one can take anyvalue of U+ to calculate ∆B. This behavior of ∆B versus U+ is shown in Figure5.55.

To check the validity of Equation (3.63), the rough pipe measurements ofNikuradse were selected and compared with his corresponding smooth pipemeasurements at the same values of u+. Table 5.7 shows samples of thiscomparison. It is clear that at the same value of u+, the value of y+

r is relatedto y+

s

√

τwr

τws

through a constant value, i.e. D. This means that Equation (3.63) is

valid for the data of Nikuradse. Samples of the calculated values of the constant



s

√

τwr

τws

D



τwr

τws

= 1.03.

D are shown in Tables 5.7, 5.8 and 5.9 and summarized in Figure 5.56 fordifferent values of the wall shear stress.


s

√

τwr

τws

D


Table 5.7: Nikuradse’s data: values of the constant D at√

τwr

τws

= 0.7176.

Different samples of Nikuradse’s data were selected to check the validityof the relationship to calculate ∆B, i.e. Equation (3.64). Figure 5.57 shows thefirst sample, a comparison between the log-law values of the smooth and therough pipes at Reynolds numbers 9200 and 680000, respectively. The lines inFigure 5.57 were fitted using the least-squares method to give a value of ∆B of5.8426. The value of ∆B was also calculated using Equation (3.64) to give avalue of 5.6993. The difference between the values is 2.5 × 10−2 % which meansthat there is good agreement between the measured and calculated values usingthe new procedure. Figures 5.58 and 5.59 show comparisons for some other setsof Nikuradse’s data at different Reynolds numbers. The above procedure tocalculate the shift ∆B was applied to these sets of data and perfect agreements


Figure 5.50: Comparison between the log-law in the case of a smooth channel atRe = 69149 and a rough channel at Re = 60122.

were obtained. Finally, one can conclude that, without any information aboutthe roughness, the shift between the log-law line in the smooth regime and thatin the rough regimes can be calculated.



Figure 5.54: Comparison between the calculated and the measured values of ∆B.


Figure 5.55: The behavior of ∆B at different values of U+.


s

√

τwr

τws

D



τwr

τws

= 2.1890.



s

√

τwr

τws

D

Rough 13.9259 293.0000 35.5000 8.1389 4.5747 1.7791Smooth 13.9200 36.0000 7.7600Rough 14.9539 431.0000 35.5000 8.6200 4.5747 1.8843Smooth 14.9500 50.0000 7.7600Rough 15.6033 550.0000 35.5000 8.4615 4.5747 1.8496Smooth 15.6000 65.0000 7.7600Rough 16.6508 815.0000 35.5000 8.7634 4.5747 1.9156Smooth 16.6500 93.0000 7.7600Rough 19.1207 2060.0000 35.5000 8.6920 4.5747 1.9000Smooth 19.1200 237.0000 7.7600Rough 19.6501 2513.0000 35.5000 8.5476 4.5747 1.8684Smooth 19.6500 294.0000 7.7600


τwr

τws

= 4.5747.

Figure 5.56: Nikuradse’s data: values of the constant D at different values of√

τwr

τws

.


Figure 5.57: Nikuradse’s data: comparison between the log-law in the case of asmooth pipe at Re = 9200 and a rough pipe at Re = 680000.

Figure 5.58: Nikuradse’s data: comparison between the log-law in the case ofsmooth and rough pipes.


Figure 5.59: Nikuradse’s data: cComparison between the log-law in the case of asmooth pipe at Re = 725000 and a rough pipe at Re = 427000.

5.7 A Procedure to Utilize the Suggested Method to Recalculate the RoughnessHeight 106

5.7 A Procedure to Utilize the Suggested

Method to Recalculate the Roughness

Height

Using the last procedure, one can calculate ∆B and the corresponding y valuefor any investigated data knowing the values of the wall shear stresses and theconstant D. From this ∆B, one can calculate the value of u+

r at any u+s from the

relation ∆B = u+r − u+

s . Introducing the values of u+r and y in Equation (3.65)

leads to

u+r − 8.5 =

1

κln

y

ks(5.31)

which in turn leads toks =

y

eκ(u+r −8.5)

(5.32)

from which one can calculate the roughness height. This procedure was applied tothe data of Nikuradse and good agreement was observed between the calculatedand measured values of the roughness heights. Table 5.10 shows samples of thesecomparisons.

Re Roughness height measured Roughness height calculated970000 0.01298 0.0100430000 0.1718 0.1600197000 0.0891 0.0800

Table 5.10: Comparison between the measured and calculated values of the rough-ness height for Nikuradse’s data.

Chapter 6

SUMMARY ANDCONCLUSIONS

Turbulent channel and pipe flows have been studied, providing useful informa-tion on wall-bounded channel and pipe flows. The first objective of this studyconcerned the consistency of the findings of Zanoun and Durst [2003] in smoothsurfaces followed by trying to deduce similarity conditions between the smoothwall-bounded flows with different dimensions. Theoretical and experimental in-vestigations towards a general solution to account for the confirmation and thesimilarity were carried out, leading to the following conclusions:

1. The value of the von Karman constant was calculated for the smooth, theasymmetric and the symmetric rough regimes. In all cases the value wasfound to be approximately equal to 1/e, confirming the findings of Zanounet al. [2003], Durst et al. [2003] and others.

2. The log-law was found to exist over the range 150 ≤ y+ ≤ 0.75 Reτ ,where Reτ = h+, the channel half-height normalized with wall variables,confirming the findings of Durst et al. [2005] and Zanoun et al. [2003]

3. Similarity conditions between the smooth wall-bounded flows with differentdimensions were investigated according to the relation

u+ = U+ +1

κln

H

h(6.1)

where h is the height of the small channel and H is that of the large channel.The similarity conditions were checked using the present smooth channeldata and also using the data of Nikuradse for smooth pipes. Good agree-ments were found in both cases.

The second part of this work provided a good basis for assessing questions re-garding asymmetric flows. Measurements were performed in asymmetric water

108

channel flow utilizing different measuring methodologies. The following conclu-sions were drawn from the results discussed in various sections of this part of thethesis:

1. The reference point, for the velocity measurements, which is the middleof the channel, can be determined accurately using a 2D-LDA measuringtechnique.

2. A new iteration method was suggested and applied successfully to calculatethe effective height of the channel.

3. It is observed that on increasing the Reynolds number, the effective heightof the channel increases, which is an indication that the y location whereU = 0 penetrates into the roughness.

The third part of this work provided a good basis for assessing questions regardingthe rough wall-bounded flows. Measurements were performed in rough channelflow utilizing different measuring methodologies. The main outcome of this partis summarized as follows:

1. A new method to measure the static pressure, using Pitot tubes, in thecase of rough wall-bounded flows was used successfully. The method waschecked for the smooth case and the agreement with other known methodswas perfect.

2. Without any information about the roughness, a new theoretical investi-gation was performed to calculate the shift between the log-law line forthe case of smooth wall-bounded flows and that for the case of rough wall-bounded flows. The new equation is

∆B =1

2κln D

τwr

τws

(6.2)

which does not need any information about the roughness. Good agreementwas found between the experimental results and those obtained using thenew equation.

The last part of this work was aimed at applying all the investigations presentedhere to the data of Nikuradse for the smooth and rough walls. This was doneand good agreement was found between the experimental data of Nikuradse andthe results after applying the new procedures.

Further work for different heights of roughness is needed to confirm theuniversality of the present investigations. The similarity conditions deduced forsmooth walls have to be studied also for rough walls and also for the case of onerough wall and one smooth wall. Finally, a universal law governing the flow inthe case of rough walls which does not depend on the roughness information hasto be investigated.

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FORTRAN PROGRAM

Input

U(y), dp/dx, , , ,r n e k

t ewr = (H - )* dp/dx

H = H + dH

H = H - dH

U = ( / )t

t rwr

0.5

U = u/u y = yu /+ +

t tn

rms between Y * dU /dy and 1/+ + +

k

If rms < C

Finish

Assuming value of H

No

Yes

or

Figure 1: Sketch of the Fortran program.

ZUSAMMENFASSUNG

In der vorliegenden Arbeit wurde die voll entwickelte turbulente Kanalstromungmit glatten Wanden, einer glatten und einer rauen Wand und mit zwei rauenWanden untersucht, um nutzliche und weiterfuhrende Informationen uber dieseArt von Stromungen zu erhalten.

Voll entwickelte, turbulente ebene glatte Kanalstromungen wurden theo-retisch und experimentell fur einen Reynoldszahl-Bereich bis zu 1.1 × 105

untersucht. Es wurde festgestellt, dass fur diese Art der Stromung der Wertder von Karman Konstante κ bei 1/e liegt, was die kurzlich von Zanoun undDurst [2003] veroffentlichten Werte bestatigt. Gleichermassen wurden Kanal-stromungen mit glatten Wanden, mit unterschiedlichen Geometrieverhaltnissen,theoretisch untersucht. Es wurden Versuche durchgefuhrt, die die neuen theo-retischen Erkenntnissen bestatigen.

Zudem wurden in vorliegender Arbeit voll entwickelte, turbulente Kanal-stromungen untersucht, bei denen eine Wand glatt und die andere Wand rauwaren. Diese Art der Stromung weist interessante Eigenschaften auf, die bishernoch nicht systematisch untersucht worden sind, trotz bestehender angemessenerexperimenteller und numerischer Moglichkeiten. Das Geschwindigkeitsmax-imum liegt aussermittig, naher an der glatten Wand und seine Lage istnicht gleich dem Null-Scherspannungspunkt der Stromung. Zudem kann dieGeschwindigkeitsverteilung in der Nahe der rauhen Wand nur dann in Wand-koordinaten aufgetragen werden, wenn die Nullspannungslage definiert werdenkann. In dieser Arbeit werden hierfur die Methoden gezeigt, die von einemzwei Komponenten LDA System Gebrauch machen, welches zur Messung desProfils der mittleren Geschwindigkeit sowie der Reynoldsspannungen und somitder genauen Lage des Null-Scherspannungspunkt verwendet wurde. UnterAnwendung der zur Verfugung stehenden theoretischen und experimentellenMitteln wurde die effektive Hohe des sowie Referenzpunkt, d.h. die Mitte, desKanals fur die Geschwindigkeitsmessungen bestimmt.

Auch die voll entwickelte, turbulente Kanalstromung mit rauen Wandenwurde untersucht. Bisweilen wurde, um den Versatz zwischen dem Logarithmis-chen Wandgesetzt fur glatte und rauhe Wande zu bestimmen, eine Informationuber die Hohe der Rauheit benotigt. In dieser Arbeit wurden theoretischeUntersuchungen durchgefuhrt, um diesen Versatz ohne die Information der Hoheder Rauheit zu bestimmen. Das Verfahren wurde durch Messungen bestatigt.Zudem wurde eine neue Methode, um den statischen Druck an rauhen Wandenzu messen mit Pitot - Rohren erfolgreich eingefuhrt.

Letztendlich wurden einige der theoretischen Ergebnisse uber die in der Lit-eratur vorhandenen Messergebnissen von Nikuradse bestatigt, sowohl fur glattewie auch fur raue Rohre.

EINLEITUNG

Stand des Wissens

Die in dieser Disseration zusammengefassten Arbeiten sind ein Fortsetzung derstromungsmechanischen Untersuchungen, die am Lehrstuhl fur

stromungsmechanik der Friedrich-Alexander-Universitat Erlangen-Nurnbergauf dem Gebiet voll entwickelter, turbulenter Kanalstrmungen durchgefuhrt wur-den. Die in dieser Arbeit beschriebenen Untersuchungen sind eine Fortsetzungder Stromungsmechanikstudien, die am Lehrstuhl fur der Stromungmechanikder Friedrich-Alexander Universitat Erlangen-Nurnberg im Bereich der vollen-twickelten, turbulenten Kanalstromungen durchgefuhrt werden. UmfangreicheStudien wurden in den neuen Publikationen von Zanoun et al. [ 2003 ] und vonDurst et al. [ 2003 ] durchgefuhrt und dargestellt. Die untersuchten Stromungenwurden in Kanalen mit hydrodynamisch glatten Oberflachen erzeugt. Durstet al. [ 2005 ] analysierten alle vorhandenen experimentellen und numerischenDaten der voll entwickelten, turbulenten, ebenen Kanalstromungen unterEinbeziehung des logarithmischen Geschwindigkeitsprofils fur Stromungen mithohen Reynoldszahlen. Dabei wurde festgestellt, dass das logarithmische Gesetzuber einer Strecke 150 ≤ y+ ≤ 0.7 Reτ gilt, wobei Reτ = h+ die halbe Kanalhohenormalisiert mit Wandvariablen darstellt. Mittels einer sehr rationalen Methodewurde die von Karman-Konstante ermittelt, κ = 0.362, siehe Durst et al. [2003 ], die in enger Ubereinstimmung zu dem Wert liegt, der von Zanoun etal. [ 2003 ] als κ = 0.37 veroffentlicht wurde. Die Untersuchungen bestatigten,dass gleichbleibende Datensatze fur die mittlere Geschwindigkeit der vollentwickelten, turbulenten, ebenen Kanalstromungen vorliegen, die alle notigenInformationen uber die mittlere Geschwindigkeit der Stromung liefern, die in derIngenieurpraxis benotigt werden. Jedoch bestehen diese gleichbleibenden Datennur fur hydrodynamisch glatte Kanalwande. Ahnliche gleichbleibende mittlereGeschwindigkeitsinformationen gibt es nicht fur Kanalstromungen mit rauhenWanden. Die in der vorliegenden Arbeit vorgestellten Untersuchungen befassensich folglich mit Kanal Stromungen mit glatten und rauhen Wanden. In dieserArbeit werden die Vermutungen von Zanoun et al. [ 2003 ] und Durst et al. [2003 ] bestatigt, die besagen, dass das logarithmische Geschwindigkeitsprofil uberfast 70 % der kompletten Grenzschicht an beiden Wanden besteht, beginnendbei y+ = 150.

Wie oben erwahnt, wurde gefunden, dass die von Karman-Konstante, κ,im logarithmischen Gesetz gleichwertig zu 1/e ist, und die additive KonstanteB nahe an 10/e, wie durch Zanoun et al. [ 2003 ] beschrieben. In eigenenUntersuchungen des Autors werden die Entdeckungen von Zanoun et al. [ 2003 ]und Durst et al. [ 2005 ] fur glatte Kanalwande bestatigt.

Ausserdem werden die Ahnlichkeitbedingungen fur glatte Wandungen mitunterschiedlichen Hohen abgeleitet. Jedoch konzentriert sich die Hauptarbeit desAutors auf voll entwickelte, turbulente Kanalstromungen in rauhen Wandungen.Aus diesem Grund zielen die hier vorgestellten Untersuchungen auf eine ein-heitliche Darstellung der mittleren Geschwindigkeitsprofile alle voll entwickelten,turbulenten, ebenen Kanalstromungen mit rauhen Wandungen, unabhangig vonder Art der Rauhigkeit. Es wird gezeigt, dass das von Zanoun et al. [ 2003 ] undDurst et al. [ 2005 ] gefundene κ = 1/e ein gleichbleibender Wert ist, auch furrauhe Wandungen.

Ausgehend von den Untersuchungen der glatten Wande, wird in der vorliegendenArbeit gezeigt, dass alle rauhen mittleren Geschwindigkeitsverteilungen durchein logarithmisches Gesetz ausgedruckt werden konnen, das abfallt durch δB inBezug auf das logarithmische Gesetz fur glatte Wandungen. δB erwies sich alsabhangig vom Impulsverlust der Stromung zu den Wandungen. Die BeziehungδB als Funktion des Verhaltnisses (τw)rough zu (τw)smooth des gleichen Kanalswird abgeleitet. Um einige der durch die theoretischen Betrachtungen erhaltenenResultate zu uberprufen, wurde eine experimentelle Testeinrichtung mit denerforderlichen Messgeraten aufgebaut, um voll entwickelte, turbulenten, ebeneKanalstromungen mit glatten und/oder rauhen Wanden zu untersuchen. Dergesamte Versuchsstand wird in dieser Arbeit beschrieben und die durchgefuhrtenMessungen werden beschrieben. Das daraus erhaltene Verhaltnis zwischender normalisierten mittleren Geschwindigkeitsprofil fur glatte Wande und dementsprechenden Profil fur rauhe Wande wird erklart. Sowohl die theoretischeAbleitungen als auch experimentellen Uberprufungen gestatten einen gutenEinblick in die allgemeinen Eigenschaften der voll entwickelten, turbulenten,ebene Kanalstromungen mit glatten und rauhen Wanden.

Es wurden wiederum Verifikationsexperimente durchgefuhrt und die Ergeb-nisse dargestellt, um die Hauptresultate der theoretischen Betrachtungen zubeweisen. Folglich gilt die vorliegende Arbeit als eine abschliesende Studie deram LSTM-Erlangen durchgefuhrten Untersuchungen von Kanalstromungen mitglatten Wandungen und als Beginn der Untersuchungen von Kanalstromungenmit rauhen Wandungen.

Ziele der Arbeit und Aufbau der Dissertation

In wandnahen Stromungen ist es von Wichtigkeit, die Ahnlichkeitseigenschaftenzwischen unterschiedlichen Wandungen in Betracht zu ziehen. Folglich war esdas Ziel dieser Dissertation, den Einfluss der rauhen Wand auf die Hohe, dasGeschwindigkeitsprofile und auch auf die Wandscherspannung zu studieren.

Es gibt einige offene Fragen bezuglich der wirkungsvollen Hohe der unter-suchten Kanalwandungen in Bezug auf den Einfluss der Rauheit, der Posi-tion der maximalen Geschwindigkeit und dem Wert der Wandscherspannung.Auch wurde die Frage nach der Verschiebung zwischen ”‘Log-Law Lines”’ beiglatten bzw. rauhen Wandungen. Diese Uberlegungen wurden in Betrachtgezogen und gleichfalls die Ergebnisse einer Literaturrecherche uber die Posi-tion der maximalen Geschwindigkeit und der Gesamtscherspannung, um dieEntscheidung zu treffen, diese Art Stromungen naher zu betrachten, um zuversuchen, einige der offenen Fragen zu beantworten. Das Ziel dieser Arbeitwar es folglich, ausfuhrliche Messungen in glatten Kanalen unterschiedlichenHohen durchzufuhren, um die vorhandenen Ahnlichkeitbedingung zwischen ih-nen abzuleiten. Auch mussten detaillierte Messungen in asymetrischem Kanalendurchgefuhrt werden, um genaue Abschatzungen der Wandscherspannung zu er-halten und eine wirkungsvolle Hohe des Kanals zu bestimmen. Es ist in hohemGrade wunschenswert, ausfuhrliche Geschwindigkeitsmessungen durchzufuhrenund die Wandscherspannung mit einer Highfidelitytechnik zu bestimmen, die u

′

und v′

Werte gleichzeitig misst. Ausserdem mussen umfangreiche und genaueMessungen im Fall rauher Kanale durchgefuhrt werden, um die Verschiebungenzwischen den Linien des logarithmischen Gesetzes zu schatzen, die von der Hoheder Rauheit unabhangig sind.

• Uberprufung der Ahnlichkeitbedingungen zwischen Kanalen mit unter-schiedlichen Hohen.

• Untersuchung des Einflusses der Rauigkeit auf das Geschwindigkeitsprofil.

• Bestimmung der Verschiebung der maximalen Geschwindigkeit in derauseren Kanalstromung.

• Feststellung der wirkungsvollen Kanalhohe.

• Abschatzung der Verschiebung zwischen den ”‘Log-Low Lines”’ in den glat-ten und rauhen Fallen ohne irgendwelche Informationen uber die Rauigkeit.

• Analyse von Nikuradses Daten mit dem neuen Verfahren.

Zusammenfassend lasst sich sagen, dass die vorliegende Arbeit abzielt auf Un-tersuchungen der Ahnlichkeitbedingungen zwischen glatten Kanalen mit unter-schiedlichen Hohen, sowohl den Effekt von Rauigkeit auf die wirkungsvollen Hoheals auch die Wandscherspannung und die Verschiebung zwischen den Linien deslogarithmischen Gesetzes abzuleiten, die den glatten und rauhen Wandungen un-abhangig von der Rauheit entspricht. Die Hauptteile der These sind untenstehenddetailliert beschrieben.

Kapitel 2 gibt eine kurze Literaturubersicht uber voll entwickelten Kanl-stromungen in glatten und rauhen Oberflachen. Es fasst die Arbeit zusammen,die sich mit dieser Art von Stromungen beschaftigen und die aufzeigen,welche Arten von Rauigkeit existieren und wie sie die unterschiedlichenStromungsparameter beeinflussen.

Kapitel 3 stellt die theoretischen Betrachtungen und die wichtigsten Gle-ichungen fur die unterschiedlichen Arten der Stromung als Grundlage derexperimentellen Studien vor. Es stellt auch die Ahnlichkeitbedingung zwischenden glatten Kanalen mit unterschiedlichen Hohen dar. Das neue Verfahren zurBestimmung der wirkungsvollen Hohe der Fuhrung bezuglich des Einflusses vonRauigkeit wird eingefuhrt. Die neue Methode zur Bestimmung der Verschiebungzwischen den glatten und rauhen ”‘Log-Low Lines”’ wird ausfuhrlich besprochen.Auserdem zeigt sie Schritt fur Schritt das neue Verfahren, dem gefolgt wird, umdie aufgeworfenen Probleme zu losen.

Kapitel 4 stellt der Grundlagen der Laser-Doppler-Anemometrie,Zweikomponenten-Laser-Doppler-Anemometrie und Hitzdraht-Anemometrieals die Messtechniken vor, die verwendet wurden, um die hier vorgestelltenUntersuchungen durchzufuhren. Der Versuchsstand und die Messtechnikenwerden in diesem Kapitel beschrieben. Es umfasst gleichfalls Betrachtungen derGleichungen, die auf die hier vorgestellten Messungen und Datenverarbeitungangewendet werden.

Kapitel 5 stellt die Resultate der drei wichtigsten Messungen dar: glatte,asymmetrische und rauhe Wand, voll entwickelte, turbulente, ebene Kanal-stromungen. Es zeigt die Ubereinstimmung mit den von Durst und Zanoungewonnenen Daten. Einflusse der Rauigkeit auf das Geschwindigkeitsprofil, aufdie maximale Geschwindigkeit und auf die Wandscherspannung werden auchbeschrieben. Die Ubereinstimmung der mit dem neuen Verfahren erzieltenResultate mit den in der Literatur vorhandenen Daten wird aufgezeigt. EineAnalyse der experimentellen Daten von Nikuradse von turbulenten Stromungenin glatten und in sandkornbeschichteten Rohren wird gegeben.

Kapitel 6 fasst die Ergebnisse der vorliegenden Untersuchungen zusammenund gibt abschliesende Bemerkungen, welche den Ansatz erlautern, der befolgtwerden muss, um Stromungen dieser Art zu studieren, um die relative Scherspan-nung zwischen den zwei Wanden richtig zu ermitteln. Eine zuverlassige und neueMethode fur die Ahnlichkeitsbestimmung zwischen den glatten Kanalen, zurBestimmung der wirkungsvollen Hohe und fur die Verschiebung zwischen denglatten und rauhen ”‘Log-Law Lines”’ ist eingefuhrt worden. Schlieslich wirdeine Zusammenfassung gegeben und Vorschlage fur weiterfuhrende Arbeitenvorgestellt.

LEBENSLAUF Osama Saleh

PERSOLICHE DATEN

Name Osama Abdelsattar Bayoumy SalehAnschrift Saalestr. 11, D-91052 ErlangenGeburtsdatum 01.05.1966

Geburtsort Sharkia-Agypten

Staatsangehorigkeit AgypterFamilienstand verheiratet, Vater von zwei Tochern und vier SohnenE-mail [email protected], [email protected] 01797383638

SCHULBILDUNG

09/1971 - 06/1977 Grundschule, Asakra-Sharkia, Agypten

09/1977 - 06/1980 Realschule; Samakeen-Sharkia; Agypten

09/1980 - 06/1983 Gymnasium, Fakoos, Sharkia, Agypten

06/1983 Zulassung Prufung, Fakoos, Sharkia, Agypten

STUDIUM

09/1983 - 08/1987 B.Sc. Physik an der Zagazig Universitat,

Zagazig, Agypten07/1993 - 10/1996 M.Sc. der Physik an der Mansoura Universtat,

Mansoura, Agypten

BERUFLICHE TATIGKEIT

04/1990 - 01/2000 Research Assistant, Photometry DepartmentResarch Institute, National Institute for Standards,

Cairo, AgyptenAb 10/2001 Wissenschaftlicher Mitarbeiter am Lehrstuhl fur

Stromungsmechanik der Friedrich-Alexander-UniverstatErlangen-Nurnberg

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