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Wake modes of rotationally oscillating circular cylinder in cross-flow and its relationship with heat transfer
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Wake modes of rotationally oscillating circular cylinder in cross-flow and its relationship with heat transfer
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Content
WAKE MODES OF ROTATIONALLY OSCILLATING CIRCULAR CYLINDER IN
CROSS-FLOW AND ITS RELATIONSHIP WITH HEAT TRANSFER
by
Prabu Sellappan
_________________________________________
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
August 2013
Copyright 2013 Prabu Sellappan
ii
Acknowledgements
First, I would like to thank my advisors, Professor Tait Pottebaum and Professor
Geoffrey Spedding, for their support and motivation. Thank you for giving me the freedom to
pursue my research and personal goals. I would especially like to thank Professor Spedding for
the critical support he provided over the last year without which this dissertation would not have
been possible.
I would like to thank Professor Larry Redekopp, Professor Veronica Eliasson, and
Professor Thorsten Becker for their helpful suggestions regarding various aspects of my research
and for being a part of my dissertation committee.
Thank you to my friends at USC, Ben Bycroft, Dean Bergman, Giacomo Castiglioni,
Stephanie Klimczak, Trevor Orr and all the others who made this whole experience more fun. I
would also like to thank Rodney Yates for his assistance with machining and fabricating the
experimental setup. I would especially like to thank Irice Castro and Samantha Graves for the
constant support, motivation, and free food. Finally, I would like to thank my family for their
support and everlasting faith in my abilities.
iii
Table of Contents
Acknowledgements ......................................................................................................................... ii
List of Tables ................................................................................................................................. vi
List of Figures ............................................................................................................................... vii
Abstract ........................................................................................................................................ xiv
Chapter 1: Introduction ............................................................................................................. 1
1.1 Motivation and problem formulation ............................................................................... 1
1.2 Wake formation ................................................................................................................ 3
1.2.1 Vortex shedding from stationary cylinders ............................................................. 3
1.2.2 Vortex shedding from forced cylinders .................................................................. 5
1.3 Heat Transfer from cylinders ........................................................................................... 8
1.3.1 Heat transfer from stationary cylinders ................................................................... 8
1.3.2 Heat transfer from forced cylinders ........................................................................ 9
1.4 Objectives and Organization .......................................................................................... 10
Chapter 2: Experimental methodology ................................................................................... 11
2.1 Introduction .................................................................................................................... 11
2.2 General approach............................................................................................................ 11
2.3 Water Tunnel .................................................................................................................. 12
2.4 Circular cylinder setup ................................................................................................... 15
2.5 Cylinder motion control ................................................................................................. 18
2.6 Cylinder heat transfer ..................................................................................................... 19
2.7 Digital Particle Image Velocimetry................................................................................ 22
2.8 Digital Particle Image Thermometry.............................................................................. 23
2.9 Phase Averaging ............................................................................................................. 24
2.10 Spectral Analysis ............................................................................................................ 24
Chapter 3: Heat transfer from cylinder and its dependence on forcing conditions ................ 25
3.1 Introduction .................................................................................................................... 25
3.2 Experimental conditions and methodology .................................................................... 25
3.3 Results ............................................................................................................................ 27
iv
3.4 Discussion ...................................................................................................................... 29
3.5 Conclusion ...................................................................................................................... 33
Chapter 4: Effects of cylinder forcing on the structure of the wake at Re = 150 ................... 34
4.1 Introduction .................................................................................................................... 34
4.2 Experimental conditions and methodology .................................................................... 34
4.3 Results ............................................................................................................................ 36
4.4 Discussion ...................................................................................................................... 37
4.4.1 Wake modes at forcing frequencies near the natural shedding frequency ............ 38
4.4.2 Wake modes at higher forcing frequencies: .......................................................... 44
4.4.2.1 ½ (2S) far wake mode ...................................................................................... 47
4.4.2.2 ½ (P+S) wake mode......................................................................................... 50
4.4.2.3 ⅓ (2P+2S) wake mode..................................................................................... 53
4.4.3 Conclusions ........................................................................................................... 56
Chapter 5: Effects of cylinder forcing on the structure of the wake at Re = 750 ................... 58
5.1 Introduction .................................................................................................................... 58
5.2 Experimental conditions and methodology .................................................................... 58
5.3 Results ............................................................................................................................ 61
5.4 Discussion ...................................................................................................................... 63
5.4.1 Wake modes at forcing frequencies near the natural shedding frequency ............ 66
5.4.2 Wake modes at higher forcing frequencies ........................................................... 80
5.4.2.1 ½ (2S) far wake mode ...................................................................................... 88
5.4.2.2 ½ (P+S) far wake mode ................................................................................... 93
5.4.2.3 ½ (P+S) wake mode; no coalescence .............................................................. 99
5.4.2.4 ⅓ (P+S) wake mode; no coalescence ............................................................ 104
5.4.2.5 ⅓ (2P) wake mode; no coalescence ............................................................... 109
5.5 Conclusions .................................................................................................................. 114
Chapter 6: Relationship between wake formation and heat transfer .................................... 116
6.1 Introduction .................................................................................................................. 116
6.2 Effect of wake mode synchronization on heat transfer enhancement .......................... 116
6.3 Tangential velocity ....................................................................................................... 119
6.4 Relationship between heat transfer and formation length at F
R
= 1.0 .......................... 120
6.5 Conclusion .................................................................................................................... 122
v
Chapter 7: Conclusions ......................................................................................................... 124
References ................................................................................................................................... 127
vi
List of Tables
Table 2.1 : Properties of water at 25.8°C. From Pottebaum (2003) ............................................. 14
Table 2.2 : Cylinder dimensions (in mm). From Pottebaum (2003) ............................................. 16
Table 2.3 : Cylinder material properties. From Pottebaum (2003) ............................................... 17
Table 2.4 : Experiment conditions and Strouhal frequencies at 25.8°C ....................................... 17
Table 2.5 : Radii (mm) used in model of cylinder internal heat transfer. Values are from
Pottebaum (2003) .......................................................................................................................... 21
Table 2.6 : Values determined for C
ΔT
.......................................................................................... 21
vii
List of Figures
Figure 1.1: Schematic of basic setup. ............................................................................................. 2
Figure 1.2: Strouhal number as a function of Reynolds number. Figure from Norberg (1994) ..... 4
Figure 1.3: Lock-on boundaries identified by Mahfouz & Badr (2000a) and Lee & Lee (2006).
Figure is from Lee & Lee (2006). F
R
is frequency ratio (f
f
/f
St
) and θ
A
is equal to θ
PP
.................... 6
Figure 1.4: Wake modes identified by Williamson & Roshko (1988) ........................................... 7
Figure 2.1: Diagram of water tunnel and basic arrangement of experiment ................................ 14
Figure 2.2: Cylinder exterior dimensions and internal structure (not to scale); dimensions are
listed in Table 2.2. From Pottebaum (2003) ................................................................................. 16
Figure 2.3: Geometry for model of heat transfer inside the cylinder. From Pottebaum (2003) ... 21
Figure 3.1: Cylinder oscillation conditions considered during heat transfer experiments ........... 26
Figure 3.2: Contours of normalized heat transfer (Nu/Nu
0
) at Re = 750 ...................................... 28
Figure 3.3: Normalized heat transfer (Nu/Nu
0
) vs. frequency ratio (F
R
) for selected oscillation
amplitudes (θ
PP
given in degrees to aid analysis). Error bars are not shown to improve clarity.
Derived from the same data set and have same relative uncertainties as Figure 3.2 .................... 29
Figure 4.1: Experimental test conditions in the parameter space ................................................. 35
Figure 4.2: Map of wake mode regions from present study with boundaries (approximate)
identified by Mahfouz & Badr (2000a) superimposed ................................................................. 36
Figure 4.3: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 3.14,
F
R
= 1.17. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of
π/24; minimum contours ±0.5, contour spacing 0.5, positive contours solid and negative contours
dashed ........................................................................................................................................... 40
Figure 4.4: Spectra of forced shedding case measured at location x/D = 2.04, y/D = 0.37. Forcing
conditions are Θ
PP
= 3.14, F
R
= 1.17 ............................................................................................ 41
Figure 4.5: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 0.70,
F
R
= 0.71. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of
π/24; minimum contours ±0.5, contour spacing 0.5, positive contours solid and negative contours
dashed ........................................................................................................................................... 43
Figure 4.6: Spectra of non lock-on case measured at location x/D = 3.25, y/D = -0.81. Forcing
conditions are Θ
PP
= 0.70, F
R
= 0.71 ............................................................................................ 44
viii
Figure 4.7: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 1.05,
F
R
= 1.75. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of
π/24; minimum contours ±0.5, contour spacing 0.5, positive contours solid and negative contours
dashed ........................................................................................................................................... 46
Figure 4.8: Spectra measured along y/D = 0.28 at stream wise locations x/D = 0.75 and x/D =
5.62. Forcing conditions are Θ
PP
= 1.05, F
R
= 1.75 ...................................................................... 47
Figure 4.9: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 2.44,
F
R
= 1.87. Two cycles divided into 48 bins, every eighth bin shown, each bin has phase width of
π/12; minimum contours ±0.5, contour spacing 0.5, positive contours solid and negative contours
dashed ........................................................................................................................................... 49
Figure 4.10: Spectra of coalescing case exhibiting ½(2S) mode in the far wake. Spectra measured
along y/D = -0.97 at stream wise locations x/D = 1.28 and x/D = 9.31. Forcing conditions are
Θ
PP
= 2.44, F
R
= 1.87 .................................................................................................................... 50
Figure 4.11: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 1.05,
F
R
= 2.00. Two cycles divided into 48 bins, every sixth bin shown, each bin has phase width of
π/12; minimum contours ±1.0, contour spacing 0.5, positive contours solid and negative contours
dashed ........................................................................................................................................... 52
Figure 4.12: Spectra of coalescing case exhibiting ½(P+S) mode in the far wake. Spectra
measured along y/D = 0.09 at stream wise locations x/D = 0.58 and x/D = 5.17. Forcing
conditions are Θ
PP
= 1.05, F
R
= 2.00 ............................................................................................ 53
Figure 4.13: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 1.74,
F
R
= 3.08. Three cycles divided into 48 bins, every third bin shown, each bin has phase width of
π/8; minimum contours ±1.0, contour spacing 0.5, positive contours solid and negative contours
dashed ........................................................................................................................................... 55
Figure 4.14: Spectra of coalescing case exhibiting ⅓ (2P+2S) mode in the far wake. Spectra
measured along y/D = -1.09 at stream wise locations x/D = 1.06 and x/D = 5.28. Forcing
conditions are Θ
PP
= 1.74, F
R
= 3.08 ............................................................................................ 56
Figure 5.1: Parameter space for experiments with unheated cylinder at Re = 750 ....................... 59
Figure 5.2: Parameter space for experiments with heated cylinder at Re = 750 ........................... 60
Figure 5.3: Map of wake mode regions for an unheated cylinder at Re = 750 ............................. 61
Figure 5.4: Map of wake mode regions at Re = 750 for a heated cylinder ................................... 62
Figure 5.5: Mean and rms normalized vorticity for an unheated, stationary cylinder. Contour
spacing 0.5, positive contours are solid and negative contours are dashed .................................. 65
Figure 5.6: Mean and rms normalized vorticity for a heated, stationary cylinder. Contour spacing
0.5, positive contours are solid and negative contours are dashed ............................................... 65
ix
Figure 5.7: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 2.09, F
R
= 1.0. One cycle divided into 48 bins, every eighth bin shown, each bin
has phase width of π/24; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 68
Figure 5.8: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
2.09, F
R
= 1.0. Contour spacing 0.5, positive contours are solid and negative contours are dashed
....................................................................................................................................................... 69
Figure 5.9: Spectra measured along y/D = -0.01 at stream wise locations x/D = 1.71 and x/D =
7.77. Cylinder is unheated. Cylinder forced at θ
PP
= 2.09, F
R
= 1.0 ............................................ 69
Figure 5.10: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 0.87, F
R
= 1.0. One cycle divided into 48 bins, every eighth bin shown, each bin
has phase width of π/24; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 70
Figure 5.11: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
0.87, F
R
= 1.0. Contour spacing 0.5, positive contours are solid and negative contours are dashed
....................................................................................................................................................... 71
Figure 5.12: Spectra measured along y/D = -0.14 at stream wise locations x/D = 1.90 and x/D =
4.25. Cylinder is heated. Cylinder forced at θ
PP
= 0.87, F
R
= 1.0 ................................................ 71
Figure 5.13: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 0.17, F
R
= 1.87. One cycle divided into 48 bins, every eighth bin shown, each bin
has phase width of π/24; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 74
Figure 5.14: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
0.17, F
R
= 1.87. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 75
Figure 5.15: Spectra measured along y/D = -0.02 at stream wise locations x/D = 1.32 and x/D =
4.83. Cylinder is unheated. Cylinder forced at θ
PP
= 0.17, F
R
= 1.87 .......................................... 75
Figure 5.16: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 0.70, F
R
= 2.89. One cycle divided into 48 bins, every eighth bin shown, each bin
has phase width of π/24; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 76
Figure 5.17: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
0.70, F
R
= 2.89. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 77
Figure 5.18: Spectra measured along y/D = -0.46 at stream wise locations x/D = 1.71 and x/D =
6.26. Cylinder is unheated. Cylinder forced at θ
PP
= 0.70, F
R
= 2.89 .......................................... 77
x
Figure 5.19: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 0.17, F
R
= 1.87. One cycle divided into 48 bins, every eighth bin shown, each bin
has phase width of π/24; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 78
Figure 5.20: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
0.17, F
R
= 1.87. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 79
Figure 5.21: Spectra measured along y/D = -0.14 at stream wise locations x/D = 1.08 and x/D =
4.60. Cylinder is heated. Cylinder forced at θ
PP
= 0.17, F
R
= 1.87 .............................................. 79
Figure 5.22: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 1.40, F
R
= 1.54. One cycle divided into 48 bins, every eighth bin shown, each bin
has phase width of π/24; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 82
Figure 5.23: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
1.40, F
R
= 1.54. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 83
Figure 5.24: Spectra measured along y/D = -0.02 at stream wise locations x/D = 1.32 and x/D =
4.83. Cylinder is unheated. Cylinder forced at θ
PP
= 1.40, F
R
= 1.54 .......................................... 83
Figure 5.25: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 1.40, F
R
= 1.54. One cycle divided into 48 bins, every eighth bin shown, each bin
has phase width of π/24; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 84
Figure 5.26: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
1.40, F
R
= 1.54. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 85
Figure 5.27: Spectra measured along y/D = 0.33 at stream wise locations x/D = 0.97 and x/D =
4.60. Cylinder is heated. Cylinder forced at θ
PP
= 1.40, F
R
= 1.54 .............................................. 85
Figure 5.28: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 1.40, F
R
= 2.72. One cycle divided into 48 bins, every eighth bin shown, each bin
has phase width of π/24; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 86
Figure 5.29: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
1.40, F
R
= 2.72. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 87
Figure 5.30: Spectra measured along y/D = -0.26 at stream wise locations x/D = 1.79 and x/D =
4.13. Cylinder is heated. Cylinder forced at θ
PP
= 1.40, F
R
= 2.72 .............................................. 87
xi
Figure 5.31: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 1.05, F
R
= 2.12. Two cycles divided into 48 bins, every sixth bin shown, each bin
has phase width of π/12; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 89
Figure 5.32: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
1.05, F
R
= 2.12. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 90
Figure 5.33: Spectra measured along y/D = -1.08 at stream wise locations x/D = 1.32 and x/D =
4.60. Cylinder is unheated. Cylinder forced at θ
PP
= 1.05, F
R
= 2.12 .......................................... 90
Figure 5.34: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 1.05, F
R
= 2.12. Two cycles divided into 48 bins, every sixth bin shown, each bin
has phase width of π/12; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 91
Figure 5.35: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
1.05, F
R
= 2.12. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 92
Figure 5.36: Spectra measured along y/D = 0.45 at stream wise locations x/D = 1.08 and x/D =
4.60. Cylinder is heated. Cylinder forced at θ
PP
= 1.05, F
R
= 2.12 .............................................. 92
Figure 5.37: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 1.40, F
R
= 2.12. Two cycles divided into 48 bins, every sixth bin shown, each bin
has phase width of π/12; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 95
Figure 5.38: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
1.40, F
R
= 2.12. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 96
Figure 5.39: Spectra measured along y/D = -1.08 at stream wise locations x/D = 1.32 and x/D =
4.83. Cylinder is unheated. Cylinder forced at θ
PP
= 1.40, F
R
= 2.12 .......................................... 96
Figure 5.40: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 1.40, F
R
= 2.12. Two cycles divided into 48 bins, every sixth bin shown, each bin
has phase width of π/12; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ......................................................................................... 97
Figure 5.41: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
1.40, F
R
= 2.12. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ........................................................................................................................................... 98
Figure 5.42: Spectra measured along y/D = -0.14 at stream wise locations x/D = 1.08 and x/D =
4.60. Cylinder is heated. Cylinder forced at θ
PP
= 1.40, F
R
= 2.12 .............................................. 98
xii
Figure 5.43: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 0.87, F
R
= 1.87. Two cycles divided into 48 bins, every sixth bin shown, each bin
has phase width of π/12; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ....................................................................................... 100
Figure 5.44: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
0.87, F
R
= 1.87. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ......................................................................................................................................... 101
Figure 5.45: Spectra measured along y/D = -0.46 at stream wise locations x/D = 1.71 and x/D =
6.26. Cylinder is unheated. Cylinder forced at θ
PP
= 0.87, F
R
= 1.87 ........................................ 101
Figure 5.46: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 0.87, F
R
= 1.87. Two cycles divided into 48 bins, every sixth bin shown, each bin
has phase width of π/12; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ....................................................................................... 102
Figure 5.47: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
0.87, F
R
= 1.87. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ......................................................................................................................................... 103
Figure 5.48: Spectra measured along y/D = -0.37 at stream wise locations x/D = 1.32 and x/D =
3.66. Cylinder is heated. Cylinder forced at θ
PP
= 0.87, F
R
= 1.87 ............................................ 103
Figure 5.49: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 0.52, F
R
= 3.16. Three cycles divided into 48 bins, every fourth bin shown, each
bin has phase width of π/8; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ....................................................................................... 105
Figure 5.50: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
0.52, F
R
= 3.16. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ......................................................................................................................................... 106
Figure 5.51: Spectra measured along y/D = -0.02 at stream wise locations x/D = 1.55 and x/D =
3.89. Cylinder is unheated. Cylinder forced at θ
PP
= 0.52, F
R
= 3.16 ........................................ 106
Figure 5.52: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 0.52, F
R
= 3.03. Three cycles divided into 48 bins, every fourth bin shown, each
bin has phase width of π/8; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ....................................................................................... 107
Figure 5.53: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
0.52, F
R
= 3.03. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ......................................................................................................................................... 108
Figure 5.54: Spectra measured along y/D = -0.14 at stream wise locations x/D = 1.08 and x/D =
4.60. Cylinder is heated. Cylinder forced at θ
PP
= 0.52, F
R
= 3.03 ............................................ 108
xiii
Figure 5.55: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder
forced at θ
PP
= 0.70, F
R
= 3.06. Three cycles divided into 48 bins, every fourth bin shown, each
bin has phase width of π/8; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ....................................................................................... 110
Figure 5.56: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
=
0.70, F
R
= 3.06. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ......................................................................................................................................... 111
Figure 5.57: Spectra measured along y/D = -0.31 at stream wise locations x/D = 1.86 and x/D =
5.35. Cylinder is unheated. Cylinder forced at θ
PP
= 0.70, F
R
= 3.06 ........................................ 111
Figure 5.58: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder
forced at θ
PP
= 0.70, F
R
= 2.89. Three cycles divided into 48 bins, every fourth bin shown, each
bin has phase width of π/8; minimum contours ±0.5, contour spacing 0.5, positive contours are
solid and negative contours are dashed ....................................................................................... 112
Figure 5.59: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
=
0.70, F
R
= 2.89. Contour spacing 0.5, positive contours are solid and negative contours are
dashed ......................................................................................................................................... 113
Figure 5.60: Spectra measured along y/D = -0.37 at stream wise locations x/D = 1.08 and x/D =
2.96. Cylinder is heated. Cylinder forced at θ
PP
= 0.70, F
R
= 2.89 ............................................ 113
Figure 6.1: Contours of normalized heat transfer (Nu/Nu
0
) at Re = 750 with wake mode
boundaries superimposed. Thick, solid black lines are wake mode boundaries. Wake mode
regions: (a) 2S lock-on region, (b) non lock-on region, (c) Near-wake 2S, far wake non locked-
on, (d) ½(2S) far wake mode, (e) ½(P+S) far wake mode, (f) ½(P+S); No coalescence, (g)
⅓(P+S); No coalescence, (h) ⅓(2P); No coalescence ................................................................ 118
Figure 6.2: Normalized heat transfer rate (Nu/Nu
0
) vs. normalized rms tangential velocity
(V
rms
/U
0
) at Re = 750 with individual wake modes identified. Data derived from same data set as
Figure 6.1 .................................................................................................................................... 120
Figure 6.3: Formation length as a function of non-dimensional oscillation amplitude (θ
PP
) at
frequency ratio F
R
= 1.0. Vertical error bars are from DPIV grid spacing resolution ................ 122
xiv
Abstract
Wake formation is an important problem in engineering due to its effect on phenomena
such as vortex induced vibrations and heat transfer. While prior work has focused on the wake
formation due to vortex shedding from stationary, stream-wise, and transversely oscillating
cylinders, limited information is available on the effect of rotary oscillations on wake formation.
The relationship between wake formation and heat transfer is also not fully understood.
Therefore, a series of experiments were conducted to determine the effect of rotationally
oscillating cylinders on wake formation and its relationship with heat transfer.
Experiments were carried out at Re = 150 and 750 in a water tunnel for oscillation
frequencies from 0.67 to 3.5 times the natural shedding frequency and peak-to-peak oscillation
amplitudes up to 320°. Experiments were performed at the lower Re using an unheated cylinder.
Two sets of experiments were performed at the higher Re, one with the cylinder unheated and the
other with the cylinder heated. Digital Particle Image Velocimetry (DPIV) was used to identify
and map wake modes (coherent vortical structures in the wake) to various regions of the
parameter space. Previously unknown wake modes that are synchronized over two and three
times the forcing frequency were also discovered.
Experiments were also performed at Re = 750 to measure the heat transfer rate for a large
number of cases in the parameter space. Significant heat transfer enhancement was observed
under certain forcing conditions and the regions of the parameter space where this occurs was
found to correspond to locked-on wake mode regions. Other factors, such as the tangential
velocity and the formation length were also found to affect the heat transfer under certain
conditions.
1
Chapter 1: Introduction
1.1 Motivation and problem formulation
Vortex shedding from bluff bodies is a fundamental physical phenomenon that is observed
frequently in nature and in many practical engineering applications. It has been observed in
flows around tall buildings, submarine periscopes, and large scale flows such as those around
mountain peaks. The alternate shedding of vortices leads to structural vibrations, which can lead
to resonance and failure of the structure in some cases (Williamson, 1996). Such vortex-induced
vibration is also known to occur in industrially important devices such as tube-bank heat
exchangers. In such cases, there is a strong coupling between the vortex shedding and heat
transfer and it is important to understand the relationship between them in order to improve heat
exchanger design. This requires an understanding of the mechanism by which vortex shedding
occurs and methods to control the vortex shedding.
One method that has proven to be effective as a control strategy for smooth cylinders
placed in cross-flow is cylinder forcing. The effect of forced oscillations is to modify the
underlying wake structure which is the connection between vortex shedding and heat transfer.
There are different ways in which the cylinder can be forced and they modify the wake structure
in different ways and to varying extents. Rotationally oscillating the cylinder is one way of
controlling the wake. Rotary oscillations, as compared to transverse or stream wise oscillations,
might be especially advantageous in the case of applications with severe space constraints.
A review of the literature shows that the effect of rotary oscillations on vortex shedding is
not fully understood and almost no information exists about the actual wake structure or heat
transfer under such forcing conditions. Therefore, the goal of this study is to understand the
effect of rotary oscillations on the vortex shedding and its effect on heat transfer from the
2
cylinder. This goal is accomplished by identifying changes to the wake structure produced by
rotary oscillations and understanding the effect of the wake structure on heat transfer.
Figure 1.1: Schematic of basic setup.
A schematic of the basic setup is shown in Figure 1.1. An important non-dimensional
parameter is the Reynolds number, defined as
,
D U
Re
0
(1.1)
where U
0
is the free stream velocity, D is the cylinder diameter, and ν is the kinematic viscosity
of the fluid. Another important parameter is the non-dimensional shedding frequency, or
Strouhal number
,
U
D f
St
St
0
(1.2)
where, f
St
is the natural shedding frequency or Strouhal frequency. The cylinder forcing can be
described in terms of the non-dimensional frequency ratio
,
f
f
F
St
f
R
(1.3)
where f
f
is the forcing frequency, and the peak-to-peak oscillation amplitude (θ
PP
).
3
1.2 Wake formation
1.2.1 Vortex shedding from stationary cylinders
Vortex shedding from stationary smooth, circular cylinders has been extensively studied
due to its simple, well defined geometric shape. The wake formed due to flow past a circular
cylinder is considered as a baseline case for more complex flows and geometries. A review of
the literature shows that the connection between vortex shedding and wake formation is not yet
fully understood. The wake formation is known to depend on Reynolds number, but different
studies have given different values for the critical Reynolds number at which the various wake
transitions occur.
The critical Reynolds number for onset of flow separation has been found to vary
between 3.2 and 7. A numerical study by Sen et al (2009) gives a value of approximately 6.29
for onset of flow separation in unbounded flows. As Re increases, the critical value for the onset
of vortex shedding is found to vary between 30 and 49. This vortex shedding leads to the
formation of the Karman vortex street. Zdravkovich (1990) gives a value of Re = 30 - 48 at
which the flow transitions to this periodic laminar shedding regime. Williamson (1996) gives a
slightly different value of Re = 49 at which this transition occurs and the laminar vortex shedding
regime exists up to Re = 140 – 194. Above Re ~ 190 the flow transitions to a three dimensional
wake regime. Further transitions are present at higher Re which leads to regimes containing even
more complicated flow structures. One early paper that looked at high Reynolds number flows is
by Roshko (1961). This experimental study performed in a pressurized wind tunnel looked at
flows around a stationary cylinder for Re = 10
6
-10
7
. Szepessy & Bearman (1992) looked at
aspect ratio effects on the vortex shedding from smooth, circular cylinders at high Reynolds
numbers (Re = 8000 – 140000). Recently, Norberg (2001) studied flows around a stationary
4
cylinder for Re = 47 - 2.2 × 10
5
and gave an empirical correlation between the Reynolds number
and Strouhal number. Empirical relationships between St and Re over limited ranges of Re have
also been proposed by Kwon & Choi (1996), Posdziech & Grundmann (2000), Norberg (1994),
Eisenlohr & Eckelmann (1989), Hammache & Gharib (1989), Williamson (1988) and Bearman
(1969), but no physical explanation for this relationship has been provided. Figure 1.2 shows the
St – Re relationship compiled from data from various authors.
Figure 1.2: Strouhal number as a function of Reynolds number. Figure from Norberg (1994)
5
1.2.2 Vortex shedding from forced cylinders
Numerous studies over the last two decades have focused on vortex shedding from
cylinders undergoing forced oscillations in various configurations. Nishihara, et al (2005) studied
cylinders undergoing stream-wise oscillations at Re = 1.7 × 10
4
and Re = 3.4 × 10
4
. Williamson
& Roshko (1988) looked at cylinders undergoing transverse oscillations at 300 < Re < 1000.
There are many other studies on flows around cylinders forced in these two configurations. But
relatively few studies exist on vortex shedding from cylinders undergoing rotary oscillations.
Choi & Choi (2002) performed a numerical study of the vortex shedding from cylinders
undergoing rotary oscillations at Re = 100. Tokumaru & Dimotakis (1991) conducted an
experimental study at Re = 1.5 × 10
4
, performing flow visualization and wake profile
measurements on circular cylinders undergoing forced rotary oscillations in a steady uniform
flow. In all studies performed at relatively high Reynolds numbers, complex three dimensional
vortex structures were seen.
Mahfouz & Badr (2000a) numerically investigated the flow in the wake of a circular
cylinder performing rotational oscillation about its own axis. They studied flows up to a
Reynolds number of 200 and forcing frequencies up to two times the natural shedding frequency.
They found the lock-on phenomenon, in which the frequency of vortex shedding is “locked-on”
or synchronized with the forcing frequency, to occur within a narrow region of the parameter
space around the natural shedding frequency. They predict the frequency range over which this
lock-on occurs to increase with the amplitude of oscillation, and also predict a threshold value
below which this frequency range is zero. At oscillation amplitudes below the threshold value
the shedding is locked on only for F
R
= 1.0. Lee & Lee (2006) experimentally investigated the
flow in the wake of a rotationally oscillating cylinder at Re = 4140. Similar to Mahfouz & Badr
6
(2000a), they studied vortex shedding at relatively low oscillation amplitudes and forcing
frequencies up to two times the natural shedding frequency. The boundaries of the lock-on
region obtained in both studies are given in Figure 1.3. Both studies focused on spectral analysis
and did not include any pertinent information about the overall wake structure. But the overall
wake structure is important since it determines the effect of the wake, such as forces on the
cylinder or the heat transfer on the downstream side of the cylinder.
Figure 1.3: Lock-on boundaries identified by Mahfouz & Badr (2000a) and Lee & Lee (2006). Figure is from
Lee & Lee (2006). F
R
is frequency ratio (f
f
/f
St
) and θ
A
is equal to θ
PP
7
Williamson and Roshko (1988) studied the wake structure of a transversely oscillating
cylinder placed in cross-flow by identifying the vortex shedding patterns in terms of the number
and relative arrangement of vortices shed over one oscillation cycle. A vortex pair, consisting of
two counter rotating vortices is defined as “P”, a single vortex is defined as “S” and the various
patterns formed are defined as a combination of single vortices and vortex pairs shed over one
cycle. The 2S wake mode consists of two single vortices of opposite sign shed over one cycle.
The P+S wake mode consists of a vortex pair and a single vortex shed per cycle. The 2P+2S
mode consists of two counter rotating pairs of vortices shed per cycle along with a single vortex
shed between each pair. The different wake modes identified by Williamson & Roshko (1988)
are shown in Figure 1.4. A similar naming convention is used in the present study and is
described in detail in chapters 4 and 5.
Figure 1.4: Wake modes identified by Williamson & Roshko (1988)
8
Thiria, Goujon-Durand & Wesfreid (2006) and Thiria & Wesfreid (2007) also observed
phenomena similar to Lee & Lee (2006) and Mahfouz & Badr (2000a). They found the wake of a
cylinder undergoing forced rotary oscillation to be dependent on the amplitude and frequency of
oscillation. They describe the shedding as “no lock-on” if the spectra contain complex frequency
content that includes the forcing frequency and a frequency linked to the far-wake instability.
They explain the “non lock-on” regime as being caused due to far-wake instabilities and
identified boundaries between lock-on and non lock-on regions. In the present study, it was
observed that this kind of mixed frequency content is actually indicative of a transition between
lock-on and a “total” non lock-on regime in which the forcing frequency is almost completely
absent from the spectra.
Thiria & Wesfreid (2007) applied linear stability analysis to a time-averaged mean flow at
Re = 150 and obtained the stability properties by numerical solution of the inviscid Orr-
Sommerfeld equation and showed that it can account for the stability properties of the wake even
under rotary forcing conditions. They showed that the wake exhibits a finite region of absolute
instability immediately behind the cylinder and this region transitions to a region of convective
instability and explained the presence of the lock-on region by showing that it is reached when
cylinder forcing conditions trigger the transition to convective instability. They also observed
that under certain forcing conditions the vortices shed from the cylinder coalesce to form larger
vortical structures.
1.3 Heat Transfer from cylinders
1.3.1 Heat transfer from stationary cylinders
Heat transfer from stationary cylinders in cross-flow has been studied extensively and is
considered well understood and information on this can be found in any standard textbook on
9
heat transfer. A review of previous numerical studies on this can be found in Bouhairie & Chu
(2007). The numerical simulation by Bouhairie and Chu looked at heat transfer from circular
cylinder into a cross-flow for Re = 200 - 15550. They computed the unsteady heat transfer using
a two-dimensional model which simulated the local and overall heat transfer rates. There have
also been experimental studies which provide empirical correlations between the heat transfer
rate and Reynolds number. A review of these studies and others can be found in Pottebaum
(2003). Through these studies, it has been shown that heat transfer from a cylinder placed in
cross-flow consists of two steps. In the first step, heat is transferred from the cylinder onto the
viscous boundary layer through conduction. In the second step, the fluid entrained in the
boundary layer is then removed through the alternate shedding of vortices. Therefore, in the case
of a cylinder, heat transfer enhancement is dependent on enhancing the removal of fluid from the
“dead-water” region downstream of the cylinder. This can be accomplished by cylinder forcing.
1.3.2 Heat transfer from forced cylinders
Heat transfer from cylinders undergoing forced oscillations have been studied for a long
time. One of the earliest studies was by Sreenivasan & Ramachandran (1961). They looked at
heat transfer from cylinders undergoing transverse oscillations in the range 2500 < Re < 15000
and found no appreciable increase in the heat transfer rate. Recently, Pottebaum (2003) and
Pottebaum & Gharib (2006) showed that transversely forced cylinders in a cross-flow do exhibit
significant heat transfer enhancement under certain forcing conditions. They also explained the
lack of heat transfer enhancement seen by Sreenivasan and Ramachandran to be due to the low
frequency ratios studied. The study by Pottebaum and Gharib was done at Re = 687 and they
showed that the heat transfer is significantly enhanced by small amplitude oscillations at
frequencies near the Strouhal frequency and its harmonics. Recently, heat transfer from cylinders
10
undergoing rotary oscillations has been studied by Mahfouz & Badr (2000b). Their numerical
study looked at flows between 40 ≤ Re ≤ 200. The frequency ratios considered were 0 ≤ F
R
≤ 2.
They found that the vortex shedding influences the thermal field in the wake region and observed
significant heat transfer enhancement in the lock-on region. But they did not look at the actual
wake structure and its relationship with heat transfer enhancement.
1.4 Objectives and Organization
The goal of this study was to investigate the effect of rotary forcing on wake formation and
its effect on heat transfer. The study was done in two phases. In the first phase, the parameter
space was surveyed to determine the heat transfer rate and its correlation, or lack thereof, with
cylinder forcing conditions. In the second phase, guided by information gleaned during the first
phase, the wake structure of a reduced set of cases were studied in greater detail using digital
particle image velocimetry and their wake modes and relationship with heat transfer were
identified.
This dissertation presents the results of experiments performed during both phases of the
study. Details of the experimental setup, methods used to acquire and process the data, and
analysis methodologies are described in chapter 2. Results from the first phase of the study, the
parameter space survey, showing the correlation between heat transfer and cylinder forcing
conditions are given in chapter 3. Results from the second phase of the study are presented in
chapters 4 and 5. In chapter 4, the results of experiments studying wake formation at Re = 150
are presented. In chapter 5, results of experiments performed at Re = 750 are presented, including
results from both heated and unheated cylinders. The relationship between wake formation and
heat transfer is explored in chapter 6. Chapter 7 contains concluding remarks and information
pertinent to future work in this topic of research.
11
Chapter 2: Experimental methodology
2.1 Introduction
The experimental setup and details of the various methods used in this study are described
in this chapter. The general approach used in this study is described in §2.2. The specifications of
the water tunnel and the conditions used in this study are described in §2.3. The cylinder setup,
along with its material properties, is described in §2.4 and the cylinder motion control is
described in §0. Details of the cylinder heat transfer and the model used are given in §2.6. The
main method used to obtain wake mode information, digital particle image velocimetry, is
described in §2.7. Details of the digital particle image thermometry system, which ultimately did
not prove successful, are given in §2.8. Two of the main analysis tools used in this study,
namely, phase averaging and spectral analysis, are described in §2.9 and §2.10, respectively.
2.2 General approach
Studies have shown that vortex shedding exists in the range 49 < Re < 10
7
and possibly at
even higher Re (Williamson, 1996). In the present study, the Reynolds number of the flow was
maintained around 750. This value of Re was chosen due to two reasons. First, most real world
flows have a high Reynolds number (Re>>1000) and are highly turbulent. But for flow around a
cylinder at Re > 1000 even the shear layers are unstable and the flow is highly turbulent, making
precise velocity measurement very difficult. Second, at low Reynolds numbers, Re = 49 – 400
(approximate values), the heat transfer rate from cylinders is very low which necessitates
maintaining a low temperature difference between the surface of the cylinder and the ambient
fluid in order to minimize the effect of natural convection. This low temperature difference
increases the relative uncertainty of measurements obtained using the thermocouple setup
12
available and reduces the resolution of temperature measurement. Therefore, in order to attain a
balance between these two constraints, a value of Re = 750 was chosen. At this Reynolds
number, the turbulence effect is minimal while still exhibiting significant characteristics of real
world flows and the heat transfer rate is also relatively high, which leads to more precise
temperature measurements.
The initial approach was to conduct a parameter space survey consisting of a large number
of test cases over a wide region of the frequency – amplitude plane at Re = 750. The heat transfer
coefficient for the cylinder was calculated for each test case in the parameter space. This
information, coupled with data about the wake structure at the same Reynolds number obtained
through velocity measurement techniques, was expected to shed light on the relationship
between them. But when the flow velocity measurements were obtained it was noticed that at the
Reynolds number considered, the velocity measurements contained a lot of cycle-to-cycle
variations due to turbulence and it was difficult to identify the exact wake structure. In order to
overcome this difficulty, velocity data was obtained at a lower Reynolds number (Re = 150) in
order to reduce the cycle-to-cycle variations and gain a better understanding of the wake
structure. Based on wake mode boundaries identified at Re = 150, initial test cases for DPIV flow
velocity measurements at Re = 750 were selected. To reiterate, present study is mainly focused
on wake formation and heat transfer at Re = 750.
2.3 Water Tunnel
The experiments were carried out in the GALCIT Heat Transfer Water Tunnel, the same
facility that was used in a previous study, Pottebaum (2003). The facility was moved to a new
location at the University of Southern California and refurbished before the experiments in the
current study were conducted. This water tunnel was designed for constant free stream
13
temperature conditions. Figure 2.1 shows a diagram of the water tunnel and basic arrangement of
experiment. The test section of the water tunnel is made of Plexiglas, to allow for visual
observation, and is 15.2 cm wide, 58 cm long, and has a maximum height of 16 cm. For all of the
experiments the water depth was maintained at 13.7 cm. The flow speed in the test section can be
varied between 1.30 cm/s and 40 cm/s. For all the experiments discussed in this study the free
stream temperature of the tunnel was set at 25.8±0.2°C, and the exact temperature varied within
that range over various experimental runs. Properties of water at 25.8°C are tabulated in Table
2.1. Similar to the setup used by Pottebaum (2003), in the current study the free-stream
temperature was maintained using an actively controlled heat input and a constant-power heat
sink. Heat input was achieved using a staggered array of 16 heated Watlow FIREROD cartridge
heaters. This array was located in the upstream settling section. The amount of heating was
actively controlled using a proportional-integral-derivative (PID) controller; with temperature
feedback provided using a RTD temperature sensor. The constant-power heat sink was located in
the downstream settling section and consisted of coils of copper tubing connected to an external
recirculating chiller (NESLAB RTE-110). A reference thermometer (Omega DP41-RTD with
Burns Engr. 392 probe) with a resolution of 0.02°C was located in the upstream settling section
and was used to measure the free-stream temperature.
14
Figure 2.1: Diagram of water tunnel and basic arrangement of experiment
Table 2.1 : Properties of water at 25.8°C. From Pottebaum (2003)
Density, ρ (kg/m
3
) 996.8
Volumetric coefficient of thermal expansion, β (1/K) -2.65∙10
-4
Viscosity, µ (kg/m·s) 8.74∙10
-4
Kinematic viscosity, ν (m
2
/s) 8.77∙10
-7
Derivative of kinematic viscosity, dν/dT (m
2
/s·K) -1.92∙10
-8
Thermal conductivity, k
w
(W/m·K) 0.612
Specific heat capacity, C
w
(J/kg·K) 4.20∙10
3
Thermal diffusivity, α = k
w
/ρ·C
w
(m
2
/s) 1.46∙10
-7
Prandtl number, Pr = ν/α 5.99
15
2.4 Circular cylinder setup
A smooth, circular cylinder with diameter 0.00942 m was used and was submerged to a
depth of 13 cm, normal to the flow direction and normal to the bottom wall of the tunnel. The
cylinder is a custom cartridge heater manufactured by Watlow. It consists of a thin coil of high
resistivity wire (80% Ni, 20% Cr) surrounded by compacted magnesium oxide insulator and
encased in an Incolloy® 800 (a stainless steel alloy) sheath. The heater was powered using a
variable-voltage DC power supply unit (Agilent E3648A), with the output voltage set to 40 V.
The cylinder also contains a type-J thermocouple embedded at the center of the heated length
along the cylinder axis. The leads of this thermocouple were connected to a Thermocouple to
Analog Converter (Omega SMCJ-J). This converted the thermocouple signal to a cold junction
compensated, linear, amplified analog signal that was recorded using a data acquisition device
(National Instruments PCI-6229) and custom LabVIEW program. The relationship between the
thermocouple temperature, T
tc
, and the measured voltage was determined through calibration
using a stationary cylinder. The uncertainty in the temperature measurements were determined to
be ±0.1°C.
The cylinder was manufactured to exactly the same specifications as one of the cylinders
used in Pottebaum (2003) and has an aspect ratio of 13.8 and, as such, was found to possess the
same material and thermal properties. Therefore the properties listed in Pottebaum (2003) are
used in the current study without any modifications. The cylinder dimensions are tabulated in
Table 2.2 and its material properties are tabulated in Table 2.3. Diagrams of the cylinder and its
internal structure are given in Figure 2.2.
16
Figure 2.2: Cylinder exterior dimensions and internal structure (not to scale); dimensions are listed in Table
2.2. From Pottebaum (2003)
Table 2.2 : Cylinder dimensions (in mm). From Pottebaum (2003)
Outer diameter (D) 9.42
Heating wire helix radius (R
1
) 2.88
Sheath inner radius (R
2
) 3.81
Heating wire helix pitch 0.6
Heated length (L
H
) 127
Lead end unheated length (L
A
) 25.4
Closed end unheated length (L
B
) 6.35
17
Table 2.3 : Cylinder material properties. From Pottebaum (2003)
Insulator Sheath
Density, ρ (kg/m
3
) 3044 8026
Thermal conductivity, k (W/m·K) 2.08 14.0
Specific heat capacity, C (J/kg·K) 879 502
Thermal diffusivity, α = k/ρC (m
2
/s) 7.77∙10
-7
3.47∙10
-6
Table 2.4 : Experiment conditions and Strouhal frequencies at 25.8°C
Free-stream velocity, U
0
(m/s) Re f
St
(Hz)
0.0140±0.0002 150±2 0.24
0.0698±0.0002 750±2 1.47
The cylinder was attached to a stepper motor through a rigid linkage. The entire cylinder
assembly was anchored to the building structure, and was isolated from the water tunnel, except
for the cylinder extending into the test section of the water tunnel. This was done to minimize
vibrations and prevent the transmission of any remaining vibrations to the water tunnel. The
Strouhal frequency, f
St
, of the cylinder was obtained by spectral analysis of the time series of the
x-component of velocity at various locations in the wake of a stationary cylinder. The tunnel
flow conditions, with the observed Strouhal frequency at those conditions, used in this study are
given in
Table 2.4. These values are slightly different from that given in the literature due to the
cylinder having a finite aspect ratio of 13.8.
18
2.5 Cylinder motion control
The cylinder was oscillated with a sinusoidal rotary motion about its own axis. The
cylinder motion was controlled using a closed loop motion control system consisting of stepper
motor (Anaheim Automation, 23MD series) with 0.225° step resolution, optical rotary encoder,
data acquisition card (National Instruments PCI-6229) and custom LabVIEW program. Motion
profiles were specified such that the cylinder motion was smooth and there were no “stop-and-
go” movements. This effectively limited the range of possible cylinder oscillation conditions.
Also, due to the design of the cylinder mounting mechanism, the maximum oscillation amplitude
that could be attained was 320° (θ
PP
= 5.58 rad) and the maximum oscillation frequency that
could be safely attained at this amplitude was 4.5 Hz. At oscillation amplitude of 90° (θ
PP
= 1.57
rad), the maximum oscillation frequency that has been safely tested is 6 Hz. At Re = 150, the
cylinder oscillation amplitudes extend up to 320° and oscillation frequencies range between 0.16
Hz ≤ f
f
≤ 0.86 Hz and these translate to non-dimensional values of 0.17 ≤ θ
PP
≤ 5.58 and 0.67 ≤
F
R
≤ 3.5. At Re = 750, due to the higher value of Strouhal frequency, the cylinder oscillation
amplitudes studied extend only up to 120° and oscillation frequencies only range between 1.03
Hz ≤ f
f
≤ 4.65 Hz. This translates to non-dimensional values of 0.09 ≤ θ
PP
≤ 2.09 and 0.70 ≤ F
R
≤
3.16. The experiments were performed such that during each run multiple test cases were
performed. For each test case, the cylinder was oscillated for 45 seconds before data acquisition
started. This was to ensure that no transient effects were captured. At the end of each test case
the vortex shedding was “reset” by forcing the cylinder to perform a very rapid traverse to its
extreme positions and the cylinder was returned to its initial position. This was to prevent any
hysteresis effect.
19
2.6 Cylinder heat transfer
For the heat transfer experiments, heating power was provided to the cylinder by a
variable-voltage DC power supply. A nominal input power, P
in
, of 60 W was used. The actual
power input was monitored by independently measuring the current through and the voltage
across the cylinder heating wire. This level of power input was chosen because it generates
temperature differences in the flow that are measurable but do not cause significant buoyancy
forces. The Richardson number, defined as the ratio of buoyancy forces to inertial forces in the
vertical momentum equation,
,
T gD
Ri
U
2
0
(2.1)
where g is gravitational acceleration, β is the volumetric coefficient of thermal expansion for
water, ΔT is the difference between the cylinder surface temperature and the free-stream
temperature, was kept below 0.05 under the conditions of this study. This was done to ensure
that the flow was in the forced convection regime.
The heat transfer coefficient for the cylinder was calculated from the temperature at the
cylinder core. The temperature at the core was measured using the embedded thermocouple. The
heat transfer coefficient was then calculated using a steady, one-dimensional model of heat
transfer inside the cylinder. This method, developed by Pottebaum (2003), uses an axisymmetric
model with no axial variation to approximate the internal structure of the cylinder. The geometry,
shown in Figure 2.3, consists of four regions (a, b, c, and d). Power is added only in region b.
Since the cylinder used in the current study was built to the specifications of the cylinder used by
20
Pottebaum, the model is used without any modification. Pottebaum gives an expression for the
cylinder heat transfer coefficient
, C T T
P
L R
h
T tc
in
d
1
2
(2.2)
where T
∞
is the free-stream temperature, L is the length of cylinder, and C
ΔT
is a constant given
by
.
R
R
ln
R R
R
k R
Rc
ln
k R
R
ln
k
R C
a
b
a b
a
b b c c
d
d
d T
2 2
2
2
1 1 1 1
(2.3)
The radii used to calculate C
ΔT
are given in Table 2.5. Although C
ΔT
can be calculated
using this expression, it can also be estimated by calibration. The values for C
ΔT
obtained
through both methods are given in Table 2.6. The heat transfer coefficient obtained using this
method is then non-dimensionalized as the Nusselt number,
.
k
hD
Nu
(2.4)
Pottebaum also gave an expression for the relative uncertainty in the heat transfer coefficient,
, h
L R P
T T
hC
h
T
d in tc
C
L
d
R
in
P
tc
T T
T
h 2 2
2
2 2
2
2 2
2
2
1
(2.5)
where δ
( )
indicates uncertainty in the subscripted quantity. The details of the model, calibration
method, and a formal error analysis can be found in Pottebaum (2003).
21
Figure 2.3: Geometry for model of heat transfer inside the cylinder. From Pottebaum (2003)
Table 2.5 : Radii (mm) used in model of cylinder internal heat transfer. Values are from Pottebaum (2003)
Table 2.6 : Values determined for C
ΔT
.
Best estimate from calibration -0.00055
Standard deviation from calibration 0.00004
Calculated from radii and material properties -0.00057
R
a
2.986
R
b
3.227
R
c
3.810
R
d
4.710
22
2.7 Digital Particle Image Velocimetry
Velocity data was obtained using DPIV. The water tunnel was seeded using silver-coated,
hollow, glass spheres (Potters Industries Inc., CONDUCT-O-FIL, SH400S33) with an average
particle diameter of 14 µm and average density of 1700 kg/m
3
. The particles were illuminated by
a horizontal laser sheet with a wavelength of 532 nm emitted by a pulsed laser system (Solo PIV,
NEW WAVE Research). The laser sheet had a thickness of ~5 mm at the mid-plane of the test
section and was located at a height of 0.065 m above the bottom wall. Images were recorded
using a black and white, CCD camera (Sony XCL X-700) connected to a frame grabber (Dalsa-
Coreco X64-CL Express). The image acquisition and laser pulsing were synchronized using a
counter/timer (National Instruments PCI-6602). The separation time between any two images
that formed an image pair was set such that, on average, particle displacements during that time
period were approximately 1/4
th
of the size of the interrogation windows used. This was done to
minimize spatial aliasing.
Since the cylinder used in the experiment is not transparent all images contain a shadow,
and no velocity information can be obtained from the shadow region. The area imaged by the
camera was set in such a way that the downstream half of the cylinder was always present in the
image, and the wake region imaged varied in length from 5 to 10 cylinder diameters. For the Re
= 150 cases, 1500 images (750 image pairs) were recorded at a rate of 13 frames per second. For
the Re = 750 cases, a minimum of 3500 images (1750 image pairs), were recorded at a rate of 27
frames per second. The images were recorded as grayscale images in RAW format with a bit
depth of 8 bits per pixel. The recorded images were processed and post-processed for each case
using standard DPIV, window shifting and outlier-detection algorithms described in Willert &
Gharib (1991), Adrian (1991), Westerweel (1994) and Raffel, et al (2007), with 32 X 32 pixels
23
interrogation windows and 50% overlap. Overall, the measured velocities have relative
uncertainties of 1.5%. From the velocity fields, derivative quantities such as vorticity are
calculated.
2.8 Digital Particle Image Thermometry
Digital particle image thermometry (DPIT) refers to the extraction of temperature field
information from the color of thermochromic liquid crystal (TLC) particles, which change their
reflected wavelength with temperature. DPIT depends upon the property of TLC that it
selectively reflects light of a particular wavelength. The TLC acts as a Bragg scattering volume
grating with a temperature dependent pitch. The apparent color of the TLC therefore depends on
the temperature and on the angle between the incident light and the observer. In this study, DPIT
was tried in certain test cases to obtain temperature profiles of the wake. In such situations, it can
be combined with DPIV to simultaneously obtain velocity information and is referred to as
digital particle image thermometry/velocimetry. DPIT/V combines the two techniques by using
the TLC particles as both temperature indicators and flow tracers. The setup and processing
methods tried were similar to the one described in Pottebaum & Gharib (2006). TLC particles
have a limited temperature bandwidth over which they exhibit a consistent color-temperature
relation, and for this study, TLC particles with a color play that spanned the range of
temperatures expected, approximately 25° C to 45°, were required. Unfortunately, TLC
formulations decay over time leading to a change in their color play, and the formulation that the
author used was rendered unstable during storage and valid temperature information could not be
extracted. Since TLC particles are custom manufactured and have a long lead time, a new batch
of TLC particles could not be obtained on time. Therefore, no information about the temperature
field inside the wake was obtained during this study. But this does not affect the measurement of
24
heat transfer coefficient from the cylinder, which is calculated using temperature data from the
embedded thermocouple. Therefore, the primary goal of this study was not affected.
2.9 Phase Averaging
Background noise and cycle-to-cycle variations are two factors that lead to difficulty in
identifying wake structures in the vorticity data. To overcome this, vorticity fields calculated
from the DPIV data are phase averaged. This is accomplished by calculating the phase of each
vorticity field obtained from the DPIV data. This is done using the known cylinder oscillation
frequency and the time between image pairs and grouping vorticity fields into bins of equal
width on the basis of their phase. The vorticity fields in each bin are then averaged. Phase
averaging is also done over multiple cylinder oscillation cycles to study vorticity structures that
are in-phase over multiple cycles. Therefore, for every case studied in the parameter space, phase
averaging was done over one, two, and three cylinder oscillation cycles.
2.10 Spectral Analysis
Spectral analysis was used to measure the frequency of the vortex shedding and to detect
the presence of any harmonics or the cylinder forcing frequency in the wake. This was
accomplished by extracting the time series of the stream-wise velocity component in the DPIV
data at various points in the spatial domain downstream of the cylinder. A Hann windowing
scheme was applied to the time series, and it was transformed to the frequency domain using the
standard Cooley-Tukey implementation (Cooley & Tukey, 1965) of the FFT algorithm in
MATLAB, and the power spectra was plotted.
25
Chapter 3: Heat transfer from cylinder and its dependence on forcing conditions
3.1 Introduction
Heat transfer from a circular cylinder placed in cross-flow and its dependence on cylinder
oscillation conditions are examined in this chapter. The results presented here form the first
phase of the overall study. The experimental conditions and the methods used to measure the
heat transfer coefficient are presented in §3.2. The results of the parameter space survey are
presented in §3.3, and the results are discussed in §3.4.
3.2 Experimental conditions and methodology
During the first phase of the study, experiments were conducted at Re = 750 over a wide
range of oscillation conditions. The parameter space was selected to span frequencies ranging
from sub-harmonic to the second harmonic of the Strouhal frequency for the cylinder at Re =
750, and rotary oscillation amplitudes ranging from zero (stationary) to the maximum limit
allowed by the setup. Figure 3.1 shows the oscillation conditions that were considered over the
course of a single run during the first phase of the study. A total of 357 cases were studied in this
run. While the results presented in this chapter are all from this single run, multiple runs
consisting of thousands of test cases were studied during experiment setup and initial testing.
The zero amplitude, non-oscillating cases were tested at the beginning of each experimental run
and were repeated again at the end.
Before the start of each experimental run, the water tunnel was cleaned, all the electrical
connections and thermocouple connections were checked, and the cylinder was securely
mounted. The active temperature control system consisting of heater and chiller were turned on
and the water tunnel was left to run overnight before the experiments were performed. During
26
the course of the experiments, the water tunnel temperature (T
∞
) was regularly monitored and
recorded before the start of each case. T
∞
was maintained within 25.8 ± 0.2 °C for all cases, with
a measurement uncertainty of 0.02 °C.
Figure 3.1: Cylinder oscillation conditions considered during heat transfer experiments
At the start of each case, the stepper motor controlling the cylinder motion was powered on
and the cylinder was forced at the prescribed frequency and amplitude, and power was supplied
to the heating coils in the cylinder. For all cases, power input (P
in
) to the cylinder was maintained
at 59.26 ± 0.02 W. A 45 second delay was imposed before data recording to allow for any
transient effects to die out. The power input to the cylinder was measured, with a measurement
uncertainty of ± 0.007 W, and recorded right before thermocouple data was recorded. The
27
thermocouple voltage was measured in chunks of 1000 samples acquired at a rate of 1 kHz, and
each chunk was averaged in order to obtain a mean voltage at a rate of 1 Hz. For all cases,
thermocouple data was recorded for a period of 120 seconds. This was done to ensure that, for
any given oscillation condition, data over a large number of oscillation cycles were captured. At
the end of data acquisition the cylinder was forced to undergo a rapid sweep to the ends of its
amplitude range and brought back to rest at its initial position and a further delay of 30 seconds
was imposed before the start of the next case. This was done to ensure that the wake structure
from one case does not affect the next case.
3.3 Results
The results of the heat transfer experiments performed during one run are shown in Figure
3.2. This data was obtained by considering the cases indicated in Figure 3.1. The results
presented in Figure 3.2 are the contours of Nusselt number (Nu) scaled by the Nusselt number of
the stationary cylinder (Nu
0
). The contour levels are spaced exponentially with the lowest
contour level being unity. Figure 3.3 presents the normalized heat transfer (Nu / Nu
0
) at selected
cylinder oscillation amplitudes as functions of the frequency ratio (F
R
). The Nusselt number for
the stationary cylinder (Nu
0
) was obtained by averaging the Nusselt numbers of all the zero
amplitude cases performed at the beginning of the run.
28
Figure 3.2: Contours of normalized heat transfer (Nu/Nu
0
) at Re = 750
29
Figure 3.3: Normalized heat transfer (Nu/Nu
0
) vs. frequency ratio (F
R
) for selected oscillation amplitudes ( θ
PP
given in degrees to aid analysis). Error bars are not shown to improve clarity. Derived from the same data set
and have same relative uncertainties as Figure 3.2
3.4 Discussion
The heat transfer rate was experimentally determined, and as such, there are uncertainties
associated with the measurement of various quantities of interest. The relative uncertainty in the
heat transfer coefficient is given by eqn. (2.5). The uncertainty in the heat transfer coefficient
depends on the heat transfer coefficient itself, and the (1-hC
ΔT
)
2
and h
2
terms amplify the
uncertainty. This implies that the uncertainty is higher in regions with a high heat transfer rate.
The main source of measurement uncertainty is the cylinder core temperature, T
tc
. This
30
temperature was measured using a thermocouple, which has an inherently noisy output and
therefore contributes the most to the overall uncertainty. The signal from the thermocouple was
time-averaged to mitigate the effect of random noise, but the uncertainty in T
tc
was still found to
be ±0.1°C. Other major sources of uncertainty include the uncertainty in T
∞
, power input P
in
,
radius R
d
, and length L. The contour levels used in Figure 3.2 were selected to account for the
relative uncertainty in the heat transfer rate. It was also found that the magnitude of the heat
transfer rate varied between runs, but the observed trends remain the same and are repeatable.
One possible reason for this variability in the heat transfer rate was the shift in room temperature
between different runs. This would have affected the resistivity of the electrical connections
leading to the shift in the measured values.
The most prominent features that are noticeable in Figure 3.2 are the regions with high heat
transfer. All of these regions are located at or near the Strouhal frequency and its harmonics. This
seems to suggest some form of synchronization between the heat transfer and the harmonics. For
cases where the forcing frequency was close to the Strouhal frequency, (F
R
= 1.0), there was a
noticeable heat transfer enhancement even at low oscillation amplitudes. An interesting feature
to note is the exact location of maximum heat transfer enhancement in this region for all
oscillation amplitudes. The maxima do not occur at the Strouhal frequency itself, but at
frequencies slightly higher than the Strouhal frequency. The reason for this is not known.
For the lowest oscillation amplitude studied the maximum heat transfer enhancement is
seen around the second harmonic (f
f
= 3f
St
). This can also be seen in Figure 3.3. For all low
amplitude cases (Θ
PP
< ~0.40 rad), the maximum heat transfer enhancement was observed
around the second harmonic. The heat transfer rate at sub-harmonic frequencies is found to be
very low and barely higher than that of a stationary cylinder.
31
This lack of heat transfer enhancement for F
R
< 1.0 is observed not only at low amplitudes
but, to some extent, at higher amplitudes as well. The region of high heat transfer rate is found to
slowly grow with increase in Θ
PP
, but for all oscillation amplitudes there is a very steep gradient
in the contours of normalized heat transfer as F
R
decreases. This lack of heat transfer
enhancement at the sub-harmonics stand in contrast to previous studies that found significant
heat transfer enhancement at the sub-harmonics when cylinders undergo transverse or stream-
wise oscillations.
Around the second harmonic (F
R
= 2.0) the region of maximum heat transfer enhancement
was observed only between Θ
PP
≈ 0.50 and Θ
PP
≈ 1.10. With further increase in amplitude, the
heat transfer rate, while still significantly higher than that of a stationary cylinder, is found to
decrease until it eventually starts increasing again at very high amplitudes. This decrease in the
heat transfer rate with increasing amplitude is unexpected as it runs counter to the general trend
of increase in heat transfer rate with increase in amplitude for any given frequency.
This phenomena also occurs around the second harmonic (F
R
= 3.0), but at lower
amplitudes. A possible explanation for this phenomenon is that the lower heat transfer rate is, in
fact, the norm and it is the occurrence of these islands of higher heat transfer enhancement at
lower amplitudes that is anomalous. This brings up the need for an explanation of this anomalous
behavior. It is postulated that the occurrence of certain wake modes might be a reason for the
occurrence of higher than expected heat transfer enhancement in these regions. This will be
discussed in detail in chapter 6.
For Θ
PP
> 1.25, there is no local peak in the heat transfer rate around the first harmonic.
Instead, at any given oscillation amplitude and F
R
> 1.5, the heat transfer rate increases steadily
with increase in F
R
. This can be clearly seen in Figure 3.3, in which the two highest amplitude
32
curves grow steadily with F
R
. The highest heat transfer enhancement is seen when Θ
PP
> 1.85
and F
R
> 2.80.
A peculiar phenomenon that was noticed during every run is the systematic shifting of the
zero amplitude, stationary cases tested at the end of each run. Compared to the string of zero
amplitude cases tested at the beginning of the run, the latter ones always had a lower heat
transfer rate. This observed shift to a lower heat transfer rate varied between runs but was never
more than 2% of Nu
0
for that run. A possible reason for this shift could be the change in
electrical resistivity of the thermocouple and power connections due to changes in room
temperature. The room temperature was not controllable and, while there were no dramatic
changes in room temperature, it varied both ways, with increasing temperatures during certain
runs and decreasing temperatures during other runs. So this does not fully explain why the
observed shift in heat transfer rate is always towards a lower rate. A possible reason could be the
evaporative loss of water, and subsequent decrease in water level, from the water tunnel during
the course of a run. The pumps used to run the water tunnel operate at a fixed flow rate. So any
drop in the volume of water in the tunnel will lead to an increase in flow velocity. While every
effort was made to minimize any changes in water level during a run, a small drop in volume
was inevitable. This drop in volume was too low to cause a measurable drop in the level of water
and any increase in flow velocity was too small to be measured by DPIV. But this would have
led to infinitesimal changes in Re, T
∞
, aspect ratio, and thermal and momentum boundary layer
thicknesses. All of these may have contributed to the observed shift. This systematic shift is still
smaller than the uncertainty in the normalized heat transfer rate and there is no issue with the
reliability of the data.
33
3.5 Conclusion
The details of the experiment to study the heat transfer from a rotationally oscillating
circular cylinder placed in cross-flow were presented in this chapter. The major finding was the
synchronization of heat transfer enhancement with certain cylinder oscillation conditions. The
regions of the parameter space with the highest heat transfer enhancement were identified.
Significant heat transfer enhancement was observed near the Strouhal frequency and its
harmonics, which suggests that the wake structure might be an important factor affecting the heat
transfer. This and various other trends observed in the data were examined. The shift in heat
transfer rates of the zero amplitude, stationary cases were studied and a possible explanation was
provided. In the future, conducting shorter runs would help in reducing this shift in the heat
transfer rate.
The boundaries of regions with heat transfer enhancement were found to share similarities
to wake mode boundaries. This reaffirms the need to understand the effect of forcing on the
wake structure in order to understand the relationship between vortex shedding and heat transfer.
This will be examined in detail in chapter 5 and the similarities in the boundaries will be
examined in chapter 6.
The measurement uncertainties and the overall reliability of the data were also examined.
A major source of error was found to be in the measurement of the cylinder core temperature.
For future work, a more precise method of measuring T
tc
would be useful in reducing the
uncertainty.
34
Chapter 4: Effects of cylinder forcing on the structure of the wake at Re = 150
4.1 Introduction
Vortex shedding from rotationally oscillating circular cylinders placed in a cross-flow and
the wake patterns that are formed are examined in this chapter. The experiments described in this
chapter were all performed at a Reynolds number of 150. This is a subset of the second phase of
the overall study. The experimental conditions and methodology are presented in §4.2. The
results of the DPIV measurements are presented in §4.3, and the results are discussed in §4.4.
4.2 Experimental conditions and methodology
The experiments described in this chapter were conducted at Re = 150. The reason for
performing experiments at this relatively low Reynolds number is given in §2.2. These
experiments are a subset of the second phase of the study. The experiments were carried out
using the same setup as described in chapter 2. The cylinder was not heated and only wake
structure information was obtained using DPIV. The details of the DPIV setup are described in
§2.7. In order to study the wake modes, phase averaging and spectral analysis are used and these
are described in §2.9 and §2.10, respectively.
Experiments were performed at 304 different oscillation conditions in the frequency –
amplitude plane. As given in §2.4, f
St
= 0.24 Hz when Re = 150. This limits the possible forcing
conditions that can be tested safely. In these experiments, cylinder oscillation amplitudes extend
up to 320° and oscillation frequencies range between 0.16 Hz ≤ f
f
≤ 0.86 Hz and these translate
to non-dimensional values of 0.17 ≤ Θ
PP
≤ 5.58 and 0.67 ≤ F
R
≤ 3.5. Figure 4.1 shows the
experimental test conditions in the parameter space.
35
The distribution of test cases shown in Figure 4.2 is non-uniform, with cases clustered in
certain regions. This distribution is based on the results of tests performed during the initial setup
and testing. Regions of the parameter space that exhibited new wake modes and regions near
wake mode boundaries contain a high density of test cases. The results described in this chapter
were obtained from 6 runs conducted on different days. All 304 cases could not be tested in one
single run because of the very large memory requirements of the DPIV technique. During each
run, cases were tested in chunks of 15 cases, with DPIV seeding particles added at the beginning
of each chunk. This was done to ensure a uniform particle density in all images.
Figure 4.1: Experimental test conditions in the parameter space
36
4.3 Results
A map of the different wake mode regions identified as functions of θ
PP
and F
R
is shown
in Figure 4.2. Figure 4.2 also shows the Mahfouz & Badr (2000a) lock-on boundaries
superimposed on the boundaries identified in the present study. The different wake mode regions
are discussed in §4.4. Figures Figure 4.3, Figure 4.5, Figure 4.7, Figure 4.9, Figure 4.11, and
Figure 4.13 show the phase averaged non-dimensional vorticity fields at certain forcing
conditions. Each selected case is representative of the wake structure that was observed in that
region. Figures Figure 4.4, Figure 4.6, Figure 4.8, Figure 4.10, Figure 4.12, and Figure 4.14
show the power spectra for the same selected cases.
Figure 4.2: Map of wake mode regions from present study with boundaries (approximate) identified by
Mahfouz & Badr (2000a) superimposed
37
4.4 Discussion
At this low Re, the flow is found to be in a laminar vortex shedding regime and contains
very few three dimensional structures (Williamson, 1996), which allows for a comprehensive
description of the vortical structures in the wake. Spectral analysis of the wake is used to study
its frequency content. The presence of only the forcing frequency as the dominant frequency in
the power spectrum is an indication of the lock-on phenomenon. Presence of peaks in the power
spectrum, at the forcing frequency and also the natural shedding frequency, is an indication of a
transition regime. The presence of the natural shedding frequency as the only dominant
frequency in the power spectrum indicates that vortex shedding is not locked to the forcing of the
cylinder.
In cases that exhibit coalescence, spectral analysis of the wake at locations downstream of
the coalescence location shows that the dominant frequencies in the power spectrum are different
from the frequencies observed before coalescence. The wake before coalescence is classified as
near wake and the wake after coalescence is classified as far wake. Coalescence is the
mechanism by which a less stable wake configuration transitions to a more stable one. When the
near wake is unstable, it transitions to a different wake structure through the combination of
smaller vortices to form larger vortical structures which are arranged in a globally stable
formation. This stable far wake is similar to the wake of an unforced cylinder. Other stable
arrangements are also possible under certain conditions, depending on the cylinder forcing and
the unperturbed outer flow. The stability of the wake is highly dependent on the spatial
arrangement of vortices in the near and far wake. This spatial arrangement or wake mode is
classified based on the Williamson and Roshko scheme described in §1.2.2 and is a function of
the forcing conditions.
38
4.4.1 Wake modes at forcing frequencies near the natural shedding frequency
When the cylinder was forced at frequencies close to the natural shedding frequency, the
“lock-on” phenomenon was observed. This generally agrees with previous studies, but with
certain key differences.
In Figure 4.2, the lock-on region is marked as “2S lock-on region”. The frequency range
over which lock-on occurs is found to increase with increase in oscillation amplitude but the
increase is not steady. This kind of non-steady increase was also observed by Lee & Lee (2006).
At the lowest amplitude studied, Θ
PP
= 0.17, Mahfouz & Badr (2000a) predict the shedding to be
locked-on only at F
R
= 1.0. Instead, the lock-on region is found to be wider at this amplitude and,
as the amplitude increases further, the lock-on frequency range increases quickly and at Θ
PP
=
0.52 it extends from 0.91 ≤ F
R
≤ 1.29. Further increase in the lock-on frequency range is seen
only above Θ
PP
= 2.09. Between peak-to-peak oscillation amplitudes 2.09 ≤ Θ
PP
≤ 4.19, the
increase in lock-on frequency range agrees closely with the numerical results of Mahfouz & Badr
(2000a). Further increase in Θ
PP
leads to changes in lock-on frequency range that are quite
different from Mahfouz & Badr (2000a).
The exact structure of the wake when locked-on to the forcing can be studied by phase-
averaging the vorticity field. This minimizes cycle-to-cycle variation and noise, thereby
revealing the underlying periodic flow. Figure 4.3 shows the phase-averaged vorticity fields for
the locked-on case, Θ
PP
= 3.14 and F
R
= 1.17. When phase-averaged over one cylinder
oscillation cycle, it is observed that two vortices, one of each sign, are shed per cycle. This is
classified as 2S wake mode. Figure 4.4 shows the spectra measured at a location about 2D
downstream. The only dominant frequency is the forcing frequency. Harmonics of the forcing
frequency are also seen in the spectra but are of low magnitudes. This is an indication that the
39
shedding is locked on to the cylinder forcing. The shed vortices are convected downstream and
dissipate due to viscous effects.
Another key phenomenon observed in this study, which was not seen in previous studies,
is the extension of the lock-on region at large amplitudes (Θ
PP
≥ 4.89) to a higher frequency
range than predicted by previous studies. This extension was seen up to a frequency range of F
R
= 2.5. The reason for this is not clearly understood, but is thought to be caused due to the
relatively compact and strong vortical structures formed at these high oscillation amplitudes
which lead to a stable wake configuration. This wake is similar to that of an unforced cylinder
and is globally stable.
40
Figure 4.3: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 3.14, F
R
= 1.17. One
cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24; minimum contours ±0.5,
contour spacing 0.5, positive contours solid and negative contours dashed
41
Figure 4.4: Spectra of forced shedding case measured at location x/D = 2.04, y/D = 0.37. Forcing conditions
are Θ
PP
= 3.14, F
R
= 1.17
Previous studies have predicted a transition regime on either side of the lock-on region in
the frequency – amplitude plane and a non lock-on region outside the transition regime. While
this is generally true, the boundaries are not uniform. At the sub-harmonic frequency range, a
very sharp transition from lock-on to non locked-on behavior is observed. Mahfouz & Badr
(2000a) observed a distinct transition region between lock-on and non lock-on regions. No such
transition region, usually indicated by the presence of coalescence, is observed in the phase
averaged vorticity fields, though spectral analysis does suggest the presence of such a transition
regime. Phase averaging of vorticity data from the non lock-on region in Figure 4.2 show the
wake to have no synchronization with the cylinder forcing. Figure 4.5 shows the phase averaged
vorticity plots and Figure 4.6 shows the spectra of the wake when Θ
PP
= 0.70, F
R
= 0.71. The
42
spectrum contains complex frequency content, including the forcing and Strouhal frequencies,
which usually indicates a transition regime. But the phase-averaged vorticity fields show no
coherent structures or the presence of coalescence, which implies that the wake is not locked-on.
At higher frequencies (F
R
> 1.29) the non locked-on behavior is found to have strong
amplitude dependence. In this frequency range, below a certain threshold amplitude value, the
flow is always non lock-on, except for a small region around the second harmonic which will be
discussed in a later section. This threshold value (Θ
PP_critical
) lies between 0.70 < Θ
PP_critical
<
1.05.
43
Figure 4.5: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 0.70, F
R
= 0.71. One
cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24; minimum contours ±0.5,
contour spacing 0.5, positive contours solid and negative contours dashed
44
Figure 4.6: Spectra of non lock-on case measured at location x/D = 3.25, y/D = -0.81. Forcing conditions are
Θ
PP
= 0.70, F
R
= 0.71
4.4.2 Wake modes at higher forcing frequencies:
Under oscillation conditions that force the cylinder at relatively high frequencies (F
R
>
~1.3) and amplitudes (Θ
PP
> Θ
PP_critical
), certain interesting phenomena are observed. These new
phenomena are easily identifiable only through phase-averaging and, as a result, previous studies
which utilized spectral analysis alone, could not observe them. This region of the parameter
space was classified as a transition region by Mahfouz & Badr (2000a), Lee & Lee (2006) and
Thiria & Wesfreid (2007). Spectral analyses of the wake in these studies, and in the present
study, indicate a complex frequency content that includes the forcing and Strouhal frequencies
and harmonics. While this kind of spectral content does indicate transition, phase averaging of
45
the vorticity fields over multiple cylinder oscillation cycles reveal new locked-on structures in
the far wake. Coalescence is observed throughout this region. Phase averaged vorticity fields of
these cases, with phase averaging performed over the cylinder oscillation cycle, show the near
wake before coalescence to be synchronized with the cylinder motion with a 2S wake mode, but
not the far wake after coalescence.
Figure 4.7 shows the phase averaged vorticity fields when cylinder is forced at Θ
PP
=
1.05, F
R
= 1.75. This is representative of the wake observed in the region marked as “Near-wake
2S, far wake non locked-on” in Figure 4.2. Over one oscillation cycle two single vortices are
shed. But the wake structure is unstable and they coalesce to form a far wake that is not locked
on to the cylinder forcing. Spectral data of the near and far wake is shown in Figure 4.8. In the
near wake, the dominant frequencies are the forcing and Strouhal frequencies. The far wake
contains much more complex frequency content in which neither the forcing frequency nor the
Strouhal frequency are dominant. This reinforces the result that the far wake is not locked on.
46
Figure 4.7: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 1.05, F
R
= 1.75. One
cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24; minimum contours ±0.5,
contour spacing 0.5, positive contours solid and negative contours dashed
47
Figure 4.8: Spectra measured along y/D = 0.28 at stream wise locations x/D = 0.75 and x/D = 5.62. Forcing
conditions are Θ
PP
= 1.05, F
R
= 1.75
4.4.2.1 ½ (2S) far wake mode
The near wake in the region labeled as “½ (2S) far wake mode” in Figure 4.2 has a 2S
wake mode. But this near wake is unstable and coalesces to form a far wake that appears to be
non locked-on when phase-averaged over a single oscillation cycle. But, vorticity fields phase
averaged over two oscillation cycles reveal a far wake that has a 2S mode. This mode, classified
as ½(2S), is synchronized over two oscillation cycles. Figure 4.9 shows the vorticity fields phase
averaged over two cycles when cylinder forcing conditions are Θ
PP
= 2.44, F
R
= 1.87. Two single
vortices, of same sign, shed from each side of the cylinder over two cycles coalesce to form a
single vortex. This leads to the formation of a 2S mode in the far wake and this wake mode is
48
locked-on over two cycles. Figure 4.10 shows the spectral content in the near and far wake. The
forcing frequency is dominant in the near wake and the Strouhal frequency is dominant in the far
wake. The occurrence of this kind of a wake mode synchronized over two oscillation cycles at a
forcing frequency close to the first harmonic of the Strouhal frequency suggests that the presence
of the harmonic might be influencing the stability of the far wake. A quasi-stable wake structure
whose stability is related to the presence of the first harmonic is achieved instead of the usual
configuration that is similar to the wake of an unforced cylinder.
49
Figure 4.9: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 2.44, F
R
= 1.87.
Two cycles divided into 48 bins, every eighth bin shown, each bin has phase width of π/12; minimum contours
±0.5, contour spacing 0.5, positive contours solid and negative contours dashed
50
Figure 4.10: Spectra of coalescing case exhibiting ½(2S) mode in the far wake. Spectra measured along y/D =
-0.97 at stream wise locations x/D = 1.28 and x/D = 9.31. Forcing conditions are Θ
PP
= 2.44, F
R
= 1.87
4.4.2.2 ½ (P+S) wake mode
The region labeled “½ (P+S) far wake mode” in Figure 4.2 also displays a near wake that
has a 2S wake mode and a far wake mode that is locked-on over two oscillation cycles. Figure
4.11 shows the vorticity fields phase averaged over two cycles when cylinder is forced at Θ
PP
=
1.05, F
R
= 2.00. The forcing frequency in this region is around the first harmonic of the natural
shedding frequency but, the amplitude is lower than the cases studied in the “½ (2S) far wake
mode” region. As previously mentioned, in this region the near wake has a 2S wake mode that is
locked on over a single cycle. The far wake displays a wake mode, classified as ½(P+S), that is
synchronized over two cycles. Two single vortices shed from the same side of the cylinder over
51
two cycles coalesce to form a larger vortex but the corresponding two single vortices shed from
the other side do not coalesce and one of them is pulled across the centerline by the larger vortex,
formed through coalescence, to form a pair and convect downstream. The other single vortex
does not cross the centerline and is convected downstream. Figure 4.12 shows the spectra for this
case measured at various points downstream of the cylinder. The near wake contains the Strouhal
frequency, forcing frequency and harmonics with the forcing frequency dominating. The far
wake contains the Strouhal and forcing frequencies, with the Strouhal frequency being the
dominant one. Thus, spectral analysis suggests that the wake is transitional but phase averaging
of the vorticity fields show the shedding to be locked-on over two cycles.
Again, the presence of the first harmonic is believed to have an influence on the stability
of the wake configuration and eventual lock-on over two cycles. The close proximity of the ½
(2S) and ½ (P+S) far wake mode regions, with one on top of the other on the frequency –
amplitude plane, suggests that the stability of the wake is highly sensitive to the forcing
conditions, especially the amplitude, and even small perturbations in the amplitude can cause it
to pick one mode over the other. A detailed stability analysis is required to better understand the
preferential selection of these wake mode regimes.
52
Figure 4.11: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 1.05, F
R
= 2.00.
Two cycles divided into 48 bins, every sixth bin shown, each bin has phase width of π/12; minimum contours
±1.0, contour spacing 0.5, positive contours solid and negative contours dashed
53
Figure 4.12: Spectra of coalescing case exhibiting ½(P+S) mode in the far wake. Spectra measured along y/D
= 0.09 at stream wise locations x/D = 0.58 and x/D = 5.17. Forcing conditions are Θ
PP
= 1.05, F
R
= 2.00
4.4.2.3 ⅓ (2P+2S) wake mode
“⅓ (2P+2S) far wake mode” region in Figure 4.2 is around the second harmonic of the
natural shedding frequency with an amplitude range that extends lower than Θ
PP_critical
. This
extension of the lock-on region below Θ
PP_critical
can be attributed to the presence of the second
harmonic which allows for an efficient infusion of vorticity into the flow. At the second
harmonic the cylinder completes one oscillation cycle in a third of the natural shedding time
period. This leads to a more efficient infusion of vorticity into the flow than at the first harmonic
where the cylinder completes one oscillation in half the time period of natural shedding and
therefore ends up opposing the tendency of vortices to roll up and shed at the natural shedding
54
frequency. This can also be interpreted as implying that Θ
PP_critical
would normally have a lower
value but the presence of the first harmonic is increasing it to its observed value. Further study of
the vortex roll-up and pinch off process at these forcing conditions will be able to explain this
further.
At high forcing frequencies, the vortices on either side of the cylinder do not have time to
grow and shed as smaller vortices as soon as they form (Lee & Lee, 2008). This leads to two
vortices, one of each sign, shedding each cycle. Thus, the near wake has a 2S wake mode. The
vortex street formed is unstable and coalesces to form a far wake that is locked-on over three
cycles. This can be seen in Figure 4.13, which shows the vorticity fields phase averaged over
three cycles when forcing conditions are Θ
PP
= 1.74, F
R
= 3.08. The far wake has a ⅓ (2P+2S)
wake mode which is the same as 2P+2S wake mode identified by Williamson and Roshko but
over three oscillation cycles.
Figure 4.14 shows the power spectra of the near and far wake for the case exhibiting ⅓
(2P+2S) mode. The forcing frequency is the dominant frequency in the near wake due to the
vortex shedding being locked-on to the forcing. In the far wake, the dominant frequency is the
Strouhal frequency. Thus, the spectra give no indication about the presence of the 2P+2S wake
mode locked-on over three cycles but indicate a transition.
55
Figure 4.13: Phase averaged non-dimensional vorticity fields when cylinder forced at Θ
PP
= 1.74, F
R
= 3.08.
Three cycles divided into 48 bins, every third bin shown, each bin has phase width of π/8; minimum contours
±1.0, contour spacing 0.5, positive contours solid and negative contours dashed
56
Figure 4.14: Spectra of coalescing case exhibiting ⅓ (2P+2S) mode in the far wake. Spectra measured along
y/D = -1.09 at stream wise locations x/D = 1.06 and x/D = 5.28. Forcing conditions are Θ
PP
= 1.74, F
R
= 3.08
4.4.3 Conclusions
The wake of a rotationally oscillating cylinder at Re = 150 was studied using phase
averaging and spectral analysis of DPIV data. Cylinder forcing was varied, with Θ
PP
varying
from 0.17 to 5.58 and F
R
ranging from 0.67 to 3.5. Phase averaging of vorticity fields revealed
locked-on wake modes that were undetectable through spectral analysis alone.
Synchronization of the wake mode with cylinder forcing occurs around the natural
shedding frequency in the parameter space, with the region of lock-on increasing non-uniformly
with oscillation amplitude. For oscillation amplitudes Θ
PP
≥ 4.89 the lock-on region extends
abruptly to F
R
= 2.5. In this lock-on region, the near wake has a 2S wake mode and is stable and
57
does not coalesce to form a far wake. At higher F
R
and above Θ
PP_critical
, a transitional region is
present in which the near wake has a 2S wake mode that is unstable and coalesces to form a far
wake. The far wake is either locked-on or non locked-on to the cylinder forcing. Regions where
the far wake is locked-on occur around the first and second harmonic of the natural shedding
frequency. Near the first harmonic, two regions of far wake mode lock-on occur. In these two
regions, the wake mode is synchronized over twice the cylinder oscillation period. Two different
wake modes, ½(P+S) and ½(2S), occur in these two regions. At the second harmonic, a
⅓(2P+2S) wake mode lock-on region occurs and is synchronized over three times the cylinder
oscillation period. This lock-on region extends below Θ
PP_critical
due to the presence of the second
harmonic. These descriptions of the various wake modes and their dependence on the forcing
conditions lead to a better understanding of the wake of a rotationally oscillating cylinder at Re =
150. The wake mode map also reveals the general location of wake mode boundaries. This
information was used as one of the factors in selecting test cases for the experiments at Re = 750,
which are described in the next chapter.
58
Chapter 5: Effects of cylinder forcing on the structure of the wake at Re = 750
5.1 Introduction
Vortex shedding and wake formation from a rotationally oscillating circular cylinder
placed in a cross-flow is examined in this chapter. The experiments described in this chapter
were all performed at a Reynolds number of 750. Two sets of experiments are performed. In the
first set of experiments, the cylinder is not heated. In the second set of experiments, the heating
coil embedded within the cylinder is supplied with a constant input power, P
in
. The results of
both sets are compared in order to understand the effect of heat transfer on the wake modes. The
experiments described in this chapter are a subset of the second phase of the overall study. The
experimental conditions and methodology are presented in §5.2. The results of the DPIV
measurements are presented in §5.3, and the results are discussed in §5.4.
5.2 Experimental conditions and methodology
The experiments described in this chapter were conducted at Re = 750. These experiments
are a subset of the second phase of the study. The experiments were carried out using the same
setup as described in chapter 2. In the first set of experiments the cylinder was not heated and
only wake structure information was obtained using DPIV. In the second set, the cylinder was
heated and, both heat transfer information and wake information was obtained concurrently. The
details of the DPIV setup used are described in §2.7. In order to study the wake modes, phase
averaging and spectral analysis are used and these are described in §2.9 and §2.10, respectively.
For the first set in which the cylinder was not heated, experiments were performed at 177
different oscillation conditions in the frequency – amplitude plane. The cases that were tested are
shown in Figure 5.1. The results were obtained from 4 runs conducted on different days. During
59
each run, testing was done in chunks of 15 cases, with DPIV seeding particles added at the
beginning of each chunk. Again, this was done to ensure a uniform particle density in all images.
Figure 5.1: Parameter space for experiments with unheated cylinder at Re = 750
During the second set, experiments were performed at 114 distinct oscillation conditions.
The cylinder was provided with a constant power input during all cases in this set. All of these
cases were tested in one run, in chunks of 15 cases. DPIV seeding particles were added between
chunks to ensure uniform particle density in all images. The cases tested in this set are shown in
Figure 5.2. The heat transfer from the cylinder was also measured simultaneously and was
compared with the heat transfer coefficient results given in chapter 3 to ensure consistency of
results. This was done to facilitate an accurate study of the relationship between heat transfer and
wake formation, which will be discussed in the next chapter.
60
Figure 5.2: Parameter space for experiments with heated cylinder at Re = 750
As given in §2.4, f
St
= 1.47 Hz when Re = 750. This limits the possible forcing conditions
that can be tested safely. Due to the higher value of Strouhal frequency, the cylinder oscillation
amplitudes studied extend only up to 120° and oscillation frequencies only range between 1.03
Hz ≤ f
f
≤ 4.65 Hz. This translates to non-dimensional values of 0.09 ≤ Θ
PP
≤ 2.09 and 0.70 ≤ F
R
≤ 3.16.
Similar to experiments at Re = 150, the distribution of test cases shown in figures Figure
5.1 and Figure 5.2 is non-uniform, with cases clustered in certain regions. This distribution is
based on the results of tests performed during the initial setup and testing. Regions of the
parameter space that exhibited new wake modes and regions near wake mode boundaries contain
a high density of test cases.
61
5.3 Results
A map of the different wake mode regions identified as functions of θ
PP
and F
R
is shown
in Figure 5.3. Figure 5.3 shows the results of experiments performed with an unheated cylinder
placed in cross-flow at Re = 750. The map of wake mode regions for a heated cylinder is shown
in Figure 5.4. The different wake mode regions are discussed in §5.4. It is evident from figures
Figure 5.3 and Figure 5.4 that the unheated cylinder and heated cylinder exhibit the same types
of wake modes. Therefore the results from both sets are discussed simultaneously, with
representative examples taken from both.
Figure 5.3: Map of wake mode regions for an unheated cylinder at Re = 750
62
Figure 5.4: Map of wake mode regions at Re = 750 for a heated cylinder
For each sample test case discussed in §5.4, three figures are presented. The first figure
for each sample test case shows the phase averaged non-dimensional vorticity fields. The second
figure shows the mean and rms normalized vorticity (ωD/U). The third figure for the same
sample test case shows the power spectra. All vorticity fields contain a shaded region close to the
cylinder which indicates the region in which no useful information could be obtained due to the
presence of a shadow in the original DPIV images. For the stationary cases only the mean and
rms normalized vorticity fields are shown, in figures Figure 5.5 and Figure 5.6, for the unheated
and heated cylinder, respectively.
63
5.4 Discussion
It was discovered that, in the parameter space studied, heating the cylinder did not lead to
the formation of any new wake modes that were not present in the wake of an unheated cylinder.
But heating the cylinder was found to change the shape of wake mode regions and location of
wake mode boundaries.
For most cases, the wake modes were identified using the phase averaged non-dimensional
vorticity fields. In certain cases, the data from the phase averaged non-dimensional vorticity
fields were found to be ambiguous and therefore the information from the mean and rms
normalized vorticity fields and the power spectra were used to conclusively determine the wake
mode.
Similar to the experiments described in the previous chapter, in the higher Re (Re = 750)
experiments discussed in this chapter, spectral analysis of the wake is used to study its frequency
content. The presence of only the forcing frequency as the dominant frequency in the power
spectrum is an indication of the lock-on phenomenon. Presence of peaks in the power spectrum,
at the forcing frequency and also the natural shedding frequency, is an indication of a transition
regime. The presence of the natural shedding frequency as the only dominant frequency in the
power spectrum indicates that vortex shedding is not locked to the forcing of the cylinder.
Spectral analysis for all cases was performed at two locations, one close to the cylinder and the
other further downstream, in the flow domain. In cases that exhibit coalescence, these locations
were chosen such that one was upstream of the coalescence point and the other one was
downstream. Following the same scheme used in the Re = 150 experiments, the wake before
coalescence is classified as near wake and the wake after coalescence is classified as far wake.
64
The wake mode is also classified based on the same scheme that was used in the lower Re
experiments.
For comparison, and as a baseline case, the wake structure when the cylinder is stationary
is given in figures Figure 5.5 and Figure 5.6. Figure 5.5 shows the mean and rms normalized
vorticity fields for an unheated cylinder. Figure 5.6 shows the mean and rms normalized vorticity
fields for a heated cylinder.
It is apparent that the mean wake structure is qualitatively similar, both with and without
cylinder heating. The only difference is the slightly elongated shear layer seen in the case of the
unheated cylinder. The shorter length of the shear layers seen in the heated cylinder wake are
caused due to the shorter vortex roll-up distance. This can be attributed to the decrease in local
viscosity at the boundary layer due to the addition of heat from the cylinder. This slight change
in viscosity would also affect the onset of the Kelvin-Helmholtz instability that initiates vortex
shedding.
65
Figure 5.5: Mean and rms normalized vorticity for an unheated, stationary cylinder. Contour spacing 0.5,
positive contours are solid and negative contours are dashed
Figure 5.6: Mean and rms normalized vorticity for a heated, stationary cylinder. Contour spacing 0.5,
positive contours are solid and negative contours are dashed
66
5.4.1 Wake modes at forcing frequencies near the natural shedding frequency
When the cylinder was forced at frequencies close to the natural shedding frequency, the
“lock-on” phenomenon occurred as expected. This “lock-on” phenomenon occurred in the wake
of the cylinder both when it was heated and also when it was unheated. But there were a few
differences in the range and shape of this “lock-on” region for the two conditions.
In Figure 5.3 and Figure 5.4, the “lock-on” region is labeled as “2S lock-on region, No
coalescence”. It is apparent that the lock-on region is narrower for the heated cylinder, compared
to the unheated cylinder. In both cases, the frequency range over which lock-on occurs is found
to increase with increase in oscillation amplitude but the increase is not uniform. The unheated
cylinder exhibits a steeper increase, which becomes more pronounced for Θ
PP
> ~1.57 rad. When
the forcing frequency was equal to the Strouhal frequency (F
R
= 1.0), both the unheated and
heated cylinder exhibit lock-on even at the lowest amplitudes that were studied.
Figure 5.7 shows the phase-averaged vorticity fields seen in the wake of an unheated
cylinder for the locked-on case, Θ
PP
= 2.09 and F
R
= 1.00. Figure 5.8 shows the mean and rms
normalized vorticity fields and Figure 5.9 shows the power spectra for the same test case. For a
heated cylinder, the phase-averaged vorticity fields are shown in Figure 5.10 for the locked-on
case, Θ
PP
= 0.87 and F
R
= 1.00. Figure 5.11 shows the mean and rms normalized vorticity fields
and Figure 5.12 shows the power spectra for the same test case. For both cases, the vorticity
fields are phase-averaged over one cylinder oscillation cycle and two vortices, one of each sign,
are shed per cycle. This is classified as 2S wake mode. The power spectra for both cases show
only one dominant frequency, at f
f
, which is also equal to f
St
since F
R
= 1.0. The presence of only
the forcing frequency as the dominant frequency is an indication that the shedding is locked on to
the cylinder forcing. This dominance is seen even far downstream from the cylinder. This is
67
expected since the vorticity fields show strong, coherent vortical structures that are locked-on to
the cylinder forcing even at stream-wise locations more than 8 cylinder diameters downstream.
68
Figure 5.7: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
=
2.09, F
R
= 1.0. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
69
Figure 5.8: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 2.09, F
R
= 1.0.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.9: Spectra measured along y/D = -0.01 at stream wise locations x/D = 1.71 and x/D = 7.77. Cylinder is
unheated. Cylinder forced at θ
PP
= 2.09, F
R
= 1.0
70
Figure 5.10: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
0.87, F
R
= 1.0. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
71
Figure 5.11: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 0.87, F
R
= 1.0.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.12: Spectra measured along y/D = -0.14 at stream wise locations x/D = 1.90 and x/D = 4.25. Cylinder
is heated. Cylinder forced at θ
PP
= 0.87, F
R
= 1.0
72
At the sub-harmonic range, there is a sharp transition from the lock-on regime to the non-
lock-on regime. This is seen in both cases, heated and unheated cylinder. While these kind of
sharp transitions have been observed at lower Re for an unheated cylinder and it is not surprising
to observe it for an unheated cylinder at Re = 750, the fact that it was also observed for a heated
cylinder is somewhat puzzling. Previous studies that looked at similar problems found that in the
case of a heated cylinder the transition regions are mostly smooth, with some test cases on the
boundary even exhibiting a periodic mode switching. A possible reason for the sharp transition
that was observed is that a narrow transition region does indeed exist at the boundary between
wake mode regions, but the test case density in the parameter space is insufficient to detect it. In
the future, increasing the test case density at the wake mode boundaries identified in the current
study would help to understand this discrepancy.
The non-lock-on region, labeled as “non lock-on region” in the wake mode maps given in
figures Figure 5.3 and Figure 5.4, extends on both sides of the lock-on region. In the sub-
harmonic range, it extends to higher amplitudes as F
R
decreases. For F
R
> 1.0, for both the
unheated and heated cylinder, the non-lock-on regions exhibit similar trends, with some minor
differences. In both cases, the maximum amplitude up to which it extends is approximately
around Θ
PP
= 0.75. Near the harmonics the extent of the non-lock-on region is much lower. This
is different from the wake mode boundaries seen at Re = 150, where, below a certain threshold
amplitude value, the flow is always non-lock-on except for a small region around the second
harmonic. At Re = 750, there is some locked-on wake mode near both the first and second
harmonic.
Figure 5.13 shows the phase-averaged vorticity fields seen in the wake of an unheated
cylinder for the non-locked-on case, Θ
PP
= 0.17 and F
R
= 1.87. Figure 5.14 shows the mean and
73
rms normalized vorticity fields and Figure 5.9 shows the power spectra for the same test case.
Figure 5.16 shows the phase-averaged vorticity fields, again of an unheated cylinder, for the non-
locked-on case, Θ
PP
= 0.70 and F
R
= 2.89. Figure 5.17 shows the mean and rms normalized
vorticity fields and Figure 5.18 shows the power spectra for the same test case. For a heated
cylinder, the phase-averaged vorticity fields are shown in Figure 5.19 for the non-locked-on case,
Θ
PP
= 0.17 and F
R
= 1.87. Figure 5.20 shows the mean and rms normalized vorticity fields and
Figure 5.12 shows the power spectra for the same test case. For all three cases, the vorticity
fields are phase-averaged over one cylinder oscillation cycle. The phase averaged vorticity plots
show no locked-on structures in the wake. Close to the cylinder, the dominant peak in the power
spectra for all three cases is at f
St
. At locations further downstream, the power spectra has
complex frequency content consisting of f
St
, harmonics and other unrelated frequencies.
74
Figure 5.13: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
= 0.17, F
R
= 1.87. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
75
Figure 5.14: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 0.17, F
R
=
1.87. Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.15: Spectra measured along y/D = -0.02 at stream wise locations x/D = 1.32 and x/D = 4.83. Cylinder
is unheated. Cylinder forced at θ
PP
= 0.17, F
R
= 1.87
76
Figure 5.16: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
= 0.70, F
R
= 2.89. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
77
Figure 5.17: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 0.70, F
R
=
2.89. Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.18: Spectra measured along y/D = -0.46 at stream wise locations x/D = 1.71 and x/D = 6.26. Cylinder
is unheated. Cylinder forced at θ
PP
= 0.70, F
R
= 2.89
78
Figure 5.19: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
0.17, F
R
= 1.87. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
79
Figure 5.20: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 0.17, F
R
= 1.87.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.21: Spectra measured along y/D = -0.14 at stream wise locations x/D = 1.08 and x/D = 4.60. Cylinder
is heated. Cylinder forced at θ
PP
= 0.17, F
R
= 1.87
80
5.4.2 Wake modes at higher forcing frequencies
At higher forcing frequencies, similar to the lower Re results, certain interesting
phenomena were observed. Again, these are easily identifiable only through phase-averaging.
Coalescence is observed in certain areas, but not all. This is quite different from the results at Re
= 150, wherein coalescence was observed everywhere in this region. At Re = 750, coalescence is
observed only at certain locations. But when it did occur, it was similar to the coalescence
observed at Re = 150, and is defined in the same way. The definitions of near and far wakes
developed in §4.4 still hold.
The region labeled “Near wake 2S, far wake non locked-on” in the wake mode maps,
given in figures Figure 5.3 and Figure 5.4, has the largest extent. Phase averaging of cases in this
region reveals that over one oscillation cycle two single vortices are shed and the vortices
coalesce to form a far wake that is not locked on to the cylinder forcing. Figure 5.22 shows the
phase-averaged vorticity fields seen in the wake of an unheated cylinder for cylinder forcing, Θ
PP
= 1.40 and F
R
= 1.54. Figure 5.23 shows the mean and rms normalized vorticity fields and Figure
5.24 shows the power spectra for the same test case. Figure 5.25 shows the phase-averaged
vorticity fields of a heated cylinder for cylinder forcing, Θ
PP
= 1.40 and F
R
= 1.54. Figure 5.26
shows the mean and rms normalized vorticity fields and Figure 5.27 shows the power spectra for
the same test case. Figure 5.28 shows, again for a heated cylinder, the phase-averaged vorticity
fields for the case, Θ
PP
= 1.40 and F
R
= 2.72. Figure 5.29 shows the mean and rms normalized
vorticity fields and Figure 5.30 shows the power spectra for the same test case. For all three
cases, the vorticity fields are phase-averaged over one cylinder oscillation cycle.
Spectral data for all three cases show the dominant frequency in the near wake to be the
forcing frequency. This implies that the vortex shedding is locked-on to the cylinder forcing. But
81
downstream of the coalescence location, in the far wake, the spectral data is more complex with
peaks at f
St
and other frequencies. This indicates that the wake is not locked-on to the cylinder
forcing at this point.
82
Figure 5.22: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
= 1.40, F
R
= 1.54. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
83
Figure 5.23: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 1.40, F
R
=
1.54. Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.24: Spectra measured along y/D = -0.02 at stream wise locations x/D = 1.32 and x/D = 4.83. Cylinder
is unheated. Cylinder forced at θ
PP
= 1.40, F
R
= 1.54
84
Figure 5.25: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
1.40, F
R
= 1.54. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
85
Figure 5.26: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 1.40, F
R
= 1.54.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.27: Spectra measured along y/D = 0.33 at stream wise locations x/D = 0.97 and x/D = 4.60. Cylinder
is heated. Cylinder forced at θ
PP
= 1.40, F
R
= 1.54
86
Figure 5.28: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
1.40, F
R
= 2.72. One cycle divided into 48 bins, every eighth bin shown, each bin has phase width of π/24;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
87
Figure 5.29: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 1.40, F
R
= 2.72.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.30: Spectra measured along y/D = -0.26 at stream wise locations x/D = 1.79 and x/D = 4.13. Cylinder
is heated. Cylinder forced at θ
PP
= 1.40, F
R
= 2.72
88
5.4.2.1 ½ (2S) far wake mode
This region is labeled as “½ (2S) far wake mode” in the wake mode maps given in figures
Figure 5.3 and Figure 5.4. The near wake in this region has a 2S wake mode, which is unstable
and coalesces to form a far wake that has a 2S wake mode over two oscillation cycles. This can
be readily seen in the phase averaged vorticity plots, with phase averaging performed over two
cycles.
Figure 5.31 shows the phase-averaged vorticity fields seen in the wake of an unheated
cylinder when Θ
PP
= 1.05 and F
R
= 2.12. Figure 5.32 shows the mean and rms normalized
vorticity fields and Figure 5.33 shows the power spectra for the same test case. Figure 5.34
shows the phase-averaged vorticity fields of a heated cylinder for the forcing conditions, Θ
PP
=
1.05 and F
R
= 2.12. Figure 5.35 shows the mean and rms normalized vorticity fields and Figure
5.27 shows the power spectra for the same test case. For both cases, the vorticity fields are
phase-averaged over two oscillation cycles.
The spectral data for both cases shows that, in the near wake, the dominant frequency is
the forcing frequency. This agrees with the phase-averaged vorticity plots which show that 4
single vortices, two of each sense, are shed over two cycles. In the far wake, the spectral data
contains a dominant peak at a frequency that is half of the forcing frequency. This agrees with
the information from the phase averaged vorticity plots which show a 2S mode locked-on over
two cycles. While both sample test cases shown here are at identical forcing conditions, there are
minor differences in the mean and rms normalized vorticity plots for the two cases. The mean
vorticity plot clearly shows that the near wake of the unheated cylinder is longer than that of the
heated cylinder. The shorter near wake of the heated cylinder is due to the decrease in viscosity
in the boundary layer caused by heat addition.
89
Figure 5.31: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
= 1.05, F
R
= 2.12. Two cycles divided into 48 bins, every sixth bin shown, each bin has phase width of π/12;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
90
Figure 5.32: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 1.05, F
R
=
2.12. Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.33: Spectra measured along y/D = -1.08 at stream wise locations x/D = 1.32 and x/D = 4.60. Cylinder
is unheated. Cylinder forced at θ
PP
= 1.05, F
R
= 2.12
91
Figure 5.34: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
1.05, F
R
= 2.12. Two cycles divided into 48 bins, every sixth bin shown, each bin has phase width of π/12;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
92
Figure 5.35: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 1.05, F
R
= 2.12.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.36: Spectra measured along y/D = 0.45 at stream wise locations x/D = 1.08 and x/D = 4.60. Cylinder
is heated. Cylinder forced at θ
PP
= 1.05, F
R
= 2.12
93
5.4.2.2 ½ (P+S) far wake mode
This region is labeled as “½ (P+S) far wake mode” in the wake mode maps given in
figures Figure 5.3 and Figure 5.4. The near wake in this region has a 2S wake mode, which is
unstable and coalesces to form a far wake that has a P+S wake mode that is locked-on over two
oscillation cycles. This can be seen only in the phase averaged vorticity plots, with phase
averaging performed over two cycles.
Figure 5.37 shows the phase-averaged vorticity fields seen in the wake of an unheated
cylinder when Θ
PP
= 1.40 and F
R
= 2.12. Figure 5.38 shows the mean and rms normalized
vorticity fields and Figure 5.39 shows the power spectra for the same test case. Figure 5.40
shows the phase-averaged vorticity fields of a heated cylinder for the forcing conditions, Θ
PP
=
1.40 and F
R
= 2.12. Figure 5.41 shows the mean and rms normalized vorticity fields and Figure
5.42 shows the power spectra for the same test case. For both cases, the vorticity fields are
phase-averaged over two oscillation cycles.
Similar to the “½ (2S) far wake mode” region, the spectral data for both cases shown here
indicate that in the near wake, the dominant frequency is the forcing frequency. This agrees with
the phase-averaged vorticity plots which show that 4 single vortices, two of each sense, are shed
over two cycles. In the far wake, the spectral data contains a dominant peak at a frequency that is
half of the forcing frequency. This suggests that the far wake is again 2S over two cycles. But the
phase-averaged vorticity plots show that only counter-clockwise rotating vortices coalesce
together to form a single vortex while the clockwise rotating vortices, whose contours are
indicated by dashed lines, do not merge together completely. Instead, as they start to merge, the
leading vortex is pulled away by the single vortex of the opposite sense formed by coalescence
during the previous cycle, to form a pair (P) of counter-rotating vortices.
94
It was also observed that for a dis-proportionate number of cases in this region, the single
vortex (S) in the P+S mode was formed by the clockwise rotating vortex, as described above.
This trend was reversed, when the cylinder oscillations were started in the opposite direction.
This suggests that the wake formation is extremely sensitive to the initial conditions. Further
experiments, with the measurement domain enlarged to include the entire cylinder and with
higher temporal resolution, are required to study this.
95
Figure 5.37: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
= 1.40, F
R
= 2.12. Two cycles divided into 48 bins, every sixth bin shown, each bin has phase width of π/12;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
96
Figure 5.38: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 1.40, F
R
=
2.12. Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.39: Spectra measured along y/D = -1.08 at stream wise locations x/D = 1.32 and x/D = 4.83. Cylinder
is unheated. Cylinder forced at θ
PP
= 1.40, F
R
= 2.12
97
Figure 5.40: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
1.40, F
R
= 2.12. Two cycles divided into 48 bins, every sixth bin shown, each bin has phase width of π/12;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
98
Figure 5.41: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 1.40, F
R
= 2.12.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.42: Spectra measured along y/D = -0.14 at stream wise locations x/D = 1.08 and x/D = 4.60. Cylinder
is heated. Cylinder forced at θ
PP
= 1.40, F
R
= 2.12
99
5.4.2.3 ½ (P+S) wake mode; no coalescence
This region is labeled as “½ (P+S); No coalescence” in the wake mode maps given in
figures Figure 5.3 and Figure 5.4. No coalescence was observed in this region. Phase averaging
vorticity fields over a single cycle showed a non-locked-on wake. But phase averaged vorticity
plots, with phase averaging performed over two cycles, show a P+S locked-on wake mode.
Figure 5.43 shows the phase-averaged vorticity fields seen in the wake of an unheated
cylinder when Θ
PP
= 0.87 and F
R
= 1.87. Figure 5.44 shows the mean and rms normalized
vorticity fields and Figure 5.45 shows the power spectra for the same test case. Figure 5.46
shows the phase-averaged vorticity fields of a heated cylinder for the forcing conditions, Θ
PP
=
0.87 and F
R
= 1.87. Figure 5.47 shows the mean and rms normalized vorticity fields and Figure
5.48 shows the power spectra for the same test case. For both cases, the vorticity fields are
phase-averaged over two oscillation cycles.
Spectral data for both cases show frequency peaks at the forcing frequency and at half the
forcing frequency. This indicates the occurrence of locked-on structures at twice the oscillation
period. Phase averaged vorticity plots show that over two cycles three vortices are shed. Two are
of the same rotational sense and the other is of the opposite sense. The leading vortex, among the
two single vortices of the same sense, is pulled away by the third vortex to form a pair. The
unheated cylinder wake and the heated cylinder wake are similar in this region, with asymmetric
shedding in one direction preferred over the other in both cases. This is also apparent in the rms
normalized vorticity plots for both cases. This implies that this wake mode is also highly
sensitive to initial conditions and further studies are required to fully understand the wake
formation in this region.
100
Figure 5.43: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
= 0.87, F
R
= 1.87. Two cycles divided into 48 bins, every sixth bin shown, each bin has phase width of π/12;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
101
Figure 5.44: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 0.87, F
R
=
1.87. Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.45: Spectra measured along y/D = -0.46 at stream wise locations x/D = 1.71 and x/D = 6.26. Cylinder
is unheated. Cylinder forced at θ
PP
= 0.87, F
R
= 1.87
102
Figure 5.46: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
0.87, F
R
= 1.87. Two cycles divided into 48 bins, every sixth bin shown, each bin has phase width of π/12;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
103
Figure 5.47: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 0.87, F
R
= 1.87.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.48: Spectra measured along y/D = -0.37 at stream wise locations x/D = 1.32 and x/D = 3.66. Cylinder
is heated. Cylinder forced at θ
PP
= 0.87, F
R
= 1.87
104
5.4.2.4 ⅓ (P+S) wake mode; no coalescence
This region is labeled as “⅓ (P+S); No coalescence” in the wake mode maps given in
figures Figure 5.3 and Figure 5.4. No coalescence was observed in this region. Phase-averaging
over three oscillation cycles shows a P+S wake mode that is locked-on to the cylinder forcing.
Figure 5.49 shows the phase-averaged vorticity fields seen in the wake of an unheated
cylinder when Θ
PP
= 0.52 and F
R
= 3.16. Figure 5.50 shows the mean and rms normalized
vorticity fields and Figure 5.51 shows the power spectra for the same test case. Figure 5.52
shows the phase-averaged vorticity fields of a heated cylinder for the forcing conditions, Θ
PP
=
0.52 and F
R
= 3.03. Figure 5.53 shows the mean and rms normalized vorticity fields and Figure
5.54 shows the power spectra for the same test case. For both cases, the vorticity fields are
phase-averaged over three oscillation cycles.
The wake of the unheated cylinder and the wake of the heated cylinder look similar in
this region of the parameter space. When phase averaging is performed over one or two cycles,
the wake structure looks similar to a non-locked-on wake. But when phase-averaged over three
cycles, the vorticity fields show that three vortices are shed over three oscillation cycles. The
spectral data for both cases show peaks at f
f
, ½ f
f
, and ⅓f
f
. The peak at a third of the forcing
frequency is the most dominant one. This indicates the occurrence of wake structures
synchronized over three times the oscillation period. The mean and rms normalized vorticity
plots indicate the trajectory of the vortices clearly.
105
Figure 5.49: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
= 0.52, F
R
= 3.16. Three cycles divided into 48 bins, every fourth bin shown, each bin has phase width of π/8;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
106
Figure 5.50: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 0.52, F
R
=
3.16. Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.51: Spectra measured along y/D = -0.02 at stream wise locations x/D = 1.55 and x/D = 3.89. Cylinder
is unheated. Cylinder forced at θ
PP
= 0.52, F
R
= 3.16
107
Figure 5.52: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
0.52, F
R
= 3.03. Three cycles divided into 48 bins, every fourth bin shown, each bin has phase width of π/8;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
108
Figure 5.53: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 0.52, F
R
= 3.03.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.54: Spectra measured along y/D = -0.14 at stream wise locations x/D = 1.08 and x/D = 4.60. Cylinder
is heated. Cylinder forced at θ
PP
= 0.52, F
R
= 3.03
109
5.4.2.5 ⅓ (2P) wake mode; no coalescence
This region is labeled as “⅓ (2P); No coalescence” in the wake mode maps given in
figures Figure 5.3 and Figure 5.4. No coalescence was observed in this region. Phase-averaging
vorticity fields over one and two oscillation cycles show a non-locked-on wake. But phase-
averaging over three oscillation cycles shows a 2P wake mode that is locked-on to the cylinder
forcing.
Figure 5.55 shows the phase-averaged vorticity fields seen in the wake of an unheated
cylinder when Θ
PP
= 0.70 and F
R
= 3.06. Figure 5.56 shows the mean and rms normalized
vorticity fields and Figure 5.57 shows the power spectra for the same test case. Figure 5.58
shows the phase-averaged vorticity fields of a heated cylinder for the forcing conditions, Θ
PP
=
0.70 and F
R
= 2.89. Figure 5.59 shows the mean and rms normalized vorticity fields and Figure
5.60 shows the power spectra for the same test case. For both cases, the vorticity fields are
phase-averaged over three oscillation cycles.
The phase-averaged vorticity plots clearly show that four vortices are shed over three
cycles. Each vortex pairs up with a vortex of the opposite sense to form a pair. Therefore, two
pairs are formed over three cycles. This is true for both the unheated cylinder wake and the
heated cylinder wake. The spectral data for both cases show peaks at f
f
, ½ f
f
, and ⅓f
f
. The
dominant peaks in both cases are at a third of the forcing frequency and at half the forcing
frequency. This is indicative of a 2P mode synchronized over three cycles.
This region is relatively small and occurs at very low oscillation amplitude in the
parameter space. Especially for the unheated cylinder, this region is extremely narrow. In both
cases, it is centered on the second harmonic. Therefore, the presence of the harmonic might be a
reason for the occurrence of this stable, locked-on wake mode.
110
Figure 5.55: Phase averaged non-dimensional vorticity fields for an unheated cylinder. Cylinder forced at θ
PP
= 0.70, F
R
= 3.06. Three cycles divided into 48 bins, every fourth bin shown, each bin has phase width of π/8;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
111
Figure 5.56: Mean and rms normalized vorticity for unheated cylinder. Cylinder forced at θ
PP
= 0.70, F
R
=
3.06. Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.57: Spectra measured along y/D = -0.31 at stream wise locations x/D = 1.86 and x/D = 5.35. Cylinder
is unheated. Cylinder forced at θ
PP
= 0.70, F
R
= 3.06
112
Figure 5.58: Phase averaged non-dimensional vorticity fields for a heated cylinder. Cylinder forced at θ
PP
=
0.70, F
R
= 2.89. Three cycles divided into 48 bins, every fourth bin shown, each bin has phase width of π/8;
minimum contours ±0.5, contour spacing 0.5, positive contours are solid and negative contours are dashed
113
Figure 5.59: Mean and rms normalized vorticity for heated cylinder. Cylinder forced at θ
PP
= 0.70, F
R
= 2.89.
Contour spacing 0.5, positive contours are solid and negative contours are dashed
Figure 5.60: Spectra measured along y/D = -0.37 at stream wise locations x/D = 1.08 and x/D = 2.96. Cylinder
is heated. Cylinder forced at θ
PP
= 0.70, F
R
= 2.89
114
5.5 Conclusions
The wake of a rotationally oscillating cylinder at Re = 750 was studied by analyzing
phase-averaged normalized vorticity fields, mean and rms normalized vorticity fields, and power
spectral data. Two sets of experiments were performed. The first set of experiments was
conducted with the cylinder unheated. During the second set of experiments the cylinder was
heated. Results from both sets of experiments show that similar wake modes are formed in both
cases, but the boundaries of the wake mode regions differ.
Wake modes synchronized to the cylinder forcing were discovered in certain areas of the
parameter space. In the region centered about the Strouhal frequency, F
R
= 1.0, the locked-on 2S
wake mode was discovered. This is similar to results of experiments performed at Re = 150. But
the extent of this 2S wake mode region was smaller for both the heated and unheated cylinder at
the higher Re. The non-locked-on region, where the vortex shedding was not synchronized to the
cylinder forcing, was observed at low amplitudes and in the sub-harmonic frequency range.
At higher forcing frequencies, F
R
> ~ 1.40, new wake modes were discovered. These
occur around the first and second harmonics. The “½ (P+S) far wake mode” and the “½ (2S) far
wake mode”, as their names imply, occur only in the far wake, with the near wake in those cases
exhibiting the 2S wake mode. There were also cases with wakes which exhibit no coalescence,
but still contain locked-on wake modes. Near the first harmonic, a ½ (P+S) wake mode, without
any coalescence, was observed. Near the second harmonic, ⅓ (P+S) and ⅓ (2P) wake modes
were discovered.
The boundaries between the ½ (P+S) far wake mode region, ½ (2S) far wake mode
region, and the ½ (P+S) wake mode region are extremely sensitive to any perturbations.
Compared to the unheated cylinder, a significant change in the shape and location of these
115
boundaries relative to one another occurs with the heat addition. This is also observed to a lesser
extent at the boundary between the ⅓ (P+S) and ⅓ (2P) wake modes. A detailed stability
analysis is required to fully understand these changes. There were other changes caused by the
addition of heat from the cylinder to the wake. The relationship between heat transfer and wake
modes will be explored further in the next chapter.
116
Chapter 6: Relationship between wake formation and heat transfer
6.1 Introduction
The relationship between wake modes and heat transfer from a rotationally oscillating
circular cylinder placed in cross-flow is analyzed in this chapter. It is based on data from
experiments performed at Re = 750 which are described in chapters 3 and 5. Different aspects of
the relationship between wake formation and heat transfer are analyzed. The effect of wake
modes on heat transfer enhancement is discussed in §6.2. The relationship between the cylinder
surface velocity, heat transfer rate, and wake modes is discussed in §6.3. In §6.4, the relationship
between the formation length and heat transfer is described.
6.2 Effect of wake mode synchronization on heat transfer enhancement
The occurrence of heat transfer enhancement, or lack thereof, in certain regions of the
parameter space suggests that cylinder forcing may be affecting it. It was previously shown in
chapters 4 and 5 that wake formation also depends on the cylinder forcing conditions. So it is
possible that wake formation and heat transfer are closely related. Figure 6.1 shows the wake
mode boundaries of a heated cylinder superimposed on the contour plot of normalized heat
transfer. It is immediately apparent that certain wake modes correlate to regions with significant
heat transfer enhancement.
There is a region of heat transfer enhancement around the Strouhal frequency. This is
expected since the vortex shedding seen in this region is highly synchronized with the cylinder
forcing leading to the formation of a locked-on 2S wake mode. There is also significant heat
transfer enhancement around the first and second harmonic and this correlates with locked-on
wake modes. Near the second harmonic, the ⅓ (P+S) and ⅓ (2P) wake modes are of particular
117
interest since they occur at very low oscillation amplitudes. This region would not normally be
expected to have a high heat transfer rate, so the occurrence of the locked-on wake modes is
assumed to be the reason for the observed enhancement. The locked-on wake modes are
generally effective at removing fluid away from the cylinder and this leads to the relatively
cooler fluid coming in contact with the cylinder. This leads to a higher than normal temperature
difference between the surface of the cylinder and the surrounding fluid, leading to an increase in
the heat transfer rate.
Around the second harmonic the correlation between wake modes and heat transfer
enhancement is not straightforward. While the “½(P+S); No coalescence” wake mode region
displays heat transfer enhancement, the ½ (2S) far wake mode region does so to a lesser extent
and the ½ (P+S) far wake mode region displays an abnormally low heat transfer enhancement for
its high amplitude. The reason could be because these two modes have locked-on structures only
in the far wake, which is located far downstream from the cylinder and is therefore not as
effective at removing fluid from the dead-water region adjacent to the cylinder. As previously
described, the 2S mode near the Strouhal frequency is efficient at removing fluid from near the
cylinder and increases the heat transfer rate. The ½ (2S) far wake mode region and the ½ (P+S)
far wake mode region both possess a locked-on 2S mode in the near wake. The reason for the 2S
wake mode in this region not being as effective at increasing the heat transfer rate is not clearly
understood. One possible reason could be that while the near wake exhibits what looks like a 2S
mode, it is unstable and not completely locked-on. It is also possible that the wake structure
immediately downstream of the cylinder is very different from a classic 2S mode, but no
information is available regarding the wake structure in this region since it is not imaged due to
limitations in the experimental setup.
118
Figure 6.1: Contours of normalized heat transfer (Nu/Nu
0
) at Re = 750 with wake mode boundaries
superimposed. Thick, solid black lines are wake mode boundaries. Wake mode regions: (a) 2S lock-on region,
(b) non lock-on region, (c) Near-wake 2S, far wake non locked-on, (d) ½(2S) far wake mode, (e) ½(P+S) far
wake mode, (f) ½(P+S); No coalescence, (g) ⅓(P+S); No coalescence, (h) ⅓(2P); No coalescence
119
6.3 Tangential velocity
Pottebaum (2003) found that in the case of transversely oscillating cylinders, the transverse
velocity is related to heat transfer enhancement in certain regions of the parameter space. So it is
possible that a similar relationship may exist in the case of rotationally oscillating cylinders too.
In order to verify this, the peak tangential velocity on the surface of the cylinder was calculated
for each oscillation condition and, since the cylinder oscillation is sinusoidal, the rms tangential
velocity, which is equal to 1/√2 times the peak tangential velocity, was calculated. The
normalized heat transfer rate (Nu / Nu
0
) is plotted versus the normalized rms tangential velocity
(V
rms
/ U
o
) in Figure 6.2. The wake mode of each case was also identified.
The locked-on modes exhibit significant heat transfer enhancement as expected and also
exhibit significant scatter within this region. But for V
rms
/U
0
greater than unity the data partially
collapses. Almost all of the cases from the “Near wake 2S, far wake non locked-on” region
follow this collapsed path. The implication of this result is, similar to that found by Pottebaum,
that away from locked-on wake mode regions the tangential velocity on the cylinder surface is an
important factor in deciding the heat transfer rate.
120
Figure 6.2: Normalized heat transfer rate (Nu/Nu
0
) vs. normalized rms tangential velocity (V
rms
/U
0
) at Re =
750 with individual wake modes identified. Data derived from same data set as Figure 6.1
6.4 Relationship between heat transfer and formation length at F
R
= 1.0
Previous studies have shown that formation length, or vortex roll-up distance, is known to
affect heat transfer. Shorter formation lengths result in more efficient removal of fluid from the
dead water region, thereby enhancing heat transfer. It is possible that such a relationship between
heat transfer and formation length exists in the case of vortex shedding from a rotationally
oscillating cylinder too. One of the methods for determining the formation length, given in
Pottebaum (2003), is to locate the peak value in the rms vorticity field and measuring the stream-
wise distance from that point to the cylinder base. This method was found to work well only for
cases with a 2S wake mode. Since the wake mode is 2S at all oscillation amplitudes when F
R
=
121
1.0, the method is applied only to cases at this frequency ratio. This allows for better
understanding the effect of cylinder forcing on formation length and heat transfer enhancement
in cases with the relatively simple 2S wake mode.
Figure 6.3 shows the formation length normalized by the cylinder diameter as a function of
the non-dimensional oscillation amplitude. Curves for three different conditions are shown,
unheated cylinder at Re = 150, unheated cylinder at Re = 750, heated cylinder at Re = 750.
Curves are formed by straight line connectors connecting consecutive data points on a scatter
plot. Vertical error bars correspond to the DPIV grid spacing used in each data set. Data from
experiments performed with unheated cylinders are provided for comparison with the formation
lengths seen in the heated cylinder experiments.
For vortex shedding from a heated cylinder, the formation length is found to initially
decrease with increase in oscillation amplitude up to approximately θ
PP
= 0.70. Further increase
in amplitude does not correspond to a decrease in formation length, but instead it is found to
stagnate. This is consistent with the trend seen in heat transfer enhancement at this frequency
ratio wherein it initially increases with θ
PP
but then stays approximately constant for θ
PP
> 0.70.
The formation length for an unheated cylinder at the same Re is found to follow the same
trend initially. But at higher amplitudes the trend reverses and the formation length is found to
increase again for θ
PP
> 1.22. It is quite unusual and the reason behind this is not known. A
similar trend reversal can also be seen in the data for an unheated cylinder at Re = 150, although
it is less pronounced and occurs at a much higher oscillation amplitude.
Since there is no heat addition in both these cases, the effect of viscosity changes in the
boundary layer can be ruled out as a cause. It is possible that it is caused by a phenomenon
entirely unrelated to heat transfer. If that is true, then it is possible that the trend reversal also
122
occurs for the heated cylinder, but there is no data available at higher oscillation amplitudes to
conclusively prove it.
Figure 6.3: Formation length as a function of non-dimensional oscillation amplitude ( θ
PP
) at frequency ratio
F
R
= 1.0. Vertical error bars are from DPIV grid spacing resolution
6.5 Conclusion
Three different aspects of the relationship between heat transfer and wake formation were
analyzed in this chapter. First, the correlation of wake mode boundaries to boundaries in the
contours of heat transfer enhancement was analyzed. It is shown that certain locked-on wake
modes correspond to heat transfer enhancement. But the location of wake mode boundaries need
123
to be further refined to exactly pinpoint mode transitions. This requires a higher density of test
cases in the parameter space.
Second, the effect of the cylinder circumferential velocity, or surface tangential velocity,
on the heat transfer rate was analyzed. It is clear that for cases that are not locked-on the
tangential velocity plays an important role in determining the heat transfer rate, especially for
cases where V
rms
/U
0
> 1.0. At the other end of the scale, even for low values of tangential
velocity, wake mode synchronization lead to significant heat transfer enhancement.
The third aspect that was analyzed is the relationship between the formation length and
heat transfer. The formation length was found to affect the heat transfer rate as expected. It was
also found to decrease with increase in oscillation amplitude. Further experiments at higher
oscillation amplitudes are required to confirm if the formation length for a heated cylinder starts
increasing again with increasing amplitude. If this is true, and since shorter formation lengths
correspond to higher heat transfer rates, it is possible that the heat transfer enhancement region
seen at F
R
= 1.0 does not extend indefinitely.
124
Chapter 7: Conclusions
The goals of this study were to understand the effects of rotary oscillations on wake
formation and its effect on heat transfer from the cylinder. These goals were accomplished
through a two-step process.
In the first step, the heat transfer rates at different cylinder forcing conditions were
determined. This was accomplished by conducting a parameter space survey with a relatively
high density of test cases. The results of this parameter space survey allowed for construction of
a map of the heat transfer enhancement regions. This information was also used to select cases
for further study using DPIV.
Certain regions of the parameter space were found to have significantly higher heat transfer
rates compared to the heat transfer rate of a stationary cylinder. The regions of heat transfer
enhancement were found to correspond to the Strouhal frequency and its harmonics.
In the second step of this study, two sets of experiments were performed. The first set of
experiments was performed at Re = 150 with an unheated cylinder and the second set of
experiments was performed at Re = 750 with both unheated and heated cylinders. In both sets
DPIV was used to obtain the velocity and vorticity fields. Phase averaging of vorticity fields
allowed for identification of the wake structure and information about the frequency content of
the wake was obtained through spectral analysis. Several new locked-on wake modes were
discovered including wake modes that were synchronized over multiple cylinder oscillation
cycles.
The regions of the parameter space in which these wake modes occur were mapped and
wake mode boundaries were identified. Previous studies, which used only spectral analysis, were
unable to identify the exact wake structures and the regions in the parameter space where they
125
occur. That limitation was overcome in this study by the use of phase averaging of vorticity
fields which revealed wake structures that would have otherwise been undetectable.
The wake modes map shows that there is clear correlation between wake modes and
cylinder forcing. It was discovered that locked-on wake modes that occur near the first harmonic
were synchronized over a period equal to twice the cylinder oscillation period and locked-on
wake modes that occur near the second harmonic were synchronized over a period equal to three
times the cylinder oscillation period.
In the second step, since the experiments at Re = 750 were performed with both an unheated
cylinder and a heated cylinder it was possible to study how the heat transfer affected the wake
structure. The various wake modes were identified and the wake mode boundaries identified. It
was found that inclusion of heat transfer in the wake formation process did not lead to any new
wake modes. It did lead to changes in the sizes and shapes of the wake mode boundaries, but the
general location of these wake mode regions in the parameter space remained more or less
constant. This suggests that heat transfer may play a more subtle role than expected. Further
experiments with a higher test grid density, especially near the existing wake mode boundaries,
are needed to fully capture the effect of heat transfer on the location of wake mode boundaries.
The wake mode boundaries were also compared with the contours of normalized heat
transfer. Regions of heat transfer enhancement were found to correspond to locked-on wake
mode regions. Some of these regions would have normally been expected to have relatively low
heat transfer rates, but the presence of these locked-on wake modes resulted in a higher-than-
expected heat transfer rate. This could be useful in designing and operating highly efficient heat
exchangers with rotationally oscillating elements.
126
The effect of the surface tangential velocity on heat transfer and wake modes was also
studied. By comparing the heat transfer rate for each forcing condition with the normalized rms
tangential velocity it was once again shown that locked-on wake modes correspond to
significantly higher heat transfer rates. Outside of regions with locked-on wake modes the
surface tangential velocity was found to be the primary factor that determined the heat transfer
rate for cases with an rms tangential velocity greater than the incoming free-stream velocity.
The relationship between formation length, heat transfer, and oscillation amplitude was
analyzed for cases with F
R
= 1.0. The formation length for an unheated cylinder subjected to an
equivalent set of forcing conditions at Re = 150 and 750 was also measured and compared to the
heated cylinder results. The results from the unheated cylinder reveal an unusual trend in
formation length. Previously it was thought that formation length always decreased with increase
in amplitude. But beyond certain amplitude the trend was found to be reversed. The reason for
this is not known, and it is not known if this trend reversal occurs for a heated cylinder too.
Further experiments with a heated cylinder at higher amplitudes are required to confirm it.
Additional work with a higher density of test cases is required in order to refine the wake
mode boundaries. The range of the parameter space should also be increased in order to explain
some of the unresolved issues discussed previously. It would also be useful to design a setup
capable of measuring the flow field adjacent to the cylinder and a temperature measurement
system that is not as noisy and imprecise as the present one. The DPIT method, despite the lack
of success in the present study, could be very useful in studying the distribution of heat inside the
wake and should be considered for future research in this topic.
127
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Abstract (if available)
Abstract
Wake formation is an important problem in engineering due to its effect on phenomena such as vortex induced vibrations and heat transfer. While prior work has focused on the wake formation due to vortex shedding from stationary, stream-wise, and transversely oscillating cylinders, limited information is available on the effect of rotary oscillations on wake formation. The relationship between wake formation and heat transfer is also not fully understood. Therefore, a series of experiments were conducted to determine the effect of rotationally oscillating cylinders on wake formation and its relationship with heat transfer. ❧ Experiments were carried out at Re = 150 and 750 in a water tunnel for oscillation frequencies from 0.67 to 3.5 times the natural shedding frequency and peak-to-peak oscillation amplitudes up to 320°. Experiments were performed at the lower Re using an unheated cylinder. Two sets of experiments were performed at the higher Re, one with the cylinder unheated and the other with the cylinder heated. Digital Particle Image Velocimetry (DPIV) was used to identify and map wake modes (coherent vortical structures in the wake) to various regions of the parameter space. Previously unknown wake modes that are synchronized over two and three times the forcing frequency were also discovered. ❧ Experiments were also performed at Re = 750 to measure the heat transfer rate for a large number of cases in the parameter space. Significant heat transfer enhancement was observed under certain forcing conditions and the regions of the parameter space where this occurs was found to correspond to locked-on wake mode regions. Other factors, such as the tangential velocity and the formation length were also found to affect the heat transfer under certain conditions.
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Sellappan, Prabu
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Core Title
Wake modes of rotationally oscillating circular cylinder in cross-flow and its relationship with heat transfer
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Publication Date
07/16/2013
Defense Date
04/24/2013
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