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Transition to turbulence in steady flow
around a cylinder near a plane boundary
by
FAN YANG
B.Eng.
This thesis is presented for the degree of
Doctor of Philosophy
of
The University of Western Australia
School of Civil, Environmental and Mining Engineering
2018
Thesis Declaration
I, FAN YANG, certify that:
This thesis has been substantially accomplished during enrolment in the degree.
This thesis does not contain material which has been accepted for the award of any
other degree or diploma in my name, in any university or other tertiary institution.
No part of this work will, in the future, be used in a submission in my name, for any
other degree or diploma in any university or other tertiary institution without the prior
approval of The University of Western Australia and where applicable, any partner
institution responsible for the joint-award of this degree.
This thesis does not contain any material previously published or written by another
person, except where due reference has been made in the text.
The work(s) are not in any way a violation or infringement of any copyright,
trademark, patent, or other rights whatsoever of any person.
This thesis contains published work and/or work prepared for publication, some of
which has been co-authored.
Signature:
Date:
Thesis Organization and Candidate Contribution
This thesis is presented as a series of papers which have been co-authored and have
been submitted or ready to be submitted for publication. The bibliographical details of
the works and where they appear in the thesis are outlined below.
Chapter 2: Yang F., An, H., Cheng, L., 2017. Drag crisis of a circular cylinder near
a plan boundary. Ocean engineering 154 (2018): 133-142.
The estimated percentage contribution by the candidate is 85%.
Chapter 3: Yang F., An, H., Cheng, L., Wang H., Zhang M., 2017. Turbulent
boundary layer transition of steady flow around a cylinder near a plane boundary.
Submitted to Journal of Fluid Mechanics.
The estimated percentage contribution by the candidate is 70%.
Chapter 4: An, H., Yang F., Cheng, L., Tong F., 2017. A Re-examination of the
Laminar Separation Bubble on a circular cylinder. Submitted to Physics of Fluid.
The estimated percentage contribution by the candidate is 30%.
Chapter 5: Yang F, An, H., Cheng, L., 2018. Large eddy simulation of End Effect of
flow around a near wall cylinder in a water channel.
The estimated percentage contribution by the candidate is 90%.
FAN YANG 06/08/2018
Candidate Signature Date
Prof. Liang Cheng 06/08/2018
Coordinating Supervisor Signature Date
i
Abstract
Three-dimensional numerical simulations and physical experimental tests have
been carried out in the thesis to investigate the flow structures around a circular
cylinder near a plane boundary. The hydrodynamic forces, flow structures and
boundary layer transitions have been analysed. Flow structures and boundary layer
transitions have been studied in the magnitude of O(105) of Reynolds number under a
few scenarios, i.e. a circular cylinder near a plane boundary in a water flume with free
stream turbulence intensity of 4% (Chapter 2), a circular cylinder near a plane
boundary in a wind tunnel testing facility with free stream turbulence intensity of 1%
(Chapter 3), and an isolated cylinder through three-dimensional (3D) large eddy
simulation (LES) (Chapter 4). To check the end effect, 3D LES has been employed to
investigate the influence of gap to diameter ratio (𝐺/𝐷), Reynolds number (𝑅𝑒) and
aspect ratio (𝐿/𝐷) on the end effect for a near wall circular cylinder (Chapter 5).
The influence of 𝐺/𝐷 on the force coefficients when the boundary layer on the
cylinder surface transits from laminar to turbulent has been quantified through
experimental testing in Large O-Tube facility. The pressure distribution and the
hydrodynamic forces on a circular cylinder placed near a plane boundary are
investigated over a range of Re = 1.1×105 ~ 4.3×105 and 𝐺/𝐷 = 0 ~ 1.0. A significant
reduction drag coefficient (𝐶𝐷) from about 0.9 to 0.35 is observed for 𝐺/𝐷 ≥ 0.5 in
the range of Re = 1.9×105 ~ 2.7×105. This is the so-called drag crisis induced by the
boundary layer transition on a circular cylinder. At smaller 𝐺/𝐷 values of 0.25 and
0.1, the drag coefficient shows much less reduction than those observed at larger 𝐺/𝐷
values. No obvious drag reduction is found at 𝐺/𝐷 = 0.01 and 0, but it has been
demonstrated that the transition happens at lower Re. Based on the observed features
of pressure distributions and force coefficients, the boundary layer transition from
laminar to turbulent and its effect on the force coefficient is inferred for all the gap
ratios (𝐺/𝐷 = 0 ~ ∞).
For the testing scenario in the wind tunnel facility, the boundary layer transition
ii
on the surface of a near wall cylinder in steady flow is investigated through
measurements of the pressure distribution around the cylinder, covering a total of 11
𝐺/𝐷 values (𝐺/𝐷 = 3, 2, 1, 0.8, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1 and 0), two boundary layer
thickness to diameter ratios (𝛿/𝐷 = 0.1 and 0.5) and 𝑅𝑒 in the range of 1.33×105 ~
4.8×105. Due to the relatively low free stream turbulence intensity, the effect of
boundary layer transition is more pronounced than the test in the O-tube facility. It is
found that the proximity of the plane boundary affects the transition process through a
number of physical mechanisms such as the changes of flow rate through the gap and
the favourable pressure gradient over the cylinder surface induced by the blockage
effect and the properties of the boundary layer on the plane boundary. Asymmetric
transition on the top and gap side is an important feature observed here. The transition
initiates from the gap side, but switches to the top side before the transition happens
on both sides at intermediate gap ratios (𝐺/𝐷 = 0.5 ~ 2) with the increase of Re
which is referred to as the side swapping. The transition process is mainly influenced
by 𝐺/𝐷 at intermediate gap ratios (𝐺/𝐷 = 0.5 ~ 2) and by both 𝐺/𝐷 and velocity
profiles of the approaching flow at small gap ratios (𝐺/𝐷 = 0.1 ~ 0.4).
The flow around a circular cylinder is simulated numerically based on a 3D LES
model in the range of 𝑅𝑒 = 1 × 105 ~ 6.0 × 105. The numerical results show that the
LES model captures the transition from the subcritical to critical and then to
supercritical regime with reasonable Reynolds number sensitivity. The drag crisis is
also captured in good agreement with published data. The flow visualization shows
the Kelvin-Helmholtz (KH) vortices forms in the separated shear layer and gradually
travel towards the separation points on the cylinder surface. In supercritical regime,
KH vortices form on the cylinder surface after boundary layer separation, with the
same sign on each side of the cylinder. KH vortices slide along the cylinder surface
and then gradually decay. The numerical visualization reveals that the laminar
separation bubble, which is captured in the critical and supercritical regimes, is the
average of the KH vortices formed on the surface of the cylinder.
Turbulent flow around a near bed horizontal circular cylinder mounted on a
vertical side wall is simulated numerically in this study. The purpose of the study is to
iii
investigate the flow characteristics around the junction area between the cylinder and
two plane boundaries. The effects of gap to diameter ratio (G/D = ∞, 0.2 & 0.5),
Reynolds number (𝑅𝑒𝐷 = 1250, 2500 & 5000) and the length to diameter (aspect)
ratio (L/D = 10, 20 & 40) are investigated. It is found that the three-dimensional flow
structures formed in the junction area between the vertical wall and the cylinder are
strong affected by the gap ratio. The end effect induced by the vertical wall is limited
within 0.7D in the spanwise direction, 0.5D upstream and 3D downstream of the
cylinder. The extent of the end effect is not sensitive to the gap ratio, Reynolds
number and the aspect ratios.
iv
Table of Contents
Abstract ......................................................................................................................... i
Table of Contents ........................................................................................................ iv
Acknowledgements .................................................................................................... vii
List of Figures ........................................................................................................... viii
List of Tables ............................................................................................................. xvi
List of Abbreviations ............................................................................................... xvii
List of Symbols ........................................................................................................ xviii
Chapter 1 Introduction .......................................................................................... 1
1.1 Background and motivations ............................................................................ 1
1.2 Research goals .................................................................................................. 1
1.3 Thesis outline.................................................................................................... 2
Chapter 2 Drag crisis of a circular cylinder near a plane boundary ................. 3
3.1 Introduction ...................................................................................................... 3
3.2 Testing facility and model setup ....................................................................... 8
3.3 Test results ....................................................................................................... 11
3.3.1 Boundary Layer and Turbulent Intensity Measurement .......................... 11
3.3.2 Pressure Distributions around the Cylinder ............................................. 11
3.3.3 Drag and lift force coefficients ............................................................... 20
3.4 Engineering implications ................................................................................ 26
3.5 Conclusions .................................................................................................... 26
Acknowledgements .............................................................................................. 27
References ............................................................................................................ 28
Chapter 3 Turbulent boundary layer transition of steady flow around a
cylindr near a plane boundary ................................................................................. 33
v
3.1 Introductions ................................................................................................... 34
3.2 Experimental setup ......................................................................................... 38
3.3 Boundary Layer and Turbulent Intensity on the Plane Boundary .................. 39
3.4 Test Results ..................................................................................................... 41
3.4.1 Experimental validations .......................................................................... 41
3.4.2 Tests with δ/D=0.1 .................................................................................. 43
3.4.3 Tests with δ/D=0.5 .................................................................................. 53
3.4.4 Wall effect on the force coefficients.......................................................... 63
3.5 Discussions ..................................................................................................... 67
3.6 Conclusions .................................................................................................... 68
Acknowledgement ................................................................................................ 70
References ............................................................................................................ 71
Appendix .............................................................................................................. 77
Chapter 4 Laminar separation bubble on a circular cylinder ......................... 79
4.1 Introduction .................................................................................................... 79
4.2 Governing Equations and numerical method ................................................. 84
4.3 Mesh dependency study and model validations ............................................. 85
4.3.1 Mesh dependency study .......................................................................... 85
4.3.2 Turbulence intensity ................................. Error! Bookmark not defined.
4.3.3 Force coefficients .................................................................................... 87
4.3.4 Vortex shedding frequency ...................................................................... 89
4.3.5 Pressure coefficients ............................................................................... 90
4.4 Numerical result and discussion ..................................................................... 91
4.4.1 Transition features ................................................................................... 92
4.4.2 Laminar separation bubbles .................................................................... 97
4.4.3 Frequency of Kelvin-Helmholtz (KH) vortices .................................... 106
vi
4.5 Conclusions .................................................................................................. 109
Acknowledgement ............................................................................................... 110
References ........................................................................................................... 110
Chapter 5 Flow around a near bed horizontal cylinder mounted on a vertical
wall ............................................................................................................................. 114
5.1 Introductions .................................................................................................. 115
5.2 Methodology ................................................................................................. 115
5.3 Mesh dependency check and model validations .......................................... 124
5.4 Numerical Results ........................................................................................ 129
5.4.1 Influence of Gap Ratio .......................................................................... 129
5.4.2 Influence of Reynolds number .............................................................. 138
5.4.3 Influence of the aspect ratio .................................................................. 141
5.5 Conclusions .................................................................................................. 142
Acknowledgement .............................................................................................. 143
References .......................................................................................................... 144
Chapter 6 Conclusions ........................................................................................ 149
6.1 Summary of conclusions .............................................................................. 149
6.2 Recommendations for future studies ............................................................ 149
vii
Acknowledgements
I would like to express my deepest appreciation to my supervisor, Prof. Liang
Cheng, who has the altitude and the substance of a genius: he continually and
convincingly conveyed a spirit of adventure in regard to research and an excitement in
regard to daily life. Without his guidance and persistent help, this dissertation would
not have been possible. It’s been a great honour to work with him and learn from him.
I am also deeply grateful to Dr. Hongwei An. His enthusiasm, encouragement and
support help me complete my work. He is a skilful supervisor, innovative colleague
and patient elder cousin for me.
My sincere thanks go to Prof. Yuxia Hu, Dr. Scott Draper, Dr. Feifei Tong, Prof.
Hanfeng Wang and Dr. Hongyi Jiang for their constructive suggestions for my work. I
also thank my fellow colleagues, Mr. Chengwang Xiong, Ms. Xiaoying Ju and Dr.
Qin Zhang for their support during the past four years.
This research was supported by an Australian Government Research Training
Program (RTP) Scholarship. The computational resources provided by the Pawsey
Supercomputing Centre is also much appreciated.
I would also like to thank the thesis examiners, Prof. B. Mutlu Sumer from
Technical University of Denmark, Prof. Zhenhua Huang from Dalian University of
Technology, and Dr. Rajet Jaimin from National Univerisity of Singapore, for their
valuable comments and suggestions in improving the thesis. In addition, the
anonymous reviewers who provided constructive comments on the journal papers
which are reproduced in the thesis are gratefully acknowledged.
Finally, my sincere thanks to my parents for allowing me to realize my own
potential. All the support they have provided me over the years was the key to make
this all possible. Also, thanks to my wife, Ms. Mengya Zhang, she is a great
inspiration to me in my life.
viii
List of Figures
Figure 2.1. A summary of the Drag coefficients measured in previous work
(Wieselsberger, 1922, Fage, 1931, Bursnall and Loftin Jr, 1951, Spitzer, 1965,
Achenbach and Heinecke, 1981, Schewe, 1983, Vaz et al., 2007) ................................ 7
Figure 2.2. A 3D drawing of the model pipe and its actuator system installed in the
O-tube test section. ........................................................................................................ 8
Figure 2.3. The configurations of the model setup in the O-tube test section. (a) The
model pipe was located above the test section floor directly; (b) Overview of the
model pipe in the O-tube; (c) The data acquisition system located in the chamber of
the model pipe. ............................................................................................................ 10
Figure 2.4. Measured boundary layer andturbulent intensity profiles. ....................... 10
Figure 2.5. The pressure coefficient for G/D = 1.0 and Re = 1.1 × 105, together with
the data presented by Bearman and Zdravkovich (1978). ........................................... 12
Figure 2.6. The distribution of averaged CP on the cylinder surface at different Re (=
1.1× 105~4.3 × 105) and different G/D (= 0 ~ 1.0) ................................................ 14
Figure 2.7. The pressure coefficient at Re = 4.8 × 105 for different gap ratios in the
work of Bearman and Zdravkovich (1978). ................................................................ 16
Figure 2.8. The critical Re value of supercritical regime at different G/D values. . 17
Figure 2.9. (a) The base pressure coefficient CPb, (b) the difference between the base
pressure and minimum pressure on the top side CPb − CP−min−top, (c) the difference
between the base pressure and minimum pressure on the gap side and CCPb −
CP−min−gap (d) wake width. ....................................................................................... 20
ix
Figure 2.10. The nine steps of input velocity (a), and the drag coefficients at different
flow velocity at (b) G/D=1, (c) G/D=0.25 and (d) G/D=0. ..................................... 21
Figure 2.11. Comparison of the present drag coefficient for G/D=1 with published
results reported in literature. ........................................................................................ 22
Figure 2.12. Variations of force coefficients with Re and G/D. (a) CD from present
work; (b) CL from present work. ............................................................................... 23
Figure 2.13. Critical conditions for drag crisis to be considered as a function of U, D
and G/D. ..................................................................................................................... 25
Figure 3.1. A sketch of the cylinder and plane wall model setup for the wind tunnel
tests. ............................................................................................................................. 38
Figure 3.2. The boundary layer and turbulent intensity profiles at x=0 with the model
pipe removed, measured at U = 10 m/s and 20 m/s. .................................................. 40
Figure 3.3. The typical pressure distributions around the cylinder at three different
flow regimes, together with the results from Cadot et al. (2015), (a), subcritical regime,
(b) critical regime with transition on the gap side and (c) supercritical regime.
Present tests were conducted with G/D= 3.0 and the results of Cadot et al. (2015)
were at G/D =∞. ...................................................................................................... 42
Figure 3.4. The temporal and spatial distributions of pressure coefficient (CP(Ut/D, θ))
on the cylinder surface for cases G/D = 0.8 and δ/D =0.1 with Re in the range of
2.0 × 105 ~ 3.47 × 105............................................................................................. 44
Figure 3.5. The temporal and spatial distributions of pressure coefficient (CP(Ut/D, θ))
on the cylinder surface for cases G/D = 0.3 and δ/D =0.1 with Re = 1.66 × 105,
3.47 × 105 and 4.40 × 105 ....................................................................................... 46
x
Figure 3.6. The varitaion of mean pressure coefficient distribution on the cylinder
surface vs. Re at G/D = 0.8 and δ/D = 0.1 (Scatter colour code: green→ #0,
blue→ #1B, red →#1T and black →#2). .................................................................... 48
Figure 3.7. The varitaion of mean pressure distribution on the cylinder surface vs. Re
with G/D = 3.0 ~ 0.0 with δ/D = 0.1 (Scatter colour code: green→ #0, blue→ #1B,
red →#1T and black →#2). The second plot is reproduced from Cadot et al. (2015)
for G/D =∞. .............................................................................................................. 51
Figure 3.8. The variation of stagnation point position (θ) and favourable pressure
gradient on the gap side cylinder surface (φ1) and that for the top side (φ2) for G/D
= 0.8 with δ/D=0.1. .................................................................................................... 51
Figure 3.9. The pressrue distribution around cylinder sitting on the plane boundary
(G/D = 0). ................................................................................................................... 52
Figure 3.10. The effect of G/D on the critical Re value for the transitions to the #1B,
#1T and #2 state. The wall boundary layer condition is δ/D= 0.1. ............................ 52
Figure 3.11. The G/D effect on pressrue distribution around cylinder at three
different Re values with δ/D = 0.1 , (a) 1.93 × 105 , (b) 3.09 × 105 , (c)
4.80 × 105. .................................................................................................................. 55
Figure 3.12. The effect of δ/D on CP and CP′ for the case with G/D = 0.8 and Re
= 3.07× 105. ................................................................................................................ 55
Figure 3.13. The temporal and spatial distributions of pressure coefficient (CP(Ut/
D, θ)) on the cylinder surface for cases G/D = 0.4 and δ/D =0.5 with Re =
2.00 × 105, 2.62 × 105 and 3.07 × 105. .............................................................. 56
Figure 3.14. The temporal and spatial distributions of pressure coefficient (CP(Ut/
D, θ)) on the cylinder surface for cases G/D = 0.3 and δ/D =0.5 with Re =
xi
1.33 × 105, 2.00 × 105, 2.67 × 105 and 3.07 × 105. ........................................... 58
Figure 3.15 The lift time histories and the corresponding frequency spectrums for Re
= 1.33× 105 with two different plane wall boundary layers and with G/D = 0.3 and
G/D = 0.8. ................................................................................................................... 58
Figure 3.16. The distribution of CP′ on the cylinder surface at G/D = 0.3, Re =
2.67 × 105 and 4.40 × 105. ...................................................................................... 60
Figure 3.17. The detial of CP′ on the cylinder surface at G/D = 0, Re = 2.67 × 105
and 4.40 × 105. .......................................................................................................... 61
Figure 3.18. The varitaion of mean pressure distribution on the cylinder surface vs.
Re with G/D = 3.0 ~ 0.0 (δ/D = 0.5) (Scatter colour code: green→ #0, blue→
#1B, red →#1T and black →#2). ................................................................................ 63
Figure 3.19. The effect of G/D on the critical Re value for the transitions to #1B, #1T
and #2 state. The wall boundary layer condition is δ/D= 0.5. ................................... 63
Figure 3.20. The effects of G/D and Re on CD and CL at different gap ratio (δ/
D=0.1). ......................................................................................................................... 65
Figure 3.21. The lift coefficients at G/D = 0.8, δ/D = 0.1 with different Re variation
sequences and pipe orientations. ................................................................................. 66
Figure 3.22. The temporal and spatial distributions of pressure coefficient on the
cylinder surface for cases with G/D = 0.8 and δ/D = 0.5 at different Re values. 77
Figure 4.1. A sketch of the laminar separation bubble in the critical turbulent flow
regime. The details about the labels are as follows; 1. Laminar boundary layer; 2.
Laminar separation point; 3. Laminar separation bubble; 4. Turbulent re-attachment; 5.
Turbulent boundary layer; 6. Turbulent separation point; 7. Wake separation bubble,
θ is the angle position on the cylinder surface. .......................................................... 81
xii
Figure 4.2. The computational mesh and a zoom-in view of the elements near the wall.
..................................................................................................................................... 86
Figure 4.3. The pressure coefficients on the cylinder surface simulated with different
mesh density at Re = 6 × 105, compared with the experimental data by Bursnall and
Loftin (1951) at Re = 5.95 × 105. The detailed information about the meshes can be
found in Table 4.1. ....................................................................................................... 87
Figure 4.4. The variation of CD, CL and St against Re, compared with previously
published data. ............................................................................................................. 89
Figure 4.5. Mean pressure coefficient CP on the cylinder surface at different Re
values. (a), subcritical regime flow, (b) critical regime flow....................................... 91
Figure 4.6. The spatial-temporal evolution of pressure coefficient on the surface of the
cylinder at (a) Re = 105, (b) 2×105 and (c) 6×105. The two vertical dash lines in (a)
enclose a typical vortex shedding period. The arrows in (b) and (c) indicate the
formation of small scale vortices sliding on the cylinder surface. Low pressure zones
associated with boundary layer transition are also labelled in (b) and (c). ................. 92
Figure 4.7. The near wake flow structure represented by iso-surface of λ2 = -1 for Re
= 105, 2×105 and 6×105, from top to bottom, while the colour contours are based on
pressure coefficient. ..................................................................................................... 94
Figure 4.8. Instantaneous vorticity (ωz) and pressure contours at Re = 105, 2×105
and 6×105 from left to right. ........................................................................................ 95
Figure 4.9. The averaged flow field represented by streamlines and pressure contours
for Re = 105, 2×105 and 6×105 from left to right. ...................................................... 95
Figure 4.10. The positions of LSB on the cylinder surface as a function of Reynolds
number. (a), the starting position of LSB; (b) the end position of the LSB on the
xiii
cylinder surface. .......................................................................................................... 98
Figure 4.11. The distribution of KH vortices at Re = 6 × 105 viewed in two different
directions. The vortices are visualized through iso-surfaces of λ2 = −1000 and
colour contours of ωz in the range of -100 to 100. The angle position cylinder
surface is also labelled in each plot. ............................................................................ 98
Figure 4.12. The KH vortices in the boundary layer on the cylinder surface at Re =
6× 105 from Ut/D = 75.91 to 75.99 with interval of ∆(UtD) = 0.02. ................... 99
Figure 4.13. The merging process of KH vortices in the boundary layer on the cylinder
surface at Re = 6× 105 from Ut/D = 75.955 to 75.975 with interval of U∆t/D =
0.005. ........................................................................................................................... 99
Figure 4.14. The variation of KH vortices with Reynolds number. (a), Re = 105; (b),
2 × 105 ; (c), 3 × 105; d, 4 × 105; e, 5 × 105; f, 6 × 105. The KH vortices are
visualized through iso-surfaces of λ2 = −1000 and colour contours of ωz in the
range of -100 to 100. The angle position on this side of the cylinder is also labelled
under each plot. The shadow area represents the region for LSB in the averaged flow
field. ........................................................................................................................... 100
Figure 4.15. The time histories of pressure on the cylinder surface for θ =
230° ~ 280° at Re = 6 × 105. ............................................................................... 103
Figure 4.16. The frequency spectrums of the pressure signals on the cylinder surface
at five selected locations at Re = 6 × 105. ............................................................. 105
Figure 4.17. The variation of fKH/fst against Reynolds number. ............................ 105
Figure 4.18. The variation of StKH against Reynolds number based on the present
proposed equation. ..................................................................................................... 105
Figure 5.1. The three-dimensional geometry of the calculation model with the gap
xiv
below the cylinder. .................................................................................................... 122
Figure 5.2. Computation mesh of Case 4 (a) x-z plane, (b) rectangle centre mesh and
(c) detailed mesh near the cylinder. ........................................................................... 126
Figure 5.3. Pressure distributions for five validation meshes. .................................. 127
Figure 5.4. Comparison of HVs in plane of y/D = 0 at ReD = 2500. a) PIV results
from Huang et al. 2014 (δ*⁄D = 0.113), b) present CFD (δ*⁄D = 0.129). .................. 128
Figure 5.5. Comparison of velocity streamlines in horizontal plane in the upstream
z/D = 0.05 &0.1. a) & c) PIV results of Huang et al. (2014), b) & d) present LES
result .......................................................................................................................... 129
Figure 5.6. Snapshot of time-averaged iso-surfaces of λ for G/D =∞ (a), 0.5 (b)
and 0.2(c). .................................................................................................................. 132
Figure 5.7. Mean values of velocity streamlines as well as contours of ωy in the plane
of y = 0 for (a) G/D = ∞, (b) G/D = 0.5 and (c) G/D = 0.2. .............................. 133
Figure 5.8. Mean values of velocity streamlines as well as contours of ωx in the plane
of x = 0 for (a) G/D = ∞, (b) G/D = 0.5 and (c) G/D = 0.2. .............................. 133
Figure 5.9. Variation of wake length with z/D in the wake and definition sketch of
wake length. ............................................................................................................... 134
Figure 5.10. Separation points on the cylinder surface in the influence of gap ratios. (a)
separation points on the up side, Sp1, and (b) separation points on the gap side, Sp2. 135
Figure 5.11. Sectional pressure distributions of the cylinder for (a) G/D = ∞, (b) for
G/D = 0.5 and (c) for G/D = 0.2............................................................................. 136
Figure 5.12. The difference between the base pressure Cpb and the minimum
xv
pressure in the gap Cp−min−gap in the influence of gap ratio. ................................. 136
Figure 5.13. Magnitude of the wall shear stress on the seabed plane boundary for (a)
G/D = 0.5 and (b) for G/D = 0.2. ........................................................................... 137
Figure 5.14. Comparison of amplification factor of the shear stress at x = 0 in the
influence of gap ratios. .............................................................................................. 137
Figure 5.15. Adverse pressure gradient in the influence of ReD. ............................. 139
Figure 5.16. Comparison of amplification factor of the shear stress at x = 0 in the
influence of ReD. (a) ReD = 1250 and (b) ReD = 5000. ...................................... 140
Figure 5.17. Adverse pressure gradient in the influence of aspect ratio (L/D). ........ 141
Figure 5.18. The sectional drag (a) and lift (b) coefficients in the influence of aspect
ratio (L/D). ................................................................................................................ 142
xvi
List of Tables
Table 3.1. The regimes of boundary layer transitions on the cylinder surface with the
increase of Reynolds number. ..................................................................................... 76
Table 4.1. A summary of the detailed information about the three meshes used in the
mesh dependency study. .............................................................................................. 85
Table 5.1. Mesh details of the five cases chosen for mesh dependence check.......... 124
Table 5.2. Force coefficients obtained with different meshes. .................................. 124
Table 5.3. Comparison of centre positions of three horseshoe vortices in the plane of y
= 0. ............................................................................................................................. 124
Table 5.4. Locations of the source and saddle points. ............................................... 124
Table 5.5. Variation of force coefficients and Strouhal number with gap ratio. ........ 131
Table 5.6. Variation of vortex centre locations with gap ratio. .................................. 131
Table 5.7. Variation of force coefficients and Strouhal number with Re. .................. 139
Table 5.8. Variation of vortex centre locations with Re. ........................................... 139
Table 5.9. Variation of force coefficients and Strouhal number with L/D. ................ 142
Table 5.10. Variation of vortex centre locations with L/D . ...................................... 142
xvii
List of Abbreviations
2D Two-dimensional
3D Three-dimensional
ADV Acoustic doppler velocimeter
BAV Bottom attached vortices
LES Large eddy numerical simulation
LSB Laminar separation bubble
HV Horseshoe vortices
KH Kelvin-Helmholtz vortices
RANS Steady Reynolds-Averaged Navier-Stokes
simulation
URANS Unsteady Reynolds-Averaged
Navier-Stokes simulation
xviii
List of Symbols
𝐶𝐷 Drag coefficient
𝐶𝐷 Mean values of drag coefficient
𝐶𝐿 Lift coefficient
𝐶𝐿 Mean values of lift coefficient
CL′ Root-mean-square of lift coefficient
𝐶𝑃 Pressure coefficient
𝐶𝑝′ Fluctuations of pressure coefficient
𝐶𝑃𝑏 Base pressure coefficient
𝐶𝑃−𝑔𝑎𝑝 Minimum values of pressure coefficient
on the gap side
𝐶𝑃−𝑡𝑜𝑝 Minimum values of pressure coefficient
on the top side
𝐷 Diameter of the cylinder
𝐹𝐷 Drag force
𝐹𝐿 Lift force
𝑓𝐾𝐻 Frequency of Kelvin-Helmholtz vortices
𝑓𝑠𝑡 Frequency of Karman vortex shedding
𝐺 Gap distance between the plane wall and
the cylinder
𝐼𝑢 Turbulence intensity
𝑘𝑠 Surface roughness
𝐿/𝐷 Aspect ratio
xix
𝑁𝑒𝑙𝑒𝑚𝑒𝑛𝑡 Total element number
𝑁𝑐𝑖𝑟𝑐𝑙𝑒 Element number on the cylinder
circumference
𝑁𝑠𝑝𝑎𝑛 Element number in the spanwise direction
of the cylinder
𝑝 Pressure
𝑝0, 𝑝∞ Reference pressure
𝑅𝑒, ReD Reynolds number
Sp Separation point
𝑆𝑡 Strouhal number
𝑆𝑡𝐾𝐻 Strouhal number for Kelvin-Helmholtz
vortices
𝑈 Free stream flow velocity
�� Mean values of free stream flow velocity
𝑈′, 𝑢′ Fluctuation values of free stream flow
velocity
WL Wake length
𝜌 Water density
𝜓 Favourable pressure gradient on the
cylinder surface
𝜏0 Magnitude of shear stress
𝜏𝑥, 𝜏𝑦 Shear stress in the 𝑥- and 𝑦- directions
𝜔 Vorticity
𝜇 Dynamic viscosity
xx
𝜎0 Amplification factor of the shear stress
𝜈 Kinematic viscosity
𝛿 Boundary layer thickness on the plane
wall
𝛿∗/𝐷 The relative displacement thickness of
boundary layer
𝜑 Wake width
𝜃 Angular position on the cylinder surface
∆t Non-dimensional computational time step
∆ 𝐷⁄ Size of the first layer mesh on the
cylinder surface
xxi
1
Chapter 1
Introduction
1.1 Background and motivations
With the increasing demand for fossil fuel and the diminishing of onshore and
near-shore resources, offshore oil and gas projects are moving further offshore and to
deep waters. Offshore pipelines are key infrastructures to transport oil and gas
products from production wells to storage/processing facilities either onshore or
offshore. The cost for installing pipelines offshore is normally very high, in the order
of a few millions/km (depending on the diameter and total length of the pipeline). The
consequence of pipeline failures is severe both economically and socially. It is
therefore crutial to develop a robust pipeline design method that ensures the integrity
of the pipeline with a sustainable cost. Prediction of hydrodynamic forces acting on
offshore pipelines under storm conditions is one of the key elements in pipeline
on-bottom stability design. An accurate prediction of hydrodynamic forces has a
significant bearing on the costs related to stabilization measures. This motivates the
research topics presented in this thesis. Since problem background and research
motivations for each topic are introduced separately, they are not addressed in detail
here.
1.2 Research goals
The present thesis mainly focuses on the following topics:
1. Boundary layer transition related drag crisis phenomenon for a near wall
circular cylinder and its effect on the hydrodynamic force coefficients and the
pressure distribution around the cylinder. The effect of free stream turbulence
intensity and the plane wall boundary layer profiles are considered in the
work.
2
2. Flow characteristics and structures around a cylinder during the boundary
layer transition from laminar to turbulent through numerical simulations,
aiming to understand and interpret the results obtained from physical model
tests in Goal 1.
3. Investigation of flow characteristics around the junction of a cylinder near a
plane boundary with a wall that is perpendicular to the cylinder, aiming to
quantify the end effects that are inevitable in physical model testing in
laboratories.
1.3 Thesis outline
This thesis comprises 6 chapters. The remainder of the thesis is organized in the
following manner.
Chapter 2 studies the drag crisis phenomenon for a circular cylinder with different
gap ratios in steady currents with relatively high turbulence intensity (Goal 1).
Chapter 3 focuses on the mechanism of the boundary layer transition on a
cylinder close to a plane boundary through wind tunnel tests. The influence of gap
ratio, boundary layer thickness and turbulence intensity on the transition process is
quantified (Goal 1).
Chapter 4 investigates the formation of the laminar separation bubble and
Kelvin-Helmholtz (KH) frequency in the wake during the transition process (Goal 2).
Due to the strong velocity gradient in the separated shear layer from the cylinder
surface, small-scale eddies form close to the wall and are named as Kelvin-Helmholtz
(KH) vortices.
Chapter 5 investigates the junction flow through three-dimensional simulations.
Quantifying the influence of the gap, Reynolds number and the aspect ratio on the
flow (Goal 3).
Chapter 6 summarizes the main outcome of this study and suggestions on future
work.
3
Chapter 2
Drag crisis of a circular cylinder near a plane
boundary†
Abstract: The pressure distribution and the hydrodynamic forces on a circular
cylinder placed near a plane boundary are investigated experimentally over a range of
Reynolds numbers (Re) of 1.1×105 ~ 4.3×105 and gap (G) to cylinder diameter (D)
ratio (𝐺/𝐷) of 0 ~ 1.0. The objective of the study is to quantify the influence of 𝐺/𝐷
on the force coefficients when the boundary layer on the cylinder surface transits from
laminar to turbulent. The hydrodynamic forces acting on the cylinder are obtained by
integrating the measured pressure around the cylinder surface. A significant drag
reduction from about 0.9 to 0.35 is observed for 𝐺/𝐷 ≥ 0.5 in the range of Re =
1.9×105 ~ 2.7×105. At smaller 𝐺/𝐷 values of 0.25 and 0.1, the drag coefficient shows
much less reduction than those observed at larger 𝐺/𝐷 values. No obvious drag
reduction is found at 𝐺/𝐷 =0.01 and 0. Based on the observed features of pressure
distributions and force coefficients, the boundary layer transition from laminar to
turbulent is inferred for all the gap ratios (𝐺/𝐷 = 0 ~ ∞).
† This chapter is presented as a paper which has been accepted as “Yang F., An, H., Cheng, L.,
2018. Drag crisis of a circular cylinder near a plane boundary. Ocean engineering, 154, 133-142.”
2.1 Introduction
Offshore pipelines are key infrastructures for transporting oil and gas products
across the seabed and are often installed on the seabed directly in medium and deep
waters. Costs for offshore pipelines are often very high, partly due to the expensive
stabilization measures which can amount up to 30% of the total cost for pipelines. The
estimated drag acting on pipelines under extreme storm conditions has a significant
bearing on the stabilization costs. The common practice for estimating hydrodynamic
4
forces on pipelines is to use Morrison equation.
The flow around a smooth circular cylinder is mainly governed by the Reynolds
number, defined as 𝑅𝑒 = 𝑈𝐷/𝜈, where 𝑈 is the free stream flow velocity, 𝐷 is the
diameter of the cylinder and 𝜈 is the kinematic viscosity of the flow. The wake flow
around a circular cylinder goes through a series of transitions from laminar to
turbulent states with the increase of Re. A pair of stable symmetric vortices is formed
in the wake of the cylinder at 𝑅𝑒 =5 ~ 47, which is referred to as the symmetrical
mode (Batchelor, 1967). As 𝑅𝑒 is increased further, regular two-dimensional vortex
shedding is observed at 𝑅𝑒 ≲ 194 (Williamson, 1989, Jiang et al., 2016). With a
further increase of Re, three-dimensional instability occurs in the wake, but flow is
still in laminar regime up to Re ≈ 300 (Williamson, 1989). The vortex street in the
wake transits to turbulence at higher Re (Bloor, 1964).
The turbulent flow in the wake is further classified into five different regimes,
namely the subcritical, critical, supercritical, upper transition and trans-critical
regimes based on the development of turbulence in the wake and around the cylinder
surface. The corresponding critical Re values for transitions to the critical,
supercritical, upper transition and trans-critical regimes are 3.0×105, 3.5×105, 1.5×106
and 4×106, respectively, as summarized by Sumer and Fredsøe (1997). However, it
should be noted that these critical Re values for different flow regimes are sensitive to
a number of factors, such as free stream turbulence, cylinder surface roughness, aspect
ratio of the model cylinder, blockage ratio of the physical experimental set-up etc.
This causes a certain level of discrepancies among the reported critical Re values in
the literature. Turbulence only exists in the wake in the subcritical regime. In this flow
regime, drag coefficient (𝐶𝐷) and normalized vortex shedding frequency show little
dependence on Re. In the critical flow regime, the boundary layer shows intermittent
transitions to the turbulent state. The transition often starts from one side while the
other side remains in the laminar state. 𝐶𝐷 experiences a dramatic reduction from
about 1.2 to 0.3. This reduction is named as the drag crisis, which is normally referred
to the sudden reduction of the drag coefficient as Re increases (Sumer et al. 1997). A
summary of the drag crisis reported in the literature is given in Figure 2.1. The scatter
5
of the data presented in Figure 2.1 is attributed to the differences in the test setups,
such as free stream turbulence, cylinder end conditions, surface roughness, blockage
ratio etc. Due to the asymmetric boundary layer transition on the two sides of the
cylinder in the critical regime, a strong non-zero mean lift force on the cylinder is
often observed (Achenbach and Heinecke, 1981, Schewe, 1983). Cadot et al. (2015)
reported a study in the range of Re = 1.25×105 ~ 3.75×105 and found that different
flow regimes could co-exist under the same Re values in the range of 3.4×105 ~
3.75×105 with intermittent transitions between different regimes frequently. In the
supercritical regime with Re ranging from 3.5×105 to 1.5×106, turbulent separations
occur on both sides of the cylinder. The energy of vortices in the wake reduces
significantly and 𝐶𝐷 remains around 0.3. Schewe (1983) found when 𝑅𝑒 is
increased from 1.30×105 to 7.2×105, the energy of the dominating frequency is
reduced by a factor of 300. In the upper transition regime with Re ranging from
1.5×106 to 4×106, the turbulence propagates towards the front stagnation point and the
boundary layer becomes fully turbulent on one side of the cylinder but remains
partially laminar on the other side. At the same time, 𝐶𝐷 experiences a recovery,
rising from 0.3 to about 0.8. With a further increase of Re, the boundary layers on the
cylinder surface become fully turbulent and the flow enters the transcritical regime.
The boundary layer transition process for an isolated circular cylinder has been
extensively investigated (e.g. Zdravkovich (1997)) but still remains as an active
research topic due to its engineering significance and academic values.
The influence of free stream turbulence intensity has been studied by many
researchers (Fage, 1929, Cheung and Melbourne, 1983, Norberg and Sunden, 1987,
Blackburn and Melbourne, 1996). It is found that a high free stream turbulence
intensity leads to an early transition to turbulence in the boundary layers on the
cylinder surface with a less obvious drag reduction during the drag crisis. Surface
roughness (𝑘𝑠) of the cylinder also affects the appearance of the drag crisis. The
critical value of 𝑅𝑒 for drag crisis decreases with the increase of 𝑘𝑠/𝐷, but the
reduction of 𝐶𝐷 becomes less obvious (Achenbach and Heinecke, 1981). The shift of
critical value of 𝑅𝑒 with the increase in 𝑘𝑠/𝐷 is due to the change of separation
6
locations induced by the surface roughness. The blockage ratio and the aspect ratio in
the experiments also affect the boundary layer transition as investigated by many
researchers (Nishioka and Sato, 1974, Hiwada et al., 1979, Ramamurthy and Lee,
1973, Richter and Naudascher, 1976, Hiwada et al., 1983, West and Apelt, 1982).
Richter and Naudascher (1976) found that when the blockage ratio is increased to 25%
and 50%, the transition to supercritical regime happens at 𝑅𝑒 = 2.0 × 105 and
1.1 × 105, respectively. This is much lower than the value observed by Bearman
(1969), which is 𝑅𝑒 = 3.9 × 105 with a blockage ratio of 8.4%.
The effect of a plane boundary on the flow around a circular cylinder has also
attracted substantial research interests in the past a few decades (Kiya, 1968, Roshko
et al., 1975, Bearman and Zdravkovich, 1978, Zdravkovich, 1985, Jensen et al., 1990,
Buresti and Lanciotti, 1992, Lei et al., 1999). For a near wall cylinder, the flow
around the cylinder and hydrodynamic forces acting on the cylinder are affected by
the gap to diameter ratio (𝐺/𝐷), plane wall boundary layer thickness to diameter ratio
(𝛿/𝐷) and the turbulence intensity in the boundary layer on the wall. Bearman and
Zdravkovich (1978) visualized flow around a circular cylinder near a plane boundary
and measured pressure distributions on the cylinder surface and the plane wall at 𝑅𝑒
= 4.5×104. It was reported that vortex shedding was suppressed for 𝐺/𝐷 < 0.3. It
was also found that 𝐶𝐷 increases with 𝐺/𝐷 up to a certain value of 𝐺/𝐷, and
remains a constant for further increase of 𝐺/𝐷, whick links to the influence of the
boundary layer thickness in the work of Bearman and Zdravkovich (1978). A
non-zero mean lift (𝐶𝐿) is observed as the cylinder moves close to a plane boundary
and 𝐶𝐿 increases dramatically as 𝐺/𝐷 asymptotes to 0. Fredsøe et al. (1987) and
Thomschke (1971) attributed the increase of 𝐶𝐿 with the reduction of 𝐺/𝐷 to the
large suction pressure on the free-stream side of the cylinder due to the blockage
effect.
The transition of the boundary layers on a near wall cylinder has not been
reported. Most of the existing studies for a near wall cylinder were carried out in the
subcritical regime. Buresti and Lanciotti (1992) measured force coefficients of a near
wall cylinder with 𝑅𝑒 = 0.9×105 ~ 2.8×105 and 𝐺/𝐷 = 0 ~ 1.5. Among the
8
In this work, the boundary layer transition on a cylinder close to a plane boundary
is investigated through physical experiments. A series of model tests carried out to
measure the hydrodynamic forces acting on a smooth circular cylinder in the ranges of
Re = 1.1×105 to 4.1×105 and 𝐺/𝐷 = 0, 0.01, 0.1, 0.25, 0.5, 0.7 and 1. The effects of
Re and 𝐺/𝐷 on the force coefficients are quantified.
2.2 Testing facility and model setup
The physical experiments reported in this paper were conducted in the O-tube
facility at the University of Western Australia. The O-tube is a fully closed circulating
water channel driven by an axial flow pump. The rectangular test section has
dimensions of 17 m × 1 m × 1 m (length × width × depth), in which various flow
conditions can be generated by controlling the pump speed. A detailed introduction
about the O-tube facility was given by An et al. (2013).
Figure 2.2 A 3D drawing of the model pipe and its actuator system installed in the
O-tube test section.
The model pipe used in the experiments has a diameter of 196 mm and a length of
998 mm. The model pipe was installed at the middle of the test section with two
supporting arms from an actuator system as shown in Figure 2.3. The actuator system
is controlled by a computer and allows the model pipe being fixed at different
locations vertically and horizontally. Two flow laminators were installed at the two
ends of the test section to smooth the flow. The distance from the laminator to the
9
model pipe was 8.5 m. The model pipe was equipped with 16 pore pressure
transducers uniformly distributed along the middle cross section. The transducers are
piezo-resistive silicon pressure sensors, which use silicon oil to transfer pressure from
stainless steel diaphragm to the pressure sensor. A filter cover was used for each
pressure sensor and the filter surface was machined to maintain a smooth curvature of
the model pipe. Figure 2.3(a) shows a sketch of the setup of the model pipe above a
plane wall. The centre of the coordinate system is set on the plane wall with 𝑥 = 0 at
the centre of the model pipe and the 𝑧-axis is along the water depth and pointing
upwards. The angular position on the cylinder surface is defined as 𝜃 with 𝜃 = 0°
on the upstream side. A photo of the model pipe installed in the test section is given in
Figure 2.3(b). The hydrodynamic forces on the model pipe was obtained by
integrating the measured pressure distributions around the pipe surface. A data
acquisition system was located in the chamber of the model pipe as shown in Figure
2.3(c), sampling rate of 30 𝐻𝑧 was used in the tests. The bed beneath the cylinder
was made of PVC panels a smooth surface.
A Nortek ADV (Vectrino II profiler) was used to measure flow velocity profiles.
The ADV measures a velocity profile within 35 mm range with a cell size of 1 mm
simultaneously. The measuring locations were 40 mm to 75 mm below the probe of
the ADV. The signal correlation of the ADV probes is an important parameter
indicating the quality of the data. In the present experiments, it was found that the first
25 points in the measurement normally have correlation value higher than 90% and
the correlation values drop with the increase of the distance to the probe for the other
10 points. In the analysis, the samples with correlation lower than 90% were removed.
In this work, different free stream flow velocity values were tested from 0.54 m/s
to 1.98 m/s. The test at each velocity increment was maintained for 150 s. The
sampling rate of the ADV was 50 Hz.
11
2.3 Test results
2.3.1 Boundary Layer and Turbulent Intensity Measurement
The mean velocity and the corresponding turbulence intensity profiles in the
boundary layer above the plane boundary of the test section (with the cylinder
removed) are presented in Figure 2.4. A simple regression analysis suggests that
velocity profiles within 𝑧 = 0 ~ 250 mm follow a logarithmic profile with 𝑧 = 0 on
the plane wall as shown in Figure 2.3 (a). The mean flow velocity approaches to the
free stream flow velocity for z > 250 mm. The turbulence intensity 𝐼𝑢 shown in
Figure 2.4 is defined by as 𝐼𝑢 = √𝑈′2 ��⁄ in which √𝑈′2 is the root-mean-square
value of the velocity fluctuation and �� is the mean value of the free stream velocity.
It can be seen from Figure 2.4(b) that the turbulence intensity increases with the
decrease of z for 𝑧 < 120 mm with 𝐼𝑢 value reaching around 10% to 11% near the
base of the test section, while for 𝑧 > 120 mm, 𝐼𝑢 remains about 4% - 5%. In this
work, the surface of the plane wall is smooth. It has been known that the roughness on
a plane wall influences the development of the boundary layer profile on the wall.
Schultz and Flack (2007) conducted measurements of the rough wall turbulent
boundary layer with different roughness height. It was found that with the increase of
roughness height, the boundary layer thickness increases and the tubulent intensity
within the boundary layer also increases.
2.3.2 Pressure Distributions around the Cylinder
The hydrodynamic pressure around the cylinder is investigated in this section to
achieve a better understanding of flow regime transitions. The pressure coefficient 𝐶𝑃
is defined as 𝐶𝑃 = (𝑝 − 𝑝0)/(0.5𝜌��2), where the reference pressure 𝑝0 is taken as
the pressure at 𝜃 = 0°, 𝜌 is water density (= 1000 kg⁄m3).
For an isolated cylinder in steady currents, the distribution of 𝐶𝑃 around a
cylinder has been well investigated and understood. The minimum value of 𝐶𝑃
(defined as 𝐶𝑃−𝑚𝑖𝑛) is around -1.5 ~ -2 in the subcritical regime, and shows a
significant reduction to around -3.5 in the supercritical regime (Zdravkovich, 1997).
12
In the critical regime, the pressure drop only happens to one side and leads to an
asymmetry distribution of 𝐶𝑃 when the transition only occurs on one side of the
cylinder surface. Based on the above feature, the distributions of 𝐶𝑃 and 𝐶𝑃−𝑚𝑖𝑛 are
used to examine the state of the boundary layer on the cylinder surface.
Figure 2.5 shows a comparison of the pressure distributions measured at Re =
1.1×105 in this work with that presented by Bearman and Zdravkovich (1978). A
reasonable agreement is observed. The discrepancy is attributed to the difference in
the boundary layer profiles in the two works.
The mean pressure distributions around the cylinder at nine different 𝑅𝑒 values
and different gap ratios are shown in Figure 2.6. The pressure distributions are
classified into two groups according to the influence level of the plane boundary on
flow regime transitions. The first group includes the cases with 𝐺/𝐷 = 1.0, 0.7 and 0.
5 where the flow regime transition is clearly identified based on the 𝐶𝑝 distribution.
In the second group (𝐺/𝐷 =0.25, 0.1, 0.01 and 0), the flow regime transition is less
obvious than that shown in the first group.
Figure 2.5. The pressure coefficient for 𝐺/𝐷 = 1.0 and Re = 1.1 × 105, together
with the data presented by Bearman and Zdravkovich (1978).
In the first group, where 𝐺/𝐷 is relatively large, the influence of Re has on the
pressure distributions is obvious. For example, in the case with 𝐺/𝐷 = 1.0 shown in
Figure 2.6(a), 𝐶𝑃𝑏 experiences only a slight change from about −2.2 to about −1.75
o
Cp
0 90 180 270 360-3
-2
-1
0G/D=1, Re=1 110
5, I
u=5 3%
Bearman et al (1978), G/D=1, Re=4 8104, I
u=0 2%
13
as 𝑅𝑒 increases from 1.1×105 to 1.9×105. The 𝐶𝑃𝑏 is defined as the average value of
the nearly constant pressure coefficient in the wake region, following Güven et al.
(1980). As Re increases from 1.9×105 to 2.7×105, the pressure distribution
experiences the following changes: (1) significant drop happens to 𝐶𝑃−𝑚𝑖𝑛, (2) an
increase of 𝐶𝑃𝑏 is observed, (3) 𝐶𝑃−𝑡𝑜𝑝 is lower than 𝐶𝑃−𝑔𝑎𝑝 and this becomes
more pronounced as 𝐺/𝐷 is reduced. The three changes are the typical
characteristics of the boundary layer transition to turbulence for an isolated circular
cylinder (Zdravkovich, 1997). Based on the features observed above, the flow with
𝐺/𝐷 = 1.0 is deemed in the critical regime for Re = 1.9×105 to 2.7×105. With a
further increase of Re from 2.7×105, the pressure distributions change little. This
indicates that the flow is in the supercritical regime. The trends observed in the cases
with 𝐺/𝐷 = 0.7 and 0.5 are very similar to that observed with 𝐺/𝐷 = 1.0.
In the second group, (Figure 2.6(d), (e), (f) and (g)), the gap ratios are relatively
small and the effect of 𝑅𝑒 on the distribution of 𝐶𝑃 is less obvious. Only a minor
change to the pressure distribution is observed from 𝑅𝑒 =1.1×105 to 1.9×105 for the
cases with 𝐺/𝐷 = 0.25 and 0.1. For 𝐺/𝐷 = 0.01, 𝑅𝑒 only affects 𝐶𝑃−𝑔𝑎𝑝
measured at 𝜃 = 270°. However, it is noticed that 𝐶𝑃−𝑔𝑎𝑝 at 𝐺/𝐷 = 0.01 is still
comparable to that at G/D ≥ 0.1. When the gap is fully closed (𝐺/𝐷 = 0), 𝐶𝑃−𝑔𝑎𝑝
recovers significantly and the overall distribution of 𝐶𝑃 is not affected by 𝑅𝑒. This
indicates that the lift force is very sensitive to 𝐺/𝐷 at small G/D, which will be
explained in the following section.
14
Figure 2.6. The distribution of averaged 𝐶𝑃 on the cylinder surface at different Re (=
1.1× 105~4.3 × 105) and different 𝐺/𝐷 (= 0 ~ 1.0)
The flow regime transition to the supercritical is identified through the feature of
𝐶𝑃 curves given in the first group (Figure 2.6, (a), (b) and (c)), based on the similar
features observed from an isolated cylinder. Such a transition feature is expected for
all gap ratios, from 0 to ∞. For 𝐺/𝐷 >1 (not tested in this work), the influence of the
plane wall is expected to be minor and the regime transition of flow around the
cylinder is similar to the isolated cylinder condition. The flow for 𝐺/𝐷 ≤ 0.25 tested
in this work are mostly in the supercritical regime, justified from 𝐶𝑃𝑚𝑖𝑛 values. No
subcritical regime flows at these low gap ratio conditions (the second group) were
captured within the tested range of 𝑅𝑒. The existing studies in the literature reported
the existence of the subcritical state at low 𝐺/𝐷 conditions. Bearman and
15
Zdravkovich (1978) reported a detailed measurement of pressure coefficients for a
cylinder with 𝐺/𝐷 = 0 ~ 3 and 𝑅𝑒 = 0.48×105. The 𝐶𝑃−𝑡𝑜𝑝 value achieved by
Bearman and Zdravkovich (1978) was -1.1 at 𝐺/𝐷 = 0 and -1.6 at 𝐺/𝐷 = 0.1,
respectively. These two values are significantly lower than the values measured in the
present study with 𝑅𝑒 = 1.1×105 for 𝐺/𝐷 = 0 and 0.1. This suggests that the flows
investigated by Bearman and Zdravkovich (1978) were in the subcritical regime and
the corresponding ones in this work are in the supercritical regime. This allows us to
believe that the transition from the subcritical to the supercritical regime exists at all
gap ratios.
The critical 𝑅𝑒 value for the transition to the supercritical regime (defined as
𝑅𝑒𝑐𝑟) is identified based on the pressure distributions shown in Figure 2.6 for most of
the 𝐺/𝐷 values, except for 𝐺/𝐷 = 0 and 0.01. A summary of 𝑅𝑒𝑐𝑟 is given in
Figure 2.8. It can be seen that 𝑅𝑒𝑐𝑟 is a constant value of 2.6×105 for 𝐺/𝐷 > 0.5
and reduces with the reduction of 𝐺/𝐷 for 𝐺/𝐷 < 0.5. It is believed that this due to
the increase of turbulence intensity as shown in Figure 2.4 for 𝑧/𝐷 < 0.6. This is
similar to the effect of the free stream turbulence level on the flow regime transition
for an isolated cylinder.
When the cylinder is close to the plane wall, the wall proximity affects the
distribution of 𝐶𝑃. Based on the fitted curves of the data points in Figure 2.6, the
angular position of 𝐶𝑝−𝑔𝑎𝑝 shifts towards downstream direction when 𝐺/𝐷 is
reduced from 1 to 0.01 for most of the 𝑅𝑒 values. This indicates that the separation
point on the gap side moves towards downstream direction since the separation point
is normally about 10° downstream of the location of the 𝐶𝑃−𝑚𝑖𝑛 as reported by Tani
(1964). It has been known that the flow separation from a cylinder surface is due to
the adverse pressure gradient (Clancy, 1975). If the pressure shows a reduction trend
along the flow direction, it tends to stabilize the flow from separation and this is
called a favorable pressure gradient. As an example, the area with adverse pressure
gradient on the gap side are marked between line 1 and 2 in Figure 2.6 (f) and the
favourable pressure gradient is labelled between line 2 and 3. The favorable pressure
gradient on the gap side is due to the wall proximity and this postpones the flow
17
Figure 2.8. The critical 𝑅𝑒 value of supercritical regime at different 𝐺/𝐷 values.
Some key features of the pressure distribution are summarized in Figure 2.9,
which includes the 𝐶𝑃−𝑚𝑖𝑛 , the difference between base pressure and minimum
pressure (𝐶𝑃𝑏 − 𝐶𝑃−𝑚𝑖𝑛) and the wake width (𝜑), which is the angle distance between
two turbulent separation points, corresponding to a near constant 𝐶𝑃 in the wake. For
an isolated cylinder, the wake width depends on states of the boundary layer on the
cylinder surface as summarized by Zdravkovich (1997). These three values (𝐶𝑃−𝑚𝑖𝑛,
𝐶𝑃𝑏 − 𝐶𝑃−𝑚𝑖𝑛 and 𝜑) are often used in identifying the transition from the subcritical
to the supercritical regime in the literature. For an isolated cylinder, 𝐶𝑃𝑏 experiences
a dramatic change when the drag crisis happens. It has been known that the negative
base pressure on the downstream side of cylinder is induced by the vortices formed
after flow separations. The vortices in the wake of the cylinder in the supercritical
regime are much weaker than that in the subcritical regime. Therefore, the base
pressure coefficient is higher than that of the supercritical flow. The increase of the
base pressure is also the reason for the reduction of the drag coefficient related to the
drag crisis. The effect of Re on the base pressure is also observed in the present test
results. For example, at 𝐺/𝐷 = 1.0, 𝐶𝑃𝑏 = -2.05 is observed at Re = 1.1×105. This is
in a close agreement with that of an isolated cylinder (𝐶𝑃𝑏 = -2.2 at Re = 1.1×105) as
reported by Achenbach (1968). An increase of the base pressure can be observed in
the range of Re = 1.1×105 to 2.7×105. The 𝐶𝑃𝑏 remains almost a constant value of
-1.25 for Re > 2.7×105. This suggests that the flow transition to the supercritical
regime happens at around 𝑅𝑒 = 2.7×105 for 𝐺/𝐷 = 1. For 𝐺/𝐷 = 0.75 and 0.5, the
G/D
Re cr
/10
5
0 0.2 0.4 0.6 0.8 11.5
2
2.5
3
18
trend of 𝐶𝑃𝑏 is very similar to that of 𝐺/𝐷 = 1 and the transition to supercritical
regime happens at the same 𝑅𝑒, but 𝐶𝑃𝑏 in the supercritical regime is about -1.4,
slightly lower than that of 𝐺/𝐷 = 1. For 𝐺/𝐷 = 0.25, the recovery of 𝐶𝑃𝑏 with Re
also happens but in a more gradual manner. For 𝐺/𝐷 = 0, 0.01 and 0.1, only very
minor increases are observed in the range of 𝑅𝑒 = 1.1×105 to 1.4×105. This also
indicates that the transition to the supercritical regime happens at lower Re values than
the minimum Re tested for 𝐺/𝐷 < 0.25. The minimum pressure coefficient 𝐶𝑃−𝑚𝑖𝑛
is also sensitive to 𝐺/𝐷 and Re as shown in Figure 2.6. As discussed by Achenbach
(1968), Roshko (1961) and Van Nunen (1974), 𝐶𝑃𝑏 − 𝐶𝑃−𝑚𝑖𝑛 represents the adverse
pressure recovery in the wake. Since the flow is asymmetrical on the top and bottom
sides of the cylinder, 𝐶𝑃𝑏 − 𝐶𝑃−𝑡𝑜𝑝 and 𝐶𝑃𝑏 − 𝐶𝑃−𝑔𝑎𝑝 are presented separately in
Figure 2.9 (b) and (c). For 𝐺/𝐷 =1, 0.7, 0.5 and 0.25, 𝐶𝑃𝑏 − 𝐶𝑃−𝑡𝑜𝑝 increases with
Re and reaches the maximum value around 2.0 at 𝑅𝑒=2.7×105 and then a minor
reduction happens with further increase of Re. Once again this demonstrates that the
transition to the supercritical regime occurs at 𝑅𝑒 = 2.7×105 for 𝐺/𝐷 ≥ 0.25. For
𝐺/𝐷 =0.1, 0.01 and 0, no clear transition is observed. This is consistent with the
observations from the pressure distributions shown in Figure 2.6. Figure 2.9 (b) and (c)
show similar features. The wake width φ decreases from around 180 to 110º for 𝐺/𝐷
= 0.5, 0.75 and 1 as Re falls into the supercritical regime, which is caused by the delay
of flow separations. For G/D ≤ 0.25, φ keeps almost constant at around 115º
regardless of Re, which suggests that the transition has had occurred in this range of
𝑅𝑒 for these 𝐺/𝐷 conditions.
21
(c) and (d), respectively. The following features are observed. Firstly, the reduction
trend of 𝐶𝐷 with the increase of flow velocity indicates the existence of drag crisis
for a near wall cylinder, except for 𝐺/𝐷 = 0. The other feature shown in Figure 2.10
is that the fluctuations of the drag forces become rather weak after the reduction of
𝐶𝐷, which suggests weak vortex shedding in this range of Re.
Figure 2.10. The nine steps of input velocity (a), and the drag coefficients at different
flow velocity at (b) 𝐺/𝐷=1, (c) 𝐺/𝐷=0.25 and (d) 𝐺/𝐷=0.
The mean drag and lift force coefficients (𝐶�� and 𝐶��) are calculated and further
analysed. Figure 2.11 shows the comparison of the present 𝐶�� for 𝐺/𝐷 = 1 with 𝐶��
for an isolated circular cylinder (Cheung and Melbourne, 1983, Schewe, 1983, Cadot
et al., 2015). The turbulence intensity values measured in the independent test cases
were 4.4%, 0.4% and 0.1%, respectively. The present result is in a good agreement
with Cheung and Melbourne (1983), but shows obvious differences from those
reported by Schewe (1983) and Cadot et al. (2015). This is mainly attributed to the
high turbulence intensity (𝐼𝑢= 4 ~ 5 %) in this study, which is very close to the value
in tests by Cheung and Melbourne (1983).
t (s)
U(m
/s)
0 500 1000 15000 5
1
1 5
2
t (s)
CD
0 500 1000 15000 2
0 4
0 6
0 8
1
1 2
1 4G/D=1
t (s)
CD
0 500 1000 15000 2
0 4
0 6
0 8
1
1 2
1 4G/D=0.25
t (s)
CD
500 1000 1500
0 4
0 6
0 8
1G/D=0
(a) (b)
(c) (d)
24
of 0.52 for Re >1.9 ×105. Therefore, it is believed that the boundary transition to
turbulence also occurs for the case with 𝐺/𝐷 = 0.1. The 𝐶�� shows a small reduction
rate and amplitude for 𝐺/𝐷 = 0.01 (from 0.64 to about 0.56 over Re =1.1×105 ~ 1.5
×105). For the case with 𝐺/𝐷 = 0, 𝐶�� keeps almost a constant of 0.71 within the
tested range of 𝑅𝑒. This appears to support conclusion that no boundary layer
transition occurs in the range of Re tested in this study for 𝐺/𝐷 = 0.0 and 0.01.
The following generalizations are made based on the above observations: (1) The
critical Re for the onset of transition decreases with the decrease of 𝐺/𝐷, (2) the
reduction rate and the reduction amplitude of 𝐶�� during transition decrease with the
decrease of 𝐺/𝐷 and (3) the critical Re for the supercritical regime also decreases
with the decrease of 𝐺/𝐷. The above features are very similar to the effect of free
stream turbulence level on the 𝐶�� values for an isolated cylinder (Cheung and
Melbourne, 1983). Therefore, the turbulence level in the wall boundary layer plays an
important role for the boundary layer transition process for a near wall cylinder.
For an isolated cylinder, 𝐶�� remains zero for flows in the subcritical and
supercritical regimes, but a significant non-zero 𝐶�� value (up to 1) exists in the
critical regime due to the different boundary layer states on the two sides of the
cylinder surface (laminar state on one side and turbulent state on the other side)
(Schewe, 1983, Cadot et al., 2015). This non-zero 𝐶�� was only observed when the
free stream turbulence level is very low. For example, 𝐼𝑢 = 0.4% and 0.1% for
Schewe (1983) and Cadot et al. (2015), respectively. The non-zero 𝐶�� was not
observed in the critical regime by Cheung and Melbourne (1983) for the flow with 𝐼𝑢
= 4.4 ~ 9.1% or So and Savkar (1981) for flow with 𝐼𝑢 = 9.5%. This indicates that
𝐶�� is not affected by 𝑅𝑒 for relatively high 𝐼𝑢 conditions, even in the Re range of
transition from the subcritical, to the critical and then to the super-critical regime. In
the present work with 𝐼𝑢 = 4 ~ 10%, it is therefore not expected that 𝐶�� would
experience a significant change due to flow regime transitions. Figure 2.12 (b) shows
the effect of 𝐺/𝐷 on 𝐶�� at different 𝑅𝑒 values. As expected, there is not a
significant change of 𝐶�� with 𝑅𝑒 when the flow regime transition happens.
However, local peaks of 𝐶�� are found in the range of Re = 2.3 × 105~ 3.9 × 105
26
2.4 Engineering implications
The present work is relevant to the engineering application of hydrodynamic
forces on a free-spanning subsea pipeline. The results show that the boundary layer
transition has a significant effect on the force coefficients on pipelines, which are the
key parameters required in the stability design of offshore pipelines.
Holloway et al. (2001) presented the measurement of tidal flow at the North West
Shelf region of Australia. A peak tidal current of 0.2 m/s was measured in the water
depth of 200 m to 400 m. In this water depth, the surface wave induced flow is less
significant due to the large water depth. Therefore, the hydrodynamic force on a
subsea pipeline is dominated by steady currents. The diameter of a subsea pipeline is
normally in the range of 3 inches (76 mm) for flow lines, to 72 inches (1.8 m) for high
capacity trunk lines (Dean, 2010). This corresponds to a Re range of 1.56×104 ~
3.0×105 for a current velocity of 0.2 m/s. On the other hand, the turbulence measured
in the present work is about 10% near the plane wall, which is close to the near seabed
turbulence intensity in tidal flow measured by Bowden (1962) (12% ~ 15%). It can be
seen that the Re range and the flow turbulence intensity investigated in this work are
well representative of those in prototype applications. Based on the 𝑅𝑒𝑐𝑟 shown in
Figure 2.8, a series of curve for 𝑈 × 𝐷 = 𝜈 × 𝑅𝑒𝑐𝑟 are plotted in Figure 2.13. The
curves provide the threshold velocity magnitudes beyond which the drag crisis is
expected for a range of pipeline diameters and gap ratios.
2.5 Conclusions
A series of model tests were conducted using the O-tube facility to investigate the
drag and lift coefficients under the effect of boundary layer transition for a circular
cylinder near a plane boundary. The pressure around the cylinder was measured in the
range of 𝑅𝑒 = 1.1×105 ~ 4.3×105 and the force coefficients were analyzed. The main
conclusions are summarized as below.
27
a) For G/D ≥ 0.5, drag crisis happens in the range of 𝑅𝑒=1.9×105 ~ 2.7 ×105.
𝐶�� reduces to about 0.35 after the drag crisis.
b) For G/D ≤ 0.25, drag crisis still exists but happens at lower Re values than
the minimum Re tested in the present study. The reduction of 𝑅𝑒𝑐𝑟 for small gap
ratios is mainly due to the high turbulence intensity close to the plane boundary. For
the smallest gap ratio tested in this work (𝐺/𝐷 = 0), 𝐶�� value is around 0.71 and it
shows a very low sensitivity to Re within the tested range of Re.
c) Reynolds number has a less effect on lift coefficient than on drag coefficient.
The general trend found in this work is that the lift coefficient reduces with the
increase of gap ratio. No flow asymmetry due to successive transition is observed in
the critical regime.
d) The wall proximity effect tends to delay the gap side separation on the
cylinder surface due to the favorable pressure gradient. This is a common feature for
flow with boundary layer in both laminar and turbulent state.
e) The testing conditions covered in this work are in the range of prototype field
conditions and the results have the potential to influence engineering practices in
future.
Acknowledgements
The authors would like to acknowledge the support from the National Key R&D
Program of China (Project ID: 2016YFE0200100), the Australian Research Council
through Discovery Early Career Research Award (DE150100428) and Linkage project
(LP150100249) and the ECR Fellowship Supporting Program from the University of
Western Australia. F. Yang would like to acknowledge the PhD scholarships provided
by the University of Western Australia.
28
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33
Chapter 3
Turbulent boundary layer transition of steady flow
around a cylinder near a plane boundary†
Abstract: The boundary layer transition on the surface of a near wall cylinder in
steady flow is investigated through measurements of the pressure distribution around
the cylinder in a wind tunnel, covering a total of 11 gap to diameter ratios (𝐺/𝐷 = 3,
2, 1, 0.8, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1 and 0), two boundary layer thickness to diameter
ratios (𝛿/𝐷 = 0.1 and 0.5) and Reynolds number in the range of 𝑅𝑒 = 1.33×105 ~
4.8×105. It is found that the proximity of the plane boundary affects the transition
process through a number of physical mechanisms such as the changes of flow rate
through the gap and the favourable pressure gradient over the cylinder surface
induced by the blockage effect and the thickness of boundary layer (or velocity
profiles) of the approaching flow. These leads to distinct flow characteristics that are
unique for the flow around a cylinder near a plane boundary. For example, the
transition initiates from the gap side, but switches to the top side before the transition
happens on both sides at intermediate gap ratios (𝐺/𝐷 = 0.5 ~ 2). The transition
switch from the gap side to the top side with the increase of Re is referred to as the
side swapping. The transition process is mainly influenced by 𝐺/𝐷 at intermediate
gap ratios and by both 𝐺/𝐷 and velocity profiles of the approaching flow at small
gap ratios. The physical mechanisms responsible for the unique flow characteristics
and force coefficients are explained with the support of quantitative evidence obtained
in this study.
† This chapter is presented as a paper which has been submitted to Journal of Fluid Mechanics as
“Yang F., An, H., Cheng, L., Wang H., Zhang M., 2017. Turbulent boundary layer transition of
steady flow around a cylinder near a plane boundary.”
34
3.1 Introduction
Steady flow around a circular cylinder has been the topic for many academic
publications due to its fundamental and engineering significance. The flow around a
smooth circular cylinder is mainly governed by the Reynolds number, defined as
𝑅𝑒 = 𝑈𝐷/𝜈, where 𝑈 is free stream flow velocity, 𝐷 is the diameter and 𝜈 is the
kinematic viscosity of the flow. Many interesting flow features have been studied such
as flow separation, boundary layer transition to turbulence, vortex shedding and
hydrodynamic forces on the cylinder. Different flow regimes have been identified,
based on the state of boundary layer on the cylinder surface. These include the
subcritical, critical, supercritical and transcritical regimes. (Roshko, 1961). In the
subcritical regime (102 < 𝑅𝑒 < 105), the drag coefficient remains almost a constant
value of 1.2 throughout the whole range of Re (Roshko, 1961, Zdravkovich, 1997)
and the Strouhal number, (defined as 𝑆𝑡 = 𝑓𝐷/𝑈 , 𝑓 is the vortex shedding
frequency) is also almost a constant value of 0.21 (Bloor, 1964). In the critical regime,
the drag coefficient 𝐶𝐷 changes dramatically from about 1.2 to 0.3 and asymmetric
boundary layer transition on the two side is often seen when 𝑅𝑒 increases from
3.0×105 to 3.5×105 (Bearman, 1969, Schewe, 1983, Schewe, 1986). With a further
increase of 𝑅𝑒, transition to turbulence happens on both sides of the cylinder surface
and flow is in the supercritical regime (3.5×105 < 𝑅𝑒 < 2×106). For higher 𝑅𝑒 value,
turbulence propagates further upstream along the cylinder surface until to the
stagnation point and this regime is named as the transcritical regime (2×106 < 𝑅𝑒 <
3.5×106) (Achenbach, 1968, Achenbach and Heinecke, 1981, Roshko, 1961). It
should be noted that the terminology used for the regime definitions are slightly
different in different work. The above definitions follow the ones given by Sumer and
Fredsøe (1997). The measurement for force coefficients and vortex shedding
frequency has been pushed to even higher Reynolds number in the last a few years.
For example, van Hinsberg (2015) and van Hinsberg et al. (2017) measured a slightly
roughened circular cylinder and a square cylinder with rounded corners in a pressure
raised wind tunnel and achieved 𝑅𝑒 = 1.2×107.
35
The boundary layer transition process on the surface of a circular cylinder is also
affected by other testing parameters, such as the surface roughness, free stream
turbulence intensity, test section blockage ratio, model aspect ratio and end conditions
(Zdravkovich, 2003). For a cylinder with surface roughness of 𝑘𝑠 (Nikuradse
roughness), the flow around the cylinder is governed by two parameters (𝑅𝑒 and
𝑘𝑠/𝐷) (Achenbach, 1971, Achenbach and Heinecke, 1981, Batham, 1973, Güven et
al., 1980, van Hinsberg, 2015, Zan and Matsuda, 2002, Zan, 2008). As shown in the
data presented by Achenbach (1971), Achenbach and Heinecke (1981), and Güven et
al. (1980), the critical Re values between different flow regimes shift to lower values
with the increse of 𝑘𝑠/𝐷 and the variation of drag coefficient and separation location
becomes less significant.
The free stream turbulence intensity (𝐼𝑢) also affects the transition process (Arie
et al., 1981, Cheung and Melbourne, 1983, So and Savkar, 1981, Zan and Matsuda,
2002). The effect of 𝐼𝑢 is similar to that of 𝑘𝑠/𝐷. An increase in free stream
turbulence intensity leads to an early transition to turbulence for the upstream
boundary layer on the cylinder surface, a reduction of the drag force in the sub-critical
flow regime and an increase of the drag force in the supercritical flow regime. For a
flow with 𝐼𝑢 = 9.1%, the drag only experienced a slight reduction from 0.82 to 0.7
within the critical regime over the range of 4×104 < 𝑅𝑒 < 2×105 (Cheung and
Melbourne, 1983).
Cadot et al. (2015) investigated the dynamic pressure on an isolated cylinder in
the range of 𝑅𝑒 = 1.25× 105 ~ 3.75× 105. Special attention was paid to the flow in
the critical regime. By analysing the pressure time histories, it was found that the flow
in the critical regime shows multi-stable states. The flow was characterized through a
conditional statistical analysis method and the different states were clearly separated.
The following notations were employed to denote the different regimes of boundary
layer transition reported in Cadot et al. (2015): #0 for no boundary layer transition
(subcritical regime flow), #1T for transition on the top side, #1B for transition on the
bottom side and #2 means transition on both sides (super-critical regime). The same
notations are adopted in this work. Cadot et al. (2015) demonstrated that the transition
36
could take two different routes, depending on the test setup. In the first route, the flow
switches randomly among three stable states (#0, #1T and #1B) in the range of 𝑅𝑒 =
3.04× 105 ~ 3.36× 105, with #1T as the dominant regime for 𝑅𝑒 = 3.40×105 ~
3.58×105 and eventually to #2 regime. The second transition route was observed when
the model cylinder was rotated by 84° along the longitudinal axis from where the
first transition route was observed. In the second transition route flow remains in #0
for 𝑅𝑒 < 3.40× 105, and then largely in #2 for 𝑅𝑒 = 3.60× 105 ~ 3.70× 105, with
#0, #1T and #1B co-existing. This was attributed to the potential difference in the
shape and roughness of the model pipe induced by the rotation of the model cylinder.
The effect of a plane boundary on the steady flow around a circular cylinder also
attracted substantial research interests in the past. Large amount of research work on
this topic has been published due to its engineering significance (Lei et al., 1999,
Buresti and Lanciotti, 1992, Jensen et al., 1990, Zdravkovich, 1985, Bearman and
Zdravkovich, 1978, Roshko et al., 1975). Sumer and Fredsøe (1997) provided a
detailed account of the work carried out to the date of their publication. For a smooth
cylinder above a smooth plane boundary, the flow is governed by four parameters,
which include Re, boundary layer thickness to cylinder diameter ratio (𝛿/𝐷), gap
ratio (𝐺/𝐷) and turbulence intensity (𝐼𝑢) of incoming flow. Bearman and Zdravkovich
(1978) visualized the flow around a circular cylinder near a plane boundary and
measured pressure distributions on the cylinder surface and on the plane boundary at
𝑅𝑒 = 4.5×104. It was found that vortex shedding was suppressed for cases with
𝐺/𝐷 < 0.3. The drag coefficient decreases as the cylinder is moved towards the
plane boundary when 𝐺/𝐷 is lower than a certain value (Roshko et al., 1975,
Zdravkovich, 1985, Jensen et al., 1990). A positive mean lift on the cylinder was
observed for 𝐺/𝐷 = 0 (Bearman and Zdravkovich, 1978). Geöktun (1975) measured
the pressure distribution and force coefficients for a near wall cylinder in the range of
0 < 𝐺/𝐷 < 5 and 𝑅𝑒 = 0.9×105, 1.53×105 and 2.5×105. Buresti and Lanciotti (1992)
quantified the force coefficients in the range of 𝑅𝑒 = 0.86×105 ~ 2.77×105 and 𝐺/𝐷
= 0 ~ 1.6. It is suspected that the upper limit of 𝑅𝑒 values tested by Buresti and
Lanciotti (1992) are in or close to the critical regime, but no discussion about the
37
effect of boundary layer transition was offered. Yang et al. (2017) measured the force
coefficients of the cylinder in the range of 𝐺/𝐷 = 0 ~ 1.0 and 𝑅𝑒 = 1.1×105 ~
4.3×105. A drag crisis was observed for G/D = 0.5 ~ 1.0 in the range of 𝑅𝑒 = 1.9×105
~ 2.7×105, where 𝐶𝐷 reduces to about 0.35. For 𝐺/𝐷 ≤ 0.25, the drag crisis still
exists but happens at 𝑅𝑒 values lower than the minimum 𝑅𝑒 tested.
With the rapid development of the supercomputing facilities, three-dimensional
numerical simulations of the flow in the range of 𝑅𝑒 = 105 ~ 106 have been reported
in the last few years (Breuer, 2000, Cao and Tamura, 2017, Cheng et al., 2017, Chopra
and Mittal, 2016, Lehmkuhl et al., 2014, Lloyd and James, 2015, Rodríguez et al.,
2015, Yeon et al., 2015). All of these works simulated the boundary layer transition
with Large Eddy Simulation (LES) models. The boundary layer transitions were well
captured in the above-mentioned simulations. Tong et al. (2017 a and b) carried out
three-dimensional LES simulations for flow around a near wall cylinder with 𝑅𝑒 =
1000 ~ 2.5×104. It was found that the transition to turbulent happens in the boundary
layer on the plane wall at 1.5×104 for 𝐺/𝐷 = 0.02. Under such a condition, the
boundary layer on the cylinder is still in laminar state. The wall boundary layer
transition has a strong effect on the pressure distribution and force coefficients for the
cylinder. This work also demonstrated that the flow around a near wall cylinder is a
complex process due to the effect from the wall boundary layer.
As mentioned above, the free stream turbulence level has strong effect on the
cylinder boundary layer transition. The free stream turbulence in the work of Yang et
al. (2017) was about 4 ~ 5%, which was relatively high in comparison with previous
independent tests for an isolated circular cylinder. It is speculated that the high
turbulence level of the incoming flow might have masked some of the important
features of the flow in the critical regime, such as the asymmetrical transition in the
critical regime. It is expected that different flow features could be observed when the
turbulence intensity is reduced. This motivates the present study. A series of wind
tunnel model tests have been carried out to quantify the influence of the plane
boundary on drag crisis through measuring pressure distributions around the cylinder.
The rest of the paper is organised as following. The details of experimental setup are
39
perpendicular to the plane wall and pointing upwards, with z = 0 at the plane boundary
level. The upstream edge of the plane boundary was 5𝐷 away from the centre of the
cylinder in flow direction, and the downstream length of the plate was 10𝐷. In the
middle section of cylinder, 36 pressure sensors were uniformly installed around the
circumference to measure the pressure distribution on the cylinder surface. Pressure
sensors were connected to a DSM3400 data-scanning system by PVC tubes with
diameter of 1 mm. The sampling rate was to 625 Hz and accuracy was at ±0.05% of
the measured pressure. The same data-scanning system was applied in the work of
Wang et al. (2017). A pitot tube was installed in the upstream of the model pipe to
monitor the free stream air velocity. The temperature during the testing program was
around 8 ~ 10℃ in the test section. The kinematic viscosity of air was taken as
1.48×10-5 m2/s in calculating 𝑅𝑒.
A total of 16 different free stream flow velocity values in the range of 11 m/s ~ 36
m/s, corresponding to 𝑅𝑒 = 1.33×105 ~ 4.8×105, were tested with two different
boundary layer profiles on the plane wall. A total of 11 gap ratios (𝐺/𝐷) ranging from 0
to 3 were tested in this study.
3.3 Boundary Layer and Turbulent Intensity on the
Plane Boundary
The velocity profile above the plane boundary was measured at x = 0 without the
model pipe presented in the test section, to quantify the boundary layer formed above
the plane wall. Two sets of tests with different boundary layer conditions were
conducted. The first type of boundary layer on the plane wall was allowed to form
naturally from the leading edge of the plane. The second type of boundary layer was
generated by perturbing the flow at the leading edge of the plane wall with a square rod
of 1 cm edge width. Both the mean velocity profile and the turbulence intensity for the
two types of plane boundary conditions are given in Figure 3.2. Two incoming
freestream velocities were tested, which were 10 m/s (𝑅𝑒 = 1.33×105) and 20 m/s (𝑅𝑒
40
= 2.67×105). The sampling frequency is 1200 HZ for 20 s. The boundary layer
thickness (𝛿) on the plane wall is determined at the level where its velocity equals to
0.99U. For the naturally developed boundary layer on a plane wall, 𝛿 gradually
increases with the distance from the leading edge (defined as L). In laminar state, 𝛿 can
be calculated as 5𝐿/𝑅𝑒𝐿0.5 based on the Blasius solution, where 𝑅𝑒𝐿 is a Reynolds
number based on L. The boundary layer transition to turbulence on the plane boundary
happens when 𝑅𝑒𝐿 is greater than a certain value. Through direct numerical
simulations, Sayadi et al. (2013) demonstrated that transition to turbulent starts at 𝑅𝑒𝐿
= 3.5×105 (H-type transition) or 6.5×105 (K-type transition). In the present work, 𝑅𝑒𝐿
= 6.65×105 corresponding to U = 10 m/s at the cylinder location. This suggests that
the naturally developed boundary layer was in a turbulence state at the cylinder
location. For the turbulent boundary layer on a plane wall, the boundary layer
thickness can be estimated as 𝛿 = 0.37𝐿/𝑅𝑒𝐿0.2 (Schlichting, 1979). Based on this,
the predicted 𝛿/𝐷 values are 0.127 and 0.110 for U = 10 m/s and 20 m/s respectively.
For the naturally developed plane boundary layer, 𝛿/𝐷 = 0.1 and 0.09 were measured
at U =10 m/s and 20 m/s, respectively, which largely agree with those predicted by the
equations proposed by Schlichting (1979). For the perturbed plane boundary condition,
𝛿/𝐷 = 0.5 and 0.49 were achieved at U =10 m/s and 20 m/s, respectively. As expected,
𝛿/𝐷 increased significantly after the perturbation.
Figure 3.2. The boundary layer and turbulent intensity profiles at 𝑥=0 with the model pipe
removed, measured at 𝑈 = 10 m/s and 20 m/s.
u(z)/U
z/D
0.8 0.9 10
0.2
0.4
0.6
0.8
1/D=0 1 U=20m/s
/D=0 5 U=20m/s
/D=0 1 U=10m/s
/D=0 5 U=10m/s
(a)
Iu
z/D
0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
1(b)
41
Turbulence intensity at level 𝑧 , defined as 𝐼𝑢(𝑧), is calculated as 𝐼𝑢(𝑧) =
√𝑈(𝑧)′2 /��(𝑧), where 𝑈(𝑧)′and �� (𝑧) are the fluctuation and mean velocities at
level z above the plane boundary. The distributions of 𝐼𝑢(𝑧) along the measured
vertical location line are presented in Figure 3.2 (b). Under the naturally developed
plane boundary layer condition, a free stream 𝐼𝑢 = 0.7% was achieved. Within the
boundary layer (𝑧/𝐷 < 0.1), 𝐼𝑢(𝑧) increases dramatically with the decrease of 𝑧
and reaches to about 12% at z = 0.01D. Under the perturbed plane boundary condition,
𝐼𝑢 was also measured at about 0.7% in the free stream and the maximum 𝐼𝑢 about
14.3% at z = 0.01D. The subsequent tests on drag crisis were conducted under
identical test conditions to those two types of plane boundary conditions.
3.4 Test Results
3.4.1 Experimental validations
Validation tests were conducted to verify the present testing setup at 𝐺/𝐷 = 3,
where the wall proximity effect is almost negligible as demonstrated by Bearman and
Zdravkovich (1978). The minimum test duration was 𝑈𝑡/𝐷 = 2000 to obtain
statistically independent results. The pressure coefficient (𝐶𝑝) on the cylinder is
normalized is defined as
𝐶𝑝 =𝑝 − 𝑝𝑟𝑒𝑓
0.5𝜌𝑈2
where 𝑝𝑟𝑒𝑓 is the reference pressure, which is taken at 𝜃 = 0 in the present
study and U is the free stream velocity.
The mean pressure distributions for three representative cases in the subcritical
(𝑅𝑒 = 1.33×105), critical (𝑅𝑒 = 3.07×105) and supercritical regimes (𝑅𝑒 = 3.87×105)
are compared with the experimental results reported by Cadot et al. (2015) in Figure
3.3 with a reasonable agreement found there. The pressure distributions for the
subcritical and supercritical cases are near symmetrical as expected. The base pressure
coefficient 𝐶𝑝𝑏 (defined as the averaged value of the near constant pressure
coefficient in the wake region (Güven et al., 1980)) shows a recovery from -2.3 to -1.5
43
3.4.2 Tests with 𝛿/𝐷=0.1
This section presents the test results under the naturally developed plane
boundary layer conditions. The model tests were conducted with wind speed being
gradually increased to the target 𝑅𝑒 values and then repeated the tests by gradually
decreasing the 𝑅𝑒 values. Hysteresis existed within a small range of 𝑅𝑒 value for
the boundary layer transition process around a cylinder (Schewe, 1983), but the key
phenomena about the cylinder boundary layer transition were not affected by the
hysteresis.
Figure 3.4. The temporal and spatial distributions of pressure coefficient (𝐶𝑝(𝑈𝑡/
𝐷, 𝜃)) on the cylinder surface for cases 𝐺/𝐷 = 0.8 and 𝛿/𝐷 =0.1 with 𝑅𝑒 in the
range of 2.0 × 105 ~ 3.47 × 105.
(a) 𝑅𝑒 = 2.0 × 105
(b) 𝑅𝑒 = 2.67 × 105
(c) 𝑅𝑒 = 3.07 × 105
(d) 𝑅𝑒 = 3.13 × 105
(e) 𝑅𝑒 = 3.27 × 105
(f) 𝑅𝑒 = 3.33 × 105
44
Figure 3.4. The temporal and spatial distributions of pressure coefficient (𝐶𝑝(𝑈𝑡/
𝐷, 𝜃)) on the cylinder surface for cases 𝐺/𝐷 = 0.8 and 𝛿/𝐷 =0.1 with 𝑅𝑒 in the
range of 2.0 × 105 ~ 3.47 × 105.
The dynamic feature of the pressure time histories is examined here through the
contour plot of 𝐶𝑝(𝑈𝑡/𝐷, 𝜃) at 𝐺/𝐷 = 0.8 in Figure 3.4. The pressure contours of
𝑅𝑒 = 2.0 × 105 (Figure 3.4(a)) shows periodically staggered patterns, which are
induced by the alternative vortex shedding from the cylinder. It is also noticed that the
periodic pressure fluctuation covers a large area along the cylinder surface (𝜃 ≈
60°~300°) . At 𝑅𝑒 = 2.67 × 105 (Figure 3.4(b)), the periodic vortex shedding
weakens significantly between 𝑈𝑡/𝐷 = 320 and 400. Such a weakening of vortex
shedding was also observed for an isolated cylinder condition by Cadot et al. (2015)
and was attributed to the partial onset of the boundary layer transition on the cylinder
surface. When 𝑅𝑒 is increased to 3.07 × 105 (Figure 3.4(c)), the occurrence of
intermittent switching between #0 and #1B is observed. This leads to strong irregular
pressure fluctuations on the gap side of the cylinder surface and much weaker
pressure fluctuations in the wake (𝜃 ≈ 90° ~ 270°) than its subcritical counterparts
shown in Figure 4(a) and (b). Similar features are also observed in Figure 3.4(d) (𝑅𝑒
= 3.13 × 105) and Figure 4(e) (𝑅𝑒 = 3.27 × 105) over a long-time duration (𝑈𝑡/𝐷
= 3000), where #1B state dominates the flow. With a slight increase of 𝑅𝑒 to
3.33 × 105 (Figure 3.4(f)), an interesting phenomenon is observed. The transition to
turbulence appears to be supressed on the gap side and appears on the top side,
leading to a swap from the #1B state to the #1T state. This dramatic change of the
boundary layer state leads to a significant change of pressure distributions around the
cylinder surface. As Re is increased to 3.47 × 105 the boundary layer transition on
the gap side gradually re-develops and forms the #2 state (Figure 3.4(g)).
The dynamic features of the pressure time histories for 𝐺/𝐷 = 0.3 (with 𝑅𝑒 =
(g) 𝑅𝑒 = 3.47 × 105
45
1.66 × 105 , 3.47 × 105 and 4.40 × 105 ) are shown in Figure 3.5. No obvious
vortex shedding is observed in the subcritical regime as shown in Figure 3.5(a).
Similar features are also observed for flow with 𝐺/𝐷 < 0.3, suggesting that the
vortex shedding is supressed for 𝐺/𝐷 ≤ 0.3 in the subcritical regime. This agrees
with the observation by Bearman and Zdravkovich (1978) and Grass et al. (1984). The
flow is in the #1T state at 𝑅𝑒 = 3.47 × 105 and #2 state at 𝑅𝑒 = 4.4 × 105.
To examine the transition between the different states with the increase of Re, the
variation of the mean 𝐶𝑝 over the entire sampling period against Re at 𝐺/𝐷 = 0.8 is
shown in Figure 3.6, where the same 𝐶𝑝 scale as that shown in Figure 3.4 is used.
The two dark stripes represent strong low-pressure regions on the cylinder surface,
which can be employed to identify the turbulent boundary layer state based on the
features shown in Figure 3.4. The flow transition to different flow states with the
increase of Re is clearly observed in Figure 3.6. The first feature observed in Figure
3.6 is that the transition to turbulence is initiated on the gap side (#1B) at around 𝑅𝑒
= 3.07 × 105 and then on the top side (#1T) at 𝑅𝑒 = 3.33 × 105. This matches the
information shown in Figure 3.4(c) and (f). The second feature observed in Figure 3.6
is that, when the transition initiates on the top side at 𝑅𝑒 = 3.33 × 105 (Figure
3.4(f)), the minimum of the mean 𝐶𝑝 on the gap side recovers to around -2.5 in the
range of 𝑅𝑒 = 3.33 × 105 ~ 3.53 × 105, suggesting that the boundary layer on the
gap side switches back to the laminar state again. The change of the transition states
from the #1B to #1T is referred as the side swapping hereafter.
The effect of 𝐺/𝐷 on the laminar to turbulent boundary layer transition on the
cylinder surface is illustrated in Figure 3.7 based on the contours of mean 𝐶𝑝, where
the variation of the mean 𝐶𝑝 against 𝑅𝑒 for 𝐺/𝐷 in the range of 0 ~ 3 is shown,
together with that for an isolated cylinder reported by Cadot et al. (2015) (Figure 3.7
(b)). The different boundary layer states are labelled by different colour codes in each
contour plot. The results for 𝐺/𝐷 = 3.0 shown in Figure 3.7 (a) indicate that the
transition starts with the #1B state at Re = 3.07×105 until Re = 3.47×105, then
switches to the #2 state for Re > 3.47×105. Although the side swapping is not
observed, there is an obvious recovery of the negative pressure on the gap side when
46
the transition on the top side is initiated at Re = 3.47×105. This is similar with the side
swapping process for the case with 𝐺/𝐷 = 0.8 as shown in Figure 3.6(b) shows the
results for an isolated cylinder reported by Cadot et al. (2015). There was a clear
recovery of the negative pressure on the top side in the range of 𝑅𝑒 = 3.6 ×
105 ~ 3.7 × 105 after the transition switched from the #1T state to #2 state, although
the #1B state was bypassed. The transition process observed at 𝐺/𝐷 = 3.0 is
somewhat similar with that of an isolated cylinder shown in Figure 3.7 (b).
Figure 3.5. The temporal and spatial distributions of pressure coefficient (𝐶𝑝(𝑈𝑡/
𝐷, 𝜃)) on the cylinder surface for cases 𝐺/𝐷 = 0.3 and 𝛿/𝐷 =0.1 with 𝑅𝑒 =
1.66 × 105, 3.47 × 105 and 4.40 × 105
The boundary layer transition from laminar to turbulent is initiated from the gap
side accompanied by the side swapping for 0.5 ≤ 𝐺/𝐷 ≤ 2. To understand the flow
mechanisms responsible for the #1B→#1T transition (side swapping), further analysis
is carried out for the case of 𝐺/𝐷 = 0.8 with reference to the stagnation point and the
favourable pressure gradient on the cylinder surface. The variations of the stagnation
(a) 𝑅𝑒 = 1.66 × 105
(b) 𝑅𝑒 = 3.47 × 105
(c) 𝑅𝑒 = 4.40 × 105
47
point and the favourable pressure gradient on the two sides of the cylinder at 𝐺/𝐷 =
0.8 are shown in Figure 3.8. The stagnation points are determined based on curve
fitting to the averaged pressure distribution on the cylinder surface. The favourable
pressure gradient on the cylinder surface in the gap is defined as 𝜑1 = ∆𝐶𝑝/(𝐿1/𝐷),
in which ∆𝐶𝑝 is defined as the difference of 𝐶𝑝 values measured at the stagnation
point and θ = 270° and 𝐿1 is the arc length between the two points. Similarly, the
favourable pressure gradient on the top side (between the stagnation point and θ = 90°
is defined as 𝜑2. The four flow states observed in this range of 𝑅𝑒 values are
labelled in Figure 3.8. For the #0 state, the stagnation point remains fixed at about
𝜃 =353.4°, which is below 𝜃 = 0 and this is attributed to the proximity of the plane
boundary. The same feature has been reported by Bearman and Zdravkovich (1978)
for 𝑅𝑒 = 4.5 × 104. It clearly shows 𝜑1 > 𝜑2 in the #0 state. Hiwada et al. (1986)
measured the separation points on the surface of a near wall cylinder at 𝑅𝑒 =
2 × 104. It was found that the gap side separation point shifts towards downstream
and the top side shifts towards upstream. This was attributed to the enhanced
favourable pressure gradient on the gap side (Zdravkovich, 2003). As demonstrated in
Figure 3.4 and Figure 3.6, the transition initiates from the gap side and this leads to a
shift of the stagnation point towards the top side as shown in Figure 3.8 in the #1B
state. The occurrence of transition from the #0 state to the #1B state is attributed to the
enhancement of the shear layer on the gap side of the cylinder surface due to the
blockage effect of the plane boundary. Since a favourable pressure gradient is known
to stabilize flow instabilities (Schmid and Henningson, 2001), a reverse transition
from the turbulent boundary layer to the laminar boundary layer on the gap side
cylinder surface occurs as the favourable pressure gradient on the gap side increases
to a certain level. The recovery of the minimum pressure (or weakening of the laminar
to turbulent transition) on the gap side of the cylinder as shown in Figure 3.4(d) and (f)
is attributed to the stabilization effect induced by the favourable pressure gradient on
the gap side. When the side swapping happens, it can be seen that 𝜑1 is reduced
dramatically from 5.62 to 3.67 and the stagnation point also shifted about 2° towards
the plane wall. This indicates that more flow is directed to the top side and this leads
48
to an increase of the local velocity near the top surface of the cylinder. This
contributes to the boundary layer transition to turbulent on the top side while the gap
side boundary layer switches back to the laminar state. This is the mechanism of the
side swapping phenomenon. Based on above observations, it is concluded that the
side swapping of boundary layer transition from laminar to turbulent is nothing but a
transitional stage of the transition to #2 state. The blockage effect induced favourable
pressure gradient on the gap side is the main reason for the side swapping.
Figure 3.6. The varitaion of mean pressure coefficient distribution on the cylinder
surface vs. 𝑅𝑒 at 𝐺/𝐷 = 0.8 and 𝛿/𝐷 = 0.1 (Scatter colour code: green→ #0,
blue→ #1B, red →#1T and black →#2).
The boundary layer transition from laminar to turbulent is initiated from the top
side of the cylinder without the side swapping for 0.1 ≤ 𝐺/𝐷 ≤ 0.4 as shown in
Figure 3.7 (f) ~ (i). The major reason for this is that more flow is directed to the top
side with the reduction of 𝐺/𝐷, leading to an enhanced shear layer on the top side.
The reduction of the flow through the gap at small gap ratios has been reported
52
comparison of 𝐶𝑝 obtained in different studies is given in Figure 3.9. It can be seen
that the minimum pressure on the top side of the cylinder is -1.4 at 𝑅𝑒 = 0.48 × 105
as given by Bearman and Zdravkovich (1978), which is significantly higher than that
measured by Geöktun (1975) at 𝑅𝑒 = 0.9 × 105 and that in the present work.
Therefore it is speculated that the top side boundary layer measured by Bearman and
Zdravkovich (1978) is in the laminar state, while those in the present work and in
Geöktun (1975) are in the turbulent state.
Figure 3.9. The pressrue distribution around cylinder sitting on the plane boundary
(𝐺/𝐷 = 0).
Figure 3.10. The effect of 𝐺/𝐷 on the critical Re value for the transitions to the #1B,
#1T and #2 state. The wall boundary layer condition is 𝛿/𝐷= 0.1.
Figure 3.10 summarizes the critical 𝑅𝑒 values for the transitions to the #1B, #1T
o
Cp
0 60 120 180 240 300 360-4
-3
-2
-1
0
1
Re=0.48105(Bearman et al. 1975)
Re=0.90105(Geoktun 1975)
Re=2.50105(Geoktun 1975)
Re=2.52105(Present)
Re /105
G/D
1 2 3 4 5
1
2
3#1B
#1T
#2
sid
esw
app
ing
53
and #2 states under the test conditions with 𝛿/𝐷= 0.1 in the range of 0.1 ≤ 𝐺/𝐷 ≤ 3.0.
It can be seen that the critical 𝑅𝑒 values for the #1B and #2 are insensitive to 𝐺/𝐷
for 𝐺/𝐷 ≥ 0.6. For 𝐺/𝐷 < 0.5, only two transition states (#1T and #2) exist. The
critical 𝑅𝑒 for transition to the #1T shows only minor variations with the reduction
of 𝐺/𝐷. However，the critical values for #2 increase from 3.73 × 105 to 4.80× 105
as 𝐺/𝐷 is reduced from 0.4 to 0.1. This is attributed to the reduction of gap side
velocity with the decrease of 𝐺/𝐷.
The influence of 𝐺/𝐷 on pressure distributions around the cylinder are
examined in Figure 3.11 where the variations of mean 𝐶𝑝 with 𝐺/𝐷 at three
different 𝑅𝑒 values are shown. At Re = 1.93 × 105 (Figure 3.11 (a)), most of the
cases are in the subcritical regime, except for 𝐺/𝐷 = 0. For the subcritical regime
flow, the minimum base pressure 𝐶𝑝 in the wake region (80° < 𝜃 < 280°) occurs at
around 𝐺/𝐷 = 0.8 and the maximum base pressure is observed at 𝐺/𝐷= 0.0. In the
range of 300° < 𝜃 < 360°, there is a monotonic increasing trend of 𝐶𝑝 with the
reduction of 𝐺/𝐷. This indicates that 𝐺/𝐷 has a significant effect on the
hydrodynamic force coefficient when the flow is in the subcritical regime and this is
in a good agreement with the observations by Geöktun (1975) and Lei et al. (1999).
Figure 3.11(b) (Re = 3.09 × 105) shows that the transition to turbulence happens on
the gap side only for 𝐺/𝐷 ≥ 0.5 and for 𝐺/𝐷 = 0 on the top side. For 𝐺/𝐷 = 0.1 ~
0.4, the boundary layer is in the laminar state on both sides of the cylinder surface.
When 𝑅𝑒 is increased to 4.80 × 105 (Figure 3.11(c)), all the cases are in the
super-critical regime. It is seen that 𝐶𝑝 is almost independent of 𝐺/𝐷 in the
super-critical regime except for 𝜃 > 240° where 𝐶𝑝 is weakly dependent on 𝐺/𝐷.
This is attributed to the fact that the wake of the cylinder becomes much narrower for
the super-critical flow (Lehmkuhl et al., 2014) than those in the subcritical and critical
regimes.
3.4.3 Tests with 𝛿/𝐷=0.5
To examine the effect of δ/D, a thicker boundary layer was triggered by a square
54
rod installed at the upstream tip of the plane boundary. The measured 𝛿/𝐷 is about
0.5 at the model cylinder location without the presence of the cylinder. It is found that
the cylinder boundary layer transition behaves differently from that under the
condition of 𝛿/𝐷=0.1 for the cases with 𝐺/𝐷 < 0.5 but less so for the cases with
𝐺/𝐷 ≥ 0.5. As an example, the distribution of the averaged 𝐶𝑝 and pressure
fluctuation (𝐶𝑝′) at 𝐺/𝐷 = 0.8 and Re = 3.07× 105 under the two boundary layer
conditions are compared in Figure 3.12. It can be seen that test results under the two
different boundary layer conditions show only minor difference. As further evidence,
the contour plot of 𝐶𝑝(𝑈𝑡/𝐷, 𝜃) at 𝐺/𝐷 = 0.8 and 𝛿/𝐷=0.5 is given in Figure 4.22.
This can be compared with Figure 3.4 and can be found the dynamic features of 𝐶𝑝
are very similar at these two different boundary layer conditions. Therefore, the
analysis is focused on the cases with 𝐺/𝐷 < 0.5 in this section.
Figure. 3.11. The 𝐺/𝐷 effect on pressrue distribution around cylinder at three different 𝑅𝑒
values with 𝛿/𝐷 = 0.1, (a) 1.93 × 105, (b) 3.09 × 105, (c) 4.80 × 105.
o
Cp
0 60 120 180 240 300 360-4
-3
-2
-1
0
1G/D = 0 0G/D = 0 1G/D = 0 2G/D = 0 3G/D = 0 4G/D = 0 5G/D = 0 6G/D = 0 8G/D = 1 0G/D = 2 0
(a) Re=1.93105
o
Cp
0 60 120 180 240 300 360-5
-4
-3
-2
-1
0
1(b) Re=3.0910
5
55
Figure 3.11. The 𝐺/𝐷 effect on pressrue distribution around cylinder at three different 𝑅𝑒
values with 𝛿/𝐷 = 0.1, (a) 1.93 × 105, (b) 3.09 × 105, (c) 4.80 × 105.
Figure 3.12. The effect of 𝛿/𝐷 on Cp and Cp′ for the case with 𝐺/𝐷 = 0.8 and Re
= 3.07× 105.
The dynamic features of the pressure distribution on the cylinder surface for
𝐺/𝐷 = 0.4 are examined in Figure 3.13, with three different 𝑅𝑒 values. At 𝑅𝑒 =
o
Cp
0 60 120 180 240 300 360-5
-4
-3
-2
-1
0
1(c) Re=4.8010
5
o
Cp
0 60 120 180 240 300 360-3
-2
-1
0 /D = 0.1/D = 0.5
o
Cp'
0 60 120 180 240 300 3600
0.1
0.2
0.3
0.4
0.5 /D = 0.1/D = 0.5
56
1.93× 105 and 2.52× 105 (Figure 3.13(a) and (b)), the flow is dominated by the #0
state. With Re further increases to 2.88× 105, the transition to turbulence is initiated
on the gap side (Figure 3.13(c)). This is in contrast to the case with 𝛿/𝐷=0.1 at the
same 𝐺/𝐷 where the transition to turbulence is initiated on the top side of the
cylinder. The initiation of the transition on the gap side for the case with 𝛿/𝐷 = 0.5
and 𝐺/𝐷 = 0.4 is attributed to the high turbulence intensity in the vicinity of the
plane boundary. Existing study (Cheung and Melbourne, 1983) showed that high
turbulence intensity leads to early boundary layer transitions (at a lower Re value).
Figure 3.13. The temporal and spatial distributions of pressure coefficient (𝐶𝑝(𝑈𝑡/
𝐷, 𝜃)) on the cylinder surface for cases 𝐺/𝐷 = 0.4 and 𝛿/𝐷 =0.5 with 𝑅𝑒 =
2.00 × 105, 2.62 × 105 and 3.07 × 105.
As 𝐺/𝐷 is further reduced, the effect from the turbulence on the wall is even
more obvious. Figure 3.14 shows the contours of 𝐶𝑝 at 𝐺/𝐷 = 0.3, 𝛿/𝐷 = 0.5 and
Re = 1.33× 105~ 3.07 × 105. The flow shows two distinct features compared with
the corresponding cases with 𝛿/𝐷 = 0.1. The first feature is that strong pressure
fluctuations are observed in the range of 100 ° < 𝜃 < 260° for Re = 1.33 ×
105and 2.0 × 105 in Figure 3.14(a) and (b). The lift time history for Re = 1.33× 105
( 𝐺/𝐷 = 0.3 and 𝛿/𝐷 = 0.5 ) is shown in Figure 3.15(a), together with the
(a) 𝑅𝑒 = 2.00 × 105
(c) 𝑅𝑒 = 3.07 × 105
(b) 𝑅𝑒 = 2.62 × 105
57
counterpart with 𝛿/𝐷 = 0.1. As discussed in Section 4.2, the vortex shedding is
supressed for Re = 1.33× 105, 𝐺/𝐷 = 0.3 with 𝛿/𝐷 = 0.1 and this is supported by
the very low fluctuation level in its lift time history. In contrast, the lift fluctuation
level is much higher under the perturbed boundary condition. The frequency
spectrums of the two cases are shown in Figure 3.15(b). No dominating peaks are
found for both cases, but the energy level under the perturbed wall boundary
condition is much higher. For comparison, the lift time histories at 𝐺/𝐷 = 0.8 are
given in Figure 3.15(c). Although the two curves show an obvious difference in time
domain, but the frequency spectrums show identical dominating peaks (𝑆𝑡 = 0.218)
with the same energy level. Figure 3.15(c) and (d) demonstrate the occurrence of
vortex shedding at 𝐺/𝐷 = 0.8 and the negligible effect from the plane wall
boundary layer. The velocity fluctuation generated from the tripping rod on the wall
boundary layer is the main reason for the strong irregular pressure fluctuation (Figure
3.14 (a)) and lift fluctuations (Figure 3.15(a)) on the cylinder.
Figure 3.14. The temporal and spatial distributions of pressure coefficient (𝐶𝑝(𝑈𝑡/
𝐷, 𝜃)) on the cylinder surface for cases 𝐺/𝐷 = 0.3 and 𝛿/𝐷 =0.5 with 𝑅𝑒 =
1.33 × 105, 2.00 × 105, 2.67 × 105 and 3.07 × 105.
(b) 𝑅𝑒 =2.0 0 ×
0
(c) 𝑅𝑒 =2.67× 105
(a) 𝑅𝑒 =1.33× 105
58
Figure 3.14. The temporal and spatial distributions of pressure coefficient (𝐶𝑝(𝑈𝑡/
𝐷, 𝜃)) on the cylinder surface for cases 𝐺/𝐷 = 0.3 and 𝛿/𝐷 =0.5 with 𝑅𝑒 =
1.33 × 105, 2.00 × 105, 2.67 × 105 and 3.07 × 105.
tU/D
CL
500 600 700 800-1.5
-1
-0.5
0
0.5
1
fU/D
E/U
2D
10-2
10-1
100
100
101
102
103
tU/D
CL
500 600 700 800-1
-0.5
0
0.5
1
fU/D
E/U
2D
10-2
10-1
100
100
101
102
103
D=0 1
D=0 5
D=0 1
D=0 5(c) 𝐺/𝐷 = 0.8
(a) 𝐺/𝐷 = 0.3 (b)
(d)
Figure 3.15 The lift time histories and the corresponding frequency spectrums for Re
= 1.33× 105 with two different plane wall boundary layers and with 𝐺/𝐷 = 0.3 and
𝐺/𝐷 = 0.8.
Some further comparison about the pressure fluctuation between the two different
plane boundary layer conditions is given here. Figure 3.16 compares the details of 𝐶𝑝′
on the cylinder surface at 𝐺/𝐷 = 0.3 under different wall boundary layer conditions.
Figure 3.16 (a) shows the results at Re = 1.33 × 105. For 𝛿/𝐷= 0.1, the peak value
of 𝐶𝑝′ is only 0.09. This is because the vortex shedding is totally supressed in this
case. However when the boundary layer is perturbed (δ/D= 0.5), the peak values of
𝐶𝑝′ are 0.71 and 0.29 on the gap and top side respectatively. This significant increase
(d) 𝑅𝑒 =3.07× 105
59
of 𝐶𝑝′ value are also attributed to the influence from the strong pressure flucuation
on the plane wall due to the perturbation at the leading edge and the main effect is on
the gap side cylinder surface. The significant difference of 𝐶𝑝′ shown in Figure 3.16
(a) demonstrates the wall boundary layer condition has a dominant effect on the flow.
Figure 3.16 (b) shows the results at Re = 4.40 × 105 , as an example for the
supercritical regime flows. The two curves share similar shapes, and the 𝐶𝑝′ value
under the perturbed boundary layer condition is about twice of that under 𝛿/𝐷= 0.1.
It can be seen that the effect of the wall boundary layer is no longer as significant as
that in the subcritical regime.
When 𝐺/𝐷 is reduced to zero, the boundary layer effect on 𝐶𝑝′ distribution is
compared at two 𝑅𝑒 values for δ/D= 0.1 and 0.5 in Figure 3.17. The first feature
observed from Figure 17 is that the 𝑅𝑒 effect on 𝐶𝑝′ distribution is very small since
the cases are all in the same flow regime with the top side boundary layer being in a
turbulent state. It seems that the high turbulence intensity in the perturbed plane wall
boundary only influences the upstream side of the cylinder surface, where large 𝐶𝑝′
values are observed.
The evolution of the averaged 𝐶𝑝 against Re at different 𝐺/𝐷 is summarized in
Figure 3.18. The following features are observed: (1) the boundary layer transition
happens at a lower 𝑅𝑒 value with the reduction of 𝐺/𝐷, mainly due to the increase
of turbulence intensity, (2) the side swapping exists at 𝐺/𝐷 = 0.4 ~ 1.0, but not for
𝐺/𝐷 < 0.4. It is noticed that the side swapping exists at 𝐺/𝐷 = 2.0 for 𝛿/𝐷 =
0.1, but not for 𝛿/𝐷 = 0.5. This is not due to the effect of the plane boundary layer
since 𝐺/𝐷 is significantly higher than 𝛿/𝐷. For 𝐺/𝐷 = 2, the blockage effect from
the plane boundary is relatively small, so the occurrence of the side swapping shows
certain uncertainty, similar to that of an isolated cylinder. (3) For 𝐺/𝐷 = 0.5~2.0,
transition to turbulence initiates from the gap side. This is the same feature as the
cases with 𝛿/𝐷 = 0.1. (4) For 𝐺/𝐷 = 0.2 ~ 0.4, the transition to turbulence also
initiates from the gap side, but this is due to the strong turbulence on the plane
boundary. (5) The pressure distribution is very similar to the corresponding one at
𝛿/𝐷 = 0.1 at 𝐺/𝐷 = 0.0, induced by the same flow mechanisms.
60
Figure 3.16. The distribution of 𝐶𝑝′ on the cylinder surface at 𝐺/𝐷 = 0.3, Re =
2.67 × 105 and 4.40 × 105.
The critical 𝑅𝑒 values for #1T, #1B and #2 are summarized in Figure 3.19 for
the cases with 𝛿/𝐷 =0.5. An important feature shown in Figure 3.19 is that the
critical Re values decreases with the reduction of 𝐺/𝐷 for 𝐺/𝐷 ≤ 0.4. This is
because the gap side cylinder surface is partially submerged in the plane wall
boundary layer and is affected by the high turbulence level at small 𝐺/𝐷 values. The
critical 𝑅𝑒 values for 𝐺/𝐷 ≤ 0.2 are not captured in the present study.
As a summary, the boundary transition processes observed in this study are
classified into five groups as summarized in Table 3.1. The first group covers the
cases with large 𝐺/𝐷 values. For example, the case with 𝐺/𝐷 = 3.0 falls into this
group. The minor blockage effect due to the wall at 𝐺/𝐷 = 3.0 is still enough to
trigger the transition from the gap side. There is no side swapping observed in this
group. The second group experiences the most complicated transition process which
o
Cp'
0 60 120 180 240 300 3600
0.2
0.4
0.6
0.8
1/D = 0.1/D = 0.5
(a) Re = 1.33105
o
Cp'
0 60 120 180 240 300 3600
0.1
0.2
0.3
0.4
0.5/D = 0.1/D = 0.5
(b) Re = 4.4105
61
goes through #0→#1B→#1T→#2. The transition of #1B→#1T (side swapping) is a
certain feature in this group. The third and the forth groups occur at small gap ratio
ratios, where the transitions go through #0→#1B→#2 or #0→#1T→#2, depending on
the boundary layer profile on the plane wall. The last group covers 𝐺/𝐷 = 0, where
the top side cylinder surface experiences the transition from laminar to turbulent state,
but the critical 𝑅𝑒 value for this group is much lower than that for 𝐺/𝐷 ≥ 0.1.
/D = 0.1/D = 0.5
o
Cp'
0 60 120 180 240 300 3600
0.2
0.4
(a) 𝑅𝑒 = 1.33 × 105
/D = 0.1/D = 0.5
o
Cp'
0 60 120 180 240 300 3600
0.2
0.4
(b) 𝑅𝑒 = 3.60 × 105
Figure 3.17. The detial of 𝐶𝑝′ on the cylinder surface at 𝐺/𝐷 = 0, Re = 2.67 ×
105 and 4.40 × 105.
64
CL(𝑡) = 0.5 ∮ 𝐶𝑝(𝜃, 𝑡)sin (𝜃)𝑑𝜃
The mean force coefficients (𝐶𝐷 and 𝐶𝐿) are shown in Figure 3.20. For 𝐺/𝐷 =
3.0 (Figure 3.20(a)), similar to previous published results, 𝐶𝐷 experiences a minor
reduction from 1.2 to 1.0 in the subcritical regime (𝑅𝑒 = 1.0 × 105 to 3.0 × 105)
and then a further reduction to 0.46 at 𝑅𝑒 =3.4 × 105. The above observation
about 𝐶𝐷 is in a good agreement with the 𝐶𝐷 values of an isolated cylinder reported
in the literature (Almosnino and McAlister, 1984, Cadot et al., 2015). This
corresponds to the so-called drag-crisis in the literatures. The mean 𝐶𝐿 achieves a
value of -0.05 in the subcritical regime. A dramatic reduction of 𝐶𝐿 to -1.15 happens
between 𝑅𝑒 = 2.67 × 105 and 3.07 × 105 due to the formation of the #1B state,
which leads to an unbalanced pressure distribution on the top and gap sides of the
cylinder and results in a strong negative mean 𝐶𝐿. With the development of the
turbulent boundary layer on the top side of the cylinder at 𝑅𝑒 = 3.32 × 105 ,
𝐶𝐿 gradually recovers to -0.12 at 𝑅𝑒 = 3.53 × 105 . For 4 × 105 < 𝑅𝑒 < 4.8 ×
105, 𝐶𝐿 stabilizes at a value of -0.2.
For 𝐺/𝐷 = 2.0 ~ 0.5 (Figure 3.20(b) ~ (f)), the flow experiences the side
swapping (#1B→ #1T). The variation trends of 𝐶𝐷 with 𝑅𝑒 are somewhat similar at
all gap ratios. The transition to turbulence has a significant effect on 𝐶𝐿. An obvious
reduction of 𝐶𝐿 is observed prior to the transition, followed by a sudden jump of 𝐶𝐿
as 𝑅𝑒 is further increased. The obvious reduction of 𝐶𝐿 observed prior to the
transition is induced by the progressive pressure reduction on the gap side when the
critical Re is approached with the increase of Re. The sudden jump of 𝐶𝐿 is due to
the side swapping. As an example, the force coefficients at 𝐺/𝐷 = 0.8 (Figure 3.20
(d)) are examined based on the pressure distributions shown in Figure 3.4 and Figure
3.6. During the side swapping process, 𝐶𝐿 experiences a dramatic change. This can
be seen in Figure 3.21 20(d), where 𝐶𝐿 changed from -0.43 to +0.54 when 𝑅𝑒 is
increased from 3.27 × 105 to 3.32 × 105.
65
Figure 3.20. The effects of 𝐺/𝐷 and 𝑅𝑒 on 𝐶𝐷 and 𝐶𝐿 at different gap ratio (𝛿/
𝐷=0.1).
Figure 3.20 (g) ~ (j) show 𝐶𝐷 and 𝐶𝐿 for 𝐺/𝐷 = 0.4 ~ 0.1. 𝐶𝐿 remains positive
for flow in the #0 state. It has been known that the boundary layer transition on the
cylinder initiates from the top side of the cylinder in this group as shown in Figure
3.18. Consequently 𝐶𝐿 shows a sudden increase when the flow transits into the #1T
Re /105
CD
,C
L
1 2 3 4 5-1 5
-1
-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-1 5
-1
-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-1 5
-1
-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-1 5
-1
-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-1 5
-1
-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-1 5
-1
-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 5-0 5
0
0 5
1
1 5 CD
CL
Re /105
CD
,C
L
1 2 3 4 50
0 5
1
1 5 CD
CL
(a) G/D =3 0
(j) G/D =0 1 (i) G/D =0 2
(h) G/D =0 3 (g) G/D =0 4
(f) G/D =0 5 (e) G/D =0 6
(d) G/D =0 8 (c) G/D =1 0
(b) G/D =2 0
(k) G/D =0 0
66
state. At the same time 𝐶𝐷 shows a sudden reduction. With further increases of Re,
𝐶𝐷 remains a constant value within the tested range of Re, but 𝐶𝐿 shows a gradual
reduction accompanied by the development of the turbulent state on the gap side.
When the flow transits into the #2 stage, 𝐶𝐿 approaches to a value close to zero. The
reduction of 𝐶𝐿 observed during the transition from the #1T state to the #2 state is
simply because the asymmetry of the flow reduces during this process. Figure 3.20 (g)
shows 𝐶𝐷 and 𝐶𝐿 for 𝐺/𝐷 = 0. Within the tested range of 𝑅𝑒, 𝐶𝐿 reduces from
1.1 to 0.9 and 𝐶𝐷 varies in the range of 0.7 ~ 0.77. No sudden change of the force
coefficients is observed for this case over the range of Re investigated in this study. It
is believed that the flow transition to turbulence for the case with 𝐺/𝐷 = 0 occurs at
a smaller Re than the smallest Re tested in this study.
Figure 3.21. The lift coefficients at 𝐺/𝐷 = 0.8, 𝛿/𝐷 = 0.1 with different Re variation
sequences and pipe orientations.
Re/105
CL
1 2 3 4 5-1
-0.5
0
0.5
1 Re increasing, regular pipe positionRe decreasing, regular pipe positionRe increasing, pipe rotated 180 degree
67
3.5 Discussions
The Side swapping is a new phenomenon observed in the present work and the
formation mechanism is elaborated in Section 4. The most important feature for the
side swapping is the zig-zag variation trend of 𝐶𝐿 with the increase of Re. Almosnino
and McAlister (1984) conducted water tunnel tests and measured hydrodynamic load
using load cells. It was found that 𝐶𝐿 experienced a reduction from 1.2 to -1.2 in the
critical regime. This significant change of 𝐶𝐿 suggests the one bubble state switched
to its mirror image and this is the same as the side swapping defined in this work.
Almosnino and McAlister (1984) also found that the side swapping disappeared when
the model cylinder was turned by 180 degrees. Side swapping can also be seen in
Wang et al. (2016a) for flow around a finite height cylinder mounted on a
perpendicular wall. From the distribution of 𝐶𝑝 at one diameter below the free end of
the cylinder, it was found that the transition happened only on the side of 𝜃 =
0 ~ 180° at 𝑅𝑒 = 2.32 × 105, and then the transition switched to the opposite side
at 𝑅𝑒 = 2.74 × 105, before the transition occurred on both sides of the cylinder at
𝑅𝑒 = 3.42 × 105. Kamiya et al. (1979) conducted wind tunnel tests and presented
pressure distributions and force coefficients in the range of = 105~106. It was
found that 𝐶𝐿 swapped signs within the critical regime and this is an indication about
side swapping. Kamiya et al. (1979) also observed the transition to turbulence
separation developed on different sides of the cylinder, depending on the way by
which Re is varied in the test (increasing or decreasing). The side swapping was not
captured in many other works with a similar range of Reynolds number, such as
Schewe (1986), Qiu et al. (2014), Cadot et al. (2015) and present work for 𝐺/𝐷 =
3.0. As summarized above, for an isolated cylinder condition or a near wall cylinder
with 𝐺/𝐷 ≥ 2.0, it seems that the occurrence of the side swapping appears to be
random, which could be related to both the geometry imperfection and the disturbance
from the incoming flow. When a plane wall is introduced parallel to the cylinder, the
side swapping become a certain feature of the flow for a large range of 𝐺/𝐷 values
(0.5 ~ 1.0) as shown in Figure 3.18, irrespective of the model positions or the way by
68
which 𝑅𝑒 is varied (increasing or decreasing). To demonstrate this, one example is
given in Figure 3.21, which shows 𝐶𝐿 v.s. 𝑅𝑒 obtained under three different testing
conditions. Under the first condition, the model pipe was installed in the regular
position (the first pressure sensor at 𝜃 = 0°) and 𝑅𝑒 was gradually increased. The
second condition was tested with the pipe in its regular position and 𝑅𝑒 gradually
reduced. In the third condition, the cylinder was rotated with respect to the
longitudinal axis (the first pressure sensor at 𝜃 = 180°). The three curves shown in
Figure 3.21 all displayed the zig-zag feature related to the side swapping process.
However, the three curves are not exactly the same, although the key testing
parameters (𝑅𝑒, 𝐺/𝐷 and 𝛿/𝐷) are all the same. The difference between the first and
the second condition (increasing and decreasing of 𝑅𝑒) is mainly due to the hysteresis
of the transition swapping. For the 𝑅𝑒 increasing condition, the side swapping
happens at a higher 𝑅𝑒 value than that with the 𝑅𝑒 decreasing condition. The first
and the third conditions were tested with the pipe in different orientations. After
rotating the pipe by 180°, the onset of critical regime is pushed to a higher Re value
(about 3.33 × 105). The difference between the first and the third group is attributed
to the model geometry imperfection and the testing setup error. Although certain
differences exist among the test results under the three conditions, the important
information shown here is that the side swapping process exists under all the tested
conditions. This demonstrates that the side swapping is a certain feature in the critical
flow regime for flow around a cylinder near a plane boundary with G/D = 0.5 ~ 1.0.
3.6 Conclusions
In this work, a series of wind tunnel tests were conducted to investigate the
boundary layer transition to turbulence for flow around a circular cylinder above a
plane wall. The testing conditions cover 𝐺/𝐷 from 0 to 3, 𝑅𝑒 from 1.33×105 to
4.8×105 and 𝛿 𝐷 = 0.1⁄ and 0.5. The analysis is focused on the influence of the
plane boundary on the boundary layer transition from the sub-critical to critical and
then to supercritical on the cylinder surface. It is found that the proximity of the plane
boundary affects the transition process through a number of physical mechanisms
69
such as the changes of flow rate through the gap and the favourable pressure gradient
over the cylinder surface induced by the blockage effect and the thickness of
boundary layer (or velocity profiles) of the approaching flow. These lead to distinct
flow characteristics that are unique for flow around a cylinder near a plane boundary.
The major conclusions derived from this study are summarized below.
1. The influence of the proximity of the plane boundary is weak at large gap
ratios (e.g. 𝐺/𝐷 > 2.0). The transition process around the cylinder is similar
to that observed for an isolated cylinder. A slight difference is that the
boundary layer transition to turbulence is always initiated on the gap side of
the cylinder surface due to the enhanced shear layer induced by the blockage
effect on the gap side.
2. The transition process is significantly affected by the proximity of the plane
boundary but less so by the boundary thickness at intermediate gap ratios (e.g.
0.5 ≤ 𝐺/𝐷 ≤ 2.0). The transition process is distinctively different from that
of an isolated circular cylinder. The boundary layer transition is initiated on
the gap side of the cylinder surface, then swaps to the top side of the cylinder
surface while the boundary layer on the gap side switches back to laminar, and
eventually occurs on both sides of the cylinder surface as Re is increased from
near the upper bound value of the subcritical regime. This transition process is
described as the #0→#1B→#1T→#2 process, while the transition from the
#1B state to the #1T state is referred to as the side swapping in this study. The
enhanced favourable pressure gradient developed on the gap side surface of
the cylinder is identified as the main culprit for the suppression of the
boundary layer transition on the gap side and the side swapping observed in
this study.
3. The influence of velocity profiles (and turbulence intensity) of the approaching
flow becomes significant at small gap ratios (e.g 𝐺/𝐷 < 0.5). The transition
process changes from the sequence of #0→#1T→#2 (0.1 ≤ 𝐺/𝐷 ≤ 0.4) to
the sequence of #0→#1B→#2 (0.1 ≤ 𝐺/𝐷 ≤ 0.3) as the boundary layer
thickness 𝛿 𝐷⁄ is increased from 0.1 to 0.5 correspondingly. The transition
70
sequence of #0→#1T (0.1 ≤ 𝐺/𝐷 ≤ 0.4) for 𝛿 𝐷⁄ = 0.1 is attributed to the
enhanced shear layer on the top side of the cylinder induced by the reduction
of the flow through the gap, while the high turbulence intensity is responsible
for the sequence of #0→#1B (0.1 ≤ 𝐺/𝐷 ≤ 0.3) with 𝛿 𝐷⁄ = 0.5.
4. When 𝐺/𝐷 is reduced to zero, the transition only occurs on the top side of the
cylinder, but the critical 𝑅𝑒 is below the minimum 𝑅𝑒 tested in this study.
5. The boundary transition on the cylinder surface has a significant effect on the
mean lift coefficient. For 𝐺/𝐷 =2.0 ~ 0.5, the flow experiences the side
swapping (#1B→ #1T). An obvious reduction of 𝐶𝐿 is observed prior to the
transition, followed by a sudden jump of 𝐶𝐿 as Re is further increased. The
obvious reduction of 𝐶𝐿 is observed prior to the transition is induced by the
progressive pressure reduction on the gap side when the critical Re is
approached with the increase of Re. The sudden jump of 𝐶𝐿 is attributed to
the side swapping. When the side swapping happens, the lift force on the
cylinder shows a significant sudden jump due to the one bubble state
switching to its mirror image.
Acknowledgement
The authors would like to acknowledge the support from the National Key R&D
Program of China (Project ID: 2016YFE0200100), Australian Research Council
through DECRA scheme (DE150100428) and Linkage scheme (LP150100249), the
Fellowship Supporting Scheme and PhD scholarships from the University of Western
Australia.
71
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76
Table 3.1. The regimes of boundary layer transitions on the cylinder surface with the
increase of Reynolds number.
Transition processes Parameters range Key Features
1 #0→#1B→#2 𝐺
𝐷 = 3 (
𝛿
𝐷= 0.1 and 0.5)
No side swapping was observed here, but
it could happen in a random manner.
2 #0→#1B→#1T→#2
(1)𝐺
𝐷= 0.5~2 (
𝛿
𝐷= 0.1)
(2)𝐺
𝐷= 0.4~2(
𝛿
𝐷= 0.5)
Cylinder boundary layer transition always
initiates from the gap side. Side swapping
happens before the two bubble state is
fully established.
3 #0→#1T→#2 𝐺
𝐷= 0.1~0.4(
𝛿
𝐷= 0.1, )
Cylinder boundary layer transition to one
bubble state always initiates from the top
side, and then enters two bubble regime
without side swapping.
4 #0→1B→#2 𝐺
𝐷= 0.1~0.3 (
𝛿
𝐷= 0.5)
Cylinder boundary layer transition to one
bubble state always initiates from the gap
side, and then enters two bubble regime
without side swapping.
5 #0→#1T
𝐺
𝐷= 0 (
𝛿
𝐷=
0.1 and 0.5)
The critical Re value for the transition is
below the lowest Re value test here.
77
Appendix
Figure 3.22. The temporal and spatial distributions of pressure coefficient on the
cylinder surface for cases with 𝐺/𝐷 = 0.8 and 𝛿/𝐷 = 0.5 at different Re values.
(a) 𝑅𝑒 = 2.0 × 105
(b) 𝑅𝑒 = 2.67 × 105
(c) 𝑅𝑒 = 3.07 × 105
(d) 𝑅𝑒 = 3.13 × 105
(e) 𝑅𝑒 = 3.27 × 105
(f) 𝑅𝑒 = 3.33 × 105
(g) 𝑅𝑒 = 3.47 × 105
78
79
Chapter 4
Laminar separation bubble on a circular cylinder†
Abstract: Steady flow around a circular cylinder is investigated based on a
dynamic Smagorinsky Large Eddy Simulation (LES) model for Reynolds number (Re)
in the range of Re = 1 × 105 ~ 6.0 × 105. The numerical results show the flow
transitions from the subcritical regime to critical and then to the supercritical regime
with a reasonable Re sensitivity. Through flow visualization, the laminar separation
bubble (LSB) is demonstrated to be comprised of spanwise vortex tubes that
propagate and evolve along the cylinder surface before they become detached from
the cylinder surface and break down into small scale structures. The generation and
breakdown process of the spanwise vortex tubes is somewhat similar to the boundary
layer transition above a flat wall. The mean LSB streamline shape appears as a
recirculation bubble because the spanwise vortex tubes on the same side of cylinder
surface are of the same sense. The formation mechanism of the Kelvin-Helmholtz
(KH) shear layers around the cylinder is discussed, and a new scaling relationship for
the KH frequency is proposed, which is valid for a wide range of Re values from the
subcritical to the supercritical regime.
† This chapter is presented as a paper which has been submitted to Physics of Fulid as “An, H.,
Yang F., Cheng, L., Tong F., 2017. A Re-examination of the Laminar Separation Bubble on a
circular cylinder.”
4.1 Introduction
Flow around a smooth circular cylinder is governed by Reynolds number, which
is defined as 𝑅𝑒 = 𝑈𝐷/𝜐, where 𝑈 is the free stream velocity, 𝐷 is the diameter of
the cylinder and 𝜐 is the kinematic viscosity of the fluid. With the increase of 𝑅𝑒, a
80
series of changes happens to the boundary layer and the wake. A large amount of
research work has been published on the transition process between different flow
regimes. A detailed summary and review of the flow were documented by Sumer and
Fredsøe (2006). For a smooth cylinder, the wake transition from laminar to turbulence
first emerges at Re ≈ 300 and turbulence in the wake propagates upstream towards
the cylinder with the increase of 𝑅𝑒. In a large range of 𝑅𝑒 (300 ~ 3 × 105), the
wake flow is turbulent but the boundary layer on the cylinder surface remains laminar.
This regime is named as the subcritical regime. Within this regime, the drag
coefficient shows very low sensitivity to 𝑅𝑒 at around 1.2. Similarly, the Strouhal
number, St (= fD/U, where f is the frequency of vortex shedding) remains around 0.21.
With a further increase of Re, the drag coefficient experiences a dramatic reduction
from 1.2 to about 0.3 within Re ≈ 3 × 105~3.5× 105, which is known as the critical
regime (Achenbach, 1968, Shih et al., 1993, Bearman, 1969). This phenomenon is
also referred to as the drag crisis. In the transition from the subcritical to the critical
regime, the boundary layer on one side of the cylinder forms a laminar separation
bubble (LSB), followed by a turbulent reattachment and then a turbulent separation as
sketched in Figure 4.1. The concept of the LSB was firstly introduced in the research
on flow around an aerofoil (Gault, 1949). The LSB registers a plateau in the
distribution of the mean pressure on the aerofoil surface. Based on this feature, Tani
(1964) reviewed the research work about the LSB on aerofoils and also demonstrated
that it exists on the surface of a circular cylinder, according to the cylinder surface
pressure measurement conducted by Fage (1929) and Yamamoto and Iuchi (1965).
Achenbach (1968) also demonstrated the existence of the LSB through pressure and
skin friction measurements on a smooth cylinder in the critical range of Re and they
found that the LSB normally covers a range of 10° ~ 15° starting at about 𝜃 ≈ 100°
(𝑅𝑒 dependent), where 𝜃 is the clockwise angular position on the cylinder surface
with 𝜃 = 0 corresponding to the stagnation point (see Figure 4.1). Due to the
limitation of flow measurement devices and the specific feature of the LSB (such as
dimensions), the thickness of the LSB has not been quantified experimentally so far.
Bearman (1969) inferred that the LSB only exists on one side of the cylinder in the
81
range of Re ≈ 3.5 × 105~3.8× 105 and this is called one bubble state. Two bubbles
were observed with one bubble on either side of the cylinder for 3.8× 105 < Re ≤
7.5 × 105 . These two states occur in the critical and supercritical regimes,
respectively. Lehmkuhl et al. (2014) visualized the LSB in detail through numerical
simulations. It was found that the LSB is a flat surface-touching circulation zone,
which covers a range of about 10° along the cylinder surface and the centre of the
LSB is about 0.003D away from the cylinder surface. The dimensions and the angular
position of the LSB are also sensitive to Re. In the trans-critical regime, LSB
disappears and the boundary layer becomes turbulent at certain locations on the
cylinder surface (Achenbach, 1968). The separation points move towards upstream
with the increase of 𝑅𝑒 and consequently the drag coefficient recovers to about
0.5~0.7 in the trans-critical regime. Strong vortex shedding was observed again in this
regime with St ≈ 0.27 (Roshko, 1961).
2 3 4
1
5
6
7
𝜃
Figure 4.1. A sketch of the laminar separation bubble in the critical turbulent flow
regime. The details about the labels are as follows; 1. Laminar boundary layer; 2.
Laminar separation point; 3. Laminar separation bubble; 4. Turbulent re-attachment; 5.
Turbulent boundary layer; 6. Turbulent separation point; 7. Wake separation bubble,
𝜃 is the angle position on the cylinder surface.
It is noted that the critical 𝑅𝑒 for different flow regimes is affected by multiple
factors, such as turbulence intensity of the incoming flow, model aspect ratio (length
to diameter), blockage ratio (test section width to diameter), surface roughness and the
accuracy of model setup. Therefore, certain discrepancies exist among published data
due to this reason. The difficulty of identifying a clear cut for a transition also adds to
the discrepancy. The flow structure can switch between two regimes intermittently.
For example, Cadot et al. (2015) conducted a detailed investigation about the pressure
82
distribution with 𝑅𝑒 in the range of 1.25× 105 ~ 3.75× 105. It was found that the
flow was mainly in the transit from the subcritical to the critical regime. Through a
series of probability analysis of the surface pressure, Cadot et al. (2015) observed that
two or three different boundary layer states co-exist in one test in the range of
3.0× 105 ~ 3.75× 105.
Due to the strong velocity gradient in the separated shear layer from the cylinder
surface, small-scale eddies form close to the wall and are named as Kelvin-Helmholtz
(KH) vortices. The KH vortices play an important role for the transition to turbulence
in the wake and on the cylinder surface. Other terminologies have also been used in
the literature, such as transitional wave, secondary vortices, Bloor-Gerard vortices,
shear layer vortices, along with KH vortices. The KH vortices in the separated shear
layer of a circular cylinder was systematically reported by Bloor (1964) through
hotwire measurements in the range of 𝑅𝑒 = 250 ~ 4500. Bloor (1964) demonstrated
that the formation region of the KH vortices moves towards the cylinder surface with
the increase of 𝑅𝑒 and plays an important role in the transition to turbulence in the
wake. Wei and Smith (1986) conducted measurements of the KH vortices using the
hydrogen-bubble technique in the range of 𝑅𝑒 = 1200 ~ 11000 and some
three-dimensional features of the KH vortices were examined. It was found that the
KH vortex tubes develop a wavy feature along the cylinder spanwise direction and roll
up with the Kármán vortices to induce cellular structures in the wake. Most of the
existing experimental work about the KH vortices was conducted with Re < 105. It is
extremely difficult to visualize the KH vortices for 𝑅𝑒 > 105, mainly due to its high
frequency and small sizes. Prasad and Williamson (1997) summarized the published
data about the frequency of the KH vortices (𝑓𝐾𝐻). It was found the KH frequency and
the Kármán vortex shedding frequency (𝑓𝑠𝑡) follow a scaling of 𝑓𝐾𝐻/𝑓𝑠𝑡 ~ 𝑅𝑒0.67.
With the rapid development of supercomputing facilities, computational fluid
dynamic (CFD) simulation of the flow around a cylinder has gradually progressed
with an increasing trend of 𝑅𝑒. The effort so far with various Reynolds averaged
Navier-Stokes (RANS) models in capturing the drag crisis phenomenon has not been
awarded with satisfactory results. In contrast, the large eddy simulation (LES) models
83
appear to have had a reasonable success. Breuer (2000) simulated flow at Re =
1.4 × 105 based on a finite-volume method with both the conventional Smagorinsky
model and a dynamic sub-grid scale model. The numerical results agree well with
experimental measurements (Cantwell and Coles, 1983). Lehmkuhl et al. (2014)
reported a series of LES results with a focus on the boundary layer transition process.
The wall-adapting local-eddy viscosity model (WALE) was used. The dramatic
reductions of the drag coefficient in the critical regime and the transition from the
critical to the supercritical regimes were well captured. Additionally, the one bubble
state was observed at Re = 2.5 × 105 and detailed instantaneous wake structures
were discussed. Yeon et al. (2015) reported an LES investigation on the drag crisis for
𝑅𝑒 ranging from 6.31 × 104 to 7.57 × 105, with a Lagrangian dynamic Subgrid-scale
(SGS) model and a finite difference scheme. A detailed mesh dependence check at
different 𝑅𝑒 values was presented and the transitions from the sub-critical to the
critical and then to the super-critical regimes were well captured. Lloyd and James
(2015) examined the drag crisis phenomenon using LES based on the OpenFOAM®.
It was found that the Dynamic mixed Smagorinsky model performed better than the
original Smagorinsky model. Lloyd and James (2015) also pointed out that the
supercritical regime flow is more sensitive to the spanwise mesh resolution than the
subcritical regime flow. Chopra and Mittal (2016) paid a special attention to the LSB
in the range of Re around 1 × 104 ~ 4 × 105 using a finite element method. The LES
model with the conventional Smagorinsky model was adopted to simulate the flow.
The results showed that the formation of the LSB is intermittent and this finding
agrees with that by Cadot et al. (2015). Cheng et al. (2017) carried out a LES
simulation with a stretched-vortex SGS model at 𝑅𝑒 = 3.9× 103~8.5 × 105, with a
focus on the property of the skin-friction in different flow regimes. Cao and Tamura
(2017) applied a dynamic mixed SGS model to predict the supercritical flow past a
square cylinder with rounded corners. The boundary layer transition with LSB was
well captured.
Although substantial amount of research work has been done on the drag crisis
and the LSB, there are still some questions to be answered. The first question is the
84
formation mechanism of the LSB. Although as discussed above, the LSB has been
reported in both experiments (Achenbach, 1968, Bearman, 1969) and numerical
simulations (Lehmkuhl et al., 2014, Chopra and Mittal, 2016), the formation
mechanism is yet not clear. The second question is the magnitude of KH frequency in
the supercritical regime. Prasad and Williamson (1997) commented that the scaling of
𝑓𝐾𝐻/𝑓𝑠𝑡 ~ 𝑅𝑒0.67 is valid for 𝑅𝑒 up to 105. The validity of this scaling for higher
𝑅𝑒 range, especially in the supercritical regime is still not known. Motivated by the
above two questions, a series numerical simulations of the flow are carried out in the
transition range of Re (subcritical to supercritical). The remaining of the paper is
organised as following. A brief description about the governing equations is given in
section 2. In section 3, the mesh dependency check and model validation are reported.
The main findings on the features of the laminar separation bubbles and the frequency
of KH vortices are discussed in section 4, followed by conclusions in section 5.
4.2 Governing Equations and numerical method
In this work, the flow is simulated with a LES closure embedded in the
OpenFOAM®, an open source CFD package. The large scale turbulene structures are
seperated from small scale structures through a filtering function in the following
manner,
𝑓(��) = ∫ 𝑓(𝑟′)𝛺
𝐺(𝑟 − 𝑟′)𝑑𝑟′ (1)
where Ω is the domain size and G is the filter kernel, r representes the original
signal of fluid (velocity components and pressure), over bar (��) denotes the large scale
variable, which will be resolved in the simulation; the prime (𝑟′) denotes the subgrid
scale which will be considered through a model.
Applying the filter function defined in Eq. (1) to the Navier-Stokes equations, we
get
𝜕𝑢𝑖
𝜕𝑡+
𝜕
𝜕𝑥𝑗(��𝑖��𝑗) = −
1
𝜌
𝜕��
𝜕𝑥𝑖+
𝜕
𝜕𝑥𝑗((𝜈 + 𝜈𝑡) (
𝜕𝑢𝑖
𝜕𝑥𝑗+
𝜕𝑢𝑗
𝜕𝑥𝑖)) (2)
And
𝜕𝑢𝑖
𝜕𝑥𝑖= 0 (3)
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In Eq. 2, 𝜈𝑡 is the sub-grid scale stress (SGS) turbulent viscosity. The value of
𝜈𝑡 is calculated based on a homogeneous dynamic Smagorinsky model (Smagorinsky,
1963, and Germano et al., 1991).
The governing equations are solved through finite volume scheme based on
OpenFOAM®, an open source CFD package. The solver based on the Pressure
Implicit with Splitting of Operators (PISO) method in OpenFOAM is used here. The
convection terms are discretised using the Gauss cubic scheme, while the Laplacian
and pressure terms in the momentum equations are discretised using the Gauss linear
scheme.
4.3 Mesh dependency study and model validations
4.3.1 Mesh dependency study
A rectangular cuboid domain with dimensions of 24D×24D×0.5πD in the x-, y-
and z-direction, respectively, is used in present work. The distance from the cylinder
to the inlet boundary is 8D. A spanwise domain length (L) of 0.5πD is selected for
present simulation, which was based on the spanwise correlation length at 1D ~ 0.5πD
of flow around circular cylinders in the range of 𝑅𝑒 = 1 × 105 ~ 5 × 105, measured
by Blackburn and Melbourne (1996). Lehmkuhl et al. (2014) used the same spanwise
dimension and Cheng et al. (2017) used 𝐿/𝐷 = 1 for cases in the critical and
supercritical regimes. For the inlet boundary, a constant velocity (U) and a zero
pressure gradient in x-direction are applied. The out flow is specified as velocity
zero-gradient along the free stream direction and a constant pressure (=0). On the top
and bottom boundary, symmetry conditions are applied. Periodic boundary conditions
are applied on the two side boundaries at the two ends of the cylinder.
Table 4.1. A summary of the detailed information about the three meshes used in the
mesh dependency study.
Mesh 𝑁𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑁𝑐𝑖𝑟𝑐𝑙𝑒 𝑁𝑠𝑝𝑎𝑛 ∆/𝐷 𝐶𝐷
1 1.29 × 106 280 32 0.00016 0.465
2 8.78 × 106 560 48 0.00008 0.314
3 1.49 × 107 1120 64 0.00004 0.307
86
A mesh dependence study is carried out to choose an appropriate mesh. A typical
two-dimensional (2D) mesh around the cylinder is shown in Figure 4.2 (a), along with
a zoom-in view around the cylinder surface in Figure 4.2 (b). The area near the
cylinder surface is discretised with a structured mesh, which allows for the mesh
density near the wall being precisely controlled to resolve the viscous sublayer on the
cylinder surface. This is critically important for capturing the transition of the
boundary layer on the cylinder surface. Unstructured mesh is used to allow the mesh
density to be reduced in the area relatively far from the cylinder boundary layer and
the wake. The three-dimensional (3D) mesh is formed by replicating the 2D mesh in
the spanwise direction of the cylinder. Three meshes with different density are tested
at 𝑅𝑒 = 6 × 105 , which is the highest 𝑅𝑒 number investigated in this study.
Detailed mesh information is given in Table 4.1. The key information about the
meshes includes the total number of elements in the whole domain (𝑁𝑒𝑙𝑒𝑚𝑒𝑛𝑡), the
number of elements on the cylinder circumference (𝑁𝑐𝑖𝑟𝑐𝑙𝑒) in a 2D slice in the x-y
plane, first layer mesh thickness on the cylinder surface (∆) and the number of
spanwise elements (𝑁𝑠𝑝𝑎𝑛). The non-dimensional computational time step size is
chosen at ∆𝑡 = 0.0002.
Figure 4.2 The computational mesh and a zoom-in view of the elements near the wall.
The computational facility Magnus supercomputer provided by Pawsey
87
(https://www.pawsey.org.au) is used for the simulations. Magnus comprises a total of
35,712 cores Intel Xeon E5-2690V3 Haswell processors running at 2.6 GHz. The
computational cost is very high in present work. Each case is conducted with 240
cores in parallel and the total wall time is 11 days. At the end of the simulation, it
reaches a non-dimensional time of Ut/D = 100 at 𝑅𝑒 = 6×105.
Firstly, the distributions of averaged pressure coefficient (𝐶𝑃 = (𝑝 − 𝑝∞)/0.5𝜌𝑈2)
obtained using the three meshes are shown in Figure 4.3, where 𝜌 is the density of
the fluid and 𝑝∞ is the reference pressure at outlet boundary. The experimental data
by Bursnall and Loftin Jr (1951) are also presented in Figure 4.3 for comparison
purpose. It can be seen that the pressure distribution around the cylinder shows a
convergent trend and a reasonable agreement with Bursnall and Loftin Jr (1951). The
Cp from mesh 3 shows local peaks around 𝜃 = 100° and 260°, which are associated
with the LSBs as indicated by Tani (1964). More discussion about LSB is given in the
next section. Therefore, Mesh 3 is used in this work for other simulations.
Figure 4.3. The pressure coefficients on the cylinder surface simulated with different
mesh density at 𝑅𝑒 = 6 × 105, compared with the experimental data by Bursnall and
Loftin (1951) at 𝑅𝑒 = 5.95 × 105. The detailed information about the meshes can be
found in Table 4.1.
4.3.2 Force coefficients
The mean drag and lift coefficients (𝐶𝐷 and 𝐶𝐿
) are examined here, which are
defined as 𝐶𝐷 = 𝐹𝐷
/0.5𝜌𝐷𝑈2𝐿 and 𝐶𝐿 = 𝐹𝐿
/0.5𝜌𝐷𝑈2𝐿, where 𝐹𝐷 and 𝐹𝐿 are the
90
boundary layer transition on the cylinder surface. A series of measurements about St
have been reported (Achenbach, 1968, Bearman, 1969, Schewe, 1983, Qiu et al., 2014,
Lehmkuhl et al., 2014). It has been understood that regular vortex shedding exists in
the subcritical regime at St ≈ 0.21 and the vortex shedding becomes less regular in
the critical and supercritical regimes. Similar features are captured by the present LES
model and the variation of St with Re is shown in Figure 4.17(c), together with the
results from a few independent studies. It is found that St ≈ 0.21 in the subcritical
regime, which is in good agreement with other works. For Re ≥ 2 × 105, 𝑆𝑡 ≈ 0.35
is predicted in the present work, which agrees well with that reported for critical
regime flow, but slightly lower than that of supercritical regime flows reported in
experimental measurements (Bearman, 1969, Schewe, 1983).
4.3.4 Pressure coefficients
The distribution of 𝐶𝑝 in the subcritical regime (𝑅𝑒 = 1.0 × 105) and the critical
regime (𝑅𝑒 = 2 × 105) are examined in Figure 4.5, together with other published data.
The Cp in the supercritical regime at 𝑅𝑒 = 6 × 105is reported in Figure 4.3. The
present result obtained at 𝑅𝑒 = 1 × 105 (Figure 4.5 (a)) is compared with the
experimental data given by Cantwell and Coles (1983) at 𝑅𝑒 = 1.41 × 105 and by
Braza et al. (2006) at 𝑅𝑒 = 1.12 × 105. As a general feature of the three groups of
data, the minimum 𝐶𝑝 was captured at 𝜃 = 180°. This is because the strong vortex
shedding leads to a strong negative pressure zone in the near wake of the cylinder.
The Cp distribution in the critical regime (Figure 4.5 (b)) shows a very different
feature from that in the subcritical regime and is characterized by asymmetric
distributions on the upper and lower surfaces of the cylinder. A strong negative
pressure is observed at around 𝜃 = 280°, which is a signature of the transition for
the boundary layer on this side. This flow phenomenon agrees well with the
experimental data given by Qiu et al. (2014) and Cadot et al. (2015).
91
Figure 4.5. Mean pressure coefficient 𝐶𝑝 on the cylinder surface at different 𝑅𝑒
values. (a), subcritical regime flow, (b) critical regime flow.
4.4 Numerical result and discussion
The numerical simulations are carried out for ten Re values over the range of Re =
1×105 ~ 6×105. The simulations are designed to cover the transitions from the
subcritical to the critical and then the supercritical regimes. The force coefficients,
pressure distribution, instantaneous & mean flow structures and the LSB are
quantified. As a very brief summary, the flow with 𝑅𝑒 = 2×105 is found to be in the
critical regime where the boundary layer on one side of the surface becomes turbulent.
The flow is respectively in the subcritical and supercritical regimes on the two sides
of 𝑅𝑒 = 2×105. Although the key features of the flow in different regimes are well
captured, the present critical regime 𝑅𝑒 is slightly lower than the values reported in
the literature. Detailed results are given below.
92
Figure 4.6. The spatial-temporal evolution of pressure coefficient on the surface of the
cylinder at (a) Re = 105, (b) 2×105 and (c) 6×105. The two vertical dash lines in (a)
enclose a typical vortex shedding period. The arrows in (b) and (c) indicate the
formation of small scale vortices sliding on the cylinder surface. Low pressure zones
associated with boundary layer transition are also labelled in (b) and (c).
4.4.1 Transition features
The dynamic feature of the pressure distribution on the cylinder is investigated
first. The distributions of instantaneous 𝐶𝑝are plotted as spatial-temporal contours in
Figure 4.6 at three representative Re values, which correspond to the flow in the
subcritical, the critical and the supercitical regimes, respectively. A strong alternate
pattern of low and high 𝐶𝑝 values is found on the top and bottom sides of the
cylinder at 𝑅𝑒 = 1 × 105, as enclosed by the dash lines in Figure 4.6 (a). This is
attributed to the periodic vortex shedding from the cylinder. Figure 4.6 (b) depicts a
typical feature of surface pressure in the critical regime (𝑅𝑒 = 2 × 105). The alternate
93
pattern observed in Figure 4.6 (a) is significantly weakened, suggesting the weakening
of the regular vortex shedding in the wake. Instead, strong negative inclined pressure
strips occur intermittently around 270° < 𝜃 < 300°, which signal the boundary layer
transition on the cylinder surface. The strong negative pressure strips are caused by
the formation of small scale vortices near the low-pressure zone and the inclination
shape suggests that the vortices are convected downstream along the cylinder surface
before they are merged into the wake. The flow feature is a typical of the so-called
one-bubble state reported in the literature. Schewe (1983) speculated that the
one-bubble state tends to stabilize the flow. The flow could remain in the asymmetric
state over a range of 𝑅𝑒 value until the boundary layer transition on the other side
occurs. Cadot et al. (2015) observed that the asymmetric flow pattern could switch to
its mirror image pattern in the critical regime intermittently at a constant Re value.
This is not observed in this study within the simulation time and efforts were not
made to extend the simulation due to the limitation in computation resource. The
instantaneous Cp contour of the supercritical flow at 𝑅𝑒 = 6 × 105 is given in Figure
4.6(c). Two low pressure areas exist at around 𝜃 = 80° and 𝜃 = 280° , which
suggests that the boundary layer transition happens on both sides of the cylinder
surface and the flow is in the supercritical regime. The overall feature of the flow is
symmetric on the two sides of the cylinder and the mean lift coefficient is zero. This is
a typical of the two-bubble state reported in the literature.
It should be noted that there is some inconsistency in the literature about the
symmetry state of the supercritical flow. For example, Bearman (1969), Schewe (1983)
and Qiu et al. (2014) reported a zero 𝐶𝐿 = 0 in the supercritical regime, but
Almosnino and McAlister (1984) and Kamiya et al. (1979) reported a non-zero 𝐶𝐿 ≠
0 and attributed this to the existence of a supercritical asymmetry in the flow.
94
Figure 4.7. The near wake flow structure represented by iso-surface of 𝜆2 = -1 for 𝑅𝑒
= 105, 2×105 and 6×105, from top to bottom, while the colour contours are based on
pressure coefficient.
Typical instantaneous near wake flow structures at different Re values are shown
in Figure 4.7 through the iso-surface of 𝜆2 = −1, which is the second eigenvalue of
the tensor Ψ2+Ω2. Here Ψ and Ω are the symmetric and the anti-symmetric parts of
the velocity-gradient tensor, respectively, as defined by Jeong and Hussain (1995). In
Figure 4.7 (a) (𝑅𝑒 = 1 × 105), the key flow feature is that a Kármán vortex street is
formed in the wake. The wake also shows strong three-dimensionality, with a large
amount of streamwise coherent vortex tubes connecting the spanwise vortices. The
Kármán vortices become less regular in Figure 4.7 (b) at 𝑅𝑒 = 2 × 105, which is
mainly due to the influence from the transition on the lower boundary layer. The plot
for the supercritical regime flow is shown in Figure 4.7 (c). The near wake of the
cylinder becomes narrower in this case due to the transition to the supercritical regime.
95
Figure 4.8. Instantaneous vorticity (𝜔𝑧) and pressure contours at 𝑅𝑒 = 105, 2×105 and
6×105 from left to right.
Figure 4.9. The averaged flow field represented by streamlines (red lines) and
pressure contours for 𝑅𝑒 = 105, 2×105 and 6×105 from left to right.
The separated shear layers from the cylinder surface are examined through the
spanwise vorticity field, which is calculated as 𝜔𝑧 = (𝜕𝑣
𝜕𝑥−
𝜕𝑢
𝜕𝑦)
𝐷
𝑈. Figure 4.8(a~c)
shows the instantaneous contours of 𝜔𝑧 in the middle cross-section of the cylinder at
the three representative Re values, along with the contour of intantanouse Cp in (d~f).
The solid arrows in Figure 4.8 point to the flow separation points at the instant. It can
be seen that the boundary layers remain laminar before the separation points for the
subcritical flow shown in Figure 4.8 (a), judged by the laminar shear layers enclosed
by the dashed rectangles. With the flow travelling further downstream, instability
96
happens to the shear layers and forms a series of small scale vortices. This is referred
to as the Kelvin-Helmholtz (KH) instability of the shear layer in the literature. The
distance between the separation point and the appearance of the KH instability is
about 0.28D on both sides of the cylinder and it does show a strong variation with
time. A strong negative pressure area exists in the wake near the cylinder in Figure 4.8
(d). This is because the shear layer at the upper side of the cylinder rolls up and forms
a large-scale vortex. The small-scale vortices in the shear layer can also be seen in the
pressure field. For 𝑅𝑒 = 2 × 105 shown Figure 4.8 (b), the separated shear layer on
the upper side of the cylinder is still in the laminar regime, similar to that at 𝑅𝑒 =
1 × 105. But the distance from the separation point (at 𝜃 = 86°) to the formation of
the KH instability reduces to about 0.09D. A dramatic difference can be seen on the
lower half of the cylinder surface, where the flow separates at 𝜃 = 268°. The
separated laminar shear layer disappears and vortices of small scales are found on the
cylinder surface downstream the separation point. The corresponding pressure
contours shown in Figure 4.8 (e) show an area of strong negative pressure on the
lower side of the cylinder surface, which is a typical feature associated with the
boundary layer transition to turbulence. Figure 4.8 (c) shows the vorticity contours at
𝑅𝑒 = 6 × 105 as an example of the supercritical regime flow. The boundary layer
transition happens on both sides of the cylinder surface. The low pressure zones shift
to around 𝜃 = 80° and 280° as shown in Figure 4.8(f).
The mean streamlines and pressure contours at three different Re values are
examined in Figure 4.9. A vortex pair is observed in the wake for each case. With the
increase of Re, the mean wake gradually becomes narrower in the cross flow direction
and shorter in the main stream direction. For the subcritical regime flow shown in
Figure 4.9(a), the mean flow is generally symmetric and a low pressure zone is
observed in the wake. For the critical flow shown in Figure 4.9(b), due to the different
boundary layers formed on the two sides of the cylinder, the mean flow field becomes
asymmetric. The mean wake becomes very narrow and retains the symmetry about
x-axis for the supercritical flow as shown in Figure 4.4(c). The low pressure zones
occur near 𝜃 = 80° and 280° , which is very similar to the typical instantaneous
97
pressure field shown in Figure 4.8(f).
4.4.2 Laminar separation bubbles
The small LSB attached to the cylinder surface is a key feature of the boundary
layer transition and is visualized through the mean flow field as demonstrated by
Achenbach (1968), Bearman (1969) and Farell and Blessmann (1983). The LSB is
found at 𝑅𝑒 = 2~6 × 105 in this study. The starting and ending position angles of
the LSB on the cylinder surface are quantified in Figure 4.10 (a) and (b), together with
those reported by Tani (1964) and Lehmkuhl et al. (2014). Since LSB is observed on
the lower side for 𝑅𝑒 = 2 × 105 in present work, so the lower side angle positions
are used. For 𝑅𝑒 = 3~6 × 105, the LSBs on the two sides of the cylinder are
symmetrically distributed in present numerical results. Tani (1964) observed the LSB
in the range of 𝑅𝑒 = 3.7 × 105~1 × 106. Within this range of Re, the position angle
covered by the LSB decreases from 15° to 10°. The starting position angle of the LSB
decreases monotonically with the increase of Re and the ending position angle only
changes slightly with Re. The present numerical results and those by Lehmkuhl et al.
(2014) follow similar trends with that by Tani (1964), although the observed LSB
occurs at a slightly lower 𝑅𝑒 value than that reported by Tani (1964).
99
Figure 4.12. The KH vortices in the boundary layer on the cylinder surface at 𝑅𝑒 =
6× 105 from 𝑈𝑡/𝐷 = 75.91 to 75.99 with interval of ∆(𝑈𝑡
𝐷) = 0.02.
Figure 4.13. The merging process of KH vortices in the boundary layer on the cylinder
surface at 𝑅𝑒 = 6× 105 from 𝑈𝑡/𝐷 = 75.955 to 75.975 with interval of 𝑈∆𝑡/𝐷 =
0.005.
100
Figure 4.14. The variation of KH vortices with Reynolds number. (a), 𝑅𝑒 = 105; (b),
2 × 105 ; (c), 3 × 105; (d), 4 × 105; (e), 5 × 105; (f), 6 × 105. The 𝐾𝐻 vortices
are visualized through iso-surfaces of 𝜆2 = −1000 and colour contours of 𝜔𝑧 in the
range of -100 to 100. The angle position on this side of the cylinder is also labelled
under each plot. The shadow area represents the region for LSB in the averaged flow
field.
To understand the formation mechanism of the LSB, instantaneous flow
structures near the LSB region are examined through visualizing the iso-surface of 𝜆2.
It is found that the value of 𝜆2 near the core of the LSB is higher than those in other
regions. According to this feature, the flow structure is observed by adjusting 𝜆2
value to an appropriate level. For example, the iso-surfaces of 𝜆2 = −1000 for
𝑅𝑒 = 6 × 105 at 𝑈𝑡/𝐷 = 80.3 are shown in Figure 4.11 (a) and (b) with views along
101
the y-axis and z-axis directions, respectively. The colour contours are based on 𝜔𝑧 in
the range of -100 to 100. The vortices circled by the dash line in Figure 4.11(b)
correspond to these shown in Figure 4.11(a). The separation line on the cylinder
surface is also shown in Figure 4.11(a). The following features are observed:
(1) The separation line shows slight undulations, suggesting the flow is only
weakly three-dimensional upstream of the flow separation.
(2) The spanwise vortex tubes near the LSB region are coherent structures
extending along the cylinder spanwise direction. The vortex tubes exist in different
length scales in both spanwise and streamwise directions.
(3) The spanwise vortex tubes are of the same sign (positive vortices) on each
side of the cylinder (for instance in Figure 4.11(a)). This is because the vortex tubes
are formed from the separated shear layer rolling towards the cylinder surface
direction.
(4) The strength of vortex tubes decay rapidly along the flow direction, judging
based on the colour contours of the vortex tubes and the length scales in the
streamwise directions.
(5) The diameter and the spanwise length of the spanwise vortex tubes increase
as they are convected along the cylinder surface towards downstream and eventually
breakdown into small scale structures (the KH vortices) as they become detached
from the cylinder surface.
The spanwise vortex tubes are formed about 0.025𝐷 behind the separation line
and are aligned approximately parallel to the cylinder axis for the case shown in
Figure 4.11. They go through an evolution process of generation→ evolution→
decay/breakdown. This process is demonstrated through an example given in Figure
4.12, where the mean flow structure in the middle cross section and a sequence of
instantaneous flow structures at 𝑅𝑒 = 6 × 105 are shown. A recirculation zone is
observed in the range of 251.2° ≤ 𝜃 ≤ 261.1° by the streamlines of the mean flow.
This is deemed as the LSB. The evolution process of the spanwise vortex tubes near
the LSB region is examined within 𝑈𝑡/𝐷 = 75.91 ~ 75.99 at an interval of 𝑈∆𝑡/𝐷
102
= 0.02 in Figure 4.12 (b) ~ (f). Such a small time interval is necessary to resolve the
evolution process of the spanwise vortex tubes. At 𝑈𝑡/𝐷 = 75.91 (Figure 4.12 (b)),
the vortices A and B are formed and slide along the cylinder surface towards
downstream direction with time. At the same time, the sizes of the vortices A and B
experience increases first and then decreases as they are convected downstream along
the cylinder surface. The interactions of the two vortices with a thin shear layer of the
opposite sign (red colour) are clearly observed in Figure 4.12. At 𝑈𝑡/𝐷 = 75.99
(Figure 4.12 (f)), the two vortices are almost fully decayed and two new vortices are
formed again upstream. This completes a vortex evolution cycle. The coherent
spanwise vortex tubes are of the same sign on each side of the cylinder surface and
they all go through the same evolution process as demonstrated by vortex A and B.
The KH vortices also experiences frequent merging process while they are convected
along the cylinder surface. An example is given in Figure 4.13 at five frames with
𝑈∆𝑡/𝐷 = 0.005. The range of cylinder surface shown here corresponds to the region
covered by LSB. Three vortices exist in Figure 4.13 (a), named as A, C and D. Vortex
A is the one discussed in Figure 4.12. The propagation and merging process of C and
D can be seen clearly here. The distance between C and D reduces gradually (Figure
4.13(a) ~ (d)) and form a new vortex C+D in Figure 4.13(e).
Figure 4.14 shows the variation of the spanwise vortex structures with 𝑅𝑒 in the
range of 𝑅𝑒 = 105~6 × 105. Angle scale is labelled beneath each plot. The region
where LSB appears in the mean flow field is covered by the blue shadow area on the
angle scales. It is observed that the spanwise vortex tubes are only slightly distorted
from two-dimensional tubes with relatively large diameter and large spanwise length
scales in the subcritical regime at 𝑅𝑒 = 105 as shown in Figure 4.14(a). As Re is
increased from 𝑅𝑒 = 2 × 105 to 𝑅𝑒 = 6 × 105 as shown in Figure 4.14(b) to
Figure 4.14(f), the diameter and spanwise length of the vortex tubes become smaller.
The generation and breakdown process of the vortex tubes observed in Figure 4.14 is
a typical process of the flow transition to turbulence, somewhat similar to the
transition to turbulence of a flat-wall boundary layer (Sayadi et al., 2013). Within the
LSB range (covered by the blue shadow colour on the angle scales), the diameter and
103
the spanwise length of the vortex tubes increase slightly as they are convected
downstream. The vortex tubes start to break down to small scale structures as soon as
they become detached from the cylinder surface (Figure 4.11). It is also observed
from Figure 4.14 that the spanwise vortex tubes appear in a similar region on the
cylinder surface in the supercritical regime (𝑅𝑒 = 3 ~ 6 × 105).
Figure 4.15. The time histories of pressure on the cylinder surface for 𝜃 =
230° ~ 280° at 𝑅𝑒 = 6 × 105.
The time history of pressure fluctuations at certain probe points on the cylinder
surface are examined in Figure 4.15 over the range of 𝜃 = 230° ~ 280° for 𝑅𝑒 =
6 × 105, which covers the LSB region. The low frequency pressure fluctuations at
𝜃 = 270° and 280° (Figure 4.15(a)) are induced by the Kármán vortex shedding. At
𝜃 = 260°, which is near the upstream tip of the LSB, high frequency pressure
fluctuations are superposed on the low frequency fluctuations. The fluctuations with
104
the high frequency is induced by the KH shear layer near the upstream tip of the LSB,
the one with the lowest frequency fluctuation is due to the Kármán vortex shedding.
At 𝜃 = 255° and 250° , the pressure fluctuations are characterised by high
amplitudes and frequencies. Further downstream at 𝜃 = 240° and 230°, both the
level of fluctuations and the frequency of Cp reduce.
Figure 4.16. The frequency spectrums of the pressure signals on the cylinder surface
at five selected locations at 𝑅𝑒 = 6 × 105.
fD/U
E
10-1
100
101
102
10310
-1
100
101
102
103
104 (a) =
E
10-1
100
101
102
10310
-1
100
101
102
103
104 (b) =
40 50 600
5
10
E
10-1
100
101
102
10310
-1
100
101
102
103
104 (c) =
E
10-1
100
101
102
10310
-1
100
101
102
103
104 (d) =
106
components is very low and the energy level shows monotonic reduction in general.
No dominating peak with high frequency component is observed. At 𝜃 = 260°
(Figure 4.16 (b)), the energy level decays in a similar way to that of 𝜃 = 270°, except
for a small hump in the frequency range of 20 < 𝑓𝐷/𝑈 < 40. The inset in Figure 4.11
(b) shows a zoom-in view of the high frequency peak at 𝑓𝐷/𝑈 = 39.8. This high
frequency component corresponds to the frequency of the KH shear layer. Figure 4.16
(c) shows the spectrum at 𝜃 = 255°, which is roughly in the middle of the LSB. It
can be seen that the KH frequency covers a wide range of high frequency components
in the range of 20.8 < 𝑓𝐷/𝑈 < 103.2. Dong et al. (2006) reported a very similar
feature for a subcritical regime flow at 𝑅𝑒 = 104. Among the higher frequency
component shown in Figure 4.16(c), the most energetic components occur in the
range of 31.6 < 𝑓𝐷/𝑈 < 47.8, which leads to an average of 39.2, almost identical to
the peak shown in Figure 4.16(b). So this frequency is taken as the dominating
frequency of the KH vortices. Further downstream at 𝜃 = 250°, (Figure 4.16 (d)), a
broadband peak covering the range of 12.2 < 𝑓𝐷/𝑈 < 30 is observed. The
dominating peak in this range is taken as the middle value 𝑓𝐷/𝑈 = 21.1. This is
about half of that at 𝜃 = 260° and 255°. The reason for the reduction of the
dominating frequency is due to the merging of vortices within the LSB as shown in
Figure 4.13. No obvious high frequency peaks are observed at 𝜃 = 240°, which is
located downstream of the LSB.
4.4.3 Frequency of Kelvin-Helmholtz (KH) vortices
The KH shear layers are formed with a clear dominating frequency. The published
measurements of the KH frequency were well summarized by Prasad and Williamson
(1997). The published data from different sources show a scaling of
𝑓𝐾𝐻/𝑓𝑠𝑡 = 0.0235× 𝑅𝑒0.67 (4)
where 𝑓𝐾𝐻 and 𝑓𝑠𝑡 are the frequency for KH shear layers and the Kármán
vortices respectively. Since most of the published data about KH frequency is in the
range of 103 < 𝑅𝑒 < 105 (Prasad and Williamson, 1997, Norberg, 1987, Maekawa
and Mizuno, 1967), Prasad and Williamson (1997) commented that Eq (4) is only
107
valid for flows in the subcritical regime for 𝑅𝑒 up to 105. The validity of Eq. (4) in
the regimes beyond the subcritical regime has rarely been studied.
Homeyer et al. (2014) conducted wind tunnel tests to investigate KH frequency
for Re up to 4.7× 105. Flow velocity near the cylinder surface was measured using a
hot-wire probe located about 0.001D away from the cylinder surface. The KH
frequency was identified at 𝜃 = 102.5° for 𝑅𝑒 > 4 × 105. It was found that it
follows the same scaling as shown by Eq (4). However it should be noted that 𝑓𝑠𝑡
was not directly measured in Homeyer et al. (2014). It was deduced instead from
𝑓𝑠𝑡 = 0.21𝑈/𝐷 based on an assumption of St =0.21 for all the tests with 𝑅𝑒 up to
4.7× 105. The assumption of St = 0.21 in the critical and supercritical regimes
appears to be subjective because St has been reported to be around 0.4 ~ 0.5 (Shih et
al., 1993) in the supercritical regime.
To test the validity of Eq. (4) in the critical and supercritical regimes, the data
obtained from this study are plotted in Figure 4.17 where the predicted St is used,
together with the existing data. It is seen that the data obtained in this study fall well
below the trend line suggested by Eq. (4). The reason for this is that the St in the
critical and supercritical regimes is substantially larger than 𝑆𝑡 = 0.21 based on
which Eq. (4) was derived. This leads to a fundamental question on the validity of Eq.
(4): does the frequency of KH vortices really scale on the Kármán vortex frequency?
Is it a coincidence that Eq. (4) works well in the subcritical regime because St is
almost a constant of 0.21 in the subcritical regime. From fundamental fluid mechanics
point of view, the KH vortices and vortex shedding are induced by two different flow
mechanisms. The KH vortices are related to the instability of separated shear layers
from the cylinder while the Kármán vortex shedding is induced by the large scale roll
up of the separated shear layers. The KH shear layer instability and the vortex
shedding stability can exist independently of each other. To revisit this problem, a
simple dimensional analysis is carried out and the result shows that
𝑓𝐾𝐻𝐷
𝑈= 𝑓(𝑅𝑒) (5)
It is seen that the left hand side of the equation is nothing but the Strouhal number
108
for the KH vortices. It is expected that Eq. (5) would work regardless of flow regimes.
To demonstrate this, the data shown in Figure 4.17 are correlated by using Eq. (5).
This leads to a new scaling relationship
𝑆𝑡𝐾𝐻 =𝑓𝐾𝐻𝐷
𝑈 = 0.004935× 𝑅𝑒0.67 (6)
The 𝑆𝑡𝐾𝐻 values obtained in this study are summarized in Figure 4.18, together
with other published data. It can be seen that the present data follows Eq. (6) very
well. Lehmkuhl et al. (2014) presented four data points about 𝑓𝐾𝐻/𝑓𝑘 in the form of
Eq. (4). The data points are also converted to 𝑆𝑡𝐾𝐻 values and plotted in Figure 4.18.
It can be seen that the first two data points (𝑅𝑒 = 2.5 × 105 and 3.8× 105) follow Eq.
(6) well, but the other two points with 𝑅𝑒 = 5.3 × 105 and 6.5× 105 are slightly
higher than those suggested by Eq. (6). Although the cause for this is unclear, we
noticed that, for 𝑅𝑒 = 6.5 × 105, the measurement of the velocity was taken in the
wake of the cylinder by Lehmkuhl et al. (2014), rather than near the tip of the laminar
separation bubble. It is shown clearly in Figure 4.16 that the frequency of the KH
shear layers could not be captured further downstream from the middle of the LSB.
The reason that Eq. (4) works well in the subcritical regime is because St is almost a
constant of 0.21 in the subcritical regime (it is not difficult to show that Eq. (4) can
easily be re-organised as 𝑆𝑡𝐾𝐻/𝑆𝑡 = 0.0235× 𝑅𝑒0.67.
An indirect support to the above argument is that the KH shear layers are not only
observed in the wake of a circular cylinder, but also widely exist in nature. It can be
triggered without the cylinder, such as wind driven waves and regular cloud patterns.
Corcos and Sherman (1976) and Smyth (2003) found that the KH instability in a shear
flow is governed by three parameters, which are Reynolds number, Richardson
number (𝑅𝑖) and Prantdl number (𝑃𝑟). 𝑅𝑖 is a measure of the ratio of the buoyancy
term to the flow shear term and 𝑃𝑟 represents the ratio of momentum diffusivity to
thermal diffusivity. In the present situation, the fluid is an incompressible and heat
transfer is not involved. Therefore, 𝑅𝑖 and 𝑃𝑟 will have no effect to the KH
instability and 𝑆𝑡𝐾𝐻 should be governed by only one parameter (𝑅𝑒). This shows that
Eq (6) is physically sound.
109
4.5 Conclusions
In this paper, the flow around a circular cylinder with 𝑅𝑒 = 1×105 ~ 6×105 are
simulated using a three-dimensional LES model based on OpenFOAM®. The
dynamic Smagorinsky model is applied in the simulations. The flow structures around
the cylinder surface and in the wake are analysed. The main conclusions from this
work are summarized as follows:
• The numerical model successfully captures the flow transitions from the
subcritical to the critical regime and then to the supercritical regime. The predicted
pressure distribution along the cylinder surface and the locations of the laminar
separation bubbles (LSB) agree well with the reported experimental results. The drag
crisis and the non-zero mean lift force in the critical regime are also captured.
• The LSB on the cylinder surface is successfully predicted and the predicted
LSB locations on the cylinder surface are in good agreement with previous
experimental measurements. It is also identified that the LSB in the averaged flow
represents Kelvin-Helmholtz (KH) vortices on the cylinder surface in the supercritical
regime.
• In the subcritical regime, the KH vortices are formed in the separated shear
layers and are convected downstream without direct interaction with the cylinder
surface. In contrast, for the supercritical regime flow, the KH vortices are formed on
the cylinder surface and propagate a certain distance along the cylinder surface before
they decay or break down to turbulent scales. Some of the KH vortices merge together
while they travel along the cylinder surface. Consequently, the frequency measured at
the upstream tip of the LSB is higher than that measured at the downstream tip.
• It is revealed based on a simple dimensional analysis that the frequency of
KH vortices is only dependent on the Reynolds number, and thus a new equation of
KH frequency is proposed as 𝑆𝑡𝐾𝐻 = 0.004935 × 𝑅𝑒0.67 . This equation is
demonstrated to be applicable to flows with 𝑅𝑒 from 103 to 6×105 based on the
collapse of available published data, and it is therefore believed that the equation is
valid in all range of Re, where KH vortices exist, regardless of flow regimes.
110
Acknowledgement
The authors would like to acknowledge the support from the National Key R&D
Program of China (Project ID: 2016YFE0200100). H. An would like to acknowledge
the support from the Australian Research Council through DECRA Schemes
(DE150100428) and the ECR supporting scheme at UWA. F. Yang would like to
acknowledge the PhD scholarships provided by UWA. The simulations were
conducted using the computational resources provided by the Pawsey
Supercomputing Centre funded by the Australian Government and the Government of
Western Australia.
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Chapter 5
Flow around a near bed horizontal cylinder mounted
on a vertical wall
Abstract: Turbulent flow around a near bed horizontal circular cylinder mounted
on a vertical side wall is simulated numerically in this study. The purpose of the study
is to investigate the flow characteristics around the junction area between the cylinder
and two plane boundaries. The effects of gap to diameter ratio (G/D = ∞, 0.5 & 0.2),
Reynolds number (𝑅𝑒𝐷 = 1250, 2500 & 5000) and the length to diameter (aspect)
ratio (L/D = 10, 20 & 40) are investigated. It is found that the three-dimensional flow
structures formed in the junction area between the vertical wall and the cylinder are
strongly affected by the gap ratio. The end effect induced by the vertical wall is
limited within 0.7D in the spanwise direction, 0.5D upstream and 3D downstream of
the cylinder. The extent of the end effect is not sensitive to the gap ratio, Reynolds
number and aspect ratios.
5.1 Introduction
Steady flow around a horizontal cylinder mounted perpendicularly to side walls at
two ends with a parallel plane boundary in proximity is often encountered in physical
model testing (in water flumes or wind tunnels) of flow around a near wall cylinder
for simulating local scour around subsea pipelines (Cheng et al., 2009) and related
hydrodynamic forces (Yang et al. 2018). A schematic illustration of the configuration
is shown in Figure 5.1. The flow is highly three-dimensional (3D) near the junction
area between the cylinder and two plane boundaries. In some situations, the highly
3-D flow is unwanted and creates difficulty in interpreting the test results, as it is very
different from the flow over the majority part of the cylinder. This is often referred to
116
as “end effect” in the literature. The highly 3-D flow is expected to exhibit the flow
characteristics of the following two flows that are widely studied: (1) flow around
junction area of a cylinder mounted on a perpendicular plane and (2) flow around a
cylinder above a parallel plane. The existing work on those two flows in the literature
will be briefly reviewed below.
Junction flow around a circular cylinder attached to a plane boundary has
attracted substantial research interests over the past decades for its importance to
engineering applications in the fields of aerodynamics, heat transfer and offshore
hydrodynamics. Junction flow is generally a complex three-dimensional phenomenon,
involving three-dimensional boundary layer separations, formation of coherent flow
structures and vortex shedding. When a cylinder is mounted on a plane boundary, the
adverse pressure gradient in front of the cylinder near the plane boundary leads to the
boundary layer to separate and forms a series of vortex tubes to wrap around the
junction of the cylinder. The vortex tubes formed around the base of the cylinder are
often referred to as horseshoe vortices (HV) in the literature (Baker, 1985, Baker,
1980, Baker, 1979). In the experimental results of Seal and Smith (1999), a
complicated intertwining, or braiding, of two initially co-rotating necklace vortices
around the juncture of a circular cylinder was reported through hydrogen bubble
visualization. The horseshoe vortices are highly unsteady and cause high surface
pressure fluctuations and are attributed to the initiation of local scour around the
cylinder founded on an erodible surface (Simpson, 2001).
Early studies by Thwaites and Street (1960) and Baker (1979, 1980, 1985) found
that the formation of HV is dependent on Reynolds number, which is defined as
𝑅𝑒𝐷 = 𝑈𝐷/𝜐 with 𝐷 being the diameter of the cylinder, 𝑈 being the approaching
flow velocity far away from the plane boundary and 𝜐 being the kinematic viscosity
of the fluid. The HV is classified into three types: steady single or multiple pairs of
vortices, regular oscillatory vortices and irregular unsteady vortices. The position of
the primary horseshoe vortex in the steady vortex system has been found to be
dependent on both of 𝑅𝑒𝐷 and the relative displacement boundary layer thickness to
diameter ratio, 𝛿∗ 𝐷⁄ (Baker, 1985, Baker, 1979). 𝛿∗ is defind by 𝛿∗ = ∫(1 −
117
𝑢(𝑦)
𝑈)𝑑𝑦 in which 𝑈 is the freestream velocity and 𝑢(𝑦) is the velocity at the level of
y. In the steady HV system, one or more steady horseshoe vortices are formed. The
number of the HVs increases with the increase of 𝑅𝑒𝐷 (Baker, 1979). When 𝑅𝑒𝐷
increases to a certain value, the HV system becomes unsteady. The steady HV was
observed at 𝑅𝑒𝐷 =1000, and chaotic HVs were observed at 𝑅𝑒𝐷 = 13000 in the
investigation reported by Thomas (1987), while the corresponding HV systems were
observed at 𝑅𝑒𝐷 = 2000 and 𝑅𝑒𝐷 = 8000 respectively by Wei et al. (2001). Greco
(1990) classified the laminar horseshoe vortex system into five sub-regimes, which
are steady, oscillation, amalgamating, breakaway and transitional regimes in the
sequence of increasing 𝑅𝑒𝐷 . When the HV system becomes unsteady, the vortices in
the upstream start to oscillate around their mean positions without direct interactions
with each other. This sub-regime is called the oscillating sub-regime. As 𝑅𝑒𝐷 is
further increased, the vortices are shed periodically and neighboring vortices interact
and combine with each other to form new vortices. This regime is referred to as the
amalgamation sub-regime. The oscillating sub-regime is observed when 𝑅𝑒𝐷is in the
range of 1700 to 1900, and the amalgamation sub-regime starts to appear when
𝑅𝑒𝐷 increases to around 2500 (Greco, 1990). The unsteady state of the horseshoe
vortex system was also investigated by Seal et al. (1995) and Wei et al. (2001)
experimentally and Kirkil and Constantinescu (2012) numerically. It was found that a
significant change to the HV system dynamics is accompanied by the change of
𝛿∗ 𝐷⁄ . When 𝑅𝑒𝐷 increased further, the transition from the laminar to the turbulent
state occurs in the junction area and the separated shear layer becomes irregular. As
𝑅𝑒𝐷 increased to around 9000, the clear separated shear layer turns to a turbulent
juncture flow, which indicated that the behaviors of the horseshoe vortex system
cannot be predicted easily. Thomas (1987) by experiments and Visbal (1991) by
numerical simulations suggest that the periodical oscillation phenomenon of the HVs
is not related to the Karman vortex shedding from the cylinder and is independent of
the influence of the incoming flow. Baker (1991) indicated that oscillations of HV
system were ascribed to the oscillation of the entire vortex system as well as the
118
instabilities of vortex core itself.
In the experiment of Wei et al. (2001), a large number of vortices was observed to
cascade chaotically when 𝑅𝑒𝐷 increases from 14000 to 24000, while Escauriaza and
Sotiropoulos (2011) observed a fast regular breakaway behavior for 𝑅𝑒𝐷 = 2 × 104.
Baker (1980) investigated the horseshoe vortex formed by the turbulent boundary
layer in the range of 𝑅𝑒𝐷 = 4 × 103 ~ 9 × 104 with different boundary layer
thickness (𝐷 𝛿∗⁄ = 4 ~ 30). The measurement of pressure distributions showed that
the distribution of the primary horseshoe vortex is not affected much by 𝑅𝑒𝐷 under
turbulent boundary layer conditions. Dargahi (1989) studied the flow around a vertical
circular cylinder by flow visualizations through hydrogen bubbles at 𝑅𝑒𝐷 =
6.6 × 103 ~ 6.5 × 104 . The quasi-periodically shedding of vortices was found,
which is similar to the results reported by Baker (1980). The number of HV increases
with the increase of 𝑅𝑒𝐷 . The dimensions of the vortex system is found independent
of 𝑅𝑒𝐷, but dependent on the diameter of the cylinder. A detailed discussion on the
junction flow is reported by Simpson (2001).
Numerical simulations of junction flows have also been carried out to capture the
dynamic features of horseshoe vortexes. Briley et al. (1985) conducted a
three-dimensional simulation to investigate the horseshoe vortex of the flow at 𝑅𝑒𝐷 =
200. The results demonstrated that the captured laminar flow structures can be
accurately computed under a good convergence rates of the grid. Steady and unsteady
Reynolds-Averaged Navier-Stokes simulations (RANS and URANDS) were used by
Apsley and Leschziner (2001) and Chen (1995). Even though the positions of vortices,
turbulence intensities, pressure distributions, shear stress as well as the turbulent
kinetic energy were provided, the one or two eddy-viscosity turbulent model applied
showed a poor agreement with the experimental data and failed to capture the
dynamic features of horseshoe vortex system. Paik et al. (2007) applied Detached
eddy-simulation to simulate turbulent horseshoe vortex system in a wing-body
junction successfully. In Paik et al. (2007), periodic oscillating HVs were observed
which was very similar to the previous experimental results given, but the results
failed to capture the location of HV cores. Detached eddy-simulation was also used in
119
the study of Escauriaza and Sotiropoulos (2011). 𝑅𝑒𝐷 = 20000 and 39000 were
investigated to study the influence of 𝑅𝑒𝐷 on the horseshoe vortex system. The
observed instantaneous flow structures were very similar to the results of Devenport
and Simpson (1990). Large-eddy simulations (LES) were conducted by Rodi (1997),
Tseng et al. (2000), Krajnovic and Davidson (2002) and Kirkil and Constantinescu
(2012). The simulations of Rodi (1997), Tseng et al. (2000) and Krajnovic and
Davidson (2002) performed computations on a square cylinder and have a general
agreement with the data and topology structures observed in experiments at 𝑅𝑒𝐷 =
40000. Tseng et al. (2000) only presented averaged magnitudes of pressure and shear
stress and compared with Dargahi (1989) for a circular cylinder. The discrepancies
were attributed to the conditions of incoming flow boundary layer conditions as well
as the mesh resolution of the boundary layer (Rodi, 1997). Kirkil and Constantinescu
(2012) preformed three simulations ranging from 𝑅𝑒𝐷 = 800, 2140 and 4460 with
relatively large boundary layer thickness and captured the laminar HV system
transiting from the steady, to oscillating, and then to the breakaway sub-regime.
The HV is a genuine feature of the junction flow and has a side effect on model
testing results in wind tunnels or water flumes. Flow boundary layer will naturally
form on the solid walls (side, top and bottom) of the wind tunnel or water flume.
When a two-dimensional slender model is mounted against the channel walls, a HV
system will be formed at each end of the model. This HV represents an undesired
three-dimensional disturbance to the flow field. For example, when the
hydrodynamics/aerodynamics of a uniform cylinder is tested, the two ends of the
cylinder are normally mounted with two end plates to avoid the three-dimensional
effect from the free ends (Stansby, 1974). The diameter of the end plates is normally
just slightly larger than the cylinder diameter, to limit the development of a thick
boundary layer on the end plates (Gerich and Eckelmann, 1982). In this way, the end
effect due to HV can be managed to some extent. However, some model tests in a
flume are not suitable for adding end plates to the cylinder. For example, when local
scour below a cylinder is investigated in a water channel, normally the ends of the
cylinder are set against the side walls of test section directly without end plates
120
attached. This is because the end plates could cause extra scour. Sumer et al. (2001)
mentioned a half HV around the end of a cylinder partially buried on the sand bed.
The end effect due to HV will lead to different shear stress distribution on the sand
bed near the ends of the model cylinder. Consequently, it leads to different sediment
transport rate which affects the time scale of the scour process. The HV formed at the
ends will also affect the pressure distribution on the cylinder. Therefore, it is
important to quantify the end effect due to HV in similar testing setup.
The effect of a plane parallel to the cylinder on the flow around circular cylinder
also attracted substantial research interests in the past a few decades (Lei et al., 1999,
Buresti and Lanciotti, 1992, Jensen et al., 1990, Zdravkovich, 1985, Bearman and
Zdravkovich, 1978, Roshko et al., 1975, Kiya, 1968). The flow structures and
hydrodynamic forces of the cylinder are affected by the gap to diameter ratio (𝐺/𝐷),
wall boundary layer thickness to diameter ratio (𝛿/𝐷) as well as the turbulence
intensity in the boundary layer. Bearman and Zdravkovich (1978) visualized flow
around a circular cylinder near a plane boundary and measured pressure distribution
on the cylinder surface and on the plane boundary at 𝑅𝑒𝐷 = 4.5×104. It was found the
vortex shedding was suppressed for all gaps less than 0.3D. Lei et al. (1999) found
that the stagnation point moves upward and the base pressure decreases as the gap to
diameter ratio increases, and the vortex shedding is suppressed at gap ratio of 0.2 ~
0.3, depending on the different wall boundary layer thickness. Yang et al. (2018)
found that the drag crisis phenomenon exists for 𝐺/𝐷 ≥ 0.5 and found that the lift
coefficient reduced with the increases of the gap ratio. In the work of Yang et al.
(2018), the boundary layer transition was largely influenced by the proximity of the
plane boundary, and side swapping phenomenon was found at large gap ratios. The
suppression of vortex shedding was observed at 𝐺/𝐷 ≤ 0.3. Tong et al. (2017)
reported a deflected flow near the shoulders between a spanning and non-spanning
section under the conditions of intermittent gap ratios, which led to large variations of
sectional forces.
The influence of the blockage ratio and aspect ratio has been revealed by West
and Apelt (1982). It was found that the blockage ratio affects pressure distributions,
121
force coefficients and vortex shedding frequency. Lei et al. (2001) suugested that the
aspect ratio of cylinder must be larger than 4D in order to simulate the
three-dimensional wake flow accurately. Sumner et al. (2004), Pattenden et al. (2005),
Rodríguez y Domínguez et al. (2006), and Sumner and Heseltine (2008) reported the
junction flow around the circular cylinder with small aspect ratio. It was found that
HV systems was weak, and as the aspect ratio increased, the strength of the system as
well as the number of vortices increased. Sumner and Heseltine (2008) reported a
downwash phenomenon for cylinders with small aspect ratios. The horseshoe vortex
system and its interaction with the wake-flow for a circular cylinder with large aspect
ratio were reported by Sahin et al. (2007) and Sahin and Ozturk (2009). It was found
that the swirling horseshoe vortices in the streamwise direction and counter-clockwise
vortices in the vertical plane downstream of the cylinder led an increased level of
scour in the junction region.
To understand the end effect due to HVs on a circular cylinder near a parallel
plane boundary, a series of numerical simulation is carried out using LES model.
Firstly, the numerical model is validated by comparing with published experimental
results and then the end effect is analyzed in detail. The paper is organized as follows.
In Section 2, the governing equations of the flow have been presented. The mesh
dependency and the validation are given in Section 3. And Section 4 presents the
simulation results, followed by conclusions in the last section.
5.2 Methodology
The governing equations are the Navier-Stokes equations
𝜕𝑢𝑖
𝜕𝑡= −
𝜕𝑢𝑖𝑢𝑗
𝜕𝑥𝑗−
1
𝜌
𝜕𝑝
𝜕𝑥𝑖+ 𝜈
𝜕2𝑢𝑖
𝜕𝑥𝑗2 (1)
and the continuity equations
𝜕𝑢𝑖
𝜕𝑥𝑖= 0 (2)
where ui are flow velocities in the three spatial directions. i = 1, 2 and 3,
corresponding to the 𝑥-, 𝑦- and 𝑧- direction, respectively. 𝜌 is the density of the
fluid. 𝑝 is the pressure fluctuation. 𝜈 is the kinematic viscosity of the
123
124
Table 5.1. Mesh details of the five cases chosen for mesh dependence check
Case 𝑁𝑜. 1 2 3 4 5
∆ 𝐷⁄ 0.004 0.002 0.001 0.001 0.001
Refined area 6D×6D 6D×6D 6D×6D 5D×5D 4D×4D
𝑁𝑒𝑙𝑒𝑚𝑒𝑛𝑡 1.69× 105 3.49× 105 5.51× 105 4.86× 105 4.22× 105
Ω 1.8 1.5 1.25 1.25 1.25
𝑁𝑐𝑖𝑟𝑐𝑙𝑒 240 320 400 400 400
𝑁𝑠𝑝𝑎𝑛 85 85 85 85 85
Boundary layer (𝛿∗ /𝐷) 0.109 0.112 0.113 0.113 0.113
Table 5.2. Force coefficients obtained with different meshes.
𝑀𝑒𝑠ℎ 𝐶𝐷 𝐶𝐿
′ 𝑆𝑡
1 1.20 0.262 0.210
2 1.12 0.151 0.210
3 1.08 0.143 0.214
4 1.07 0.146 0.217
5 1.03 0.203 0.217
Table 5.3. Comparison of centre positions of three horseshoe vortices in the plane of y
= 0.
Vortex Centre position (𝑥/𝐷, 𝑧/𝐷)
HV1 𝐻𝑉2 𝐻𝑉3
Huang et al. (2014) (-0.96, 0.05) (-1.20, 0.03) (-1.45, 0.02)
This work (-0.94, 0.06) (-1.11, 0.06) (-1.27, 0.05)
Table 5.4. Locations of the source and saddle points.
Position of Source Point (x/D,
y/D)
Position of Saddle Point (x/D,
y/D)
Huang et al. (2014) (-0.85, 0) (-1.15, 0)
This work (-0.82, 0) (-1.22, 0)
5.3 Mesh dependency check and model validations
A circular cylinder mounted on a perpendicular wall is simulated for mesh
dependency and validation purposes. The domain geometry is shown in Figure 5.1
and has dimensions of 28D in the flow direction, with 8D in the upstream side, 16D in
the cross-flow direction and 5D in the spanwise direction (𝐿/𝐷 = 10 due to the
symmetry boundary condition). The computational domain in the x-y plane is
discretized with structured 4-nodes quadratic elements. The two-dimensional mesh is
125
extruded along the spanwise direction to form the three-dimensional mesh. Five
different meshes are tested to examine the convergence of the numerical model with
the key information including the total element number (𝑁𝑒𝑙𝑒𝑚𝑒𝑛𝑡), the expansion rate
of mesh along the cylinder (Ω), element number on the cylinder circumference
(𝑁𝑐𝑖𝑟𝑐𝑙𝑒), size of the first layer mesh on the cylinder surface (∆ 𝐷⁄ ) as well as the
element number in the spanwise direction of the cylinder (𝑁𝑠𝑝𝑎𝑛). Details of the
meshes are given in Table 5.1. The first layer mesh size around the cylinder and that
on the perpendicular wall are kept the same to resolve the boundary layer flow. In
present work, mesh dependency is conducted in two steps. The first step focuses on
the first layer mesh on the wall surface, and the second is the size of the square region
in Figure 5.2. In Mesh 1~3, the first layer mesh on the wall surface is refined from
0.004D to 0.002D and then to 0.001D, correspondingly the total number of mesh
elements increasing from 1.69 × 105 to 3.49 × 105 and then to 5.51 × 105. The
non-dimensional computational time step is set at ∆t = 0.0005. 𝑅𝑒𝐷 was set to 2500
for the purpose of comparing with experimental data by Huang et al. (2014). The
averaged distributions of pressure coefficient (𝐶𝑝 = 𝑝 (𝜌𝑈2 2⁄ )⁄ ) are given in Figure
5.3, in which 𝜌 is the density of the flow. It can be seen that a good convergence
trend is achieved for Mesh2 and Mesh3, only a slight difference in the range of θ = 60
~ 300 in Figure 5.3 (a), which indicates the benefit of the reduction of ∆ 𝐷⁄ .
For capturing the HVs, a square region around the cylinder (6D×6D) is refined.
Three different sizes of the refined area were examined in the mesh dependency in
order to accurately capture the detailed information of the flow structure around the
cylinder and the HV structure at the junction in the meantime to save computational
costs. Mesh4 (5D×5D) and Mesh5 (4D×4D) are summarized in Table 5.1. Figure 5.3
(b) presents the convergence trend of pressure distributions for different refined areas.
The pressure distribution of Mesh4 (5D×5D) is almost overlapping with the results of
Mesh3 (6D×6D), while 𝐶𝑝 of Mesh5 (4D×4D) is obviously larger than 𝐶𝑝 of Mesh4
and Mesh3. The minimum pressure for Mesh5 is 12.6% higher than the results of
Mesh3 and Mesh4. This indicates that the square size of 4D×4D for Mesh5 is not big
enough for the simulation, comparing with results of Mesh3 and Mesh4.
126
Figure 5.2. Computation mesh of Case 4 (a) x-z plane, (b) rectangle centre mesh and
(c) detailed mesh near the cylinder.
The forces and vortex shedding frequency are also checked in Table 5.2. In the
present study, the drag and lift coefficients are defined as 𝐶𝐷 = 𝐹𝐷 (𝜌𝐿𝐷𝑈2 2⁄ )⁄ and
𝐶𝐿 = 𝐹𝐿 (𝜌𝐿𝐷𝑈2 2⁄ )⁄ , in which 𝐹𝐷 and 𝐹𝐿 are the force component in the 𝑥- and
𝑦- directions, respectively and L is the cylinder length. The normalized vortex
shedding frequency (Strouhal number) is defined as 𝑆𝑡 = 𝑓𝐷 𝑈⁄ , where 𝑓 is the
frequency of the vortex shedding. The mean drag coefficients 𝐶𝐷 for Mesh3 and
Mesh4 are 1.08 and 1.07 correspondingly, which are within the measured range of 1.0
~ 1.2 in the subcritical range of Reynolds number (Niemann and Hölscher, 1990,
Braza et al., 1986, Schewe, 1983). The corresponding Strouhal number are 0.214 and
0.217 respectively, which also agree with the range of 0.207 ~ 0.22 in the work of
Roshko (1954). By comparing with the pressure distributions, the mean drag
coefficients, the root-mean-square lift coefficient (CL′ ) as well as Strouhal number, the
results of Mesh4 agree with the results of Mesh3 very well. Mesh4 is used in this
work for further simulations.
8D 20D
5D
(a)
(b) (c)
(a)
127
Figure 5.3. Pressure distributions for five validation meshes.
Huang et al. (2014) successfully captured the different flow modes of HV through
a particle image velocimetry (PIV) system in the range of 𝑅𝑒𝐷 = 500 ~ 6000 and
𝛿∗ ⁄ 𝐷 = 0.083 ~ 0.288. The mean values of flow structures at 𝑅𝑒𝐷 = 2500 and 𝛿∗ ⁄
𝐷 = 0.113 were given in Huang et al. (2014)’s work. A comparison between present
numerical results and those by Huang et al. (2014) is provided in Figure 5.4 ~ 5.5. In
present work, 𝛿∗ ⁄ 𝐷 = 0.129 is captured at 𝑥 = 0 (with the cylinder absent), which
is slightly higher than that measured 𝑦 Huang et al. (2014). Figure 5.4 shows the
velocity vectors and streamlines of HV structure in the vertical plane 𝑦 = 0 upstream
of the cylinder. Figure 5.4 (a) shows the PIV result reported by Huang et al. (2014).
Three HVs were observed in the experiment. In this work, the vortices are named as
HV1, HV2, and HV3, respectively, with HV1 closest to the cylinder. Figure 5.4 (b)
shows the results from the present study. The three HVs were also captured in this
work which agrees well with the flow structure observed by Huang et al (2014). The
center positions of three main vortexes in the plane of 𝑦 = 0 are located at
(𝑥/𝐷, 𝑧/𝐷) = (-0.94, 0.06), (-1.11, 0.06) and (-1.27, 0.05), respectively. In the
experiment, they were observed at (𝑥/𝐷, 𝑧/𝐷) = (-0.96, 0.05), (-1.20, 0.03) and (-1.45,
o
Cp
0 60 120 180 240 300 360-1 5
-1
-0 5
0Mesh 1
Mesh 2
Mesh 3
o
Cp
0 60 120 180 240 300 360-1 5
-1
-0 5
0Mesh 3
Mesh 4
Mesh 5
(b)
128
0.02). It can be seen that the vortices captured by the present CFD model are slightly
further away from the base wall (𝑧 = 0), comparing with the experimental results.
Figure 5.5 shows comparisons of velocity streamlines in two planes in the upstream
with 𝑧/𝐷 = 0.05 and 0.1. Due to the existence of the HV system, a source point and
a saddle point are observed in the plane of 𝑧/𝐷 = 0.05. This is captured by both the
experiment and the present numerical simulation. The flow structure in the plane of
𝑧/𝐷 = 0.1 are beyond the HVs, therefore the source point and saddle point are absent.
Table 5.4 shows the comparison of locations of source and saddle points. It can be
seen that the results in this work agree quite well with the results of Huang et al.
(2014). The comparisons shown above demonstrate that the present numerical model
can capture the key features of HV system and can be applied for further investigation
on the end effect of a near wall cylinder.
Figure 5.4. Comparison of HVs in plane of 𝑦/𝐷 = 0 at 𝑅𝑒𝐷 = 2500. a) PIV results
from Huang et al. 2014 (δ∗ ⁄ D = 0.113), b) present CFD (δ∗ ⁄ D = 0.129).
129
Figure 5.5. Comparison of velocity streamlines in horizontal plane in the upstream
z/D = 0.05 &0.1. a) & c) PIV results of Huang et al. (2014), b) & d) present LES
result
5.4 Numerical Results
The numerical results are reported in this section. The main focus of the
discussions will be on three-dimensional flow structures in the junction area between
the cylinder and the plane walls and their influences on other physical quantities such as
pressure on the cylinder surface, the shear stress on the wall parallel to the cylinder at
different 𝐺/𝐷 conditions.
5.4.1 Influence of Gap Ratio
The influence of 𝐺/𝐷 on the three-dimensional flow feature is examined by
analyzing the simulation results of Case 1, 2, and 3 with 𝐺/𝐷 = ∞, 0.5, and 0.2,
respectively, at 𝑅𝑒𝐷 = 2500. It has been known that the vortex shedding from the
cylinder will be suppressed for a cylinder near a plane boundary with 𝐺/𝐷 <
0.3 (Bearman and Zdravkovich, 1978). The cases with 𝐺/𝐷 = 0.5 and 𝐺/𝐷 = 0.2 are
chosen to represent the flow regimes with and without vortex shedding, respectively,
under the influence of the parallel wall. The hydrodynamic forces for the three gap
130
ratios are presented in Table 5.5. The mean drag coefficient (𝐶𝐷 ) increases from 1.08 at
𝐺/𝐷 = ∞ to 1.18 at 𝐺/𝐷 = 0.5 and then reduces to 1.09 at 𝐺/𝐷 = 0.2. The mean lift
coefficient (𝐶𝐿 ) increases from 0.0 to 0.15 and 0.27 as 𝐺/𝐷 is reduced from ∞ to 0.5
and 0.2 correspondingly. The root-mean-square value of the lift, which is normally
used as a measure of strength of vortex shedding, reaches a value of 0.177 at 𝐺/𝐷 =
0.5 and reduces to a very small value of 0.011 at 𝐺/𝐷 = 0.2, suggesting the
suppression of vortex shedding at 𝐺/𝐷 = 0.2. The 𝑆𝑡 increases slightly as 𝐺/𝐷 is
reduced from ∞ to 0.5 and disappears at 𝐺/𝐷 = 0.2. The above results agree well with
the experimental data in the subcritical flow regime reported by Jensen et al. (1990).
In order to examine the general 3-D vortex structure around the cylinder, the λ2
criteria (Jeong and Hussain, 1995) is used to visualize the flow structure around the
cylinder. Figure 5.6 presents two selected 3-D views of time-averaged iso-surfaces of
λ2 = 1 for 𝐺/𝐷 = ∞, 0.5, and 0.2. The right column of Figure 5.6 shows an overall
view of the 3-D flow structures with the side wall (perpendicular to the cylinder) at the
bottom and the parallel wall on the right, and the left column presents a projected view
of that shown in the right column in the opposite direction to the z-axis with the parallel
wall on the bottom side. It is seen from Figure 5.6 that HV tubes are developed at the
junction of the cylinder with the side wall and wrap around the cylinder surface facing
the flow at a certain distance. For 𝐺/𝐷 = ∞ shown in Figure 5.6 (a), four vortex tubes
are observed in the junction area with the side wall. Three of the four vortex tubes are
actually the horseshoe vortices (HVs) and one of them is the bottom attached vortex
(BAV). The HVs, namely as HV1, HV2 and HV3 and the BAV are illustrated by the ωy
contours along the middle section of 𝑦 = 0 in Figure 5.7. The following changes of the
flow structures near the junction area of the cylinder with the side wall are observed as
𝐺/𝐷 is reduced in Figure 5.6 and Figure 5.7: (1) the vortex tubes on the gap side
between the cylinder and the parallel wall are squeezed, merged and weakened due to
their interactions with the boundary layer above the parallel wall (Figure 5.6 (b)) and
the shear layer developed on the gap side of the cylinder surface (Figure 5.6 (c)), (2) the
vortex tubes move closer to the cylinder as 𝐺/𝐷 is reduced (Figure 5.7) and (3) a Luff
vortex tube is formed along the spanwise direction (Figure 5.6 (c)) and HV3 disappear
131
(Figure 5.7 (c)) at 𝐺/𝐷 = 0.2. Locations of the center of HVs in the plane 𝑦 = 0 are
quantified in Table 5.6. It shows center locations the vortices move towards the
cylinder with the decrease of 𝐺/𝐷.
Table 5.5. Variation of force coefficients and Strouhal number with gap ratio.
Case 𝑁𝑜. 𝐺/𝐷 𝑅𝑒𝐷 𝐿/𝐷 CD CL
𝑆𝑡 𝐶𝐿′
1 ∞ 2500 5 1.08 0 0.217 0.143
2 0.5 2500 5 1.18 0.15 0.253 0.177
3 0.2 2500 5 1.09 0.27 N/A 0.011
Table 5.6. Variation of vortex centre locations with gap ratio
Case 𝑁𝑜. Vortex Centre position (𝑥/𝐷, 𝑧/𝐷)
HV1 HV2 HV3
1 (-0.92, 0.06) (-1.10, 0.06) (-1.28, 0.05)
2 (-0.91, 0.06) (-1.08, 0.06) (-1.26, 0.05)
3 (-0.83, 0.06) (-1.00, 0.05)
.
132
Figure 5.6. Snapshot of time-averaged iso-surfaces of λ for G/D=∞ (a), 0.5 (b) and
0.2(c).
133
Figure 5.7. Mean values of velocity streamlines as well as contours of ωy in the plane
of 𝑦 = 0 for (a) 𝐺/𝐷 = ∞, (b) 𝐺/𝐷 = 0.5 and (c) 𝐺/𝐷 = 0.2.
Figure 5.8. Mean values of velocity streamlines as well as contours of ωx in the plane
of x=0 for (a) 𝐺/𝐷 = ∞, (b) 𝐺/𝐷 = 0.5 and (c) 𝐺/𝐷 = 0.2.
The influence of the parallel wall on the 3-D flow structure in the junction area is
further quantified in Figure 5.8 by examining the ωx contours in the plane 𝑥 = 0. It is
seen that the flow structures on positive 𝑦 side are similar to those visualized in the 𝑦
= 0 plane in Figure 5.7, except that HV3 for 𝐺/𝐷 = ∞ & 0.5 is not captured by the ωx
135
near the side wall for G/D = ∞, 0.5 and 0.2. It is seen that Cp values experiences large
rates of variations for 𝑧/𝐷 < 0.3 and rarely change for 𝑧/𝐷 > 0.3, especially near
the peaks at 𝜃 = 80o and 290o. To further quantify this, the variation of Φ = 𝐶𝑝𝑏 −
𝐶𝑝−𝑚𝑖𝑛−𝑔𝑎𝑝 with z/D is plotted in Figure 5.14, where 𝐶𝑝𝑏 is the base pressure and
defined as the average of the pressure on the cylinder surface between 𝜃 = 120o
and 260o , 𝐶𝑝−𝑚𝑖𝑛−𝑔𝑎𝑝 is the minimum pressure at around 𝛼 = 290o. for all three
gap ratios. It is seen that the influence of end effect is limited at 𝑧/𝐷 < 0.35 for G/D
= ∞ and 0.5, and 𝑧/𝐷 < 0.5 for G/D = 0.2. This result is consistent with findings
derived from the flow structures shown in Figure 5.8.
Figure 5.10. Separation points on the cylinder surface in the influence of gap ratios. (a)
separation points on the up side, Sp1, and (b) separation points on the gap side, Sp2.
z/D
0.5 1 1.5 28
85
90
95 G/D= Re=2500 L/D=10
G/D=0.5 Re=2500 L/D=10
G/D=0.2 Re=2500 L/D=10
z/D
0 0.5 1 1.5 2250
255
260
265
270
(a)
(b)
136
Figure 5.11. Sectional pressure distributions of the cylinder for (a) 𝐺/𝐷 = ∞, (b) for
𝐺/𝐷 = 0.5 and (c) for 𝐺/𝐷 = 0.2.
Figure 5.12. The difference between the base pressure 𝐶𝑝𝑏 and the minimum
pressure in the gap 𝐶𝑝−𝑚𝑖𝑛−𝑔𝑎𝑝 in the influence of gap ratio.
Cp
0 60 120 180 240 300 360-1.5
-1
-0.5
0
Cp
0 60 120 180 240 300 360-1.5
-1
-0.5
0
Cp
0 60 120 180 240 300 360-1.5
-1
-0.5
0 z/D=0 01
z/D=0 05
z/D=0 1
z/D=0 3
z/D=0 5
z/D
0 0.4 0.8 1.2 1.6 20.05
0.1
0.15
0.2
G/D= Re=2500 L/D=10
G/D=0.5 Re=2500 L/D=10
G/D=0.2 Re=2500 L/D=10
(a) (b)
(c)
138
the shear stress, the amplification factor of the shear stress, 𝜎0, is checked, where 𝜎0
is defined as 𝜎0 = 𝜏0 𝜏𝑟𝑒𝑓⁄ , in which 𝜏0 is the magnitude of the shear stress, and
𝜏𝑟𝑒𝑓 is the magnitude of reference shear stress in an area far away from the sidewall
plane boundary and the cylinder. Shear stress 𝜏0 is defined as 𝜏0 = √𝜏𝑥2 + 𝜏𝑧
2 in
which 𝜏𝑥 = 𝜇(𝜕𝑢 𝜕𝑦⁄ ) and 𝜏𝑧 = 𝜇(𝜕𝑤 𝜕𝑦⁄ ) are the mean shear stress in the 𝑥- and
𝑧- directions respectively, and 𝜇 is the dynamic viscosity. Figure 5.13 shows the
contours of 𝜎0 as well as the streamlines on the parallel wall for 𝐺/𝐷 = 0.5 and 0.2.
For 𝐺/𝐷 = 0.5 in Figure 5.13 (a), 𝜎0 underneath the cylinder is obviously higher
than the other area. This is mainly due to the blockage effect of the cylinder. The
variation of shear stress in the spanwise direction under the cylinder is clearly due to
the existence of HVs at the junction. The area influenced by the side wall is roughly
about -0.5 < 𝑥/𝐷 < 1.7 and 𝑧/𝐷 < 0.5. Similar to the results of 𝐺/𝐷 = 0.5, strong
shear stress is also observed underneath the cylinder for 𝐺/𝐷 = 0.2. The range of
end effect is roughly at -0.5 < 𝑥/𝐷 < 3 and 𝑧/𝐷 < 0.7 in Figure 5.13 (b). To
quantify the influence of the side wall, the shear stress distributions along a few lines
with constant x/D values underneath the cylinder are extracted in Figure 5.14. The
range of end effect on 𝜎0 is limited to 𝑧/𝐷 < 0.4 for 𝐺/𝐷 = 0.5 in Figure 5.14 (a)
and the highest 𝜎0 is observed at around 𝑧/𝐷 = 0.12 which is due to horseshoe
vortexes in the junction. As for 𝐺/𝐷 = 0.2, the range of end effect on 𝜎0 is limited
to 𝑧/𝐷 < 0.7. The fluctuations of 𝜎0 observed between 𝑧/𝐷 = 0.3 ~ 0.6 are good
indications of the complex three-dimensionality in the junction area.
5.4.2 Influence of Reynolds number
In the present work, the influence of 𝑅𝑒𝐷 on the flow and the end effect was
investigated at which are 𝑅𝑒𝐷 =1250, 2500 and 5000 with 𝐺/𝐷 = 0.5. Due to the
limitation of computational resources, relatively small 𝑅𝑒𝐷 values were employed.
139
Table 5.7. Variation of force coefficients and Strouhal number with Re.
Case 𝑁𝑜. 𝐺/𝐷 𝑅𝑒𝐷 𝐿/𝐷 CD CL
𝑆𝑡 𝐶𝐿′
1 0.5 1250 5 1.19 0.13 0.258 0.131
2 0.5 2500 5 1.18 0.15 0.253 0.177
3 0.5 5000 5 1.21 0.15 0.255 0.243
Table 5.8. Variation of vortex centre locations with Re.
Case 𝑁𝑜. Vortex Centre position (𝑥/𝐷, 𝑧/𝐷)
HV1 HV2 HV3
1 (-0.92, 0.08) (-1.14, 0.07)
2 (-0.91, 0.06) (-1.08, 0.06) (-1.26, 0.05)
3 (-1.20, 0.05)
Figure 5.15. Adverse pressure gradient in the influence of 𝑅𝑒𝐷 .
z/D
0 0.4 0.8 1.2 1.6 20.05
0.1
0.15
0.2
0.25
G/D=0.5 Re=1250 L/D=10
G/D=0.5 Re=2500 L/D=10
G/D=0.5 Re=5000 L/D=10
141
Figure 5.17. Adverse pressure gradient in the influence of aspect ratio (𝐿/𝐷).
The influence of 𝑅𝑒𝐷 on the extent of end effect is quantified through the variation
of Φ with 𝑧/𝐷 in Figure 5.15 and the amplification factor of shear stress 𝜎0 in Figure
5.16. The extent of the end effect decreases with 𝑅𝑒𝐷 and is limited at 𝑧/𝐷 = 0.4, 0.35
and 0.25 for 𝑅𝑒𝐷 = 1250, 2500 and 5000 correspondingly.
5.4.3 Influence of the aspect ratio
The influence of the aspect ratio, 𝐿 𝐷⁄ on the flow and the end effect is examined
in this study. Many studies have been carried out to investigate the influence of 𝐿 𝐷⁄
on HV systems for a single cylinder without the parallel wall (Sumner and Heseltine,
2008, Rodríguez y Domínguez et al., 2006, Pattenden et al., 2005, Sumner et al., 2004).
The largest 𝐿 𝐷⁄ that has been previously studied was 𝐿 ⁄ 𝐷 = 6 (Sahin et al. (2007)
and Sahin and Ozturk (2009)). In the present work, three different 𝐿 𝐷⁄ , which are 10,
20 and 40, are employed for the case with 𝐺/𝐷 = 0.5. The hydrodynamic forces as well
as the averaged positions of HVs in the plane of 𝑦 = 0 are presented in Table 5.9 and
Table 5.10. It is seen that they are not sensitive to L/D, possibly because the minimum
L/D investigated is too large. This is further confirmed by the variation of Φ with L/D
in Figure 5.17. The sectional drag, 𝐶𝐷−𝑆 and lift 𝐶𝐿−𝑆 coefficient are plotted in Figure
5.18. The influence range of end effect is limited at 𝑧/𝐷 = 0.5 for all three aspect ratios.
z/D
0 0.4 0.8 1.2 1.6 20.05
0.1
0.15
0.2
0.25
0.3
G/D=0.5 Re=2500 L/D=10
G/D=0.5 Re=2500 L/D=20
G/D=0.5 Re=2500 L/D=40
143
wall on the flow structure is investigated at three gap ratios (G/D = ∞, 0.5 and 0.2),
three Reynolds numbers (𝑅𝑒𝐷=1250, 2500 and 5000) and three aspect ratios (𝐿/𝐷=10,
20 and 40). The main conclusions from the present work are summarized as follows:
1. The three-dimensional horse-shoe vortex structures formed in the junction
area between the cylinder and the side wall are strongly affected by the proximity of
the parallel wall. The vortex tubes on the gap side between the cylinder and the
parallel wall are squeezed, merged and weakened due to their interactions with the
boundary layer above the parallel wall and the shear layer developed on the gap side
of the cylinder surface. The vortex tubes move closer to the cylinder as G/D is
reduced and a Luff vortex tube is formed along the spanwise direction.
2. The extent of the end effect is examined by a number of quantities such as
pressure distribution on the cylinder surface and shear stress on the parallel wall. It is
found that the extent of the end effect is not sensitive to gap ratio, Reynolds number
and the aspect ratio within the parameter ranges studied. The end effect is largely
contained within 0.7D from the side wall.
Acknowledgement
The authors would like to acknowledge the support from the National Key R&D
Program of China (Project ID: 2016YFE0200100), Australian Research Council
through DECRA scheme (DE150100428) and Linkage scheme (LP150100249), the
Fellowship Supporting Scheme and PhD scholarships from the University of Western
Australia.
144
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Chapter 6
Conclusions
6.1 Summary of conclusions
Three-dimensional large eddy simulations and physical experimental tests have
been carried out to investigate the flow structures around a circular cylinder near a
plane boundary. The range of 𝑅𝑒 is 1 ×105 ~ 6×105. The hydrodynamic forces, flow
structures and boundary layer transitions have been analysed. Major investigated
topics are as follows
(1) The plane boundary effect on the transition of the boundary layer on a
cylinder surface and its impact on hydrodynamic load on the cylinder. This
work was done through physical experiments. The effect of boundary layer
thickness and free stream turbulence level were discussed in detail (Chapter 2
and 3).
(2) The fundamental features of the boundary layer transition on the cylinder
sureface, including the Laminar Separation Bubbles and the KH vortices. This
work was done through 3D LES (Chapter 4).
(3) The end effect in the physical experiments and this work was done through
3D LES (Chapter 5).
The detailed conclusions have been already summarized in corresponding
chapters and will not be repeated here.
6.2 Recommendations for future studies
1. In the physical experimetns, the analysis was mainly based on the pressure
information on the cylinder surface. Some detailed measurement about the
flow around the cylinder and in the wake will help to improve the
150
understanding about the boundary layer transition process on the cylinder
surface.
2. For the condition of 𝐺/𝐷 = 0, the transition on the cylinder surface
should happen at a Reynolds value below the mimimum value that could
be tested in this work. It is worthwhile conducting further tests to examine
cylinder boundary layer transition on the cylinder surface at 𝐺/𝐷 = 0.
3. Due to the limitation of computational cost, the boundary layer transition
was only simulated for the isolated cylinder condition. When
computational rescource is available, it is worthwhile to visualize the
boundary layer transition around a near wall cylinder.
4. The transition of the boundary layer on the plane boundary and its effect
on the hydrodynamic of circular cylinder has not been well understood
and this is recommended for further investigation.
