WAKE CHARACTERISTICS OF YAWED CIRCULAR CYLINDERS … · Siti Fatin Mohd. Razali Candidate‟s Name...

WAKE CHARACTERISTICS OF YAWED

CIRCULAR CYLINDERS AND SUPPRESSION

OF VORTEX-INDUCED VIBRATION USING

HELICAL STRAKES

by

SITI FATIN MOHD RAZALI

This thesis is presented for the degree of

Doctor of Philosophy

of

The University of Western Australia

School of Civil and Resource Engineering

November 2010

DECLARATION FOR THESIS CONTAINING PUBLISHED WORK

AND/OR WORK PREPARED FOR PUBLICATION

This thesis contains published work and/or work prepared for publication, some of

which has been co-authored. The bibliographical details of the work and where it

appears in the thesis are outlined below.

1. Zhou, T., Razali, S.F.M., Zhou, Y., Chua, L.P. and Cheng, L. (2009). Dependence

of the wake on inclination of a stationary cylinder. Experiments in Fluids. 46,

1125-1138. (Chapter 2).

The estimated percentage contribution of the candidate is 40%.

2. Zhou, T., Wang, H., Razali, S.F.M., Zhou, Y. and Cheng, L. (2010). Three-

dimensionality vorticity measurements in the wake of a yawed circular

cylinder. Physics of Fluids. 22, 015108: 1-15. (Chapter 3).


3. Wang, H., Razali, S.F.M., Zhou, T., Zhou, Y. and Cheng, L. (2011). Streamwise

evolution of an inclined cylinder wake. Accepted for publication in

Experiments in Fluids. (Chapter 4).


4. Razali, S.F.M., Zhou, T., Rinoshika, A. and Cheng, L. (2010). Wavelet analysis of

the turbulent wake generated by an inclined circular cylinder. Journal of

Turbulence. 11, No. 15: 1-25. (Chapter 5).


5. Razali, S.F.M., Zhou, T. and Cheng, L. (2010). On the study of the wake behind

two yawed cylinders in side-by-side arrangement. To be submitted. (Chapter

6).


6. Zhou, T., Razali, S.F.M., Hao, Z. and Cheng, L. (2011). On the study of vortex-

induced vibration of a cylinder with helical strakes. Accepted for publication in

Journal of Fluids and Structures. (Chapter 7).


Siti Fatin Mohd. Razali

Candidate‟s Name Signature Date

Prof. Tongming Zhou

Coordinating Supervisor‟s Name

Signature Date

Prof. Liang Cheng

Co-Supervisor‟s Name Signature Date

i

ABSTRACT

The research carried out in this thesis has been focused on the characteristics of

the wakes of a single cylinder and two cylinders arranged side-by-side at various

cylinder yaw angles (= 0°-45°). The independence principle (IP) was examined. As

helical strake is one of the most proven and widely used control measures in

engineering applications to suppress vortex-induced vibration (VIV) of cylindrical

structures, the mechanism of VIV mitigation using helical strakes is also investigated

for a single cylinder. The main findings are summarized as follow:

In Chapter 2, the dependence of the wake characteristics at streamwise locations

of x/d = 10-40 on cylinder yaw angles is examined. It was found that the spanwise mean

velocity W , which represents the three-dimensionality of the wake flow, increases

monotonically with . The root-mean-square (rms) values of the streamwise (u) and

spanwise (w) velocities and the three vorticity components decrease significantly with

the increase of , whereas the transverse velocity (v) does not follow the same trend.

The vortex shedding frequency decreases with the increase of . The Strouhal number

StN, obtained by using the velocity component normal to the cylinder axis, remains

approximately a constant within the experimental uncertainty when < 40, indicating

the validity of IP over this range of yaw angles.

In Chapter 3, phase-averaging technique is used to analyze the velocity and

vorticity signals and to obtain the coherent and incoherent contours at a streamwise

location of x/d = 10. When 15, the maximum coherent concentrations of the three

vorticity components do not change with . However, when is increased to 45, the

maximum concentrations of the coherent transverse and spanwise vorticity components

decrease by about 33% and 50% respectively while that of the streamwise vorticity

increases by about 70%, suggesting the generation of the secondary streamwise vortices

and that the strength of the Kármán vortex shed from the yawed cylinder decreases and

the three-dimensionality of the flow is enhanced.

In Chapter 4, the streamwise evolution of a yawed cylinder wake at downstream

locations x/d = 10-40 from the cylinder is examined by using phase-averaging

technique. At x/d = 10, the effects of on the three coherent vorticity components are

negligibly small for ≤ 15°. When increases further to 45°, the maximum of

coherent spanwise vorticity reduces by about 50%, while that of the streamwise

ii

vorticity increases by about 70%. Similar results are found at x/d = 20, indicating the

impaired spanwise vortices and the enhancement of the three-dimensionality of the

wake with increasing . The streamwise decay rate of the coherent spanwise

vorticity is smaller for a larger . This is because the streamwise spacing between the

spanwise vortices is bigger for a larger , resulting in a weak interaction between the

vortices and hence slower decaying rate in the streamwise direction.

In Chapter 5, wavelet method is used to examine the dependence of the velocity

and vorticity characteristics at different wavelet levels on as compared with that

obtained at = 0°. It was found that the IP is only applicable for < 40°. The energy

spectra for the intermediate- and large-scale structures decrease in terms of their

maximum energy and disperse extensively over an enlarged frequency band with the

increase of . At = 45, the large-scale vortex dislocations may occur due to the

increase of the three-dimensionality in the wake region. Although the large-scale

structures are the dominant contributors to the Reynolds stresses at all inclination angles

and followed by the intermediate-scale structures, the wake vorticity is mostly

dominated by the small and intermediate-scale structures and has the smallest values at

large-scale structures.

In Chapter 6, the wake structures behind two yawed cylinders in side-by-side

arrangement for two centre-to-centre cylinder spacings T* = 3.0 and 1.7 are studied. For

large cylinder spacing T* = 3.0, there exist two vortex streets and the cylinders behave

as independent and isolated ones. The lateral distance between the two centres of vortex

streets for the T* = 3.0 wake is about 3d. When < 40°, the IP is applicable in terms of

the Strouhal number. The coherent streamwise vorticity contours *~x for T

* = 3.0 is

only about 10% of that of *~z in a cross-flow. With the increase of , *~

x increases

while *~z decreases. At = 45°, *~

x is about 67% of *~z , indicating the existence of the

secondary axial vortices or the occurrence of the vortex dislocation with an enhanced

three-dimensionality with larger . For intermediate cylinder spacing T* = 1.7, the IP is

applicable to the wake when < 40°. The values of the spanwise velocity *W is not

close to zero, even at = 0°, which suggests that the vortex in the wake is inclined and

the wake may not behave a perfect two-dimensionality. The vorticity contours for T* =

1.7 have a more organized pattern at = 0° while become scattered and smaller with

the increase of yaw angles. The vorticity contours for T* = 1.7 are much less organized

iii

and the weakest in comparison with those for T* = ∞ and 3.0, indicating that the vortex

motion in the wake is not stable.

In Chapter 7, the mechanisms of VIV mitigation using helical strakes is studied.

It was found that the helical strakes can reduce VIV by about 98%. Unlike the bare

cylinder which experiences lock-in over the reduced velocity in the range of 5-8.5, the

straked cylinder does not show any lock-in region and the vortices shed from the straked

cylinder are weakened significantly. This is because the strakes do not necessarily

suppress vortex shedding, but the wake is strongly disorganized as the helical strakes

prevent the vortex from being correlated along the span, hence is not able to excite VIV

significantly. The correlation length of the vortex structures in the bare cylinder is much

larger than that obtained in the straked cylinder wake. As a result, the straked cylinder

wake agrees more closely with isotropy than the bare cylinder wake.

iv

ACKNOWLEDGEMENTS

I owe my deepest gratitude to my supervisor Prof. Tongming Zhou for his

continuous support, guidance, priceless discussion, understanding and highly

responsibility throughout my PhD study. I deeply appreciate his effort to provide his

time enabled me to understand this subject.

I am grateful to my co-supervisor Prof. Liang Cheng for his help and guidance

and providing me the opportunity to pursue my PhD study at the University of Western

Australia (UWA) and some financial support

It is an honour for me to acknowledge the financial support from the Ministry of

Higher Education Malaysia (MOHE), the Universiti Kebangsaan Malaysia (UKM) and

the Ad-Hoc Scholarship from School of Civil and Resource Engineering and Graduate

Research School, UWA for me to pursue my PhD study.

I would like to thank the postgraduate students and staff of School of Civil and

Resource Engineering and Centre of Offshore Foundation Systems, UWA for providing

me a good and friendly study environment to maximize my learning efficiency in the

past years.

I am indebted to my husband Mohd. Thabrani Marwan and my daughter Nurina

Huda for their continuous love and patience for me to finish my PhD journey. They are

my motivation and have made available their support in a number of ways.

Finally, this thesis would not have been possible without encouragement from

my parents Mohd. Razali Mohd. Yunus and Wan Rihayah Wan Ahmad, my brother and

my sisters.

v

TABLE OF CONTENTS

ABSTRACT i

ACKNOWLEDGEMENTS iv

TABLE OF CONTENTS v

LIST OF TABLES ix

LIST OF FIGURES x

LIST OF SYMBOLS xvi

ABBREVIATIONS xx

CHAPTER 1: INTRODUCTION 1

1.1 Overview ................................................................................................................... 1

1.2 Research Motivations ................................................................................................ 5

1.3 Research Objectives .................................................................................................. 7

1.4 Thesis Outlines .......................................................................................................... 7

CHAPTER 2: DEPENDENCE OF THE WAKE ON YAW ANGLES OF A

STATIONARY CYLINDER 9

2.1 Introduction ............................................................................................................... 9

2.2 Experimental Details ............................................................................................... 12

2.3 Results and Discussion ............................................................................................ 15

2.3.1 Mean streamwise and spanwise velocity distributions ............................... 15

2.3.2 Rms velocity and vorticity distributions across the wake ........................... 16

2.3.3 Dependence of the vortex shedding frequency on ................................... 19

2.3.4 Autocorrelation coefficients ........................................................................ 21

2.4 Conclusions ............................................................................................................. 23

CHAPTER 3: THREE-DIMENSIONAL VORTICITY MEASUREMENTS

IN THE WAKE OF A YAWED CIRCULAR CYLINDER 33

3.1 Introduction ............................................................................................................. 33

3.2 Experimental Details ............................................................................................... 37

3.3 Velocity and Vorticity Signals ................................................................................ 39

3.4 Phase-Averaged Velocity and Vorticity Fields ....................................................... 44

3.4.1 Phase-averaging .......................................................................................... 44

3.4.2 Structural averaging .................................................................................... 46

3.4.3 Coherent vorticity fields .............................................................................. 46

vi

3.4.4 Incoherent vorticity fields ........................................................................... 48

3.4.5 Topology of Reynolds stresses ................................................................... 49

3.5 Coherent and Incoherent Contributions to Velocity and Vorticity Variances ........ 51

3.6 Conclusions ............................................................................................................. 53

CHAPTER 4: STREAMWISE EVOLUTION OF A YAWED CYLINDER

WAKE 67

4.1 Introduction ............................................................................................................. 67

4.2 Experimental Details ............................................................................................. 69

4.3 Time-Averaged Velocities and Wake Periodicity ............................................... 71

4.3.1 Distributions of time-averaged streamwise and spanwise velocities..... 71

4.3.2 Power spectral density functions of velocities and vorticities ............... 72

4.4 Phase-Averaged Results ........................................................................................ 73

4.4.1 Brief introduction to phase-averaging technique .................................... 73

4.4.2 Phase-averaged vorticity fields ................................................................. 75

4.4.3 Phase-averaged velocity fields ................................................................. 77

4.4.4 Phase-averaged Reynolds shear stresses .................................................. 79

4.5 Coherent Contributions to Reynolds Stresses and Vorticity Variances ............ 80

4.6 Conclusions ............................................................................................................ 84

CHAPTER 5: WAVELET ANALYSIS OF THE TURBULENT WAKE

GENERATED BY A YAWED CIRCULAR CYLINDER 101

5.1 Introduction ........................................................................................................... 101


5.3 Decomposition of Experimental Signals into Various Wavelet Levels ................ 107

5.4 Results and Discussion .......................................................................................... 109

5.4.1 Spectra of measured and wavelet components of the velocity signals ..... 110

5.4.2 Strouhal numbers from the measured and wavelet components of

velocity signals .......................................................................................... 112

5.4.3 Contributions to velocity variances from different wavelet components . 113

5.4.4 Contributions to vorticity variances from different wavelet components . 115

5.4.5 Autocorrelation coefficients of velocity components at different

wavelet levels ............................................................................................ 117

5.4.6 Flow visualization of the yawed cylinder wake ........................................ 119

5.5 Conclusions ........................................................................................................... 120

vii

CHAPTER 6: ON THE STUDY OF THE WAKE BEHIND TWO YAWED

CYLINDERS IN SIDE-BY-SIDE ARRANGEMENT 131

6.1 Introduction ........................................................................................................... 131


6.2.1 Experimental arrangement ........................................................................ 134

6.2.2 Velocity and vorticity signals from the X-probes ..................................... 134

6.2.3 Phase-averaging method ........................................................................... 136


6.3.1 Mean streamwise and spanwise velocity profiles ..................................... 138

6.3.2 Power spectra of velocity and vorticity signals......................................... 141

6.3.3 Phase-averaged vorticity fields ................................................................. 143

6.3.4 Spanwise vortex patterns using the phase-averaged velocity

components ............................................................................................... 148

6.3.5 Incoherent vorticity fields ......................................................................... 149

6.3.6 Phase-averaged velocity components ....................................................... 149

6.3.7 Phase-averaged Reynolds shear stresses ................................................... 151

6.3.8 Incoherent Reynolds shear stresses ........................................................... 152

6.3.9 Coherent and incoherent contributions to Reynolds stresses .................... 153

6.3.10 Coherent and incoherent contributions to vorticity variances................... 155

6.4 Conclusions ........................................................................................................... 157

CHAPTER 7: ON THE STUDY OF VORTEX-INDUCED VIBRATION

OF A CYLINDER WITH HELICAL STRAKES 183

7.1 Introduction ........................................................................................................... 183



7.3.1 Vortex shedding frequency for the bare and straked cylinders ................. 188

7.3.2 Vortex-induced vibration for the bare and straked cylinders .................... 189

7.3.3 Energy spectra in the streamwise and spanwise directions of the

stationary cylinders ................................................................................... 191

7.3.4 Cross-correlations in the stationary cylinder wakes.................................. 194

7.3.5 Flow visualization of the stationary cylinder wakes ................................. 195

7.3.6 Isotropy assessments in the wake of stationary cylinders ......................... 196

7.4 Conclusions ........................................................................................................... 198

viii

CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS 209

8.1 Summary of Findings ............................................................................................ 209

8.2 Recommendations for Future Research ................................................................ 213

REFERENCES 215

ix

LIST OF TABLES

Table 2-1. Values of U0, L and rms velocities on the wake centreline for different

downstream locations and yaw angles .......................................................... 25

Table 3-1. Flatness factors of the vorticity components at = 0° and 45° ..................... 56

Table 3-2. The vortex shedding angle , vortex shedding frequency f0, convection

velocity Uc, wavelength and the inclination angle between the main

vortex stretching direction and the x-axis for different cylinder yaw

angles............................................................................................................. 56

Table 3-3. The maximum contour values of the coherent vorticity and Reynolds

stresses for different yaw angles ................................................................... 56

Table 4-1. Position and the measured velocities of each X-probe contained in the

vorticity probe ............................................................................................... 86

Table 4-2. Maximum velocity deficit and wake half-width ............................................ 86

Table 4-3. The maximum values of the coherent vorticity, velocity and Reynolds

stresses contours for different yaw angles at x* = 10, 20 and 40 .................. 87

Table 4-4. Averaged contributions from the coherent motion to Reynolds stresses

and vorticity variances for different yaw angles at x* = 10, 20 and 40 ......... 88

Table 5-1. Summary of the experimental conditions on wake centreline ..................... 122

Table 5-2. Central frequencies f0 and their respective frequency bandwidth of the v

signal of wavelet levels 1-8 at y/d ≈ 0.5 for = 0°, 15°, 30° and 45° ........ 122

Table 6-1. The vortex shedding angle, maximum velocity defect, convection

velocity, half-width, normalized Strouhal number and wavelength at

different yaw angles for cylinder spacings T* = ∞, 3.0 and 1.7 .................. 159

Table 6-2. The maximum values of the coherent vorticity, velocity and Reynolds

stresses contours at different yaw angles for T* = ∞, 3.0 and 1.7 ............... 160

Table 7-1. Velocity ratio 22 / uv at different downstream locations in

the stationary bare and straked cylinder wakes ........................................... 200

x

LIST OF FIGURES

Figure 2-1 Definition of the coordinate system and the sketches of the vorticity

probe ........................................................................................................... 26

Figure 2-2. Normalized mean streamwise velocity distribution for different yaw

angles.. ......................................................................................................... 26

Figure 2-3. Normalized mean spanwise velocity distribution for different yaw

angles.. ......................................................................................................... 27

Figure 2-4. Normalized velocity gradients for different yaw angles at x/d = 20 ............. 27

Figure 2-5. Distributions of u′ for different yaw angles ................................................... 28

Figure 2-6. Distribution of v′ for different yaw angles .................................................... 28

Figure 2-7. Distributions of w′ for different yaw angles .................................................. 29

Figure 2-8. Distributions of x′, y′ and z′ at x/d = 10 for different yaw angles ........... 29

Figure 2-9. Dependence of the cross-correlation coefficient zw , between the

spanwise velocity w and the spanwise vorticity z on at different

streamwise locations .................................................................................... 30

Figure 2-10. Spectral density function v of the transverse velocity component at

x/d = 10 for different yaw angles ................................................................. 30

Figure 2-11. Comparison of StN with other experimental and numerical results ............ 31

Figure 2-12. Autocorrelation coefficient u for different yaw angles ............................ 31

Figure 2-13. Autocorrelation coefficient v for different yaw angles ............................ 32

Figure 2-14. Autocorrelation coefficient w for different yaw angles ............................ 32

Figure 3-1. Sketches of the coordinate system and the 3-dimensional vorticity probe ... 57

Figure 3-2. Time traces of the fluctuating transverse velocity v and the three

vorticity components measured at y/d = 0.5 ................................................ 58

Figure 3-3. Probability density function of vorticity components at = 0° and 45° ...... 59

Figure 3-4. Spectra of three vorticity components at (a): = 0° and (b) = 45° ........... 59

Figure 3-5. Normalized velocity gradients for different at x/d = 10 ............................ 60

Figure 3-6. Autocorrelation coefficient i

for different ............................................ 60

Figure 3-7. Phase-averaged coherent vorticity contours at different . .......................... 61

xi

Figure 3-8. Phase-averaged sectional streamlines at different . .................................... 62

Figure 3-9. Phase-averaged incoherent vorticity contours at different ........................ 63

Figure 3-10. Phase-averaged velocities contours at different . ..................................... 64

Figure 3-11. Phase-averaged Reynolds shear stresses at different ............................... 65

Figure 3-12. Time-averaged Reynolds normal stresses and their coherent and

incoherent contributions at different . ....................................................... 66

Figure 3-13. Lateral distribution of coherent contributions to vorticity variances .......... 66

Figure 4-1. Definition of the coordinate system and the sketches of the vorticity

probe.. .......................................................................................................... 89

Figure 4-2. Time-averaged u- and w-component velocities at different yaw angles ....... 89

Figure 4-3. Sketch of the inclined vortices in a yawed cylinder wake ............................ 90

Figure 4-4. The power spectra density function u, v and w of the three velocity

components for different yaw angles ........................................................... 90

Figure 4-5. The power spectra density function x ,

y and z of the three

vorticity components for different yaw angles ............................................ 91

Figure 4-6. Signal v from the X-probe B of the vorticity probe at x* = 10 and y

* ≈

0.5. The thicker line represents the filtered signal vf. .................................. 91

Figure 4-7. Phase-averaged vorticity components at x* = 20 for different yaw

angles ........................................................................................................... 92

Figure 4-8. Phase-averaged vorticity components at x* = 40 for different yaw angles ... 93

Figure 4-9. Phase-averaged velocities at x* = 20 for different yaw angles ...................... 94

Figure 4-10. Phase-averaged velocities at x* = 40 for different yaw angles .................... 95

Figure 4-11. Phase-averaged Reynolds stresses at x* = 20 for different yaw angles ....... 96

Figure 4-12. Coherent and incoherent contributions to time-averaged Reynolds

stresses for different yaw angle at x* = 20. .................................................. 97

Figure 4-13. Coherent and incoherent contributions to time-averaged Reynolds

stresses for different yaw angle at x* = 40. .................................................. 98

Figure 4-14. Coherent contribution to vorticity variances for different yaw angles

at x* = 10, 20 and 40 .................................................................................... 99

Figure 5-1. Definition of the coordinate system and the sketches of the vorticity

probe. ......................................................................................................... 123

xii

Figure 5-2. Basic concept of the wavelet multiresolution analysis ............................... 123

Figure 5-3. Spectra of the v-signal measured at y/d ≈ 0.5, and a correspondence

between the central frequencies (or levels i) and the dimensions of

the vortex structures for different cylinder yaw angles ............................. 124

Figure 5-4. Comparison between (a-d) spectra of the v-signal measured at y/d ≈ 0.5

for = 0°, 15°, 30° and 45° respectively; and (e-h) spectra of various

wavelet levels at y/d ≈ 0.5 for = 0°, 15°, 30° and 45° respectively ........ 125

Figure 5-5. Peak frequency fN on the energy spectra and the ratio 0

/ NN ff for

various wavelet levels at y/d ≈ 0.5 ............................................................. 126

Figure 5-6. Velocity variances 2u , 2v and 2w of the measured

signals for different cylinder yaw angles ................................................... 126

Figure 5-7. Velocity variances 2iu , 2

iv and 2iw at various wavelet

levels for = 0° and 45° ............................................................................ 127

Figure 5-8. Vorticity variances 2x , 2

y and 2z of the measured

signals for different cylinder yaw angles ................................................... 128

Figure 5-9. Vorticity variances 2,ix , 2

,iy and 2,iz at various wavelet

levels for = 0 and 45 ............................................................................ 129

Figure 5-10. Autocorrelation coefficients of v at y/d ≈ 0.5 for various wavelet

levels at different cylinder yaw angles ...................................................... 130

Figure 5-11. Flow visualization of wake with the existence of secondary vortex

and streamwise vortex when = 45° ........................................................ 130

Figure 6-1. Two cylinders side-by-side arrangement with definitions of the

coordinate system and the sketches of the vorticity probe ........................ 161

Figure 6-2. The experimental setup of two cylinders with side-by-side

arrangement ............................................................................................... 162

Figure 6-3. Time traces of v signals measured at y* = 0.5 for different angles in

arbitrary scales. (a) T* = 3.0; (b) T

* = 1.7. The thicker line represents

the filtered signal vf. ................................................................................... 163

Figure 6-4.Comparisons of time-averaged velocities at different yaw angles for

T* = 3.0 and 1.7. ......................................................................................... 164

xiii

Figure 6-5. Normalized velocity gradients at different yaw angles for (a) T* = 3.0;

(b) 1.7 ......................................................................................................... 164

Figure 6-6. The power spectra of u, v and w components for different cylinder

spacings at (a, c): = 0°; (b, d): 45° .......................................................... 165

Figure 6-7. The power spectra of x, y, z components for different cylinder

spacings at (a, c): = 0°; (b, d): 45° .......................................................... 165

Figure 6-8. Strouhal number StN for different cylinder spacings ................................... 166

Figure 6-9. Phase-averaged vorticity components at different yaw angles for

T* = 3.0. ..................................................................................................... 167

Figure 6-10. Phase-averaged vorticity components at different yaw angles for

T* = 1.7. ..................................................................................................... 168

Figure 6-11. Coherent vorticity contours *~z from phase-averaged *~u and *~v at

different yaw angles for T* = 3.0. .............................................................. 169

Figure 6-12. Coherent vorticity contours *~z from phase-averaged *~u and *~v at

different yaw angles for T* = 1.7. .............................................................. 169

Figure 6-13. Phase-averaged sectional streamlines at different yaw angles for

T* = 3.0. ...................................................................................................... 170

Figure 6-14. Phase-averaged sectional streamlines at different yaw angles for

T* = 1.7.. ..................................................................................................... 170

Figure 6-15. Phase-averaged incoherent vorticity contours at different yaw angles

for T*

= 3.0.. ............................................................................................... 171

Figure 6-16. Phase-averaged incoherent vorticity contours at different yaw angles

for T*

= 1.7 ................................................................................................. 172

Figure 6-17. Phase-averaged velocities components at different yaw angles for

T* = 3.0.. .................................................................................................... 173

Figure 6-18. Phase-averaged velocities components at different yaw angles for

T* = 1.7 ...................................................................................................... 174

Figure 6-19. Phase-averaged Reynolds stresses at different yaw angles for T* = 3.0.. . 175

Figure 6-20. Phase-averaged Reynolds stresses at different yaw angles for T* = 1.7. .. 176

Figure 6-21. Phase-averaged incoherent Reynolds stresses at different yaw angles

for T* = 3.0. ................................................................................................ 177

xiv

Figure 6-22. Phase-averaged incoherent Reynolds stresses at different yaw angles

for T* = 1.7 ................................................................................................. 178

Figure 6-23. Time-averaged Reynolds stresses and their coherent and incoherent

contributions at different yaw angles for T* = 3.0 ..................................... 179


contributions at different yaw angles for T* = 1.7 ..................................... 180

Figure 6-25. Coherent contribution to vorticity variances for different yaw angles. .... 181

Figure 7-1. Geometry of the cylinder with three-strand helical strakes and

definition of the coordinate system. ........................................................... 201

Figure 7-2. Experimental setup of the straked cylinder ................................................. 201

Figure 7-3. Velocity spectra in the wake of the stationary bare and straked

cylinders on the centreline at x/d =5 for Re = 20430. ................................ 201

Figure 7-4. Vortex shedding frequency from the bare cylinder normalized by the

natural frequency. ...................................................................................... 202

Figure 7-5. Comparison of vibration response of a bare cylinder. ................................ 202

Figure 7-6. Comparison of vibration response between bare cylinder and straked

cylinder. ..................................................................................................... 203

Figure 7-7. Energy spectra obtained on the centreline at different streamwise

locations for Re = 20430 in the stationary bare and straked cylinder

wakes. ........................................................................................................ 203

Figure 7-8. Energy spectra obtained at different spanwise locations in the

stationary bare and straked cylinder wakes at x/d =5 for Re = 20430. ...... 204

Figure 7-9. Cross-correlation coefficients in the stationary bare and straked

cylinder at x/d =5 for Re = 20430. ............................................................. 204

Figure 7-10. Cross-correlation coefficients of the velocity components in the

stationary bare and straked cylinder wakes obtained at x/d =5 for

Re = 20430. ................................................................................................ 205

Figure 7-11. Flow visualization of the bare and straked cylinder wakes. ..................... 205

Figure 7-12. Velocity spectra calv and m

v and the ratio calv

mv / between the

measured and calculated transverse velocity spectra in the stationary

straked cylinder wake at various downstream locations and Reynolds

numbers. ..................................................................................................... 206

xv



mv / between the

measured and calculated transverse velocity spectra in the stationary

bare cylinder wake at various downstream locations and Reynolds

numbers. ..................................................................................................... 207

xvi

LIST OF SYMBOLS

Re Reynolds number

Kinematic viscosity

Yaw / Inclination angle

Vortex shedding angle

Inclination angle between the main vortex stretching direction

and the x-axis

d Cylinder diameter

dw Hot wire diameter

t Time

l Length

P Pitch

h Height

k Stiffness

* Superscript asterisk denotes normalization

Difference value

U∞ Free-stream velocity

Uc Convection velocity of vortices

U0 Velocity defect

UN Velocity component normal to the cylinder axis

St Strouhal number

St0 Strouhal number at = 0°

StN Strouhal number normalized by velocity component normal to

the cylinder axis

f0 Peak/ central frequency

fc Cut-off frequency

fn Natural frequency

fs Sampling frequency

fk Kolmogorov frequency

fN Peak/ central frequency normalized by diameter and velocity

component normal to the cylinder axis

0Nf Peak/ central frequency normalized by diameter and velocity

component normal to the cylinder axis at = 0°

xvii

x Streamwise/ longitudinal direction (or location)

y Transverse / lateral direction (or location)

z Spanwise direction (or location)

U (or V and W) Measured velocity component

u (or v and w) Velocity fluctuation component

x (or y and z) Vorticity component

a′ Superscript prime denotes root-mean-square

a Over-bar denotes time-averaging (average value)

a Double over-bar denotes structural average

2a Variance of an instantaneous quantity a

ka Phase average of an instantaneous quantity a, where k

represents the phase

a~ Tilde denotes coherent structure

ar Subscript r denotes incoherent structure

af Filtered signal

Standard deviation

L Wake half-width

Wavelength

T Longitudinal Taylor microscale

R Taylor microscale Reynolds number

Kolmogorov length scale

uk Kolmogorov velocity

Mean turbulent energy dissipation rate

k1 Longitudinal wavenumber

SW Maximum velocity gradients on the distributions of yW /

SU Maximum velocity gradients on the distributions of yU /

yU / (or yW / ) Mean velocity gradient

N Total number

Ts Record duration

T0 Average vortex shedding period

Tu Integral time scale

s second

3D Three-dimensional

xviii

2D Two-dimensional

A/D Analogue-to-digital

a Autocorrelation coefficient of a velocity signal a

)(, zba Cross-correlation coefficient between two velocity signals a

and b in term of the spanwise variation

)(, baR Cross-correlation coefficient between two velocity signals a

and b in term of time delay

La Integral length scale

21,aaL Correlation length

Pa Probability density function

Fa Flatness factors

a Spectral density function of a signal

Phase

r Longitudinal separation between the two points (Taylor‟s

hypothesis eq.)

ar The location at which a has the first zero-crossing

Time delay

0 Time at which the first zero crossing occurs

i Wavelet level

G High-pass wavelet filter

H Low-pass wavelet filter

G~

High-pass synthesis filter

H~

Low-pass synthesis filter

i

vD High-pass subband

i

vA Low-pass residue

ivd Detail signal

iva Approximation signal

T Lateral spacing between two cylinders

Vr Reduced velocity

m Structural mass including the enclosed fluid mass

mf Displaced fluid mass

s Structural damping factor

xix

SG Response parameter

Ks Stability parameter

A Vibration amplitude

Hz Hertz

m metre

cm centimetre

mm millimetre

m micrometre

dB decibel

xx

ABBREVIATIONS

CFD Computational fluid dynamics

DWT Discrete wavelet transform

FFT Fast Fourier transform

IP Independence principle

LVDT Linear variable differential transformer

VIV Vortex-induced vibration

eq. Equation

pdf Probability density function

rms Root-mean-square

sect. Section

vs Versus

1

CHAPTER 1

INTRODUCTION

1.1 Overview

Throughout the world there are at present about more than 180,000 km of subsea

pipelines. Pipeline systems evolve into two main types which include that situated on

the sea floor and that arranged in vertical configurations. Both types are used to

transport oil and gas. Examples of the vertical pipeline systems include risers to carry

electric power lines, drilling tools, cold water pipes for ocean thermal energy and

dredge pipes for deep sea mining (Wilson, 2002). Flows past pipelines can be best

represented by flows past circular cylinders. As a fluid particle flows towards the

leading edge of the cylinder, the pressure in the fluid increases from the free-stream

pressure to the stagnation pressure. The high pressure at the stagnation point “pushes”

the fluid particles to follow the cylinder surface as the boundary layers develop along

the surface. However, the pressure is not sufficiently high to force the fluid particle to

flow closely along the cylinder surface at high velocities (or Reynolds numbers Re).

Around the widest section of the cylinder, the fluid particles near the cylinder surface

cannot follow the cylinder and start to separate from the surface and form two shear

layers. Since the innermost portion of the shear layers moves much slower than the

outermost portion of the shear layers, they roll into the near wake alternatively and

regularly from both sides of the cylinder. This process is known as vortex shedding.

Vortex shedding is an unsteady flow that is created at the back of the cylinder

and formed periodically at both sides of the structure. A few flow regimes have been

identified for vortex shedding previously, depending on the magnitude of Reynolds

number Re, which is defined as Re = U∞d/, where U∞ is the free-stream velocity; d is

cylinder diameter and is kinematic viscosity of the fluid. When Reynolds number is in

the range of 5 Re 45, the flow separates from the cylinder surface to form a

symmetric pair of vortices in the near wake. As Re is increased further, the wake

becomes unstable and one of the vortices breaks away forming a periodic laminar wake

with staggered vortices of opposite sign. The wake becomes turbulent when Re is in the

range of 150 and 300, regardless of the boundary layer on the cylinder remains laminar.

2

The wake flow is completely turbulent when Reynolds number is in the range of 300

and 3105, which is called as subcritical regime. The boundary layer over the cylinder

surface remains laminar and separates from the cylinder surface at about 80° aft the

stagnation point. The vortex shedding is strong and periodic. When Reynolds number is

in the range of 3105 to 3.510

6, the boundary layer over the cylinder surface is in the

transition from laminar to turbulent. In this regime, the separation points move aft to

140° from the stagnation point and the wake becomes narrower and disorganized. The

laminar separation bubbles and three-dimensional effects disrupt the regular vortex

shedding process and resulting in the generation of smaller size vortices and the

broadening of the vortex shedding frequency. Lastly, when Re is increased over 3.5106

(supercritical flow regime), a new vortex shedding is formed with the both sides of

boundary layer are completely turbulent. When vortex is shed from a bluff body,

alternating low pressure vortexes are created along the surface of the structure. The

structure will tend to move towards the low pressure side resulting in the fluctuating

forces acting on the element in both transverse and in-line directions to the current. The

forces will result in large vibrations in the same manner as the vortex is shedding from

the structure, which will influence the life-span of the structure. This phenomenon is

called as vortex-induced vibration.

Vortex-induced vibration (VIV) is widely recognized in the offshore industry as

one of the main causes of fatigue damage to structures, such as marine piles, submarine

periscopes and braced members of offshore structures, exposing to flows and vacillating

in both the in line and cross flow directions. The cross flow oscillations can be excited

at flow velocities greater than the in-line motion (King, 1977b). VIV should be avoided

in engineering applications. This is because: (1) VIV will increase the fluid dynamic

loading to the structures; (2) it will also influence the stability of the structures; and (3)

the vibration of the structures will accelerate the fatigue failure etc. The above factors

will influence both capital investment of the structures and expenses for the

maintenance.

Despite of all previous investigations on vortex shedding from a single cylinder

with normal incidence of velocity where the flow is perpendicular to the cylinder, there

are also many studies concentrate on the vortex shedding behaviours from a yawed

cylinder (e.g. Hanson, 1966; Van Atta, 1968; King, 1977b; Ramberg, 1983;

Kozakiewicz et al., 1995; Lucor and Karniadakis, 2003; Marshall, 2003; Thakur et al.,

2004). These studies are important as in the engineering applications such as suspension

3

bridges, marine pipeline and risers; the flow may not be perpendicular to the bluff body.

In these cases, the vortex shedding from the yawed cylinder structures may be different

with that of a cylinder in a normal incidence flow. The definition of the yaw angle of

a cylinder is the angle between the free-stream flow and the plane which is

perpendicular to the cylinder. Therefore, = 0° is corresponding to the cross-flow case

while = 90° is corresponding to the axial flow case. In a yawed cylinder case, it was

suggested that the Strouhal number St are approximately independent of the yaw angle

when normalized by the free-stream velocity component normal to the cylinder axis UN

(≡ U∞ cos ). This is known as a cosine law or independence principle (IP). However,

there are several studies showed deviations from the IP, mostly at large yaw angles

(Van Atta, 1968; King, 1977a; Ramberg, 1983). They suggested that the IP may not be

suitable for all yaw angles. For example, based on the study of Van Atta (1968), the

vortex shedding frequency tends to decrease with the increase of , for < 35o, while

the decreasing vortex shedding frequency is slower for larger yaw angles. Therefore, the

IP is still a doubt principle in matter of its use in yaw circular cylinder.

In many engineering applications, multiple cylindrical structures are often found

in close proximity. Some examples include: (1) oil platform, which contains column

groups; (2) pile groups in marine structures; (3) floating structures either coupled or

interlinked; and (4) group of tall chimneys and tall buildings, etc. Thus, the study of the

flow in the wakes of multiple cylinders is relevant to many engineering applications and

will lead to improve understanding on the vortex shedding phenomenon on such

multiple bluff-body configurations. A critical review on wakes generated by multiple

cylinders of various arrangements has been given by Zdravkovich (1977). He stated that

there are mainly three groups of possible arrangements of two parallel cylinders,

depending on the approaching flow directions. The first group is cylinders which are in

a tandem arrangement, one behind the other at any longitudinal spacing; the second

group is cylinders placed side-by-side facing the flow at any transverse spacing and the

last group is staggered arrangement, a combination of the two previous arrangements.

Numerical and experimental studies of flow around cylinders in a side-by-side

arrangement have been done by many researchers (Gowda and Deshkulkarni, 1987;

Zhou et al., 2002; Jester and Kallinderis, 2004; Alam and Zhou, 2007a). There are many

major parameters that may be responsible for vortex shed from two cylinders arranged

side-by-side with normal velocity incidence (i.e. the flow is perpendicular to the

cylinders) such as initial conditions, pressure distribution and Reynolds number

4

(Zdravkovich, 1977; Williamson, 1985; Zhou et al., 2002). Besides the parameters

mentioned above, the vortex structures are also influenced by the centre-to-centre

cylinder spacing T. Various flow patterns and behaviours have been identified as the

cylinder spacing T is varied. For example, the variations of T from close proximity

spacing to large spacing could contribute to the formation of a single or multiple wakes.

The T variations are divided into three regimes classification, which are the large

cylinder spacings regime (T ≥ 2d), intermediate cylinder spacing regime (1.2d < T < 2d)

and small cylinder spacing regime (T 1.2d). It is expected that the turbulence wake for

the large cylinder spacing regime may form two vortex streets that are strongly coupled

forming symmetrical wake while that for the intermediate cylinder spacing regime may

form two vortex streets and forming deflected flow, one is wide and the other is narrow.

The small cylinder spacing regime may have the similar wake characteristics as those

for the single cylinder, generating a single vortex street (Sumner et al., 1999; Zhou et

al., 2002; Chen et al., 2003).

Many methods for VIV reduction have been invented by researchers all over the

world. They were suggested to inhibit the formation of vortices or disrupt their

structured formation through the application of the so-called suppression devices.

Zdravkovich (1981) reviewed that there are three categories for interfering vortex

shedding which are surface protrusions, shrouds and near wake stabilisers. Surface

protrusion devices can be grouped into two subdivisions; omnidirectional response type

and unidirectional response type. It suppresses vortex shedding by affecting separation

lines and/or separating shear layers. The examples of surface protrusion devices are

helical wires, helical strake, rectangular plates, bumpy cylinder, fins and spheres with

various arrangements and geometries. However, not all alternatives give satisfied results

in order to suppress the vortex shedding. Slightly change in geometry and arrangement

may give contrary results with an enhancement of oscillation that exceed that found for

the bare cylinders (Zdravkovich, 1981). Helical strake is one of the effective

omnidirectional devices, which is not influenced by fluid velocity direction. The strakes

are screw-like protrusions that are wrapped around the cylinders. It is commonly known

that helical strake is used on cylindrical element to break down large vortices which can

cause VIV and substantial vibrations of the structure, especially when the harmonic load

is at or near the structural natural frequency. However, there are other explanations

about the working mechanism of helical strakes on suppressing VIV. One explanation is

that the strakes cut off the correlation of vortices along the cylinder‟s length and

5

reducing the net transverse force on the cylinder induced by the flow (Thiagarajan et al.,

2005). There is also a belief that the three-dimensionality of separating flow introduced

by the helical strakes destructs the vortex shedding (Bearman and Branković, 2004).

Previous studies also showed that the efficiency of the helical strakes is generally

depending on its geometry, i.e. the height of the protrusion and the pitch of the strakes.

A drawback of the helical strakes is that they may increase drag on pipe sections

significantly. They will also increase the fabrication and installation cost of the

cylindrical structures. Flow visualization done by Constantinides and Oakley (2006)

affirms that the strakes can control the separation of the flow. The helical separation

point at the tip of the strakes induces a three-dimensional flow behind the cylinders

which breaks up the vortex shedding and thus, mitigates VIV. On the other hand, bare

cylinder vibrates with higher amplitude due to the coherent vortex shedding generated

where separation occurred uniformly on the span of the cylinder (Constantinides and

Oakley, 2006). Another mechanism of the helical strakes use to suppress VIV is by

restricting the interaction between two shear layers that are formed due to separation

and acts as obstacle from keeping the shear layers to connect and form the typical

vortices as found in bare cylinders (Constantinides and Oakley, 2006). Many researches

have shown the effectiveness of helical strakes on cylinders in mitigating VIV.

Branković and Bearman (2006) observed that cylinder with helical strakes responds

over a narrow range of reduced flow velocity and its maximum amplitude is decreased

by more than 60%, compared with a bare cylinder. Howells (1998) showed that the

helical strakes can reduce VIV fatigue damage by over 80%. This result was supported

by Vandiver et al. (2006) who performed a study to investigate the effectiveness of

helical strakes in VIV suppressing. They tested experimentally four pipes, which are

bare pipe, pipe with 40% and 70% strakes coverage and pipe with fully strakes

coverage. The study clearly proved that the strakes are effective in mitigating VIV.

1.2 Research Motivations

Since vorticity is an important characteristic of turbulence (Tennekes and

Lumley, 1972), it is important that the turbulent wake characteristics are studied by

measuring all three vorticity components simultaneously. There have been some

published experimental data on vorticity in turbulent near- and intermediate-wake of a

circular cylinder in a cross-flow ( = 0°) (e.g. Marasli et al., 1993; Zhou et al., 2003;

6

Yiu et al., 2004). However, there is no attempt to measure simultaneously the three-

dimensional vorticity components in a yawed cylinder wake. Therefore, the motivation

of the study in the first part of this thesis (Chapter 2) is to provide quantitative

measurements of three-dimensional (3D) velocity and vorticity components in the wake

of a yawed cylinder using a 3D vorticity probe consisting of eight-hot wires. Based on

these measurements, the effects of the yaw angle on the wake characteristics can be

examined in details.

Despite of previous literatures of the yawed cylinder wake studied both

experimentally and numerically (e.g. Lucor and Karniadakis, 2003; Marshall, 2003;

Thakur et al., 2004; Alam and Zhou, 2007b; Zhao et al., 2008), there is no

investigation on the flow structure and its streamwise evolution dependence on yaw

angles. This issue was the motivation of the study in Chapters 3 and 4. In this study, a

phase-averaging technique developed by Kiya & Matsumura (1985) is used to study the

topology of the coherent and incoherent flow structures in the cylinder wake. This

technique is used to analyze the velocity and vorticity signals to obtain the coherent and

incoherent contours and to examine their dependence on cylinder yaw angle. It provides

the quantitative information on the flow structures characteristics and the three-

dimensional velocity and vorticity fields at different downstream locations so that the

effects of the yaw angle on the wake structures and its streamwise evolution can be

examined in details.

In Chapter 5, a discrete wavelet transform (DWT) method is used to analyze the

measured signals by decomposing them into time and frequency domains with wavelet

method. It is a linear decomposition process to convert a signal into a sum of a number

of wavelet components at different scales. The motivation of this chapter is to study the

effect of the yaw angle of a single circular cylinder on the characteristics of large-,

intermediate- and small-scale turbulent structures and to compare them with the

measured large-scale structures.

Although there are some published experimental data on turbulence

characteristics in the wake behind a pair of circular cylinders arranged side-by-side in a

cross-flow (e.g. Bearman and Wadcock, 1973; Zdravkovich, 1977; Williamson, 1985;

Mahir and Rockwell, 1996; Zhou et al., 2001; Zhou et al., 2002), there is no study on

the three-dimensional vorticity characteristics in the wake behind yawed cylinders

arranged side-by-side. Therefore, to gain some more fundamental understanding of the

flow structures in the intermediate wake region, the three-components velocity and

7

vorticity fluctuations in the wake of yawed cylinders arranged side-by-side for cylinder

spacings T = 3.0d and 1.7d are measured using a multiple hotwires vorticity probe.

Based on the data, the effects of the yaw angle and the cylinder spacing on the

turbulence characteristics and vortical structures are examined in details. These aspects

are presented in Chapter 6.

Even though helical strakes are proven effective for suppressing VIV especially

for low mass-damping values, the mechanism of VIV suppression has not yet been fully

understood (Constantinides and Oakley, 2006). Therefore, Chapter 7 aims to shed some

light on the vortex characteristics and to enhance understanding on the mechanisms of

helical strakes have on VIV suppression and to investigate the effect of helical strakes

on vortex shedding from circular cylinders. Comparisons are made between bare

cylinder and the straked cylinder results where the bare cylinder experiment is

performed to provide a benchmark for the straked cylinder system. From the

investigations, the effectiveness of helical strakes on vortex shedding suppression is

evaluated.

The outcomes of this research present reliable deterministic and statistical

methods to predict the VIV effect for wake of a yawed cylinder, two yawed cylinders in

side-by-side arrangement and a helical strakes cylinder.

1.3 Research Objectives

The research objectives of this study are as follows:

i) To investigate the effect of cylinder yaw angle ( = 0°-45°) to the wake

structures and vortex shedding of a single cylinder;

ii) To examine the flow structures downstream of a yawed cylinder;

iii) To investigate the effect of cylinder yaw angle ( = 0°-45°) to the wake

structures and the vortex shedding of two cylinders arranged side-by-side with

centre-to-centre cylinder spacings T = 3.0d and 1.7d ; and

iv) To study the effectiveness of helical strake on a circular cylinder in the reduction

of VIV and to compare the results with those of a bare cylinder.

1.4 Thesis Outlines

This thesis comprises eight chapters. The organization of this thesis is as

follows:

8

In Chapter 1, the background of the study, motivations and objectives of the

study and the outlines of the thesis are presented.

In Chapter 2, three-dimensional velocity and vorticity characteristics of a

stationary cylinder in the streamwise range of x/d = 10-40 and yaw angle of = 0°-45°

are investigated.

In Chapter 3, further investigation on the dependence of the wake vortical

structures on the cylinder yaw angles in the intermediate wake region (x/d = 10) is

carried out by using phase-averaging technique.

In Chapter 4, the streamwise evolution of the yawed cylinder wake at three

downstream location, i.e. x/d = 10, 20 and 40 from the cylinder is investigated by using

phase-averaging technique.

In Chapter 5, wavelet method is used to examine the validity of the

independence principle (IP) of a yawed circular cylinder wake with = 0°-45° at x/d =

10. The velocity and vorticity characteristics at different wavelet scales were also

examined.

In Chapter 6, the wake structures behind two yawed circular cylinders ( = 0°-

45°) in side-by-side arrangement (centre-to-centre cylinder spacings T = 3.0d and 1.7d)

are studied using phase-averaging method.

In Chapter 7, the mechanisms of the reduction of vortex-induced vibration

(VIV) using helical strakes is studied by using hot wire technique and comparisons are

done between the results of the bare cylinder wake and those of the staked cylinder

wake.

In Chapter 8, general conclusions of this study and suggestions for future work

are presented.

9

CHAPTER 2

DEPENDENCE OF THE WAKE ON YAW ANGLES OF A

STATIONARY CYLINDER

2.1 Introduction

Vortex shedding is a phenomenon when a fluid flows over a bluff body at a

sufficiently high velocity, or Reynolds number Re (≡ U∞d/, where U∞ is the free-stream

velocity in the streamwise direction, d is the cylinder diameter and is the kinematic

viscosity of the fluid). When vortex is shed from the bluff body, the latter is subjected to

time-dependent drag and lift forces. The forces will result in vibrations if the vortex shedding

frequency is close to the natural frequency of the bluff body, which may influence the fatigue

life of the structure. This phenomenon is called vortex-induced vibration (VIV). VIV is

widely recognized in the offshore industry as one of the main causes of fatigue damage to

structures, such as marine piles and pipelines, submarine periscopes and braced members of

offshore structures exposing to flows and oscillating in both the in-line and cross-flow

directions. VIV should be avoided in engineering applications due to its influence on the

stability and fatigue life of the structures.

Vortex shedding from a single circular cylinder with normal incidence of velocity (i.e.

the flow is perpendicular to the bluff body) is well documented. The phenomenon has been

investigated extensively by many researchers both in air and water. Previous investigations

have shown that there are three types of vortical structures in the near wake, namely, the

Kármán alternating vortices with predominantly spanwise vorticity (Roshko, 1961), the

longitudinal smaller scale rib-like vortices wrapping around and connecting the consecutive

spanwise structures (Williamson, 1992) and the Kelvin-Helmholtz vortices in the shear layer

(Bloor, 1964). In practical engineering applications, such as the flow past cables of a

suspension bridge (Matsumoto et al., 1992; 2001; Alam and Zhou, 2007b), subsea pipelines

and risers etc., the direction of the flow may not be perpendicular to the bluff body. In these

cases, the fluid velocity in the axial direction of the structure may not be negligible and

vortex shedding from these yawed structures may be different from that of a cylinder in a

cross-flow. Wake flows of a stationary yawed cylinder have been studied by a number of

10

investigators both experimentally in a wind tunnel or water flume (e.g. King, 1977a;

Ramberg, 1983; Kozakiewicz et al., 1995; Lucor and Karniadakis, 2003; Thakur et al., 2004)

and numerically (e.g. Chiba and Horikawa, 1987; Marshall, 2003) for inclination or yaw

angles ranging from 0° to 75°. They have also been studied when a rigid cylinder is subjected

to vibration (e.g. King, 1977b; Ramberg, 1983; Lucor and Karniadakis, 2003). The yaw angle

() is defined as the angle between the free-stream and the plane which is perpendicular to

the cylinder axis so that = 0° corresponds to the cross-flow case while = 90° corresponds

to the axial flow case. The results on cylinder base pressure and vortex shedding frequency

have shown that the yawed cylinders behave in a similar way to the normal-incidence case

through the use of the component of the free-stream velocity normal to the cylinder axis. For

example, the force coefficients and the Strouhal number, when normalized by the velocity

component normal to the cylinder axis, are approximately independent of . This is often

known as the independence principle (IP) or the cosine law in the literature.

Several theoretical and experimental studies have verified the IP (Hoerner, 1965;

Schlichting, 1979). Most experimental and numerical studies have shown deviations from the

predictions based on the IP (Hanson, 1966; Surry and Surry, 1967; Van Atta, 1968; King,

1977b; Ramberg, 1983; Kozakiewicz et al., 1995; Lucor and Karniadakis, 2003; Marshall,

2003; Thakur et al., 2004), especially at large yaw angles. Hanson (1966) studied vortex

shedding from vibrating inclined hot wires in air flow for low Reynolds numbers.

Independence principle was validated at < 68°. Deviation from the IP for vortex shedding

frequencies is found at = 70°. Van Atta (1968) showed that for the non-vibrating yawed

cylinders, the decrease in the vortex shedding frequency follows approximately the IP at ≤

35°, whereas for larger the decrease of vortex shedding frequency with the increase of is

slower than that predicted by the IP. Similar results were reported by Thakur et al. (2004).

Surry and Surry (1967) found that the Strouhal number based on the normal velocity

component is approximately constant for Re = 4,000-63,000 up to = 40°-50°. They also

found that the energy in the Strouhal peak disperses and decreases significantly with the

increase of . By measuring the lift and drag forces of a yawed cylinder in a steady current

and waves, Kozakiewicz et al. (1995) showed that the IP can be applied to stationary

cylinders in the vicinity of a plane wall in the subcritical range at = 0°-45°. Based on flow

visualization in a yawed cylinder wake, they explained that although the approaching flow is

at an angle to the cylinder ( 0°), the streamlines around the surface of the cylinder are

roughly perpendicular to the cylinder axis, justifying the use of the normal velocity

11

components in calculations of shedding frequency and steady drag forces. They found that

the wake flow characteristics began to change at = 55°, and at = 70° the water particles

moved along the cylinder axis rather than perpendicular to the cylinder axis. From the flow

visualization experiments, they inferred that the IP is valid till = 55°. Ramberg (1983)

studied the effect of the yaw angle ( = 60°) and the end conditions on vortex shedding for

stationary and forced vibrating circular cylinders with aspect ratios of 20–90 at Re = 160–

1,100. He found that the results were very sensitive to end-conditions especially at low

Reynolds numbers. He showed that slantwise vortex shedding at angles other than the

cylinder yaw angles is intrinsic to stationary yawed cylinders in the absence of end effects.

As a result, the IP is not valid in the case of stationary yawed cylinders because the shedding

frequency is always greater than that predicted by the IP, while the shedding angle, the

vortex-formation length, the base pressure and the wake width are all less than expected.

However, for vibrating yawed cylinders at Reynolds number in the range of 160-460, vortex

shedding parallel to the cylinder axis is found. Frequency lock-in between the wake and the

cylinder motion was accompanied by vortex shedding parallel to the cylinder axis and the

lock-in vortex wakes can be described successfully by means of the IP. These results were

supported by Lucor and Karniadakis (2003) using a direct numerical simulation of flow past a

yawed cylinder with infinite length, who showed that at high yaw angles (e.g. > 60°), the

vortices shed from the cylinder orientated at a yaw angle somewhat less than the cylinder

yaw angle. They also showed that slantwise shedding at angles other than the cylinder yaw

angle is intrinsic to stationary yawed cylinders in the absence of end effects. The base

pressure is lower than the value predicted by the IP and hence the drag coefficient is higher

than the value predicted the IP. Despite the previous experimental and numerical studies of

yawed cylinder wakes on vortex shedding patterns and base pressure etc., there is no

information about the effect of on 3-dimensional (3D) velocity and vorticity distributions

across the wake of a yawed circular cylinder. Most of the previous numerical and

experimental works have shown, at least in the subcritical region, that the vortex shedding

frequency and forces of a yawed cylinder can be obtained by using the velocity component

normal to the cylinder axis (Lucor and Karniadakis, 2003). Nevertheless, there are some

controversial reports about the range of over which the prediction of vortex shedding

frequency based on the IP is reliable.

In the present study, the effects of on the velocity and vorticity distributions across

the wake, the vortex shedding frequencies and wake characteristics are examined using a 3-

12

dimensional vorticity probe consisting of 8-hot wires at a free-stream velocity U∞ of about

8.5 m/s, corresponding to Re = 7,200, which is high enough to classify the flow as in the

turbulent regime. Since vorticity is an important characteristic of turbulence (Tennekes and

Lumley, 1972), it is important that the turbulent wake characteristics are studied by

measuring all three vorticity components simultaneously. However, the measurement of

vorticity has proven to be a big challenge (e.g. Tsinober et al., 1992; Marasli et al., 1993;

Wallace and Foss, 1995; Antonia et al., 1998; Cavo et al., 2007), especially when all three

components of the vorticity vector are needed in high Reynolds number flows. A few types of

multiple hot wire probes have been developed over the past decades to measure the 3D

vorticity components simultaneously (e.g. Tsinober et al., 1992; Antonia et al., 1998; Gulitski

et al., 2007). Gulitski et al. (2007) used a probe consisting of 20 wires to measure all velocity

and temperature derivatives without invoking Taylor‟s hypothesis in an atmospheric surface

layer. The 3D vorticity probe developed by Antonia et al. (1998) consists of 4 X-probes (i.e.

with a total number of 8 hot wires). The most significant advantage of the probe is its ease in

making the wires and the simple calibrating process, which involves only the same

calibration procedures as for a single X-probe. This kind of probe has been successfully used

in grid and wake flows to measure the 3D vorticity components simultaneously (Antonia et

al., 1998; Zhou et al., 2003). There have been some published experimental data on vorticity

in turbulent near- and intermediate-wake of a circular cylinder in a cross-flow ( = 0°) (e.g.

Marasli et al., 1993; Zhou et al., 2003; Yiu et al., 2004). However, there is no attempt to

measure simultaneously the 3-dimensional vorticity components in yawed cylinder wakes.

Therefore, the main objective of the present paper is to provide quantitative measurements of

3D velocity and vorticity components in the wake of a yawed cylinder. Based on these

measurements, the effects of the yaw angle on the wake characteristics can be examined in

details.

2.2 Experimental Details

The experiments were conducted in a closed loop wind tunnel with a test section of

1.2 m (width) 0.8 m (height) and 2 m long. The free-stream across the tunnel is uniform to

within 0.5%. The free-stream turbulence intensity is less than 0.5%. Three-components

vorticity are measured simultaneously in the cylinder wake at downstream locations x/d = 10,

20 and 40 for = 0°, 15°, 30° and 45°, respectively. The cylinder is a 115 cm long smooth

stainless steel cylinder with an outer diameter of 12.7 mm, i.e. an aspect ratio of about 86,

13

which is close to the value used by Ramberg (1983) for studying the end effects in yawed

cylinder wakes. The cylinder is aligned horizontally at the centre of the test section and

supported rigidly at the ends by two aluminium plates. A vorticity probe was moved across

the wake in the y-direction to measure simultaneously the 3D vorticity components. The

probe consists of 4 X-probes (i.e. X-probes A, B, C and D), as shown in Figure 2-1. In the

present paper, the coordinate system is defined such that the x-axis is in the same direction as

the incoming flow, the y-axis is perpendicular to the x-axis in the vertical plane through the

cylinder and the z-axis is normal to both x and y axes. Two X-probes (B and D) aligned in the

x-y plane and separated in the z-direction measure velocity fluctuations u and v; the other two

(A and C) aligned in the x-z plane and separated in the y-direction measure velocity

fluctuations u and w. The separations between the centres of the two opposite X-probes

(either B and D or A and C) are about 2.7mm. The separation between the two inclined wires

of each X-probe was about 0.7 mm. The hot wires were etched from Wollaston (Pt-10% Rh)

wires. The active length was about 200dw, where dw (2.5 m) is the hot wire diameter. The

wires were operated with in-house constant temperature circuits at an overheat ratio of 0.5.

The vorticity probe was calibrated at the centreline of the wind tunnel against a Pitot-static

tube connected to a MKS Baratron pressure transducer (least count = 0.01 mm H2O). The

angle calibration was performed over 20°. The included angle of each X-probe was about

110° and the effective angle of the inclined wires was about 35°. Output signals from the

anemometers were passed through buck and gain circuits and low-pass filtered at a cut-off

frequency fc of 5,200 Hz, which is high enough for the purposes to detect the vortex shedding

frequency. The filtered signals were subsequently sampled at a frequency fs of 10,400 Hz

using a 16 bit A/D converter. The record duration Ts was about 60s.

Since an X-probe measures the two velocity components at the centre of the probe,

the measured velocity components can be significantly in error if the velocity gradients are

large. In the present study, as the measurement locations are from x/d = 10-40, the mean

velocity gradient is not significant. Using a range of hot wire yaw factors corresponding to

the present experimental conditions, the errors in neglecting the velocity gradients are

estimated to be about 4% and 5% for u′ and v′ (or w′), respectively, where u’, v’ and w’ are

the velocity fluctuations in the x, y and z directions, respectively, and a superscript prime

denotes root-mean-square (rms) values. The binormal cooling effect on each X-probe has

also been neglected, which only gives rise to an error of 1-3% for u′ and v′ (or w′) when the

local turbulence intensity is about 10%. The present u′/U ranges from 20% -10% for x/d=10

14

to 40, where U is the local streamwise mean velocity and an over-bar denotes time-

averaging. Therefore, the error in neglecting the binormal cooling effect is estimated to be

about 36%. Experimental uncertainties in U and u′ (or v′ and w′) were inferred from the

errors in the hot wire calibration data as well as the scatter (20 to 1 odds) observed in

repeating the experiment for a number of times. The uncertainty for U (≡ Uu ) was about

±3%, while the uncertainties for u′, v′ and w′ were about ±7%, ±8% and ±8%, respectively.

The vorticity components are calculated from the measured velocity signals via.

z

v

y

w

z

v

y

wx

, (2-1)

x

w

z

u

x

w

z

uy

, (2-2)

y

uU

x

v

y

uU

x

vz

)()(

. (2-3)

where w and u in Eqs. (2-1) and (2-3), respectively, are velocity differences between X-

probes A and C (Figure 2-1b and c); v and u in Eqs. (2-1) and (2-2), respectively, are

velocity differences between X-probes B and D. The velocity gradients in the streamwise

direction w/x and v/x are obtained by using a central difference scheme to the time

series of the measured velocity signals, e.g. v/x ≈ xiviv /)]1()1([ , where the

streamwise separation x is estimated based on Taylor‟s hypothesis given by x = -Uc(2t).

Uc is the convection velocity of vortices and t ( 1/fs) is the time interval between two

consecutive points in the time series of the velocity signals. Zhou and Antonia (1992) showed

that Uc is in the range of 0.87-0.89U∞ for measurement locations x/d = 10-40. The value of Uc

at location y is obtained by performing an ensemble average given by

n

i

cic yUn

yU1

)(1

)(

where Uci (y) is the velocity at the centre of the ith

vortex and n is the total number of vortices

detected at the location y (Zhou and Antonia, 1992). A central difference scheme in

estimating w/x and v/x would have the advantage to avoid phase shifts between the

velocity gradients involved in Eqs. (2-2) and (2-3) (Wallace and Foss, 1995), where the phase

shift is any change that occurs in the phase of one quantity, or in the phase difference

between two or more quantities (Ballou, 2005).

15

2.3 Results and Discussion

2.3.1 Mean streamwise and spanwise velocity distributions

Before the vorticity components are calculated, it is necessary to make sure that the

velocity components are measured correctly. It has been checked that the mean velocity in

the streamwise direction measured from the 4 Xprobes agree favourably. Therefore, only

the values averaged from the 4 X-probes are shown for each x/d station. The mean velocity

distributions U measured at x/d = 10-40 for different are shown in Figure 2-2 (a-c). The

distributions are symmetric about the centreline. The velocity defect, which is the difference

between the free-stream velocity U∞ and the measured local mean velocity U , is the largest

for = 0°. With increasing , the velocity defect decreases, which is consistent with that

shown by Surry and Surry (1967). This result reflects a decreased maximum velocity defect

U0 and wake half-width L with the increase of , which is defined as the lateral distance from

the centreline to the point where the velocity defect is 0.5U0. The values of U0 and L for

different downstream locations and yaw angles are listed in Table 2-1. The values of U0 and

L for = 0° agree very well with those given by Antonia and Mi (1998) for a circular

cylinder wake. If the mean velocity distributions shown in Figure 2-2 are replotted in the

form of the velocity defect (U∞-U )/U0 as a function of y/L, there is excellent agreement of

the data at different x/d and (results are not shown here). However, this result does not

necessarily mean that self-preservation of the mean velocity has been achieved for all these

cases. Previous studies showed that self-preservation is not attained until / 280x d (e.g.

Zhou and Antonia, 1995). The spanwise velocity W provides a measure of the flow three-

dimensionality (Matsumoto et al., 1992). A larger magnitude in W implies a higher degree of

three-dimensionality. The spanwise velocity in general acts to impair the 2-dimensionality of

the wake and hence the vorticity strength (Marshall, 2003; Alam and Zhou, 2007b). The

values of W for different are shown in Figure 2-3 (a-c). For = 0°, W is nearly zero

across the wake at all downstream locations, which is expected in a two-dimensional plane

wake. When increases, W increases significantly around the centreline and yet remains

symmetric about y/d = 0. This trend is consistent with that reported by Alam and Zhou

(2007b). The significant larger values of W reported in the latter are due to the measurement

location, which is much closer to the cylinder than in the present study. The magnitude of W

is at least 3 times larger than that of V (results are not shown here) at the same , indicating

16

a strong three-dimensionality of the yawed circular cylinder wake. With the increase of the

downstream locations, the magnitude of W decreases. The values of W are positive for y/d <

2 and negative for y/d > 2. The mechanism of such a change in W has been explained by

Alam and Zhou (2007b). Using flow visualization, Kozakiewicz et al. (1995) showed that the

streamlines that approach the leading surface of the cylinder bend to the cross cylinder

direction and then some of the dye moves in a helical track along the cylinder axis for a short

distance while the rest continues to move in the streamwise direction. Similar results are also

obtained by Zhao et al. (2008) for an yawed cylinder wake using direct numerical

simulations. These results indicate an enhanced velocity component in the spanwise direction

as the yaw angle increases, which is consistent with the trend shown by the present

distribution of W in Figure 2-3. As the mean velocity gradients are related with the mean

vorticity components, it would be interesting to examine the dependence of the former on .

The mean velocity gradients are obtained by first applying polynomial curve fits to the

measured time-averaged velocity distributions. The gradients are then calculated from the

fitted curves. The variations of /W y at x/d = 20 for different are shown in Figure 2-4.

The trends of the velocity gradients at other downstream locations are comparable with that

shown in Figure 2-4 and are therefore not given here. The values of /W y at = 0° are

close to zero, consistent with the two-dimensionality of the flow. The maximum mean

velocity gradient /W y occurs at around y/d = 0.75, implying a strong mean streamwise

vorticity )//( zVyWx for the yawed cylinder wakes. Clearly, with the increase of

, the magnitude of /W y increases. For comparison, the distributions of /U y at x/d =

20 for different are also included in the figure. The magnitude of /U y is comparable

with that of /W y . Its values decrease with the increase of , indicating a decreased mean

shear /U y with the increase of . Due to the small magnitude of V and hence the small

gradients of V in all three directions, it is expected that the spanwise mean vorticity

)//( yUxVz is comparable in magnitude with the streamwise mean vorticity.

2.3.2 Rms velocity and vorticity distributions across the wake

The root-mean-square (rms) values of the longitudinal velocity component u at x/d =

10-40 for different are shown in Figure 2-5 (a-c). The distributions of u′/U∞ are symmetric

about y/d = 0 and show twin peaks, especially at = 0°. With the increase of , the

17

magnitude of u′ decreases significantly, which is consistent with that shown by Surry and

Surry (1967), especially when is changed from 15° to 45°. The decay of u′ in the

streamwise direction for all is significant. Both the decay rate and the magnitude of u′ for

= 0° agree well with those published previously (e.g. Zhou et al., 2003). If the rms values are

renormalized by the maximum wake velocity deficit U0 and y is normalized by the wake half-

width L for the purpose to examine the self-preservation, the distribution of u′/U0 versus y/L

at x/d = 40 is still much smaller than that at x/d = 20 (results are not shown here), indicating

that self-preservation is not attained for all cases. In contrast to the distribution of u′, the

distributions of v′ at a fixed downstream location show reasonable agreement for different

(Figure 2-6a-c) although there is a significant decrease of v′ in the streamwise direction.

There is only one peak occurring at the wake centreline for all cases. This result is in

agreement with that shown by Alam and Zhou (2007b) for of 10° and 35°. The

distributions of w′ are shown in Figure 2-7 (a-c). Similar to that of u′, the distributions of w′

have twin peaks. With the increase of , w′ decreases consistently.

Based on the measured velocity signals, the three-dimensional vorticity components

are calculated using Eqs. (2-1) to (2-3). Their rms values at x/d = 10, normalized by U∞ and d,

are shown in Figure 2-8 (a-c) for different . It can be seen that with the increase of , the

rms values of all three vorticity components decrease, even though the change from = 0° to

15° is not significant. This is consistent with the increase of the spanwise mean velocity W ,

which in general acts to impair the two-dimensionality of the wake and reduce the vorticity

strength. The numerical simulations by Lucor and Karniadakis (2003) and Zhao et al. (2008)

showed that with the increase of , the vortex shedding angle is less well defined and the

vortices are distorted. Locally the axial vortices may or may not be parallel to the cylinder.

The decrease of velocity and vorticity rms values with the increase of may be related with

the instability of the wake vortices and the decrease of the vortex strength. The magnitude of

z′ is comparable with that of y′ and they are about 20% larger than that of x′ for all yaw

angles over the range of y/d = ±1. It needs to be noted that this result may not necessarily

mean that the transverse and spanwise vorticity components are stronger than that of the

streamwise one due to the spatial resolution of the probe. The spatial resolution y or z of

the probe in the y or z-direction is about 2.7mm. When the velocity gradients in the

streamwise direction xv / and xw / involved in Eqs. (2-2) and (2-3) are calculated, to

avoid phase shift between the two velocity gradients in the expression of y and z, the

18

spatial separation x in the streamwise direction is calculated from the time delay by using

Taylor's hypothesis via x = -Uc(2t). This will result in a separation in the streamwise

direction x ≈ 1.47 mm, which is only half of y or z (≡ 2.7 mm). Therefore, the spatial

resolution of the probe in the x-direction is better than that both in the y and z-directions. This

may result in larger measured values of both y and z than x. The trends of the rms

vorticity components at other downstream locations are similar to those shown in Figure 2-8

(x/d = 10) and are therefore not shown in this paper.

By examining the dependence on of the cross-correlation coefficient zw , between

the spanwise velocity w and the spanwise vorticity z, some light may be shed on the axial

velocity deficit reported by Marshall (2003) using computations of quasi-two-dimensional

theory. The cross-correlation coefficient zw , is defined as

''/))()()((, zzzw wtWtwz

, (2-4)

where the angular brackets denote time-averaging. For a yawed cylinder wake, the incoming

velocity can be decomposed into two components, the normal and the axial to the cylinder.

The normal velocity generates Kármán vortices and the axial component generates the axial

vortices. Computations using a quasi-two-dimensional theory by Marshall (2003) showed

that the cross-stream vorticity is shed from the cylinder as thin sheets and to wrap around the

Kármán vortices, which in turn induce an axial velocity deficit, i.e., with axial velocity lower

than the surrounding fluid, within the wake vortex cores. He proposed two mechanisms by

which the axial velocity might act to destabilize the vortex street for large yaw angles. The

first mechanism is the Kelvin-Helmholtz type instability of the cross-stream vorticity sheets

in the cylinder near wake, partially inhibited by stretching from the cylinder wake vortices

and by shear in the cross-stream plane acting orthogonally to the shear induced by the vortex

sheet. The second mechanism is the instability of the downstream Kármán vortices due to the

presence of axial velocity deficit within the vortex cores. Recent direct numerical simulations

by Zhao et al. (2008) supported Marshall‟s (2003) theoretical predictions for the existence of

the axial velocity deficit within the wake vortex cores. These authors also showed that the

axial velocity deficit exists for all yaw angles considered and its magnitude increases with the

increasing . Figure 2-9 shows the cross-correlation coefficient zw , at y/d = 0.75 (i.e. the

location with maximum mean shear as shown in Figure 2-4) for different downstream

locations and at different . The values of zw , are negative. This result is consistent with

19

the theoretical prediction of Marshall (2003) that the wake vortices exhibit an axial velocity

deficit. When = 0°, zw , is very close to zero ( 0.04 to 0.05) at all measurement

locations, which is consistent with 2D characteristics. At x/d = 10, when increases, zw ,

increases monotonically, changing from 0.06 to about 0.17, indicating an enhanced axial

velocity deficit, which is consistent with the direct numerical simulations of Zhao et al.

(2008). At x/d = 20 and 40, the increase of zw , with is also evident even though the rate

of increase is smaller than that at x/d = 10. It is apparent that zw , decreases consistently in

magnitude for all yaw angles with downstream distance, which is consistent with the trend

expected by Marshall (2003). The decrease of the correlation coefficient is related with the

decay of the vortices in the streamwise direction.

2.3.3 Dependence of the vortex shedding frequency on

The vortex shedding frequency can be determined by examining the peak frequency

on the power spectral density function, which is obtained by calculating fast Fourier

transform (FFT) of the velocity signals. It is well known that the transverse velocity

component v is more sensitive to the organized structures than the other two components.

Therefore, only the spectra of the transverse velocity component v are shown. The velocity

spectra obtained at y/d = 0.5 for different at x/d = 10 are shown in Figure 2-10. The

normalized spectral density function is defined such that

0

2)/()( Uvdxxv , where x

represents the frequency, which is normalized by U∞ and d, i.e. x = fd/UN and UN (≡ U∞ cos

) is the velocity component normal to the cylinder axis. With this normalization, the peak

frequency f0, or after normalization f0d/UN, on the spectrum corresponds to the Strouhal

number StN (≡ f0d/UN). The peak frequency for other downstream locations (x/d = 20 and 40)

is the same as that for x/d = 10 and was not shown. For all the cases, there is a clear peak on

the energy spectra, which corresponds to the Strouhal number. For = 0°, St0 = 0.195, which

is the same as our previous result (Zhou et al., 2003). There are second and third harmonics at

= 0° and 15° due to the effect of the alternate vortex shedding from both sides of the

cylinder at y/d 0.5. The harmonic peaks diminish as the yaw angle increases. At the

downstream location x/d = 20, the second harmonic peak disappears completely except for

= 0°. A compilation of some existing results on St0 (Surry and Surry, 1967; Norberg, 2003) in

a cross-flow have shown that for Re 300, St0 of circular cylinder wakes depends

significantly on Re, while for Re > 300in the subcritical flow regime, the change in St0 is not

20

very significant. Ideally, to examine the effects of on StN, the experiments should be done

at the same ReN. In the present study, Re = 7,000, and ReN at = 45° is about 5,100.

According to the compilation of St0 on Re (Surry and Surry, 1967; Norberg, 2003), the

change of St0 over this Reynolds number range is not significant. Therefore, we have used a

constant value for St0 at different Reynolds numbers (ReN = 5,1007,000). Our results for all

three downstream locations have shown pronounced peaks, which correspond to the Strouhal

numbers StN of 0.195, 0.202, 0.206 and 0.224 for = 0°, 15°, 30° and 45°, respectively,

increasing monotonically with . However, for = 15° and 30°, the StN values are only about

4% and 6% higher than that obtained at = 0°. These differences may be disguised by the

experimental uncertainty of StN, which is estimated to be within 8% and StN can be

considered as a constant. The increase of StN from = 0° to 45° (by about 15%) cannot be

ascribed to experimental uncertainty. It may reflect a genuine departure from the IP for large

. In this case, significant spanwise velocity component W has been measured (Figure 2-3).

Marshall (2003) suggested that two mechanisms by which the axial velocity might act to

destabilize the vortex street and lead to the breakdown of the IP for sufficiently large yaw

angles. At low frequencies, the spectra level out, but at decreasing levels with the increase of

(Figure 2-10), indicating that the energy contained by the lower frequency structures,

which correspond to larger size structures, decreases and so a reduced vortex strength. At

high frequencies, the spectra collapse well for all with a common slope of about 2 for 4f0

f 15f0, which is larger than that of the energy spectra for an isotropic flow (5/3). For

even higher frequencies, say f ≥ 30f0, the energy spectra drops at a much higher rate of

around 3.8. From Figure 2-10, it is also noted that for = 0°, the peak is sharp and narrow.

The ratio between the peak height and the height of the plateau on the spectra over the range

of f* = 0.006-0.06 is as high as 110. Hereafter, an asterisk denotes normalization by d or/and

U∞. When is increased, the height of the peak decreases and the ratio between the peak

height and the plateau reduces to 90, 50 and 30 for = 15°, 30° and 45°, respectively,

indicating a reduced vortex shedding intensity. The width of the peak region on the spectra,

as indicated by the arrows, broadens as is increased. This result indicates the dispersion of

the vortex shedding frequency which may be caused by the breakdown of the large organized

structures as is increased. These results are consistent with the vorticity contours obtained

by Lucor and Karniadakis (2003) and Zhao et al. (2008), who showed that, as is increased,

the shape of the vortices is distorted and locally the axial vortices may or may not be parallel

21

to the cylinder axis. It is also found that vortices right behind the cylinder are convected not

only in the incoming flow direction, but also in the cylinder axial direction. This results in

dispersion of the large-organized vortical structures.

Figure 2-11 shows the comparison of the present Strouhal numbers with the

experimental data obtained by Ramberg (1983) for flows past an yawed cylinder of 90d long,

experimental results by Surry and Surry (1967), the fitted curve from Van Atta (1968) and the

numerical results of Thakur et al. (2004). Instead of showing the measured StN, the ratio

StN/St0 is shown in Figure 2-11, where St0 represents the Strouhal number at = 0°. If IP

prediction works satisfactorily, the ratio StN/St0 should be equal to 1 for all . The

experimental uncertainty of the present StN is estimated to be around 8%. The results show

that if a tolerance of 8% is applied, most of the data support the IP till = 40°. This means

that the IP works reasonably well if the yaw angle is smaller than about 40°. For > 40°,

the results show a systematically increased departure from the IP. At = 60°, the departure

from the IP prediction is as high as 40%.

2.3.4 Autocorrelation coefficients

Autocorrelation coefficients can be used to highlight the differences of the organized

structures at different yaw angles and it has been known that the transverse velocity

component v is a more sensitive indicator of the large-scale structures than either u or w. The

autocorrelation coefficient is defined as 2/)()( arxaxaa , where a represents

the velocity components u, v and w and r is the longitudinal separation between the two

points, which is calculated using Taylor‟s hypothesis by converting the time delay to a

streamwise separation via r = U∞/fs. The autocorrelation coefficients for u, v and w at

different downstream locations and different yaw angles are shown in Figure 2-12 to Figure

2-14, respectively. For all , u shows apparent periodicity (Figure 2-12). Their periods are

consistent with the vortex shedding frequency for each . Clearly, with the increase of , the

vortex shedding frequency decreases. At x/d = 10 (Figure 2-12a), u drops quickly from 1 to

about 0.3. The autocorrelation coefficient for = 0° persists for a long distance. Even at

tU∞/d = 200, the periodicity is still apparent. But with the increase of , the magnitude of u

decreases and approaches zero at a faster rate. For example, at = 15°, u approaches zero at

about tU∞/d = 60. This has been decreased to 40 and 14 for = 30° and 45°, respectively.

This result indicates that the vortices shed from the yawed circular cylinder decay more

22

rapidly as compared with that at = 0° and the decay is more apparent with the increase of .

At further downstream location, e.g. x/d = 20 (Figure 2-12b), the magnitude of u for all

decreases significantly, even though they still show clear periodicity. Their values approach

zero even faster than those at x/d = 10. This result indicates that at this downstream location,

the vortices have decayed significantly. At x/d = 40 (Figure 2-12c), all u distributions show

no periodicity, even for = 0°. These results are consistent with that reported by Zhou et al.

(2003), who showed using a phase-averaging technique that the Kármán vortex street broke

down at x/d = 40. The decay of the wake vortices in the streamwise direction at different

needs further investigation by using either phase-averaging or flow visualization techniques.

The autocorrelation coefficients v at x/d = 10 for different are shown in Figure

2-13a. Their magnitudes are much larger than that of u at the same streamwise location,

indicating that the v signals are more sensitive to the large-scale organized structures than u.

The periodicity reflected by this figure for each yaw angle is the same as that in Figure 2-12a.

For = 0°, v persists for a long distance. Its magnitude is about 0.17 at tU∞/d = 200. When

is increased to 15°, v does not change significantly for tU∞/d 10. But from tU∞/d > 10, v

starts to depart from that at = 0° and approaches zero at tU∞/d = 75. This has been further

reduced to tU∞/d =50 and 20 when is increased to 30° and 45°, respectively. This result

suggests that with the increase of , the organized large-scale structures decay faster and lose

their initial characteristics quickly, as shown by Lucor and Karniadakis (2003) and Thakur et

al. (2004) in their numerical simulations. At x/d = 20 (Figure 2-13b), the magnitude of v

decreases significantly with apparent periodicity for all yaw angles. The v signals lose their

autocorrelation at tU∞/d = 70, 30 and 20 for = 15°, 30° and 45° respectively. At x/d = 40

(Figure 2-13c), the magnitude of v decreases further as compared with that at x/d = 10 or 20

and yet still keeps apparent periodicity with a magnitude of about 0.1 for all yaw angles.

Compared with Figure 2-12c, which shows no periodicity at x/d = 40 usingu, Figure 2-13c

indicates that the v signals are indeed more sensitive to the organized structures than u. It is

expected that at further downstream locations, the organized structures decay completely and

there would be no periodicity on v observed. Antonia et al. (2002) have shown in a circular

cylinder wake that at x/d = 70, there is no periodicity in the distribution of v, indicating that

the large-organized structures completely disappeared at this location.

Significant differences can be observed between the autocorrelation coefficient w of

the spanwise velocity component w and both u (Figure 2-12) and v (Figure 2-13). The

23

autocorrelation coefficient w is very small, even at x/d = 10 (Figure 2-14a). At this location,

w increases with the increase of . It shows apparent periodicity for 30°. This result is

consistent with the increase of W as increases (Figure 2-3). Since the increase of W

implies the enhancement of the 3-dimensionality of the wake, the periodicity of w for

30° also indicates a stronger 3-dimensionality as increases. The values of w approach zero

quickly, say before tU∞/d = 20. At a further downstream location (x/d = 20) (Figure 2-14b),

the values of w reduce significantly, even for = 45° and approach zero quickly. At x/d = 40

(Figure 2-14c), the autocorrelation coefficient w is nearly zero for all yaw angles, which is in

agreement with that of u at this location, suggesting that the large-scale organized structures

generated in the near wake have decayed significantly at x/d = 40.

2.4 Conclusions

The dependence of the wake characteristics on cylinder yaw angle bas been studied using a

multi-hot wire vorticity probe over the region of x/d = 10-40 for of 0°, 15°, 30° and 45°,

respectively. The results are summarized as follows:

1. The mean streamwise and spanwise velocity components U and W increase with .

The increase of the streamwise mean velocity U indicates a reduction of the wake

width whereas the increase of the spanwise mean velocity W represents that the three-

dimensionality of the wake flow is enhanced.

2. The root-mean-square (rms) values of the longitudinal (u) and spanwise (w) velocities

and the three vorticity components decrease with the increase of . This result reflects

the reduction of the vortex strength as increases.

3. The vortex shedding frequency decreases with the increase of . The Strouhal number

(StN), obtained by using the velocity component normal to the cylinder axis, also

increases monotonically with . However, when is smaller than 40°, StN keeps

approximately a constant within the experimental uncertainty (±8%), supporting the

use of the IP. The intensity of vortex shedding is decreased and the frequency is

dispersed with the increase of .

4. The autocorrelation coefficients of the u and v signals show apparent periodic

characteristics of the vortex structures. With the increase of , the autocorrelation

24

coefficients u and v reduce and approach zero quickly. In contrast, the

autocorrelation coefficient w increases with at x/d = 10 and 20, indicating the

enhancement of the three-dimensionality of the wake.

5. The values of the cross-correlation coefficient zw , between the spanwise velocity w

and the spanwise vorticity z are negative for all yaw angles. This result is consistent

with the theoretical prediction by Marshall (2003) that an axial velocity deficit exists

in the wake vortex cores. With the increase of , the magnitude of zw , increases,

indicating an enhanced axial velocity deficit, consistent with the direct numerical

simulation results of Zhao et al. (2008). In the streamwise direction, zw , decreases

apparently, which is related with the decay of the vortices.

25

Table 2-1. Values of U0, L and rms velocities on the wake centreline for different

downstream locations and yaw angles.

x/d U0 (m/s) L (mm) u′ (m/s) v′(m/s) w′(m/s)

0° 1.417 10.1 1.506 2.227 1.109

10 15° 1.165 10.4 1.484 2.364 1.06

30° 0.855 10.10 1.348 2.329 1.025

45° 0.631 8.64 1.18 2.204 0.942

0° 1.604 12.93 1.06 1.305 0.849

20 15° 1.446 12.7 1.055 1.356 0.843

30° 1.119 11.72 0. 96 1.38 0.827

45° 0.761 11.3 0.89 1.372 0.781

0° 1.266 19.6 0.642 0.67 0.609

40 15° 1.172 19.04 0.642 0.688 0.55

30° 0.971 17.08 0.62 0.708 0.55

45° 0.702 15.57 0.589 0.704 0.54

26

(a) Coordinate system

y

z

x

V

U

W

U

z

x

y

(a) Coordinate system

y

z

x

V

U

W

U

z

x

y

y

z

x

V

U

W

y

z

x

V

U

W

U

z

x

y

(c) Side view of the probe

U

4,7

C(u,w)

B,D

y

3,8

1

2

5

x

A(u,w)

U

4,7

C(u,w)

B,D

y

3,8

1

2

5

x

A(u,w)

(b) Front view of the probe

8 74

z

y

y

1

2

6

3

5

z

A(u,w)

C(u,w)

B(u,v)

D(u,v)

8 74

z

y

y

1

2

6

3

5

z

A(u,w)

C(u,w)

B(u,v)

D(u,v)

Figure 2-1 Definition of the coordinate system and the sketches of the vorticity probe.

Note: X-probe A contains wires 1 and 2 to measure u and w; X-probe B contains wires 3 and 4 to

measure u and v; X-probe C contains wires 5 and 6 to measure u and w; X-probe D contains wires 7

and 8 to measure u and v.

0.80

0.85

0.90

0.95

1.00

1.05

-2 -1 0 1 2 3 4

0

15

30

45

(a)

y/d

U/U

0.80

0.85

0.90

0.95

1.00

1.05

-2 -1 0 1 2 3 4

0

15

30

45

(b)

y/d

U/U

0.80

0.85

0.90

0.95

1.00

1.05

-2 -1 0 1 2 3 4 5

0

15

30

45

(c)

y/d

U/U

Figure 2-2. Normalized mean streamwise velocity distribution for different yaw angles. (a)

x/d = 10; (b) 20; (c) 40.

27

-0.05

0

0.05

0.10

-2 -1 0 1 2 3 4

0

15

30

45

(a)

y/d

W/U

-0.05

0

0.05

0.10

-2 -1 0 1 2 3 4

0

15

30

45

(b)

y/d

W/U

-0.05

0

0.05

0.10

-2 -1 0 1 2 3 4 5

0

15

30

45

(c)

y/d

W/U

Figure 2-3. Normalized mean spanwise velocity distribution for different yaw angles. (a) x/d

= 10; (b) 20; (c) 40.

-0.15

-0.10

-0.05

0

0.05

0.10

0.15

-2 -1 0 1 2 3 4

W*/y

*

U*/y

*

y/d

U

* /y* ,

W

* /y*

Figure 2-4. Normalized velocity gradients for different yaw angles at x/d = 20. The arrows

indicate the direction of increasing . : = 0°; ---: 15°; - : 30°; : 45°.

28

0

0.05

0.10

0.15

0.20

-2 -1 0 1 2 3 4

0

15

30

45

(a)

y/d

u'/U

0

0.05

0.10

0.15

-2 -1 0 1 2 3 4

0

15

30

45

(b)

y/d

u'/U

0

0.02

0.04

0.06

0.08

-2 -1 0 1 2 3 4 5

0

15

30

45

(c)

yld

u'/U

Figure 2-5. Distributions of u′ for different yaw angles. (a): x/d = 10; (b): 20; (c): 40.

0

0.1

0.2

0.3

-2 -1 0 1 2 3 4

0

15

30

45

(a)

y/d

v'/U

0

0.05

0.10

0.15

0.20

-2 -1 0 1 2 3 4

0

15

30

45

(b)

y/d

v'/U

0

0.03

0.06

0.09

-2 -1 0 1 2 3 4 5

0

15

30

45

(c)

y/d

v'/U

Figure 2-6. Distribution of v′ for different yaw angles. (a): x/d = 10; (b): 20; (c): 40.

29

0

0.05

0.10

0.15

-2 -1 0 1 2 3 4

0

15

30

45

(a)

y/d

w'/U

0

0.04

0.08

0.12

-2 -1 0 1 2 3 4

0

15

30

45

(b)

y/d

w'/U

0

0.02

0.04

0.06

0.08

-2 -1 0 1 2 3 4 5

0

15

30

45

(c)

y/d

w'/U

Figure 2-7. Distributions of w′ for different yaw angles. (a): x/d = 10; (b): 20; (c): 40.

0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 2 3 4

0

15

30

45

(a)

y/d

' xd

/U

0

0.2

0.4

0.6

0.8

1.0

1.2

-2 -1 0 1 2 3 4

0

15

30

45

(b)

y/d

' yd

/U

0

0.2

0.4

0.6

0.8

1.0

1.2

-2 -1 0 1 2 3 4

0

15

30

45

(c)

y/d

' zd

/U

Figure 2-8. Distributions of x′, y′ and z′ at x/d = 10 for different yaw angles. (a): x′; (b):

y′: (c): z′.

30

0

0.05

0.10

0.15

0.20

0 15 30 45

x/d = 10

x/d = 20

x/d = 40

(deg)

w,

z

Figure 2-9. Dependence of the cross-correlation coefficient zw , between the spanwise

velocity w and the spanwise vorticity z on at different streamwise locations.

10-5

10-3

10-1

101

10-2

10-1

100

101

0

15

30

45

Slope: –3.8

Slope: –2

fd/UN

v

Figure 2-10. Spectral density function v of the transverse velocity component at x/d = 10 for

different yaw angles. The arrows represent the direction of increasing yaw angle. : =

0°; ---: 15°; - : 30°; : 45°.

31

0.8

1.0

1.2

1.4

1.6

0 10 20 30 40 50 60

Re=3600 (Surry and Surry, 1967)

Re=25000 (Surry and Surry, 1967)

Re=63000(Surry and Surry, 1967)

Present

Thakur et al. (2003)

Van Atta (1968) (fitted curve)

Ramberg (1968) (Re=1100)

Ramberg (1968) (Re=900)

-8%

+8%

(degree)

St N

/St 0

Figure 2-11. Comparison of StN with other experimental and numerical results. Present:

. Surry and Surry (1967): + (Re = 3,600); (25000); (63,000). Thakur et al. (2004):

■. Ramberg (1983): □ (Re = 1,100); 〇 (900). Van Atta (1968): . The horizontal

short dashed lines represent the range of experimental uncertainty.

-1.0

-0.5

0

0.5

1.0

0.1 1 10 100

0

15

30

45

(a)

tU/d

u

0

0.5

1.0

0.1 1 10 100

0

15

30

45

(b)

tU/d

u

0

0.25

0.50

0.75

1.00

0.1 1 10 100

0

15

30

45

(c)

tU/d

u

Figure 2-12. Autocorrelation coefficient u for different yaw angles. (a): x/d = 10; (b): 20;

(c): 40. : = 0°; - : 15°; ---: 30°; : 45°.

32

-1.0

-0.5

0

0.5

1.0

0.1 1 10 100

0

15

30

45

(a)

tU/d

v

-1.0

-0.5

0

0.5

1.0

0.1 1 10 100

0

15

30

45

(b)

tU/d

v

-1.0

-0.5

0

0.5

1.0

0.1 1 10 100

0

15

30

45

(c)

tU/d

v

Figure 2-13. Autocorrelation coefficient v for different yaw angles. (a): x/d = 10; (b): 20;

(c): 40. : = 0°; - : 15°; ---: 30°; : 45°.

-0.5

0

0.5

1.0

0.1 1 10 100

0

15

30

45

(a)

tU/d

w

-0.5

0

0.5

1.0

0.1 1 10 100

0

15

30

45

(b)

tU/d

w

-0.5

0

0.5

1.0

0.1 1 10 100

0

15

30

45

(c)

tU/d

w

Figure 2-14. Autocorrelation coefficient w for different yaw angles. (a): x/d = 10; (b): 20;

(c): 40. : = 0°; - : 15°; ---: 30°; : 45°.

33

CHAPTER 3

THREE-DIMENSIONAL VORTICITY MEASUREMENTS

IN THE WAKE OF A YAWED CIRCULAR CYLINDER

3.1 Introduction

When a fluid flows over a circular cylinder at a Reynolds number exceeding 47-

49 (Provansal, 1987; Norberg, 1994; Williamson, 1996c), a laminar 2-dimensional (2D)

periodic wake of staggered vortices of opposite sign is formed. The vortices can be

either parallel or oblique to the cylinder axis, depending on the end boundary conditions

which also caused vortex dislocation in the laminar shedding regime (Williamson,

1996b). These forms of vortex shedding result in a 20% disparity in the relationship

between the Strouhal number and Reynolds number in the laminar shedding regime

(Williamson, 1988). When Re is increased to above 180, there exists a transition of the

2D wake to 3D wake. It is now well known that this transition is associated with two

discontinuous changes in the wake formation (Williamson, 1996b), which may be

manifested by discontinuous changes in the Strouhal number ~ Reynolds number

relationship, or the change in base suction coefficient (Williamson and Roshko, 1990;

Norberg, 1994). The first discontinuity, defined as Mode A shedding by Williamson

(1992), is hysteric and occurs at Re = 180-190, depending on the experimental

conditions (Williamson and Roshko, 1990; Norberg, 1994; Leweke and Provansal,

1995; Williamson, 1996c; Thompson et al., 2006). It is associated with the inception of

the vortex loops and the formation of the streamwise vortex pairs due to deformation of

the primary vortices. The spanwise wavelength of the loops is around 3-4d. Using

highly accurate numerical methods and Floquet stability analysis, Henderson and

Barkley (1996) predicted the onset of Mode A instability occurring at Re = 188.5 with a

spanwise wavelength of 4d, which is very close to the experimental results above

(Provansal, 1987; Williamson, 1988; Norberg, 1994; Williamson, 1996c, 1996b). The

second discontinuity occurs over Re = 230-260, defined as Mode B shedding

(Williamson, 1988), which involves a gradual transfer of energy from Mode A to Mode

B. It is associated with the formation of the rib-like small-scale streamwise vortex pairs

with a spanwise wavelength of about 1d (Williamson, 1996b). Within the transition,

34

large-scale spot-like vortex dislocations caused by local shedding-phase dislocation

along the span of the cylinder have been identified (Williamson, 1992). The above

phenomena, namely vortex dislocation, parallel and oblique shedding and cellular

shedding at low Re, are also observed at high Re turbulent wakes. Prasad and

Williamson (1997) showed that at Re = 5,000, parallel and oblique shedding can be

triggered by manipulating the cylinder end conditions. Some evidences of vortex

dislocation over the span at Re = 10,000 were also provided by Norberg (1992). Even at

Re = 130,000, the experimental results for cylinder aspect ratio of 7 by Szepessy and

Bearman (1992) strongly suggested the existence of vortex dislocation in their wakes.

Therefore, Prasad and Williamson (1997) suggested that vortex dislocation may be a

feature of the flow for Re up to 200,000. When vortices are shed from a cylindrical

structure, the latter is subjected to time-dependent drag and lift forces. The forces will

result in vibrations if the vortex shedding frequency is close to the natural frequency of

the bluff body, which may influence the fatigue life of the structure. As in nearly all the

engineering applications such as marine piles and pipelines and braced members of

offshore structures, the Reynolds numbers are much larger than the above critical

values, turbulent vortex shedding is a widely existing phenomenon.

In some engineering applications, such as flow past cables of a cable-stayed

bridge (Matsumoto et al., 1992; Matsumoto et al., 2001; Alam and Zhou, 2007b),

subsea pipelines and marine risers, raker piles on deep-water oil terminals (King,

1977a), etc., the incoming flow approaches the cylindrical structures obliquely. In these

cases, the fluid velocity in the axial direction of the structure may not be negligible,

which may influence the vortex structures and enhance the three-dimensionality of the

flow downstream. However, the study of yawed cylinder wakes is far less extensive

than that of a cylinder in a cross-flow. In the present study, the yaw angle () is defined

as the angle between the free-stream velocity and the plane which is perpendicular to

the cylinder axis so that = 0° corresponds to the cross-flow case while = 90°

corresponds to the axial-flow case. A number of studies have been conducted on the

vortex shedding from a stationary yawed cylinder both experimentally and numerically

at yaw angles ranging from 0° to 75°. It was found that the force coefficients and the

Strouhal number, when normalized by UN (≡ U∞ cos ) the velocity component normal

to the cylinder axis, are approximately independent of . This is often known as the

independence principle (IP), or the cosine law. One implication of the IP is that the yaw

angle does not influence the spanwise coherence of the vortex shedding (King, 1977a).

35

There are a number of studies on the validity of the IP. Van Atta (1968) showed that for

non-vibrating yawed cylinders, the decrease in the vortex shedding frequency followed

approximately the IP for 35°, whereas this decrease was slower than that predicted

by the IP for larger . Kozakiewicz et al. (1995) showed that the IP could be applied to

stationary cylinders in the vicinity of a plane wall in the subcritical range, and explained

based on the flow visualization results that, although the approaching flow was at an

angle to the cylinder (i.e. ≠ 0°), the streamlines around the surface of the cylinder

were roughly perpendicular to the cylinder axis, providing a validation for the use of the

normal velocity component in determining the normalized shedding frequency or the

Strouhal number and steady drag. These authors found that the wake characteristics

began to change at = 55°. They inferred that the IP was valid till = 55°. The

numerical simulations by Zhao et al. (2008) verify the IP satisfactorily in terms of

Strouhal numbers up to = 60°. They also showed that with the increase of , the

vortex shedding angle is less well defined as compared with that in a cross-flow and the

vortices are distorted. Ramberg (1983) studied the effect of the yaw angle up to = 60°

and the end conditions on vortex shedding for both stationary and forced vibrating

circular cylinders with aspect ratios of 20-90 at Re = 160-1100. His results were very

sensitive to the end conditions especially at low Reynolds numbers, and the slantwise

vortex shedding at an angle other than the cylinder yaw angle was intrinsic to the

stationary yawed cylinders in the absence of the end effects, leading to the violation of

the IP. Based on numerical simulations, Lucor & Karniadakis (2003) found that the

angle of vortex shedding from a yawed stationary cylinder was somewhat less than the

cylinder‟s yaw angle, which violated the IP. While the previous studies mostly focused

on vortex shedding frequencies, base pressure, and drag and lift forces, there has been

no detailed study on the effect of on the three-dimensionality of the wake flow behind

a yawed circular cylinder. This knowledge is of both fundamental and practical

importance to our understanding of the physics of vortex shedding, three-dimensionality

of the velocity and vorticity fields, the change of the coherent structures and the vortex

dislocation in the yawed cylinder wake at subcritical Reynolds numbers. A three-

dimensional detached eddy simulation (DES) was conducted by Yeo and Jones (2008)

for three different spanwise aspect ratios (l/d = 10, 20 and 30, where l and d are the

length and diameter of the cylinder respectively) using both slip wall and periodic wall

conditions for the spanwise wall boundary conditions. They found that the slip wall

spanwise boundary conditions influenced the vortex flow structures significantly over

36

quite a large spanwise range from the upper end of the cylinder, which was not the case

if the periodic spanwise wall boundary conditions were used. It can also be inferred

from their iso-surface contours of the second invariant that the vortex flow structures

downstream of the cylinder was generally parallel to the cylinder except at locations

where intense vorticity, low pressure and swirling flow were found. These locations

were not fixed but shifted along the cylinder axis, which was associated with the

spanwise velocity component of the flow.

In the present study, the effects of on the three-dimensional velocity and

vorticity fields, the strength of the coherent structures and the vortex shedding

frequency are examined using a three-dimensional vorticity probe, consisting of 8-hot

wires, at U∞ of about 8.5 m/s, corresponding to Re = 7,200. Since vorticity is an

important characteristic of turbulence (Tennekes and Lumley, 1972), it would be ideal

to measure all three vorticity components simultaneously across the wake. This is,

however, impractical due to possible blockage to the flow and the tremendous

difficulties involved in dealing with a great number of three-dimensional vorticity

probes. Alternatively, a phase-averaging technique developed by Kiya & Matsumura

(1985) can be used to study the topology of the coherent and incoherent flow structures

in the cylinder wake. This technique has also been used by Zhou et al. (2002) to

investigate the wake of two interfering cylinders. It requires only one moving

measurement probe and one fixed single-wire or X-wire for reference signals, thus

providing a good compromise between the sophistication of the vorticity probe and the

need to measure the flow field for the study of large-scale vortical structures. The

present work provides a relatively complete set of data for the three-velocity and

vorticity components at a streamwise location of x/d = 10 for different cylinder yaw

angles. The analysis of the measurements is conducted by using the phase-averaging

technique to the velocity and vorticity signals to obtain the coherent and incoherent

contours and to examine their dependence on cylinder yaw angle. This study also

quantified the contributions of the phase-averaged coherent and incoherent structures to

the velocity and vorticity components and based on these results, the effects of the yaw

angle on the wake structures can be examined in details. After an introduction in Sect.

3.1, the experimental setup is given in Sect. 3.2. The velocity and vorticity signals for

cylinder yaw angles of 0° and 45° are examined in Sect. 3.3. The phase-averaging and

structural averaging of the vorticity and Reynolds stresses and their dependence on

37

are discussed in Sects. 3.4 and 3.5. Based on the above discussion, conclusions are

drawn in Sect. 3.6.


The experiments were conducted in a closed loop wind tunnel with a test section

of 1.2 m (width) 0.8 m (height) and 2 m long. The free-stream across the tunnel is

uniform to within 0.5%. The free-stream turbulence intensity is less than 0.5%. All

three-components of the vorticity vector are measured simultaneously in the cylinder

wake at downstream location x/d = 10 for = 0°, 15°, 30° and 45°, respectively. The

cylinder is a 115 cm long smooth stainless steel cylinder with an outer diameter of 12.7

mm. It is aligned horizontally at the centre of the test section and supported rigidly at

the ends by two square aluminium plates to eliminate the side wall boundary layer

effects. The coordinate system is defined such that the x-axis is in the same direction as

the incoming flow. The y-axis is perpendicular to the x-axis in the vertical plane through

the cylinder and out of the paper towards the reader, and the z-axis is normal to both x

and y axes (Figure 3-1a). A vorticity probe was moved across the wake in the y-

direction to measure simultaneously the three-dimensional vorticity components.

Another X-probe located at y = 4-7d was used in conjunction with the vorticity probe to

provide a phase reference for the measured velocity and vorticity signals (Figure 3-1b).

The vorticity probe consists of four X-probes (i.e. X-probes A, B, C and D), as shown in

Figure 3-1 (c, d). Two X-probes (B and D) aligned in the x-y plane and separated in the

z-direction measure velocity fluctuations u and v; the other two (A and C) aligned in the

x-z plane and separated in the y-direction measure velocity fluctuations u and w. The

separations between the centres of the two opposite X-probes (either B and D or A and

C) were about 2.7mm. The separation between the two inclined wires of each X-probe

was about 0.7 mm. The hot wires were etched from Wollaston (Pt-10% Rh) wires. The

active length was about 200dw, where dw (2.5 m) is the wire diameter. The angle

calibration was performed over 20°. The included angle of each X-probe was about

110° and the effective angle of the inclined wires was about 35°. The signals were low

pass filtered at a frequency of 5,200 Hz and subsequently sampled at a frequency fs =

10,400 Hz using a 16 bit A/D converter. The record duration Ts was about 20s. As in the

present study, the measured minimum vortex shedding frequency at = 45° is 102 Hz,

this record duration corresponds to 2040 cycles, which is about 14 times of the

suggested signal length (Sakamoto et al., 1987) for studying the vortex shedding. The

38

number of independent samples N ≡Ts/2Tu suggested by Tennekes and Lumley (1972) is

around 12,320, where Tu 0

0)(

du is an integral time scale, u is the longitudinal

velocity autocorrelation coefficient of u and 0 is the time at which the first zero

crossing occurs.

The vorticity components are calculated from the measured velocity signals viz.

z

v

y

w

z

v

y

wx

, (3-1)

x

w

z

u

x

w

z

uy

, (3-2)

y

uU

x

v

y

uU

x

vz

)()(

. (3-3)

where w and u in Eqs. (3-1) and (3-3), respectively, are the velocity differences

between X-probes A and C (Figure 3-1c and d); v and u in Eqs. (3-1) and (3-2),

respectively, are the velocity differences between X-probes B and D. The velocity

gradients in the streamwise direction w/x and v/x are obtained by using a central

difference scheme to the time series of the measured velocity signals, e.g. v/x ≈

xiviv /)]1()1([ , where the streamwise separation x is estimated based on

Taylor‟s hypothesis given by x = -Uc(2t) as Uc is the convection velocity of vortices

and t ( 1/fs) is the time interval between two consecutive points in the time series of

the velocity signals. A central difference scheme in estimating w/x and v/x would

has the advantage to avoid phase shifts between the velocity gradients involved in Eqs.

(3-2) and (3-3) (Wallace and Foss, 1995).

Experimental uncertainties in U and u′ (or v′ and w′) were inferred from the

errors in the hot wire calibration data as well as the scatter (20 to 1 odds) observed in

repeating the experiment for a number of times. Hereafter, a superscript prime denotes

root-mean-square values. The uncertainty for U was about ±3%, while the uncertainties

for u′, v′ and w′ were about ±7%, ±8% and ±8%, respectively. The uncertainty for wire

separation x was about ±4% and that for y or z was about ±5%. Using these values,

the uncertainties for vorticity components were estimated by the method of propagation

of errors (e.g. Kline and McClintock, 1953; Moffat, 1985, , 1988). The resulting

maximum uncertainties for x, y and z were about ±14%, ±13% and ±13%,

39

respectively. It needs to be noted that these estimations of uncertainties do not include

the spectral attenuation caused by the unsatisfactory spatial resolution in x, y and z

of the 3D vorticity probe (Antonia et al., 1998). The correction of the spectral

attenuation is based on the assumption of isotropy of the flow. Unfortunately, the

spectral attenuation effect cannot be corrected in the present study due to the lack of

isotropy at x/d = 10 for the shedding of large organized vortex structures. The vortex

shedding frequency is identified using the fast Fourier transform algorithm with a

window size of 211

. The frequency resolution on the power spectra is about 2.5 Hz.

Taking into the uncertainties in hot wire calibration, incoming velocity measurements

and misalignment, etc., the uncertainty in the Strouhal number (will be defined later) is

conservatively estimated to be not more than 8%.

3.3 Velocity and Vorticity Signals

Figure 3-2 shows the filtered time traces of the fluctuating velocity v measured

at y/d = 0.5 for = 0° and 45°, respectively, together with signals for x, y and z. In

the figure, an asterisk denotes normalization by d and U∞. Previous studies have shown

that y/d = 0.5 is the location of the vortex centres (Zhou et al., 2002). The v-signals at

both yaw angles show apparent large-scale quasi-periodical fluctuations with relatively

fixed frequency, indicating the occurrence of the Kármán vortices. The fluctuation of

the velocity signals is not as significant as the vorticity signals. This result clearly shows

that the vorticity signals represent more small-scale structures. As indicated by the

dashed vertical lines, there is in general a correspondence between the large negative z

fluctuations and zero-crossing point (positive dv/dt) of the v-signal. Based on

conditional analysis results, Zhou and Antonia (1993) found a correspondence between

the centre of spanwise vortices (negative sign) and zero v of positive dv/dt. Therefore,

the zero values of v at positive dv/dt, marked by dots in the v-signals (Figure 3-2), are

identified with the possible Kármán vortex centre. Similarly, the x and y signals also

exhibit considerable fluctuations at the vortex centres, though at relatively small-scales

in terms of time-wise or longitudinal spatial distance, reflecting the three-dimensionality

of the Kármán vortices. At = 0°, the fluctuations of the vorticity signals are quite

significant. However, at = 45°, all the three vorticity signals show apparent enhanced

intermittency which can be reflected by the flatness factors )/( 224 iiiF of

the vorticity components. The values of i

F for the two angles are given in Table 3-1.

40

This result indicates that the shedding of vortices at high yaw angles are not as strong as

that in a cross-flow ( = 0°). We have checked that with the increase of from 0° to

45°, the change of vorticity signals is gradual. Zhao et al. (2008) and also Lucor &

Karniadakis (2003) have shown that as is increased, the shape of the vortices is

distorted and locally the axial vortices may or may not be parallel to the cylinder axis. It

is also found by these authors that the vortices right behind the cylinder are convected

not only in the incoming flow direction, but also in the cylinder axial direction, which is

consistent with that of Yeo and Jones (2008). The enhanced spanwise velocity as is

increased helps to dislocate the large-scale structures and enhance the three-

dimensionality of the flow, resulting in dispersion of the large-scale vortical structures.

This result may indicate that with the increase of , the intermittency of the vortex

structures increases, which can be revealed also by the probability density function of

the vorticity signals.

The probability density function (pdf) of the three vorticity components for =

0° and 45° are shown in Figure 3-3. Also shown in the figure is the Gaussian

distribution. The pdf is defined such that

1idP

i . The measured distributions of

the vorticity pdf are apparently non-symmetric. They exceed the Gaussian distribution

for both large and small amplitude fluctuations. The distributions of x

P and y

P show

a pronounced peak at around zero. It can be seen that for small vorticity fluctuations,

with the increase of , the shapes of the pdfs of x and y contracted and their

amplitudes increase. For large vorticity fluctuations, the tails of both x

P and y

P

spread out slightly and follow a single exponential distribution as is increased from 0°

to 45°, indicating an increased y

F and z

F . The distribution of z

P exhibits twin

peaks (Figure 3-3c). The peak of negative vorticity is due to the spanwise vortices of

negative sign that occur at y/d > 0, whereas that of positive vorticity is attributed to the

spanwise vortices of positive sign, whose centre is at y/d < 0. As is increased from 0°

to 45°, the twin peaks in z

P increase slightly and shift to larger values of '/ zz while

the magnitude of z

P around zero also increases. For = 0°, an expression xePi

~

fits the exponential tail adequately (where = 1.3 and x stands for the negative vorticity

fluctuations normalized by the rms values), which is close to that reported by Kida and

Murakami (1989) using DNS, where a power of 2 was reported. When is increased

41

to 45°, the slope of the tails for x

P and y

P decreases slightly even though it still

shows a single exponential distribution. The slightly increased peaks for all the three

vorticity components in the vicinity of the origin indicate the increased flatness factors

with the increase of the yaw angle, which is consistent with the results shown in Figure

3-2.

Figure 3-4 shows the vorticity spectra ,x

y and ,z

measured at the wake

vortex centre (y/d = 0.5) for = 0° and 45° respectively. The vorticity spectrum i

has

been normalized to the decibel scale using the maxima of z at = 0° and 45°

respectively, while the frequency is normalized by UN and d, i.e. x = fd/UN and UN (≡

U∞ cos ) is the velocity component normal to the cylinder axis. With this

normalization, the peak frequency f0 on the spectrum, or after normalization, fd/UN,

corresponds to the Strouhal number StN (≡ f0d/UN). For all cases, there is a clear peak on

the vorticity spectra, which corresponds to the Strouhal number. For = 0°, St0 = 0.195,

which is the same as our previous results (Zhou et al., 2003; Zhou et al., 2009). The

peak on the spectrum z is sharp and narrow, which occurs over a narrow range (f

*)

of f* = 0.146-0.223. There is only a minor peak on

x . The peak heights relative to the

heights of the plateaux on the vorticity spectra x and

z over the range of f* =

0.004-0.12 are 2.6 dB and 14 dB, respectively. When is increased to 45°, the peak on

the spectrum z is much less sharp and the peak region (f

*) is enlarged to f

* = 0.090-

0.34. In contrast to = 0°, the peak on x is increased and becomes more apparent,

even though the peak region is enlarged. The peak heights relative to the heights of the

plateaux on the vorticity spectra x and

z over the range of f* = 0.004-0.1 are 6 dB

and 10.6 dB, respectively, indicating a reduced spanwise vortex shedding intensity and

an enhanced streamwise vortex shedding intensity. These results indicate the dispersion

of the turbulent energy over the vortex shedding frequency, as well as the enhancement

of the three-dimensionality of the flow. Williamson (1996b) stated that after the onset of

the turbulent shedding and with the increase of Reynolds number, there seems to be an

increased disorder in the fine-scale three-dimensionality associated with the secondary

and essentiality streamwise-oriented vortices of type mode B. The study of Mansy et al.

(1994) showed that the increase in the secondary or streamwise vortices probably is at

the expense of the primary spanwise vortices, which is consistent with the trend shown

42

in Figure 3-4. The present results are also consistent with the vorticity iso-contours

obtained by Lucor & Karniadakis (2003) and Zhao et al. (2008), who showed that, as

is increased, the shape of the spanwise vortices is distorted and reveals a helical style.

The large-scale organized structures found in the wake of a cross-flow are broken down

and the vortex shedding angle of the spanwise vortices is much less well defined. They

may or may not be parallel to the cylinder axis. It is also found that vortices right behind

the cylinder are convected not only in the incoming flow direction, but also in the

cylinder axial direction. This results in the dispersion of the large-organized vortical

structures. The peak frequencies on the spectra of i

suggest that with the increase of

, the Strouhal number StN changes gradually from 0.195 at = 0° to 0.202, 0.206 and

0.224 for = 15°, 30° and 45°, respectively, increasing monotonically with (Figures

for = 15° and 30° are not shown here). However, as the StN values at = 15° and 30°

are only about 4% and 6% higher than that obtained at = 0°, these differences may be

disguised by the experimental uncertainty of StN, which is estimated to be not more than

8% and StN can be considered as a constant. The increase of StN from = 0° to 45° (by

about 15%) cannot be ascribed to the experimental uncertainty. It may reflect a genuine

departure from the IP for large . In this case, significant mean spanwise velocity

component W has been measured as is increased (Zhou et al., 2009), which was

conjectured as the main mechanism by which the axial velocity might act to destabilize

the vortex street and cause the vortex dislocation in the spanwise direction, leading to

the breakdown of the IP for sufficiently large yaw angles (Chiba and Horikawa, 1987).

The spanwise velocity W provides a measure of three-dimensionality of the

flow (Matsumoto et al., 1992). A larger magnitude in W implies a higher degree of

three-dimensionality and a stronger instability of the vortex filament (Szepessy and

Bearman, 1992). The variations of /W y at x/d = 10 for different are shown in

Figure 3-5. The values of /W y for = 0° are close to zero, consistent with the quasi-

two-dimensionality of the flow. This result also implies that the vortex shedding is

parallel with the cylinder axis (Hammache and Gharib, 1991). Clearly, with the increase

of , the magnitude of /W y increases. The maximum mean velocity gradient

/W y occurs at around y/d = 0.5-0.65, depending on ; with the increase of , the

location of the maximum mean shear /W y approaches to the centreline. This result

43

implies an increased mean streamwise vorticity )//( zVyWx as is

increased. For comparison, the distributions of /U y at x/d = 10 for different are

also included in the figure. The magnitude of /U y is comparable with that of

/W y at large yaw angles. Its values decrease with the increase of , indicating a

decreased mean spanwise vorticity )//( yUxVz with the increase of . Due

to the small magnitude of V and hence the small velocity gradients of V in the x and z-

directions, the magnitudes of xV / and zV / in the expressions of z and x

should be negligible and the spanwise mean vorticity z is comparable in magnitude

with the streamwise mean vorticity x .

The present measurements allow the shedding angle downstream of the cylinder

to be evaluated using the method proposed by Hammache and Gharib (1991). They

showed that for oblique shedding, the shedding angle is related with the vorticity

components x and z viz.,

zx /tan , (3-4)

which can then be simplified further as

UW SS /tan , (3-5)

where SW and SU are the maximum velocity gradients on the distributions of yW /

and yU / . The shedding angle ( UW SS /(tan 1 ) at different cylinder yaw angle

can then be calculated from the values of SW and SU (Figure 3-5). The shedding angles

for different cylinder yaw angles are listed in Table 3-2. The values of agree very well

with the corresponding cylinder yaw angle , indicating that the vortex is parallel to the

cylinder in the present study.

Autocorrelation coefficient can be used to highlight the differences of the

organized structures at different yaw angles. The autocorrelation coefficient is defined

as 2/)()( arxaxaa , where a represents the vorticity components x, y

and z and r is the longitudinal separation between the two points, which is calculated

using Taylor‟s hypothesis by converting the time delay to a streamwise separation via

r = UN/fs. The autocorrelation coefficients for x, y and z at different cylinder yaw

44

angles are shown in Figure 3-6. For ≤ 15°, the autocorrelation coefficient x

is very

small and there is nearly no periodicity. When is increased to 30°, x

starts to reveal

some periodicity, especially at 45°. This result is consistent with the increase of W as

increases since the increase of W causes the instability of the primary vortex shedding,

leading to the enhancement of the 3-dimensionality of the wake (Szepessy and

Bearman, 1992). The distributions of y show no periodicity at all . This is

consistent with the spectrum of y (Figure 3-4), where no peak on y is identified. The

periodicity on the distribution of z

is very apparent. This is true as z mainly

represents the primary vortices. At = 0°, z

persists for a long distance. Even at

tU∞/d = 200, the periodicity on z

is still very apparent. However, with the increase of

, the magnitude of z

decreases and approaches zero at a faster rate. For example, at

= 15°, u approaches zero at about tU∞/d = 60. This has been decreased to 40 and 14

for = 30° and 45°, respectively. This result indicates that the vortices shed from a

yawed circular cylinder decay more rapidly as compared with that at = 0° and the

decay is more apparent with the increase of . Comparing with Figure 3-6a, where x

increases with , it seems that the increase in the secondary or streamwise vortices,

reflected by x, is at the expense of the primary spanwise vortices, reflected by z,

which is in agreement with Mansy et al. (1994).

3.4 Phase-Averaged Velocity and Vorticity Fields

3.4.1 Phase-averaging

The phase-averaging method is similar to that used by Kiya & Matsumura (1985) and

Zhou et al. (2002). Briefly, the v-signal measured by the reference X-probe was band-

pass filtered with the central frequency set at f0. The two phases of particular interest

were identified on the filtered signal vf, viz.

Phase A: 0fv

and ,0

dt

dv f

(3-6)

Phase B: 0fv

and .0

dt

dv f

(3-7)

45

The two phases correspond to time tA,i and tB,i (measured from an arbitrary time origin),

respectively. The phase was then calculated from vf, viz.

,,,

,

iAiB

iA

tt

tt

where iBiA ttt ,, (3-8)

,,1,

,

iBiA

iB

tt

tt

where 1,, iAiB ttt (3-9)

The interval between phases A and B was made equal to 0.5/f0 by compression or

stretching, which was further divided into 30 equal intervals. The difference between

the local phase at each y-location of the vorticity probe and the reference phase of the

fixed X-wire was used to produce phase-averaged sectional streamlines and contours of

the coherent vorticity in the (, y)-plane. The phase-average of an instantaneous

quantity B is given by

N

iikk B

NB

1,

1 (3-10)

where N is the number detected and k represents the phase. For convenience, the

subscript k will be omitted hereinafter. The variable B can be written as the sum of a

time-averaged component ,B a coherent fluctuation ~

and a remainder r (Hussain,

1986), viz.

rBB ~

(3-11)

where B stands for instantaneous vorticity or velocity signals and the fluctuation b is

given by

b = ~

+ r (3-12)

The coherent fluctuation ~

( ) reflects the effect from the large-scale coherent

structures while the remainder r mainly refers to the incoherent structures. The

following equation may be derived from Eq. (3-12):

222 ~r (3-13)

46

3.4.2 Structural averaging

Once the coherent components of velocity and vorticity are extracted, the

coherent contributions to the vorticity variances can be obtained in terms of the

structural average. For each wavelength, the phase-averaged structure begins at j1

samples (corresponding to = -π) before = 0 and ends at j2 samples (corresponding to

= π) after = 0. The structural average, denoted by a double over-bar, is defined by

2

121

22 ~

1

1~ j

jjj (3-14)

The value for j1 (= j2) is 24 so that the value (j1 + j2 + 1) corresponds approximately to

the average vortex shedding frequency. The structurally averaged quantities can provide

a measurement of the contribution from the coherent structures to the vorticity

variances. Similarly, Eq. (3-14) may also be used to calculate the contribution of

2r , the incoherent structures make to the vorticity variances. The structural

average 2 of the vorticity variances may be derived from Eqs. (3-13) and (3-14),

viz.

2 = 2~ + 2

r (3-15)

It has been verified that the difference between 2 and 2 in general is within 5%,

suggesting that for vorticity, the selected data for phase-averaging/structural averaging

is representative to the flow.

3.4.3 Coherent vorticity fields

Figure 3-7 presents the iso-contours of the phase-averaged or coherent vorticity

** ~ ,~yx and *~

z for different yaw angles. The phase , ranging from 2π to +2π, can be

interpreted in terms of a longitudinal distance; = 2π corresponds to the average vortex

wavelength ( Uc/f0), which is given in Table 3-2 for different cylinder yaw angles.

The values of Uc for different cylinder yaw angles are estimated with the values of

uU ~ at the vortex centres. These values are also included in Table 3-2. At = 0°, the

wavelength of the spanwise vortices is about 4.2d, in agreement with that found

previously (Zhou et al., 2003). With the increase of cylinder yaw angle, the wavelength

increases systematically. However, if a factor of cos () is multiplied with , the value

47

(cos ) is approximately constant for different angles. This result indicates that the

spanwise vortices shed from the yawed cylinder are parallel to the cylinder axis, which

is consistent with the shedding angle obtained using Eq. (3-5) (Table 3-2). In Figure

3-7 and other figures for iso-contours that follow, to avoid any distortion of the physical

space, the same scales are used in the - and y*-directions. The *~

z contours (Figure

3-7i-l) display the well-known Kármán vortex street for all yaw angles. The vortex

centres, identified by the maximum concentration of *~z , coincide well with the foci

(marked by “+” in the figure) determined from the phase-averaged sectional streamlines

shown in Figure 3-8, which were constructed based on the velocity signals measured by

one of the two X-wires aligned in the xy plane. The saddle points determined from the

sectional streamlines are marked by “”. The spanwise vortices shrink slowly with the

increase of , changing from y/d = 0.4 at = 0° to y/d = 0.2 at = 45° (Figure 3-7i, l),

indicating a reduced wake region. These results agree well with those reported

previously (Lucor and Karniadakis, 2003). The maximum concentration of *~z at = 0°

is 0.8 (the negative value of the vorticity contours represents the vorticity above the

centreline). This value is slightly lower than that reported previously at a comparable

Reynolds number (Zhou et al., 2003) due mainly to the difference in spatial resolution

of the probes between the two studies. For 15°, the maximum concentration of *~z

does not change. When is increased further, the maximum concentration of *~z

decreases. The maximum concentration of *~z at = 45° decreases by about 50%

compared with that at = 0°. This result indicates that with the increase of the yaw

angle, the spanwise vortex strength decreases, which is in agreement with that shown in

Figure 3-4.

The *~x contours (Figure 3-7a-d) are much weaker than those of *~

z even

though they also exhibit organized patterns, especially at large yaw angles. The size of

the longitudinal vortices is much smaller than that of the spanwise structures. The

strength of *~x for = 0° at the vortex centre is only about 1/5 of that of *~

z . The

maximum longitudinal vortex concentrations are shifted away from the Kármán vortex

centres. The change of *~x for 15° is not apparent. When is further increased to

30°, the contours of the longitudinal vortices become more organized and the maximum

longitudinal vorticity concentration increases by about 70%. This result indicates the

48

generation of the secondary axial vortices or the occurrence of vortex dislocation

(Williamson, 1996b) and an enhanced three-dimensionality with the increase of . As

the calculation of x is related with the velocity signal for w, the increase of *~x should

be caused by the increase of w at large yaw angles (see Figure 3-10 below). The *~y

contours at = 0° (Figure 3-7e) are comparable to that of *~x . With the increase of ,

the organized patterns of the *~y contours become less apparent and the maximum

concentration of *~y decreases consistently. The reduction of *~

y at = 45° is

comparable with that of *~z , which is about 33% compared with that at = 0°. Overall,

from the contours of *~x and *~

z , it can be seen that with the increase of , the strength

of the former increases while it decreases for the latter. At = 45°, the magnitudes of

the coherent vorticity components *~x and *~

z tend to be comparable, which are about

0.25 to 0.4. This result further confirms that with the increase of , the three-

dimensionality of a yawed cylinder wake tends to be enhanced, consistent with that

shown by Zhou et al. (2009). This result is also consistent with that of Mansy et al.

(1994) who argued that the increase in the streamwise circulations occurs probably at

the expense of the primary spanwise vortex circulation. The maximum contour values

of the coherent vorticity for different are summarised in Table 3-3.

3.4.4 Incoherent vorticity fields

Figure 3-9 shows the incoherent vorticity contours *2 xr , *2 yr and

*2 zr for different yaw angles. The magnitudes of the maximum incoherent vorticity

concentrations at = 0° agree well with our previous published results (Zhou et al.,

2003). They occur at approximately the same location, which is slightly downstream of

the vortex centres for all cylinder yaw angles. Note that the *2 xr contours running

along the diverging separatrix through the saddle point (marked in the figure by “”)

wrap around the consecutive spanwise structures of opposite sign. Presumably, these

contours represent the signature of the streamwise or rib-like structures of the wake. It

seems that the incoherent vorticity contours *2 xr are stretched along an axis which

inclines to the x-axis at an angle in the range between 60° (when = 0°) and 25°

(when = 45°) (shown in Table 3-2). The magnitude of the maximum incoherent

49

vorticity concentration decreases with the increase of . At = 0°, the *2 xr

contour level through the saddle point is about 0.62, corresponding to a magnitude of

79.0*2 xr , which is the same as the maximum magnitude of the coherent

spanwise vorticity component *~z (Figure 3-7i). With the increase of , the *2 xr

contour level through the saddle point decreases. However, as the spanwise coherent

vorticity *~z decays faster, the magnitude of *2 xr exceeds *~

z for > 15°. The

values of *2 xr through the saddle points for different are compared with the

maximum concentration of *~z in Table 3-3. Also given are the values of *2 yr

through the saddle points. It can be seen that the magnitudes of *2 xr and

*2 yr exceed the spanwise coherent vorticity at large yaw angles. These results

seem to support the speculation of Hayakawa & Hussain (1989) that the strength of the

rib-like structures in the cylinder wake is about the same as that of the spanwise

structures, even in the yawed cylinder wake.

3.4.5 Topology of Reynolds stresses

The phase-averaged velocity components u, v and w for different yaw angles are

shown in Figure 3-10. The *~u contours (Figure 3-10a-d) are symmetric about y-axis

and anti-symmetric about the x-axis. The maximum concentration occurs at about y/d =

0.75. The maximum value at = 0° agrees well with those reported previously in

cylinder wake of a cross-flow (Matsumura and Antonia, 1993; Zhou et al., 2002). When

is increased to 15°, there is no apparent change in *~u , which is consistent with the

trend revealed by the root-mean-square (rms) values of u (Zhou et al., 2009). When is

further increased to 30°, the maximum value of *~u decreases by about 20%, and the

range of the vortices also reduces, changing from y/d = 3.0 at = 0° and 15° to about

y/d = 2.7 at = 30° The decreasing in the maximum *~u concentration becomes more

apparent at = 45° with a reduction in the wake range to y/d = 2.5. This trend of *~u

indicates a reduction of the vortex shedding strength with the increase of in a yawed

cylinder wake. The *~v contours (Figure 3-10e-h) are anti-symmetric about = 0. The

maximum value occurs at the centreline located in the alleyways between adjacent and

opposite-signed spanwise vorticity contours. The maximum concentration of *~v

50

occurring at = 0° agrees well with those published previously (Matsumura and

Antonia, 1993; Zhou et al., 2002). With the increase of to 30°, the maximum contour

of *~v decreases by about 7%. When is further increased to 45°, the maximum value

of *~v reduces to 0.24, which is about 14% lower than that for = 0°. These results

indicate that with the increase of , the intensity of the vortices decreases, which is

consistent with that revealed by the vorticity contours (Figure 3-7). The magnitude of

*~v is more than 2 times larger than that of ,~*u indicating that the coherent structures

contribute more to v than to u component, especially at large yaw angles.

In contrast to the coherent contours of *~u and *~v , the magnitudes of

*~w contours (Figure 3-10i-l) are much smaller, indicating that the coherent structures

contribute much less to w than that to the other two velocity components. This result is

consistent with the quasi-two-dimensional flow, especially when 15°. Over this

range of angles, the *~w contours show much less regular patterns than that at higher .

When is increased further, the organized patterns of *~w become more apparent, with

the maximum concentration being increased by about 50% at = 30° and by 100% at

= 45° compared with that at = 0°. At = 45°, the maximum contour of *~u and *~w

are comparable. The *~w contours are anti-symmetric about y/d = 0 with the maximum

concentration occurring at y/d = ±0.75. These results indicate that with the increase of

, secondary axial organized structures may be generated (Hammache and Gharib,

1991; Szepessy and Bearman, 1992; Matsumoto et al., 2001; Norberg, 2001). These

structures may impair the quasi-two-dimensional structures and enhancing the three-

dimensionality of the flow. This result is consistent with that shown by the increase of

the spanwise mean velocity W and the increase of the autocorrelation coefficient of the

spanwise velocity fluctuations for large values of (Zhou et al., 2009). The maximum

contour values of the coherent phase-averaged velocity components u, v and w are also

summarised in Table 3-3.

The phase-averaged Reynolds shear stresses **~~ vu , ** ~~ wu and ** ~~ wv for different

are shown in Figure 3-11. The **~~ vu contours display a clover-leaf pattern about the

vortex centre, which is the result of the coherent velocity field associated with the

vortical motion in a reference frame translating at Uc (Zhou et al., 2002). They are both

anti-symmetric about = 0 and about y/d = 0 which implies that the distribution of uv

51

should pass the origin of the coordinate system, due to the cancellation of positive and

negative **~~ vu . The maximum value of **~~ vu contours do not change when is

increased from 0° to 15°. However, when is increased to 30°, the **~~ vu contours

decrease by about 40%. When is further increased to 45°, the wake region shrinks and

the maximum contour value decreases significantly by 50%. This result is consistent

with that given by Zhou et al. (2009), who showed that with the increase of , the

spanwise vortex shedding weakens. The contours of ** ~~ wu also reveal apparent

organized patterns (the negative contours in the upper part of the wake), though with

much lower maximum concentration compared with that of **~~ vu . With the increase of

, the maximum concentration of ** ~~ wu increases gradually, and at = 45°, it is

increased by 80% compared with that at = 0°. This result confirms the generation of

the secondary vortex induced by the spanwise mean velocity shear yW / (Zhou et al.,

2009). The ** ~~ wv -contours display an alternately signed pattern similar to those of

**~~ vu , though there is a considerable phase difference between them (Figure 3-11c, d, k

& l), which increases with increasing , reaching about π at = 45°. This is expected

since the *~u - and *~w -contours are similar in pattern but opposite in sign, i.e. with a

phase difference, which grows with (Figure 3-10c, d, k & l). The maximum

magnitude of ** ~~ wv at = 0° is much smaller than that of **~~ vu but comparable with

that of ** ~~ wu , and is furthermore essentially independent of , probably because, with

increasing , the maximum magnitude reduces in *~v but increases in *~w (Figure 3-10).

The maximum magnitudes of **~~ vu , ** ~~ wu and ** ~~ wv for different are summarised in

Table 3-3.

3.5 Coherent and Incoherent Contributions to Velocity and

Vorticity Variances

After the coherent components of the velocity and vorticity signals are extracted,

the coherent contributions to Reynolds stresses and vorticity variances can be obtained

in terms of their structural average. The coherent and incoherent contributions of the

velocity variances for different yaw angles are shown in Figure 3-12. The time-averaged

values are also included. For the u component (Figure 3-12a-d), the coherent

contribution displays a peak at around y/d = 0.75. Due to the symmetry of the flow,

52

another peak in the distribution is expected at y/d = -0.75. This distribution results in

double peaks on the u velocity variance across the wake, which is consistent with Figure

3-10 (a-d). The main contribution to the velocity variance is from the incoherent

component, especially at the wake centreline, where the coherent component is nearly

zero. For = 0°, the coherent component 2*~u is about 50% of the incoherent one 2*~ru

at y/d = 0.75, which is consistent with that reported previously (Zhou et al., 2002). This

value is comparable with those for = 15° (60%) and 30° (57%). When = 45°, the

coherent component reduces to 30% of the incoherent one, suggesting that with the

increase of , the large-scale organized structures become less intensified, which may

be related with the breakdown of the large-scale structures. This result may also be

related with the increase of the spanwise velocity component, which is generally

regarded as the deterioration of the two-dimensionality of the flow (Hammache and

Gharib, 1991; Zhou et al., 2009). It is apparent that the coherent structures contribute to

the v velocity component more significantly than to the other two velocity components

at all yaw angles (Figure 3-12e-h). The coherent contribution to 2*v for all yaw angles

dominates in the central part of the wake (-1 y/d 1). When is increased, the

coherent contribution to 2*v decreases while the incoherent contribution increases. For

example, at = 45°, the ratio of the coherent component to the incoherent component is

about 1.8 at the wake centreline. This ratio is smaller than those at other yaw angles on

the wake centreline, which are 2.9, 3.9 and 2.9 for = 0°, 15° and 30°, respectively. As

the v velocity component is sensitive to the large organized structures, this result

indicates that with the increase of , the large-scale organized structures break down

and become less intensified as compared with that at = 0°. The coherent contributions

to 2*w are very small at = 0° and 15° (Figure 3-12i-l). Nearly 96% contribution to the

time-averaged 2*w is from the incoherent structures. This result is consistent with the

quasi-two-dimensionality of the wake flow for small yaw angles. When increases to

30° and 45°, the coherent contribution to 2*w increases, changing to 15% and 22%,

respectively. The coherent component 2*~w also displays a peak at about y/d = 0.75.

This peak location is consistent with that revealed by 2*~u (Figure 3-12a-d) and the

mean velocity gradients (Figure 3-5).

53

The coherent contributions to the vorticity variances for different are shown in

Figure 3-13. At = 0° (Figure 3-13a), the coherent contribution to the longitudinal

vorticity component is essentially negligible (less than 1%). This result is consistent

with that shown by Zhou et al. (2003) and is also consistent with that revealed by the

longitudinal vorticity contours (Figure 3-7a). As pointed out earlier, the detection of the

organized structures is based on the spanwise structures, which may result in negligibly

small values of *~x due to the cancellation of the opposite-signed vorticity in phase-

averaging. It can still be seen that with the increase of , the coherent contribution to x

increases, changing from about 1% at = 0° to 6% at = 45°. This result is in

agreement with the phase-averaged longitudinal vorticity contours (Figure 3-5a), which

shows that with the increase of , the maximum longitudinal vorticity concentration

also increases. The coherent contribution to the lateral vorticity variance 2~y is one

order smaller than that for x (Figure 3-13b). The largest coherent contribution is to the

spanwise vorticity 2~z (Figure 3-13c), which is about 24% on the centreline. With the

increase of , the coherent contribution toz decreases. This result indicates that with

the increase of , the intensity of the spanwise vorticity components decreases, resulting

in a reduction in the coherent vorticity variance. It seems that the increase in the

streamwise vorticity is at the expense of the reduction in the spanwise vorticity,

supporting the result of Mansy (1994). The present results shown in Figure 3-13

indicate that vorticity mostly resides in relatively smaller scale structures. Note the fact

that the coherent contribution is calculated based on the detection of the spanwise

vortical structures, the contribution from the longitudinal structures to the vorticity

variances may not be taken into account in the coherent variances ;~ 2 rather this

contribution may be included in the incoherent variances 2r .

3.6 Conclusions

An eight-hot wire vorticity probe has been used to measure simultaneously the three-

dimensional velocity and vorticity components in the intermediate wake (x/d = 10) of a

stationary circular cylinder at a yaw angle in the range of 0°-45°. The phase-averaged

technique is used to analyze the large-scale organized vortex structures of the wakes.

The main results are summarised as follows:

54

1. The instantaneous vorticity signals of x, y and z are comparable in

magnitude and show relatively large-scale fluctuations near the spanwise vortex

centres. When increases, the vorticity signals exhibit higher intermittency.

2. There is an apparent peak on the spanwise vorticity spectrum, which

corresponds to the shedding frequency of the Kármán vortex. When is small,

the peak is narrow and sharp. When is increased to 45°, the peak region of the

spanwise vorticity spectra is enlarged and the peak height of z is reduced

while that of x is increased, indicating a dispersion of the turbulent energy

over the vortex shedding frequency, as well as the enhancement of the three-

dimensionality of the flow at large yaw angles.

3. Based on the phase-averaging technique, the coherent contours of the three

vorticity components show apparent Kármán vortex shedding phenomena. At

= 0°, the maximum concentration of the spanwise vorticity component is about 6

times of the longitudinal and transverse ones. For 15°, the maximum

concentration of the three vorticity components does not depend on . However,

when increases to 45°, the maximum coherent concentrations of the transverse

and spanwise vorticity components at the vortex centre decrease by about 33%

and 50% respectively while the contours of the longitudinal vortices become

more organized with the maximum concentration being increased by 70%. This

result indicates the generation of the secondary axial vortices, which is

consistent with that found by Matsumoto et al. (2001). It also supports the

argument of Mansy et al. (1994) that the increase in the streamwise circulations

is at an expense of the primary spanwise vortex circulation. The maximum

coherent concentrations of the streamwise and spanwise vorticity components at

the vortex centre are comparable in magnitude at = 45°. These results indicate

that the three-dimensionality of the wake flow has been enhanced significantly

when is increased to 45°.

4. The incoherent vorticity contours *2 xr are stretched along an axis inclining

to the x-axis at an angle in the range of 60°-25° when is changed from 0° to

45°. The magnitudes of *2 xr and *2 yr (Figure 3-9) through the

saddle point are comparable with the maximum magnitude of the coherent

55

spanwise vorticity component *~z at all cylinder yaw angles (Figure 3-7). This

result supports the speculation of Hayakawa & Hussain (1989) that the strength

of the rib-like structures in the cylinder wake is about the same as that of the

spanwise structures, even in the yawed cylinder wakes.

5. The contours of the three velocity components exhibit apparent vortex patterns

at all yaw angles. With the increase of , the maximum concentrations of u and

v contours decrease while that of *~w become more apparent with the maximum

concentration being increased by about 50% at = 30° and by 100% at = 45°

compared with that at = 0°. Correspondingly, the coherent contributions to the

velocity variances 2u and 2v decrease while that to 2w increases.

6. The coherent contribution to 2~x at = 0° is negligible. With the increase of ,

the coherent contribution to 2~x increases slightly. In contrast, the coherent

contributions to 2~y and 2~

z decrease with increasing . These results suggest

that the strength of the Kármán vortex shed from the yawed cylinder decreases

and the three-dimensionality of the flow is enhanced when is increased.

56

Table 3-1. Flatness factors of the vorticity components at = 0° and 45°

i

F

x y

z

0° 5.83 6.6 6.02

45° 7.46 8.94 6.24

Table 3-2. The vortex shedding angle , vortex shedding frequency f0, convection velocity Uc, wavelength and the inclination angle between the

main vortex stretching direction and the x-axis for different cylinder yaw angles.

(°) (°) f0 (Hz) Uc/U∞ /d /d (cos ) (°)

0 0.53 132 0.82 4.16 4.16 60

15 16.8 131.2 0.83 4.23 4.09 57

0 33 121.7 0.87 4.8 4.16 45

45 45.6 101.6 0.89 5.86 4.14 25

Table 3-3. The maximum contour values of the coherent vorticity and Reynolds stresses for different yaw angles.

(°) *~x

*~y *~

z *~u *~v *~w **~~ vu ** ~~ wu

** ~~ wv *2 xr

*2 yr

0 –0.15 –0.12 –0.8 0.12 –0.28 –0.04 –0.018 –0.005 –0.004 0.79 0.82

15 –0.15 –0.1 –0.8 0.14 –0.28 –0.04 –0.018 –0.006 –0.004 0.77 0.86

30 –0.25 –0.1 –0.6 0.10 –0.26 –0.06 –0.012 –0.007 –0.006 0.71 0.77

45 –0.25 –0.08 –0.4 0.08 –0.24 –0.08 –0.009 –0.009 –0.004 0.63 0.69

57

U

= 0

= 0

Tunnel wall

Tunnel wall

End plate

45z

x

Yaw angle

U

= 0

= 0

Tunnel wall

Tunnel wall

End plate

45z

x

Yaw angle

(a) Cylinder arrangement and

coordinate system (plan view)

(b) Wake profile and probe

arrangements (side view)

U

4,7

C

B, D

y

3,8

1

2

6

5

X

A

U

4,7

C

B, D

y

3,8

1

2

6

5

X

A

(c) Enlarged 3D vorticity probe (side view) (d) Enlarged 3D vorticity probe (front view)

8D

A

z

y

y

1

6

B

3

5

C

z

4 7

2

8D

A

z

y

y

1

6

B

3

5

C

z

4 7

2

Moveable 3D

Vorticity Probe

Fixed referenceX-probe

U

Moveable 3D

Vorticity Probe


U

Figure 3-1. Sketches of the coordinate system and the 3-dimensional vorticity probe.

58

-2

0

2

-2

0

2

-2

0

2

-0.4

0

0.4

0 5 10 15

z*

y*

x*

v*

t/T

(a)

-2

0

2

-2

0

2

-2

0

2

-0.4

0

0.4

0 5 10 15

z*

y*

x*

v*

t/T

-2

0

2

-2

0

2

-2

0

2

-0.4

0

0.4

0 5 10 15

z*

y*

x*

v*

t/T

(a)

-2

0

2

-2

0

2

-2

0

2

-0.4

0

0.4

0 5 10 15

z*

y*

x*

v*

t/T

(b)

-2

0

2

-2

0

2

-2

0

2

-0.4

0

0.4

0 5 10 15

z*

y*

x*

v*

t/T

(b)

Figure 3-2. Time traces of the fluctuating transverse velocity v and the three vorticity

components measured at y/d = 0.5. The dots on the traces of the v signals indicate the

possible centres of the spanwise vortices. (a) = 0°; (b) = 45°.

59

10-5

10-3

10-1

101

-8 -6 -4 -2 0 2 4 6 8

x(0)

x(45)

Gaussian

y=0.6e1.3x (a)

x/x'

P

x

10-5

10-3

10-1

101

-8 -6 -4 -2 0 2 4 6 8

y(0)

y (45°)

Gaussian

(b)y=0.6e

1.3x

y/y'

P

y

10-5

10-4

10-3

10-2

10-1

100

-8 -6 -4 -2 0 2 4 6 8

z(0)

z(45)

Gaussian

(c)

y=0.6e1.3x

z/z'

P

z

Figure 3-3. Probability density function of vorticity components at = 0° and 45°. (a) x;

(b) y and (c) z.

-40

-30

-20

-10

0

0.01 0.1 1 10

f

(a) = 0

y

z

x

fd/UN

i(dB

)

-40

-30

-20

-10

0

0.01 0.1 1 10

f

(b) = 45

y

z

x

fd/UN

i(dB

)

Figure 3-4. Spectra of three vorticity components at (a): = 0° and (b) = 45°.

60

-0.20

-0.15

-0.10

-0.05

0

0.05

0.10

0.15

0.20

-1 0 1 2 3

W*/y

*

U*/y

*

y/d

U

* /y

* ,

W* /

y*

Figure 3-5. Normalized velocity gradients for different at x/d = 10. The arrows

indicate the direction of increasing . ---: = 0°; - : 15°; : 30°; : 45°.

-0.2

0.2

0.6

1.0

0.01 0.1 1 10 100 1000

= 0

15

30

45

(a)

tUN/d

x

-0.2

0

0.2

0.4

0.6

0.8

0.01 0.1 1 10 100 1000

= 0

15

30

45

(b)

tUN/d

y

-0.4

0

0.4

0.8

0.01 0.1 1 10 100 1000

= 0

15

30

45

(c)

tUN/d

z

Figure 3-6. Autocorrelation coefficient i

for different . (a): x ; (b)

y ; (c) z .

: = 0°; ---: 15°; - : 30°; : 45°.

61

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

-1

0

1

2

3

-1

0

1

2

3

-1

0

1

2

3 (a) (b) (c) (d)

(e) ( f) (g) (h)

( i) ( j ) (k) ( l)

= 0o

= 15o

= 30o

= 45o

-0.15 0.1

-0.12 0.06

0.8 -0.8

-0.15 0.15

0.08 -0.1

-0.80.9

-0.25 0.25

0.08 -0.1

-0.60.6

-0.25 0.25

-0.08 0.08

0.4 -0.4

Figure 3-7. Phase-averaged coherent vorticity contours at different . (a-d) *~x ; (e-h) *~

y ; (i-l) *~z . Centre and saddle points are marked by plus and

cross, respectively. (a-d) Contour interval = 0.05; (e-h) 0.02; (i-l) 0.1.

62

(a) (b) (c) (d)

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

-1

0

1

2

3

Figure 3-8. Phase-averaged sectional streamlines at different . Centre and saddle points are marked by plus and cross, respectively. (a) = 0°; (b) =

15°; (c) = 30°; (d) = 45°.

63

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

-1

0

1

2

3

-1

0

1

2

3

-1

0

1

2

3(a) (b) (c) (d)

(e) ( f) (g) (h)

( i) ( j ) (k) ( l)

= 0o

= 15o

= 30o

= 45o

1.1

1.6

2.0

1.1

1.6

2.0

0.9

1.3

1.7

0.7

1.0

1.3

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

-1

0

1

2

3

-1

0

1

2

3

-1

0

1

2

3(a) (b) (c) (d)

(e) ( f) (g) (h)

( i) ( j ) (k) ( l)

= 0o

= 15o

= 30o

= 45o

1.1

1.6

2.0

1.1

1.6

2.0

0.9

1.3

1.7

0.7

1.0

1.3

Figure 3-9. Phase-averaged incoherent vorticity contours at different . (a-d) *2 xr ; (e-h) *2 yr ; (i-l) *2 zr . Centre and saddle points are

marked by plus and cross, respectively. The thick dash line represents the stretching direction of the contours in *2 xr and is the angle of the

stretching with respect to x-direction. Contour interval = 0.1.

64

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

0

1

2

3

0

1

2

3

0

1

2

3

= 0o

= 15o

= 30o

= 45o

(a) (b) (c) (d)

(e) ( f ) (g) (h)

( i) ( j ) (k) ( l )

0.12-0.12

0.28-0.28

-0.04 0.04

-0.14 0.14

-0.28 0.28

-0.04 0.04

-0.1 0.1

-0.26 0.26

-0.06 0.07

-0.08 0.08

-0.24 0.24

-0.08 0.09

Figure 3-10. Phase-averaged velocities contours at different . (a-d) *~u ; (e-h) *~v and (i-l) *~w . Centre and saddle points are marked by plus and cross,

respectively. (a-d) Contour interval = 0.02; (e-h) 0.04; (i-l) 0.01.

65

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

0

1

2

3

0

1

2

3

0

1

2

3

= 0o = 15

o = 30

o = 45o

(a) (b) (c) (d)

(e) ( f ) (g) (h)

( i) ( j ) (k) ( l )

-0.018 0.018 -0.018 0.021 -0.012 0.015 -0.009 0.009

-0.0050.004 -0.006 0.002 -0.007 0.001 -0.0090.001

-0.0040.01 -0.0040.01 -0.0060.01 -0.0040.004

Figure 3-11. Phase-averaged Reynolds shear stresses at different . (a-d) **~~ vu ; (e-h) ** ~~ wu and (i-l) ** ~~ wv . Centre and saddle points are marked by

plus and cross, respectively. (a-d) Contour interval = 0.003; (e-f) 0.001; (g-l) 0.002.

66

(a) (b) (c) (d)

(e) ( f ) (g) (h)

(i) (j) (k) (l)

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

0.00

0.01

0.02

0.03

0.08

0.06

0.04

0.02

0.00

0.020

0.015

0.010

0.005

0.000

Time-averaged

Coherent

Incoherent

y* y* y* y*

Figure 3-12. Time-averaged Reynolds normal stresses and their coherent and incoherent

contributions at different . (a, e, i) = 0°; (b, f, j) = 15°; (c, g, k) = 30°; (d, h, l)

= 45°.

Figure 3-13. Lateral distribution of coherent contributions to vorticity variances. (a)

22 /~xx ; (b)

22 /~yy ; (c) 22 /~

zz .

0

0.05

0.10

0.15

0.20

0.25

0 1 2 3

(c)

= 45

= 30

= 15

= 0

y/d

22

/~

xx

2

2

/~

yy

2

2

/~

zz

0

0.05

0.10

0.15

0.20

0.25

0 1 2 3

(b)

= 45

= 30

= 15

= 0

y/d

0

0.02

0.04

0.06

0 1 2 3

(a)

= 45

= 15

= 30

= 0

y/d

0

0.001

0.002

0.003

0.004

0.005

0 1 2 3

(c)

= 45

= 30

= 15

= 0

y/d

22

/~

xx

2

2

/~

yy

2

2

/~

zz

0

0.05

0.10

0.15

0.20

0.25

0 1 2 3

(b)

= 45

= 30

= 15

= 0

y/d

0

0.02

0.04

0.06

0 1 2 3

(a)

= 45

= 15

= 30

= 0

y/d

0

0.001

0.002

0.003

0.004

0.005

0 1 2 3

(c)

= 45

= 30

= 15

= 0

y/d

22

/~

xx

2

2

/~

yy

2

2

/~

zz

0

0.05

0.10

0.15

0.20

0.25

0 1 2 3

(b)

= 45

= 30

= 15

= 0

y/d

0

0.02

0.04

0.06

0 1 2 3

(a)

= 45

= 15

= 30

= 0

y/d

0

0.001

0.002

0.003

0.004

0.005

0 1 2 3

(c)

= 45

= 30

= 15

= 0

y/d

22

/~

xx

2

2

/~

yy

2

2

/~

zz

0

0.05

0.10

0.15

0.20

0.25

0 1 2 3

(b)

= 45

= 30

= 15

= 0

y/d

0

0.02

0.04

0.06

0 1 2 3

(a)

= 45

= 15

= 30

= 0

y/d

0

0.001

0.002

0.003

0.004

0.005

0 1 2 3

(c)

= 45

= 30

= 15

= 0

y/d

0

0.02

0.04

0.06

0 1 2 3

(a)

= 45

= 15

= 30

= 0

y/d

22

/~

xx

2

2

/~

yy

2

2

/~

zz

0

0.05

0.10

0.15

0.20

0.25

0 1 2 3

(b)

= 45

= 30

= 15

= 0

y/d

0

0.02

0.04

0.06

0 1 2 3

(a)

= 45

= 15

= 30

= 0

y/d

0

0.001

0.002

0.003

0.004

0.005

0 1 2 3

(c)

= 45

= 30

= 15

= 0

y/d

22

/~

xx

2

2

/~

yy

2

2

/~

zz

0

0.05

0.10

0.15

0.20

0.25

0 1 2 3

(b)

= 45

= 30

= 15

= 0

y/d

0

0.02

0.04

0.06

0 1 2 3

(a)

= 45

= 15

= 30

= 0

y/d

0

0.001

0.002

0.003

0.004

0.005

0 1 2 3

(c)

= 45

= 30

= 15

= 0

y/d

0

0.001

0.002

0.003

0.004

0.005

0 1 2 3

(b)

= 45

= 30

= 15

= 0

y/d

67

CHAPTER 4

STREAMWISE EVOLUTION OF A YAWED CYLINDER

WAKE

4.1 Introduction

Flow around a circular cylinder is of both fundamental and practical significance in

many branches of engineering. The related papers amount to thousands, mostly

focused on the flow behind a two-dimensional circular cylinder placed normal to uniform

incoming flow. This flow is characterized by periodic alternating Kármán vortex shedding

downstream of the cylinder when Reynolds number Re (≡ U∞d/, where U∞ is the

incoming velocity, d is the cylinder diameter and is the kinematic viscosity of the

fluid) exceeds 47 (Provansal, 1987; Norberg, 1994; Williamson, 1996c, 1996b).

Previous investigations have unveiled three types of vortical structures in the cylinder

near wake, i.e. the Kelvin-Helmholtz (K-H) vortices in the shear layers, alternating

spanwise Kármán vortices which dominate the near wake, longitudinal rib-like vortices

wrapping around and connecting the consecutive Kármán vortices. The Kármán vortex

shedding and its streamwise evolution have significant effects on mass and heat transport

in the wake and also the aerodynamic forces acting on the cylinder (Bisset et al., 1990;

Antonia, 1991; Matsumura and Antonia, 1993; Nishimuraa and Taniike, 2001).

In many engineering applications, such as the cables of a suspension bridge and

subsea pipelines, the direction of the incoming flow may not be perpendicular to the

cylinder axis. The incoming flow velocity results in a projection along the cylinder

axis, which may affect the vortex formation and its streamwise evolution, and thus

is not negligible (Marshall, 2003). In this work, the yaw angle () is defined as the

angle between the incoming flow velocity and the direction normal to the cylinder axis.

Thus, = 0° and 90° correspond to the cross flow and axial flow cases, respectively. It

has been found previously that the yawed cylinder near wake was dominated by

vortices aligned parallel to the cylinder axis given a sufficiently far distance from

cylinder ends (Ramberg, 1983; Matsuzaki et al., 2004; Thakur et al., 2004). The

aerodynamic forces on and the frequency of vortex shedding from a yawed cylinder

behave similarly to their counterparts in the wake of a cylinder normal to incident

68

flow. If normalized by the component, U∞ cos , of U∞ normal to the cylinder

axis, the force coefficient and the Strouhal number StN (≡ f0d/U∞ cos , where f0 is

the vortex shedding frequency) are approximately independent of . This is the well

known independence principle (IP) (Van Atta, 1968).

Van Atta (1968) and also Thakur et al. (2004) noted that the IP was valid for ≤

35°, and beyond = 35°, the decrease in f0 with the increasing was slower than

that predicted by the IP. For Re = 4,000-63,000, Surry and Surry (1967) suggested the

IP could be applied up to = 40°-50°. Smith et al. (1972) measured f0 for a yawed

cylinder at 2,000 < Re < 10,000, and suggested that the IP was applicable only

for < 45°. It was also noted that the peak corresponding to the predominant

frequency in the velocity spectrum was sharp and narrow-banded for cross-flow cases,

but became less pronounced and wide-banded for yawed cylinder cases. No pronounced

peak was observable at = 60°, indicating the lack of periodic coherent structures at

this yaw angle. Based on flow visualization, Ramberg (1983) suggested that the IP

was valid till = 55°. More recently, based on the measurement of the instantaneous

lift and drag forces on a fixed yawed cylinder, Kozakiewicz et al. (1995) proposed that the

IP was applicable for = 0°-45° in the subcritical regime.

The flow behind a yawed cylinder is highly three-dimensional because of the

presence of spanwise flow (U∞ sin ) along the cylinder span (e.g. Alam and Zhou,

2007b). This spanwise flow is directed away from the upstream end to the

downstream end. The interference between this spanwise flow and the near wake is

enhanced with increasing (Matsuzaki et al., 2004). Ramberg (1986) demonstrated

based on flow visualization that there were two regions in a yawed cylinder wake.

One was near the upstream end of the cylinder, where vortices were aligned at an

angle greater than the cylinder yaw angle. The other was near the mid-span of the

cylinder, where the vortices were oriented at approximately the cylinder yaw angle.

The spanwise flow may also vary along the streamwise direction. Smith et al. (1972)

measured the velocity direction behind a yawed circular cylinder of = 30° (Re =

7,400). They found that at x* = 0.5, 1 and 2 the flow was directed along the span near

the wake centre line. In this paper, an asterisk denotes normalization by U∞ and/or d.

Further away from the centreline, the flow direction might be opposite to the direction

of spanwise flow. This flow in the opposite direction vanished at about x* = 5-10.

69

Utilizing a vorticity probe, Zhou et al. (2010) measured the three-

dimensional vortical structure in a yawed cylinder wake at x* = 10. They found that

at = 0°, the maximum concentration of the spanwise vorticity component is about 6

times of the longitudinal and transverse ones; while at = 45°, the three vorticity

components at vortex centre is comparable in magnitude. Although the effect of the

yaw angle on the three-dimensional vortical structure was quantitated at x* = 10

(Zhou et al., 2010), its streamwise evolution has not been revealed yet.

Previous investigations have improved tremendously our understanding of the

yawed cylinder wake, in particular on vortex shedding patterns and frequencies.

Nevertheless, there is no detailed information on how the yaw angle alters the flow

structure and its streamwise evolution. This work aims to fill in this gap and provide the

quantitative information on how the yaw angle of cylinder influences the flow structure

and the three-dimensional velocity and vorticity fields at different downstream locations.

A three-dimensional vorticity probe developed by Antonia et al. (1998), consisting of

four X-probes, was used to measure the three-dimensional velocities and vorticities

simultaneously. The coherent structures have been detected from the data through the

use of a phase-averaging technique (Kiya and Matsumura, 1988; Matsumura and

Antonia, 1993; Zhou et al., 2002). These structures and their contributions are

quantified and discussed in the context of the vortex pattern of velocities and

Reynolds stresses. The effects of the yaw angle on the wake and their streamwise

evolution are examined in details.


The experiments were conducted in a closed-loop low-speed wind tunnel

with a 2.0 m long test section of 1.2 m (width) × 0.8 m (height). The free-stream

velocity in the test section was uniform to 0.5% and the longitudinal turbulence

intensity was less than 0.5%. A stainless steel circular cylinder was installed

horizontally at the centre of the test section and supported rigidly at both ends by two

aluminium end plates. The cylinder diameter (d) is 12.7 mm, and the space between

the two end plates is 1.1 m, resulting in an aspect ratio of about 86. Four yaw

angles were tested, i.e. = 0°, 15°, 30° and 45°, respectively. Figure 4-1 (a) shows the

installation of the cylinder and the definition of the coordinates system. All

measurements were performed at the mid-span of the cylinder along y direction.

70

All measurements were accomplished at a free-stream velocity (U∞) of 8.5 m/s,

corresponding to a Reynolds number Re of 7,200. For each yaw angle, measurements

were conducted at three downstream locations, i.e. x* = 10, 20 and 40, to study the

streamwise evolution of the wake.

The vorticity probe consists of four X-probes (X-probe A, B, C and D), as

shown in Figure 4-1 (b). The position of each X-probe and the velocity components

they measured are listed in Table 4-1. As an example, the X-probe B consisting of

wires 3 and 4 aligning in the (x, y)-plane measures velocity components u and v. The

separations between the centres of the two opposite X-probes, i.e. y or z in Figure

4-1 (b), are 2.7 mm. For each X-probe, the clearance between the two wires was about

0.7 mm. The hot wires of the vorticity probe were etched from Wollaston (Pt-10% Rh)

wires. The active length was etched to about 200dw, where dw = 2.5 μm is the hot

wire diameter. The hot wires were operated on constant temperature circuits at an

overheat ratio of 1.5. All output signals from the anemometers were offset, low-pass

filtered with a cut-off frequency of 5,200 Hz, and sampled at a frequency fs = 10,400

Hz using a 16 bit A/D converter. The sampling duration for each measurement point

was 60 s.

The vorticity components can be calculated based on the measured velocity

signals via.

z

v

y

w

z

v

y

wx

, (4-1)

x

w

z

u

x

w

z

uy

, (4-2)

y

uU

x

v

y

uU

x

vz

)()(

. (4-3)

where w and u in Eqs. (4-1) and (4-3) are differences between the velocity signals

measured by X- probes A and C; v and u in Eqs. (4-1) and (4-2) are differences

between the velocity signals measured by X-probes B and D. The velocity gradients

in the streamwise direction w/x and v/x in Eqs. (4-2) and (4-3), respectively,

are obtained by using a central difference scheme to the time series of the measured

velocity signals, e.g. ∂v/∂x ≈ v/x ≈ xiviv /)]1()1([ , where x is estimated

based on Taylor‟s hypothesis given by x = -Uc(2t). A central difference scheme in

71

estimating w/x and v/x has the advantage to avoid phase shifts between the

velocity gradients involved in Eqs. (4-2) and (4-3) (Wallace and Foss, 1995). Uc is

the vortex convection velocity, which is in the range of 0.87-0.89U∞ for x* = 10-40

(Zhou and Antonia, 1992). t ( 1/fs) is the time interval between two consecutive

points in the time series of the velocity signals. The uncertainties for time-averaged

velocities U (a single over bar denotes conventional time-averaging) were about

±3%, while the uncertainties for fluctuation velocities u, v and w were about ±7, ±8

and ±8%, respectively. More details of this vorticity probe and the experiment

uncertainties were given in Zhou et al. (2009).

4.3 Time-Averaged Velocities and Wake Periodicity

4.3.1 Distributions of time-averaged streamwise and spanwise velocities

It has been confirmed that the time-averaged streamwise velocities measured

from the four X-probes agree quite well. Therefore, only the mean values measured

from the X-probes are presented. Similarly, the mean values of the spanwise velocities

measured by the X-probes A and C are given. Figure 4-2 shows the *U and *W at

different downstream locations for = 0°, 15°, 30° and 45°. Only the data at y*

≥ 0 are presented in view of the symmetry of the flow. The result for = 0° at x* =

10 corresponds well with that reported by Zhou et al. (2002), providing a validation

of the present measurement. For all downstream positions, *U is the smallest for =

0°. From = 0° to 45°, *U increases gradually as shown in Figure 4-2 (a-c).

The maximum velocity deficit (1- min*U ) and mean velocity half-widths L

*

are summarized in Table 4-2. The wake half-width grows less rapidly from x* = 10

to 40 with an increase in from 0° to 45°. For example, L* for = 0°, at x

* = 40 is 1.93

times larger than that at x*= 10, but is only 1.81 for = 45°. The faster growing wake

half-width at = 0° than that at ≠ 0° is consistent with the stronger vortex

shedding at = 0°, which enhances momentum transport in the wake.

The near wake of a circular cylinder with α = 0° is inherently three-dimensional

at Re > 140, caused by the occurrence of streamwise riblike vortices (Williamson,

1996c). However, the time-averaged spanwise velocity *W is still zero across the wake

because of the cancellation of the effects of the riblike structures. The *W provides a

72

measure for the time-averaged three dimensionality of the flow. A larger

*W corresponds to a higher degree of three-dimensionality (Marshall, 2003; Alam and

Zhou, 2007b). In the wake of cylinder placed normal to the incoming flow, *W is zero

across the wake. Figure 4-2 (d-f) shows the value of at different downstream positions

for = 0° to 45°. As expected, *W is approximately zero across the wake at all

x* for = 0°, suggesting the two-dimensionality of the wake at = 0°. With the

increase of , *W increases significantly near the centreline, but decreases

remarkably and becomes negative in other region (Figure 4-2d-f). For example, *W at

x* = 10 is positive for y

* < 2.2 at = 15° and 30° and for y

* < 1.75 at = 45°, and

negative for a larger y*. At all streamwise positions, the maximum *W at the

centreline increases monotonously as increases from 0° to 45°. A sketch of the

vortical structures in a yawed cylinder wake is shown in Figure 4-3 (a) to understand

the dependence of *W on . As noted by Marshall (2003) and Zhou et al. (2009), the

vortices in a yawed cylinder wake are oriented at approximately the same angle as the

cylinder when the latter is not too large. As the vortices are inclined, as shown in

Figure 4-3 (a), the vortices of both signs may result in a positive spanwise velocity

near the centreline and a negative one away from the centreline. This is consistent

with that shown in Figure 4-2 (d-f). Obviously, the maximal *W at the centreline

depends on two factors, i.e. and the vortex strength. The latter may not be

suppressed considerably when is not too large (Thakur et al., 2004), thus enlarging

the maximum of gradually from = 0° to 45° (Figure 4-2d-f). As becomes even

larger, the vortex strength will be restrained. This is expected since at = 90° no

vortex shedding occurs. Consequently, the magnitude of *W may increase first

and then decreases when is changed continuously from 0° to 90°. With the increase

of x*, the magnitude of *W decreases, though still appreciable at x

* = 40, indicating

the persistence of the three-dimensionality.

4.3.2 Power spectral density functions of velocities and vorticities

Figure 4-4 presents the power spectral density functions, u, v, w, of the

three velocity components for = 0° and 45°, measured at y* = 1.0 and x

* = 10 and

40. Note that u, v and w are normalized to decibel scale by the maximum of v.

For = 0°, both u and v at x* = 10 display one pronounced peak at f0

* = 0.195.

73

The peak in u is about 4 dB lower than that in v. A much less pronounced peak

occurs in w, only -14.2 dB. A minor peak is discernible in u and v at = 0.38

(Figure 4-4a) apparently the second harmonic of f0*. This minor peak vanishes at x

*

= 40. The major peak in u or w disappears, while that in v still exists at x* = 40

(Figure 4-4b), suggesting a faster decay of the coherent components in u and w.

The peak in u, v and w at x* = 10 is broad and less pronounced at =

45° (Figure 4-4c) than that at = 0°. The observation is consistent with Zhou et al.‟s

(2009) report that the sharp and narrow-banded peak is replaced by an order of

magnitude weaker and wide-banded peak as changes from 0° to 30°. The present

results indicates that this trend continues at least up to = 45°. The peak in u, v

and w occurs at f0* = 0.158. The dependence of the vortex shedding

frequency on has been discussed in Zhou et al. (2010), thus not repeated here. It is

worth mentioning that the peaks in u and w at x* = 10 exhibit a similar strength for

= 45° (Figure 4-4c), in distinct contrast with that for = 0°. Apparently, the inclined

vortices (Figure 4-3a) will have projections in both x and z directions, i.e. coherent u

and w, which may have a similar strength and the same periodicity for = 45°. The

power spectral density functions, x ,

y , z , of the three vorticity components are

shown in Figure 4-5. The x ,

y and z are normalized by the maximum of

z . A pronounced peak occurs in z , regardless of , at the same frequency as

in u, v and w (Figure 4-4). For = 45°, a relative weak peak is discernible in x

at x* = 10, which occurs at the same f0

* as in

z (Figure 4-5c). The observation is

ascribed to the inclination of vortices, which will be reflected inx .

4.4 Phase-Averaged Results

4.4.1 Brief introduction to phase-averaging technique

Vortices shed from a circular cylinder are characterized by a marked

periodicity in the near and intermediate wake. This remarkable periodicity persists

even for the cylinder yaw angle of 45°. Figure 4-6 presents the v-signals from the X-

probe B of the vorticity probe at the lateral location of y* ≈ 0.5 and x

* = 10. The filtered

signal vf is denoted by a thicker line in Figure 4-6. A fourth-order Butterworth filter

was used. The phase shift about 0.5% of the vortex shedding period caused by

74

filtering is very small, since the shift depends largely on the filtering frequency set at

vortices shedding frequency (Matsumura and Antonia, 1993; Zhou et al., 2002). Both

the raw and filtered v-signals exhibit a marked periodicity for all tested yaw angles, as

shown in Figure 4-6, suggesting a noticeable periodic vortices shedding in the wake.

Signals at x* = 20 and 40 are similar to those at x

* = 10, and thus are not shown here.

It is worth mentioning that, the scales of horizontal and vertical coordinates are

arbitrary, but the same scales are used for all the four tested yaw angles. The vortex

shedding period increases gradually as the yaw angle increases from 0° to 45°, as

shown in Figure 4-6. The detailed discussion about the behaviour of the vortex

shedding process for different yaw angles has been given in Section 3.2, in the context

of power spectral density function of v-signals. The phase of the vortex shedding

process could be identified on the filtered vf, as denoted by the thicker line in Figure

4-6, namely

Phase A: 0fv

and ,0

dt

dv f

(4-4)

Phase B: 0fv

and .0

dt

dv f

(4-5)

This two phases correspond to time tA,i and tB,i (measured from an arbitrary time origin),

respectively. The phase was then calculated from the filtered signals as:

,,,

,

iAiB

iA

tt

tt

iBiA ttt ,,

(4-6)

,,1,

,

iBiA

iB

tt

tt1,, iAiB ttt (4-7)

The interval between phases A and B was made equal to 0.5T0, half of the average

vortex shedding period, by compression or stretching. It was farther divided into 30

equal intervals. Phase-averaging was then conducted on the measured signals, not on the

filtered signals. The phase-average of an instantaneous quantity B is given by

N

iikk B

NB

1,

1 (4-8)

where k represents phase. N is about 1200 for all the presently tested cases. Following

the triple decomposition proposed by Reynolds and Hussain (1972), a variable B can be

viewed as the sum of the time-averaged component B and the fluctuation component β,

75

which can be further decomposed into coherent fluctuation and incoherent

fluctuation βr.

Once the coherent components of u, v and w fluctuations are extracted, the

coherent contributions to the conventional Reynolds stresses can be given in terms of

the structural average. The conditionally averaged structure begins at k1 (corresponding

to = -π) and ends at k2 (corresponding to = π). The structural average is defined by

2

11 2

1

1

k

kk k

(4-9)

where β, like γ denotes either u, v and w.

4.4.2 Phase-averaged vorticity fields

Figures 4-7 and 4-8 present the contours of phase-averaged vorticity

components, i.e. *~x , *~

y and *~z at x

* = 20 and 40 for all four tested yaw angles. The

contours of *~x , *~

y and *~z at x

* = 10 were given in Zhou et al. (2010), thus will not be

repeated here. Phase (from -2π to 2π) in the figures can be interpreted in terms of a

streamwise distance and = 2π corresponds to the Kármán wavelength Uc/fs, where Uc

is the vortex convection velocity. The same scales are used in - and y*-directions in

order to avoid any distortion of the physical space. The flow direction is left to right in

all the figures given in the (, y*)-plane.

At x* = 20, the coherent spanwise vorticity displays a marked periodicity for all

examined, similar to those shown by Zhou et al. (2010) at x* = 10. The maximum of

*~z for = 0°, 15° and 30° is identical and equals to 0.3, but decreases to 0.25 for =

45°, suggesting that the effect on spanwise vortices is appreciable only for > 30°.

At x* = 10, the maximal concentration of *~

z at = 45° remains only 50% of that at =

0° (Zhou et al., 2010). While as x* reaches 20, the maximal concentration of *~

z at =

45° remains 83.3% of that at = 0° (Figure 4-7i and l). This observation suggests that

the effect of yaw angle on the coherent spanwise vortices becomes less obvious with the

increasing of x*. The dependence of *~

z on conforms to Marshall‟s (2003) report that

the vorticity strength degrade gradually with the increase of . Note that the outermost

contour levels of *~z are 0.1 and -0.1, the same for all plots (Figure 4-7i-l). The area of

the outermost contour is smaller at = 45° than others, indicating a weakened vortex

76

strength and a reduced wake width, in good agreement with Lucor and Karniadakis‟s

(2003) report.

The phase-averaged lateral vorticity *~y (Figure 4-7e-h) fail to display any

coherence at x* = 20, for all tested . Similar observation was made by Zhou et al.

(2010) at x* = 10. Based on flow visualization and numerical simulation, Ramberg

(1983), Marshall (2003) and Matsuzaki et al. (2004) observed lateral coherent

structures near the upstream end of a yawed cylinder, which was caused by the

spanwise velocity. This lateral coherent structure is not observed presently because

measurements were conducted at the mid-span of the cylinder, which is essentially

unaffected by the end effects.

The coherent streamwise vorticity *~x (Figure 4-7a-d) exhibits organized

structures, especially at large , albeit with a much weaker strength than *~z . In

distinct contrast to *~z , *~

x is enhanced gradually from = 0° to 45°. For example,

the maximal magnitude of *~x is 0.04 at = 0°, about 13% of the corresponding *~

z

(Figure 4-7a and i), but becomes 0.12 at = 45°, about 50% of the corresponding *~z

(Figure 4-7d and l). That is, *~x is considerably better organized with increasing .

The observation is ascribed to the inclination of the vortex roll in a yawed cylinder

wake, which enhances the streamwise component of vortices. Considering the

distribution of *~z and *~

x at x* = 10 (Zhou et al., 2010), the systematically weakening

of spanwise vortices with the increase of is less obvious at x* = 20, though the gradual

enhancement of *~x is still remarkable. The observation, that the increase in the

streamwise circulation is at an expense of the primary spanwise circulation (Mansy et

al., 1994), is discernible at a small x*, e.g. x

* = 10, but almost vanishes with x

* increases

to 20. The phase of the concentrated *~x corresponds well with that of *~

z . For

example, the centres of positively signed structures occur at = ±π in both *~x -

and *~z -contours, while those of negatively signed structures are at = 0 and ±2π.

As a matter of fact, the observation conforms well to our schematic sketch of an

inclined vortex roll in Figure 4-3 (b). An inclined positively signed vortex roll can

result in a positive vortex in the (x, y)-plane, i.e. z > 0, and meanwhile cause a

positive vortex in the (y, z)-plane, i.e. x > 0. Similarly, for a negatively signed

77

vortex roll, the corresponding z- and x-components are both negative (not shown).

As evident in Figure 4-3 (b), both *~x and *~

z concentrations may be considered as

the projections of the inclined vortex roll in the (y, z)- and the (x, y)-planes,

respectively. Naturally, *~x and *~

z are in-phased.

The effect of on *~z at x

* = 40 is insignificant. The range of the maximum

*~z for all is 0.04-0.06 (Figure 4-8i-l). Considering *~

z at x* = 10 (Figure 3-7i-l

similar as in Zhou et al. 2010) and 20 (Figure 4-7i-l), it can be concluded that the

decay of *~z in the streamwise direction is slower for a larger . For instance, the

maxima of *~z at x

* = 40 remains 5% and 12.5% of their counterpart at x

* = 10 for

= 0° and 45°, respectively. The *~y -contours display no large-scale concentrations

and the maximum *~y remains about 10% of that at x

* = 10 (Figure 4-8e-h).

Similarly to that at x* = 10 and 20, *~

x at x* = 40 increases in strength and becomes

more organized with increasing . The maximum concentrations of *~x , *~

y and *~z

are summarized in Table 4-3. It is worth mentioning that, although the position of the

centres of *~z does not change significantly, the vortex strength and its size impair

considerably from x* = 20 to 40, since the outmost contour value in Figure 4-8 (i-l) is

only 1/5 of that in Figure 4-7 (i-l).

4.4.3 Phase-averaged velocity fields

The variation in *~x , *~

y and *~z with is connected to the dependence on

of the phase-averaged coherent velocities, i.e. *~u , *~v and *~w . Figures 4-9 and 4-10

present the iso-contours of *~u , *~v and *~w at x* = 20 and 40 for the four . The

maximum of is *~v considerably larger than that of *~u or *~w at all streamwise

positions, irrespective of , internally consistent with the more pronounced peak in

v than that inu or w.

At x* = 20, for all tested , the *~u contours display approximate up-down

antisymmetry about the vortex centre (the lower half is not shown), while the *~v

contours present antisymmetry about = 0 (Figure 4-9a-d, e-h). Both *~u and *~v

display qualitatively the same structure as those at x* = 10 (see Figure 3-10 similar as

in Zhou et al. 2010) and those reported by Matsumura and Antonia (1993) and Zhou et

78

al. (2002) for = 0°. The maximal concentrations of *~u and *~v are not altered at ≤

15°. They decrease when is increased from 30° to 45°, as shown in Figure 4-9 (a-h).

The maximum *~u and *~v for = 45° remain to be about 60% and 83% of their

counterparts for = 15°, suggesting that affects *~u more than *~v , as noted by Zhou

et al. (2010) at x* = 10.

At = 0° and 15°, the maximum contours of *~w is much smaller compared

with that of *~u and *~v , since the coherent motion contributes much less to *~w than to

*~u and *~v . This may suggest that inclination of the coherent spanwise vortices is

limited for ≤ 15°, thus resulting in negligible small *~w . In contrast to *~u and *~v , *~w

is enhanced and may be considered to be increasingly organized with increasing ,

showing up-down antisymmetry about the vortex centre (Figure 4-9i-l). The maximum

of *~w almost doubles from = 0° to 45°. In fact, it is comparable to the maximum of

*~u at = 45°. The contours of *~w is analogous but opposite in sign to those of *~u .

For instance, *~u > 0 and *~w < 0 at = 0 and y* > 0. Furthermore, the extremum in the

contours of *~u and *~w occurs at almost the same location, especially for = 30° and

45°. The observations indicate an inherent connection between *~u and *~w . Figure 4-3

(c) shows two schematic inclined vortices and the corresponding coherent velocities.

For the counterclockwise vortex of z > 0, *~u is negative in Quadrants 1 and 2 but

positive in Quadrants 3 and 4. Meanwhile, *~v is negative in Quadrants 2 and 3 but

positive in Quadrants 1 and 4. These distributions of *~u and *~v are consistent with

those at = 0° (Balachandar et al., 1997). Similarly, the distributions of *~w are also

shown in Figure 4-3 (c). For the vortex of z > 0, *~w is positive in Quadrants 1 and 2

and negative in Quadrants 3 and 4, opposite to *~u . The coherent velocity components

of the clockwise vortex of z < 0 are opposite in sign to those of the positively signed

vortex (Figure 4-3c), thus not discussed here. In conclusion, the phase- averaged

velocities shown in Figure 4-9 correspond well with those shown in Figure 4-3 (c).

The enhanced *~w with increasing was ascribed to the generation of the

secondary axial organized structures, i.e. the streamwise vortices, in some

investigations (Hammache and Gharib, 1991; Szepessy and Bearman, 1992;

Matsumoto et al., 2001). Based on the present measurements and the sketches in

79

Figure 4-3, *~u , *~v and *~w result from the projection of the inclined vortex roll in

the x-, y- and z-direction, respectively. Similarly, the streamwise projection of the

inclined vortex roll (Figure 4-3b) may also contribute to *~x (Figure 4-7a-d).

At x* = 40, the effect of on *~u , *~v and *~w (Figure 4-10) is insignificant,

consistent with the phase-averaged vorticity contours at the same x* (Figure 4-8).

The extremum of *~u is almost the same for all (Figure 4-10a-d). A comparison

with the data at x* = 10 (Zhou et al., 2010) and 20 (Figure 4-9a-d) unveils that *~u

and also *~v decay more rapidly downstream for a small , as observed earlier for

*~z . At x

* = 40, the concentrations of *~u and *~v are still discernible and show a

similar pattern to that at x* = 10 and 20. This is not the case for *~w (Figure 4-10i-l).

It can be concluded that the marked effect of on the flow diminishes downstream

and becomes negligibly small at x* = 40. The maximum levels of *~u , *~v and *~w at

x* = 10, 20 and 40 are summarized in Table 4-3.

4.4.4 Phase-averaged Reynolds shear stresses

The phase-averaged Reynolds shear stresses, i.e. **~~ vu , ** ~~ wu and ** ~~ wv , at

x* = 20 are presented in Figure 4-11 for all . Those at x

* = 40 are not shown

because the organized patterns have disappeared at this location. For all tested , the

* *u v contours (Figure 4-11a-d) are almost zero at the position of vortex centre, and

present a local extremum in each quadrant. This distribution is similar to the pattern of a

clover-leaf about the vortex centre (the lower half is not shown). This pattern results

from the coherent motion in a reference frame moving at Uc. For the vortex of *~z <

0, *~u > 0 and *~v <0 in Quadrant 1 yields **~~ vu < 0; *~u > 0 and *~v >0 in Quadrant

2 leads to **~~ vu > 0. The sign of **~~ vu in Quadrant 3 and 4 can be determined

accordingly. Obviously, **~~ vu is related to *~z . The maximum **~~ vu changes little,

about 0.0025, from = 0° to 30°, but reduces to 0.0015 at = 45°. This observation

conforms to the dependence of *~z on , as shown in Figure 4-7 (i-l). The

approximate antisymmetry of **~~ vu about = 0 implies a small net coherent

contribution to uv , due to the cancellation of positive and negative **~~ vu , which will

be further discussed later.

80

The ** ~~ wu contours (Figure 4-11e-h) display negatively signed concentrations

at y* ≈ 0.75-1.1. The concentrations shift toward the centreline with increasing .

As shown in Figure 4-9 (a-d, i-l), *~u and *~w exhibit similar distribution but

opposite signs. Consequently, ** ~~ wu is negative in most wake regions, especially for

a large . Although considerably smaller in magnitude than that of **~~ vu or ** ~~ wv ,

** ~~ wu appears to improve its periodicity from = 0° to 45°, apparently linked to the

enlarged projection of the vortex roll onto the (x, z)-plane.

At x* = 10, the maximum magnitude of the negative ** ~~ wu increases from

0.005 to 0.009 when is increased from 0° to 45°, while that of the positive ** ~~ wu

decreases significantly, from 0.004 to 0.001 (Figure 3-11 similar as in Zhou et al.

2010). However, at x* = 20, the dependence of the maximum level of ** ~~ wu on

becomes less pronounced (Figure 4-11e-h).

The contours of ** ~~ wv (Figure 4-11i-l) are less organized, compared with

**~~ vu and ** ~~ wu , probably due to the fact that both *~u and *~w are less organized

than *~v (Figure 4-9). Again, with increasing , ** ~~ wv appears better organized. The

maximum contour levels of **~~ vu , ** ~~ wu and ** ~~ wv are summarized in Table 4-3.

4.5 Coherent Contributions to Reynolds Stresses and Vorticity

Variances

The coherent contributions to Reynolds stresses can be quantified in terms of

the structure average and are shown in Figures 4-12 and 4-13 for x* = 20 and 40. All

profiles are either quite symmetrical or antisymmetrical about y* = 0 for all . At

x* = 10, 2*~u displays a pronounced peak at y

* ≈ 0.75, and is nearly zero at y

* = 0

(Zhou et al., 2010). At x* = 20, this peak shifts to y

* ≈ 1.25, and is less pronounced for

all (Figure 4-12a, g, m and s). The major contribution to 2*u is from the incoherent

component. The maximum 2*~u is almost independent of the yaw angle for ≤

30°, and becomes appreciably smaller at = 45°. This observation is consistent with

the dependence of *~u on (Figure 4-9), and is inherently connected to *~z (Figure 4-7i-

l). At x* = 40, 2*~u is more broadly distributed than at x

* = 10 and 20. The maximum

81

2*~u is similar in magnitude for all and considerably smaller than that at x* = 10

and 20, internally consistent with the streamwise variation of *~u (Figure 3-10a-d

similar as in Zhou et al. (2010), Figure 4-9a-d and 4-10a-d). The result suggests the

weakened coherent structures and a reduced effect as vortices are advected

downstream.

The maximum 2*v , 2*~v and 2*~

rv occur at the centreline for all , and are

larger than their counterparts of the streamwise and spanwise components (Figure

4-12b, h, n and t). Zhou et al. (2010) noted that the coherent contribution to 2*v at

x* = 10 is larger than the incoherent contribution over y

* = -1 to 1 for all . This is

not so at x* = 20 (Figure 4-12b, h, n and t). Two factors may be responsible for this

change, i.e. the downstream decay and moving away from the centreline of the

vortices. At x* = 40, the coherent contribution to 2*v almost completely vanishes except

near the centreline (Figure 4-13b, h, n and t). Note that at x* = 40, 2*~v increases

with an increase in (Figure 4-13b, h, n and t), suggesting a smaller downstream

decay rate for a larger

At x* = 10, Zhou et al. (2010) showed that 2*~w increases when is changed

from 30° to 45° and revealed a pronounced peak at y* ≈ 0.75. At x

* = 20-40, the

coherent contribution to 2*~w is essentially zero for all (Figure 4-12c, i, o, u and 4-

13c, i, o, u), suggesting that the coherent contribution to 2*w decays much faster

than to 2*u and 2*v .

The Reynolds shear stresses at x* = 20 are generally smaller in magnitude

than the Reynolds normal stresses. As shown in Figure 4-12 (d, j, p, v, f, l, r and x),

the **~~ vu and ** ~~ wv , are approximately antisymmetrical about y* = 0; so are their

coherent and incoherent contributions. The coherent and incoherent components of

** ~~ wu are opposite in sign, especially for beyond 15°; and the latter follows the

sign of ** ~~ wu . This observation suggests that the coherent motions tends to suppress the

production of ** ~~ wu , that is the ** ~~ wu is dominated by the incoherent motions. At x* =

40, the coherent contributions to all the Reynolds stresses, except to 2*u and 2*v ,

are approximately zero across the wake (Figure 4-13).This result reconfirms the

82

persistence of the spanwise coherent structures, as shown in Figure 4-8 (i-l), in view

of the association of *~u and *~v with *~z .

The coherent contributions to the variances of the three vorticity components

are shown in Figure 4-14 for different . At x* = 10,

22/~

xx is negligibly small,

less than 1%, at = 0°, as observed by Zhou et al. (2003). For = 30°-45°,

22/~

xx increases significantly, with its maximum reaching 5% (Figure 4-14a).

The dependence of 22

/~xx on at x

* = 20 (Figure 4-14b) is qualitatively the same

as that at x* = 10. It is worth mentioning that the maximum

22/~

xx reduces

markedly at x* = 20, compared with that at x

* = 10, in conformity with the

streamwise evolution of the phase-averaged longitudinal vorticity (Figures 4-7a-d

and 4-8a-d). The 22

/~yy is negligibly small for = 0° - 45° (Figure 4-14d, e and

f), which is not surprising in view of the result in Figures 4-7 and 4-8. The 22

/~zz

is by far larger than 22

/~xx and

22/~

yy at x* = 10 and 20, regardless of . At

x* = 10,

22/~

zz is essentially unchanged from = 0° to 15°, except near the

centreline, that is, the effect on spanwise vorticity may be neglected for ≤ 15°.

However, 22

/~zz retreats from = 15° to 45° (Figure 4-14g). At x* = 20, the

difference in 22

/~zz between different diminishes (Figure 4-14h) relative to that

at x* = 10.

22/~

zz decays rapidly downstream for ≤ 30°. On the other hand,

22/~

zz drops rather slowly for = 45°, that is, *~z decays slowly for a larger yaw

angle, as observed in Figures 4-7 (i-l) and 4-8 (i-l). At x* = 40,

22/~

zz is

negligibly small across the wake for all (Figure 4-14i), as 22

/~xx and

22/~

yy .

The streamwise variation of the coherent contribution may be quantified by

the dependence of the ratio qssq /~~ on x*, where q and s may represent u, v and w.

83

This ratio also depends on y*. Following Zhou et al. (2002), we define an averaged

contribution mqssq )/~~( , across the wake, from the coherent vortical structures:

** /~~)/~~( dyqsdysqqssq m

(4-10)

In view of the symmetry of the wake, the lower integral limit (-∞) may be replaced

by 0. The values of mqssq )/~~( at x* = 10, 20 and 40 are given in Table 4-4.

The coherent contribution to 2v is greater than that to 2u , this contribution

to 2w being smallest, regardless of . The difference is pronounced at x* = 10 and

20, but vanishes at x* = 40 where all of them approach zero because of the breakup

of vortices. At x* = 10, the coherent contribution to 2u or 2v differs little from =

0° to 15°. This contribution reduces gradually with a further increase in . On the

other hand, the coherent contribution to 2w increases constantly with increasing ,

due to the enhanced *~x and the three-dimensionality of the flow. The muvvu )/~~(

behaves quite similarly to muu )/~( 22 or mvv )/~( 22 , changing little from = 0° and

15° and decreasing more appreciably from = 30° to 45°. Note that muwwu )/~~(

exceeds 100% for all , as may be inferred from Figure 4-12 (e, k, q and w).

The downstream decrease in muu )/~( 22 is much slower than mvv )/~( 22 and

mww )/~( 22 . For example, muu )/~( 22 at x* = 40 remains at 15.2%, 18.7%, 25.7% and

28.6% of that at x* = 10 for u= 0°, 15°, 30° and 45°, respectively. However, the

corresponding mvv )/~( 22 is only 3.1%, 4.0%, 6.0% and 7.5%, and mww )/~( 22 is only

3.3%, 2.6%, 1.1% and 1.4%. Furthermore, muu )/~( 22 and mvv )/~( 22 decay more

rapidly at a small . The observation is linked to the streamwise evolution of *~z , as

shown in Figures 4-7 (i-l) and 4-8 (i-l). As noted earlier, the vortex shedding

frequency decreases gradually with increasing , that is, the vortices at a large

are longitudinally separated farther, implying a weak interaction between them and

hence a slow decay or longer survival.

84

The coherent contribution is much larger to 2

z than to 2

x and 2

y ,

especially at x* =10 and 20 because the vortex roll is predominantly spanwise. At x

* =

10, mzz )/~(22

is almost unchanged from = 0o to 15

o, but reduces appreciably

from = 30° to 45°. Meanwhile, mxx )/~(22

increases monotonically. A similar

observation is made at x* = 20. That is, with increasing , the streamwise

component of the vortex roll is enhanced at the expense of the weakening spanwise

component. The streamwise decay of 22

/~zz is less rapid for a large than for a

small one, consistent with that shown in Figures 4-7 and 4-8. For example, mzz )/~(22

at x* = 20 remains 29% and 55% of that at x

* = 10 for = 0° and 45°, respectively.

4.6 Conclusions

The yawed cylinder wake has been experimentally investigated using an

eight-wire vorticity probe. The three-dimensional velocity and vorticity components

are measured simultaneously at the streamwise position of x* = 10, 20 and 40, and the

yaw angle of = 0°-45°. The phase-averaged technique is utilized to extract the large-

scale coherent structures. The dependence of the coherent structures on and its

streamwise evolution are quantified. The major results are summarized below:

1. Both *~x and *~

z may be considered to result from the projection of the inclined

vortex roll in the (y, z)- and (x, y)-planes. At x* = 20, *~

x is enhanced

progressively with increasing , its maximum magnitude rising from 0.04 at

= 0° to 0.12 at = 45°. On the other hand, *~z is almost independent of for

≤ 30°, but reduces in magnitude for = 45°. The result reconfirms Mansy et

al. ‟s (1994) argument that the increase in the streamwise circulation was at an

expense of the primary spanwise circulation. At x* = 40, the organized motion

is still appreciable in terms of *~z . Its maximum concentration, which is

almost independent of , remains about 5.0%, 6.3%, 10.0% and 12.5% of that

at x* = 10 for = 0°, 15°, 30° and 45°, respectively, indicating a faster

streamwise decay of *~z for smaller . The observation is ascribed to the larger

85

streamwise spacing and hence weakened interactions between vortices for

larger .

2. The coherent velocities, *~u , *~v and *~w , depend on . With an increase in ,

the magnitude of *~u and *~v at x* = 20 retreats, while that of *~w grows,

with its maximum at = 45° nearly doubling that at = 0°. As x* reaches 40,

this dependence is hardly noticeable.

3. The coherent contributions to Reynolds stresses and vorticity variances are

approximately independent of for ≤ 15°. At x* = 10, the coherent

contribution to 2

x increases from = 15° to 45°. The opposite is observed

for the coherent contribution to 2

z . The result is essentially connected to the

fact that, with being increased, the streamwise component of the vortex roll

is enhanced at the expense of the spanwise component. Meanwhile, the

coherent contributions to 2u and 2v decrease, while those to 2w increase.

The coherent contribution remains dependent on at x* = 20 but not at x

* = 40.

4. The coherent contributions to 2u , 2v and 2

z decay less rapidly in the

streamwise direction, given a larger . At = 0°, the magnitude of

muu )/~( 22 , mvv )/~( 22 and mzz )/~(22

at x* = 40 are about 15.2%, 3.1% and

1.6% of those at x* = 10; at = 45°, however, the corresponding values are

28.5%, 7.5% and 8.7%, respectively. The result is connected to the dependence

on of the longitudinal spacing between the vortex rolls.

86

Table 4-1. Position and the measured velocities of each X-probe contained in the

vorticity probe.

X-probe Wires consisted Position Velocities

measured

A

B

C

D

1, 2

3, 4

5, 6

7, 8

(x, z)-plane

(x, y)-plane

(x, z)-plane

(x, y)-plane

u, w

u, v

u, w

u, v

Table 4-2. Maximum velocity deficit and wake half-width.

x*

0° 15° 30° 45°

min*1 U

10 0.17 0.14 0.10 0.07

20 0.19 0.17 0.13 0.09

40 0.15 0.14 0.11 0.08

*L

10 0.80 0.82 0.80 0.68

20 1.02 1.00 0.92 0.89

40 1.54 1.50 1.34 1.23

87

Table 4-3. The maximum values of the coherent vorticity, velocity and Reynolds stresses contours for different yaw angles at x* = 10, 20 and 40.

x* 10 20 40

0° 15° 30° 45° 0° 15° 30° 45° 0° 15° 30° 45°

*~x 0.15 0.15 0.25 0.25 0.04 0.06 0.1 0.12 0.01 0.02 0.025 0.03

*~y 0.12 0.10 0.10 0.08 0.04 0.06 0.04 0.04 0.01 0.02 0.025 0.025

*~z 0.8 0.8 0.6 0.4 0.3 0.3 0.3 0.25 0.04 0.05 0.06 0.05

| *~u | 0.12 0.14 0.10 0.08 0.05 0.05 0.04 0.03 0.008 0.008 0.008 0.008

| *~v | 0.28 0.28 0.26 0.24 0.12 0.12 0.12 0.10 0.020 0.023 0.025 0.025

| *w | 0.04 0.04 0.06 0.08 0.015 0.015 0.02 0.025 0.004 0.004 0.004 0.004

| **~~ vu | 0.018 0.018 0.012 0.009 0.0025 0.0025 0.0025 0.0015 --- --- --- ---

| ** ~~ wu | 0.005 0.006 0.007 0.009 0.0007 0.0005 0.0009 0.0007 --- --- --- ---

| ** ~~ wv | 0.004 0.004 0.006 0.004 0.001 0.0015 0.001 0.0015 --- --- --- ---

88

Table 4-4. Averaged contributions from the coherent motion to Reynolds stresses and vorticity variances for different yaw angles at x* = 10, 20 and 40

x* 10 20 40

0° 15° 30° 45° 0° 15° 30° 45° 0° 15° 30° 45°

2 2( )mu u (%) 22.3 24.2 21.0 16.4 9.0 9.2 9.1 7.4 3.4 4.7 5.4 4.7

2 2( )mv v (%) 64.1 64.3 61.5 54.7 24.6 25.7 24.3 18.2 2.0 2.6 3.7 4.1

2 2( )mw w (%) 3.0 3.8 9.0 14.5 0.5 0.7 1.3 1.8 0.1 0.1 0.1 0.2

( )muv uv (%) 37.7 38.9 28.6 27.7 24.1 15.8 27.5 21.6 1.4 1.9 1.5 1.1

( )muw uw (%) 105.5 102.6 123.0 112.3 45.0 59.1 60.4 55.5 5.9 5.4 7.7 8.2

( )mvw vw (%) 24.1 34.6 33.4 32.6 18.5 24.3 16.8 11.1 2.9 2.8 2.4 1.9

2 2( )x x m (%) 0.48 0.76 2.5 2.7 0.10 0.24 0.78 1.19 0.05 0.06 0.09 0.17

2 2( )y y m (%) 0.13 0.17 0.16 0.20 0.07 0.09 0.07 0.07 0.03 0.04 0.04 0.05

2 2( )z z m (%) 13.2 14.5 11.4 5.3 3.8 4.27 4.23 2.94 0.21 0.28 0.39 0.46

89

Figure 4-1. Definition of the coordinate system and the sketches of the vorticity probe.

(a) definition of coordinate system; (b) Sketch of the vorticity probe.

Figure 4-2. Time-averaged u- and w-component velocities at different yaw angles. (a, d)

x* = 10; (b, e) 20; (c, f) 40.

90

x

y

z

+ +

-

Wake centerline

+ Positive vortex

Negative vortex

The (x, y)-plane

The (y, z)-plane

z > 0

x > 0

(a)

(b)

u > 0~

v < 0~

x

y

z

O w < 0~

Quadrant 1

u > 0~

v > 0~

w < 0~

Quadrant 2

u < 0~

v > 0~

w > 0~

Quadrant 3

u < 0~

v < 0~

w > 0~

Quadrant 4

u < 0~

v > 0~

x

y

z

O w > 0~

Quadrant 1

u < 0~

v < 0~

w > 0~

Quadrant 2

u > 0~

v < 0~

w < 0~

Quadrant 3

u > 0~

v > 0~

w < 0~

Quadrant 4

(c)

Negative vortexPositive vortex

O

Yawed-cylinder

Positive vortex roll

Figure 4-3. Sketch of the inclined vortices in a yawed cylinder wake. (a) inclined vortex

street, (b) projections of a positive vortex roll in the (x, y)- and the (y, z)-planes, (c)

distributions of the three phase-averaged velocity components in a inclined vortex

0.1 1 0.1 1

-40

-30

-20

-10

0

f *

x* = 10 x* = 40

f *

(a) (b)

(c) (d)

-40

-30

-20

-10

0

u

v

w

Figure 4-4. The power spectra density function u, v and w of the three velocity

components for different yaw angles. (a, b) = 0°; (c, d) 45°.

91

0.1 1 0.1 1-20

-15

-10

-5

0

-20

-15

-10

-5

0

f *

x* = 10 x* = 40

f *

x

y

z

(a) (b)

(c) (d)

Figure 4-5. The power spectra density function x ,

y and z of the three vorticity

components for different yaw angles. (a, b) = 0°; (c, d) 45°.

v

v

v

v

t

tA,i tB,i tA,i+1

o

o

o

o

Figure 4-6. Signal v from the X-probe B of the vorticity probe at x* = 10 and y

* ≈ 0.5.

The thicker line represents the filtered signal vf.

92

2 1 0 -1 -2 2 1 0 -1 -2 2 1 0 -1 -2 2 1 0 -1 -2

0

1

2

3

-1

0

1

2

3

-1

0

1

2

3

-1

y *

y *

y *

(a) (b) (c) (d)

(e) (f) (g) (h)

( i) (j) (k)

-0.04 0.04

-0.04 0.04

0.3 -0.3

-0.06 0.06

-0.06 0.04

0.3 -0.3

-0.1 0.08

-0.04 0.04

0.3 -0.3

-0.12 0.12

-0.04 0.04

-0.250.25(l )

Figure 4-7. Phase-averaged vorticity components at x* = 20 for different yaw angles. (a-d) *~

x ; (e-f) *~y ; (i-l) *~

z . (a-h) Contour intervals = 0.02; (i-l)

0.05.

93

(a) (b) (c) (d)

(e) (f) (g) (h)

( i) (k) (l)-0.04 0.04 -0.060.05 -0.050.05

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

0

-1

1

2

3

0

-1

1

2

3

0

-1

1

2

3

y *

y *

y *

0.05 -0.05

2 1 0 -1 -2

( j)

Figure 4-8. Phase-averaged vorticity components at x* = 40 for different yaw angles. (a-d) *~

x ; (e-f) *~y ; (i-l) *~

z . (a) Minimum contour value = -0.01,

maximum contour value = 0.01, contour intervals = 0.005; (b) -0.02, 0.02, 0.005; (c) -0.025, 0.025, 0.005; (d) -0.03, 0.03, 0.005; (e) -0.01, 0.01, 0.005;

(f) -0.02, 0.02, 0.005, (g) -0.025, 0.025, 0.005, (h) -0.025, 0.025, 0.005, (i-l) contours intervals = 0.01.

94

2 1 0 -1 -2 2 1 0 -1 -2 2 1 0 -1 -2 2 1 0 -1 -2

0

1

2

3

0

1

2

3

0

1

2

3

y *

y *

y *

(a) (b) (c) (d)

(e) (f) (g) (h)

( i) (j) (k) (l)

-0.050.05 -0.05 0.05 -0.040.04 -0.03 0.03

-0.12 0.12 -0.12 0.12 -0.12 0.12 -0.10 0.10

-0.0150.01 -0.015 0.01 -0.015 0.02 -0.02 0.025

Figure 4-9. Phase-averaged velocities at x* = 20 for different yaw angles. (a-d) *~u , (e-f) *~v , (i-l) *~w . (a-d) Contours intervals = 0.01, (e-f) 0.02 and (i-

l) 0.005.

95

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

2 1 0 -1 -2

0

1

2

3

0

1

2

3

0

1

2

3

y *

y *

y *

(a) (b) (c) (d)

(e) (f) (g) (h)

( i) (k) (l)( j)

0.008 -0.008 0.008 -0.008 0.008 -0.008 0.008 -0.006

-0.02 0.02 -0.023 0.023-0.025 0.025 -0.025 0.025

-0.004 0.004 -0.004 0.004 -0.004 0.004 -0.004 0.004

Figure 4-10. Phase-averaged velocities at x* = 40 for different yaw angles. (a-d) *~u , (e-f) *~v , (i-l) *~w . (a-d) Contour interval = 0.002, (e-f) 0.005 and

(i-l) 0.002.

96

2 1 0 -1 -2 2 1 0 -1 -2 2 1 0 -1 -2 2 1 0 -1 -2

0

1

2

3

0

1

2

3

0

1

2

3

y *

y *

y *

(a) (b) (c) (d)

(e) (f) (g) (h)

( i) (j) (k) (l)

-0.002 0.0025 -0.002 0.0025 -0.002 0.0025 -0.001 0.0015

-0.0007 0.0003 -0.00050.0003 -0.00090.0003 -0.00070.0001

-0.0005 0.0010.0015 -0.0005

0.001 -0.001 0.0015-0.001

Figure 4-11. Phase-averaged Reynolds stresses at x* = 20 for different yaw angles. (a-d) **~~ vu ; (e-h) ** ~~ wu ; (i-l) ** ~~ wv . (a-d) Contour interval = 0.003;

(e-f) 0.002; (i-l) 0.0005.

97

0.008

0.004

0

-0.004

-0.008

0.002

0

-0.002

-0.004

0.002

0.001

0

-0.001

-0.002

0.015

0.010

0.005

0

0.08

0.06

0.04

0.02

0

0.03

0.02

0.01

0

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5

= 0o

= 15o

= 30o

= 45o

y* y* y* y*

Time-averaged

Coherent

Incoherent

(a)

(b)

(c)

(d)

(e)

( f )

(g)

(h)

( i )

(j )

(k)

(l ) (r) (x)

(q) (w)

(p) (v)

(o) (u)

(n) ( t )

(m) (s)

Figure 4-12. Coherent and incoherent contributions to time-averaged Reynolds stresses

for different yaw angle at x* = 20.

98

Time-averaged

Coherent

Incoherent

= 0o

= 15o

= 30o

= 45o

0

0.003

0.006

0

0.005

0

0.002

0.004

-0.002

-0.001

0

0.001

0.002

-0.002

-0.001

0

0.001

-0.0012

-0.0008

-0.0004

0

0.0004

0.0008

0.0012

(a)

(b)

(c)

(d)

(e)

( f )

(g)

(h)

(i )

(j )

(k)

(l ) (r) (x)

(q) (w)

(p) (v)

(o) (u)

(n) ( t )

(m) (s)

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5

y* y* y* y*

Figure 4-13. Coherent and incoherent contributions to time-averaged Reynolds stresses

for different yaw angle at x* = 40.

99

0.00

0.02

0.04

0.06

0.00

0.01

0.02

0.0

0.1

0.2

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

(a) (b) (c)

(d) (e) ( f )

(g) (h) ( i )

x* = 10 x* = 20 x* = 40

y* y* y*

Figure 4-14. Coherent contribution to vorticity variances for different yaw angles at x* =

10, 20 and 40: (a-c) 22

/~xx ; (d-f)

22/~

yy ; (g-i) 22

/~zz .

101

CHAPTER 5

WAVELET ANALYSIS OF THE TURBULENT WAKE

GENERATED BY A YAWED CIRCULAR CYLINDER

5.1 Introduction

When fluid flows over a circular cylinder at a Reynolds number Re (≡ U∞d/,

where U∞ is the free-stream mean velocity in the streamwise direction, d is the cylinder

diameter and is the fluid kinematic viscosity) larger than 47-49 (Provansal, 1987;

Norberg, 1994; Williamson, 1996b), vortices are shed from the cylinder. Due to the

change of the pressure distribution on the surface of the structure, time-dependent drag

and lift forces will be generated. In engineering applications, these forces are highly

undesirables as they can induce vibration on the structure when the frequency of the

vortex shedding and the natural frequency of the structure are close, which is known as

the vortex-induced vibration (VIV). Nowadays, VIV is a widely recognized

phenomenon in offshore industry and is considered as one of the main causes of fatigue

damage to the offshore structures. Some examples include marine piles, umbilical

pipelines, braced members of offshore structures, etc. These structures face an external

fluid flow which may induce structural oscillation both in the flow and cross-flow

directions due to vortex shedding. Therefore, the study of VIV and vortex shedding

from a circular cylinder has both significant theoretical as well as practical applications.

There have been extensive studies on the vortex shedding regimes classified

based on Re. (Williamson, 1996b). When Re is smaller than 140-194, laminar two-

dimensional vortex shedding occurs (Provansal, 1987; Norberg, 1994; Williamson,

1996b). The shed vortices may be parallel or oblique to the cylinder, depending on the

end conditions. The end conditions may also induce vortex dislocations at the ends of

the cylinder (Williamson, 1996a). Due to the different shedding patterns, either parallel

or oblique, a difference of about 20% in the relationship between the Strouhal number

St and Re in this regime has been found (e.g. Norberg, 1994; Williamson, 1996a). When

Re is further increases, there exists a wake transition from two- to three-dimensional

involving two shedding modes; Mode A and Mode B (Karniadakis and Triantafyllou,

1992; Williamson, 1992; Thompson et al., 1996). The two modes are demonstrated by

102

two discontinuous changes in the St ~ Re relationship (e.g. Williamson, 1996b). The

first discontinuity (Mode A) is visible in the wake when Re ≈ 180-190 (Williamson,

1992; Norberg, 1994; Leweke and Provansal, 1995; Thompson et al., 1996; Williamson,

1996c), which is characterized by regular streamwise vortices with spanwise

wavelength of approximately 3 to 4d. It is associated with the origination of vortex

loops and the streamwise vortex pairs formation resulted from the deformation of the

primary vortices. According to Henderson & Barkley (1996), by using highly accurate

numerical methods and Floquet stability analysis, the onset of Mode A instability is

predicted at Re ≈ 188.5. When Re ≈ 230-260 (Williamson, 1988), the second

discontinuity (Mode B) appears which involves a gradual transfer of energy from Mode

A to Mode B. Mode B has more irregular array of rib-like small-scale streamwise

vortical structures with the mean spanwise wavelength of about 1d (Williamson,

1996b). In the transition process, large-scale spot-like vortex dislocations resulted by

local shedding-phase dislocation along the span of the cylinder are found in both low

and high Re turbulent wakes (Williamson, 1992). Eisenlohr and Eckelmann (1989) have

shown that the formation of the streamwise vortices superimposed on the large-scale

wake vortices is connected to the vortex splitting, namely vortex dislocations. In the

range of Reynolds number 103-10

4, Kelvin-Helmholtz (K-H) vortices which are linked

with the instability in the free-shear layer are formed at the surface of the structure

(Bloor, 1964; Chyu et al., 1995). The instability of K-H vortices appears as intermittent

patches in the hot wire measurement (Rajagopalan and Antonia, 2005) and eventually

lead to small-scale vortices which could influence on the Kármán vortices (Filler et al.,

1991; Sheridan et al., 1992).

In many practical engineering applications such as the flow past cables of a

suspension or a cable-stayed bridge, subsea pipelines and risers etc., the direction of the

flow may not be perpendicular to the structure. In these cases, the incoming flow

velocity can be decomposed into two components. One is perpendicular to the structure

while the other is parallel to it. The axial-flow may have significant effect to the wake

structures, especially when the yaw angle is large. Hereafter, the yaw angle is defined

as the angle between the free-stream flow direction and the normal to the cylinder so

that = 0° represents the cross-flow case while = 90° represents the axial-flow case.

Initiative study has been conducted by Hanson (1966) and followed by a number of

researchers both experimentally (e.g. Surry and Surry, 1967; Van Atta, 1968; King,

1977a; Ramberg, 1983; Kozakiewicz et al., 1995; Lucor and Karniadakis, 2003; Thakur

103

et al., 2004; Alam and Zhou, 2007b) and numerically (e.g. Chiba and Horikawa, 1987;

Marshall, 2003; Zhao et al., 2008) at yaw angles in the range of 0° to 75°. Generally, it

was found that when is small, say smaller than 35°, the results on cylinder base

pressure and vortex shedding frequency of a yawed cylinder behave in a similar way to

the normal-incidence case through the use of the component of the free-stream velocity

normal to the cylinder axis. This is often known as the independence principle (IP) or

cosine law in the literature (e.g. Hanson, 1966; Surry and Surry, 1967; Van Atta, 1968;

King, 1977a; Ramberg, 1983; Lucor and Karniadakis, 2003; Zhou et al., 2009). Zhou et

al. (2009) have shown that with the increase of , the spanwise mean velocity W which

represents three-dimensionality of the wake increases. This in turn results in a decrease

in the strength of the vortex shedding. They also found that with the increase of , the

root-mean-square (rms) values of the of the streamwise u and spanwise w velocities and

all three vorticity components (x, y and z) decrease significantly, whereas the

transverse velocity (v) does not follow that trend. Although the Strouhal numbers of the

numerical results by Zhao et al. (2008) follow the IP closely, they agree well with the

experimental data only when 30° and slightly smaller than the experimental data

when > 30°. A maximum discrepancy of about 20% was found between the numerical

results and that from the experiments at = 60°.

As turbulent structures are characterized by eddies of various sizes, namely

large-scale structures, intermediate-scale structures and small-scale structures, by

examining these various scales of the turbulent structures, deeper understanding of the

wake flows could be obtained. A number of methods for examining the large-scale

vortex structures have been proposed previously. These include the vorticity-based

technique (e.g. Hussain and Hayakawa, 1987; Zhou and Antonia, 1993), the window

average gradient method (e.g.Antonia and Fulachier, 1989; Bisset et al., 1990; Zhang et

al., 2000), the proper orthogonal decomposition (POD) (e.g. Sullivan and Pollard, 1996;

Neumann and Wengle, 2004; Sen et al., 2007), the linear stochastic estimation (LSE)

(e.g. Bonnet et al., 1994; Sullivan and Pollard, 1996) and the phase-averaging technique

(e.g. Kiya and Matsumura, 1988; Zhou et al., 2002). The first two methods could not

extract the turbulent structures of scales except the large-scale vortices. POD is a

method that decomposes signals into a set of eigenmodes. The first eigenmode has the

maximum turbulence energy which is defined as a coherent structure while the second

eigenmode has smaller energy than the first one and more than the remaining and so on

(Sullivan and Pollard, 1996; Sen et al., 2007). According to Sullivan & Pollard (1996),

104

LSE provides essentially the same information of the coherent structure as POD and

therefore can be viewed as a weighted sum of an infinite number of POD eigenmodes.

Similarly, the phase-averaging technique is widely used to study the vortical structures

when a dominant frequency can be identified. Both coherent and incoherent turbulent

structures can be quantified using this method (Kiya and Matsumura, 1988; Zhou et al.,

2002).

Another useful method in analyzing turbulent structures is the wavelet method.

Unlike POD and the phase-averaging methods that decompose signals into coherent and

incoherent structures according to their energy and phase values respectively, wavelet

method is used by many researchers to decompose signals according to their scales or

frequencies (e.g. Mallat, 1989; Xu et al., 2006; Srinivas et al., 2007; Yoshimatsu and

Okamoto, 2008). By decomposing a measured signal into a number of scales or

frequencies, wavelet technique allows the measured time-frequency data to be analyzed

in time-domain, frequency-domain and a combination of both, which cannot be done by

the other techniques. Recently, orthogonal wavelet multiresolution analysis method has

been applied by a number of researchers to examine the turbulent wake structures (e.g.

Mallat, 1989; Li et al., 2001; Rinoshika and Zhou, 2005a; Rinoshika et al., 2006; Zhou

et al., 2006). This technique is used to decompose the measured velocity and vorticity

signals into a number of wavelet levels characterized by their central frequencies, which

represent the different scales of turbulent structures. Higher wavelet levels correspond

to lower frequency bands or larger scale structures while lower wavelet levels

correspond to higher frequency bands or structures in small-scales. The analysis would

give detailed results of various scales of the wake in terms of the three-dimensional

vorticity characteristics. More information on this method is given in Sect. 5.3.

In the present study, experiments were conducted using a three-dimensional

vorticity probe similar to that used by Antonia et al. (1998) and Zhou et al. (2003) in

order to measure all three velocity and vorticity components simultaneously. An

Uvi_Wave (version 3.0) wavelet toolbox (Sanchez et al., 1996) is used to analyze the

measured signals by decomposing them into time and frequency domains with wavelet

method. The objective of the current investigation is to study the effect of the yaw angle

of the cylinder on the characteristics of three-dimensional vorticity by examined the

effect of on the power spectral density, Strouhal number, velocities and vorticity

variances of the wake by using wavelet analysis. The motivation in the use of the

wavelet application is to make in depth understanding on the large-, intermediate- and

105

small-scale turbulent structures and to compare them with the measured large-scale

structures when the cylinder yaw angle is changed.


The experiments were conducted in a closed loop wind tunnel with a test section

of 1.2 m (width) 0.8 m (height) 2 m (length). Across the tunnel, the free-stream flow

was uniform within 0.5%. The free-stream turbulence intensity was not more than 0.5%.

For all the experiments, the free-stream velocity U∞ was about 8.5 m/s, resulting in a

Reynolds number of 7200. The three velocity components u, v and w were measured at

the downstream location x/d = 10 for four yaw angles, namely = 0°, 15°, 30° and 45°.

In the present study, the coordinate system is defined such that the x-axis is in the

incoming flow direction, the y-axis is perpendicular to the x-axis in the vertical plane

through the cylinder while the z-axis is normal to the both x and y axes. The cylinder

was a smooth stainless steel tube with a diameter of 12.7 mm and 1.15 m long. It was

aligned horizontally at the centre of the test section and supported rigidly by two

aluminum plates at both ends to minimize the end effects (Figure 5-1a).

The probe used to measure the three velocity components is sketched in Figure

5-1 (c, d). It consists of 4 standard X-probes with X-probes A and C being used to

measure u and w at two locations separated by a finite distance y = 2.7 mm while X-

probes B and D are used to measure u and v at two locations separated by a finite

distance z = 2.7 mm. These separations correspond to about 33, where is the

Kolmogorov length scale estimated using 4/13 )/( , is the mean energy

dissipation rate and can be obtained using 2)/(15 xu based on the

isotropic assumption and the Taylor‟s hypothesis. The experimental conditions on the

wake centreline for different yaw angles are summarized in Table 5-1. The separation

between the inclined wires of each X-probe was about 0.6 mm which corresponds to

about 7. The 2.5 μm diameter hot wires were etched from Wollaston (Pt-10% Rh)

wires to an active length about 0.5 mm, which corresponds to about 200 times wire

diameter. The wires were operated with in-house constant temperature circuits at an

overheat ratio = 0.5. The wires were calibrated for velocity and yaw angle using a Pitot-

static tube connected to a MKS Baratron pressure transducer (least count = 0.01 mm

H2O) at the centreline of the tunnel while the angle calibration was performed over ±

20° both before and after the experiments to ensure high quality of the data. The

106

included angle of each X-wire was about 110° and the effective angle of the inclined

wires was about 35°. The output signals from the anemometers were passed through

buck and gain circuits and low-pass filtered at a cut-off frequency fc= 5,200 Hz which is

high enough to examine the vortex shedding frequency. This frequency is much smaller

than the Kolmogorov frequency fk (≡ U/2π), which is estimated to be about 14 kHz

(Table 5-1) on the centreline of the cylinder wake. However, as the contribution from

the high frequency turbulent structures to vorticity is very small (Sect.5.4.4), the low

cut-off frequency will not cause significant errors in the measurement of vorticity. The

filtered signals were subsequently sampled at a frequency fs = 10,400 Hz by using a 16

bit A/D converter. The sampling frequency is 2fc in order to satisfy the Nyquist

criterion. The duration of each record was about 20 seconds and the minimum number

of effective acquired independent samples N ≡Ts/2Tu for this experiment is 12,320.

Here, Ts is the total record length time and Tu

du )(0

0 is the integral time scale;

where u is the longitudinal autocorrelation coefficient, is the time delay and 0 is the

time at which the first zero crossing occurs (Tennekes and Lumley, 1972; Pearson and

Antonia, 2001).

The three vorticity components in the x, y and z directions are computed based

on the measured velocity signals u, v and w at different locations, viz.

,z

v

y

w

z

v

y

wx

(5-1)

,x

w

z

u

x

w

z

uy

(5-2)

y

uU

x

v

y

uU

x

vz

)()( (5-3)

where w and u in Eqs. (5-1) and (5-3), respectively are the velocity differences

between X-probes A and C (Figure 5-1b, c); v and u in Eqs. (5-1) and (5-2),

respectively are the velocity differences between X-probes B and D. The spatial

separation x is estimated based on Taylor‟s hypothesis given by Uc(2t), where Uc =

0.87 U∞ is the average convection velocity of vortices and t ≡ 1/fs is the time interval

between two consecutive points in the time series of velocity signals. Experimental

uncertainties in velocity and vorticity measurements were estimated from errors in the

107

hot wire calibration data as well as the scatter (20 to 1 odds) observed in repeating the

experiment a number of times. The uncertainty for U was about ±3%, while

uncertainties for 2/12 u , 2/12 v and 2/12 w were about ±7%, ±8% and ±8%,

respectively, where angular brackets represent time-averaging. The uncertainty for

streamwise separation x is about ±4% and that for y and z are about ±5%. Using

these values, the uncertainties for vorticity components were estimated by the method of

propagation of errors (e.g. Kline and McClintock, 1953; Moffat, 1985, , 1988). The

resulting maximum uncertainties for 2/12 x , 2/12 y and 2/12 z are about

±14%, ±13% and ±13%, respectively. It needs to be noted that these estimations of

uncertainties do not include the spectral attenuation caused by the unsatisfactory spatial

resolution in x, y and z of the 3D vorticity probe (Antonia et al., 1998). The

correction of the spectral attenuation is based on the assumption of isotropy of the flow.

Unfortunately, the spectral attenuation effect cannot be corrected in the present study

due to the lack of isotropy at x/d = 10 for the shedding of large organized vortex

structures.

5.3 Decomposition of Experimental Signals into Various Wavelet

Levels

Orthonormal discrete wavelet transform (DWT) is a linear decomposition

process to convert a signal into a sum of a number of wavelet components at different

scales. It uses G and H filters, which are high-pass and low-pass wavelet filters,

respectively, to perform wavelet transform on any one-dimensional input signal, v(x).

Both G and H filters are constructed from the filter family. In the present study, the

Daubechies filter family with an order of 20 is chosen as the wavelet basis (Daubechies,

1992) since it has a good frequency localization and characterized by its smoothness,

making it suitable for the turbulent signals analysis (Rinoshika and Zhou, 2007). The

outputs give results of a high-pass subband i

vD from the G branch and low-pass residue

i

vA from H filter branch, where v is the symbol of the input signal and i represent the

number of the wavelet level (i.e. the scale). The original signal can be reconstructed by

using the inversion of DWT. It is performed using the synthesis filter set ( G~

and H~

filter) with the same cell iterated in order to get the approximation signal for the

following stage.

108

The wavelet multiresolution analysis is used to represent a signal or image in

various resolutions and at the same time preserves all its essential components (Li et al.,

2002; Rinoshika et al., 2006). The aim is to decompose a non-linear function into the

sum of a number of wavelet components at various central frequencies. The basic

concept of the wavelet multiresolution analysis is shown in Figure 5-2. It first performs

a discrete time wavelet transform process followed by the independent inverse

transform in each generated subband which results in a number of detail signals and an

approximation signal. A detail signal ivd is the difference between two successive low

resolution representations of signals which is 1 iv

iv

iv aad and an approximation

signal iva is a low resolution representation of the original signal at wavelet level i. The

constructions of the original signal can be done by the summation of all wavelet

components in wavelet multiresolution analysis.

In the present study, the measured velocity component U (or V and W) can be

written as the total of a time-averaged velocity component and a velocity fluctuation, as

follows

,uUU (5-4)

where an over-bar denotes time-averaging. In order to extract the turbulent structures

into various scales, the wavelet multiresolution analysis is used to decompose the

fluctuation velocity component u into a number of wavelet levels, each corresponds to

different central frequencies and directly linked to turbulent structures scales (Li et al.,

2002; Rinoshika and Zhou, 2005a, 2005b; Rinoshika et al., 2006; Zhou et al., 2006).

Any information in the original turbulent structures will be sustained regardless of the

number of the wavelet levels, as each wavelet level represents the essential information

of original turbulent structures but only in its certain range of frequencies (or band).

Thus, the original fluctuation velocity data can be reconstructed by the summation of all

fluctuation velocity of each wavelet level in wavelet multiresolution analysis, viz.

,1

n

i

iuu (5-5)

where iu is the fluctuation velocity component of u at i-th wavelet level, and n is the

total number of wavelet levels. Therefore the measured velocity component of U (x, t)

also can be written as

109

n

i

iuUU1

(5-6)

To get the wavelet components of the instantaneous vorticity component of x (or y

and z), multiresolution analysis is done to the measured vorticity signals obtained by

using Eqs. (5-1) to (5-3). The summation of the vorticity vector of all wavelet levels

i, in wavelet multiresolution analysis has the same value as the original vorticity

signal, viz.

n

i

i

1

, , (5-7)

where i, is the vorticity component at i-th wavelet level, and n is the total number of

wavelet levels.


In the present study, the velocity and vorticity signals are decomposed into 17

orthonormal wavelet levels where higher wavelet levels correspond to lower frequency

bands or larger scale structures while lower wavelet levels correspond to higher

frequency bands or structures in small-scales. However, as the contributions to the

measured velocities at y/d ≈ 0.5 from wavelet levels higher than 8 are very small, where

each wavelet level is lower than 3%, 0.5% and 2.5% for U, V and W signals

respectively, the present discussion only concerns wavelet levels i = 1-8 where i = 1-3

represent the small-scale structures; i = 4 and 5 represent the intermediate-scale

structures; i = 6 represents the organized large-scale structures located at the vortex

shedding frequency band; and i = 7-8 represent the large-scale structures. A

correspondence between the central frequencies (or wavelet levels) and the dimensions

of the vortex structures can be found in Figure 5-3 (a-d) for = 0° to 45°. In Figure 5-3,

the x-axis is shown in longitudinal wavenumber k1 (≡ 2πf/UN) where UN (≡ U∞ cos ) is

the free-stream velocity component normal to the cylinder axis. The scales

corresponding to the longitudinal integral length

dUL uu )(≡ 0

0 and the Taylor

microscale T [≡ u′/(du/dx)′, where a prime denotes the root-mean-square value] are also

indicated in the figure. The unit for the k1 is m-1

, therefore smaller values of k1 represent

the larger turbulent structures. When = 0° for example, the small-scale structures refer

110

to structures with dimensions smaller than 2.13 mm, the intermediate-scale structures

are about 2.13-8.73 mm whilst the large-scale structures are larger than 8.73 mm.

5.4.1 Spectra of measured and wavelet components of the velocity signals

In order to use the wavelet multiresolution analysis, a fundamental or dominant

frequency of turbulent structures has to be determined. This frequency is selected at the

Kármán vortex shedding frequency in the present study. Figure 5-4 (a-h) presents the

power spectra of velocity v for = 0°, 15°, 30° and 45° measured at y/d ≈ 0.5, which

correspond roughly to the vortex centre, and the spectra at their respective wavelet

levels i = 1-8. The wavelet frequency bands are also indicated in the figure. The v

velocity component is chosen here because it is known that this velocity component is

more sensitive to the organized structures than the other two components u and w. Fast

Fourier Transform method is used to obtain the power spectra for the measured v signals

(Figure 5-4a-d) and the signals of wavelet levels 1 to 8 that resulting from wavelet

multiresolution analysis (Figure 5-4e-h) at yaw angles of 0°, 15°, 30° and 45°.

The power spectra of the measured v signal (Figure 5-4a-d) exhibit a strong peak

at the frequencies f0 = 129 Hz, 129 Hz, 119 Hz and 109 Hz for = 0°, 15°, 30° and 45°,

respectively. The strong peak on the distribution of the energy spectra is a consequence

of the occurrence of the large-scale Kármán vortex structures. In comparison among the

peaks of the energy spectra at different yaw angles, the maximum energy of the

measured v spectra decreases with the increase of , changing from v = 0.12 at = 0°

to v = 0.02 for = 45°. This result seems to suggest that the intensity of the vortex

shedding does not follow IP. By using a phase-averaging technique, Zhou et al. (2010)

also showed that the magnitudes of the maximum contours of the coherent velocities u,

v and w and vorticities x, y and z do not follow the IP either. The figure also shows

significant differences of the measured spectra at different yaw angles. For example, at

= 0°, the peak is sharp and restricted to a narrow frequency band, indicating a strong

and more periodic vortex shedding. However, when the yaw angle increases, the energy

at the peak decreases significantly and disperses over a wide frequency range. These

results agree well with those reported by Surry and Surry (1967), who showed that the

energy at the vortex shedding frequency decreases and disperses significantly with the

increase of .

Wavelet multiresolution analysis is performed in order to get a better view of the

turbulence structures of various frequencies. The signals of velocity component v at y/d

111

≈ 0.5 are decomposed into 17 wavelet levels. Each wavelet level represents a local

frequency band of the measured signal. Figure 5-4 (e-h) shows the spectra of the

wavelet levels 1 to 8 for = 0°, 15°, 30° and 45° respectively. All spectra show a clear

peak, representing their corresponding central frequency f0 and frequency bandwidth

(wavelet level), which are summarized in Table 5-2. The spectra of wavelet level 6 for

all (Figure 5-4e-h) show the same peak location as the measured signals for the yaw

angles considered (Figure 5-4a-d). This is because the peak of the measured signal is

located in the frequency band 6 (please see Figure 5-4a-d). Therefore, the justifications

for the central frequency for all yaw angles are true when the central frequency obtained

from wavelet level 6 of the wavelet analysis is compared with the shedding frequency

obtained from the measured signals, i.e. both shedding frequency for = 0° and central

frequency of wavelet level 6 for = 0° (Figure 5-4a, e) are about 129 Hz, hence

validating the present data analysis technique.

Energy spectra of v from small-scale structures (wavelet levels 1-3) are very

small and are not really affected by the cylinder yaw angles. For wavelet levels 4-7, the

energy spectra start to show some differences in magnitude as is increased. While the

peaks on the spectra at = 0° and 15° (Figure 5-4e, f) are more pronounced and limited

to a narrow frequency band, the magnitudes of the energy spectra decrease with the

increase of , indicating a more periodicity at small yaw angles. The dispersion of the

energy spectra of wavelet levels 4-6 at = 30° and 45° over a frequency band is also

apparent, i.e. the energy spectra at these two angles are largely dispersed over a large

frequency band rather than concentrated only in a small range of frequency. At these

yaw angles, the peak energy is dominated by wavelet level 8, which represents the even

larger vortex structures compared with wavelet level 6. This result indicates that the

strength of the Kármán vortex shedding at larger yaw angles is reduced. The numerical

simulations by Lucor and Karniadakis (2003) and Zhao et al. (2008) showed that with

the increase of , the vortex shed from the yawed cylinder is less well defined with the

vortex filament being distorted. They stated that the axial vortices in the form of vortex

cells may or may not be parallel to the cylinder, resulting in a reduction of the vortex

strength.

However, at the frequency band of wavelet level 7, the energy spectra are the

same for all angles except for = 45° which have the highest magnitudes. The ratio of

v at wavelet level 6 to v at wavelet level 7 decreases from 5.3 to 2.7, 1.9 and 0.9 for

112

= 0° to 15°, 30° and 45°, respectively. It was found that the peak of v at wavelet level 7

has increased by 30% at = 45° as compared with that for = 0°. The increase of the

peak of v at wavelet level 7 when = 45° may be because of the downstream induction

of the vortices in higher frequency cells (wavelet level 6) to the vortices in lower

frequency cells (wavelet level 7) (Williamson, 1989). These results may indicate that

the large-scale vortex dislocations occur due to the increase of the three-dimensionality

in the wake region (Williamson, 1992, , 1996c). A „vortex dislocation‟ is a region

constituting a whole set of „split‟ or divided vortices that represent 2π phase variation

between the neighbouring frequencies (Williamson, 1992). The occurrence of the vortex

dislocations is also supported by the appearance of the quasi-periodic spectrum in the

wavelet level 6 when = 45° (Figure 5-4h), while the spectra of the wavelet level 6 for

smaller yaw angles show only one apparent peak. The quasi-periodic spectrum is

reflecting the presence of different frequency cells where vortex dislocation may occur

(Williamson, 1989).

5.4.2 Strouhal numbers from the measured and wavelet components of velocity

signals

It may be interesting to examine how the central frequencies at different wavelet

levels are influenced by . For this purpose, the central frequencies in Figure 5-4 (e-h)

are normalized by using d and UN . The central frequencies at wavelet level 6 for all yaw

angles have the same values as that correspond to the Strouhal number StN calculated

from the measured signals at these angles. The StN is defined as:

.cos/ UfdStN (5-8)

They are 0.195, 0.20, 0.205 and 0.230 for = 0°, 15°, 30° and 45°, respectively.

Comparison of the measured StN in the present study with previous experimental and

numerical results obtained by other authors can be found in Zhou et al. (2009). It was

stated that the IP works reasonably well for lower than 40 (Zhou et al., 2009). The

normalized central frequencies fN at wavelet levels 1-8 for the four yaw angles are

plotted in Figure 5-5. It can be seen that fN decreases with the increase of the wavelet

levels which is obvious as the frequency band increases with wavelet level. At each

wavelet level, it is interesting to note that fN for = 0° to 30° have a quite similar values

except for = 45° which is larger than that at other angles. To illustrate the differences

between fN at different yaw angles more clearly, the ratio 0

/ NN ff at various wavelet

113

levels are also included in Figure 5-5, where 0Nf represents the central frequency at =

0° of various wavelet levels. Evidently, at wavelet level 6, the ratios are the smallest.

For an ideal case when IP applies perfectly, the ratio 0

/ NN ff should be 1. For = 15°

and 30° at wavelet level 6, the ratios 0

/ NN ff are about 1.04 and 1.07, which are close

to 1. These differences are within the experimental uncertainty for fN, which is estimated

to be about 8% and fN can be considered as a constant. Meanwhile, the increase of the

values for 0

/ NN ff from = 0° to 45° (by about 19%) cannot be ascribed to the

experimental uncertainty. It may reflect a genuine departure from the IP for large .

This result is consistent with that reported by Van Atta (1968) and Thakur et al. (2004)

who found that for yawed cylinders, the Strouhal number with high yaw angle ( ≥ 35°)

is larger than that predicted by the independence principle. At other wavelet levels, the

ratio 0

/ NN ff becomes larger with the increase of . At = 15° for example, the

maximum ratio 0

/ NN ff is less than 1.06 while it is 1.20 and 1.56 at = 30° and 45°,

respectively. It seems that there is a 30% jump of the ratio when is increased from 30°

to 45°. This result suggests that for < 40°, the IP can be applied to the wake while for

larger value of , the ratio0

/ NN ff is out of the experimental uncertainty.

5.4.3 Contributions to velocity variances from different wavelet components

Figure 5-6 presents the three velocity variances u, v and w across the wake at

different . The variances are normalized by the free-stream velocity U∞. Generally, the

velocity variances decrease with the increase of y/d for all . It is clear that the

magnitudes of fluctuating lateral velocity 2v are about two and four times larger

than that of 2u and 2w , respectively. This is apparent as the transverse velocity

v is more sensitive to the large-scale organized turbulent structures than the other two

velocity components. The results also show that with the increase of , the velocity

variances 2u , 2v and 2w decrease.

The velocity variances of the wavelet components 2

i are used to examine

the wake characteristics of the measured velocity in its local frequency subbands and to

quantify the contribution of each wavelet component to the measured velocity variances

with different yaw angles. The wavelet velocity variances are calculated using

114

(5-9)

where k is the total number of data points and βi represents wavelet components of

fluctuation velocities u, v or w and i is the wavelet level. Figure 5-7 shows the wavelet

components of the velocity variances of u, v and w for wavelet levels 1 to 8 at = 0°

and 45°. Results for = 15° and 30° are not shown here as they reveal the similar trends

as those when is increased from 0° to 45°. Velocity variances of u, v and w for

wavelet levels 1-8 contribute about 93-99% to the total of the measured velocity

variance at y/d ≈ 0.5 for all yaw angles. Those at other wavelet levels do not show

significant contributions to the measured velocity variances and therefore are not given

in the figure.

For velocity variances 2

iu (Figure 5-7a, b), the wavelet level 6 which

corresponds to the organized large vortex structures, makes the highest contribution to

2u (45%) compared with other wavelet levels and followed by wavelet levels 5

(21%) and 4 (11%), indicating that the organized large- and intermediate-scale

structures contain more energy in the streamwise direction than the smaller structures.

The magnitude of 2

iu at all wavelet levels decreases with the increase of . At

wavelet level 6, 2

iu peaks at y/d ≈ 0.5, where the maximum mean shear yU / and

yW / also occurs (Zhou et al., 2009).

In regards of 2

iv (Figure 5-7c, d), about 71% of the contribution to 2v

is from the wavelet level 6 on the centreline. This is followed by wavelet levels 5 (14%)

and 4 (6%). Compared with 2

iu shown in Figure 5-7 (a, b) and 2

iw shown in

Figure 5-7 (e, f), the magnitude of 2

iv at wavelet level 6 is much higher, indicating

that the organized large-scale structures contribute more to the measured transverse

velocity than to the other two velocity components. At wavelet level 7, the magnitude of

2

iv for = 45° at y/d 1 is about 4 times higher than that at other angles. It seems

that at wavelet level 7, the transverse velocity v for = 45° has higher fluctuations due

to the presence of the vortex dislocations and thus increases the energy of the turbulent

wake (Williamson, 1992). The effect of this phenomenon can be seen from the

measured v component spectra (Figure 5-4d), where the velocity spectra increase

monotonically in the frequency band of wavelet level 7 as is increased from 0° to 45°.

k

j

iik 1

22 ,1

115

Among the three velocity variances, the spanwise component 2

iw (Figure

5-7e, f) has the smallest magnitude. Compared with the magnitude of 2iu and

2iv , 2

iw at wavelet level 6 has relatively smaller magnitude for = 0° (Figure

5-7e). This result is consistent with the quasi-two-dimensionality of the wake flow for

small yaw angles. The fact that the most significant contributions to w are from the

smaller scale vortex structures, i.e. lower wavelet levels such as wavelet levels 4 and 5,

indicates that the fluctuation of 2

iw is mainly related with intermediate-scale

structures. With the increase of to 45° (Figure 5-7f), the magnitude at wavelet level 6

for y/d 1 increases apparently. This result may indicate the effect from the presence of

vortex dislocations in the wake region (Williamson, 1992). It may also suggest the

generation of the secondary axial vortex structures (see the flow visualization result in

Figure 5-11) which in return enhance the three-dimensionality of the turbulent wake at

high yaw angles, consistent with that reported by Matsumoto (2001) who showed that at

large yaw angles, additional axial vortex structures were generated in yawed cylinder

wakes. Zhou et al. (2009) found that with the increase of , the mean spanwise velocity

W and hence the spanwise mean velocity gradient yW / also increases. They also

found that the magnitude of yW / is comparable with that of yU / for large yaw

angles. This result indicates that the mean streamwise vorticity )( z/Vy/Wx

and the mean spanwise vorticity )( y/Ux/Vz are expected to have comparable

magnitudes for large values of . It is noteworthy that at wavelet level 6, the

distribution of 2

iw peaks at y/d = 0.6-0.7, which is coincidence with the peak

location of yW / (Zhou et al., 2009). For other frequency bands which also make

large contributions to 2w , the magnitude of 2

iw at these wavelet levels

decreases with the increase of .

5.4.4 Contributions to vorticity variances from different wavelet components

The time-averaged vorticity variances of the measured signals are shown in

Figure 5-8 for different yaw angles, where the vorticity components in the x, y and z

directions are calculated from Eqs. (5-1) to (5-3). Hereafter, a superscript asterisk

denotes normalization by d and U∞. It is shown that for y/d 1, *2 z has the largest

magnitude and followed by *2 y and *2

x components for all yaw angles.

116

However, the values of the vorticity variances here do not necessarily mean that z and

y are stronger than x. The differences may be caused by the unsatisfactory spatial

resolution of the probe. In order to avoid phase shifting between the two velocity

gradients in the expression of y and z, the spatial separation x in the streamwise

direction is calculated from the time delay by using Taylor‟s hypothesis via x = -

Uc(2t) (Wallace and Foss, 1995). Therefore, this will give a separation in the

longitudinal direction x = 1.47mm, which is only half of either y or z (≡ 2.7 mm).

Thus, the spatial resolution of the probe in the streamwise direction is better than either

in the lateral and spanwise directions which will result in larger measured values for

both z and y than x. From the figure, all components of vorticity variances decrease

with the increase of , which may reflect that at large yaw angles the strength of the

vortices decreases which is in agreement with that reported by Lucor and Karniadakis

(2003).

The vorticity variances of the wavelet components >< 2,i are calculated by

(5-10)

where k is the total number of data points and i, is vorticity components in the x, y or

z direction and i is the wavelet level. The values of vorticity variances are normalized by

d and U∞. Figure 5-9 shows the vorticity variances >< 2,ix , >< 2

,iy and >< 2,iz of

the wavelet components of levels 1-8 for = 0° and 45°. Although figures for = 15°

and 30° are not shown here, as they show quite similar features with the trend when is

changed from = 0° to 45°, the discussion on those angles are also included. It can be

seen from the figure that the vorticity variances for levels 1 to 8 contribute about 97-

99% to the total of the measured vorticity variances at y/d ≈ 0.5 for all yaw angles.

For >< 2,ix and >< 2

,iy (Figure 5-9a-d), the main contributions are from

wavelet levels 2-5 even though they decrease with the increase of . This result

indicates that the longitudinal and lateral vorticities reside in relatively small and

intermediate-scale structures. For = 45° and at wavelet level 6, 2,ix has larger

values than that for other yaw angles at y/d < 0.6. This may be due to the effect of

increased magnitude of the spanwise mean velocity at large yaw angles, which results in

the generation of secondary large-scale structures in the near wake region at y/d 1.

k

j

iik 1

2

,

2

, ,1

117

Zhou et al. (2010) have shown that with the increase of , the contours of the coherent

longitudinal vortices x become more organized and the maximum coherent x

concentration increases. This result is also consistent with the flow visualization (Figure

5-11) which shows the generation of the secondary streamwise vortices. For small yaw

angles, the dominant contributors to 2,iz (Figure 5-9e, f) are from the small-scale

structures (wavelet levels 2 and 3) and followed by followed the intermediate-scale

structures (wavelet levels 4 and 5). However, the values of 2,iz for wavelet level 6

at y/d < 0.5 have the higher contribution to 2z than the other wavelet levels for all

yaw angles. This result suggests that at vortex shedding frequency band (wavelet level

6), z is largely generated especially at the central region of the wake and becomes

weaker after that location. With the increase of , the magnitude of 2,iz from the

wavelet level 6 decreases (see Figure 5-9e, f) which is consistent with Zhou et al. (2010)

who showed that the intensity of the coherent spanwise vortices z decreases with the

increase of . This result seems to support Mansy et al. (1994) who stated that the

increase of the streamwise vortices (Figure 5-9a, b) is at the expense of the spanwise

vortices (Figure 5-9e, f).

5.4.5 Autocorrelation coefficients of velocity components at different wavelet

levels

The autocorrelation coefficient can be used to examine the periodicity of the

turbulence structures. It is defined as

></>)+()(=< 2, iiii rxx , (5-11)

where i represents the velocity component u, v or w at i-th wavelet level and r is the

separation between the two points, which is calculated using Taylor‟s hypothesis by

converting a time delay to a streamwise separation via r = U∞/fs. It is well known

that v is a more sensitive indicator to the large-scale turbulent structures. Also, as the

trends of the autocorrelation coefficients of u and w are similar to that of v at various

wavelet levels, in the present paper, only results for velocity component v are given.

The autocorrelation coefficient of the transverse velocity v of the wavelet component

levels 3, 5, 6 and 7 at y/d ≈ 0.5 which correspond roughly to the vortex centre, are

shown in Figure 5-10. The wavelet level 3 represents the small-scale structures; 5

represents the intermediate-scale structures; 6 represents the organized large-scale

118

structures at the vortex shedding frequency band; and 7 represents the large-scale

structures. For small-scale turbulence structures (Figure 5-10a), there are no significant

differences in iv, values among various yaw angles. The autocorrelation coefficients of

the small-scale structures decrease to zero at tU∞/d = 3. This is true as the small-scale

structures generated in the wake tend to be isotropic and the initial conditions for these

structures are not important. Therefore, the effect of on autocorrelation coefficients of

the small-scale structures for all velocity components is negligible. While for

intermediate-scale structures (Figure 5-10b), the magnitudes of iv, appear to be

affected by . With the increase of , the magnitudes of iv, decrease and approach to

zero at a faster rate. For example, the turbulence structures show apparent

autocorrelation even at tU∞/d = 200 for = 0° while this has been reduced to tU∞/d = 20

for = 45°. This result may be caused by the introduction of the spanwise velocity (w

component) which deteriorates the stability of the vortex filament and hence generates

the vortex cells (Hammache and Gharib, 1991; Szepessy and Bearman, 1992). These

cells are generated quite randomly, and they lose the self-similarity quickly. Wavelet

level 6 (Figure 5-10c) mainly represents the large-scale organized structures where the

magnitudes iv, of that wavelet level are significant for all and the period of iv,

increases slightly with . Although the periodicity of iv, for large-scale structures

persists for a long distance at = 0°, the periodicity decreases to zero quickly with the

increase of . This result indicates that the vortices shed from a yawed circular cylinder

decay and lose their initial characteristics quickly as compared with that at = 0° and

the decay is more apparent with the increase of . For wavelet level 7 (Figure 5-10d),

which corresponds to the larger scale structures, iv, is not largely affected by except

for = 45°. It seems that at = 45°, iv, has larger magnitude and more organized

periodicity than that at other yaw angles. The iv, magnitudes approach to zero quickly

after 2 periods. The effect of increasing to the large-scale structures of wavelet level 7

also can be seen in Figure 5-7 (c, d), where the velocity variances 2

iv for this

wavelet level at = 45° and y/d 0.5 are about 3 times larger than that at = 0°, which

is consistent with the increase in energy spectrum of v (Figure 5-4h, wavelet level 7).

These results indicate that the large-scale structures of wavelet level 7 at = 45° are

119

intensified which may be resulted from the induction of the vortices from wavelet level

6 due to the occurrence of the vortex dislocations.

5.4.6 Flow visualization of the yawed cylinder wake

Flow visualization was also conducted to reveal the generation of the secondary

axial and streamwise vortices when yaw angle is increased to 45°. It was conducted in a

wind tunnel with a cross section of 0.4 m 0.3 m using the smoke wire method for =

0°, 15°, 30° and 45°. A cylinder of 8 mm in diameter was used without the end plates at

both ends. The velocity was 0.3m/s, corresponding to a Reynolds number of about 200.

Figure 5-11 shows that when the cylinder yaw angle is increased to 45° (the figures for

= 0° to 30° are not shown here), some smoke moves along the cylinder axial direction

immediately downstream of the cylinder. In the streamwise direction, the Kármán

vortex is not as apparent as in a cross-flow ( = 0°). In the process when the large-scale

structures evolve downstream, they also rotate in clockwise direction, indicating the

generation of the secondary streamwise vortices. This result is in agreement with the

result shown in Figure 5-9, where 2,ix at vortex shedding frequency band (i.e.

wavelet level 6) for y/d < 0.6 suddenly increases with the increase of to 45°. Based on

the flow visualization results, the streamwise vortices at = 45° are more visible and

more organized compared with that at smaller yaw angles. This result also agrees with

that reported by Zhou et al. (2010) who found that the coherent streamwise contours

become more organized and the maximum streamwise vorticity concentration increases

by about 70% when is increased to 45° compared with that at = 0°. It can also be

seen that when the streamwise vortex strength increases, the spanwise vortex strength

decreases with the increase of , which support the speculation of Mansy et al. (1994)

that the increase of the streamwise vortices is at the expense of the spanwise vortices.

The secondary axial vortex tends to propagate along the cylinder axial direction and

helps to break down the large-scale structures downstream, which in return enhances the

three-dimensionality of the turbulent wake (Zhou et al., 2010). It is also apparent that

the large-scale vortex dislocation occurs in the wake. According to Maekawa & Abe

(2002), the vortex dislocations in the wake may also contribute to the disruption of the

large-scale vortices in the turbulence wake as they travel downstream.

120

5.5 Conclusions

A wavelet multiresolution technique has been used to examine the wake velocity and

vorticity characteristics obtained using an eight-hot wire 3D vorticity probe for = 0°,

15°, 30° and 45°, respectively. The results are summarized as follows:

5. With the increase of , the energy spectra for the intermediate- (wavelet levels 4

and 5) and large-scale structures (wavelet level 6) decrease in terms of their

maximum energy and disperse extensively over an enlarged frequency band. It is

found that at = 45°, the large-scale vortex dislocations may occur as reflected

by the increase in the energy spectrum at wavelet level 7.

6. The ratios 0

/ NN ff are far from 1 except for the wavelet level 6 located at the

vortex shedding frequency band for 30°. This result indicates that IP may be

implemented for < 40° while it is not suitable for larger values of .

7. From the results of 2

iu , 2

iv and 2

iw at various wavelet levels, it can

be seen that the most significant contributions to the measured velocity

variances are from the organized large-scale structures at the vortex shedding

frequency band and followed by the intermediate-scale structures. With the

increase of to 45°, the magnitude of 2iu at all wavelet levels decreases

monotonically while for 2

iv , there is an increase in wavelet level 7 which

may be related with the occurrence of vortex dislocations. The magnitude of

2

iw from level 6 at y/d 1 increases apparently when = 45° compared

with that at = 0°. This result indicates the effect of the presence of vortex

dislocations in the wake region (Williamson, 1992). It may also be related with

the generation of the secondary axial vortex structures which in return enhance

the three-dimensionality of the turbulent wake at high yaw angles.

8. Generally, the three vorticity variances at various wavelet levels decrease with

the increase of except that of 2,ix at wavelet level 6, which increases by

about 40% on the centreline for = 45°. In contrast, 2,iz decreases by

about 30% for = 45° compared with that at = 0°. This result seems to

support that the increase of the streamwise vortices is at the expense of the

spanwise vortices (Mansy et al., 1994). The three vorticity components are

121

mostly dominated by small and intermediate-scale structures and have the

smallest values at the large-scale structures. However, the magnitude of 2,iz

for wavelet level 6 at y/d < 0.5 has higher contribution to 2z than the other

wavelet levels for all yaw angles. This result suggests that at vortex shedding

frequency band (wavelet level 6), z is largely generated especially at the central

region of the wake and becomes weaker after that location.

9. The results of autocorrelation coefficients of the velocity component v at

different wavelet levels suggest that for small-scale turbulence structures, there

are no significant differences in the magnitude for different yaw angles while for

other frequency bands especially at wavelet level 6, the differences in

autocorrelation coefficients are significant among different yaw angles. At this

wavelet level, the periodicity in iv, persists for a long distance for small and

decreases to zero quickly with the increase of . It may imply that the vortices

shed from a yawed circular cylinder decay at a faster rate and consequently lose

their initial characteristics quickly when is increased. For wavelet level 7, the

magnitude of iv, at = 45° increase significantly and its periodicity becomes

more apparent compared with that at = 0°, indicating the occurrence of the

vortex dislocations.

10. The flow visualization result shows the generation of the secondary axial and

streamwise vortices at = 45°. The former propagate along the cylinder axial

direction and disrupt the formation of the Kármán type vortex structures,

resulting in the increase of three-dimensionality of the cylinder wake. The vortex

dislocation may also be responsible for the disruption of the large-scale

structures.

122

Table 5-2. Central frequencies f0 and their respective frequency bandwidth of the v

signal of wavelet levels 1-8 at y/d ≈ 0.5 for = 0°, 15°, 30° and 45°.

Level

0°

15°

30°

45° Frequency

bandwidth (Hz)

1 2,931 2,979 2,979 2,896 1,880-5,200

2 1,530 1,490 1,430 1,470 940-3,000

3 823 797 736 779 470-1,500

4 396 396 384 361 230-800

5 196 193 203 203 110-400

6 129 129 119 109 60-200

7 71 51 61 66 30-100

8 30 30 30 33 10-55

Table 5-1. Summary of the experimental conditions on wake centreline.

U u′ Lu T R fk uk

(m/s) (m/s) (m2s

-3) (mm) (mm) (mm) (Hz) (m/s)

0° 7.24 1.50 82.0 0.080 5.87 2.49 250 14,400 0.187

15° 7.45 1.48 79.6 0.081 6.03 2.49 246 14,700 0.186

30° 7.80 1.35 65.1 0.085 6.66 2.50 224 14,600 0.177

45° 7.97 1.17 52.5 0.090 7.11 2.42 188 14 ,100 0.167

Note: U is the local velocity of the wake. R ≡ u′/ is the Taylor microscale Reynolds

number where T ≡ u′/(du/dx)′ is the longitudinal Taylor microscale. is the mean

turbulent energy dissipation rate. 4/13 )/( is the Kolmogorov length scale.

dUL uu )(≡ 0

0 is the longitudinal integral length scale where u is the longitudinal

autocorrelation coefficient, is the time delay and 0 is the time at which the first zero

crossing occurs. fk ≡ U/2π is the Kolmogorov frequency and uk ≡ / is the

Kolmogorov velocity.

123

(a) Cylinder arrangement and

coordinate system (top view)




8D

A

z

y

y

1

6

B

3

5

C

z

4 7

2

8D

A

z

y

y

1

6

B

3

5

C

z

4 7

2

U

4,7

C

B, D

y

3,8

1

2

6

5

X

A

U

4,7

C

B, D

y

3,8

1

2

6

5

X

A

U

= 0

= 0

Tunnel wall

Tunnel wall

End plate

45z

x

Inclination angle

U

= 0

= 0

Tunnel wall

Tunnel wall

End plate

45z

x

Inclination angle

Moveable 3-D

Vorticity Probe


U

Moveable 3-D

Vorticity Probe


U

Figure 5-1. Definition of the coordinate system and the sketches of the vorticity probe.

G(z)

H(z) G(z)

H(z)

2

2

2vD

2 2

vd

2

va 2

: keep one sample out of two 2

v(x)

2

2

vA 2

2

)(~

zH

)(~

zH

)(~

zG 1

vd

2 : put one zero between each sample

1vD

Figure 5-2. Basic concept of the wavelet multiresolution analysis.

124

10-7

10-5

10-3

10-1

1 10 100 1000 10000

Small scale structures

Intermediate scale structuresLarge scale structures

Lu

i=6

i=1

i=4

i=3

i=7 i=5

i=2

i=8

(a) = 0o

k1 (m

-1) = 2f/U

N

(k

1)

10-7

10-5

10-3

10-1

1 10 100 1000 10000



Lu

i=6

i=1

i=4

i=3

i=7 i=5

i=2

i=8

(b) = 15o

k1 (m

-1) = 2f/U

N

(k

1)

10-7

10-5

10-3

10-1

1 10 100 1000 10000



Lu

i=6

i=1

i=4

i=3

i=7 i=5

i=2

i=8

(c) = 30o

k1 (m

-1) = 2f/U

N

(k

1)

10-7

10-5

10-3

10-1

1 10 100 1000 10000



Lu

i=6

i=1

i=4

i=3

i=7

i=5

i=2

i=8

(d) = 45o

k1 (m

-1) = 2f/U

N

(k

1)

Figure 5-3. Spectra of the v-signal measured at y/d ≈ 0.5, and a correspondence between

the central frequencies (or levels i) and the dimensions of the vortex structures for

different cylinder yaw angles.

125

0

0.04

0.08

0.12

0.16

1 10 100 1000 10000

12 11 10 9 8 7 6 5 4 3 2 1

(a) = 0o

Level / Frequency band

f(Hz)

0

0.04

0.08

0.12

0.16

1 10 100 1000 10000

12 11 10 9 8 7 6 5 4 3 2 1

i=1i=2

i=7

i=6

i=5

i=4i=3

i=8

e) = 0o

f(Hz)

0

0.04

0.08

0.12

0.16

1 10 100 1000 10000

(b) = 15o


12 11 10 9 8 7 6 5 4 3 2 1

f(Hz)

0

0.04

0.08

0.12

0.16

1 10 100 1000 10000

5 4 3 2 1 12 11 10 9 8 7 6

i=1i=2

i=8

i=7

i=6

i=5

i=4 i=3

f) = 15o

f(Hz)

0

0.04

0.08

0.12

0.16

1 10 100 1000 10000

12 11 10 9 8 7 6 5 4 3 2 1

(c) = 30o


f(Hz)

0

0.04

0.08

0.12

0.16

1 10 100 1000 10000

5 4 3 2 1 12 11 10 9 8 7 6

i=1i=2i=3

i=8

i=7

i=6

i=5

i=4

g) = 30o

f(Hz)

0

0.04

0.08

0.12

0.16

1 10 100 1000 10000

(d) = 45o


12 11 10 9 8 7 6 5 4 3 2 1

f(Hz)

0

0.04

0.08

0.12

0.16

1 10 100 1000 10000

12 11 10 9 8 7 6 5 4 3 2 1

i=1i=2i=3

i=8

i=7i=6

i=5 i=4

h) = 45o

f(Hz)

Figure 5-4. Comparison between (a-d) spectra of the v-signal measured at y/d ≈ 0.5 for

= 0°, 15°, 30° and 45° respectively; and (e-h) spectra of various wavelet levels at y/d

≈ 0.5 for = 0°, 15°, 30° and 45° respectively.

126

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8

= 0o

30o

15o

45o

fN

15

/fN

0

fN

30

/fN

0

fN

45

/fN

0

Wavelet Level, i

f N=

fd

/(U

co

s

)

Figure 5-5. Peak frequency fN on the energy spectra and the ratio 0

/ NN ff for various

wavelet levels at y/d ≈ 0.5.

0

0.01

0.02

0.03

0.04

0 1 2 3 4

= 0o

15o

30o

45o

(a)

y/d

<u

2>

/U2

0

0.02

0.04

0.06

0.08

0 1 2 3 4

= 0o

15o

30o

45o

(b)

y/d

<

2>

/U2

0

0.005

0.010

0.015

0.020

0 1 2 3 4

= 0o

15o

30o

45o

(c)

y/d

<w

2>

/U2

Figure 5-6. Velocity variances 2u , 2v and 2w of the measured signals

for different cylinder yaw angles.

127

0

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0 1 2 3 4

Level 1Level 2Level 3Level 4Level 5Level 6Level 7Level 8

(a) = 0o

y/d

<u

i2>

/U2

0

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0 1 2 3 4


(b) = 45o

y/d

<u

i2>

/U2

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4


(c) = 0o

y/d

<v

i2>

/U2

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4


(d) = 45o

y/d

<v

i2>

/U2

0

0.001

0.002

0.003

0.004

0.005

0.006

0 1 2 3 4


(e) = 0o

y/d

<w

i2>

/U2

0

0.001

0.002

0.003

0.004

0.005

0.006

0 1 2 3 4


(f) = 45o

y/d

<w

i2>

/U2

Figure 5-7. Velocity variances 2iu , 2

iv and 2iw at various wavelet levels

for = 0° and 45°.

128

0

0.2

0.4

0.6

0.8

0 1 2 3 4

= 0o

15o

30o

45o

(a)

y/d

<

x2>

*

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 1 2 3 4

= 0o

15o

30o

45o

(b)

y/d

<

y2>

*

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 1 2 3 4

= 0o

15o

30o

45o

(c)

y/d

<

z2>

*

Figure 5-8. Vorticity variances 2x , 2

y and 2z of the measured signals

for different cylinder yaw angles.

129

0

0.04

0.08

0.12

0.16

0 1 2 3 4


(a) = 0o

y/d

<

x,i

2>

*

0

0.04

0.08

0.12

0.16

0 1 2 3 4


(b) = 45o

y/d

<

x,i

2>

*

0

0.1

0.2

0.3

0 1 2 3 4


(c) = 0o

y/d

<

y,i

2>

*

0

0.1

0.2

0.3

0 1 2 3 4


(d) = 45o

y/d

<

y,i

2>

*

0

0.1

0.2

0.3

0.4

0 1 2 3 4


(e) = 0o

y/d

<

z,i

2>

*

0

0.1

0.2

0.3

0.4

0 1 2 3 4


(f) = 45o

y/d

<

z,i

2>

*

Figure 5-9. Vorticity variances 2,ix , 2

,iy and 2,iz at various wavelet levels

for = 0 and 45.

130

-1.0

-0.5

0

0.5

1.0

0.01 0.1 1 10 100

= 0o

15o

30o

45o

(a) Level 3

tU/d

R,i

-1.0

-0.5

0

0.5

1.0

0.01 0.1 1 10 100

= 0o

15o

30o

45o

(b) Level 5

tU/d

R,i

-1.0

-0.5

0

0.5

1.0

0.01 0.1 1 10 100

= 0o

15o

30o

45o

(c) Level 6

tU/d

R,i

-1.0

-0.5

0

0.5

1.0

0.01 0.1 1 10 100

= 0o

15o

30o

45o

(d) Level 7

tU/d

R,i

Figure 5-10. Autocorrelation coefficients of v at y/d ≈ 0.5 for various wavelet levels at

different cylinder yaw angles.

x

z

= 45°

U

Secondary

axial vortex

Streamwise vortex

Vortex dislocations

x

z

= 45°

U

Secondary

axial vortex

Streamwise vortex

Vortex dislocations

Figure 5-11. Flow visualization of wake with the existence of secondary vortex and

streamwise vortex when = 45°.

131

CHAPTER 6

ON THE STUDY OF THE WAKE BEHIND TWO YAWED

CYLINDERS IN SIDE-BY-SIDE ARRANGEMENT

6.1 Introduction

When a fluid flows over a bluff body at a sufficiently high velocity or Reynolds

number Re (≡ U∞d/, where U∞ is the free-stream velocity in the streamwise direction, d

is the cylinder diameter and is the kinematic viscosity of the fluid), vortex shedding

occurs, which results in a time-dependent pressure distribution on the solid surface.

Therefore, the bluff body is subjected to the time-dependent in-line and transverse

forces. These forces will induce vibrations to the bluff body when the vortex shedding

frequency is close to its natural frequency. This phenomenon is known as vortex-

induced vibration (VIV). VIV is one of the most important aspects that should be

avoided in engineering applications as it will contribute to the stability and fatigue

damage to the structures. Thus, the studies in understanding the flow over bluff bodies

are important in engineering applications, such as fluid flow across heat exchangers, air

flow around high-rise buildings or yaw-cables of bridges. These studies would have

benefits in understanding the vortex structures to minimize many problems associated

with the VIV and to improve the efficiency in the maintaining operation of the structure.

Vortex shedding from a single cylinder wake is well documented both in air and

water flows. Many issues in this area have been resolved (e.g. Prasad and Williamson,

1997; Williamson and Govardhan, 2004). However, in the case when a fluid flows over

an identical neighbouring cylinder, the VIV caused by fluid-cylinder interaction is a far

more complicated problem than that of the single cylinder case. Therefore, to

investigate the problems, many researchers have contributed to better understanding in

the flow around a pair of cylinders in a side-by-side arrangement (e.g. Bearman and

Wadcock, 1973; Zdravkovich, 1977; Williamson, 1985; Mahir and Rockwell, 1996;

Zhou et al., 2001; Zhou et al., 2002).

There are many major parameters that may be responsible for vortex shed from

two cylinders arranged side-by-side with normal velocity incidence (i.e. the flow is

perpendicular to the cylinders) such as initial conditions, pressure distribution and

132

Reynolds number (Zdravkovich, 1977; Williamson, 1985; Zhou et al., 2002). Besides

the parameters mentioned above, the vortex structures are also influenced by the centre-

to-centre cylinder spacing T* (hereafter, an asterisk denotes normalization by the

diameter d and/or free-stream velocity U∞). Various flow patterns and behaviours have

been identified as the cylinder spacing T* is varied. For example, the variations of T

*

from close proximity spacing to large spacing could contribute to the formation of a

single or multiple wakes. At large cylinder spacing (T* ≥ 2), two coupled vortex streets

have been observed with a definite phase relationship (Chen et al., 2003). Based on

Williamson‟s (1985) results, the two streets may occur in phase or in anti-phase. The

two in-phase streets are antisymmetrical about the flow centreline but symmetrical for

the anti-phase case. The in-phase streets eventually merged downstream to form a single

street, while the anti-phase streets remained distinct farther downstream. He found that

for 2 < T* < 6, the vortex shedding is predominant in anti-phase. At intermediate

cylinder spacing, 1.2 < T*

< 2.0, the flow behind two cylinders produces two bi-stable

wakes, one is narrow and the other is wide (Chen et al., 2003). The flow was flip-

flopping and randomly changes over from one side to the other which is caused by the

bi-stable deflected flow between the cylinders (Ishigai et al., 1972; Bearman and

Wadcock, 1973; Kim and Durbin, 1988). By using phase-averaging method, Zhou et al.

(2002) found that the longitudinal and lateral spacing between vortices for T* = 1.5 are

larger than those for T* = ∞ and 3.0. Large lateral spacing implies a weak interaction

between vortical motions that may be responsible for the long lifespan and the stability

of the vortex street. They proposed that for T* = 1.5, the vortices developed from the

shear layer instability in the wide wake leading to the early disappearance of the narrow

wake. The dominant frequency in the narrow wake is about triple of that in the wide

wake (Bearman and Wadcock, 1973), which is caused by the flip-flopped flow due to

the squeezing effect and amalgamation of the vortices generated behind the cylinders

(Chen et al., 2003). They suggested that the vortices in the narrow wake have tendency

to pair and absorb the vortex from the wide wake.

Although many studies have been conducted in both large and intermediate

cylinder spacings, none of them measured the three-dimensional vorticity

simultaneously. What is more, the previous studies focused only on the cross-flow case

where the cylinder is perpendicular to the on-coming flow. While in practical

engineering applications, this is not always the case. More often, the pipelines are

yawed to the on-coming flow. In the present study, the yaw angle is defined as the angle

133

between the incoming flow direction and the direction which is perpendicular to the

cylinder axis, so that = 0° corresponds to the cross-flow case while = 90°

corresponds to the axial case (see Figure 6-1a). The flow structures and vortex shedding

for a single cylinder have been studied previously by a number of investigators (e.g.

Surry and Surry, 1967; Marshall, 2003; Alam and Zhou, 2007b; Zhou et al., 2009,

2010). It has been shown that the vortex shedding frequency of a yawed cylinder

behaves in a similar way to normal-incidence case through the use of the component of

the free-stream velocity normal to the cylinder axis. If the force coefficients and the

Strouhal number are normalized by the velocity component normal to the cylinder axis

(StN ≡ f0d/UN, where f0 is the vortex shedding frequency and UN ≡ U∞ cos ), the values

are approximately independent of . This is often known as the independence principle

(IP) or the Cosine Law in the literature. Several theoretical and experimental studies

have verified the IP (Hoerner, 1965; Schlichting, 1979). However, many studies also

shown some deviations from the predictions based on the IP (Hanson, 1966; Surry and

Surry, 1967; Van Atta, 1968; King, 1977b; Ramberg, 1983; Kozakiewicz et al., 1995;

Lucor and Karniadakis, 2003; Marshall, 2003; Thakur et al., 2004), especially at large

yaw angles. It has been suggested that the IP was valid for 35°-40° (Van Atta, 1968;

Smith et al., 1972; Thakur et al., 2004; Zhou et al., 2009, 2010) while for larger , the

IP is not valid because the shedding frequency is greater than that predicted by the IP

and the vortices‟ slant angle is less than the cylinder yaw angle (Lucor and Karniadakis,

2003).

In this paper, the physical phenomena in the wake of two yawed cylinders with

cylinder spacings T* = 3.0 and 1.7 are investigated where the cylinder spacings T

* = 3.0

and 1.7 represent the large (T* ≥ 2) and intermediate (1.2 < T

* < 2) cylinder spacing,

respectively. It is expected that the turbulence wake for T* = 3.0 may form two vortex

streets that are strongly coupled forming symmetrical wake while that for T* = 1.7 may

form two vortex streets and forming deflected flow, one is wide and the other is narrow.

In the experiments, the three-dimensional vorticity in the wake was measured

simultaneously using a three-dimensional multi-hotwires vorticity probe at a free-stream

velocity (U∞) of 8.5 m/s, corresponding to a Reynolds number Re = 7200, at which the

flow can be classified as in the turbulent regime. As turbulent flow is three-dimensional,

it is characterized by the vorticity fluctuations (Tennekes and Lumley, 1972). Therefore,

it is important to measure all three vorticity components simultaneously using a

vorticity probe to provide more completed data for studying turbulence characteristics

134

than using simpler probe geometries. Although there are some published experimental

data on turbulence characteristics in the wake behind a pair of circular cylinders

arranged side-by-side in a cross-flow, there is no study on the three-dimensional

vorticity characteristics in the wake behind yawed cylinders arranged side-by-side.

Therefore, to gain some more fundamental understanding of the flow structures in the

intermediate wake region, the three-components velocity and vorticity fluctuations in

the wake of yawed cylinders arranged side-by-side for cylinder spacings T* = 3.0 and

1.7 are measured using a multiple hotwires vorticity probe. Based on the data, the

effects of the yaw angle and the cylinder spacing T *

on the turbulence characteristics

and vortical structures can be examined in details.


6.2.1 Experimental arrangement

The measurements were conducted in a closed loop wind tunnel with test section

of 1.2 m (width) 0.8 m (height) and 2 m in length. The free-stream velocity in the test

section is uniform to within 0.5% and the free-stream turbulence intensity is less than

0.5%. Two similar circular cylinders arranged side-by-side with a diameter d = 12.7 mm

were used to generate the wake flow. The cylinders were installed horizontally at the

centre of the test section and supported rigidly at both ends by two aluminium end

plates to minimize the side effects to the wake flow. Two cases of different lateral

spacings between the cylinders, which are T* = 3.0 and 1.7, were tested. For each case,

four yaw angles, namely = 0°, 15°, 30° and 45°, were tested at three downstream

locations x* = 10, 20 and 40. However, in this paper, only results for x

* = 10 are shown

and discussed. The arrangement of the cylinders and the definition of the coordinate

system are shown in Figure 6-1 and 6-2. As shown in the Figure 6-1, the coordinate

system is defined as the x-axis is in the similar direction as the incoming flow located at

the centre of the transverse spacing of both cylinders. The y-axis is perpendicular to the

x-axis in the vertical plane coming through the cylinders (another cylinder cannot be

seen in the plan view) and out of the paper while the z-axis is normal to both x- and y-

axes.

6.2.2 Velocity and vorticity signals from the X-probes

In the present study, a vorticity probe was moved across the wake along the y-

direction to measure simultaneously the three-dimensional vorticity component

135

simultaneously. Another X-probe was fixed at a location y = 4-7d (depending on the

measurement location in the x-direction) as a reference probe, used to provide a phase

reference for the measured signals by the moveable vorticity probe. To minimize the

end effect, the probes were located at the middle span of the cylinder diameter in y-

direction. The vorticity probe consists of 4 X-probes (X-probes A, B, C and D) as

shown in Figure 6-1 (c, d). Both X-probes A (wires 1 and 2) and C (wires 5 and 6)

which aligned in the x-z plane and separated in the y-axis (Δy) measure u and w velocity

signals while X-probes B (wires 3 and 4) and D (wires 7 and 8) which aligned in the x-y

plane and separated in the z-axis (Δz) measure u and v velocity signals. The separations

between the centres of the two opposite X-probes (Δy and Δz) were about 2.7mm. The

clearance between two hotwires for each X-probe was about 0.7mm. The hotwires of

the vorticity probe were etched from Wollaston (Pt-10% Rh) wires. The active length

was etched to about 200dw, where the hotwire diameter dw is about 2.5μm. The s were

operated on constant temperature circuits at an overheat ratio of 1.5. The angle

calibration was carried out at ±20°. The angle of each X-probe was about 110° and the

effective angle of the inclined wires was about 35°. All output signals were low-pass

filtered at a cut-off frequency fc = 5200 Hz, and sampled at a sampling frequency fs =

10400 Hz using a 16 bit A/D converter. The record duration for each measurement point

was about 20s. From the velocity signals obtained through the experiments, the vorticity

components can be calculated viz,

,z

v

y

w

z

v

y

wx

(6-1)

,x

w

z

u

x

w

z

uy

(6-2)

,)()(

y

uU

x

v

y

uU

x

vz

(6-3)

where Δw and Δu in Eqs. (6-1) and (6-3), respectively, are velocity differences between

those velocities measured by X-probes A and C; Δv and Δu in Eqs. (6-1) and (6-2),

repectively, are velocity differences between those velocities measured by X-probes B

and D. The velocity gradients in the streamwise direction in Eqs. (6-2) and (6-3), i.e.

Δw/Δx and Δv/Δx, respectively, are obtained by using a central difference scheme to the

time series of the measured velocity signals, e.g. Δv/Δx ≈

,/)]1()1([/ xivivxv where Δx is estimated based on Taylor‟s hypothesis

136

given by x = -Uc(2t). Uc is the vortex convection velocity and t ( 1/fs) is the time

interval between two consecutive points in the time series of the velocity signals. A

central difference scheme in estimating Δw/Δx and Δv/Δx is useful to avoid phase shifts

between the velocity gradients involved in Eqs. (6-2) and (6-3) (Wallace and Foss,

1995). The experimental uncertainty for U was about ±3%, while the uncertainties for

u′, v′ and w′ were about ±7%, ±8% and ±8%, respectively. Hereafter, a single over bar

denotes conventional time-averaging and a superscript prime denotes root-mean-square

values. More details of the vorticity probe were given in Zhou et al. (2009).

6.2.3 Phase-averaging method

Eduction of the large-scale organized structures from the wake by using phase-

averaging method has been popular among the late researcher. Here, the phase-

averaging method is similar to that used by Kiya & Matsumura (1985), Zhou et al.

(2002) and Zhou et al. (2010). The v and vr signals obtained from the moveable vorticity

probe and fixed reference probe, respectively, were band-pass filtered with the central

frequency set at f0 using a fourth-order Butterworth filter. The f0 is identified from the

frequency at which a significant peak occurred in v spectrum. The spectrum was

obtained from the velocity signal using fast Fourier transform method. The low-and

high-pass frequencies were chosen to be the same value as f0 so that band-pass width of

the filtered signal would be zero. This would allow a better focus on the large-scale

organized structure. Figure 6-3 presents examples of filtered and measured v-signals

from X-probes B in arbitrary scales. The same scales are used for all yaw angles for

both T* = 3.0 and 1.7. The thicker line denotes the filtered signal vf. The v-signals for T

*

= 3.0 are more periodic than those for T* =1.7. Both cylinder spacings have a decrease

in vortex shedding period with the increase of yaw angles especially when it reaches

45°. Details on this behaviour will briefly discuss in the next section.

From the filtered signals, two phases A and B can be identified on the vf, viz.

Phase A: 0fv and ,0dt

dv f (6-4)

Phase B: 0fv and .0dt

dv f (6-5)

Both phases A and B correspond to the time tA,i and tB,i (measured from the an arbitrary

time origin), respectively. The phase was then calculated from the filtered signal vf,

viz.

137

,,,

,

iAiB

iA

tt

tt

where iBiA ttt ,, (6-6)

,,1,

,

iBiA

iB

tt

tt where 1,, iAiB ttt (6-7)

The interval between phases A and B was made to be equal to 0.5/f0 by compression or

stretching and further divided by 30 equal intervals. The difference between the local

phase of the vorticity probe at each y-location and the reference phase of the fixed X-

probe was then used to obtain phase-averaged contours of vorticity in the (, y) plane

and the sectional streamlines. After that the phase-averaging was conducted on the

measured signal, not the filtered signals. The phase average of an instantaneous quantity

B is given by

N

iikk B

NB

1,

1 (6-8)

where N is the number detected which is 200 000 and k represents the phase. For

convenience, the subscript k will be omitted after this. The variable B can be written as

the summation of the time-averaged component B and the fluctuation component

where B stands for the instantaneous vorticity of velocity signals, viz.

. BB (6-9)

The fluctuation component can be further decomposed into a coherent fluctuation ~

and a remainder r (Hussain, 1986) as follow:

= ~

+ r (6-10)

The coherent fluctuation )(~

represents the large-scale coherent structures while

the remainder r represents the incoherent structures of the wake. By squaring to both

sides of the Eq. (6-10), the following equation may be derived:

.~ 222 r (6-11)

Once the deduction of coherent components of velocity (or vorticity) is obtained, the

coherent contributions to the velocity (or vorticity) variances can be given in terms of

the structural average denoted by a double over-bar, viz.

138

,~~

1

1~~ 2

121

k

kkk (6-12)

where and denote velocity signals (u, v or w) or vorticity signals (x, y or z ). For

each wavelength, the phase averaged structure begins at k1 samples (corresponding to

= -π) before = 0 and ends at k2 samples (corresponding to = π) after = 0.


6.3.1 Mean streamwise and spanwise velocity profiles

In this study, the experimental results of T* = 3.0 and 1.7 are compared with T

*

= ∞ (single cylinder) obtained by Zhou et al. (2009; 2010) for statistical and phase-

averaging methods, respectively. In the latter paper, the dependence of single cylinder

wake on the yaw angles ( = 0°, 15°, 30° and 45°) was examined. Figure 6-4 shows the

time-averaged streamwise and spanwise velocity of the wake at = 0°, 15°, 30° and 45°

for different cylinder spacings T* = 3.0 and 1.7. The *U profiles are generally

symmetric about the flow centreline y* = 0. The trend of *U profile at = 0° (Figure

6-4a, b) are comparable with that reported by Zhou et al. (2002). The variation in the

magnitude of *U between both studies may be because of the difference in Reynolds

numbers which was about 5800 for the former study. The *U profile displays a single

peak around y* = 0 for T

* = 1.7 which suggests the occurrence of a single vortex street

at this separation, similar to that for T* = ∞. The *U profile for T

* = 3.0 displays a peak

around y* = 1.5 (the other peak is expected to be around y

* = -1.5) representing two

vortex streets for T* = 3.0. It is apparent that *U for T

* = 3.0 at y

* > 0 has a quite

similar trend with that for T* = ∞ at y

* > -1.5. This suggests that each vortex street of the

former wake is similar with the single vortex street of the latter wake and the distance

between both vortices for the former wake is about y = 3d. For all spacings, *U

magnitudes at = 0° are the smallest. With the increase of , the magnitude of *U

increases gradually.

The maximum velocity deficit U0* and the mean velocity half-width L

* for

various yaw angles and cylinder spacings are summarized in Table 6-1. There is a

significant reduction in U0* with the increase of , regardless of the cylinder spacing. It

can be seen there is no much difference in the half-width of the mean velocity L* values

139

when varies from 0° to 45° for both T* = 3.0 and 1.7. The maximum difference is

only about 5% which may be subjected to the uncertainty in the experiments. This result

is in contrast with that for T* = ∞, where there is 15% decrease in L

* for T

* = ∞ when

changes from 0° to 45° which cannot be attributed to the experimental uncertainty. It

may imply that the L* values for T

* = 3.0 and 1.7 are almost constant between various

yaw angles.

Time-averaged spanwise velocity component *W is one of the methods used to

assess the three-dimensionality of the wake (Matsumoto et al., 1992). A larger value of

*W corresponds to the higher degree of three-dimensionality. The spanwise velocity

acts to impair the two-dimensionality of the wake and hence vorticity strength

(Marshall, 2003; Alam and Zhou, 2007b). The values of *W at various for T* = 3.0

and 1.7 are shown in Figure 6-4 (c, d). The profiles of *W are generally symmetric

about y* = 0 for T

* = 3.0 (Figure 6-4c). It is shown that, at = 0° and T

* = 3.0, *W is

almost zero across the wake. The trend is similar to that for single cylinder wake

observed by Zhou et al. (2009). This result is reasonable as for flow in normal direction

to the cylinders, the flow is expected to have two-dimensional plane wake

characteristics. With the increase of , the magnitude of *W (Figure 6-4c, d) increases

gradually for y* < 3 while decreases in other region. The maximum magnitude of *W

occurs at y* = 1.5 for T

* = 3.0 and increases monotonously as varies from 0° to 45°

suggesting a stronger three-dimensionality with the increase of .

At the intermediate cylinder spacing T* = 1.7, the *W profiles (Figure 6-4d) are

lack of symmetric about y* = 0 especially at = 0°. In contrast to T

* = 3.0 and ∞, the

values of *W is not equal to zero even at = 0°. It may suggest that the vortex in the

wake is inclined and the wake may not have a perfect two-dimensional plane. These

may be caused by the vortex evolution. The numerical simulation study by Chen et al.

(2003) suggest that the wake transition to turbulence in separating shear layer occurs at

x* = 1.5. Across the downstream wake location x

* = 10, due to the shear layer

instability, the vortices are formed in the wide wake. The vortices in the narrow wake

which caused by the wake deflection, are probably disappeared before x* = 10 (Zhou et

al., 2002). They suggest that the vortex evolution or regeneration is not completed yet at

this streamwise region. With the increase of to 15° (Figure 6-4d), the symmetrical

about y* = 0 is enhanced. When further increases to 30° and 45° the symmetrical trend

140

is apparent at about y* = -0.2 and -0.3, respectively. At the centreline for T

* = 1.7, the

maximum magnitudes of *W increases significantly as increases suggesting

enhanced three-dimensionality of the wake.

As a larger *W magnitude implies a higher degree of three-dimensionality

hence a stronger instability of the vortex filament (Zhou et al., 2010), the variation of

the normalized velocity gradient **/ yW implies the magnitude of the mean

streamwise vorticity )//( zVyWx . The **/ yW variation profiles for T

* =

3.0 and 1.7 at different are shown in Figure 6-5. At = 0°, the **/ yW values for

both T* are close to zero indicating the quasi-two-dimensionality of the wake flow. This

result implies that the vortex shedding is parallel with the cylinder axis (Hammache and

Gharib, 1991; Zhou et al., 2010). The absolute magnitudes of **/ yW for both T

*

increase with the increase of . This result suggests a larger x with the increase of .

The maximum **/ yW magnitudes are around y

* = 2.0-2.2 and 1.2-1.4 for T

* = 3.0

and 1.7, respectively. While the distribution of **/ yW provides a measure to the x ,

the **/ yU profile provides a measure to the spanwise vorticity

)//( yUxVz . The **/ yU profiles at different for both T

* are also

shown in the Figure 6-5. With increasing , the **/ yU magnitudes decrease

gradually. This result indicates smaller values of z at larger . In comparison for T* =

∞, 3.0 and 1.7, the the **/ yW and **

/ yU magnitudes for T* = 1.7 are the largest

for all yaw angles while those for T* = ∞ and 3.0 are comparable except those for T

* =

3.0 which are almost zero at around y* ≈ 1.3 instead at the wake centreline.

By using the shedding angle measurement which is proposed by Hammache

and Gharib (1991), the of the vortex shedding can be evaluated. They suggested that

for oblique shedding, the is related to the x and z components, where tan () =

x / z which can be simplified further as tan () = SW / SU, where SW and SU are the

maximum velocity gradients on the **/ yW and **

/ yU profiles, respectively.

Thus, the shedding angle [≡ tan-1

(SW / SU) can then be calculated form the values of SW

and SU from Figure 6-5a. The shedding angles for different yaw angles for T* = 3.0

141

and 1.7 are given in Table 6-1. It is shown that the values for T* = 3.0 is slightly larger

than the cylinder yaw angles , indicating that the vortex is not really parallel to the

cylinder. While for T* = 1.7, the values are quite close to the , suggesting the

shedding is oblique and the vortex is almost parallel to the cylinder axis.

6.3.2 Power spectra of velocity and vorticity signals

The power spectra of the velocity and vorticity signals at = 0° and 45°,

measured around y* = 1.5 are shown in Figures 6-6 and 6-7. The results at = 15° and

30° are not shown here as the spectra at these angles follow the trend when varies

from 0° to 45°. All power spectra are normalized to decibel scale by using the

maximum of v or z at = 0° and 45°, respectively. The x-axis in the figures is

normalized to fN (≡ fd/UN). This normalization allows the peak frequency of the

spectrum f0 to correspond to the Strouhal number StN (≡f0d/UN). The values of StN at

different yaw angles for various cylinder spacings are stated in Table 6-1.

For T* = 3.0 at = 0° (Figure 6-6a), each power spectra of all velocity

components u, v and w shows a discernible peak which corresponds to StN = 0.2. The

peak energy of v is the highest, followed by the u and w. The StN for T* = 3.0 is

comparable to that for a single cylinder at all yaw angles (see Table 6-1 for

comparisons). As an example, at = 0°, StN = 0.2 for T* = 3.0 while 0.195 for T

* = ∞.

The differences of StN for both T* = 3.0 and ∞ are around 5-11% for < 40°. In both T

*

= ∞ and 3.0 wakes, the second harmonic of vortex shedding is pronounced in u and v.

The small peak may be because of the occurrence of the second harmonic of vortex

shedding (Zhou et al., 2010). With the increase of to 45° (Figure 6-6b), the peak

spectra is broadening over the frequency axis. The second harmonic peak also vanishes

at this yaw angle. Zhou et al. (2009) has shown that for the single cylinder wake, when

is smaller than 40°, the StN keeps approximately constant within the experimental

uncertainty. For T* = 3.0, the peak of v at = 15°, 30° and 45° is evident which

corresponds to StN = 0.212, 0.220 and 0.246, respectively. The comparisons of the StN

between various yaw angles for T* = 3.0 and 1.7 are shown in Figure 6-8. The ratio

StN/St0 (St0 represents the Strouhal number at = 0°) and the error bar are used in the

figure to have a better visualization on the ratio StN/St0 and the range of experimental

uncertainty. The experimental uncertainty of StN of the present study is estimated to be

around 6%. The results are evident that if a tolerance of 6% is applied, the data

142

support the IP for < 40°. While for larger , the difference of StN/St0 value from 1 is

far from the experimental uncertainty, suggesting genuinely departure from the IP.

These variations indicate that the independence principle may also applicable to the T*

= 3.0 wake when < 40°.

For T* = 1.7, the velocity spectra at = 0° (Figure 6-6c) shows a dominant peak

occurring at a vortex shedding frequency f0 which corresponds to StN = 0.118.

Unfortunately, the authors cannot detect two peaks in the spectra near 0.16 and 0.24, as

observed by Chen et al. (2003) in their study for T* = 1.7. The two peaks represent the

Strouhal numbers for the wide and narrow wakes. The wide wake has a low Strouhal

number while the narrow wake has a high Strouhal number. The single peak observation

may be because at the downstream location x* = 10, the vortex regeneration or evolution

may not completed yet. The other peak cannot be detected in the present study as the

vortex in the narrow wake is probably diminished before this downstream location

(Zhou et al., 2002). It may be argued that the hotwire measurements may detect a

double peak frequency, given that the two lateral vortices are in very close proximity to

each other. This argument is not valid here as the StN of the single peak spectra of this

study (0.118) is not half of the total measured values as that obtained by Chen et al.

(2003). The similar observation has been made by Zhou et al. (2002), who reported a

single dominant frequency across the wake, which corresponding to StN = 0.11. A

possible transition from the wide and narrow regimes to a single vortex street may occur

at the present downstream location (Zhou et al., 2002). With the increase of from 0°

to 15°, 30° and 45°, the spectra exhibit a peak at vortex shedding frequency which

corresponds to StN = 0.122, 0.119 and 0.133, respectively. The comparison of the StN for

T* = 1.7 at = 15°, 30° and 45° with that at = 0° are given in Figure 6-8. It is shown

that the StN at = 15° and 30° are within the experimental uncertainty. Therefore, the

StN can be considered as a constant. The difference (12.7%) of StN at = 45° from that

at = 0° may not be ascribed to the experimental uncertainty. It may reflect a genuine

departure in the IP when is large. This observation is similar to that for the single

cylinder results which may be because of the occurrence of the vortex evolutions from

the wide and narrow regimes to a single vortex street. Therefore, this result suggests that

the IP is applicable in the intermediate cylinder spacing wake i.e. T* = 1.7 when <

40°.

143

The spanwise vorticity spectrum for T* = 3.0 at = 0° (Figure 6-7a) shows a

significant peak at the vortex shedding frequency, implying large spanwise vortices in

the wake. Similar to the single cylinder result, the transverse vorticity spectrum y

does not show any peak while the spanwise vorticity spectrum x shows a small peak

at f0. The peak heights relative to the heights of the plateaux of the x and

z

vorticity spectra over the range of fN = 0.004-0.1 are 3 dB and 15 dB, respectively. With

the increase of to 45° (Figure 6-7b), the peak of the z spectrum is changed from

sharp and narrow to broad and wide. At this yaw angle, the peak height relative to the

height of the plateaux of the x and

z vorticity spectra over the range of fN = 0.004-

0.1 are 5 dB and 11 dB, respectively, suggesting a decrease in spanwise vortex shedding

intensity and an enhanced streamwise vortex shedding intensity. This result has the

same trend as that for T* = ∞ by Zhou et al. (2009), which can be related to the Mansy

et al.(1994) quantitative study, where the authors found that the increase of the

streamwise vortices is at the expense of the primary spanwise vortices.

The spanwise vorticity spectrum for T* = 1.7 at = 0° (Figure 6-7c) shows a

very minor peak at the same f0 as that of the velocity spectra (Figure 6-6c). However,

the peak‟s magnitude of the former is not as large as that of the latter. This result may

indicate that at the cylinder spacing of T* = 1.7, the vortex shed from the two cylinders

is not as strong as that from both an isolated cylinder and two cylinders of large spacing

(e.g. T* = 3.0). The lack of an apparent sharp peak in Figure 6-7 (c) is consistent with

the trend of coherent vorticity contours (as shown in Figure 6-10 later). The peaks of

y and z are getting closer with the increasing . This result shows that the

transverse and spanwise vorticities are likely to have similar magnitudes and strength

when is increased, which will be confirmed later.

6.3.3 Phase-averaged vorticity fields

Figures 6-9 and 6-10 present the contours of the phase-averaged vorticity

components *~x ,

*~y and *~

z for T* = 3.0 and 1.7 at four yaw angles = 0°, 15°, 30°

and 45°. The contours for T* = ∞ were shown in Zhou et al. (2010) and will not repeat

here. The contours are in the (, y*) plane, where the phase (from 2π to 2π) can be

interpreted in terms of a streamwise distance and y* (from 0 to 4.8 for T

* = 3.0 and from

-1 to 4.8 for T* = 1.7) is the spanwise distance which normalized by diameter. = 2π

144

corresponds to the Kármán wavelength (≡ Uc/f0) or /2 (≡ Uc/2f0) for T* = 3.0 and 1.7

contours, respectively, where Uc is the vortex convention velocity. The values for Uc are

estimated with the values of uU ~ at the vortex centres (Zhou and Antonia, 1992). The

Uc and values for different yaw angles and cylinder spacings are also included in

Table 6-1. In comparison of for different cylinder spacings in Table 6-1, it is evident

that the Kármán wavelength for T* = 1.7 for all yaw angles are about 50-77% greater

than those for T* = 3.0 and ∞. The flow direction is from left to right for all phase-

averaged contours (Figures 6-9 to 6-22).

For T* = 3.0, the wavelength of the spanwise vortices is about 4.45d which is

larger than that for T* = ∞. At < 40°, the wavelength value is about the same, only

about 3% difference than that obtained at = 0°. This difference is within the

experimental uncertainty, and can be considered as a constant value. However, with the

increase of to 45°, the wavelength increases significantly by about 22% from that at

= 0°. If a factor of cos() is multiplied with , the cos() values at = 15°, 30° and

45° are about 7%, 12% and 14%, respectively, lower than that obtained at = 0°.

The phase-averaged *~z contours for T

* = 3.0 (Figure 6-9i-l) display remarkable

periodicity, resulting from the Kármán vortex street for all yaw angles. The contours are

symmetry with respect to the centreline y* = 0. At all yaw angles, the vorticity contours

display two distinct vortex streets and the magnitudes of the contours are smaller with

those for T* = ∞ (Zhou et al., 2010). Here, only a single vortex street is shown as the

wake is symmetric about y* = 0. Another vortex street is located below the centreline.

The spanwise vortex centres at = 0° for T* = 3.0 are around y

* = 1.3 and 1.9 for

positive and negative vortices, respectively, while those for T* = ∞ are around y

* = -0.2

and 0.4. From the figure, it seems that if the whole set of the phase-averaged *~z

contours for T* = 3.0 wake is shifting downward about y

* = 1.5, the vortex contours are

quite similar with those for T* = ∞ (figure shown by Zhou et al. 2010). This trend is

relevant in all phase-averaged vorticity and velocity components as comparisons are

done between Figures 6-9, 6-15, 6-17 and 6-19 for T* = 3.0 with that for T

* = ∞ in Zhou

et al. (2010). These results affirm that the wake of two cylinders for T* = 3.0 acts as a

two single-cylinder wake since it has similar vortex patterns as those for T* = ∞. The

maximum of coherent spanwise vortices at = 0° for T* = 3.0 is -0.7 (Figure 6-9i).

Even though there is a small decrease in the maximum contour value of *~z when

145

varies from 0° to 30°, the variation is not very apparent. When further increases to

45°, the maximum value of *~z decreases by about 50%. This observation suggests that

the effect of yaw angle on the coherent spanwise vortex is greater when > 30°. This

may be caused by the increase of the spanwise velocity *W as increases (Figure 6-4).

The *~x contours for T

* = 3.0 exhibits organized patterns at all yaw angles.

However, their strength are much weaker compared to those of *~z . At = 0°, the size

of the longitudinal vortices is much smaller than that of the spanwise vortices. The

maximum magnitude of *~x at this yaw angle is only about 7% of that of *~

z , which is

in agreement with the two-dimensionality of the flow. With the increase of yaw angles,

the *~x contours exhibit more apparent organised patterns and the maximum contour

value increases monotonously. This is consistent with that shown in Figure 6-7 (a, b) in

that the peak height of x increases with increasing . At = 45°, the maximum

contour value of *~x (Figure 6-9d) is about 67% of that of *~

z (Figure 6-9l). This result

indicates the existence of the secondary axial vortices or the occurrence of vortex

dislocation with an enhanced three-dimensionality when increases, consistent with the

result for a single cylinder wake (Zhou et al., 2010). Zhou et al. (2010) suggest that the

increase of streamwise vorticity is caused by the increase of the spanwise velocity w at

large yaw angles (see Figure 6-17i-l). This is because the x calculation is related with

the velocity component w where the increase of w contributes to the increase of *~x . The

maximum value of *~y at = 0° is about 2 times of that of *~

x , which is still relatively

smaller than that of *~z . In comparison between the *~

x and *~z contours for T

* = 3.0,

with the increase of , the strength of the former increases while it decreases in the

latter. The behaviour of the streamwise and spanwise vortex is consistent with the result

of single cylinder wake by Zhou et al. (2010). This result seems to support Mansy et al.

(1994), who stated that the increase of the streamwise vortices is at the expense of the

spanwise vortices. There is no apparent trend to be associated with the *~y contours

when increases from 0° to 45°. At = 45°, the maximum value of *~y is about half of

that of *~x . Overall, with the increase of , the magnitudes and the pattern of *~

x and

*~z contours tend to be comparable while those of *~

y are nearly unchanged. The

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maximum contour values of coherent vorticity at different yaw angles are summarized

in Table 6-2.

The coherent vorticity contours for T* = 1.7 cylinder spacing are shown in

Figure 6-10. All vorticity contours for this spacing obviously have distinctly different

vortex patterns from the aforementioned wakes for T* = ∞ and 3.0. Evidently, the T

* =

1.7 wake regime has a larger vortex wavelength compared to that for T* = ∞ and 3.0

wake regimes. It can be seen that a single vortex street was shed from the top cylinder in

the range from y* ≈ -1 to 3. The vortex structures in the wake are deflected downward

below the wake centreline creating a wide wake. The lower part of the wake regime y* <

-1 may not be so important as it is believed that the vortex structures shed from the

bottom cylinder are already vanished before x* = 10 (Zhou et al., 2002; Chen et al.,

2003). Zhang and Zhou (2001) in their study of three side-by-side cylinders (x* 10 and

T* = 1.5) observed a wide wake behind the central cylinder and two narrow wakes on

each side of the wide wake. They found that at x* ≈ 5, the vortex structures in the

narrow wake on each side of the wide wake vanished while the vortex structures in the

wide wake started to roll up. The vortex structures in the wide wake were very weak

initially but became stronger with the streamwise direction which can be related to the

shear layer instability (Zhou et al., 2002). The formation of the vortex street in the wide

wake at large downstream region is also illustrated and briefly discussed by Chen et

al.(2003) through their numerical simulations for T* = 1.7 at low Reynolds number (Re

= 750). They suggested the mechanisms involved in the vortex evolution of the wide

and narrow wakes across the downstream region. Initially, the flow was deflected

upwards creating a narrow wake around the top cylinder. The vortices in the narrow

wake tend to pair and absorb the vortex from the bottom cylinder. Due to strong

interaction between both vortices, the vortex shed from the bottom cylinder collapsed,

thus encouraged the growth of the vortex from the top cylinder. As the vortex from the

narrow wake (top cylinder) tend to prevent the merging of the following vortex from the

wide wake (bottom cylinder), another vortex was created behind the bottom cylinder

(now is called as a narrow wake). However, the vortex also quickly collapsed. At the

same time, the vortex from the top cylinder become stronger and the flow deflected

toward the bottom cylinder creating a wide wake. Finally, a single vortex street was

formed in the wide wake at downstream region i.e. x* = 10, which is consistent with the

present results as shown in the Figure 6-10. These mechanisms also explained the bi-

stable deflected and flip-flopped flows with randomly changes from one side to the

147

other as observed by Ishigai et al. (1972), Bearman and Wadcock (1973) and Kim and

Durbin (1988). Further observation by Zhou et al. (2002) for two side-by-side cylinders

for T* = 1.5 at x

* = 10 shows that a row of weak vortices and another peculiar flow

pattern are apparent. Such a peculiar flow pattern is diminished after x* = 20, suggesting

that the vortex regeneration or evolution is probably completed (Zhou et al., 2002). The

two-cylinder case for T* = 1.7 may bear a resemblance to that for T

* = 1.5 as they are in

the same regime of the intermediate cylinder spacing (T* = 1.2-2.0).

While the vorticity contours for T* = 1.7 have a relatively more organised vortex

pattern at = 0°, it becomes scattered with the increase of the yaw angle. The

maximum value for *~z contours is 0.14, 0.08, 0.08 and 0.06 at = 0°, 15°, 30° and

45°, respectively. The values reveal approximately a decreasing trend as is increased

especially when = 45°. The unorganized vortex pattern of *~z contours may suggest

that the vortex in the wake region of x* = 10 is still not stable. This may support the

flow visualization results by Williamson (1985) behind two side-by-side cylinders for

T* = 1.5, who showed that the vortex regeneration or evolution may not complete yet at

x* = 10. These is also another speculation that the vortex in the narrow wake have a

coalescence with the vortices in the wide wake (Zhou et al., 2002). In comparison

between *~x , *~

y and *~z contours, it is obvious that the magnitudes of all coherent

vorticity components at each yaw angle are comparable (ranging from 0.06 to 0.12),

indicating a more three-dimensional characteristics of the flow. This result is consistent

with that shown in Figure 6-4. With increasing , the maximum value of *~y and *~

z

contours tend to be comparable especially when = 30° and 45°. This is consistent with

the results shown in Figure 6-10 (b, d) that with the increase of to 45°, the velocity

spectra of transverse and spanwise components are almost comparable. This confirms

that the transverse and spanwise vorticities are likely to have similar magnitudes and

strength when is increased to large yaw angles. When = 45°, the maximum value of

*~x , *~

y and *~z contours is similar (≈ 0.06). This shows that at large yaw angle, the

vortices have perfectly three-dimensional vortices as the strength of vortices for all

components is equal to each other. This result should also be related to the large values

of W for T* = 1.7. While there is an apparent dependence of *~

z on , which decays by

50% when is increased from 0° to 45°, the increase of *~x with for T

* = 1.7 is not as

apparent as that for T* = 3.0. This result indicate a strong interaction between the wake

148

structures when T* = 1.7. It is also evident that vortex contours at = 45° has a phase

variation of about = 0.5π compared with those at < 40°. The maximum contour

values of *~x , *~

y and *~z at different yaw angles for various cylinder spacings are

summarized in Table 6-2.

6.3.4 Spanwise vortex patterns using the phase-averaged velocity components

Figures 6-11 and 6-12 present the iso-contours of phase-averaged vorticity *~z

constructed from the phase-averaged *~u and *~v at various yaw angles for T* = 3.0 and

T* = 1.7, respectively. The corresponding sectional streamlines for T

* = 3.0 and T

* = 1.7

are shown in Figures 6-13 and 6-14, respectively.

For T* = 3.0, two distinct vortex streets are evident in Figure 6-11 at all yaw

angles (only one street at y* ≥ 0 are shown here). The contour pattern at = 0° and its

corresponding sectional streamlines (Figures 6-10a, 6-12a) have a good agreement with

those shown by Zhou et al. (2002). With increasing to 45°, the maximum contours for

*~z decreases by about 38% from that at = 0°. At = 45°, the contours seem to be

deflected and the vortex centre of the contours shifts to the right by about = 0.2π. The

maximum magnitudes of the *~z contours in Figure 6-11 agree quite well with those

shown in Figure 6-9. Although both results are from different methods, where the

phase-averaged *~z of the former is from the phase-averaged *~u and *~v while the latter

is from the instantaneous z signal, they provide quite similar results. The difference

between these methods is the streamwise separation x as presented in Eq. (6-3), where

the former method did not measure the x as the latter method.

For T* = 1.7, a single vortex street is displayed at all yaw angles (Figure 6-12)

confirming the earlier suggestion. The contour pattern at = 0° and its corresponding

sectional streamlines (Figures 6-12a, 6-14a) is consistent with that shown by Zhou et al.

(2002) for T* = 1.5. This is expected as both results are in the intermediate spacing

regime. Even though T* = 1.7 wake has a single street as that for T

* = ∞, the former

exhibit a largely different vortex pattern from that of the latter. At y* ≥ 0, only the

negative contours are shown while only the positive contours are apparent at y* < 0.

With the increase of from 0° to 45°, the maximum magnitude of the *~z contours

decreases gradually to about 56% of that at = 0°. In comparison of the *~z contours in

Figures 6-10 and 6-12, they display distinctly different contour patterns. The maximum

149

contours magnitudes are also largely different where the maximum contours magnitudes

for the *~z contours in Figure 6-10 are about 25-39% smaller than those in Figure 6-12.

As discussed before, the difference between both method are the non-existence

streamwise separation x variable in the latter calculation. This shows that the x value

may have influence to the results.

6.3.5 Incoherent vorticity fields

The phase-averaged incoherent vorticity contours *2 xr , *2 yr and

*2 zr at different yaw angles for T* = 3.0 and T

* = 1.7 are shown in Figures 6-15

and 6-16, respectively. It is evident that for T* = 3.0, the maximum contour values of all

incoherent vorticity components decrease gradually with the increase of . If the

*2 xr , *2 yr and *2 zr contours for T* = 3.0 (Figure 6-15) are shifted

downward by about y* =1.5, the contours display qualitatively the same contour patterns

as those for T* = ∞ (see Zhou et al. 2010), suggesting that the wake for large cylinder

spacing i.e. T* = 3.0 behaves as an independent and isolated cylinder wake. The

*2 xr contours for T* = 3.0 running along the diverging separatrix and wrapping

around the main spanwise structures. These contours may represent the existence of the

streamwise or riblike structures as have been suggested by Zhou et al. (2010). The

maximum magnitudes of *2 xr , *2 yr and *2 zr contours for T* = 3.0

decrease gradually with the increase of .

The incoherent vorticity contours *2 xr , *2 yr and *2 zr for T* = 1.7

at all yaw angles (Figure 6-16) do not show very apparent organized contour patterns

which are in contrast to those for T* = ∞ and 3.0. The contour patterns for the former are

scattered and the maximum magnitude of the contours become smaller with the increase

of .

6.3.6 Phase-averaged velocity components

The phase-averaged coherent velocities *~u , *~v and *~w for T* = 3.0 and 1.7 at

different yaw angles are shown in Figures 6-17 and 6-18, respectively. The velocity

contours for T* = 3.0 (Figure 6-17) show remarkably organized patterns. The maximum

contour value of *~v is the largest among all the three velocity components for all yaw

angles. While the *~v contours (Figure 6-17e-h) show apparent antisymmetry about =

150

0, the *~u contours (Figure 6-17a-d) display up-down antisymmetry about the vortex

centre (the bottom half is not shown). Both *~u and *~v at = 0° for T* = 3.0 exhibit

qualitatively the similar structures as those shown in Zhou et al. (2002). If the velocity

contours for T* = 3.0 (Figure 6-17) are shifted downward by about y

* =1.5, the contours

display qualitatively the same structures as those for T* = ∞ (see Zhou et al. 2010) albeit

with slightly different contour values. This further confirms that the wake of the two

side-by-side cylinders for large cylinder spacing i.e. T* = 3.0 behaves as an independent

and isolated cylinder (Sumner et al., 1999). The *~u and *~v contours at = 0° to 30° for

T* = 3.0 seem to have a quite similar maximum value for both positive and negative

contours. With the increase of to 45°, the maximum value of *~u and *~v contours

decreases by about 50% and 20%, respectively, as compared with that of = 0°. This

suggests that *~u is more dependent to yaw angles. At = 0°, the maximum value of *~w

contours is the smallest compare to that of *~u and *~v contours. The *~w contours

display up-down antisymmetry about the wake centreline and have one loop of vortex

structure. With the increase of to 30°, the vortex structures split and eventually form

two loops of vortex structures. The maximum value of *~w also increases gradually with

the increase of to 30°. When = 45°, the maximum value of *~w contours is about 2

times of that at = 0° and is comparable to that of *~u . The enhanced *~w with the

increase of yaw angles is related to a higher degree of three-dimensionality of the wake.

The *~u , *~v and *~w phase-averaged contours at various yaw angles for T* = 1.7

are shown in Figure 6-17. The *~u and *~v contours at = 0° for T* = 1.7 are

comparable with those for T* = 1.5 shown in Zhou et al. (2002). The comparable

patterns are expected because they are in the similar intermediate cylinder spacing (T* =

1.2-2.0) regime. It is evident that the *~u , *~v and *~w contours display discernible

structures at = 0°. However, with the increase of to 15°, the *~u , *~v and *~w vortex

contours become less organized and there is a large reduction by about 38-71% in the

contours maximum value. Although at = 45°, the maximum value of *~u , *~v and *~w

is about 38%, 22% and 50% of that at = 0°, there is minor decreasing trend in the

dependence of *~u , *~v and *~w on . For example, the maximum value of *~v contours

varies from 0.09 when = 0° to 0.05, 0.06 and 0.02 when = 15°, 30° and 45°,

respectively. The inconsistent of decreasing in the maximum values of vortex contours

151

may suggest that the vortex in the wake region of x* = 10 is unstable. When increases

from 0° to 45°, all velocities contours for T* = 1.7 have a large decrease in the

maximum values with *~v has the largest reduction compared with *~u and *~w . This

result is in contrast to that of T* = ∞ and 3.0, where more organised and enhanced

spanwise velocity contours are found. The velocity contours at = 45° has a variation

in phase of about 0.5π compared with other yaw angles. This observation is in contrast

with that for T* = 3.0 and ∞, where for both spacings, there is no apparent phase

variation between yaw angles. Also, large for both spacings contributes to smaller

maximum values in the *~u and *~v contours with *~u being more affected than *~v .

Besides that, the *~w contours for T* = 3.0 and ∞ have larger magnitude with the

increase of , suggesting higher three-dimensional vortex in the wake. This result

suggests that T* = 1.7 wake may have a totally different mechanism from that for T

* =

3.0 and ∞. The maximum levels of *~u , *~v and *~w contours for T* = ∞, 3.0 and 1.7 are

summarized in Table 6-2.

6.3.7 Phase-averaged Reynolds shear stresses

The phase averaged Reynolds shear stresses **~~ vu , ** ~~ wu and ** ~~ wv at different

yaw angles are shown in Figures 6-19 and 6-20 for T* = 3.0 and 1.7, respectively. The

**~~ vu , ** ~~ wu and ** ~~ wv contours for T* = 3.0 (in the range of y

* > 1.5) display similar

patterns with those for T* = ∞ (in the range of y

* > 0) as shown in Zhou et al. (2002) and

Zhou et al. (2010). The results show that the flow for T* = 3.0 at x

* = 10 behaves like

two distinct wakes with each one behaving wake characteristics similar to the wake of a

single cylinder. The **~~ vu contours display a clover-leaf pattern about the vortex centre

especially for = 0°-30°, which is led from the coherent motion of vortices with Uc as a

reference frame (Zhou et al., 2002). The contours of **~~ vu (Figure 6-19a-d) are

antisymmetry about = 0 and y* = 1.5. The maximum value of **~~ vu contours does not

change when is increased from 0° to 15°. However, when is further increased to 30°

and 45°, the maximum value decreases to about 22% and 67%, respectively, from that

when = 0°. When = 45°, the wake region shrinks and the contour is further declined

to the right. As **~~ vu is related to the *~

z , this result confirm the dependence of the

latter on where the spanwise vortices is weakened with the increase of . The

negative contours of ** ~~ wu at = 0° (Figure 6-19e) display organized patterns while

152

those of the positive contours exhibit a small clover-leaf pattern. The maximum contour

magnitudes of ** ~~ wu are considerably smaller than that of **~~ vu and ** ~~ wv . Apparently,

with the increase of , the positive contours of ** ~~ wu become smaller and almost

diminish. At = 45°, the positive ** ~~ wu contours have a reduction in their maximum

value by about 75% than that when = 0°, while the maximum value of the negative

** ~~ wu contours is increased by about 2.5 times than that when = 0°. The positive

contours of ** ~~ wv reveal a large clover-leaf pattern which is antisymmetrical about =

0 especially when = 0°-30°. The negative ** ~~ wv contours at = 0° have very small

magnitude, thus are not shown in the figure. However, with the increase of , the

negative contours appear slowly and grow larger while the positive contours shrinks

gradually leading to contours distortion when = 45°. The maximum magnitude of

** ~~ wv contours decreases from 0.006 to 0.004 with the increase of from 0° to 45°.

For T* = 1.7, the Reynolds shear stresses **~~ vu , ** ~~ wu and ** ~~ wv (Figure 6-20)

reveal a quite different pattern with those for T* = 3.0 and ∞. At = 0°, large **~~ vu ,

** ~~ wu and ** ~~ wv contours are evident where the contours are quite antisymmetry about

= 0. The magnitude of the maximum contours for the above terms are much smaller

than that for T* = 3.0. However, with increasing , the contours become more

asymmetrical about = 0. Although the **~~ vu , ** ~~ wu and ** ~~ wv contours at = 45° has

a large decrease in their maximum values by about 75-93% compared with those at =

0°, there is no identifiable trend of contour patterns and the maximum contour values

when changes. A phase variation of about 0.5π also is evident in the shear stress

contours when increases to 45°. This supports the previous results in Figures 6-10 and

6-18 that at x* = 10, the vortical motion is still in unstable mode. The vortex

regeneration or evolution process may contribute to the unstable wake. The maximum

magnitudes of **~~ vu , ** ~~ wu and ** ~~ wv contours are summarized in Table 6-2.

6.3.8 Incoherent Reynolds shear stresses

The phase-averaged incoherent Reynolds shear stresses **~~

rr vu , ** ~~

rr wu

and ** ~~

rr wv contours for T* = 3.0 and 1.7 are shown in Figures 6-21 and 6-22,

respectively. A weak vortical motion (i.e. a small value of **~~ vu ) is corresponding to a

strong incoherent motion (i.e. a large value of **~~

rr vu ) (Zhou et al., 2002). For T* =

153

3.0, the **~~

rr vu contour pattern at = 0° of this study (Figure 6-21a) is comparable

with that shown in Zhou et al. (2002). The extremum values of **~~

rr vu for the

former are larger than those for the latter. This is because of the Re of the former being

larger than that of the latter. The **~~

rr vu contours seem to stretch in the direction of

the diverging separatrix. With the increase of from 0° to 45°, the extremum value of

**~~

rr vu decreases gradually by about 58% of that when = 0°. At = 45°, the

positive contours of **~~

rr vu seem to dominate the region of y* < 1.5. While the

maximum values of ** ~~

rr wv contours for T* = 3.0 increase with increasing , there

is no trend to be associated with those of ** ~~

rr wu . The appearance of the negative

contours of ** ~~

rr wv when is increased from 0° to 30° indicates the generation of

the secondary spanwise structures, which become more apparent as is increased to

45°.

The strength of the incoherent motion for T* = 1.7 can be interpreted from the

**~~

rr vu , ** ~~

rr wu and ** ~~

rr wv contours as shown in Figure 6-22. Generally

the maximum values of **~~

rr vu and ** ~~

rr wu contours decrease with increasing

. In comparison of the **~~

rr vu contours for T* = 1.7 with that for T

* = 1.5 in Zhou

et al. (2002), the **~~

rr vu contours for the former have two sides about the wake

centreline where the upper side (y* > 0) is dominated by the negative contours while the

below side (y* 0) is dominated by the positive contours. The

**~~rr vu contours for

the latter at y* 0 is dominated by both the positive and negative contours.

6.3.9 Coherent and incoherent contributions to Reynolds stresses

By using structural average, the coherent and incoherent contributions to the

Reynolds normal and shear stresses could be quantified. The coherent and incoherent

contributions to Reynolds stresses for different yaw angles are shown in Figures 6-23

and 6-24, for T* = 3.0 and 1.7, respectively.

All distributions of the Reynolds stress profiles for T* = 3.0 at all yaw angles are

either symmetrical or antisymmetrical about wake centreline y* = 0. The distribution of

the Reynolds stresses for T* = 3.0 at = 0° agrees qualitatively with those of Zhou et al.

(2002), thus validating the present measurement. The coherent contribution to the

154

Reynolds stresses for T* = 3.0 is generally smaller than that for T

* = ∞ as shown in

Zhou et al. (2010). This may be because for T* = 3.0, the interaction between the two

vortex streets in the wake accelerate the vortex decay process (Zhou et al., 2002).

The profiles of the Reynolds normal stress 2*u for T* = 3.0 (Figure 6-23a-d)

display one peak while the coherent component 2*~u exhibits two peaks around y* ≈

0.75 and 2.25. Due to the symmetry of the flow, another two peaks are expected to be

located at y* ≈ -0.75 and -2.25. The existence of the four peaks on the 2*~u distribution

is because of the occurrence of the two vortex streets in the wake for T* = 3.0 (Zhou et

al., 2002). The figure shows that the coherent contribution to 2*u decreases slightly

with the increase of . This trend is consistent with the dependence of *~u on yaw

angles (Figure 6-17a-d) which is also related to *~z (Figure 6-9i-l). The coherent

contribution to 2*v for T* = 3.0 is quite significant for all yaw angles, indicating a

strong coherent motion in the transverse direction of the wake. It is apparent that

2*~v / 2*v is the largest and followed by 2*~u / 2*u and 2*~w / 2*w , regardless of the yaw

angles. This is expected as the v component is more sensitive to the organized structures

than u and w components. The magnitudes of 2*v (Figure 6-23e-h) are quite similar for

= 0°-30° even though they decrease slightly with the increase of , and becomes

appreciably smaller when increases to 45°. The coherent contribution to 2*w (Figure

6-23i-l) is very small, seems to be zero when < 40°. There is a slight increase in the

magnitude of the coherent component 2*~w when increases from 30° to 45°, which

peaks at around y* = 0.75 and 2.25, respectively. This result indicates the generation of

the secondary streamwise vortex structures in the yawed cylinder wake when is large.

The Reynolds shear stresses for T* = 3.0 (Figure 6-23m-x) are very small in

magnitude relative to the Reynolds normal stresses. The coherent component of **vu

has remarkable magnitude compared to that of incoherent component of **vu for y*

2, while the **~~ vu magnitude is approximately zero for y* > 2 at all yaw angles. This

observation indicates the existence of the strong spanwise coherent structures at y* 2,

in view of the association of *~u and *~v with *~

z . The ** ~~ wu magnitude has negative

sign for all yaw angles. The negative sign is related to the opposite signs of coherent

155

velocity components *~u and *~w which is shown in Figure 6-17 (a-d and i-l). The

magnitude of ** ~~ wu becomes larger with increasing , suggesting comparable

magnitudes of coherent velocity components *~u and *~w especially when = 45° and

the generation of the secondary streamwise vortical structures. The coherent

contribution to **wv decreases with increasing , suggesting a reduction of the

spanwise vorticity which cannot be compensated by the increase in the streamwise

vorticity.

The coherent contributions to Reynolds normal stresses for T* = 1.7 (Figure

6-24a-l) are approximately zero at all yaw angles. From the figure, it is evident that the

distribution of 2*~u does not present any peak which is contradict with that of T* = 3.0

and ∞. With the increase of , the time-averaged 2*u decreases gradually. The coherent

contributions to 2*v and 2*w are the largest when = 0° compared with that of larger

. The inconsistent trend in the magnitude of the time-averaged components 2*v and

2*w for = 15° to 45° suggests the instability of the wake structures at those . This

result is consistent with that in Figure 6-18, where the velocities contours are quite

organized when = 0° and become less organized with increasing .

The Reynolds shear stress profiles for T* = 1.7 (Figure 6-24m-x) shows that the

coherent contribution components **~~ vu , ** ~~ wu and ** ~~ wv are negligibly small for all

. However, the coherent contributions to the Reynolds shear stress are the largest when

= 0° which are consistent with those for Reynolds normal stress. The time-averaged

uw has negative sign in its magnitudes when = 0° while the uw magnitude increases

and shifts to the positive side at y* 0 with the increasing of to 15° and at y

* 1 when

= 30° and 45°. There is no absolute trend to be associated with the dependence of

**~~ vu and ** ~~ wv on yaw angles. These results seem to suggest the destructing effect by

the secondary spanwise flow on the wake structures when the gap between the two

cylinders is in the intermediate regime.

6.3.10 Coherent and incoherent contributions to vorticity variances

The structural average is done to measure the coherent contribution to the

vorticity variances for T* = 3.0 and 1.7. The results are shown in Figure 6-25. It is worth

156

to mention that the coherent contributions to vorticity variances 22

/~xx ,

22/~

yy

and 22

/~zz for T

* = 3.0 are qualitatively similar with that for T

* = ∞ by Zhou et al.

(2010). These are expected as the wake behind two cylinders for T* = 3.0 has two

distinct vortex streets with every street behaves like a vortex street found in a single

cylinder wake. Although each street for T* = 3.0 has the same qualitative trend to the

single cylinder street, the result shows that the maximum magnitudes of 22

/~xx ,

22/~

yy and 22

/~zz for the former are smaller, about 60%, 64% and 48%,

respectively, compare to those for the latter. The magnitude of 22

/~xx at = 0° for

T* = 3.0 is very small which is only about 0.2% and have a broad distribution along

lateral distance y*. However, with the increase of , the coherent contribution to the

longitudinal vorticity increases rapidly and exhibits a significant peak especially when

= 45°. The distribution also shows that the longitudinal vortex is largely occur at the

range of y* = 0.8-3.0 especially at = 15° to 45° which is consistent with that shown in

Figure 6-9 (a-d). The maximum magnitude of 22

/~xx at = 45° is equal to 3%

which is smaller than that for T* = ∞ (5.3%) by Zhou et al. (2010). The coherent

contribution to the lateral vorticity variance 22

/~yy is quite similar and very small

(less than 1%) for all yaw angles. This suggests that the transverse vorticity 22

/~yy

does not depend on the variation of yaw angles. With the increase of from 0° to 30°,

the coherent contribution to spanwise vorticity 22

/~zz decreases gradually. With

further increase of to 45°, the magnitude of 22

/~zz decreases significantly by about

60% of that when = 30°. These results confirm that the spanwise vortex is weaker

when > 30° while the lateral vortex is independent on .

Despite of the expected results that found in the coherent contributions to

vorticity variances 22

/~xx ,

22/~

yy and 22

/~zz for T

* = 3.0, the results of those

for T* = 1.7 are very different from those for T

* = 3.0 and ∞. The results show only the

wide wake part (y* ≥ -1). Generally,

22/~

xx , 22

/~yy and

22/~

zz have

appreciably small magnitudes for all yaw angles which are less than 1%. The maximum

157

magnitudes of 22

/~xx and

22/~

zz in the range of = 0° to 45° for T* = 1.7 are

quite small, which are only about 30% and 7% relative to those for T* = 3.0 while there

is no large differences in the magnitudes of 22

/~yy for both cylinder separations.

These results indicate that the strength of coherent vorticity of the x and z components

for two cylinders arranged side-by-side with T* = 1.7 are weaker relative to those with

T* = 3.0 and ∞. There is no significant trend that can be related to the dependence of

22/~

xx and 22

/~yy on . Thus, the coherent contributions to the streamwise and

lateral vorticity variances for T* = 1.7 are not dependent on . While for

22/~

zz ,

there is a sudden decrease in the magnitude when increases from 0° to larger yaw

angles. This result is consistent with that shown in Figure 6-9 (i-l), where the spanwise

vortices have the most organized structures and the highest in contour magnitude when

= 0°. With increasing , the spanwise vortices structures are become less organized

and weakened.

6.4 Conclusions

Experiments of two cylinders in side-by-side arrangement with centre-to-centre

cylinder spacing of T* = 3.0 and 1.7 at yaw angles = 0°, 15°, 30° and 45° have been

conducted in order to evaluate the wake behind the cylinders. It has been found that the

characteristics of the turbulent wake depend on the cylinder spacing.

For the large cylinder spacing T* = 3.0, a single peak is observed in the v which

corresponds to the Strouhal number StN. It is shown that for < 40°, the independence

principle is applicable to the wake. The velocity profiles suggest that both vortex streets

of the T* = 3.0 wake is similar to the single vortex street of the T

* = ∞ wake. The lateral

distance between both vortices for the T* = 3.0 wake is about 3d. The phase-averaged

velocity and vorticity contours for T* = 3.0 are comparable with those for T

* = ∞ when

the whole set of the contours is shifted downward about y* = 1.5. This result confirms

that the wake of the two side-by-side cylinders for large cylinder spacing i.e. T* = 3.0

behaves as an independent and isolated cylinder. The coherent streamwise vorticity

contours *~x for T

* = 3.0 is only about 10% of that of coherent spanwise vorticity

contours *~z . With the increase of ,

*~x increases while *~

z decreases. At = 45°, *~x

158

is about 67% of *~z . This result indicates the existence of the secondary axial vortices or

the occurrence of the vortex dislocation with an enhanced three-dimensionality with

larger . Although the wake behind two cylinders for T* = 3.0 has two distinct vortex

street and each street has the same qualitative trend to the single cylinder street, the

results show that the maximum magnitudes for 22

/~xx ,

22/~

yy and 22

/~zz for

the former are smaller which are about 60%, 64% and 48%, respectively, than those for

the T* = ∞ wake. The *~

z strength is weaker while the *~x strength increases rapidly

with increasing , especially when > 30°. On the other hand, the *~y magnitude is

independent on .

For the intermediate cylinder spacing T* = 1.7, only a single peak is detected in

the v. This is because at x* = 10, the vortex structures regeneration or evolution may

not be completed yet, whereas those in the narrow wake is probably diminished before

the downstream location. The independence principle (IP) is applicable to the wake

when < 40°. The values of spanwise velocity *W are not equal to zero, even at =

0°. It may suggest that the vortex in the wake is inclined and the wake may not have a

perfect two-dimensional plane, which may be because of the occurrence of the vortex

evolution. The vorticity contours for T* = 1.7 have a more organized pattern at = 0°

while become scattered and smaller with the increase of yaw angles. With increasing ,

there is a decreasing trend in the maximum value of vorticity contours. The less

organized *~z contours for T

* = 1.7 compared with those for T

* = ∞ and 3.0 indicate

that the vortex motion in the wake is still not stable, suggesting that the vortex evolution

or regeneration process may not be completed yet. At = 45°, the vortices show

apparent three-dimensionality as the strength of all vortices are same to each other.

Furthermore, the vorticity contours at = 45° have phase variation of about 0.2π

compared with that at smaller yaw angles. The maximum coherent vorticity *~x and *~

z

contours for T* = 1.7 are about 30% and 7% of those for T

* = 3.0 while there is no

apparent difference in the magnitudes of 22

/~yy for both T

*. These results suggest

that the coherent vorticity components for T* = 1.7 are mush weaker than both of T

* =

3.0 and single cylinder.

159

Table 6-1. The vortex shedding angle, maximum velocity defect, convection velocity,

half-width, normalized Strouhal number and wavelength at different yaw angles for

cylinder spacings T* = ∞, 3.0 and 1.7.

T*

0° 15° 30° 45°

(°)

∞ 0.5 16.8 33.0 45.6

3.0 4.6 19.4 37.8 52.2

1.7 3.9 12.6 25.8 42.9

U0*

∞ 0.17 0.14 0.10 0.07

3.0 0.20 0.19 0.13 0.10

1.7 0.39 0.37 0.29 0.20

Uc*

∞ 0.82 0.83 0.87 0.89

3.0 0.91 0.88 0.90 0.95

1.7 0.86 0.76 0.83 0.86

L*

∞ 0.80 0.82 0.80 0.68

3.0 2.57 2.53 2.53 2.69

1.7 1.38 1.43 1.39 1.39

StN

∞ 0.195 0.202 0.206 0.224

3.0 0.205 0.212 0.220 0.246

1.7 0.118 0.122 0.119 0.133

*

∞ 4.16 4.23 4.80 5.86

3.0 4.45 4.30 4.54 5.44

1.7 7.29 6.44 8.05 9.14

* (cos )

∞ 4.16 4.09 4.16 4.14

3.0 4.45 4.15 3.93 3.85

1.7 7.29 6.22 6.97 6.46

160

Table 6-2. The maximum values of the coherent vorticity, velocity and Reynolds stresses contours at different

yaw angles for T* = ∞, 3.0 and 1.7.

T* ∞ 3.0 1.7

0° 15° 30° 45° 0° 15° 30° 45° 0° 15° 30° 45°

*~x 0.15 0.15 0.25 0.25 0.05 0.15 0.20 0.20 0.12 0.06 0.12 0.06

*~y 0.12 0.10 0.10 0.08 0.10 0.10 0.12 0.08 0.08 0.04 0.08 0.06

*~z 0.8 0.8 0.6 0.4 0.70 0.60 0.60 0.30 0.14 0.08 0.08 0.06

*~u 0.12 0.14 0.10 0.08 0.12 0.12 0.10 0.06 0.065 0.040 0.045 0.025

*~v 0.28 0.28 0.26 0.24 -0.24 0.20 0.20 0.16 0.09 0.05 0.06 0.02

*~w 0.04 0.04 0.06 0.08 0.02 0.03 0.04 0.04 0.030 0.010 0.025 0.015

**~~ vu 0.018 0.018 0.012 0.009 0.018 0.018 0.014 0.006 0.0036 0.0009 0.0020 0.0004

** ~~ wu 0.005 0.006 0.007 0.009 0.0020 0.0030 0.0045 0.0030 0.0012 0.0003 0.0007 0.0003

** ~~ wv 0.004 0.004 0.006 0.004 0.006 0.007 0.006 0.004 0.0028 0.0004 0.0014 0.0002

161

(a) Cylinders arrangement and

coordinate system (top view)


8D

A

z

y

y

1

6

B

3

5

C

z

4 7

2

8D

A

z

y

y

1

6

B

3

5

C

z

4 7

2

U

4,7

C

B, D

y

3,8

1

2

6

5

X

A

U

4,7

C

B, D

y

3,8

1

2

6

5

X

A



T Moveable 3-D

Vorticity Probe


U

U0T Moveable 3-D

Vorticity Probe


UU

U0

U

= 0

= 0

Tunnel wall

Tunnel wall

End plate

45z

x

Yaw angle

U

= 0

= 0

Tunnel wall

Tunnel wall

End plate

45z

x

Yaw angle

Figure 6-1. Two cylinders side-by-side arrangement with definitions of the coordinate

system and the sketches of the vorticity probe.

162

Figure 6-2. The experimental setup of two cylinders with side-by-side arrangement.

Cylinders

End plate

Fixed reference X-probe

Moveable 3-D vorticity probe

Pitot tube

163

= 45

= 30

= 15

= 0

tA,i

tB,i

tA,i+1

= 0, , 2

t

v

v

v

v

(a) T* = 3.0

t

v

v

v

v

= 15

= 30

= 45

= 0

(b) T* = 1.7

tA,i

tB,i

tA,i+1

= 0, , 2

Figure 6-3. Time traces of v signals measured at y* = 0.5 for different angles in arbitrary

scales. (a) T* = 3.0; (b) T

* = 1.7. The thicker line represents the filtered signal vf .

164

T* = 3.0 T

* = 1.7

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4

= 0153045

(a)

y/d

U*

0.6

0.7

0.8

0.9

1.0

-1 0 1 2 3 4

(b)

y/d

U*

-0.05

0

0.05

0.10

0.15

0 1 2 3 4

(c)

y*

W*

-0.05

0

0.05

0.10

0.15

-1 0 1 2 3 4

(d)

y*

W*

Figure 6-4.Comparisons of time-averaged velocities at different yaw angles for T

* = 3.0

and 1.7. (a, b) *U ; (c, d) *W . The open circle with a cross at the centre

x represents

cylinder.

-0.2

-0.1

0

0.1

0.2

0.3

0 1 2 3 4

= 0153045

U*/y*

W*/y*

(a) T* = 3.0

y/d

U

*/y*

, W

*/y*

-0.2

-0.1

0

0.1

0.2

0.3

-1 0 1 2 3 4

U*/y*

W*/y*

(b) T* = 1.7

y/d

U

*/y*

, W

*/y*

Figure 6-5. Normalized velocity gradients at different yaw angles for (a) T

* = 3.0; (b)

1.7. The arrows indicate the direction of increasing yaw angles. The open circle with a

cross at the centre

x represents cylinder

x

x

x

x

x

x

165

T* = 3.0 T

* = 1.7

-50

-40

-30

-20

-10

0

0.001 0.01 0.1 1 10

u

v

w

(a)

fN*

u,

v,

w (

dB

)

-50

-40

-30

-20

-10

0

0.001 0.01 0.1 1 10

(c)

fN*

u,

v,

w (

dB

)

-50

-40

-30

-20

-10

0

0.001 0.01 0.1 1 10

(b)

fN

u,

v,

w (

dB

)

-50

-40

-30

-20

-10

0

0.001 0.01 0.1 1 10

(d)

fN

u,

v,

w (

dB

)

Figure 6-6. The power spectra of u, v and w components for different cylinder spacings

at (a, c): = 0°; (b, d): 45°.

T* = 3.0 T

* = 1.7

-50

-40

-30

-20

-10

0

0.001 0.01 0.1 1 10

x

y

z

(a)

fN*

x

,

y

,

z

(dB

)

-50

-40

-30

-20

-10

0

0.001 0.01 0.1 1 10

(c)

fN*

x

,

y

,

z

(dB

)

-50

-40

-30

-20

-10

0

0.001 0.01 0.1 1 10

(b)

fN

x

,

y

,

z

(dB

)

-50

-40

-30

-20

-10

0

0.001 0.01 0.1 1 10

(d)

fN

x

,

y

,

z

(dB

)

Figure 6-7. The power spectra of x, y, z components for different cylinder spacings

at (a, c): = 0°; (b, d): 45°.

166

0.8

0.9

1.0

1.1

1.2

1.3

1.4

0 10 20 30 40 50

T* = 3.0T* = 1.7

-6%

+6%

(degree)

St N

/St 0

Figure 6-8. Strouhal number StN for different cylinder spacings. The horizontal short

dashed lines represent the range of experimental uncertainty.

.

167

Figure 6-9. Phase-averaged vorticity components at different yaw angles for T* = 3.0. (a-d) *~

x ; (e-h) *~y ; (i-l) *~

z . (a-d) Contour interval =

0.05; (e-h) 0.02; (i-l) 0.10. = 2π corresponds to the (≡ Uc/f0). The open circle with a cross at the centre

x represents cylinder.

= 0° = 15° = 30° = 45°

y*

Phase

y/d

-2-10120

1

2

3

4(a)-0.05 0.05

Frame 001 03 Nov 2010 Frame 001 03 Nov 2010

Phase

y/d

-2-10120

1

2

3

4(b)-0.15 0.10


Phase

y/d

-2-10120

1

2

3

4(c)0.15-0.20


Phase

y/d

-2-10120

1

2

3

4 (d)-0.20 0.20


y*

Phase

y/d

-2-10120

1

2

3

4(e)-0.100.08


Phase

y/d

-2-10120

1

2

3

4(f)0.10 -0.10


Phase

y/d

-2-10120

1

2

3

4(g)0.06 -0.12


Phase

y/d

-2-10120

1

2

3

4 (h)-0.080.06


y*

Phase

y/d

-2-10120

1

2

3

4(i)-0.700.50


Phase

y/d

-2-10120

1

2

3

4(j)0.50 -0.60


Phase

y/d

-2-10120

1

2

3

4(k)0.50 -0.60


Phase

y/d

-2-10120

1

2

3

4 (l)-0.300.30


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02

Frame 001 07 Jul 2010 Frame 001 07 Jul 2010

Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

168

Figure 6-10. Phase-averaged vorticity components at different yaw angles for T* = 1.7. (a-d) *~

x ; (e-h) *~y ; (i-l) *~

z . (a-l) Contour interval

= 0.02. = 2π corresponds to the /2 (≡ Uc/2f0). The open circle with a cross at the centre


= 0° = 15° = 30° = 45°

y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(a)

-0.08 0.12


Phase

y/d

-2-1012

-1

0

1

2

3

4(b)-0.06 0.06


Phase

y/d

-2-1012

-1

0

1

2

3

4(c)-0.12 0.12


Phase

y/d

-2-1012

-1

0

1

2

3

4(d)-0.06 0.06


y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(e)

0.08 -0.06


Phase

y/d

-2-1012

-1

0

1

2

3

4(f)0.04 -0.04


Phase

y/d

-2-1012

-1

0

1

2

3

4(g)0.08 -0.08


Phase

y/d

-2-1012

-1

0

1

2

3

4(h)0.06 -0.04


y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(i)-0.14 0.12


Phase

y/d

-2-1012

-1

0

1

2

3

4(j)-0.06 0.08


Phase

y/d

-2-1012

-1

0

1

2

3

4(k)-0.08 0.08


Phase

y/d

-2-1012

-1

0

1

2

3

4(l)-0.06 0.04


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

x

x

x

169

Figure 6-11. Coherent vorticity contours *~z from phase-averaged *~u and *~v at different yaw angles for T

* = 3.0. (a-d) Contour interval =

0.1. = 2π corresponds to the (≡ Uc/f0). The open circle with a cross at the centre


Figure 6-12. Coherent vorticity contours *~z from phase-averaged *~u and *~v at different yaw angles for T

* = 1.7. (a-d) Contour interval =

0.04. = 2π corresponds to the /2 (≡ Uc/2f0). The open circle with a cross at the centre


= 0° = 15° = 30° = 45° y*

Phase

y/d

-2-10120

1

2

3

4(a)

-0.8 0.5


Phase

y/d

-2-10120

1

2

3

4(b)-0.7 0.5


Phase

y/d

-2-10120

1

2

3

4(c)

0.4 -0.6


Phase

y/d

-2-10120

1

2

3

4 (d)-0.3 0.3


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

= 0° = 15° = 30° = 45°

y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(a)

-0.36 0.20

Frame 001 31 Oct 2010 Frame 001 31 Oct 2010

Phase

y/d

-2-1012

-1

0

1

2

3

4(b)

-0.28 0.24


Phase

y/d

-2-1012

-1

0

1

2

3

4(c)

-0.24 0.16


Phase

y/d

-2-1011.99999

-1

0

1

2

3

4(d)

-0.20 0.12


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

170

Figure 6-13. Phase-averaged sectional streamlines at different yaw angles for T*

= 3.0. = 2π corresponds to the (≡ Uc/f0). The open

circle with a cross at the centre


Figure 6-14. Phase-averaged sectional streamlines at different yaw angles for T* = 1.7. = 2π corresponds to the /2 (≡ Uc/2f0). The open

circle with a cross at the centre


= 0° = 15° = 30° = 45° y*

Phase

y/d

-2-10120

1

2

3

4(a)


Phase

y/d

-2-10120

1

2

3

4(b)


Phase

y/d

-2-10120

1

2

3

4(c)


Phase

y/d

-2-10120

1

2

3

4 (d)


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

= 0° = 15° = 30° = 45°

y*

x

y

-2-1012

-1

0

1

2

3

4

5

x

y

-2-1012

-1

0

1

2

3

4

5

x

y

-2-1012-1

0

1

2

3

4

5

x

y

-2-1012-1

0

1

2

3

4

5

2 1 0 -1 -2 2 1 0 -1 -2 2 1 0 -1 -2 2 1 0 -1 -2

0

1

2

3

4

-1

(a) (b) (c) (d)

Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

(a) (b) (c)

171

Figure 6-15. Phase-averaged incoherent vorticity contours at different yaw angles for T* = 3.0. (a-d) *2 xr ; (e-h) *2 yr ; (i-l)

*2 zr . (a-d) Contour interval = 0.1; (e-l) 0.2. = 2π corresponds to the (≡ Uc/f0). The open circle with a cross at the centre

x

represents cylinder.

= 0° = 15° = 30° = 45° y*

Phase

y/d

-2-10120

1

2

3

4(a)1.3


Phase

y/d

-2-10120

1

2

3

4(b)

1.3


Phase

y/d

-2-10120

1

2

3

4(c)1.1


Phase

y/d

-2-1012.000030

1

2

3

4 (d)0.80


y*

Phase

y/d

-2-10120

1

2

3

4(e)2.2


Phase

y/d

-2-10120

1

2

3

4(f)2.0


Phase

y/d

-2-10120

1

2

3

4(g)1.6


Phase

y/d

-2-10120

1

2

3

4 (h)1.2


y*

Phase

y/d

-2-10120

1

2

3

4(i)2.4


Phase

y/d

-2-10120

1

2

3

4(j)2.4


Phase

y/d

-2-10120

1

2

3

4(k)2.0


Phase

y/d

-2-10120

1

2

3

4 (l)1.4


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

172

Figure 6-16. Phase-averaged incoherent vorticity contours at different yaw angles for T* = 1.7. (a-d) *2 xr ; (e-h) *2 yr ; (i-l)

*2 zr . (a-l) Contour interval = 0.1. = 2π corresponds to the /2 (≡ Uc/2f0). The open circle with a cross at the centre

x represents

cylinder.

= 0° = 15° = 30° = 45°

y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(a)0.9


Phase

y/d

-2-1012

-1

0

1

2

3

4(b)

0.8


Phase

y/d

-2-1012

-1

0

1

2

3

4(c)

0.9


Phase

y/d

-2-1012

-1

0

1

2

3

4(d)

0.8


y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(e)

1.4


Phase

y/d

-2-1012

-1

0

1

2

3

4(f)

1.4


Phase

y/d

-2-1012

-1

0

1

2

3

4(g)

1.2


Phase

y/d

-2-1012

-1

0

1

2

3

4(h)

1.1


y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(i)

1.3


Phase

y/d

-2-1012

-1

0

1

2

3

4(j)

1.3


Phase

y/d

-2-1012

-1

0

1

2

3

4(k)1.2


Phase

y/d

-2-1012

-1

0

1

2

3

4(l)

1.1


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

x

x

x

173

Figure 6-17. Phase-averaged velocities components at different yaw angles for T* = 3.0. (a-d) *~u ; (e-h) *~v ; (i-l) *~w . (a-d) Contour interval

= 0.02; (e-h) 0.04; (i-l) 0.01. = 2π corresponds to the (≡ Uc/f0). The open circle with a cross at the centre


= 0° = 15° = 30° = 45° y*

Phase

y/d

-2-10120

1

2

3

4(a)-0.08 0.12


Phase

y/d

-2-10120

1

2

3

4(b)-0.08 0.12


Phase

y/d

-2-10120

1

2

3

4(c)-0.08 0.10


Phase

y/d

-2-10120

1

2

3

4 (d)-0.04 0.06


y*

Phase

y/d

-2-10120

1

2

3

4(e)-0.20 0.24


Phase

y/d

-2-10120

1

2

3

4(f)-0.20 0.20


Phase

y/d

-2-10120

1

2

3

4(g)-0.20 0.20


Phase

y/d

-2-10120

1

2

3

4 (h)-0.16 0.16


y*

Phase

y/d

-2-10120

1

2

3

4(i)-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(j)-0.03 0.02


Phase

y/d

-2-10120

1

2

3

4(k)-0.04 0.04


Phase

y/d

-2-10120

1

2

3

4 (l)-0.04 0.04


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

174

Figure 6-18. Phase-averaged velocities components at different yaw angles for T* = 1.7. (a-d) *~u ; (e-h) *~v ; (i-l) *~w . (a-d) Contour interval

= 0.005; (e-h) 0.01; (i-l) 0.005. = 2π corresponds to the /2 (≡ Uc/2f0). The open circle with a cross at the centre


= 0° = 15° = 30° = 45°

y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(a)0.065 -0.030


Phase

y/d

-2-1012

-1

0

1

2

3

4(b)0.040 -0.005


Phase

y/d

-2-1012

-1

0

1

2

3

4(c)-0.0100.045


Phase

y/d

-2-1012

-1

0

1

2

3

4(d)0.025 0.005


y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(e)-0.08 0.09


Phase

y/d

-2-1012

-1

0

1

2

3

4(f)-0.05 0.05


Phase

y/d

-2-1012

-1

0

1

2

3

4(g)0.06-0.06


Phase

y/d

-2-1012

-1

0

1

2

3

4(h)

-0.02 0.02


y*

Phase

y/d

-2-1012

-1

0

1

2

3

4 (i)-0.030 0.030


Phase

y/d

-2-1012

-1

0

1

2

3

4(j)-0.010 0.010


Phase

y/d

-2-1012

-1

0

1

2

3

4(k)0.025-0.020


Phase

y/d

-2-1012

-1

0

1

2

3

4(l)0.015 -0.010


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

x

x

x

175

Figure 6-19. Phase-averaged Reynolds stresses at different yaw angles for T* = 3.0. (a-d) **~~ vu ; (e-h) ** ~~ wu ; (i-l) ** ~~ wv . (a-d) Contour

interval = 0.002; (e-f) 0.0005; (g-l) 0.001. = 2π corresponds to the (≡ Uc/f0). The open circle with a cross at the centre

x represents

cylinder.

= 0° = 15° = 30° = 45° y*

Phase

y/d

-2-10120

1

2

3

4(a)-0.014 0.018


Phase

y/d

-2-10120

1

2

3

4(b)-0.012 0.018


Phase

y/d

-2-10120

1

2

3

4(c)-0.010 0.014


Phase

y/d

-2-10120

1

2

3

4 (d)-0.004 0.006


y*

Phase

y/d

-2-10120

1

2

3

4(e)-0.0020 0.0015


Phase

y/d

-2-10120

1

2

3

4(f)-0.0030 0.0015


Phase

y/d

-2-10120

1

2

3

4(g)-0.0045 0.0005


Phase

y/d

-2-10120

1

2

3

4 (h)-0.0030 0.0005


y*

Phase

y/d

-2-10120

1

2

3

4(i)0.006


Phase

y/d

-2-10120

1

2

3

4(j)-0.001 0.007


Phase

y/d

-2-10120

1

2

3

4(k)-0.003 0.006


Phase

y/d

-2-10120

1

2

3

4 (l)-0.002 0.004


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

176

Figure 6-20. Phase-averaged Reynolds stresses at different yaw angles for T* = 1.7. (a-d) **~~ vu ; (e-h) ** ~~ wu ; (i-l) ** ~~ wv . (a-d) Contour

interval = 0.0003; (e-h) 0.0001; (i) 0.0004; (j-l) 0.0002. = 2π corresponds to the /2 (≡ Uc/2f0). The open circle with a cross at the centre


= 0° = 15° = 30° = 45°

y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(a)

-0.0036 0.0030


Phase

y/d

-2-1012

-1

0

1

2

3

4(b)-0.0006 0.0009


Phase

y/d

-2-1012

-1

0

1

2

3

4(c)

-0.0009 0.0018


Phase

y/d

-2-1012

-1

0

1

2

3

4(d)-0.0003 0.0003


y*

Phase

y/d

-2-1012

-1

0

1

2

3

4(e)-0.0012 0.0011


Phase

y/d

-2-1012

-1

0

1

2

3

4(f)-0.0003 0.0003


Phase

y/d

-2-1012

-1

0

1

2

3

4 (g)-0.0003 0.0007


Phase

y/d

-2-1012

-1

0

1

2

3

4(h)0.0003 -0.0001


y*

Phase

y/d

-2-1012

-1

0

1

2

3

4 (i)0.0028 0.0024


Phase

y/d

-2-1012

-1

0

1

2

3

4(j)

0.00040.0006


Phase

y/d

-2-1012

-1

0

1

2

3

4(k)

0.0010 0.0014


Phase

y/d

-2-1012

-1

0

1

2

3

4(l)0.0002


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

x

x

x

177

Figure 6-21. Phase-averaged incoherent Reynolds stresses at different yaw angles for T* = 3.0. (a-d)

**~~rr vu ; (e-h)

** ~~rr wu ; (i-l)

** ~~

rr wv . (a-l) Contour interval = 0.001. = 2π corresponds to the (≡ Uc/f0). The open circle with a cross at the centre

x represents

cylinder.

= 0° = 15° = 30° = 45° y*

Phase

y/d

-2-1012.000030

1

2

3

4(a)

-0.011 0.012


Phase

y/d

-2-10120

1

2

3

4(b)-0.009 0.011


Phase

y/d

-2-1011.999980

1

2

3

4(c)

-0.006 0.009


Phase

y/d

-2-1012.000030

1

2

3

4 (d)0.005 -0.003


y*

Phase

y/d

-2-1012.000030

1

2

3

4(e)

-0.004 0.001


Phase

y/d

-2-10120

1

2

3

4(f)-0.003 0.002


Phase

y/d

-2-1011.999980

1

2

3

4(g)0.002-0.004


Phase

y/d

-2-1012.000030

1

2

3

4 (h)0.001 -0.004


y*

Phase

y/d

-2-1012.000030

1

2

3

4(i)

0.007


Phase

y/d

-2-10120

1

2

3

4(j)0.007 0.001


Phase

y/d

-2-1011.999980

1

2

3

4(k)-0.0040.007


Phase

y/d

-2-1012.000030

1

2

3

4 (l)0.008 -0.005


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

178

Figure 6-22. Phase-averaged incoherent Reynolds stresses at different yaw angles for T* = 1.7. (a-d)

**~~rr vu ; (e-h)

** ~~rr wu ; (i-l)

** ~~rr wv . (a-

l) Contour interval = 0.001. = 2π corresponds to the /2 (≡ Uc/2f0). The open circle with a cross at the centre


= 0° = 15° = 30° = 45°

y*

Phase

y/d

-2-1012.00003

-1

0

1

2

3

4(a)-0.0090.011


Phase

y/d

-2-1012.00003

-1

0

1

2

3

4(b)

-0.0090.009


Phase

y/d

-2-1011.99998

-1

0

1

2

3

4(c)

-0.0050.008


Phase

y/d

-2-1011.99999

-1

0

1

2

3

4(d)

-0.0040.003


y*

Phase

y/d

-2-1012.00003

-1

0

1

2

3

4(e)

-0.009-0.007


Phase

y/d

-2-1012.00003

-1

0

1

2

3

4(f)

-0.0060.002


Phase

y/d

-2-1011.99998

-1

0

1

2

3

4(g)

-0.0040.005


Phase

y/d

-2-1011.99999

-1

0

1

2

3

4(h)

-0.0030.003


y*

Phase

y/d

-2-1012.00003

-1

0

1

2

3

4(i)0.016


Phase

y/d

-2-1012.00003

-1

0

1

2

3

4(j)0.011


Phase

y/d

-2-1011.99998

-1

0

1

2

3

4(k)

0.017


Phase

y/d

-2-1011.99999

-1

0

1

2

3

4(l)

0.012


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


Phase

y/d

-2-10120

1

2

3

4(i)

++ +

X X

-0.02 0.02


(π) (π) (π) (π)

x

x

x

x

x

x

179

= 0° = 15° = 30° = 45°

2*2*

2*,

~ ,ru

uu

0 1 2 3 4 50

0.01

0.02

0.03

0.04

(a)


0 1 2 3 4

0

0.01

0.02

0.03

0.04

(a)


0 1 2 3 4

0

0.01

0.02

0.03

0.04

(b)


0 1 2 3 4

0

0.01

0.02

0.03

0.04

(c)


0 1 2 3 4

0

0.01

0.02

0.03

0.04

(d)


2*2*

2*,

~ ,rv

vv

0 1 2 3 4 5

0

0.02

0.04

0.06 (e)


0 1 2 3 4

0

0.02

0.04

0.06 (e)


0 1 2 3 4

0

0.02

0.04

0.06 (f)


0 1 2 3 4

0

0.02

0.04

0.06 (g)


0 1 2 3 4

0

0.02

0.04

0.06 (h)


2*2*

2*,

~ ,r

ww

w

0 1 2 3 4 5

0

0.01

0.02

0.03

(i)


0 1 2 3 4

0

0.01

0.02

0.03

(i)


0 1 2 3 4

0

0.01

0.02

0.03

(j)


0 1 2 3 4

0

0.01

0.02

0.03

(k)


0 1 2 3 4

0

0.01

0.02

0.03

(l)


**

**

**

,~

~ ,r

rv

uv

uv

u

0 1 2 3 4 5

-0.004

0

0.004

0.008

(m)

Frame 001 09 Aug 2010 Frame 001 09 Aug 2010

0 1 2 3 4

-0.004

0

0.004

0.008

(m)


0 1 2 3 4

-0.004

0

0.004

0.008

(n)


0 1 2 3 4

-0.004

0

0.004

0.008

(o)


0 1 2 3 4

-0.004

0

0.004

0.008

(p)


**

**

**

,~

~ ,r

rw

uw

uw

u

0 1 2 3 4 5

-0.004

0

0.004

0.008

(q)


0 1 2 3 4

-0.004

0

0.004

0.008

(q)


0 1 2 3 4

-0.004

0

0.004

0.008

(r)


0 1 2 3 4

-0.004

0

0.004

0.008

(s)


0 1 2 3 4

-0.004

0

0.004

0.008

(t)


**

**

**

,~

~ ,r

rw

vw

vw

v

0 1 2 3 4 5

0

0.004

0.008

0.012

0.016

(u)


0 1 2 3 4

0

0.004

0.008

0.012

0.016

(u)


0 1 2 3 4

0

0.004

0.008

0.012

0.016

(v)


0 1 2 3 4

0

0.004

0.008

0.012

0.016

(w)


0 1 2 3 4

0

0.004

0.008

0.012

0.016

(x)


y* y

* y

* y

*


contributions at different yaw angles for T* = 3.0. Time-averaged: ; Coherent: ;

Incoherent: . The open circle with a cross at the centre


x

x

x

x

180

= 0° = 15° = 30° = 45°

2*2*

2*,

~ ,ru

uu

0 1 2 3 4 5

0

0.01

0.02

0.03

0.04

(a)


-1 0 1 2 3 4

0

0.01

0.02

0.03

0.04

(a)


-1 0 1 2 3 4

0

0.01

0.02

0.03

0.04

(b)


-1 0 1 2 3 4

0

0.01

0.02

0.03

0.04

(c)


-1 0 1 2 3 4

0

0.01

0.02

0.03

0.04

(d)


2*2*

2*,

~ ,rv

vv

0 1 2 3 4 5

0

0.02

0.04

0.06 (e)


-1 0 1 2 3 4

0

0.02

0.04

0.06 (e)


-1 0 1 2 3 4

0

0.02

0.04

0.06 (f)


-1 0 1 2 3 4

0

0.02

0.04

0.06 (g)


-1 0 1 2 3 4

0

0.02

0.04

0.06 (h)


2*2*

2*,

~ ,r

ww

w

0 1 2 3 4 5

0

0.01

0.02

0.03

(i)


-1 0 1 2 3 4

0

0.01

0.02

0.03

(i)


-1 0 1 2 3 4

0

0.01

0.02

0.03

(j)


-1 0 1 2 3 4

0

0.01

0.02

0.03

(k)


-1 0 1 2 3 4

0

0.01

0.02

0.03

(l)


**

**

**

,~

~ ,r

rv

uv

uv

u

-1 0 1 2 3 4 5

-0.004

0

0.004

0.008

0.012

(m)


-1 0 1 2 3 4

-0.004

0

0.004

0.008

0.012

(m)


-1 0 1 2 3 4

-0.004

0

0.004

0.008

0.012

(n)


-1 0 1 2 3 4

-0.004

0

0.004

0.008

0.012

(o)


-1 0 1 2 3 4

-0.004

0

0.004

0.008

0.012

(p)


**

**

**

,~

~ ,r

rw

uw

uw

u

-1 0 1 2 3 4 5

-0.004

0

0.004

0.008

0.012

(q)


-1 0 1 2 3 4

-0.004

0

0.004

0.008

0.012

(q)


-1 0 1 2 3 4

-0.004

0

0.004

0.008

0.012

(r)


-1 0 1 2 3 4

-0.004

0

0.004

0.008

0.012

(s)


-1 0 1 2 3 4

-0.004

0

0.004

0.008

0.012

(t)


**

**

**

,~

~ ,r

rw

vw

vw

v

-1 0 1 2 3 4 5

0

0.004

0.008

0.012

0.016

(u)


-1 0 1 2 3 4

0

0.004

0.008

0.012

0.016

(u)


-1 0 1 2 3 4

0

0.004

0.008

0.012

0.016

(v)


-1 0 1 2 3 4

0

0.004

0.008

0.012

0.016

(w)


-1 0 1 2 3 4

0

0.004

0.008

0.012

0.016

(x)


y* y

* y

* y

*


contributions at different yaw angles for T* = 1.7. Time-averaged: ; Coherent: ;

Incoherent: . The open circle with a cross at the centre


x

x

x

x

x

x

x

x

181

T* = 3.0 T

* = 1.7

2

2/

xx

-0.01

0

0.010

0.020

0.030

0.040

0 1 2 3 4

= 0 = 15 = 30 = 45

(a)

2

2/

xx

-0.005

0

0.005

0.015

0.010

0.020

-1 0 1 2 3 4

(b)

2

2/

yy

-0.005

0

0.005

0.010

0.015

0.020

0 1 2 3 4

(c)

2

2/

yy

-0.005

0

0.005

0.010

0.015

0.020

-1 0 1 2 3 4

(d)

2

2/

~z

z

-0.05

0

0.050

0.100

0.150

0.200

0 1 2 3 4

(e)

2

2/

~z

z

-0.005

0

0.005

0.010

0.015

0.020

-1 0 1 2 3 4

(f)

y

* y

*

Figure 6-25. Coherent contribution to vorticity variances for different yaw angles. (a-b)

22/~

xx ; (c-d) 22

/~yy ; (e-f)

22/~

zz . The open circle with a cross at the centre


x

x

x

183

CHAPTER 7

ON THE STUDY OF VORTEX-INDUCED VIBRATION OF

A CYLINDER WITH HELICAL STRAKES

7.1 Introduction

When vortices are shed from a bluff body, the latter is subjected to time-

dependent drag and lift forces. The lift force oscillates at the vortex shedding frequency

while the drag force oscillates at twice the vortex shedding frequency (Sumer and

Fredsøe, 1997). If the cylinder is flexibly-mounted, these forces may induce vibration of

the cylinder. The lift force may induce cross-flow vibrations and the drag force may

induce in-line vibrations. This phenomenon is called vortex-induced vibration (VIV)

(Blevins, 2001). Vortex-induced vibration of bluff structures is one of the key issues in

riser and pipeline designs. This is because VIV will increase not only the dynamic load

to the structures but will also influence the structural stability. The vibrations may cause

structural failure or accelerate the fatigue failure. The above factors may result in an

increase in capital investment of the structures and the expenses for maintenance and

replacement.

The VIV of a cylindrical structure depends on various parameters, such as

Reynolds number Re, Strouhal number St (≡ f0d/U∞, where f0 in the vortex shedding

frequency, d is the cylinder diameter and U∞ is the free-stream velocity in the

streamwise direction) and the reduced velocity Vr (U∞/fnd, where fn is the natural

frequency of the structure) etc. The mechanism of vortex shedding of a circular cylinder

and the oscillation related to it have been studied for more than a century and yet our

understanding is still far from complete even though much progress has been made

during the past several decades, both numerically and experimentally due to the

advances of the technologies in experiments and computing powers (Roshko, 1954;

Bearman, 1969; Sarpkaya, 1979; Zdravkovich, 1981; Williamson and Govardhan, 2004;

Carberry et al., 2005; Elston et al., 2006). The ultimate objective of studies on VIV is

the understanding, prediction, and prevention of VIV preferably without drag penalty.

Since VIV phenomenon is very complex to model, its prediction is challenging because

commercially available predictive tools have not been extensively calibrated with valid

184

experimental data. In many situations, these tools provide different answers for the same

problem. As a result, the level of confidence in VIV analysis is low and conservative

approaches by using high factors of safety are used in design of the pipelines and risers.

With offshore facilities being installed in increasingly deeper waters, such conservative

designs have also become increasingly economically untenable. VIV prevention or

mitigation is therefore critically important, as it will yield solutions that enable safe,

cost-effective and reliable deep-water project developments.

Suppression of vortex shedding and VIV of a bluff body has been one of the

most active topics of research and patenting in fluid dynamics for many decades due to

its significance in engineering applications. Previous investigations on passive control

devices to suppress VIV have contributed significantly to the fields of buildings, bridges

and marine engineering and yet there are difficulties in achieving an optimal balance

between performance, cost and simplicity. To prevent VIV lock-in, one can either

change the natural frequency of the structure by structural modification or to inhibit the

formation of vortices or to disrupt their structural formation, through the application of

the suppression devices. Suppression of vortex shedding and hence VIV can be

achieved by active methods (where external energy is supplied), passive methods

(where no external energy is supplied to control the flow), or a combination of the

above two methods. Helical strake is one of the most proven and widely used control

measures in various industrial and offshore applications to suppress VIV. It normally

contains three-start strakes helically wound on the surface of the structural member in a

definite fashion. The efficiency of VIV suppression using helical strakes depends on the

dimensions of the strakes, i.e. the height of the protrusion and the pitch of the helical.

Previous studies have shown that the optimal height of strakes for VIV suppression is

about 0.05 0.2d and a pitch of about 3.6 5d has generally been accepted as optimal,

though recent tests have shown that a pitch of 15d may be just as effective (Jones and

Lamb, 1992). Basically, it is believed that they adversely affect the shear layer to roll up

and to disrupt the spanwise vortex formation and shedding process (Scruton and

Walshe, 1963). As fluid flows past a cylinder with helical strakes, the strake chops up

the flow and creates vortices at various places along the cylinder. These vortices are out

of phase with one another and produce destructive interference to the dominant vortex

shedding. Due to partial cancellation of the out-of-phase lift forces at different spanwise

positions, the lift coefficient for the straked cylinder is much smaller than that of a bare

cylinder, resulting in a significant reduction in the amplitude of VIV. Even though

185

helical strakes are found effective in suppressing VIV for high damping values, the

effectiveness of VIV suppression using helical strakes for low mass-damping values is

reduced. Although Bearman and Branković (2004) did not find evidence of regular

Kármán-type vortex shedding close to the stationary cylinder attached with helical

strakes of 0.12d in height and 5d in pitch, an undulation of the wake was observed at

about 2 cylinder diameters downstream. However, when the straked cylinder is free to

respond for Vr larger than about 5 and the mass and damping are low, the flow pattern

changes significantly. The straked cylinder not only experienced the lock-in but also

exhibited 2S and 2P vortex shedding modes, which is similar to that for a bare cylinder.

These results are different from those reported by Constantinides and Oakley (2006) for

strakes of 0.25d in height and 15d in pitch, due mainly to the difference in strake

height. The latter authors showed that the helical strakes can suppress VIV nearly

completely over the lock-in range of the bare cylinder. They found that the vibration

amplitude of the straked cylinder increased monotonically with the increase of reduced

velocity, which was attributed to the effect of wake galloping. There was no organized

vortex observed in the straked cylinder wake. Their flow visualization suggested that

the separation point was at the tip of the strake for most of the coverage. The

helical separation point induces a three-dimensional flow behind the cylinder

breaking the vortex coherence. In certain locations where the strake tip is aligned

with the flow, either upstream or downstream, separation is partially controlled by the

cylinder surface. The other mechanism that helical strakes use to mitigate VIV is by

limiting the interaction between the two shear layers that are formed due to

separation (Constantinides and Oakley, 2006). The helical strakes act as

obstacles preventing the shear layers to communicate and form the typical vortices

we find in bare cylinder wakes.

Even though helical strakes are proven effective for suppressing VIV especially

for low mass-damping values, the mechanism of VIV suppression has not yet been fully

understood (Constantinides and Oakley, 2006). Therefore, the present study aims to

shed some light on the vortex characteristics and to enhance our understanding on the

mechanisms of helical strakes have on VIV suppression.


The experiments were conducted in a low speed section of the boundary layer

wind tunnel of the University of Western Australia. The dimensions of the test section

186

used are 2,230 mm high by 2,870 mm wide. An aluminium cylinder with a diameter d

of 80 mm and a length l of 1,600 mm was used either as a bare cylinder or after the

installation of the helical strakes, as a straked cylinder. The three-start strakes have a

pitch P of 10d and height h of 0.12d (Figure 7-1), which is in the range of the most

effective dimension in VIV suppression (e.g. Kumar et al., 2008). The aspect ratio of

the cylinder l/d is 20. The mass of the bare cylinder is 3.035 kg and the mass of the

cylinder with strakes is 3.496 kg. At each end of the cylinder, a steel „O‟ ring fits snug

around the outer diameter and provides an attachment for two steel eye-loops from

which the springs are attached. Four steel springs, two at either end of the cylinder,

suspend the cylinder approximately 1,000 mm off the wind tunnel floor. The springs

were selected based on their stiffness and length. They must be stiff enough and long

enough to remain in tension when measuring the amplitude of vibrations. All four

springs were of an equal stiffness k = 1,680 N/m and equal original length of 250 mm.

The stiffness was measured by applying a known load to the springs and measuring the

subsequent deflection. Once the springs were attached to the cylinder, the top and

bottom springs stretched to a length of 340 mm and 330 mm, respectively.

The amplitude of the vibrations was measured using a linear variable differential

transformer (LVDT) laser. The laser was cantilevered off the frame‟s vertical member

and is positioned 130 mm above the cylinder. The vibration signals from the laser were

sampled into a computer in the form of voltages, which have been calibrated that 1 Volt

= 10 mm, at a frequency of 50 Hz by using Labview 7.1. The natural frequencies of the

bare cylinder and the straked cylinder are obtained from the free-decay tests.

Displacements from the cylinders after an initial displacement were recorded using the

LVDT at a frequency of 50 Hz. Spectrum was then obtained by applying fast Fourier

transform (FFT) to the time series of the measured signals. A strong peak on the

distribution of the spectrum corresponds to the natural frequency, which is identified as

6.6 Hz and 6.37 Hz, for the bare and straked cylinders, respectively.

Velocity fluctuations u in the streamwise (x-) and v in the transverse (y-)

directions are measured using X-wire probes. For the purpose to examine the spanwise

cross-correlations, two X-wire probes located at x/d = 5 and 10 and y/d = 0.5 were used

with one probe fixed and the other moving along the cylinder length direction. The

separation between the two probes was in the range of 10 mm – 250 mm. For the

purpose to examine the evolution of the vortices in the streamwise direction,

measurements using a single X-wire probe were conducted in the range of x/d = 5-40 at

187

four different free-stream velocities U∞, i.e. 1.92, 3.83, 5.74 and 7.65 m/s,

corresponding to Re = 10,240; 20,430; 30,610 and 40,800; respectively. The hot wires

were operated with in-house constant temperature circuits at an over heat ratio of 1.5.

Each of the two wires in the X-wire probe had a diameter of 5 m. The wire separation

was about 1 mm. The output signals from the anemometers were low-pass filtered

through the buck and gain circuits at a cut-off frequency fc = 0.5–9.2 kHz, depending on

the measurement location and free-stream velocity. The values of fc were determined by

examining the spectrum of ∂u/∂t and identifying the onset of electronic noise (Antonia

et al., 2002), which were normally close to the Kolmogorov frequency fK (≡ U/2π,

where is the Kolmogorov length scale and can be calculated using

4/13 )/( ). Hereafter, the angular brackets denote time-averaging. The mean

energy dissipation rate is defined as ijij ss 2 , where

2/)( ,, ijjiij uus is the rate of strain, is the kinematic viscosity and jiji xuu /, .

Strictly, the mean energy dissipation rate should be obtained by measuring all the

12 terms in its expression. However, this is not normally feasible unless using a multi-

hot wire probe (e.g. Vukoslavčević et al., 2009). On the other hand, even though we can

measure the 12 terms by using the multiple hot wire probes, there also exist limitations

on spatial resolution of the probe on the measured velocity gradients, which result in an

underestimation of (e.g. Antonia et al., 1998). Therefore, a very common method

for experimentalists is to use 2)/(15 xu by assuming isotropy and using

Taylor‟s hypothesis by replacing ∂x with Ut, where t ( 1/fs) is the sampling time

interval between two consecutive points and fs is the sampling frequency.

The filtered signals were sampled at a frequency fs = 2fc into a PC using a 1-bit

A/D converter (National Instrument). The sampling period was 60–120 seconds,

depending on the Reynolds numbers and measurement locations. Experimental

uncertainties in U and u′ (or v′) were inferred from the errors in the hot wire calibration

data as well as the scatter (20 to 1 odds) observed in repeating the experiment for a

number of times. The uncertainty for U was about ±3%, while the uncertainties for u′

and v′ were about ±5% and ±6%, respectively. The experimental uncertainty for St

(Uf0/d) depends on that of the velocity and the vortex shedding frequency

measurements. The vortex shedding frequency is identified using the fast Fourier

transform (FFT) algorithm with a window size of 211

. Its uncertainty depends on the

188

sampling frequency and the window size when doing FFT (Xu and Zhou, 2004). In the

present study, we estimated that the uncertainty for St was around 6%.

To further examine the differences of the vortex structures in the wakes of the

bare cylinder and the straked cylinder, flow visualization in a wind tunnel of 25 cm

(height) 40 cm (width) and 2 m (length) was also conducted using a smoke wire at a

Reynolds number of about 300. The diameter of the cylinder for this experiment was

12.7 mm. The dimensions of the strakes are comparable to the first experiment, i.e.

0.12d in height and 10d in pitch.


7.3.1 Vortex shedding frequency for the bare and straked cylinders

The vortex shedding frequency at various flow velocities were examined by

performing FFT to the velocity signals measured in the wake using the hot wire

anemometers. Before the cylinder starts to vibrate, vortex shedding from the cylinder is

apparent with a peak frequency corresponding to a Strouhal number of 0.21, which

agrees well with the consensus values published in the literature. Figure 7-3 gives an

example of the spectrum obtained on the centreline of the bare cylinder wake at x/d = 5

for Re = 20430. The peak height relative to the height of the plateau at low frequency is

about 22 dB. After the cylinder starts to vibrate, the energy spectrum peaks at the

natural frequency of the cylinder over the whole lock-in range. The peak frequencies on

the energy spectra at different reduced velocities are replotted in the form of the ratio

f0/fn versus reduced velocities (Figure 7-4). The ratio increases linearly with Vr until it

reaches the lock-in regime at a reduced velocity of about 5. Over the lock-in regime, the

ratio f0/fn keeps a constant value of 1. The straight line in the figure has a slope of 0.21

which corresponds to the Strouhal number. In contrast to the bare cylinder, there is only

a minor peak on the spectrum for the straked cylinder (Figure 7-3). Various locations in

the streamwise (x) direction as well as in the transverse (y) direction were tested and yet,

no apparent peak on the energy spectra which is as apparent as that for the bare cylinder

was identified. The peak height of the straked cylinder relative to the height of the

plateau at low frequency is 4 dB, which is only about 1/6 of that in the bare cylinder

wake. It can also be seen that the peak region on the spectrum of the straked cylinder is

broadened, suggesting that the intensity of the vortices shed from the straked cylinder is

significantly reduced compared with that from the bare cylinder.

189

7.3.2 Vortex-induced vibration for the bare and straked cylinders

In order to validate the current experimental setup, the dynamic response of a

single bare cylinder is tested first. The response of the cylinder can be described in

terms of the vibration amplitude A/d versus the reduced velocity Vr. The cylinder was

free to oscillate in y-direction. It was found that for each reduced velocity, the vibrations

take about 50 seconds to stabilise, after which the amplitude of vibration does not

change. A test of about 60 seconds was recorded after stable patterns of vibrations were

observed. The amplitudes of vibration were determined by averaging the 10% highest

peaks recorded on the time history of the displacement after the vibration becomes

stable (Hover et al., 2001; Franzini et al., 2009). The vibration response at different

reduced velocities were compiled and displayed in Figure 7-5 for the bare cylinder. It

can be seen that the maximum peak amplitude is about 0.51, occuring at a reduce

velocity of about 6.5, which is within the consensus range found in the literature (Sumer

and Fredsøe, 1997). Experimental results by Feng (1968) in air and Khalak and

Williamson (1996) and Franzini et al. (2009) (l/d = 24) in water are also included for

comparison. It can be seen that the response amplitudes in the present study follow that

of Feng‟s results closely, especially in the upper branch. The lock-in region in the two

studies is also very similar. This is reasonable as the mass damping parameter m*s in

the two studies are comparable, with m*s being about 0.26 in the present study and

0.255 in Feng (1968), where m* (≡ m/mf, with m being the structural mass including the

enclosed fluid mass and mf being the displaced fluid mass) is the mass ratio and s is the

structural damping factor, which can be obtained using )/ln()2/1( 1 nns yy from a

free decay test. The mass damping parameters m*s in Khalak and Williamson (1996)

and Franzini et al. (2009) are 0.013 and 0.0092, respectively, which are much smaller

than that in the present study and in Feng (1968). Correspondingly, the maximum

vibration amplitudes in Khalak and Williamson (1996) and Franzini et al. (2009) are

much larger than that in the latter two studies. There exist two apparent branches, i.e.

the upper branch and the lower branch. Also in the former studies with low mass

damping parameters, the onset of cylinder vibration occurs earlier and diminishes

later in amplitude than the latter two studies.

The maximum vibration amplitude Amax/d obtained in the present study can be

compared with that obtained using the empirical relations (e.g. Sumer and Fredsøe,

1997; Sarpkaya, 2004). Sumer and Fredsøe (1997) proposed that for stability parameter

190

Ks larger than 2, where Ks can be quantified using Ks = 4πm*s, the maximum vibration

amplitude can be estimated using

sKd

A 7.1max for Ks ≥ 2. (7-1)

Based on the mass ratio m* and the structural damping factor s, the stability parameter

Ks in the present study is about 3.27. This value results in a maximum amplitude Amax/d

of 0.52 for the bare cylinder, which agrees very well with the value obtained in the

experiment. By introducing another parameter, the so-called response parameter SG

(≡2π3St

2m

*s), Sarpkaya (2004) proposed another relationship for quantifying the

maximum vibration amplitude Amax/d, viz.

GSdA 35.012.1/max (7-2)

For a rigid circular cylinder, the dimensionless mode factor = 1.0. With SG = 0.712 in

the present study, the maximum amplitude Amax/d using Eq. (7-2) is estimated to be

about 0.53, which also agrees very well with the present experimental value (0.51). The

above results for the single cylinder have justified the present experimental arrangement

for the single cylinder-spring system.

The vibration response of the cylinder attached with helical strakes is then

tested. It is found that the strakes can suppress the vibration significantly. With the

strakes attached, the cylinder does not vibrate apparently over the same reduced velocity

range of the bare cylinder. In contrary to the response of the bare cylinder which

vibrates severely over the lock-in range, it seems that the cylinder attached with strakes

does not experience a lock-in range. Actually, the peak on the energy spectra measured

using hot wire probes downstream of the straked cylinder does not coincide with the

natural frequency of the straked cylinder (= 6.4 Hz), indicating that lock-in phenomena

does not occur. The vibration amplitudes increase with Vr, even though with very small

magnitude compared with that for the bare cylinder. The vibration amplitudes of the

cylinder with strakes are shown in Figure 7-6. Even at a reduced velocity of 15, the

cylinder does not show a reduction in vibration amplitude, as that observed for a bare

cylinder. This trend is consistent with that reported by Constantinides and Oakley

(2006) and Ding et al. (2004). They suggested that the increased vibration amplitude

with the increase of reduced velocity might be attributed to wake galloping rather than

the classical VIV. Also shown in the figure is the vibration amplitude of the bare

cylinder. Apparently, the strakes can reduce the vibration amplitude by 98%. The

191

present result on the vibration amplitude for the straked cylinder is in contrast with that

reported by Bearman and Branković (2004) cylinder attached with helical strakes,

probably due to the low damping ratios used in the latter study. These authors found that

VIV occurs over a narrower range of reduced velocity. The response characteristics,

especially for the lowest mass ratio, are surprisingly similar to those for a bare cylinder

with what may be a lower branch appearing. They also found that for the straked

cylinder, lock-in phenomena occurs and the vibration exhibited 2S and 2P modes with a

behaviour similar to that for the bare cylinder, although the amplitudes were smaller.

7.3.3 Energy spectra in the streamwise and spanwise directions of the stationary

cylinders

In order to understand the physical mechanism of how the strakes mitigate VIV

on a bare cylinder, energy spectra measured using hot wire probes are analyzed. In

Figure 7-7, the energy spectra obtained in the stationary bare and straked cylinder wakes

at x/d = 5, 10, 20 and 30 on the centreline for Re = 20,430 are compared in order to

examine the evolution of the vortex structures in the streamwise direction. It can be seen

that for the bare cylinder (Figure 7-7a), the peaks of the spectra are sharp and occur at

approximately a fixed frequency in the streamwise direction, which, after normalization

by U∞ and d, gives a Strouhal number St of 0.21, consistent with the consensus values

published in the literature. This result indicates that the vortex shed from the bare

cylinder is regular. Zhou et al. (2003) showed that the vortex structures generated from

the bare cylinder broke down and the intensity of the vortices reduced significantly in

the streamwise direction. At about x/d = 40, the vortex structures nearly disappear. The

present result agrees with this. At x/d = 40, there is no peak discernable (spectrum for

this location is not shown to avoid crowding the figure) and at x/d = 30, only a very

minor peak exists. The peak frequency variation at different locations, as indicated by

f in the figure, is about 0.2 Hz, which is only about 2% of the averaged vortex

shedding frequency f0. This value should be within the experimental uncertainty for the

detection of f0 and therefore f0 can be regarded as a constant value. However, on the

spectra of the straked cylinder wake (Figure 7-7b), the peak location is not a constant

value at different downstream locations. The Strouhal number St, based on the averaged

value of f0 at different locations, is about 0.14. This value is much smaller than that for a

circular cylinder ( 0.21) at the tested Reynolds number range but closer to the value of

St found in a flat plate wake (Roshko, 1954; Castro and Rogers, 2002). Since the strakes

resemble flat plates perpendicular to the flow, they increase the effective diameter of the

192

cylinder, which correspond to a lower consensus Strouhal number of about 0.164

(Roshko, 1954). The peak varies at different locations. The maximum difference of the

peak frequency f between x/d = 5 and 30 is about 2.3 Hz. This value corresponds to a

variation of 32% of the averaged peak frequency, which is too big to be attributed to the

experimental uncertainty in measuring fs. Rather, it may indicate a dislocation of the

vortices in the streamwise direction, which may be caused by the spanwise motion of

the fluid downstream of the straked cylinder since parts of the wake will be transported

laterally and arrive in the neighbouring region. This effect will also enhance the three-

dimensionality of the flow. Flow visualization using computational fluid dynamics

(CFD) (Constantinides and Oakley, 2006) showed significant spanwise motion and

swirling in the near wake of the strake. These authors found that the spanwise motion

in the wake is caused by the fluid being channeled by the strakes and also by the

separated fluid that induces a spanwise flow.

The variation of vortex shedding frequency along the cylinder axial direction

can also be examined. Figure 7-8 shows the spectra measured in the stationary bare and

straked cylinder wakes at different locations along the cylinder length. For the bare

cylinder, the peaks are sharp and occur at a fixed frequency along the cylinder length,

i.e. f0 ≈ 10 Hz, or with a wavelength L/d = 4.8. This result indicates that vortices are

shed from the bare cylinder at a fixed frequency even though they may be shed in cells

along the cylinder length at slightly shifted phase (see discussion for Figure 7-9 below).

In contrast, the peaks on the energy spectra in the straked cylinder wake do not occur at

a fixed frequency (Figure 7-8b). The averaged frequency f0 is about 8.6 Hz which

corresponds to a wavelength L/d of 5.55. The maximum difference among the peaks

along the cylinder length direction is about 3.06 Hz, which corresponds to a variation of

39% of the averaged vortex shedding frequency. The variations of the vortex shedding

frequency in the energy spectra was also shown in the wake of wavy plate model by

Bearman and Owen (1998). The variations of near-wake width (Owen and Bearman,

2001) and the separation points (Constantinides and Oakley, 2006) along the cylinder

axis may contribute to the inconsistency of the peaks in the energy spectra along the

straked cylinder axis. The near-wake width and the separation points depend on the

cylinder surface i.e. the tip of the strakes and the alignment of the strake tip with the

flow (Constantinides and Oakley, 2006). The peak in Figure 7-8b at fd/U∞ = 0.21 (for

z/d = 1.25) corresponds to the separation at the cylinder surface while that at fd/U∞ =

0.14 (for z/d = 0.125 and 3.125) correspond to separation from the tips of the strakes,

193

which resemble the wake of a flat plate. This mechanism of the straked cylinder is in

contrast to the bare cylinder for which the flow separation points along the span are

more or less fixed, resulting in a coherent vortex shedding (Constantinides and Oakley,

2006). Using the spanwise vorticity iso-surface, Constantinides and Oakley (2006)

showed the separation lines and the disorganized vorticity in the straked cylinder

wake. The wake has a strong three-dimensional character without any coherent vortex

structures. This result may indicate that the strakes have disrupted the large-scale

structures from the cylinder. Bearman and Branković (2004) suggested that the three-

dimensionality of the separating flow introduced by the strakes could destroy the

regular vortex shedding and hence suppress the VIV of the cylinder.

Previous studies have shown that there is a phase shift among the vortex

structures shed from a bare cylinder (e.g. Roshko, 1954; Szepessy, 1994; Alam et al.,

2002; Fujisawa et al., 2004; So et al., 2005; Xu et al., 2008). To examine the phase shift

between the vortex structures, time series of the fluctuating quantities such as the

pressure, velocity or vorticity at a point may not show the phase variation along the

cylinder axis clearly. For this purpose, the cross-correlation coefficient of velocity or

pressure measured at two points along the cylinder axis as a function of time delay can

be used. This method has been used previously by a number of researchers (e.g.

Szepessy, 1994; Fujisawa et al., 2004; So et al., 2005; Xu et al., 2008). The cross-

correlation between two velocity signals measured at two points separated by a distance

z along the cylinder axial direction as a function of time delay is defined as

21

21

),(),()( 21

,vv

vv

tzzvtzvR

(7-3)

where 1v

and 2v are the standard deviations of the velocity components v1 and v2,

respectively. The cross-correlation coefficients measured in the stationary bare and

straked cylinder wakes for Re = 20,430 as a function of time delay are shown in Figure

7-9. It can be seen that there is a very small phase difference among the dominant

vortex structures for the bare cylinder (Figure 7-9a). The maximum phase difference is

about 0.25d over a spanwise length of 3.125d, which corresponds to 0.1π. This

behaviour of the vortical structures in the bare cylinder wake may imply a large

dynamic force induced on the cylinder and hence a large response in VIV. In contrast,

the phase variation of the velocity signals in the straked cylinder wake is apparent at all

probe separations (Figure 7-9b). The maximum phase variation is about 1.8d, which

194

corresponds to 0.65π. This result implies a significant phase variation of the vortical

structures along the spanwise direction in the vortex shedding process. It may suggest

that helical strakes prevent the vortex from being correlated along the spanwise

direction, in agreement with Bearman and Branković (2004). The variation in phase is

because of the different geometry of the helical strakes along the cylinder spanwise

direction (i.e. position of the helical strakes and the tip location of the strakes). The

above results confirm that the strakes do not necessarily suppress vortex shedding

from the cylinder but strongly disorganise the vortices, resulting in an apparent

reduction in VIV.

7.3.4 Cross-correlations in the stationary cylinder wakes

It has been demonstrated that in the turbulent wake regimes, vortices are shed in

cells in the cylinder axis direction (e.g. Williamson, 1996b). These cells are out-of-

phase. The averaged length of the cells is termed as the correlation length (Sumer and

Fredsøe, 1997). It can be quantified by measuring the spanwise variation of the cross-

correlation coefficients defined as

21

21

)()()( 21

,

aa

aa

zzazaz

(7-4)

where a and b represent velocity component u or v at two points separated along the

length of the cylinder by a distance z, and represents standard deviations of a1 and

a2. The cross-correlation can be used to quantify the three-dimensional flow

characteristics of the vortical structures in the wake of the cylinder in the spanwise

direction. Using two X-probes, with one probe being stationary to measure u and v at

location 1, and the other being movable to measure u and v at location 2 separated in the

spanwise direction by z, the cross-correlation coefficient between velocity component

u or v of the two probes can be calculated. The correlation length is then calculated

using the following integral:

dzzLz

aaaa )(0

2121 0 ,,

(7-5)

where z0 is the separation at which 21,aa first zero crossing.

The cross-correlation coefficients between a1 and a2 are shown in Figure 7-10. It

can be seen that all the cross-correlation coefficients decrease with the increase of probe

separation. In both wakes, )(21, zvv is larger than )(

21, zuu , indicating that v is more

195

representative to the large-scale structures than u. The magnitude of )(21, zvv in the

bare cylinder wake is much larger than that in the straked cylinder wake, indicating that

the vortical flow structures in the wake of the bare cylinder are larger than that in the

straked cylinders. Indeed, if Eq. (7-5) is used to determine the correlation length, it is

found that 21 ,vvL in the bare cylinder wake is about 2.1d. This value agrees well with

that found by Sumer and Fredsøe (1997) for the same range of Reynolds number as in

the present study. In contrast, the correlation length 21 ,vvL for the straked cylinder is

only about 0.44d, which is much smaller than that obtained in the bare cylinder wake.

This result suggests that the strakes have successfully disrupted the vortical structures in

the spanwise direction and thus enhancing the three-dimensionality of the flow. This

result is in agreement with that proposed by Bearman and Branković (2004), who

argued that the strakes do not necessarily suppress vortex shedding but they prevent

the shedding from becoming correlated along the span.

7.3.5 Flow visualization of the stationary cylinder wakes

To further examine the differences of the large-scale structures in the wakes of

the bare cylinder and the straked cylinder, flow visualization in a wind tunnel was

conducted using a smoke wire at a Reynolds number of about 300. For the bare cylinder

wake (Fig. 11a), the large organized vortices can be seen clearly with two rows of

counter-rotating vortex structures connected by the streamwise structures. The

formation length of the wake is about 2d (Unal and Rockwell, 1988). For the straked

cylinder wake (Fig. 11b), the separation point occurs either at the tip of the strakes or on

the surface of the cylinder, generating two shear layers. Small-scale Kelvin-Helmholtz

vortices are formed in the shear layers. However, these structures do not interact with

each other until x/d is larger than about 10. This flow characteristics are similar to the

wake of a circular cylinder fitted with surface bumps (Owen and Bearman, 2001). This

result also supports the numerical simulations on circular cylinder fitted with three-

strand helical strakes by Contantinides and Oakley (2006), who found that the strakes

limit the interaction between the two shear layers and hence prevented the formation of

the large organized vortices. In the near wake, swirling flow towards the leeward side is

also found. Flow visualization on the x-z plane (photos are not shown here) also reveals

that the smoke moves along the cylinder spanwise direction, indicating a strong velocity

component in the spanwise direction. It is expected that this velocity component helps

destroying the organized structures, dislocating the vortex structures which is in

196

agreement with the result shown in Figure 7-8b, forming a much stronger 3D wake than

that for a bare cylinder, resulting in a significant reduction in vibration response for the

straked cylinder.

7.3.6 Isotropy assessments in the wake of stationary cylinders

As the strakes have effectively disrupted the large organized structures in the

wake of the straked cylinder, it is therefore expected that the energy containing

structures in the straked cylinder wake should be smaller than that in the bare cylinder

wake. As a result, the flow field of the former should agree more favourably with

isotropy than the latter. This is true as the ratio 22 / uv in the former wake is

closer to 1 than the latter, irrespective of the downstream locations. Isotropy requires

that 22 / uv be 1. The ratio at different downstream locations for stationary

body of bare cylinder and straked cylinder wakes are given in Table 7-1. It can be seen

clearly that for the bare cylinder wake, 22 / uv is far from 1. Even in the far

wake, Browne et al. (1987) (at x/d = 420) and Zhu and Antonia (Zhu and Antonia,

1999) (at x/d = 240 found that the ratio 22 / uv is only around 0.73, indicating

that the effect of the large organized structures persists for a long time. A measure to

check the departure from isotropy for various turbulent scales is provided by comparing

the measured spectrum of transverse or spanwise velocity component with that

calculated based on the isotropic assumption. For isotropic turbulence, the relationship

between the streamwise and transverse (or spanwise) velocity spectra is (Zhu and

Antonia, 1995):

)(2

1)()(

1

111k

kkk uu

cal

w

cal

v

(7-6)

where the superscript cal represents “calculation” and k1 is the wavenumber. In Eq.

(7-6), the longitudinal velocity spectra u measured on the centreline at different

streamwise locations are used as the input for the calculation of v or u. The

distributions of mv , cal

v and the ratio m

v

cal

v at different streamwise locations and

different Reynolds numbers in the straked and bare cylinder wakes are shown in Figures

7-12 and 7-13, respectively, where the superscript m represents “measured” and the

superscript asterisk denotes normalization by Kolmogorov length scale and/ or

velocity scale UK ( /). When isotropy is satisfied, m

v

cal

v should be 1 at all scales.

At x/d = 10 of the straked cylinder wake (Figure 7-12a), the departure from isotropy for

197

scales k1* < 0.01 is apparent. The peak at k1

* = 0.004 is due to the vortex shedding. With

the increase of Re, the peak decreases and shifts towards the lower wavenumber region.

For wavenumbers higher than 0.01, there is satisfactory agreement between

measurement and calculation, indicating satisfactory agreement with isotropy at these

scales. The upturn at around k1* = 1 is caused by the imperfect spatial resolution of the

X-wire probe (Zhu and Antonia, 1995). At x/d = 20 (Figure 7-12b), the dependence of

the ratio on Reynolds number is comparable to that at x/d = 10. The magnitude of the

ratio at the vortex shedding frequency is still very apparent. Similar to that at x/d = 10,

the peak shifts to lower wavenumbers as Re increases. At x/d = 30 and 40, vortex

shedding is not as apparent as that at x/d = 10 and 20. The peaks in Figure 7-12 (c, d)

are much less apparent. The distribution of m

v

cal

v is quite flat and close to the

horizontal line of 1mv

calv at all wavenumbers. The results shown in Figure 7-12

indicate that the turbulent structures in the straked cylinder wakes agree well with

isotropy except in the region x/d 20, which is consistent with the results shown in

Table 7-1.

The vortex shedding phenomena in the bare cylinder wake at x/d 20 is very

apparent (Figure 7-13a, b). As a result, the ratio m

v

cal

v at these two locations departs

significantly from the isotropic value for scales larger than the most energy containing

structures. It is true that at these two locations, the values of the ratio 22 / uv

are 1.89 and 1.4, respectively (Table 7-1). With the increase of Reynolds number, the

peak reduces and shifts to the lower values of the wavenumbers, indicating that the

influence from the large organized structures to the smaller scale turbulent structures

become less dominated as Reynolds number increases. With the evolution of the

downstream locations, the influence from the large-scale structures persists, as reflected

by the apparent departure from 1mv

calv at the small wavenumber region

(corresponding to the large-scale structures). Even at x/d = 40 where the large-scale

structures disappear due to vortex break down (e.g. Zhou et al., 2003), the ratio m

v

cal

v

is far from 1 for k1* < 0.04, reflecting the departure from isotropy even when there is no

apparent large organized structures detected at this location. From Figures 7-12 and 7-

13, it can be inferred that the turbulent structures in the straked cylinder comply more

with isotropy than that in the bare cylinder wake. With the evolution of the downstream

198

locations, vortex structures break down and the ratio m

v

cal

v in the straked cylinder

wake complies more favourably with isotropy than in the bare cylinder wake.

7.4 Conclusions

In the present study, the effect and mechanism of three-strand helical strakes

with a dimension of 10d in pitch and 0.12d in height on VIV suppression are studied

experimentally in a wind tunnel. For the bare cylinder, lock-in occurs over the range of

reduced velocity of 5~8.5 with the maximum vibration amplitude of 0.51. This value

agrees well with that estimated using the empirical relation. However, after three-start

helical strakes are attached to the cylinder, vortex shedding is not apparent and lock-in

phenomena of the cylinder does not occur. The vibration amplitude of the cylinder is

suppressed by 98%.

The peak frequency on the energy spectra is very stable for the bare cylinder

both in the streamwise and in the spanwise directions. In contrast, the peak frequency on

the spectra of the straked cylinder wake varies significantly in both directions. The

separation points from the straked cylinder may contribute to the variation of the peak

frequency in the spanwise direction, which are controlled either by the strake tips or by

the cylinder surface. The strakes increase the effective diameter of the cylinder since

they behave like flat plates perpendicular to the flow, which corresponds to a lower

consensus value of St.

The generation of spanwise flow by the strakes could explain the streamwise

variations of frequency, since parts of the wake will be transported laterally and arrive

in a neighbouring region, causing a variation of frequency in the streamwise direction.

These results suggest that the strakes do not necessarily suppress vortex shedding, but

the wake is strongly disorganized as the helical strakes prevent the vortex from being

correlated along the span, hence is not able to excite VIV significantly. Flow

visualization results for the straked cylinder support the expectation that helical strakes

are able to disrupt the formation of the vortex structures and reduce the strength of the

vortices resulting in a significant reduction in vibration response for the straked

cylinder.

The cross-correlation coefficient in the straked cylinder wake is much smaller

than that in the bare cylinder wake. The correlation length of the vortical structures in

the former is only about 1/4 of the latter. As a result, the straked cylinder wake agrees

199

better with isotropy than that in the bare cylinder wake. This is verified by the

comparison of the measured transverse velocity spectra with the calculation based on

isotropy.

200

Table 7-1. Velocity ratio 22 / uv at different downstream locations in the

stationary bare and straked cylinder wakes.

x/d

22 / uv 5 10 20 30 40

Straked cylinder 0.96 1.33 1.23 1.07 0.99

Bare cylinder 1.89 1.4 0.78 0.71 0.7

201

Pitch length

Heightd

y

z

Pitch length

Heightd

y

z

Figure 7-1. Geometry of the cylinder with three-strand helical strakes and definition of

the coordinate system.

Figure 7-2. Experimental setup of the straked cylinder.

-60

-40

-20

0

20

40

0.001 0.01 0.1 1 10 100

straked cylinder

bare cylinder

fd/U

v(f

) (d

B)

Figure 7-3. Velocity spectra in the wake of the stationary bare and straked cylinders on

the centreline at x/d =5 for Re = 20430.

202

0

1

2

3

4

0 5 10 15

Slope = 0.21

Vr

f 0/f

n

Figure 7-4. Vortex shedding frequency from the bare cylinder normalized by the natural

frequency.

0

0.2

0.4

0.6

0.8

1.0

0 5 10 15

Upper branch

Lower branch

Vr

A/d

Figure 7-5. Comparison of vibration response of a bare cylinder. ▼: Present; : Khalak

and Williamson (1997); : Franzini et al. (2009); : Feng (1968).

203

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25V

r

A/d

Figure 7-6. Comparison of vibration response between bare cylinder and straked

cylinder. Present bare cylinder: ▼; Present straked cylinder: ; Straked cylinder by

Ding et al. (2004): ; Straked cylinder by Constantinides and Oakley (2006): .

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.01 0.1 1 10

x/d = 5

10

20

30

(a)

f=0.2Hz

fd/U

v(f

)

0

0.05

0.10

0.15

0.01 0.1 1 10

x/d = 5

10

20

30

(b)

f=2.2Hz

fd/U

v(f

)

Figure 7-7. Energy spectra obtained on the centreline at different streamwise locations

for Re = 20430 in the stationary bare and straked cylinder wakes. (a) Bare cylinder; (b)

Straked cylinder.

204

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.01 0.1 1 10

z/d = 0.1251.252.53.125

(a)

fd/U

v(f

)

0

0.05

0.10

0.15

0.01 0.1 1 10

z/d = 0.1251.252.53.125

fd/U

(b)f=3.20Hz

v(f

)

Figure 7-8. Energy spectra obtained at different spanwise locations in the stationary

bare and straked cylinder wakes at x/d =5 for Re = 20430. (a) Bare cylinder wake; (b)

Straked cylinder wake.

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

-10 -5 0 5 10

z/d = 3.125

z/d = 2.5

z/d = 1.25z/d = 0.125

(a) Bare cylinder wake

tU/d

Rv

1,v

2(

)

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

-10 -5 0 5 10

z/d = 3.125

z/d = 2.5

z/d = 1.25

z/d = 0.125

(b) Straked cylinder wake

tU/d

Rv

1,v

2(

)

Figure 7-9. Cross-correlation coefficients in the stationary bare and straked cylinder at

x/d =5 for Re = 20430. (a) Bare cylinder wake; (b) Straked cylinder wake.

205

-0.2

0.2

0.6

1.0

0 0.1 0.2 0.3 0.4

u

1,u

2

(straked cylinder)

v

1,v

2

(straked cylinder)

u

1,u

2

(bare cylinder)

v

1,v

2

(bare cylinder)

z/d

u

1,u

2

(z),

v

1,v

2

(z)

Figure 7-10. Cross-correlation coefficients of the velocity components in the stationary

bare and straked cylinder wakes obtained at x/d =5 for Re = 20430.

(a) Bare cylinder

(b) Straked cylinder

Figure 7-11. Flow visualization of the bare and straked cylinder wakes.

206

10-2

100

102

104

106

10-4

10-3

10-2

10-1

100

Re = 10240

20430

30610

40800

calv

mv

(a) x/d=10

k1*

u(k

1*),

v(k

1*),

vm

/vca

l

10-2

100

102

104

106

10-4

10-3

10-2

10-1

100

Re = 10240

20430

30160

40800

calv

mv

(b) x/d = 20

k1*

u(k

1*),

v(k

1*),

vm

/vca

l

10-3

10-1

101

103

105

10-4

10-3

10-2

10-1

100

Re = 10240

20430

30610

40800

calv

mv (c) x/d = 30

k1*

u(k

1*),

v(k

1*),

vm

/vca

l

10-3

10-1

101

103

105

10-4

10-3

10-2

10-1

100

Re = 10240

20430

30610

40800

calv

mv

(d) x/d = 40

k1*

u(k

1*),

v(k

1*),

vm

/vca

l



mv / between the measured

and calculated transverse velocity spectra in the stationary straked cylinder wake at

various downstream locations and Reynolds numbers.

207

10-2

100

102

104

106

10-4

10-3

10-2

10-1

100

Re = 10240

20430

30610

40800

calv

mv

(a) x/d = 10

k1*

u(k

1*),

v(k

1*),

vm

/vca

l

10-3

10-1

101

103

105

10-4

10-3

10-2

10-1

100

Re = 10240

20430

30610

40800

calv

mv

(b) x/d = 20

k1*

u(k

1*),

v(k

1*),

vm

/vca

l

10-3

10-1

101

103

105

10-4

10-3

10-2

10-1

100

Re = 10240

20430

30610

40800

m

v

cal

v (c) x/d = 30

k1

*

u(k

1*),

v(k

1*),

vm

/vca

l

10-3

10-1

101

103

105

10-4

10-3

10-2

10-1

100

Re = 10240

20430

30610

40800

mv

calv (d) x/d = 40

k1*

u(k

1*),

v(k

1*),

vm

/vca

l



mv / between the measured

and calculated transverse velocity spectra in the stationary bare cylinder wake at various

downstream locations and Reynolds numbers.

209

CHAPTER 8

CONCLUSIONS AND RECOMMENDATIONS

8.1 Summary of Findings

Several experimental studies on the turbulent structures in the wakes of a

circular cylinder and a pair of circular cylinders have been carried out and discussed in

this thesis. There are two types of circular cylinders, namely, bare cylinder and helical

straked cylinder. The results from the experimental studies of both types of cylinders are

compared to evaluate the mechanisms induced by the helical strakes as a passive

method in mitigating the vortex-induced vibration (VIV). General conclusions are

summarised as follows.

In Chapter 2, three-dimensional vorticity in the wake of a yawed stationary

circular cylinder was measured simultaneously using a multi-hot wire vorticity probe

over a streamwise range of x/d = 10-40. It was shown that the U and W magnitudes

increased with , indicating a reduction of the wake width and the increase of three-

dimensionality of the wake flow. The vortex is weakened with increasing as shown in

the decrease of all three vorticity components as is increased. The IP is applicable to

the wake when < 40. When is smaller than 40, the Strouhal number StN is

approximately constant within the experimental uncertainty. The intensity of vortex

shedding is reduced and the frequency region is dispersed with increasing . This is

caused by the break-down of the large organized structures at large . The

autocorrelation coefficients u and v of the u and v velocity signals show apparent

periodicity for all yaw angles. With increasing , u and v decrease and approach

zero quickly. In contrast, the autocorrelation coefficient w of w increases with in the

near wake, implying an enhanced three-dimensionality of the wake.

Chapter 3 provided analysis and discussion on the dependence of the wake

vortical structures on cylinder yaw angle (= 0-45) in the intermediate region (x/d =

10) using a phase-averaged technique. The technique is used to analyze the velocity and

vorticity signals and to obtain the coherent and incoherent contours. For all yaw angles,

the phase-averaged velocity and vorticity contours display apparent Kármán vortices.

210

When is smaller than 15, the maximum coherent concentrations of the three vorticity

components do not vary with. However, when = 45, there are large reductions of

about 33% and 50% in the maximum concentrations of the coherent transverse and

spanwise vorticity components respectively while that of the streamwise vorticity

increases by about 70%. This result suggests that at large yaw angles, the strength of the

Kármán vortex shed from the yawed cylinder decreases and the three-dimensionality of

the flow is enhanced. The maximum coherent concentrations of u and v contours

decrease by more than 20% while that of w increases by 100%. Correspondingly, the

coherent contributions to the velocity variances 2u and 2v decrease, while that

of 2w increases. These results indicate the generation of the secondary axial

vortices in yawed cylinder wakes when is larger than 15. The incoherent vorticity

contours 2xr are stretched along an axis inclining to the x-axis at an angle in the

range of 60-25 for = 0-45. The magnitudes of *2 xr and *2 yr through

the saddle points are comparable with the maximum concentration of the coherent

spanwise vorticity *~z at all cylinder yaw angles, supporting the previous speculation

that the strength of the rib-like structures in the cylinder wake is about the same as that

of the spanwise structures, even in the yawed cylinder wakes.

Further investigation on a yawed circular cylinder wake in the aspect of

streamwise evolution (x/d = 10-40) by using phase-averaging method was presented

in Chapter 4. Similar wake characteristics to those at x/d = 10 were found at x/d = 20.

At x/d = 20 when increases from 0° to 45°, the maximum of coherent streamwise

vorticity increases by about 2 times while that of the spanwise vorticity decreases by

about 17% compare with that of 0°. The results may suggest the impaired spanwise

vortices and the enhancement of the three-dimensionality of the wake with increasing

.. At x/d = 40, the maximum concentration of the coherent spanwise vorticity remains

about 5.0%, 6.3%, 10.0% and 12.5% of that at x* = 10 for = 0°, 15°, 30° and 45°,

respectively, indicating a slower streamwise decay rate of coherent spanwise vorticity

for larger . The slower decaying rate in the streamwise direction at larger can be

ascribed to the larger streamwise spacing between spanwise vortices at larger

resulting in a weak interaction between the vortices. For all yaw angles tested, the

coherent contribution to 2v is remarkable at x/d = 10 and 20, and

significantly larger than that to 2u and 2w . This contribution to all three

211

Reynolds normal stresses becomes negligibly small at x/d = 40. The coherent

contribution to 2u and 2v decays slower as moving downstream for a larger ,

consistent with the slow decay of the coherent spanwise vorticity for a larger .

In Chapter 5, the three-dimensional velocity and vorticity characteristics at the

downstream locations x/d = 10 of a yawed circular cylinder with (= 0-45) was

analyzed using the wavelet multiresolution method. The wavelet multiresolution method

is a linear decomposition process to convert a signal into a sum of a number of wavelet

components at different scales. The velocity and vorticity signals are decomposed into

17 wavelet levels where higher wavelet levels correspond to lower frequency bands or

larger scale structures while lower wavelet levels correspond to higher frequency bands

or small-scales structures. It is shown that IP is applicable for < 40 as the ratio of

normalized central frequency for the wavelet level 6 (located at the vortex shedding

frequency band) at larger to that of = 0 0

/ NN ff are far from 1. The maximum

energy of the energy spectra for intermediate- (wavelet levels 4 and 5) and large-scale

structures and decrease and the spectra disperse widely over an enlarged frequency band

with increasing . At = 45, the existence of vortex dislocations in the wake region is

apparent in the velocity variances results where with the increase of to 45°, the

2

iv magnitude for wavelet level 7 and 2

iw magnitude for level 6 at y/d 1

increase apparently compared with that at = 0°.The result may also be related with the

generation of the secondary axial vortex structures which in return enhance the three-

dimensionality of the turbulent wake at large yaw angles. The dominant contributors to

the Reynolds stresses at all yaw angles are the large-scale structures and followed by the

intermediate-scale structures while the three vorticity components are mostly dominated

by the small- and intermediate-scale structures and has the smallest values at large-scale

structures. The generation of the secondary axial and streamwise vortices at = 45° are

shown in the flow visualization results where the former propagate along the cylinder

axial direction and disrupt the formation of the Kármán type vortex structures, resulting

in enhanced three-dimensionality of the wake. It may suggest that the occurrence of the

vortex dislocation also responsible to the disruption of the large-scale structures.

Chapter 6 demonstrated an investigation on the wake flow behind two yawed

cylinders arranged side-by-side which has been done by measuring all three velocity

and vorticity components using an eight-hotwire vorticity probe in the intermediate

region (x/d = 10) at a Reynolds number of 7200. The stationary cylinders were tested at

212

four yaw angles (), namely, 0°, 15°, 30° and 45° for two centre-to-centre cylinder

spacings T, which are 3.0d and 1.7. For large cylinder spacing T* = 3.0, it was shown

that the IP may be implemented to the wake. A single peak is observed in the transverse

velocity spectrum which corresponds to the Strouhal number. The Strouhal number for

T* = 3.0 is larger than that of a T

* = ∞ (a single cylinder) at all yaw angles. The distance

between vortices for T* = 3.0 in lateral direction is about 3d. As the phase-averaged

velocity and vorticity contours for T* = 3.0 at y

* ≥ 1.5 are qualitatively comparable with

those for T* = ∞ at y

* ≥ 0, it may confirm that the wake for T

* = 3.0 behaves as that of

an independent and isolated cylinder. Although the wake behind two cylinders for T* =

3.0 has two distinct vortex street and each street has the same qualitative trend to the

single cylinder street, the maximum magnitudes for the coherent contribution to the

three vorticity variances components for T* = 3.0 are smaller than those for T

* = ∞.

When is increased from 0° to 45°, *~x magnitude increases rapidly while the *~

z

magnitude decreases. This result suggests that the T* = 3.0 wake experienced the

occurrence of the vortex dislocation or the existence of the secondary axial vortices at

large . For the intermediate cylinder spacing T* = 1.7, a single peak is observed in the

velocity spectrum. The peak represents a single vortex street in the wake and not a wide

and a narrow wakes, as been observed in some previous studies, mostly at around x/d

5. It is suggested that in the intermediate region (x/d = 10) the vortex structures

regeneration or evolution may not be completed yet, while those in the narrow wake is

probably diminished before the downstream location. The IP is applicable to the wake

when < 40°. The *W magnitudes are not equal to zero for all yaw angles. The result

indicates that the vortex in the wake is in inclined position and the wake may have an

impaired two-dimensional wake even at = 0°, which may be caused by the vortex

evolution. The vorticity contours for T* = 1.7 have a more organized pattern at = 0°

while become scattered and weaker in strength with the increase of . When = 45°,

the maximum magnitudes of all three vorticity contours are similar suggesting that the

wake has perfectly three-dimensional vortices. The vorticity contours for T* = 1.7 are

less organized and are the weakest compared with those for T* = ∞ and 3.0.

A study on the mechanism of VIV mitigation using helical strakes was presented

in Chapter 7. A rigid circular cylinder of diameter d = 80 mm attached with three-strand

helical strakes of a dimension of 10d in pitch and 0.12d in height was tested in a wind

tunnel. It was shown that while the bare cylinder experienced a lock-in phenomenon

213

over the range of reduced velocity of 5~8.5 with maximum vibration amplitude of 0.51,

the straked cylinder did not experience any lock-in. The vibration amplitude of the

straked cylinder is suppressed by about 98% compared to that of the bare cylinder. The

peak frequency of the energy spectra of the bare cylinder is apparent while that of the

straked cylinder varies significantly in streamwise and the spanwise direction. It may be

caused by the separation points from the straked cylinder which controlled either by the

strake tips or by the cylinder surface. It may also suggest that the generation of spanwise

flow by the strakes affect the frequency variations in the streamwise direction. Some

parts of the wake will be transported laterally and arrive in a neighbouring region hence

causing a variation of frequency in the streamwise direction. These results suggest that

the strakes do not necessarily suppress vortex shedding, but the wake is strongly

disorganized as the helical strakes prevent the vortex from being correlated along the

span, hence is not able to excite VIV significantly. The dominant frequency of the

vortex structures along the axial direction differs by about 36% of the averaged peak

frequency over a length of 3.125d. This is supported by the phase difference between

the velocity signals measured at two locations separated in the spanwise direction. The

correlation length of the vortex structures in the bare cylinder is much larger than that

obtained in the straked cylinder wake. Therefore, the straked cylinder wake agrees

better with isotropy than the bare cylinder wake, which is supported by the comparison

of the measured transverse velocity spectra with the calculation based on isotropy.

8.2 Recommendations for Future Research

According to the literature review and the present study, several

recommendations for future work are drawn as below:

i) This study did not study the effect of Reynolds numbers on the turbulent

structures. Such comparisons are recommended to improve understanding in the

vortex shedding mechanisms in the wake behind a yawed cylinder.

ii) The effect of yaw angles on the suppression of vortex-induced vibration using

straked cylinder need to be studied. Future work may include this as it may be

interesting to investigate the wake characteristics of yawed straked cylinder.

iii) The wake characteristics of two straked cylinders in side-by-side arrangement

also could be done in the future. No work has been done in this context. They

may be different from that of the bare cylinders.

214

iv) All experiments in this thesis have been done in a wind tunnel. These results in

air may be different from that in a water tunnel, especially when VIV

suppression is concerned. Therefore, it may be interesting to include this in the

future study.

215

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WAKE CHARACTERISTICS OF YAWED CIRCULAR CYLINDERS … · Siti Fatin Mohd. Razali Candidate‟s Name...

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