ABSTRACT 
Title of Dissertation: SIMULTANEOUS MEASUREMENTS OF THE VELOCITY 
AND VORTICITY VECTOR FIELDS IN THE TURBULENT 
NEAR WAKE OF A CIRCULAR CYLINDER 
Phuc Ng9c Nguy~n, Doctor of Philosophy, 1993 
Dissertation co-directed by: Dr. Barsam Marasli, Assistant Professor 
Dr. James M. Wallace, Professor 
Department of Mechanical Engineering 
Hot-wire measurements of all components of the instantaneous velocity and vor-
ti city vectors in the wake of a circular cylinder are presented. The experiments were 
performed at x / d = 20 and 30, at R ed = 2000, using a miniature 12-sensor probe 
for the simultaneous velocity-vorticity measurements and a 4-sensor probe for t he 
velocity-only measurements. 
A calibration and a data reduction method for multi-sensor hot-wire probes are 
introduced. The calibration technique is independent of the number of sensors and 
requires minimal information about probe geometry. The data reduction scheme 
involves the solution of an overdetermined nonlinear algebraic system of equations 
in a least-squares sense. 
The measurements indicate that the Karman vorti ces are accurately resolved . 
Statistical characteristics of the velocity aJ1 d vorticity fields in this flow, including 
moments, probabil ity distributions and one-dimensional spectra components are doc-
umented for the first time. Conditional analysis of t he measurements at x/ d = 30 
with respect, to the passage of the Karman vortices are presented. The vortex center 
and the saddle regions are identified and characterized. The instanteaneous veloc-
ity and vort icity signals are decomposed into mean , coherent and incoherent parts 
using t he trip le decomposition technique. A significant percentage of the incoherent 
Buctua t ions are observed to be phase-locked to the Karman vortices. The enstropy 
balances are computed for three decomposed parts. The generation of incoherent 
enstrophy due to incoherent vortex stretching is detected to be the most dominant 
term and is balanced by the viscous dissipa t ion. 
SIMULTANEOUS MEASUREMENTS OF THE VELOCITY AND 
VORTICITY VECTOR FIELDS IN THE TURBULENT 
NEAR WAKE OF A CIRCULAR CYLINDER 
by 
Ph{1c Ng9c Nguyen 
Dissertation submitted to the facu lty of the Graduate School 
of The Uni versity of Maryland in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
I,, F 'I 
1993 
/ 
f /\ \ _ _.; 
r 
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Advisory Committee: 
A 
. p f B· . ain ]Vfar"'sl1· Co-Chairman 
ss1stant ro essor ars <• , 
Professor James M. Wallace, Co-Chairman 
Associate Professor James H. Duncan 
Associate Professor Ugo Piornelli 
· o · ) · · l V C· labrese Associate Professor n .IC 1a1c . a< 
~~\ .. , . i : i { I I .. ,., ' ; j <.1 \-' /' '. I 
"\ <,' ~-' io · ..; 
r)n '· / 
"'·~ ! ~·· 
Jf/~) 
ACKNOWLEDGEMENTS 
I want to thank my advisors , Drs. Barsam Marasli and Jim Wallace, for dedicated 
guidance and support, and for working closely with me to complete this ambitious 
project. Their commitment to turbulence research is quite inspirational. I also would 
like to thank other members of the committee, Drs. Ugo Piomelli, Jim Duncan , and 
Richard Calabrese for fruitful suggestions on several occasions. I am very grateful 
for all the help from my fellow students and fri ends, Dr. Lawrence Ong, Dr. Seong-
Ryong Park, and Mr. John Wright. Their willing support for the experiments has 
been very helpful, and sometimes crucial. Lawrence and Seong-Ryong have even 
showed me how to be a graduate student sometimes. 
I would like to thank my sponsors, Drs . Frank Peterson and Bruce Douglas 
at David Taylor Model Basin , and Dr. Pat Purtell at the Office of Naval Research. 
Their support during these changing times is much appreciated. I also want to thank 
my previous mentors, Dr. Demetri Telionis at VPI & SU, and Dr. Orn Sharma at 
Pratt & Whitney. They have helped shape my interest in fluid mechani cs . 
I am grateful to my family for understanding. Most deserving are my older sisters, 
who have carried some of my responsibility. My in-laws have also been very helpful , 
especially in lending me a computer for the thesis work. Final ly, I want to thank my 
wife, Ti, for sharing all the good and bad times during the course of the PhD work . 
Without her, this project might not get done. She has even li stened a few times to 
my monologues about some of the "neat things" in turbulence . 
.. 
]] 
TABLE OF CONTENTS 
Chapter 
1 
2 
Introduction 
1.1 Characteristics of the Velocity Field 
1.2 Multi-Component Velocity and Vorticity Measurement Techniques 
1.3 Characteristics of the Vorticity Field . 
1.4 Objectives and Approach .. . . . . . 
Equipment and Experimental Program 
2.1 Wind Tunnel and Calibration J et 
2.2 Hot-Wire Probes 
2.3 Other Equipment 
2.4 Experimental Procedure 
2.4.l Preliminary Experiments 
2.4 .2 Velocity Measurements 
2.4.3 Vorticity Measurements 
3 Calibration and Data Reduction Methods 
3.1 Calibration Method . . . .. .. . . . · · · 
3.2 Data Reduction :!Vlethod for Velocity Measurement 
3.3 Data Reduction Method for Vorticity Measurement 
3.4 Solution Uniqueness .. 
3.5 Measurement Accuracy 
3.6 Summary .... . .. . 
4 Characteristics of the Velocity Field 
4.1 Statistical Properties . .... ... . 
Ill 
1 
2 
7 
11 
1.5 
17 
17 
18 
18 
19 
19 
21 
23 
25 
25 
28 
29 
31 
33 
34 
37 
37 
4.2 Velocity Spectra . . .. . .. . .... . .. ... .. ... . 
4.3 Probability Density Functions of the Velocity Components 
4.4 Joint Probability Analysis of the Velocity Components . 
4.5 Summary . . . . . . . . . . . . . . . . . .... . . · 
5 Characteristics of the Vorticity Field 
5.1 Statistical Properties . . . .. 
5 .2 Streamwise Velocity Grad ients 
5.3 Vorti city Spectra . .. .. . . 
5.4 Probabili ty Density Functions of the Vorticity Components 
5 .5 Joint Probability Analysis of the Vort icity Components 
5.6 JPDF Analysis of Vort icity and Velocity Grad ients . 
5.7 Summary .. . . .. . ... . . .... . ... . .. . 
6 Condit ional Analysis 
7 
6.1 Mot ivat ion . . . . . 
6.2 Description of the Condit ioning Detect ion Scheme 
6.3 Conditional An alysis of the Velocity Vector Field . 
6.4 Conditional Analysis of the Vorticity Vector F ield 
6.5 Phase-Locking with the Vortex Shedding 
6.6 Summary .. . . .... .. .. .... . . 
Enstrophy Balances 
7.1 Motivation . . . . 
7.2 Distributions of Kinet ic Energy Production 
7.3 Derivat ion of the Enstropby Equat ions 
7.4 The Mean En strophy Balance . . 
7 .5 The Coherent Enstrophy Balance 
JV 
42 
43 
44 
47 
49 
49 
51 
55 
57 
58 
60 
65 
67 
67 
68 
71 
76 
79 
81 
83 
83 
84 
85 
92 
93 
7 .6 The Incoherent Enstrophy Balance . 
7.7 Summary ... . . . ........ . 
8 Conclusions and Recommendations 
References 
V 
94 
96 
98 
206 
LIST OF TABLES 
Number 
1 Data reduction of velocity components (in m/s) for the calibration 
data of the 12-sensor probe . . . . . . . . . . . . . . . . . . . . . . . 35 
2 Data reduction of velocity gradients (in s-1 ) for the calibration data 
of the 12-sensor probe . . . . . . . . . . . . . . . . . . . . . . . . . . 36 
VI 
LIST OF FIGURES 
Nurnber 
1.1 Definition of the coordinate axes . .... .. ..... .. ...... 
. 104 
1.2 Sketch of the double-roller model for the vortical structure of Grant [21] 105 
1.3 Velocity vector plot in the x-y plane from Browne et al. [9] ..... . 106 
1.4 Sketch of the model of the wake vortical structure from Hussain & 
Hayakawa [26] . . . . . . . . . . . . . . . . . . . . 
1.5 Plots of the vort ical lines at various sect ion cuts through t he wake 
calcula ted by Meiburg & Lasheras [39] . . 
2.1 Scale drawing of the wind tunnel . 
2.2 Scale drawing of the calibration jet 
2.3 Schematic sketch to illustrate the angle of attack . 
2.4 Schematic sketch of the Kovasznay-type 4-sensor probe 
2.5 Schematic sketch of the plus-shape 4-sensor probe 
2.6 Schematic sketch of the 9-sensor vorticity probe . 
2 .7 Schematic sketch of th e 12-sensor vorticity probe 
2.8 Smoke flow visualizat ion of the Karman vortex li nes showing parallel 
107 
108 
109 
110 
111 
112 
113 
114 
115 
shedding . . . . . . . . . . . . . . . . . . . · · . . . . . . . . . . . . . 116 
3.1 Schematic sketch showing the three components of the cooling velocity 
for a hot-wire sensor . .... .... . .. . .. ... . 
3.2 Calibration curve of one sensor in the 12-sensor probe . 
3.3 Division of calibration angle space into calibration zones 
3.4 Comparison of solution techniques for the plus probe: (a) 4 sensors 
117 
118 
119 
used, and (b) 3 sensors used . . . . . . . . . . . . . . . . . . . . . . 120 
3.5 Solution characteri stics of the 12-sensor probe velocity components 121 
.. 
VI I 
4.1 Mean velocity and moments of the fluctuating velocity components at 
x / d=30 ... . ........ . . .. .. . . .... ........ .. 122 
4.2 Comparison of the velocity component rms values at two streamwise 
locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 
4.3 Comparison of the velocity component rms values from 12-sensor probe 
measurements to V-probe, and X-probe values at x/ d = 30 . . . . . . 124 
4.4 Comparison between the 4-sensor and 12-sensor measurements of the 
Reynolds shear stresses at x / d = 30 . . . . . . . . . . . . . . . . . . . 125 
4.5 Comparison of 12-sensor measurements of the Reynolds shear stresses 
to V-probe, and X-probe values at x/d = 30 126 
4.6 Power spectra of the streamwise velocity 127 
4.7 Power spectra of the transverse velocity . 128 
4.8 Power spectra of the spanwise velocity . 129 
4.9 Probability density function (PDF) of the streamwise velocity 130 
4.10 PDF of the transverse velocity 131 
4.11 PDF of the spanwise velocity 132 
4.12 Definition of the quadrants . . 133 
4.13 Joint probability density function (JPDF) (top) and covariance inte-
grand (bottom) contours for the streamwise and transverse velocity 
components . . . . . . . . . . . · · . . . · · . · · . . . . . . . 134 
4.14 Quadrant analysis for the correlation of the streamwise and transverse 
velocity components . . . . . . . . . . . · . . . · · · · · . . . . . . . 135 
4.15 JPDF (top) and covariance integrand (bottom) contours for the stream-
wise and spanwise velocity components . . . . . . . . . . . . . . . . . 136 
4.16 Quadrant analysis for the correlation of the streamwise and spanwise 
velocity components . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 
Vlll 
4.17 JPDF (top) and covariance integrand (bottom) contours for the trans-
verse and spanwise velocity components . . . . . . . . . . . . . . . . 138 
4.18 Quadrant analysis for the correlation of the transverse and spanwise 
velocity components . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 
5.1 Mean vorticity and moments of the fluctuating vorticity components 
atx/ d=30 .. ...... .. ...... . . . . .. .. . ..... . . 140 
5.2 Comparison of the vorticity component rms values at two streamwise 
locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 
5.3 Comparison between the central-difference scheme (C-D) and the 
backward-difference (B-D) scheme . . . 142 
5.4 Correlation vw2 with time delay using central-difference (C-D) and 
backward-difference (B-D) schemes . . . . . . . . . . . . . . . . 143 
5 .5 Comparison of the w2 time series from the central-difference ( C-D) 
and backward-difference (B-D) schemes . . . . . . . . . . . . . . 144 
5.6 Comparison of the w2 spectra from the central-difference (C-D) and 
backward-difference (B-D) schemes . . . . . . . . . . . . . . . . 145 
5. 7 Signal attenuation and phase shift of the two differencing schemes 146 
5.8 Spectra of the streamwise vorticity component 14 7 
5.9 Spectra of the transverse vorticity component 148 
5.10 Spectra of the spanwise vorticity component . 149 
5.11 PDF of the vorticity components in the freestream and the wake cen-
terline 150 
5.12 PDF of the streamwise vorticity 151 
5.13 PDF of the transverse vorticity 152 
5.14 PDF of the spanwise vorticity 153 
IX 
5.15 Joint probabili ty density function (JPDF) (top) and covariance inte-
grand (bottom) contours for the streamwise and transverse vorticity 
components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 
5.16 Quadrant analysis for the correlation of the streamwise and transverse 
vorticity components . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 
5.17 Schematic sketches of the vortex loops calculated by Mei burg & Lasheras 
[39] for the laminar wake of a flat plate with a corrugated trailing 
edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 
5.18 JPDF (top) and covariance integrand (bottom) contours for the stream-
wise and spanwise vorticity components . . . . . . . . . . . . . . . . 157 
5.19 Quadrant analysis for the correlation of the streamwise and spanwise 
vorticity components . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 
5.20 JPDF (top) and covariance integrand (bottom) contours for the trans-
verse and spanwise vorticity components . . . . . . . . . . . . . . . . 159 
5.21 Quadrant analysis for the correlation of the transverse and spanwise 
vorticity components . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 
5.22 Schematic sketches of vortex stretching (top) and reorientation due to 
shearing (bottom) by velocity gradients . . . . . . . . . . . . . . . . . 161 
5.23 JPDF (top) and covariance integrand (bottom) contours for Wx and 
du/ dx . ...... ... · · · · · · · · · · · · · · · · · 
5.24 Quadrant analysis for the correlation of W x and du / dx . 
5 .25 JP D F (top) and covariance integrand (bot tom) contours for wy and 
du/ dy ... . .... . · · · · · · · · · · · · · · · · · · 
5.26 Quadrant analysis for the correlation of Wy and dtt/ dy . 
5.27 JPDF (top) and covariance integrand (bottom) contours for W z and 
du/ dz .... .. . .. · · · · · · · · · · · · · · · · · · 
5.28 Quadrant analysis for the correlation of W z and du/ dz . 
X 
162 
163 
164 
165 
166 
167 
5.29 JPDF (top) and covariance integrand (bottom) contours for wx and 
dv / dx .. . .. .. ..... . . . ... . ....... . 
5.30 Quadrant analysis for the correlation of Wx and dv / dx . 
5.31 JP DF (top) and covariance integrand (bottom) contours for wy and 
dv / dy ...... . .. . . . ..... . . .. .. . . . . 
5 .32 Quadrant analysis for the correlation of wy and dv / dy . 
5.33 JPDF (top) and covariance integrand (bottom) contours for W z and 
dv / dz .. ... . ... . . ...... . .. · · · · · · · 
5.34 Quadrant analysis for the correlation of W z and dv / dz . 
5.35 JPDF (top) and covariance integrand (bottom) contours for w,c and 
dw / dx . . . . . ....... · · · · · · · · · · · · · · · 
5.36 Quadrant analysis for the correlation of wx and dw / dx 
5.37 JPDF (top) and covariance integrand (bottom) contours for w11 and 
dw / dy . .. . . .... . .... . .... . .. · . · · . 
5 .38 Quadrant analysis for the correlation of wy and dw / dy 
5.39 JPDF (top) and covariance integrand (bottom) contours for w2 
d'lv / d z ....... .. .... . I • • • • o o ' • ' o o • • 
5.40 Quadrant analysis for the correlation of W z and dw / dz . 
6.1 Spect ra of W z signals at r, == 0.12 . · · · · · · · · · · 
6.2 T ime series for the three parts of the normalized Wz 
6.3 Schematic sketch of the vortex shedding and the saddle regions . 
6.4 Conditional averages of u fluctuations · · · · · · 
6.5 Conditional averages of th e total u2 fluctuations 
6.6 Conditional averages of v fluctuat ions . · · · · · 
6.7 Conditional averages of the total and incoherent v2 fluctuations 
6.8 Condit ional averages of w fluctuations · · · · · 
6.9 Conditional averages of the total w2 fluctuations 
XI 
and 
168 
169 
170 
171 
172 
173 
174 
175 
176 
177 
178 
179 
180 
181 
182 
183 
184 
185 
186 
187 
188 
6.10 Conditional averages of total and incoherent k fluctuations . 189 
6.11 Conditional averages of total and incoherent uv fluctuations 190 
6.12 Conditional averages of W x fluctuations . . . . . 191 
6.13 Conditional averages of the total w; fluctuations 192 
6.14 Conditional averages of wy fluctuations . . . . . 193 
6.15 Conditional averages of the total w; fluctuations 194 
6.16 Conditional averages of W z fluctuations . . . . . 195 
6.17 Conditional averages of the total and incoherent w; fluctuations 196 
6.18 Conditional averages of the total and incoherent enstrophy 
fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 
6.19 Conditional averages of the total dissipation rate fluctuations . 198 
6.20 Phase locking of the incoherent velocity flow fi eld with the vortex 
shedd ing ..... . .. .. ... ..... ..... . 199 
6.21 Phase locking of the incoherent vorticity flow field with the vortex 
shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 
6.22 Phase locking of the incoherent Reynolds shear stress fi eld < u'v' > 
with the vortex shedding . . . . . . . . . . . . . . . . . 
7.1 Distributions of the rate of mean kineti c energy production by the 
mean velocity gradient . . . . . . . . 
7.2 Enstrophy distributions at x/d = 30 . 
7.3 Balance of the mean enstrophy at x /d = 30 . 
7.4 Balance of the incoherent enstrophy at 1:/ d = 30 
.. 
X II 
201 
202 
203 
204 
205 
NOMENCLATURE 
Aj calibration coeffici ents for each sensor 
Ci sensor y-coordinate relative to the center of the sensing area 
d; sensor z-coordinate relative to the center of the sensing area 
d diameter of cylinder 
e sensor voltage reading 
f ; error function for each sensor in Eq. (3.11) 
F sum of the squares of errors for each sensor in Eq. (3.12) 
l s vortex shedding frequency 
G gradient vector defined in Eq. (:U9) 
7i Hessian matrix defined in Eq. (3.20) 
k turbulent kinetic energy 
L 0 wake half-width 
,C locking parameter defined in Eq. (6.20) 
Ne number of calibration points 
Np number of periods in total sampling interval 
Np number of phases in a period 
N,5 number of sensors 
P(u) probabili ty distribution fun ct ion for variable u 
P( u, v) joint probability distribution function for variables 7.l and v 
P(e) polynorrual curve-fit of the sq uare of the sensor cooling velocity 
Q centerline velocity at exit of calibration jet 
r 1_ 2 magnitude of phase locking defined in Eq. (6.21) 
R e Reynolds number 
S;j symmetric part of the stress tensor 
Xlll 
t time 
u instantaneous streamwise velocity 
u total fluctuating streamwise velocity 
Uo centerline velocity deficit 
U· 
' 
velocity vector 
u= freestream velocity 
UB binormal cooling velocity 
U;ff effective cooling velocity defined in Eq. (3.1) 
UN normal cooling velocity 
UT tangential cooling velocity 
V total fluctuating transverse velocity 
w total fluctuating spanwise velocity 
X strearnwise direction 
y transverse, or lateral direction 
z spanwise direction 
Greek symbols 
a, /3 
( 
c 
'Y 
empirical parameters in Eq. (4.1) for mean strearnwise velocity 
dissipation rate 
enstrophy 
y position normalized by wake half-width 
pitch angle 
spectrum of variable u 
momentum thickness 
yaw angle 
instantaneous strearnwise vorticity 
total fluctuating streamwise vorticity 
XIV 
wy total fluctuating transverse vorticity 
wz total fluctuating spanwise vorticity 
Special notations 
11,* rms value of the total fluctuating 11, 
11, 1 incoherent part of an instantaneous turbulent variable U 
<11,> coherent part of an instantaneous turbulent variable U 
U time average of U 
<11,> phase average of 11, 
xv 
1 Introduction 
Wake flows can be found behind vehicles moving in air or water, or, in general, behind 
bodies moving in a fluid medium. Except at very low Reynolds number, wake flows 
in most practical applications are turbulent. Experimental measurements have been 
useful in a great number of applications. Wake measurements of velocity and static 
pressure have been used to determine the drag of a body moving in a fluid [27]. 
In the vibration control of lifting surfaces, one needs to measure the turbulent wake 
characteristics such as wake thickness, magnitude of the turbulent kinetic energy and 
its spatial and spectral distributions. Fluctuating velocity measurements by Huang 
et al. [24 J in airfoil wake flow, and pressure measurements by Gershfeld et al. [20] 
indicated that minor geometrical changes of the trailing edge results in measurable 
and distinctly different characteristics of the turbulent wake and the trailing edge 
pressure spectra. Therefore, understanding the effects of changes in these parameters 
and their interelation are necessary to predict the flow past lifting surfaces and to 
design them for minimum vibration. 
Most prediction procedures involve some empiricism to model the Reynolds stresses 
in the time-averaged momentum equations. One of the goals of this dissertation is 
to provide high quality measurements of the Reynolds stresses and the physical 
generation/transport mechanisms for these stresses in order to improve turbulence 
models. Schlichting [55] used a one-dimensional integral method to calculate the 
mean streamwise velocity profile in the region where the streamwise pressure gradi-
ent is negligible, and obtained good agreement with measurements . Huang et al. [24] 
calculated the wake of lifting surfaces using a modified boundary-layer method. Th is 
method was able to calculate the entire wake, but the predicted center of the wa.1«! 
needed to be shifted to agree with the measurements. Nguyen & Gorski [44] calcu-
1 
lated . the wake flows of lifting surfaces using the Reynolds-averaged Navier- Stokes 
equations. This method , in general, predicted accurately the mean streamwise ve-
locity and the Reynolds stress distributions, but it did poorly in the flow region 
immediately behind the trailing edge due to an inadequate turbulence model. Miner 
et al. [40] attributed this to the poor near-wall modelling based on comparison with 
a direct simulation of channel flow . To date, no large-eddy simulat ion or direct nu-
merical simulation has been attempted for a turbulent plane wake. Compared to 
the direct simulation by Kim & Moin [31] on the turbulent channel flow , the wake 
simulations would involve growing boundaries and would be very expensive in com-
putational resources . Therefore, experiments have been the only means so far used 
to investigate the fundamental physics of wake flows. 
1.1 Characteristics of the Velocity Field 
One of the early fundamental questions about wake flows was 111 what Reynolds 
number range does the flow become turbulent. In 1954, Roshko [53] found that for 
Reynolds number based on cylinder diameter , R ed, greater than 300, the wake is fully 
turbulent after 40-50 diameters downstream of the circular cylinder. For the region 
0-40 diameters, the discrete energy of the vortex shedding gradually feeds the cont in-
uous spectrum. The nondimensional vortex shedding frequency, called the Strouhal 
number S = J.~df U00 where f s is the shedding frequency, d the cylinder diameter 
and U
00 
the freestream velocity, increases with the Reynolds number , reaching an 
asymptotic value of approximately 0.21 above R ed ~ 700. For R ec1 between 150 
and 300, he characterized the fl ow as in "transit ion" in which turbulent velocity 
fluctuations exists along with periodic formation of the K,:irman vortices. He also 
suggested that these turbulent velocity fluctuations are ini t iated by the "larninar-
turbulent t ransition in the free layers" developing at the separation points on the 
cylinder. Williamson [68] found that there are two stages in the transition to the 
2 
three-dimensional structures of the near wake. Each stage corresponds to a discon-
tinuity in the Strauhal-Reynolds number relationship, and both stages have distinct 
characteristics. The first discontinuity, at R ed ~ 180, is associated with the inception 
of vortex loops and is hysteretic. These vortex loops are developed from the Karman 
vortices, and become stretched afterward to evolve into a pair of counter-rotating 
quasi-streamwise vortices. The second discontinuity, at R ed ~ 250, corresponds with 
a change to a finer-scale streamwise vortex structure, and is not hysteretic. 
In wake experiments, one of the first concerns is to ensure the two-dimensionality 
of the mean flow . Eisenlohr & Eckelmann [16] investigated end effects on cylinder 
wake flows. They used smoke-wire flow visualization to detect oblique vortex shed-
ding in the wake. At R ed = 130, i.e. in the transition range, with an aspect ratio 
(AR) of 60, flow visualization showed that vortex shedding can be inclined with re-
spect to the cylinder up to 30°. When end-plates and end-cylinders were attached to 
the circular cylinder , the vortex shedding became parallel to the body. For AR > 100 
and higher Reynolds numbers, they anticipated that, based on the data of Roshko 
[53], vortex shedding would be parallel to the body without the need of end-plates 
and end-cylinders . 
Early research concentrated on characterizing the mean variation in the stream-
wise and transverse directions in wake flows. Andreopoulos & Bradshaw [2], using 
a temperature-conditional sampling technique, observed 3 layers in the transverse 
direction of the near-wake region of a flat plate: a turbulent inner-layer, and 2 inter-
mittent outer layers. They noted that the wake interaction involves significant fine-
scale mixing shear layers, and recommended using shear stress transport to model 
the outer layer, and eddy-viscosity to model the inner layer. Based on the measure-
ments, they suggested an empirical relat ionship which shows logarithmic growth of 
the wake centerline velocity with respect to the streamwise direction. 
Pot [49], Alber [1 J and Ra.maprian et al. [51 J studied t he streamwise variation of 
3 
the mean streamwise velocity of a turbulent wake. Alber presented a semi-empiri cal 
relationship in which the wake centerline velocity is a logarithmic function of the 
streamwise direction. Ramaprian et al. measured the streamwise and transverse 
velocity components of a tapered flat plate wake, and suggested that this turbulent 
wake has 3 regions in the streamwise direction. In the near-wake (x/0 < 25, where 0 
is the momentum thickness), mixing of the inner shear-layer occurs. In the middle-
wake (25 < x / 0 < 350), mixing of the outer layer occurs, and self-similarity of the 
mean velocity exists . This means that the mean velocity profiles are the same at every 
streamwise stat ion when normalized by the local wake half-width and velocity-deficit. 
In the far wake (x > 350 0), all large scale mixing is completed and the wake reaches 
an "asymptotic" state where self-preservation exists . According to Ramaprian et 
al. , this state means that "the turbulent structure becomes independent of the 
initial conditions", and a far-wake model can be used to predict the flow. Schlichting 
(55] showed that in the far wake region where the pressure gradient is negligible, 
the momentum equation simplifies significantly. When a mixing length turbulence 
model was used for the Reynolds shear stress uv, the momentum balance yielded a 
mean streamwise velocity profile that is self-similar. These predictions agreed well 
with the data for x/ d=80 to 208. 
The strearnwise velocity at the wake centerline also exhibits a shape eq ui valent to 
the universal velocity profile for the sublayer and the "log" regions in the turbulent 
boundary-layer of a flat plate. The data of Pot [49] and Ramaprian et al. [51 J showed 
the logarithmic growth of the centerline velocity along the strE.arnwise direction where 
Alber's formula matched the data quite well, Haidari & Smith [22] used hot-film 
anemometry in water to study the near-wake of a tapered flat plate. They found 
that the centerline velocity varied linearly with x + for O < x+ < 100 , and varied as 
log x+ for x + > 270 ( x+ is the streamwise distance from the trailing-edge normalized 
with the wall shear stress at a certain distance upstream of the trailing-edge). Here, 
4 
one momentum thickness {) is approximately 90 x+. These patterns are similar to 
those in the viscous sublayer , and in the so-called wake region of a turbulent boundary 
layer, respectively. They suggested a linear relationship between the x + for the wake 
streamwise position, and the y+ for the boundary-layer normal position. With this 
relationship , they showed that the velocity at the wake centerline and the velocity of 
a boundary layer collapse to the same empirical relationships, for both the very-near 
wake and the "log" region. 
Narasimha & Prabhu [42] investigated the relaxation process necessary to reach 
an equilibrium state of turbulent wakes and suggested universal relationships for the 
wake half-width L0 and the velocity defi cit U o . The assumption for these relation-
ships is that the equilibrium wakes are independent of initial conditions originated 
from the wake generators. Wygnanski et al. [69] , however , found th at the initial con-
ditions dictate how the wakes approach the self-preserving st ate. They studied the 
wakes behind different two-dimensional wake generators such as circular cylinders, 
flat plates, airfoils, and screens. They systemati cally measured the mean velocity 
profiles and the Reynolds stresses at different streamwise stations. Their res ults 
indicated that u,0 , L0 , and the Reynolds stresses, u2 and uv, depend on the wake 
generator shapes , but the mean streamwise velocity does not . They also found that 
the nondimensional quantities (U00 /uo)2 and (Lo/0) 2 varied linearly with respect to 
the streamwise direction as predicted by the self-preservation arguments, but the 
slopes and intercepts of these lines were different for different wake generators . 
Yamada et al. [70] made measurements at various streamwise stat ions for a cir-
cular cylinder , from x /d=30 to 475. The mean streamwise velocity, and the various 
Reynolds stress components were found to approach the self-preserved data curves 
of Townsend [60] as the downstream distance was increased. At the x / d= 30 station, 
the rms of the transverse velocity component was highest , and the strearnwi se and 
spamvise components were approximately the same. In the far-wake, the stream-
5 
wise component was highest, and the other two components were approximately the 
same. Since Yamada et al. used X-wires for their measurements the t hree velocity 
components were not measured simultaneously. 
Fabris [17) used a 4-sensor probe to measure simultaneously the three velocity 
components and the temperature in a circular cylinder wake at x / d=200 and 400. 
T he skewness factor was negative across the wake for the streamwise velocity, and 
anti-symmetric for the transverse velocity. From the positive sign of the transverse 
velocity component skewness, it can be deduced that the author was reporting data 
from the upper wake. The spamvise component skewness was reported to be very 
close to zero, though it was not shown. The flatness factor was approximately 2.9 for 
about 4 diameters from the wake centerline, but was quite high beyond this region. 
The intermittency factor was also shown to be self-similar. 
To achieve a more complete description of turbulent wakes, researchers have used 
multi-point measurements which reveal the dynamics of the turbulent structures. 
Browne et al. [9) used a temperature-conditional sampling technique to detect large-
scale structures in the far wake region of a circular cylinder. They decomposed the 
fluctuating velocity components into a "random" part , and a "coherent" part which 
corresponds to the detected events. At x /d = 420, the cont ribution from the random 
motion is much larger than that of the coherent motion. 
Considerable physics of turbulent wakes have been discovered from velocity mea-
surement using mostly single-sensor or X-array probes. However, all three compo-
nents of the velocity or vorticity vectors have never been measured in the near-wake 
to st udy the three-dimensional fl.ow structures . The following sect ion will describe 
the development of multi-component velocity and vorticity probes. The use of these 
powerful techniques will undoubtedly reveal some of the missing pieces of the turbu-
lence puzzle. 
6 
1.2 Multi-Component Velocity and Vorticity Measurement 
Techniques 
Single-sensor and X-wire probes have been widely used in turbulence research. In 
the X-array, or V-array hot-wire measurement technique (Bradshaw [8]), two simul-
taneous algebraic equations are obtained from the two sensor readings which relate 
the two unknown velocity components to the cooling velocity of each sensor. These 
types of probes can either be used to measure the u and v components or the u 
and w components. The measuring volume is assumed to experience uniform flow. 
Commercial X-wire probes often have about 1 mm spacing between the two sensors. 
This spacing can be too large to properly assume uniform flow in the measuring 
volume for highly turbulent flows. Reducing the wire spacing can increase the mea-
surement accuracy, but the effect of the third component still cannot be accounted 
for by X-wire probes. 
Only in the last decade have three-component velocity measurements become 
a common tool in turbulence research. As mentioned above, Fabris [17] used a 
4-sensor probe to measure velocity in the far-wake region of a cylinder, with one 
sensor used for temperature measurement. Four coupled nonlinear equations were 
solved to obtain the flow temperature and the 3 velocity components. Lekakis et 
al. [36] made measurements in a turbulent pipe flow with a 3-sensor velocity probe 
which has a third sensor orthogonal to an X-wire probe. They developed a probe 
calibration method and a fast data reduction scheme that can also account for small 
non-orthogonalities in the geometry. They pointed out that a unique solution can 
only be achieved when the flow angle is within the cone of acceptance of the probe. 
This uniqueness problem is due to the nonlinearity of the cooling equations which 
can have multiple solutions. Vukoslavcevic & Wallace [62] developed a 4-sensor 
Kovasznay-type probe, to be discussed below, that was able to measure all three 
velocity components in a turbulent boundary-layer. The probe had the configuration 
7 
of two orthogonal X-probes. Samet & Emav [54] developed a 4-sensor probe to 
measure 3 components of velocity in a turbulent jet flow . The probe consisted of 
a pair of orthogonal V-probes with the inner prongs shorter than the outer prongs. 
Least-squares surfaces were fitted to the calibration data, and the solutions were 
found as intersections of the surfaces. Two values of the streamwise velocity were 
calculated by this method, and they were averaged to yield the final value. They 
obtained either unique solutions or non-convergence; the later occured when the 
angles of attack of the velocity vector to the probe axis were outside the range of 
calibration . Dobbeling et al. [1 4] also used a 4-sensor probe for measurements in 
a turbulent jet flow. The probe was configured as a pair of orthogonal V-probes 
with the inner prongs longer than the outer prongs. They showed that using a 4-
sensor probe to measure 3 components of velocity can yield a unique set of physical 
solutions, whereas a 3-sensor probe yields multiple sets of physical solutions . 
Measurement of vorticity in turbulent flows has been rare due to the difficulty 
in the estimation of velocity gradients . The following discussion will briefly describe 
the development of vorticity measurement techniques ( readers will find the definition 
of the coordinate axes in Fig. 1.1 useful). Here, the focus will be on methods 
that estimate vorticity components from velocity gradient measurements. A more 
comprehensive review of different techniques is given in Wallace [64] and Foss & 
Wallace [19]. 
In 1950, Kovasznay [33] developed a 4-sensor probe to measure the three velocity 
cornponents and the streamwise component of the vorticity vector. This probe had 
only four supporting prongs so that all sensors are electrically connected together 
through the prongs. The probe was thought to be uniquely sensitive only to the 
streamwise vorticity component, but Kistler [32] found that second-order cooling 
from the cross-stream velocity components can strongly affect the measured stream-
wise vorticity. Kastrinakis et al. [29] confirmed this finding with extensive analyt ical 
8 
and experimental studies, and proposed a Kovasznay-type vorticity probe without 
common prongs. From this suggestion, Vukoslavcevic & Wallace [62] developed a 
4-sensor probe with eight prongs. Four equations were derived to solve for the three 
velocity components and the streamwise vorticity component. They showed that 
gradients of the streamwise velocity, which are neglected in these equations, can be 
very high for near-wall boundary-layer flows thereby causing significant errors in the 
measured cross-stream velocity components and, in turn, the streamwise vorticity. 
Foss [1 8] developed a four-sensor probe to measure the spanwise vorticity com-
ponent w 2 • The probe consisted of two parallel sensors and two X-sensors. The 
parallel sensors measured au/ay, and the X-sensors measured U and V. Taylor's 
hypothesis was used to estimate av/ ox. The effect of neglecting the span wise ve-
locity component W was corrected . Lang & Dimotakis [34] used four pairs of laser 
beams to measure the spanwise vorticity component in shear layers. The streamwise 
velocity gradient was measured directly so the use of Taylor's hypothesis was not 
necessary. Hussain & Hayakawa [26] used an array of X-probes to make multi-point 
measurements of w2 in the turbulent wake of a circular cyli nder. The X-probes were 
separated in the transverse direction. Each X-probe measured the U and V velocity 
components, and au/ ay was estimated from two adjacent pairs of X-probes. Tay-
lor's hypothesis was used for av/ax. Hayakawa & Hussain [23] also measured the 
transverse vorticity component wy using a similar technique. With this arrangement, 
wy and w2 were measured at different times during their experiments . 
Eckelmann et al. [15] developed a five-sensor probe to measure simultaneously 
wy and w2 • A unique arrangement of a single sensor, a V-probe and an X-probe 
was used to get au/ az and au /ay . Recently, Kim & Fiedler [30] developed a six-
sensor probe to measure Wx and w2 • Taylor's hypothesis was used to estimate the 
streamwise gradients with both of these probes. 
There have been some efforts to measure all three components of vorticity by using 
9 
different combinations of X-probes. Antonia et al. [3] measured at cl.ff . t t· 1 eren 1mes 
the spanwise and normal components using two pairs of X-probes separated in the 
appropriate lateral directions. The two velocity gradients of the streamwise vorticity 
component were not measured simultaneously. Theoretically, it is possible to measure 
simultaneously all three components of the vorticity vector with four X-probes. Two 
probes in the x - z planes separated in the y direction can measure the gradients 
8U/8y, and aw jay. Two additional probes in the x - y planes separated in the z 
direction can measure the gradients au I a z' and av I a z . The stream wise gradients 
aV/ox, and olV/ax can be obtained from the Taylor's hypothesis. However, putting 
all these X-probes in a small measuring volume is quite a problem. Bisset et al. [7] 
arranged seven X-probes, with total sensing area of approximately 9 mm by 14 mm, 
to measure simultaneously the large-scale parts of the vorticity components. They 
used the probe to study the vortical structures in the far-wake of a circular cylinder. 
Recent breakthroughs have allowed reasonably accurate vorticity measurement 
without having to use a great number of X-probes. Vukoslavcevic et al. [63] devel-
oped a miniature 9-sensor probe to measure simultaneous ly the velocity and vorticity 
vectors. The probe had three arrays with the sensors arranged in a "T" shape in each 
array. Each array had a common prong to minimize the sensing area, which fits in-
side a circle with a diameter of 2.5 mm. Nine coupled equations were derived for the 
nine sensors. These equations for the cooling of each individual sensor were Taylor 
series expansions, to first order, around the probe centroid , thus incorporating the 
cross-stream velocity gradients. An iterat ive technique was used to solve for the three 
components of velocity and the six velocity grad ient components in the spanwise and 
transverse directions. The streamwise velocity gradients were estimated using Tay-
lor's hypothesis. Thus they were able to measure simultaneously the velocity and 
vorticity vectors. Balint et al. [5] used this probe in a turbu lent boundary-layer, and 
were able to resolve all but the smallest turbulent scales at Re0=2685. 
10 
In this dissertation, an improved version of the miniature nin · · e-sensor vorticity 
probe, constructed by P. Vukoslavcevic (private communication), was used to study 
the vortical structures in the near wake of a circular cylinder. This new probe has 
twelve sensors to help obtain the unique solutions to the cooling equations. More 
details are provided in Sections 2.2 and 3.3. 
1.3 Characteristics of the Vorticity Field 
For the turbulent wake of a circular cylinder, there were only a few experiments 
which documented the basic statisti cs of the fluctuating vorticity such as moments, 
spectra and probability distributions. Antonia et al. [3] measured separately the 
vorticity components at x /d = 420 using a pair of X-probes. Therms of wx was the 
largest maximum value, and the other two components were about the same. The 
skewness factors for wx and wy were quite high, which could indi cate an accuracy 
problem since the skewness factors of these vorticity components should be near zero 
due to the symmetry of the mean flow. Reasonable agreement was seen when the 
rms values and the probability distributions were compared to calcu lations based 011 
local isotropy. 
Bisset el al. [7] simultaneously measured the large-scale approximations of the 
vorticity vectors in the far-wake of a circular cylinder, using a very large probe made 
from seven X-probes as described earlier. The rms values of Wx and wy were about 
the same throughout the wake. The W z rms component was higher than the other 
two by approximately 20% in the region ±Lo/2 around the wake centerline. With a 
cold wire in the array probe, they measured temperature and used a temperature-
conditional sampling technique to detect the turbulent structures at :r; / d = 420. 
The existing literature yields a fairly consistent picture of the vortical structures 
for the turbulent plane wake. However, a complete description of the dynamics of 
the turbulent structures has not yet emerged . This would require time-varying infor-
11 
mation across the entire spatial domain of the turbulent wake for a wide Reynolds 
number range. The vorticity measurements in this project will shed some light on 
this very complicated problem as vorticity is the fundamental variable in the dynam-
ics of the "coherent" structures . This view is shared by many investigators such as 
Willmarth [67], and Hussain & Hayakawa [26]. 
Grant [21] made two-point velocity measurements in the turbulent far-wake to 
form correlation coefficients. He proposed a model of the vortical structure which 
is consistent with the spatial distribution of these correlation coefficients. These 
structures consisted of counter-rotating vortex pairs with their axes separated in the 
spanwise direction and approximately aligned in the direction of the mean shear (see 
Fig. 1.2). The existence of these structures, sometimes called the "double-roller" 
vortices, were also proposed by Mumford [41], and Payne & Lumley [47]. The other 
type of structure observed in the far-wake is the spanwise vortex. T his vortex is 
part of the "secondary vortex street", which has scale much larger than the original 
Karman vortex street according to Cimbala et al. [12]. The spanwise vorti ces 
and their dynamics have a lso been studied by Mumford [41], Townsend [61], and 
Browne et al. [9]. The "double-roller" structures consist of mainly the wx and wy 
components, and the spanwise structures consist of mainly the W z component of the 
vorticity vector. 
To study the secondary spanwise vortices, Browne et al. [9] measured in the 
far-wake of a slightly heated circular cylinder with an assembly of X-wires and tem-
perature sensors as mentioned above. Using relatively large temperature change 
between consecutive samples as a detection criterion of the flow structures, they 
constructed a velocity vector plot (see Fig. 1.3) in the plane of the mean shear (x - y 
plane). T he vortex centers and the saddles were clearly seen on both sides of the 
wake. The authors called the region that connects one spanwise vortex to the next 
the "converging separatrix" where fluid converges toward the saddle. At about 90° 
12 
from the "converging separatrix" is the "diverging separatrix" where fluid diverges 
away from the saddle toward the freestream or toward the wake centerline. These 
diverging separatrices are approximately aligned with the mean shear , a character-
istic which the double-roller structures also have. Therefore, it is possible that the 
double-roller structures are related to the diverging separatrix flow regions. With 
the Wx and wy vorticity data, it is possible to further characterize these structures as 
will be discussed later. 
In the near-wake of a circular cylinder, Hussain & Hayakawa [26] measured the 
large-scale spanwise vorticity by a rake of X-probes positioned in the plane of the 
mean shear. They specified the trigger signal as the passing of a large amplitude 
spanwise vortex. For the measurement range of :r /cl=lO to 40, they found that the 
spanwise vortex center was less than one wake half-width from the centerline. From 
analysis of other data and their own, Hussain & Hayakawa proposed a mechanism 
for the dynamics of the turbulent plane wake (see F ig. 1.4). In this mechanism, the 
wake can be viewed as "a superposition of two coherent vorticity layers consisting 
of large-scale , nominally span wise vortex rolls of opposite circulation". Each "layer" 
would consist of vortica l structures with the same sense of rotation separated by sad-
dles . The diverging separatrices, or "braids", oriented in the direction of the mean 
shear connect the top of one structure to the bottom of another structure in the 
same layer. Their proposed mechanism lacked the W x and wy data. Later, Hayakawa 
& Hussain [23] measured both the large-scale W y and W z components , though not si-
multaneously. From the wy contours , they claimed to see the "apparent signatures" 
of the "braids". From the simultaneous measurements of all three vort icity com-
ponents in this dissertation, it is possible to relate the braids to the double-roller 
vortices described earlier. 
Cantwell & Coles [10) invest igated the entrainment and transport processes in 
the turbulent near wake of a circular cylinder. They measured in the region with in 8 
13 
diameters downstream of the cylinder using two X-wire probes mounted on a pair of 
whirling arms. This arrangement increases the velocity along the sensor axis which 
decreases the angles of attack of the velocity vector on the sensor, thus making 
very-near wake measurement possible. They used a pressure reading on the cylin-
der surface to detect the vortex shedding and deduced a pattern of moving vortex 
centers and saddles by phase-averaging the measured streamwise and transverse ve-
locity components. They found that a substantial part of the turbulence production 
is concentrated near the saddles and suggested that the mechanism of turbulence pro-
duction is "probably vortex stretching at intermediate scales" . Their data showed 
that the normal Reynolds stresses due to the random motion have maxima near each 
vortex center, and the magnitude of the random Reynolds shear stress is maximum 
near the saddle between the vortices. 
Meiburg & Lasheras [39] investigated experimentally and numerically the laminar 
wake of a flat p late subjected to periodic spanwise pertubations. Computational 
results from the inviscid vortex technique agreed qualitatively with the laser-induced 
flow visualization. They showed that the redistribution, reorientation, and stretching 
of vorticity lead to the formation of counter-rotating pairs of streamwise vortices, 
which superimpose on the spanwise Karman vortices. The streamvvise vortices were 
shown to be located in the braids connecting the spanwise vortices with opposite 
rotation on the two sides of the wake. This picture is slightly different from those of 
Browne et al. [9] and Hussain & Hayakawa [26] where the braids connect structures 
on the same side of the wake. But, in the x- y plane, the filaments of these streamwise 
vortices calculated by Meiburg & Lasheras [39] were approximately aligned with the 
mean shear (shown in Fig. 1.5) , which agreed with the model of Grant [21]. These 
uncertainties underscore the need to obtain simultaneous measurements of the three 
vorticity components together with the velocity components. 
14 
1.4 Objectives and Approach 
The a bove literature review points out that simultaneous three-component velocity 
measurement is lacking for wake flows. Some data existed in the far-wake, but these 
are either non-simultaneous, or are very coarse in spatial resolution. Simultaneous 
three-component vorticity data for near-wake flow are non-existent . Therefore , the 
main obj ectives of this work is to measure simultaneously and with reasonably good 
spatial resolution the three components of the velocity and vorticity vectors in the 
cylinder wake, and to analyze these measured data in order to determine: 
l. Basic properties such as the moments, spectra and probability distribution 
fun ctions of the velocity and vorticity vectors. 
2. Locations of the saddle and the vortex center and the wake flow characteri stics 
at these locations. 
3. Mechanisms of the different processes of vorticity transport in the wake. 
The following steps were taken to achieve the above obj ectives : 
l. Simultaneous measurements of velocity and vorticity in the turbulent near-
wake of a circular cylinder. The experimental program had three phases . The 
first phase, which involved preliminary experiments, was mainly for identifying 
problems with the experimental setup and for optimizing the experimental con-
ditions. The second phase was for velocity component measurement, and the 
third phase was for simultaneous velocity and vorticity measurement. These 
experiments are described in Chapter 2. 
2. Development of the data calibration and redu ct ion software for measuring the 
velocity and vorticity vector fields using multi-sensor hot-wire probes . Thi s is 
addressed in Chapter 3. 
15 
3. Characterization of the velocity and vorticity fields of the turbulent plane wake 
with various statistical properties including moments up to the fourth order 
' 
probability distributions, and spectra. These are discussed in Chapters 4 & 5. 
4. Investigation of the vortical structure dynamics in the turbulent plane wake 
with a joint probability distribution analysis of the velocity components, the 
vorticity components, and the generation terms in the vorticity transport equa-
tions. This is discussed at the ends of Chapters 4 & 5. 
5. Characterization of the vortical structures by conditional analyses of the Reynolds 
stresses, the turbulent kinetic energy, the enstrophy and its cornponents, and 
the energy dissipation rate relative to the passage of the Karman vortex. This 
is discussed in Chapter 6. 
6. Computation of the mean and fluctuating enstrophy balances by decomposing 
the instantaneous signal into its components. These balances give quantitative 
description of the different vorticity transport processes. This is discussed in 
Chapter 7. 
16 
2 Equipment and Experimental Program 
2.1 Wind Tunnel and Calibration Jet 
The experiments were performed at the low speed wind tunnel of the Turbulence 
Research Laboratory at the University of Maryland. This tunnel is an open-return 
type (see Fig. 2.1 ) with a total length of 15.2 m. A centrifugal blower takes air 
from the test room and outputs to an adjacent room which contains the tunnel inlet. 
The blower can produce flow up to 5.5 m/s, with freestream turbulence level of 
0.5 %. The inlet section is 4.40 m by 2.20 m, and covered by filters. Downstream of 
the filters are 17.5 cm long, 5 cm diameter tubes to break down large-scale eddies. 
The tunnel contracts to the test section with a contraction ratio of 11.2 to 1 over 
a distance of 5.0 m. A honeycomb 7.6 cm long with 0.94 cm hexagonal cells, and 
four stainless steel screens help reduce the flow turbulence at the beginning of the 
test section. The test cross section is 1.26 m by 0.68 m with corner fillet s to reduce 
secondary flows and is 8.8 m long with the top and bottom walls diverging to account 
for pressure gradients created by the thickening boundary-layers. The measurements 
were performed in the last one meter before the tunnel exit. 
The traversing mechanism is at the tunnel exit and allows motion in the vertical 
and spanwise directions with accuracies of 0.03 mm and 0.015 mm, respectively. 
The multi-sensor hot-wire probes were calibrated in an axisymrnetric jet tunnel (see 
Fig. 2.2) bui lt by P. Vukoslavcevic. This jet can produce stable, uniform, and 
low turbulence flow in the range 0.2 - 10 m/s. The total length of the cali brating 
tunnel is about 2.0 m . Air enters the jet tunnel through a fi lter , is accelerated by 
a centrifugal blower through a long tubular fl.ow straightener and exits from a 4:1 
nozzle. A pitch/yaw mechanism is attached to the nozzle exit and can rotate through 
±20° with 5° increments . The pitch and yaw angles are defined in Fig. 2.3. 
17 
2.2 Hot-Wire Probes 
Four different types of probes were used in the present project: single-sensor, 4-
sensor, 9-sensor and 12-sensor. The preliminary measurements were made with a 
rake of single sensors. The rake consists of 20 wires approximately 2 mm apart. 
All the following multi-sensor probes were built by P. Vukoslavcevic. The 4-sensor 
probes were used to measure simultaneously all three components of velocity. Two 
different probes were used: (1) a Kovasznay-type 4-sensor probe (see Fig. 2.4) similar 
to one first used by Vukoslavcevic & Wallace [62], and a plus-shaped 4-sensor probe 
(see Fig. 2.5). The Kovasznay-type consists of 2 pairs of orthogonal X-probes, and 
the plus-shaped consists of 2 pairs of orthogonal V-probes (hereinafter referred to as 
the square probe and plus probe, respectively). The 9-sensor and 12-sensor probes 
were used to measure all three components of velocity and vorticity. The 9-sensor 
probe in Fig. 2.6 had been used successfully in boundary-layer measurements as 
reported by Vukoslavcevic et al. [63] and Balint et al. [5]. Each of the 3 arrays of 
this probe has a common prong connected to other prongs in a T-shape. The 12-
sensor probe in Fig. 2.7 consists of 3 arrays arranged at the corners of an equi lateral 
triangle. Each array is a plus probe with no common prongs. The 12-sensor probe 
was developed to help obtain more accurate measurements with the extra sensors. 
This issue is discussed in more detail in the next chapter. All probes have 2.5 micron 
Tungsten sensor wires and prong spacings of approximately 0 .. 5 mm. T he 12-sensor 
probe sensing area can fit inside a circle of 2.4 mm diameter. 
2.3 Other Equipment 
The anemometer, custom-made by AA Labs, has 12 constant-temperature channels, 
with built-in circuitry for gain , offset and low-pass filters, and "low noise", "low 
drift" options. 
Data acquisition was performed with a 12-cha.nnel, 12-bit A/D converter with 
18 
simultaneous sample and hold circuitry connected to a PDP 11 ; 23 . . microcomputer. 
The system is capable of no-gap data transfer directly to hard disk t . 
a a maximum rate 
of approximately 30000 samples/sec. Data processing were performed on ., . t k 
cc ne ,wor 
of Sun Sparcstations. During the experiments, the tunnel speed was co t · 1 n 111UOUS y 
monitored by a Pitot tube connected to a Barocel pressure transducer (model 1174). 
2.4 Experimental Procedure 
The experimental procedure consisted of 3 phases: preliminary wake meas urement 
t ests, velocity measurements with the 4-sensor probes , and vorticity measurements 
with the 9-sensor and 12-sensor probes. 
2.4.1 Preliminary Experiments 
These experiments were performed to establish the optimum conditions for wake 
measurements. Early in the experimental program, it was found that the measure-
m ent station (streamwise location of probe tip) should be about 0.5 m upstream to 
be free of the blockage effect of the traversing mechanism. Two Pitot tubes were 
used to measure the frees tream velocity, one fixed on one side of the tunnel, the 
other riding on the traverse. Measurements started with the two probes separated 
by only a few centimeters. This position had the most blockage from the travers-
ing mechanism. The probe on the traverse was then moved a short di stance in the 
spanwise direction away from the stationary probe. Measurements were taken for 
many spanwise locations across the tunnel width. The streamwise distance from the 
probe tips to the traversing mechanism were increased gradually until the variation 
between the two probe readings were within 1 % of the measured velocity. 
The streamwise velocity component was measured with the single-sensor rake. 
Several velocity profiles were taken in the range x / d = 70 - 220 downstream of the 
cylinder. These measurements indicated that, in thi s region, the velocity defi cit is 
19 
from 5% to 10% of the freestream velocity(~ 3 m/s) and the maximum mean shear 
varies only from 10 s- 1 to 20 s- 1. These values of mean shear are of the same order 
as the noise level of the 9-sensor vorticity probe. Therefore, it was decided that all 
vorticity field measurements should be carried out in the region x/ d < 50. 
The single-sensor rake experiments also indicated some unsteadiness in the tunnel 
speed. It was determined that, prior to any measurements, the tunnel must be left 
running for at least 3 hours without any disturbance to stabilize the flow. Low speeds 
(below 3 m/s) tended to be more unstable than high speeds. To obtain stable flow 
and large velocity gradients, it was decided that measurements should be carried out 
at the highes t speed possible, approximately 5 m/s for thi s tunnel. 
Preliminary measurements of the streamwise velocity components were all for 
the horizonta l cylinder orientation. The tunnel width is larger than its height; there-
fore the hori zontal position provides a larger cylinder length to diameter ratio, or 
aspect ratio (AR) , than the verti cal position. Larger AR is desirable to achieve 
two-dimensionality of the flow. For the horizontal orientation, however , the free-
stream flow region of the wake is less than that for the verti cal orientation. This 
is a consequence of the fact tha t the tunnel was originally designed for boundary-
layer measurements, and the traversing mechanism is strategically placed where the 
boundary-layer is thi ckest . The AR for a vertical orientation of the cylinder was ap-
proxiately 107 for which Roshko [53] indicated that two-dimensionality of the wake 
:flow can be achieved with R ed > 500. To obtain more confidence for multi-component 
velocity measurements , flow visualization was performed with the verti cal cylinder 
arrangement. Fig . 2.8 shows the parallel shedding of the Karman vortex behind the 
cylinder which indicates that the flow was two-dimensional. Therefore, all definitive 
measurements were performed with the verti cal cylinder orienta tion. 
20 
2.4.2 Velocity Measurements 
Velocity measurements were performed after the preliminary experiments indicated 
the optimum experimental conditions. Both 4-sensor probes were used during this 
phase. The tachometer connected to the jet fan was first calibrated against a Pitot 
tube over the range 2 m/s - 7 m/ s, so that the jet velocity could be obtained from 
th f . 
e an mput voltage. Then, the dynamic range of the sensors was determined by 
finding the maximum and minimum voltage drops for all sensors. The anemometer 
outputs were offset and amplified to maximize the range of the sensor response to 
nearly the extent of the A/D range. The temperature drift had been studied earlier, 
and determined to be negligible provided the entire calibration and data acquisition 
process could be done in under 3 hours. This was possibly due to several factors: 
(l) the anemometer channels have low drift components, (2) the anemometer bridges 
Were turned on at least two hours before the calibration process, and (3) the hot-wire 
overheat ratio was set relatively high at 1.4. The probe was pitched and yawed in the 
core region of the calibration jet flow with 5 second sampling times at each position. 
The pitch-yaw data were obtained at two speeds, one at the wind tunnel freestream 
velocity, the other about 3 wake deficits lower. Measurements were also obtained 
for a range of speeds at zero pitch and yaw. This limited set of calibration data 
Was found to be adequate as the velocity variation in wake flow is not as great as in 
boundary-layer flow . Although more data would be better, a compromise is required 
between the number of calibration points and the total calibration time. With the 
existing manual pitch/yaw mechanism, the calibration process takes approximately 
90 minutes. The long calibration time is undesirable because of the voltage drift. 
Tests were performed with both 4-sensor probes before definitive measurements 
Were obtained. The effect of moving the probes from the calibration jet to the 
Wind tunnel was studied and found to be negligible. A wake profile was taken at 
21 
x / d = 120 using the square probe. This was performed to determine the quality 
of the experimental conditions, and to test the 4-sensor data reduction software. 
Results from these measurements showed that the flow was stable, and that the data 
reduction method produced physically realistic measurements if the angles of the 
velocity vectors were within ±20°. 
The definitive velocity measurements were performed at x/d=20 and 30 using 
the plus probe. These stations were chosen to maximize the signal-to-noise ratio 
without the extremely high angles of attack of locations closer to the cylinder. This 
probe was used for comparison with the 12-sensor probe measurements, because the 
4
-sensor probe resembles one array of the vorticity probe. The cylinder, a 1/4 inch 
drill rod, was fastened to the top and bottom walls of the tunnel and held under 
tension by external fasteners. The streamwise displacement was achieved with an 
adjustable probe extender mounted on the traversing mechanism. For all definitive 
measurements reported in the following chapters, the experimental conditions were: 
• downstream locations x /d = 20 and 30 
• cylinder diameter d = 6.3 mm and L/d = 107 
• tunnel freestream speed U00 = 5.00 m/s 
• Reynolds number Red = 2000 
• data sampling period = 90 seconds 
• data sampling frequency = 2000 Hz 
• hot-wire effective overheat ratio = 1.4 
Th · . · d f. · studies which showed that 
e samplmg period was determme 10m previous 
after 90 . h l ·t d vorticity components varied 
seconds the variances of all t e ve oc1 Y an 
22 
within 0.3 %. The 2000 Hz sampling frequency was limited by the total throughput 
of the A/ D converters. 
2.4,3 Vorticity Measurements 
The vorticity measurements began with preliminary experiments to establish the 
flow quality and probe capability, and to develop the data reduction software. 
The first preliminary experiments were performed in the calibration jet with the 
9-sensor vorticity probe described by Vukoslavcevic et al. [63]. Data were obtained 
with pitch and yaw variation as described in the previous section. As the flow 
was nominally irrotational all velocity gradients should be nearly zero. Any non-
zero velocity gradients represent the noise level of the overall system which includes 
the probe and measurement system, the calibration and data reduction methods, 
and any low level rotationality of the calibration flow. Several experiments were 
performed in the jet to estab li sh the accuracy of velocity and vorticity measurements. 
One conclusion from these measurements was that some of the measured spurious 
gradients at the extreme angles (±20° pitch/yaw) can have values higher than the 
anticipated maximum mean shear in the wake region x /d = 50 - 120 . 
The next preliminary experiments were wake measurements with the 9-sensor 
probe at x/ d = 120. Repeated measurements confirmed the earlier indications that 
the maximum mean shear is of the same order of magnitude as the spurious gradients 
for this x / d location. Even though the fluctuating gradients can be an order of 
magnitude larger than the mean shear, it was decided that a confidence test would 
be how well the directly measured mean shear agrees with the derivative of the mean 
streamwise velocity profile. This test was not satisfied for these measurements. The 
skewness factors of the velocity and vorticity components also shed light on the 
accuracy of the measurements. The skewness factors of the spanwise velocity and 
the streamwise and transverse vorticity components should all be approximately zero 
23 
across the turbulent wake due to the two-dimensionality of the mean flow. These 
m easured skewness factors showed a variation of ±0.7, which is quite high. These 
experiments thus indicated that wake measurements at this x / d station need more 
accuracy than the 9-sensor probe can provide due to the low signal-to-noise ratio. 
Further measurements in nominally irrotational , uniform flow provided by the 
calibration jet confirmed that the 12-sensor probe can yield greater accuracy than 
the 9-sensor probe. The spurious velocity gradients were lower than those measured 
with the 9-sensor probe for the same conditions. Definitive measurements were 
performed in the wake with the 12-sensor probe at x /d = 20, and 30 for the same 
conditions as the velocity measurements with the plus probe. More details about 
the accuracy of the plus probe and the 12-sensor probe are given in Section 3.5. 
24 
3 Calibration and Data Reduction Methods 
3.1 Calibration Method 
In the probe calibration method developed by Marasli et al. [37], Jorgensen 's law [2S] 
is used to express the effective cooling velocity of each sensor as a nonlinear function 
of 3 velocity components for any inclined sensor in uniform flow, i.e. 
(3 .1) 
where UN, Ur, and VB are the normal, tangential and binormal components of 
the cooling velocity with respect to the sensor (see Fig. 3.1) , and Cr and CB are 
the t angential and binormal cooling coefficients, respectively. For each sensor , the 
effective cooling velocity is expressed as a 4th order polynomial , P (e), of the voltage 
drop across the sensor: 
(3.2) 
One can transform UN, Ur, and VB from a coordinate system attached to the 
sensor to a more convenient system, where the x-direction is along the probe axis, 
giving 
(3.3) 
Ur (3.4) 
(3.5) 
where n i , t i and bi (i = 1, 2, 3) are the coefficients of the coordinate transformation 
which can be determined by measuring the angles the sensors make with the probe 
axis. However, accurate measurement of these angles fo r a miniature, multi -sensor 
25 
probe is very difficult. It is more convenient to combine these geometric coefficients 
with the cooling velocity coeffi cients and determine them by direct calibration. 
Substituting Eqs. (3.3) to (3.5) into Jorgensen's cooling law yields 
(3.6) 
The coefficients Aj (j = l, 10) can be calculated for each sensor by calibrating the 
probe in a known irrotational flow with preset flow angles . For flow speed Q at given 
pitch I and yaw 'ljJ angles (illustrated in Fig. 2.3), the induced velocity components 
a t the sensor are: 
u Q cos I cos 'ljJ , (3.7) 
V Q sin, cos 'ljJ , (3 .8) 
HI Q sin ~fa . (3 .9) 
To get the 10 coefficients A.i for each sensor, more than 10 calibration points are 
required to account for the range of speeds and angles of attack the probe encounters 
in a turbulent flow. For each sensor , given the bridge voltage ei corresponding to 
the known calibration velocity components U; , Vi and W; (i = 1, Ne), where Ne is 
number of calibration points, the unknown coefficients Aj can be determined by a 
linear least- square method from the fo llowing linear system of equations: 
Ne e e2 e3 e4 
---;, w 2 U V u w vw u 2 V " A 1 
e e2 e3 e4 e5 ev 2 ew 2 eu v eu w ev w A2 eu 2 
e2 e3 e4 e5 e6 e2v2 e2w2 e2uv e2uw e2vw A3 e2 u 2 
e3 e4 e5 e6 e7 e3v2 e3 w 2 e3u v e3uw e3v w A4 e3 u 2 
e4 es e6 e7 es e4v2 e4 w 2 e4u v e4·uw e4vw A5 e4 u 2 
v2 ev 2 c2v2 e3v2 e4v 2 v 4 v2w2 u v3 uv2w v3w A6 u 2v2 
w 2 ew 2 e2w2 e3w2 e4w2 v2 w2 w4 u v w
2 u w3 vw3 A1 u.2w2 
UV eu v e 2u v e3uv e4uv uv3 u v w 2 u 2v2 u 2v w u v 2w A s u3v 
u w euw e2u w e3uw e4uw u v2w uw3 u2v w u 2w 2 u vw2 A 9 u3w 
vw e vw e2vw e3vw e4v w v3w vw3 uv 2w u vw2 v2w2 A 10 u 2vw 
26 
where the lower case characters are used here to fit the above matrix in the available 
space, and 
N e 
e=I:e;, 
i==l 
Ne 
e2 = ~ e2 etc ,, . 
i==l 
(3.10) 
A typical calibration data set and the calibration curve P( e) for one of the sensors 
is shown in Fig. 3.2. In this data set, the pitch and yaw angles were varied within 
±20° at two different speeds, Q=5 m/s, and 3.2 m/s. In addition, data were obtained 
at zero pitch and yaw for speeds over the range 2.6-5. 7 m/s. In Fig. 3.2, the ordinate 
values of the symbols correspond to the right-hand-side of Eq. (3.6), where all the 
velocity components are the known induced values given by Eqs. (3. 7) to (3.9). The 
solid line is t he fourth order polynomial curve fit which represents the left-hand-side 
of Eq. (3.6). The collapse of all points on one curve indicates that Jorgensen's 
cooling equation represents well the response of a hot-wire in a combined pitch and 
yaw orientation. Lekakis et al. [35] previously reached the same conclusion. This 
calibration method has the addit ional advantage that it is independent of probe 
geometry, and sensor orientation. The technique can be used for both velocity and 
vorticity measurements, where for the latter, only the spacings between the prongs 
are required for the data reduction. 
The calibration space is divided into 9 zones as shown in Fig. 3.3 for more 
accurate calculations of the calibration coefficients . Four zones are at the corners of 
the pitch-yaw plane having angles higher than ±10°. Another zone is at the center 
having angles smaller than ±10°. The last 4 zones fill in the remaining space. For 
the x / d = 30 measurement set, about 95% of the samples fall within the zone having 
angles less than ±10°; less than 1 % fall within each of the other zones . In the 
solution procedure of turbulent fl.ow samples, the global ("global" refers to the entire 
calibration set) cal ibration coefficients were used first to estimate the pitch and yaw 
angles . These angles determined which calibration zone the turbulent sample is in 
and the appropriate zonal calibration set was used for better accuracy. The solution 
27 
.• .~--
was flagged if the calculated flow angles jump outside the appropriate zone. The use 
of these methods for wake velocity measurements has been briefly documented by 
Nguyen et al. [44}. 
3.2 Data Reduction Method for Velocity Measurement 
The following method is applied to a 4-sensor probe , but it can be used for any probe 
with 3 or more sensors. The method assumes that the spatial resolution of the probe 
is sufficiently small so that the gradients across the sensing volume can be neglected. 
In order to reduce the uniqueness problem (addressed in Section 3.4) associat ed with 
the triple-sensor response equations reported by Dobbeling et al. [1 41, the present 
d a t a reduction scheme uses the information from all four sensors simultaneously and 
solves the overdetermined nonlinear system of equations in a least-square sense. For 
each sensor, one can rewrite Eq. (3 .6) as 
Here, the subscript j denotes a sensor. In thi s equa tion , A;j (i = 1, 10) and P.i (e.i ) 
are known from the calibration and the anemometer output voltage; U = (U, V, vV) 
is the unknown velocity vector. Thus , we have a system of 4 nonlinear algebraic 
equations with 3 unknowns. The goal of the solution scheme is to <letermine U 
which minimizes the error defined as 
N, 
F Lf.T, (3.12) 
j=l 
where N s is the number of sensors , which is four. 
A variety of iterative t echniques are available for the solution of nonlinear alge-
braic systems. Th e simples t one is Newton's method. With an initi al guess for the 
unknown vector, the incremental correction 6.U to the solution U is obtained from 
(3.13) 
28 
The solution at the nth iteration step is updated using 
(3.14) 
where the superscripts denote the iteration level. The procedure is repeated until 
convergence is achieved within a specified tolerance. For the results presented here 
th
e solution was usually obtained within five iterations. Another solution method 
' 
called Broyden-Fletcher-Goldfarb-Shanno [49], was also tried and was found to be 
less dependent on the initial guesses than Newton's method, but it took almost twice 
th
e calculation time. For the plus probe (in Fig. 2.5), an initial guess was obtained 
by treating the probe as 2 V-arrays which only need the pitch-only and yaw-only 
calibration data for calibration coefficients. 
The calibration data were tested to see whether the calibration velocities were 
recovered from the bridge voltages. Most velocities were usually recovered to within 
less than 0.5% and none were more than 1 % of the flow speed. Calculated turbulent 
velocity solutions were rejected if the computed angles of the velocity vectors were 
outside tl 1·1 · h /d 30 t t t· th 1e ca 1 )rat10n range. For t e x = measuremen s a 10n, ere were 
practically no rejection. 
3
·
3 Data Reduction Method for Vorticity Measurement 
When a multi-sensor probe is used in a shear flow, the velocity vector seen by each 
sensor will be different, i.e., the flow is not uniform across the sensing volume. The 
effect of the velocity gradients on multi-sensor probe performance has been demon-
strated by Vukoslavcevic & Wallace [62) and Park & Wallace [46). Vukoslavcevic 
et al. [63] used a 9-sensor probe to measure the spanwise and cross-stream velocity 
gradients in addition to the three components of the instantaneous velocity vector 
lil a turbulent boundary-layer. They calculated the streamwise gradients from the 
temporal gradients using Taylor's hypothesis and thus were able to obtain all three 
components of the instantaneous vorticity vector as well. The 12-sensor probe is 
29 
an ext · f' ens1on o the 9-sensor probe and the extra sensors were added to widen the 
cone of acceptance and to eliminate some of the uniqueness problems. Dracos et al. 
[l3) presented gradient measurements from a turbulent grid flow obtained with a 12-
sensor probe; however, their calibration and reduction algorithm were quite different 
from the present one. 
The data reduction for the 12-sensor probe follows a very similar procedure to 
that of the 4-sensor probe presented earlier, except now the gradients across the 
sensing volume are taken into account. Following Vukoslavcevic et al. [63], one can 
expand the velocity measured by each wire, Uj, in a Taylor series to first order 
around the centroid of the frontal area of the probe, i.e., 
uj === u + c · au au (3.15) 
J ~ +dj-;:;-, 
uy uz 
where the right-hand-side is evaluated at the probe centroid. The constants Cj and di 
are the vertical and horizontal distances from the centroid of each wire to the centroid 
of the probe, respectively. Thus, in this procedure, instead of the velocity, all velocity 
gradients are assumed to be uniform across the sensing volume. Substitution of Eq. 
(3.15) into Eq. (3.11) yields 12 nonlinear algebraic equations given by 
f 2 au au i =: -Pi+ U + 2ciU-;;- + 2d1U-;;-
uy uz 
-A,; [v' + 2c;v!~ + 2d;V!~] 
-A,; [w' + 2c;W~: + 2d;W~~l 
-Asj [uv + c· (u 8v + vau) + d1 (u~v + v~u) ] 
J ~ ~ uz ~ 
-A.; [uw + c; ( u~: + w!~) + d; (u~~ + w!~)] 
-A10; [vw + c; ( v~: + w~~) + d; ( v: + w~~)] 
wi th the 9 unknowns 
D 2: uk == ( au au av av aw, avv) . 
u , v , w , ay , oz , oy , oz , oy a z 
30 
0, (3.16) 
(3.17) 
This nonl' . 
meai system can also be solved m a least-square sense usmg Newton's 
method h . h 
' w eie t e error, defined by Eq. (3.12), is minimized. The incremental 
correction to th 1 t· h . . 
linear system 
e sou 10n at t e nth 1terat10n step can be computed from the g x g 
(3.18) 
where 
(3.19) 
is 
th
e gradient vector consisting of the derivatives of F with respect to each of the 
unknowns, and 
(3.20) 
is 
th
e Hessian matrix composed of all possible second derivatives of F with respect to 
each of the unknowns. An initial guess for the velocity components was obtained by 
treating h 4 . . eac . -w1re array as two mdependent V-arrays . The initial gradients were 
computed by differencing the velocities from the three 4-wire arrays . Convergence, 
within a ·ti d h" d · l · ti · · Tl · } spec1 e tolerance, was usually ac 1eve wit 1m ve 1terat10ns. 1e p1tc 1 
and y 
aw angles encountered by each sensor were checked a posteriori in order to 
Verify the integrity of the solution. The solutions that were outside the calibration 
ran ge Were rejected. 
The present calibration and data reduction scheme can easily be applied to probes 
Wi th different geometries. The only geometrical information needed is the distance 
from the center of each wire to the centroid of the frontal sensing area of the probe. 
Then b · · f 9 · · um er of sensors need not be fixed at 12, but a mm1mum o 1s necessary. 
Solution Uniqueness 
For the 4-sensor probe, using only 3 sensors can result in non-convergence for certain 
Probe orientations whereas using all 4 sensors results in convergence to one solution 
31 
for all the calibration angles within ±20°. To illustrate the convergence character-
istics of the velocity measurements, error contours are drawn in the U _ V plane 
demonstrating the minimum error region containing the solution. Here, the error F 
in Eq. (3.12) is expressed in terms of U and V. This is done by eliminating w from 
the four expressions in Eq. (3.11 ). One can further eliminate U or V to obtain F 
as a function of only one variable (see for example Vukoslavcevic et al. [63]). For a 
calibration point at +20° pitch and -20° yaw, Fig. 3.4 shows the error contours in 
the U - V plane with the inner contours having lower values than the outer ones. 
The data reduction converges in 3 iterations to the physical solution represented by 
the unique minimum when all 4 sensors are used , as seen in Fig. 3.4(a). When 
only 3 sensors are used, the topology of the solution surface changes. Instead of the 
unique minimum, a valley containing many local minima develops, as depicted in Fig. 
3.4(b ). In this case, the solution algorithm jumps from one local minimum to the 
next, as a clear indication of the nonuniqueness . These convergence characteristics 
are consistent with the findings of Dobbeling et al. [14]. 
For the 12-sensor data, there are 9 unknowns which are the 3 velocity components 
at the probe centroid, and the 6 velocity gradients as shown in the previous section. 
Since the equation s are nonlinear, it is not possible to eliminate enough unknowns 
to depict the error as a contour plot as was done for the velocity-only measurements. 
However, the uniqueness study for the 4-sensor measurements indi cates that , for 
relatively high angles of attack (greater than ±15°), neglecting the sensor with the 
lowest effective cooling velocity within one array can lead to nonuniqueness problem. 
Conversely, for all the calibrations points within ±20°, neglecting the sensor with the 
highest effective cooling velocity in one array leads to unique solution which is close 
to that obtained with all 4 sensors . For the 12-sensor data, using all sensors converges 
in 5 iterations whereas neglect ing the 3 sensors having lowes t cooling velocity in each 
array leads to divergence. This, however, does not prove uniqueness of the 12-sensor 
32 
data. 
For a turbulent sample in the centerline region at x /d - 30 one d' · 
1 - , - 1mens1ona 
error curves for the 3 velocity components are shown in Fig. 3.5. In each plot, the 
remaining 8 variables are held at their converged values. In Fig. 3.5(a), the error 
curve for Uhas two minima; therefore, U can have two possible solutions. However 
' 
only one minimum has positive U; the other does not have a physical meaning. Plots 
(b) and (c) for velocity components V and W show only one minimum, indicat ing 
one solution each. The error function F in Eq. (3.12) are fourth-order polynomials 
in U, V, and W. The components V and W have only one minimum because the 
fourth-power term is the most dominant. Similar plots for the velocity gradients show 
only one minimum for each since F in Eq. (3.12) is parabolic in these gradients. 
3.5 Measurement Accuracy 
To estimate the accuracy of the data reduction system and the probe, one can apply 
the data reduction method on the calibration data. This means giving the data 
reduction system the calibration sensor voltages and the calibration coefficients to 
calculate the induced calibration flow velocity components. Any nonzero velocity 
gradients computed from the calibration data will be spurious since the probe was 
calibrated in a steady, uniform, nominally irrotational flow. For this purpose, zone l 
in Fig. 3.3, which has angles between 10° - 20°, was chosen. Note that every 
zone needs the additional zero-angle calibration points for accurate calculation of 
the calibration coefficients. Table 1 shows the calculated velocity components, and 
the errors between the measured values and the known, induced values from Eqs. 
(3.7 - 3.9). These errors are normalized by the calibration speed Q. Most of the 
large angle points have higher errors than the small angle points, as expected . All of 
the calibration points have relati vely low errors which are less than or equal to 1 % 
of the calibration speed . Table 2 shows the calculated spurious velocity gradients for 
33 
tl1e Same calibration points. Aga1'n her·e th l l · 
, , e arge ang e pomts usually have higher 
spurious gradients than the small angle points. Most of the spurious v I 
a ues are much 
smaller than the maximum mean shear which is approximately 65 s- 1. 
From Table 1, the average error is 0.008 m/s for U, 0.005 m/s for V and 0.007 
m / s for W. When normalized with u0 =0.82 m/s at x /d = 30, these uncertainty 
values are 1.0 % for U, 0.6 % for V, and 0.9 % for W. For the global calibration set 
' 
these uncertainty values are 1.7 % for U, 1.5 % for V and 1.8% for W. From Table 
2, the largest average spurious velocity gradients are about 5.2 s-1 , or 5.4 % of the 
ratio u 0 / Lo for x /d = 30. For the global calibration set , this value is about 14.7 s-1. 
These average spurious gradients can give an estimate of the uncertainty limits for 
the measured vorticity. 
3.6 Summary 
The di scussion in the above sections has demonstrated the following: 
l. The present calibration and data reduction method is independent of the probe 
geometry and sensor orientation. For vorticity measurements, the only geo-
metrical information needed is the distance from the center of each wire to the 
centroid of the front al sensing area of the probe. 
2. The error minimization method , when used with more sensors than the number 
of unknowns, can produce unique solutions of the measured velocity for angles 
of attack within ±20°. 
3. For calibration angles within ±20° , neglecting the sensor with the lowes t cool-
ing velocity from the plus probe results in divergence during data reduction. 
Conversely, neglecting the sensor with the highest cooling velocity results in 
convergence to practically the same solution as when all sensors are used. 
4. The zonal calibration scheme can improve measurement accuracy. 
34 
# u V w %error of (U,V,W) pitch yaw 
1 4.582 1.671 0.860 -0.1 0.0 0.0 20. 10. 
2 4.505 1.635 1.279 0.1 0.0 -0.1 20. 15. 
3 4.385 1.592 1.695 0.0 -0.1 -0.1 20. 20. 
4 4.508 1. 211 1. 697 0.0 0.1 0.0 15. 20. 
5 4.639 1. 243 1.284 0.1 0.0 0.0 15. 15. 
6 4. 720 1. 267 0.858 0.1 0.1 -0.1 15. 10. 
7 4.806 0.847 0.867 -0.2 0.0 0.1 10. 10. 
8 4.719 0.834 1.287 -0.1 0.0 0.0 10. 15. 
9 4.595 0.810 1. 698 0.0 0.0 0.0 10. 20. 
10 4.948 -0.005 -0.005 -0.3 -0.1 -0.1 0. 0. 
11 4.983 -0.005 -0.007 0.1 -0.1 -0.1 0. 0. 
12 3.159 -0.002 0.006 -0.2 -0.1 0.2 0. 0. 
13 2.941 1. 066 0.558 0.1 -0.1 0.2 20. 10. 
14 2.877 1.055 0.831 0.0 0.2 0.3 20. 15. 
15 2.769 1.036 1.110 -1.0 0.5 0.8 20. 20. 
16 2.875 0.767 1.094 -0.1 -0.1 0.3 15. 20. 
17 2.973 0.790 0.815 0.5 -0.1 -0.2 15. 15. 
18 3.022 0.797 0.555 0.1 -0.4 0.1 15. 10. 
19 3.067 0.541 0.539 0.0 0.0 -0.3 10. 10. 
20 3.015 0.528 0.810 0.5 0.0 -0.2 10. 15. 
21 2.930 0.514 1.078 0.4 0.0 0.0 10. 20. 
22 3.172 -0 .001 0.000 0.2 0.0 0.0 0. 0. 
23 3.161 0.002 -0.006 0.1 0.1 -0.2 0. o. 
24 2.578 0.006 0.005 0.0 0.2 0.2 0. 0. 
25 2.760 0.010 - 0.004 -0.2 0.4 -0.2 0. 0. 
26 2.980 0.001 -0.005 -0.4 0.0 -0.2 0. 0. 
27 3.184 0.002 -0.004 -0.4 0.1 -0.1 0. 0. 
28 3.401 0.001 -0.003 0.0 0.0 -0.1 0. 0. 
29 3. 664 -0.002 0.006 0.0 -0.1 0.2 0. 0. 
30 3.868 -0.001 - 0.002 0.0 0.0 -0.1 0. 0. 
31 4.035 -0.003 -0.001 0.1 -0.1 0.0 0. 0. 
32 4.308 -0 .003 - 0.002 0.0 -0.1 0.0 0. 0. 
33 4.529 - 0.005 - 0.004 -0 .2 - 0.1 -0.1 0. 0. 
34 4.715 -0.002 - 0.001 0.0 0.0 0.0 0. 0. 
35 4.949 -0.006 -0.003 0.0 -0.1 -0.1 0. 0. 
36 5.139 0.000 0.000 0.0 0.0 0.0 0. 0. 
37 5.347 0.000 0.000 0.1 0.0 0.0 0. o. 
38 5.553 0.005 0.005 0.2 0.1 0.1 0. 0. 
39 5.721 O.Oll 0.010 0.0 0.2 0.2 0. 0. 
Table 1: Data reduction of velocity components (in m/s) for the calibration data of 
the 12-sensor probe. The errors are non-dimensionalized by the calibration speed. 
35 
# dU/dy dU/dz dV/dy dV/dz dW/dy dW/dz pitch yaw 
1 -1. -1. -2. -1. -1. 3. 20. 10. 
2 -2. -2. -1. 2. 2. 1. 20. 15. 
3 7. -7. -7. 5. -7. 1. 20. 20. 
4 -2. -4. 5. 5. 5. 0. 15 . 20. 
5 -5. -1. 6. 1. 9. -1. 15. 15. 
6 -3. 0. 0. -2. 3 . 0. 15. 10. 
7 1. -1. -5. -2. -7. 2. 10. 10. 
8 -1. -1. 2. 0. 0. 0. 10. 15. 
9 -2 . -1. 2 . 2. 1. -2. 10. 20. 
10 2. -6. 3 . -9. -6. 0. 0. 0. 
11 2. -6. 3. -9. -6. 0. 0. 0. 
12 0. -5. 2 . -1. 2. - 2 . 0. 0. 
13 7 . 2. 8. 3. -1. -6. 20. 10. 
14 -2. 13. 10. -7. 4. -6. 20. 15. 
15 -2. 28. 3. - 26 . 0. -6. 20 . 20. 
16 7. 14. -7. -14 . -9. 2. 15. 20. 
17 3. 8. 1. -2. -4. -1. 15. 15. 
18 5. 1. 2 . 7 . -7 . -2. 15. 10. 
19 6. 2 . -4 . 6 . -4 . 1. 10. 10. 
20 4 . 5. -5 . 0. - 1. 2 . 10 . 15. 
21 7. 5. -1 2. - 8 . - 6 . 6 . 10. 2 0 . 
22 0. -5. 4. - 2 . o. - 2. 0. 0. 
23 -2. -3. 0. 3 . 0. -2. 0. 0. 
24 -4. -9. -1. 19. 22 . 2 . 0 . 0. 
25 -4. -8 . 0. 12. 8 . 1. 0 . 0. 
26 -3. -5 . 1. 6 . 3. 0. 0. o. 
27 -2 . -2 . 1. 3 . 0. - 1. 0. 0. 
28 -1. -1. 1. 0. -1. - 2 . 0 . 0. 
29 0. 1. 1. - 2 . - 2 . -2. o. 0. 
30 0. 1. 0. -4. -3. -2. 0. 0. 
31 -1. 1. 1. -5. -5. -1. 0 . 0. 
32 -1. 0. 1. -5. - 6. -1. 0 . 0 . 
33 -2. 0. 1. -4 . -6 . 0. 0 . 0. 
34 -2 . 0. 1. -4 . -6. 0. 0. 0 . 
35 -2. 1. 0. - 2 . -4. 1. 0 . 0. 
36 -1. 1. 0. 0 . - 2 . 1. 0 . 0 . 
3 7 0. 2. -1. 3 . 2 . 1. 0 . 0 . 
38 1. 4. -2. 7. 7. 1. 0. 0. 
39 3 . 5 . - 4. 9 . 13. 0. 0 . 0 . 
Table 2: Data reduction of velocity gradients (in s- 1 ) for the calibration data of the 
12-sensor probe. 
36 
4 Characteristics of the Velocity Field 
4.1 Statistical Properties 
In this section, the mean streamwise velocity, the moments of the fluctuating veloc-
ity components and the Reynolds shear stresses are presented. These measurements 
were obtained with the 12-sensor probe. Any 4-sensor measurements used for com-
parison were obtained with the plus probe. For the x/d = 30 measurement station 
' 
the wake half-width L 0 , used to normalize the wake posit ion, is 8.5 mm, and the cen-
terline wake deficit u0 , used to normalized all Reynolds stresses, is 0.82 m/s. Unless 
otherwise noted, all measured variables are nondimensionalized by u0 or L 0 , which 
are obtained from 
( 4.1 ) 
where TJ = y / L 0 , CY = 0.88, and /3 = cosh- 1 v'2 - CY = 0.00137. The values for 
CY and /3 are empirically determined to give the best-fi t curve to the measurements . 
Generally, it is found that the value for CY decreases for measurement locations further 
downstream of the cylinder and settles on the value 0.78 for the self-preserving wake 
(Marasli et al. [36]). The measured U profile is qui te symmetric, as shown in F ig. 
4. 1 (a) along with the best-fit curve. Both the V and vV distributions are nearly 
zero and are not shown in the plot. From the above curve-fit of U and the continui ty 
equa tion, V is calculated to have maximum value of only 2% of the wake deficit, 
whi ch is of the same order as the measurement uncertainty. 
F ig. 4.l(b) shows that the maximum nns value of v is slightly higher than that 
of u due to the vortex shedding in the near-wake region; the maximum nns value 
of w is considerably lower. The average turbulence intensity in the frees trearn is 
about 1/20 of those near t he centerline, which indicates a high signal-to-noise ratio. 
37 
The x / d = 30 measurement station is in the region where the energy of the Karman 
vortex is still quite distinct. Roshko [53] found that the Karman vortex persists up 
to x / d = 50 - 70. Downstream of this region, Yamada et al. [70] measured all 
three components of the velocity vector and found that the rms for the u component 
is highest, with the other two about the same. For the current measurements the ) 
u and w components have small double peaks approximately 0. 7 Lo from the wake 
centerline, while the rms value of v peaks at the wake centerline. The nns values of 
the velocity components from the 4-sensor velocity probe have the same trends and 
approximately the same magnitudes, and therefore are not shown. 
The skewness factors of the velocity components in Fig. 4.l(c) show symmetry 
of the u component, antisymmetry of the v component, and nearly zero values of 
the w component. The u skewness is negative on both sides of the wake because 
the wake is always spreading outward; low momentum fluid from the centerline is 
being transferred to the wake edges. At the centerline, the u skewness value is about 
+O.l , consistent with observation of Fabris [17] in the far-wake region. This positive 
skewness value could indicate that a small transfer of high momentum fluid from 
the wake edges is occuring. The skewness value decreases to a minimum of about 
- 3.0 at approximately 'T/ = ±3 then goes back to zero in the freestream region. The 
v skewness is negat ive in the lower wake and positive in the upper wake. This is 
consistent with the outward spreading of the wake, which is characterized by negative 
lateral velocity in the lower wake, and positive lateral velocity in the upper wake. The 
w skew · k ·t d f ·om tl1e two-dimensionality ness 1s nearly zero across the wa e as expec e 1 
of the mean flow. 
Th fl t l · t ·n Fig 4.l(d) are about 3.0 e a ness factors for all three ve oc1 ty componen ;s 1 · 
around the wake centerline, indicating an almost Gaussian distribution in this region. 
The w k 1 h' ·l l es due to the intermittent 
a e edge regions, however , have extreme Y 1g 1 va u 
turbulence there. Also, the maximum v flatness factor is higher than that of the u 
38 
fiatness factor and occurs nearer the wake edges. A similar pat tern is noted for t he 
skewness factor. This pattern was also observed in the measurements at x /d = 400 
of Fabris (1 7]. Presently, a satisfactory explanation has not been found. In general, 
the skewness and flatness data at x /d = 30 agree well with the measurements of 
Fabris which were much further downstream. 
Fig. 4.2 shows the comparison of the nondimensional rms values ( denoted by 
superscript *) at x / d = 20 and 30. For x / d = 20, L 0 = 6.5 mm and u 0 = 0.93 
rn/s, and the u* and w * values are only slightly higher than those at x/ d = 30. The 
v* values, however , are significantly higher , reaching as much as 25% higher in the 
centerline region. This is likely due to the Karman vortices. The differences between 
the nondimensional nns values a t the two downstream locations indicate tha t the 
wake is not yet in the self-preserving region, which Wygnanski et al. [69] have found 
to occur more than 300d downstream of the cylinder. The Karman vortex decays in 
the region O - 50d according to Roshko [53], as is most evident in the large decrease 
in the v* values. Higher moments at x / d = 20 have similar characteristics as those 
a t x / d = 30 thus are not shown. 
To illustrate the effect of accounting for velocity gradients and fo r the third 
velocity component in the probe sensing area, the 12-sensor rms data at x /d = 
30 are compared wi th the V-probe values from one array of the 12-sensor probe 
(hereafter referred to as the "V-probe data" ) in Fig. 4.3. For the V-probe data, 
the u component represents the average of the two orthogonal V-probes. All three 
rms components are slightly higher than those from the 12-sensor probe. The 12-
sensor probe has a larger sensing area, but it accounts for the velocity gradients. 
The 12-sensor data should be more accura te. The small difference between the two 
probes indicates that neglecting the t hird component and the velocity gradients have 
a small effect in the rms data of plane wake flow. However, Park & Wallace [46], 
and Vukoslavcevic et al. [63] found that neglecting the velocity gradients, as must 
39 
be done with the V-probes and the plus probe, can lead to large errors in near-wall 
regions of the boundary-layer. 
The 12-sensor rms data at x /d = 30 are a lso compared with X-probe measure-
ments of Yamada et al. (70] in Fig. 4.3. The Yamada et al. data are at the same 
x /d location, but at Red = 4000, and were measured with commercial X-probes 
which have sensor spacing of approximately 1 mm, i.e. about twice that of t he 12-
sensor probe. The X-probe does not account for the velocity gradients and the third 
velocity component, but t hese effects have been shown to be small in the previous 
paragraph. What remain are the Reynolds number and the sensor spacing effects, 
which cannot be separated out. The Kolmogorov scale in Yamada et al. was smaller 
than that in the current measurements due to a higher Reynolds number , but they 
were not able to provide estimates of the Kolmogorov scale. The 12-sensor probe 
can measure a ll the necessary velocity gradients in the expression for the dissipation. 
For R ed = 2000 and x / d = 30, the Kolmogorov scale is approximately 0.2 - 0.4 mm 
across the wake. 
In Fig. 4.3, the u rms component from the 12-sensor probe is slight ly lower, and 
the v and w components are significantly lower than the X-probe data. Analysis 
of a direct numerical database by Suzuki & Kasagi [56] indicated that the sensor 
spacing has a strong effect on the measured rms velocity components. They showed 
that for an X-probe configuration, the v and w rms values are overestimated signif-
icantly when the sensor spacing is increased. Park & Wallace [46], and Tagawa et 
al. [57] came to the same conclusion when they evaluated the X-probe response in 
a simulated wall turbulence flow. These previous studies indicate that the rms data 
obtained with the miniature 12-sensor probe are likely to be closer to the true values. 
The Reynolds shear stresses are shown in F ig. 4.4. The - u v stress is negative 
m the lower wake, zero on the centerline, and posit ive in the upper wake. This is 
consistent with the u and v skewness factors which have the same signs in the lower 
40 
wake, and opposite signs in the upper wake. The difference between the 4-sensor and 
12-sensor probe measurements is also illustrated For the -u 
· v component, the two 
sets of da ta are about the same. The -uw stress, which should be zero everywl . . 
1e1e, 1s 
about 25% of the peak -uv stress at the centerline for the 4-sensor data F th 
· or e 12-
sensor data, it is considerably smaller everywhere across the wake Th -
· e -vw stress 
]. s about zero for both sets. The non-zero uw from the 4-sensor probe is sp · I 
unous. t 
is inconsistent with the nearly zero values of W and w skewness, which reflects the 
equal probability of negative and positive w fluctuations. Also, flow visualization in 
Fig. 2.8 showed that t he vortex shedding was quite parallel to the cylinder, which 
indicates two-dimensionality of the mean flow, and implies no preference for either 
negative or positive w flu ctuations. The cause of this error must therefore be the 
assumption for the 4-sensor probe that t he velocity is uniform within the sensing 
area of the probe. T his assumption is least applicable in the centerline region where 
the turbulence is highest . 
In the comparison in F ig. 4.5, the -uv values of Yamada et al. are almost twice 
the 12-sensor data near the centerline, consistent with the higher 11, and v rms values. 
Again, the differences might also be a.ttributed to their higher Reynolds number. The 
V-probe -u,v data a re about the same as those from the 12-sensor probe, which is 
consistent with t he rms comparison. For the - ttw component , the V-probe a.nd 12-
sensor data are about the same, but the -vw stress for the V-probe reaches 35% 
of the peak - uv stress. X-probe data from Yamada et al. were not available for 
comparison, and to t he author's knowledge, there are no other published X-probe 
data for t hese stresses. In summary, the comparisons for a ll the Reynolds stresses 
indicate that the 12-sensor probe data show the greatest self-consistency. 
41 
4.2 Velocity Spectra 
The spectra for all components of the velocity vector are presented and discussed 
in this section . Six cross-stream locations covering both the lower and upper layers 
were chosen for thi s purpose. The spectra are normalized by the local variance of 
the individual component so that the area under each is unity. The di scussion 1s 
representative of all the 21 wake locations where data were taken. 
The u and v spectra in Figs . 4.6 and 4. 7 show strong peaks at the vortex shedding 
frequency, l s = 168 Hz. This frequency corresponds to a Strouhal number of 0.21, in 
excellent agreement with the value reported by Roshko [53] at thi s Reynolds number. 
The w spectra in Fig. 4.8 do not have any di stin ct peak above the background; this 
indi cates tha t the shed vortex is parallel to the cylinder. The Karman vortex, if it is 
shed para llel to the cylinder , is in the z-direction so it can only influence fl ow in the 
x - y plane. The proper detection of the shedding frequency is another indication of 
the accuracy of the measurements made by the 12-sensor probe. 
The peaks for the 1t spect ra di sappear in the centerline region, and reach max-
imum values around T/ = ±l from the centerline. On the other hand , the peaks 
for the v spectra are most prominent in the centerline region , and diminish toward 
the wake edges . This pattern of the power peaks for these two compon ents can be 
predicted by linear stabili ty theory (see for example Mattingly & Crirninale [381). 
The w spectra show little change across the wake. All w spectra have rela tively 
flat di stributions up to about the shedding frequency, then drop off at higher fre-
quencies. This pattern is also observed for the v spectra at the wake edges . For 
the v spectra near the centerline region , the neighboring frequencies of J~ have high 
energy content. This is because the background v- spectrum has a local maximum 
at a frequency slightly smaller than j ~ which corresponds to the locally neutral fre-
quency from linear stability th eory (Marasli et al. [361). All frequencies above f s 
are damped , which is consistent with the fact that the Karma.n vortex street decays 
42 
... 
wi th downstream distance. 
4.3 Probability Density Functions of the Velocity 
Components 
The prob bTt d 
a 1 1 Y ensity functions (PDFs) for all three components of the fluctuating 
velocity are presented in Figs. 4.9 to 4.11. Each velocity component is normalized 
by the wake deficit u 0 • The area under the PDF curve is unity by definition. The 
six wake l t· . h . . 
oca ions m t ese figures are for the same locat10ns as m the spectra plots 
for f 
ease O comparison. The following discussion is representative of a11 positions 
measured. 
In Fig. 4.9, the PDFs for the u components are skewed at the wake edges and 
quite symmetric in the centerline region. This symmetry reflects the near-zero skew-
ness in the wake centerline. For both the upper and lower wake, the PDFs have most 
Probable +u values but have longer -u tails indicating penetration of low momentum 
fl . d 
tu from the centerline region into the outer edges of the wake. This corresponds to 
th
e negative skewness values in this region. The PDFs for the wake edge (r, = - 1.65) 
have Very high peaks near the mean value. This is why the flatness values are so 
hi h. 
g lI1 these regions. High fluctuations, about 2 times the wake deficit, rarely occur. 
Near th . · h 'd d' ··b t· e wake centerlme, the peak values are much lower, wit w1 er 1stIJ u 10ns 
aroun<l the mean, which is also indicated by the high nns values there. This pattern 
reflects the high degree of mixing occuring in the centerline region. The PDFs are 
closer to the Gaussian distributions in the centerline region than in the wake edges. 
In Fig. 4.10, the PDFs for v show the antisymmetry expected from the skewness 
dist ·b · l I I fl ution. For the lower wake, the PDFs have most probable +v va ues )Ut 1ave 
longe l · tl · ·· For r -v tails, which means that v has negative skewness va ues m 11s regwn. 
th . 
e Upper wake, the opposite pattern occurs. At both wake edges, large v fluctuations 
Occur much less frequently than small fluctuations which is why the flatness values 
43 
. ; . ------
are high there. In the centerline region (ry = 0.12) the PDF · ·t . 
' is qm e symmetn c so 
that the skewness value is approximately zero. 
In Fig. 4.11, the PDFs for ware symmetric for all the positions across the wake 
' 
indicating a two-dimensional mean flow. For the wake edges where the turbulence 
is more intermittent, the PDFs have asymmetric tails which explain the non-zero 
skewness values here. 
4.4 Joint Probability Analysis of the Velocity Components 
The joint probability density functions (JPDFs) of two variables indicate the proba-
bility of the two variables having specified values at the same time. The orientation 
of the JPDF contours reflects the preferred simultaneous states of the two variable 
fields. Thus, some t urbulence structure informat ion can be deduced, even from 
single-point measurements . JPDF analysis was utilized by Wallace & Brodkey [65] 
to examine the structure of the correlation uv in a turbulent channel flow. Ong [45] 
used JPDF analysis in the buffer region of a turbu lent boundary-layer to detect the 
presence of inclined vortices . T he covariance of two variables, v1 and v 2 , can be 
related to the JPDF by 
( 4.2) 
The JPDFs for a ll combinations of the velocity components have been determined 
for six locations across the wake. Numerical values of the contributions from all four 
quadrants, Ql, Q2, Q3 and Q4, of the two velocity component hodograph plane are 
also presented. The quadrants are defined in Fig. 4.12. For all JPDFs, the velocity 
components are normalized by the centerline velocity deficit U o . 
Fig. 4. 13 shows the JPDF, P(tt,v), and the covariance integrand contours, 
uvP( u , v ) . For the lower wake, the JPDF contours show a preferred orientation 
of 45° from the horizontal axis u. The covariance integrand contours show t his even 
more clearly with the contribution coming mainly from Ql and Q3 in the outer part 
44 
of the lower wake. The Q3 activity has lower probability but larger· fl t . t· h 
· uc ua IOns t an 
the Ql activity, and, in fact, contributes more to the covariance than Ql . . 
· as seen 1n 
Fig. 4.14. In the centerline region ('TJ == 0.12) the contours are symmetric in v but 
not in u, as expected. The upper wake has a preferred Reynolds stress orientation 
of _ 45° from the horizontal axis, and most of the activity is in Q4 and Q2. The Q
2 
activity has lower probability but larger fluctuations than the Q4 activity, and is the 
greater contributor to the uv covariance, as also seen in Fig. 4.14. These patterns 
mean that the covariance uv is positive in the lower wake, near zero in the centerline 
region, and negative in the upper wake. The quadrant contribution distributions in 
Fig. 4.14 quantify these qualitative trends. The total covariance uv (the sum of the 
quadrant contributions) is the Reynolds shear stress given in Fig. 4.4. 
The preferred ±45° angles of the JPDF contours indicates a tendency to maximize 
t' on of uv The following illustration explains why. For a unit two-dimensional genera 1 · 
vector with angle a, the x and y components are, respectively, 
cos a, 
sin a. 
The product of these two components is 
v 1v 2 = cos a sin a. 
Since the right-hand-side can be expressed as 
cosa sin a = !sin(2a), 
2 
( 4.3) 
(4 .4) 
( 4.5) 
( 4.6) 
. . . 1 t to maxi mi zing the function sin( 2o:). This · · · g tl1e product V1 v2 is eqmva en 1nax1m1z1n 
4i::o function is maximum at goo which makes a = <> . . 
_ . t deficit transport of the streamw1se The shear stress uv reflects the momen um 
h t . wal·e 1'lie streamwise mornentum ]. · to t e ou e1 c , • momentum from the center me regwn 
45 
defi cit can be seen clearly in the negative skewness of u away from the centerline . 
T his momentum deficit transport from the centerline toward the wake edges also 
can be seen in the negative skewness value of v in the lower wake , and positive 
skewness value in the upper wake . The combination of the u and v skewness values 
is consist ent with the sign change of the uv covariance across the wake . 
The quadrant contribution distribution in Fig. 4.14 gives more details about the 
momentum deficit transport. In the lower wake, the larger contributions in Ql and 
Q3 quadrants means that +u is most likely to occur with +v, and -u is most likely 
t o occur with - v. In the upper wake, +u is most likely to occur with - v, and - u 
with + v, which are the Q4 and Q2 quadrants, respectively. 
This pattern of momentum transport can be better visua lized with the aid of 
a velocity vector plot covering the lower and upper wakes . The vector plot in F ig. 
1.3 shows the vortex centers and saddle regions in both layers. Around the saddle 
regions , the diverging legs and converging legs indicate flow a.way or toward the 
saddle , respectively. In the lower wake , the converging legs would correspond to Ql 
and Q3 flow activi ty, and the di verging legs would correspond to Q2 and Q4. Since 
Ql and Q3 are preferred in this region, the converging legs a.re the domin ant uv 
generators in the lower wake, Following similar arguments, it can be seen tha t the 
converging legs are also the domin ant uv generators in the upper wa.ke . 
For P( u, w), Fig. 4.15 shows that the JPDF and covari ance integrand contours 
a re symmetric with respect to w , which must be the case for a two-dimensional 
mean flow. The symmetry in w results in the almost zero uw covariance across the 
wake. Due to the negative skewness of u across the wake, the maximum probability 
contours are on the +u side. The negative skewness also implies th at the largest 
contribution to the total covariance should come from Q2 and Q3. T his observation 
is confirmed in the quadrant plots of Fig. 4.1 6. 
46 
For P(v, w), Fig. 4.17 shows that the JPDF and covariance integrand contours 
are symmetric wit h respect to w, for the same reason as for P( u, w ), and the total 
covariance still is almost zero across the wake due to this symmetry. The skewness 
of v changes sign across the wake, which is reflected in the maximum probability 
contours. The covariance integrand contours show high probability of low-fluctuation 
activity in Ql and Q4 in the lower wake, and in Q2 and Q3 in the upper wake. This 
pattern implies that the large fluctuations should produce high correlation values in 
the opposite quadrants. This is confirmed in the quadrant plots of Fig. 4.18 with 
larger magnitudes in Q2 and Q3 for the lower wake, and Ql and Q4 in the upper 
wake. 
4.5 Summary 
The discussion in the above sections has demonstrated the following: 
1. At x /d = 20 and 30, the maximum rms of the transverse velocity component is 
higher than the streamwise, followed by the spanwise component. This trend 
is consistent with X-probe measurements of Yamada et al. [70] at x /d = 30. 
2. The small prong spacing of the 4-sensor and 12-sensor probes results in more 
accurate measurernents. 
3. The skewness and fl atness values of the velocity components show the proper 
characteristics of a turbulent plane wake. 
4. The 12-sensor probe accounts for velocity gradients in the measuring volume 
and can measure the Reynolds shear stresses more accurately than the 4-sensor 
probe. 
5. The velocity spectra show peaks at the vortex shedding frequency .fs for the 
streamwise and transverse components. T he spa.nwise velocity spectra donot 
have any peak at .fs, indicating parallel shedding. 
47 
6. The JPDFs of the streamwise and transverse velocity components show pre-
ferred orientations at +45° from the centerline for the lower wake and at -45° 
for the upper wake. The other correlations are about zero across the wake due 
to flow symmetry. 
7. The converging legs of t he saddle region are the dominant uv generators. These 
flow regions are along the direction normal to the mean shear. 
48 
5 Characteristics of the Vorticity Field 
5.1 Statistical Properties 
The statistical properties described in this section include the mean vorti city com-
ponents and moments of the fluctuating vorti city components. These properties at 
x /d = 30 are shown in Fig. 5.1, and are normalized by the ratio ua/La = 96.5 
s- 1 • The mean vorticity components fl x and fly are nearly zero as expected for a 
two-dimensional wake flow. The directly measured nz distribution is very close to 
that obtained by differentiating the curve-fit of U given by Eq. (4.1). Both the 
magnitude and the symmetry of nz are in good agreement , thus providing a rather 
stringent test of the accuracy of the vorticity measurements . As expected , the lower 
half of the wake has positive and the upper wake negative nz. 
The nondimensional rms values of the vorti city components ( denoted by super-
script *) have sufficiently high signal-to-noise ratios, as indicated by the ratio of the 
centerline to the frees tream value of about 200:1, and peak near the wake centerline. 
The peak magnitudes of all components are quite close to each other, with w; the 
largest , followed by w:, and then w;. The relatively low value for w; clearly shows 
that the Karman vortex does not the dominate the ens trophy a.t x / d = 30. 
The skewness a.nd flatness values shown a.re limited to within 17 = ±2.5, because 
the intermittent region of the wake produces extremely high values. The freestrea.m 
values at approximately r1 = ±5 are also shown for comparison; the skewness values 
are close to zero as they should be. The freestrea.m flatness values are near the 
Gaussian value of 3 which is also reasonable . The skewness factors for W x and wy are 
nearly zero across the wake within the fully rotational flow region, consistent with 
the two-dimensional nature of the mean flow . \i\Tithin the interrnittent region they 
take on non-zero values but return to nearly zero in the freest.ream. The W z skewness 
49 
,.! ,, 
·' ,, 
·' 
factor is positive for the lower wake, crosses zero at the centerline, and is negative for 
the upper wake, following the sign of !1z, The flatness factors are approximately the 
same for all three components, and are about 5.0 in t he fully rotational centerline 
region. 
There are some vorticity data in the literature for comparison . Antonia et al. [3] 
m easured separately the 3 components of vorticity at x/ cl == 420 and Red = 1170, 
with various combinations of X-probes . Their maximum vorticity nns values, when 
normalized by u 0 / L 0 , are in the range 1.4 - 1.6 in the centerline region. Bisset 
et al. [7] measured simultaneously the 3 vorticity components using a very large 
probe ( described in Section 1.2) for the same location and condition as in Antonia 
et al. . Their maximum nondimensional rrns values are in the range 1.0 - 1.3 in 
the centerline region. Both of these rms results give significantly lower values than 
the present rms data, which are more than 2.2 for all components. T he relatively 
high rrns values in present data are due to: (1) the closer wake location, (2) higher 
R eynolds number, (3) higher spatial resolution therefore lower signal attenuation of 
the miniature 12-sensor probe. 
In Fig. 5.2, therms values at x/cl = 20 are compared with t hose at x/d = 30. 
For x / cl = 20, the normalizing length scale L0 and velocity scale 'U o are 6.5 mm, and 
0.93 m/s, respectively. Even though the dimensional values are all higher than those 
at x / d = 30, the normalized rms values at x / cl = 20 are all lower due to the large 
normalization scale u 0 / L 0 . The results are not expected to collapse on one curve 
since the measurement stations are far upstream of the self-preserving region. All 
components at x /d = 20 have about the same peak value of approximately 2.1. The 
difference between the two streamwise locations is greatest for Wx, followed by w11 , 
and then W z . 
Other moments and characteristics at x /cl = 20 have similar patterns as those at 
x /d = 30 and therefore are not shown. The expected trends of the mean vort icity 
50 
. :..~ .if--· ·-
components and the skewness values at the two wake locations give further confidence 
that th t· · e vor 1c1ty measurements are reasonably accurate. 
5.2 Streamwise Velocity Gradients 
The transve · d · · · h · rse an spanw1se vort1c1ty components m t e prev10us section have 
st
reamwise velocity gradient terms that cannot be measured directly with the 12-
sensor t· · vor 1c1ty probe. In fact, due to severe blockage effects, the streamwise gra-
dients cannot be measured accurately by any hot-wire anemometry technique. For 
th
e current measurements, Taylor's hypothesis is used to estimate these streamwise 
gradients. In effect, the relationship assumes that the turbulent flow is frozen over 
a short time interval so the streamwise gradient for a turbulent variable can be cal-
culated from its temporal variation. Mathematically, the hypothesis is expressed 
as 
a a 
-at+ Uc- ~ 0 ox . (5.1) 
This relationship is somewhat like a substantial derivative operator, with the 
Variable Uc being the local convect ion velocity. For the current measurements, the 
inst t ' l h · · 'd 1 d · an aneous streamwise velocity is used for Uc· Taylor s 1ypot es1s is w1 e Y use m 
Processing turbulence measurements. Cenedese et al. [11 J tested Taylor's hypothesis 
by Laser-Doppler measurements in a rectangular pipe flow. They fou nd that the 
hypothesis is still applicable in high turbulence flows, but the integral scales have 
large errors. Piomelli et al. [4 7] have compared boundary-layer measurements with 
large-eddy and direct numerical simulation databases and found that the hypothesis 
holds r bl . b a l easona y well even mto the u11er ayer. 
The t. . · · t are computed from the mea-1ansverse and spanw1se vorticity componen s 
sured signals as 
n au 1 aw 
y === - + ---
oz Uc ot ' 
(5.2) 
51 
au 
ay 
1 av 
(5.3) 
For the time derivatives, av/at and aw/at, central-differencing (C-D) yields more 
accurate vorticity results than backward-differencing (B-D). When the measured 
data were first analyzed, the B-D scheme was used since the scheme is sometimes 
used by other researchers [3,26], and it gives double the temporal resolut ion compared 
to the C-D scheme. However, it also introduces a phase misalignment of one-half 
time step, as discussed below. During calculation of the ens trophy production, the 
B-D scheme was discovered to give unrealistic results. The term vwzdDz/dy , is 
responsible for producing the fluctuating enstrophy from the gradient of the mean 
spanwise vorticity. The term represents a loss for the mean enstrophy and should 
have negative values across the wake. The B-D scheme gave significant positive 
values for the wake centerline region, as shown in Fig. 5.3 . The C-D scheme, on the 
other hand, gives negative values for the term throughout the wake. The dramatic 
sign change is probably due to (1) low magnitude of the correlation vwz as shown in 
Section 7.6, and (2) low sampling rate. 
The difference between results from the two schemes is due to the phase rnisalign-
ment of variables v and W z in the correlation vwz. There is also phase misalignment 
within W z itself, between the estimated value of 8v/8x from Taylor 's hypothesis a.nd 
the directly measured value of fJu/ ay. The B-D scheme calculates the tin1e derivative 
from two co 1- I t]1e current time a.nd the previous. The C-D scheme nsec11, ;zve sa.mp es, -
uses the s· 1 b c d ftei· the current time. In effect then , the B-D scheme amp es e1ore an a 
is like t l C D I ·th t di'fferent features: (1) the B-D scheme evaluates 1e - SC 1eme WI WO 
the time d · t· I If t· tep earlier and (2) the B-D scheme evaluates the 
enva ·1ve one- 1a une s · ' 
timed · t· . t ·at·l1er than two. Feature (1) indicates that when enva 1ve over one tHne s ep 1 ' 
the sa 1. . h. l ·h as in the current measurements, the one-ha.If mp mg rate 1s not _ 1g 1 enoug , 
time st h . - b _ t vo variables such as v and W z can give erroneous 
ep p ase difference etween ,v 
res It h t i e correlation VWz was calculated with va.riable 
u ,s. This was observed w en 1 
52 
time delay to evaluate the effect of feature (1) as described below. 1:<eature (2) of the 
B-D scheme is an advantage because it doubles the usable frequency range of the 
data, but it must be sacrifi ced here in order to maintain the phase alignment. 
Fig. 5.4 shows the production term vwzdDz/dy versus time delay for T/ = 0.12. 
The calculation was done for a range of ±2 time steps (±1 ms) for both the C- D 
and B-D schemes. The plot clearly shows that after the B-D curve is shifted by 
one-half time step in the d irection of negative time, it falls almost on top of the 
C-D curve. This time-delay calculation was repeated for a few other wake locations, 
and the results were simila r. This indicates that the positi ve (erroneous) ens trophy 
production is due to the phase difference of the B-D scheme. 
Since B-D evaluates the t ime derivative over only one time step, it does not 
attenuate the turbulent signal as much as the C- D scheme. Time series data of Wz 
are shown for both schemes in Fig. 5.5. The two match quite well except that the 
B-D curve sometimes has higher peaks and resolves higher frequencies. The latter 
difference shows up more clearly in the spectra in Fig. 5.6. For Wz, the two spectra 
are virtually identical up to approximately twice the shedding frequency. Beyond 
thi s, the B-D curve is clearly higher than the C-D curve. Therefore, it is clear that 
the B-D scheme resolves more of the high frequency fluctuations, but the half time 
step phase difference between t he estimated value of 8v / ox from Taylor's hypothesis 
and the directly measured value of 8u/8y makes the B-D scheme inaccurate. The 
high frequency values of the B-D scheme compared to the C- D scheme are erroneous 
as shown below. 
The signal attenuation and phase shift of the two differencing schemes can be 
calculated a,na lytically. A t ime dependent variable can be expressed in a Fourier 
series as 
N 
v(t) - '°' 
- L anc.pn(t), ( 5.4) 
n=l 
53 
...... # ..,. . .... ~ • 
Where a - .8 · . 
n - ouner coefficient for n = I to N components. Each Fourier component 
can be expressed as 
(5.5) 
Whe · yCf 
re z == -1 , K = 2nf, and f = frequency, and differentiation of r.p(t) gives 
<p'(t) == iK<p(t). 
(5 .6) 
It can be shown easily that differentiation with the C-D scheme gives 
<p0' D(t) _ . sin K/it 
, - ZK-- (t) K/it <p . (5.7) 
Where lit - 1· . b E 
- samp mg rnterval. Comparison etween q. 
that a d"fi 
(5 .6) and Eq. (5. 7) shows 
mo 1 ed wave number, "'CD, can be defined as 
KcD 2 K~ 
K£1t . (5.8) 
PhysicaU t} · · 1 b 1 Y, 11s relationship means that the C-D scheme attenuates a s1gna y t 1e 
factor ~ . . · · . 1 b . 
1<6.t • However, there is no phase shift because this is a 1ea num er. 
Sirnilarly, it can be shown easily that differentiation with the B-D scheme gives 
<pkD(t) == iK [sinK!it + i COSK!it - l ] r.p(t). 
K£1t Kf1t 
(5.9) 
A . gain co . 
' mpanson between Eq. (5 .6) and Eq. (5.9) shows that a modified wave 
nurnber 
' "'BD, can be defined as 
KaD == ~ [ sin K!it cos K11t l]. (5.10) 
- K, - + i-----
Kb.t Kf1t 
.Physically, this relationship means that the B-D scheme attenuates a signal, and 
introduces a phase shift because "'BD has a complex component. The magnitude of 
the · 
signal attenuation can be shown to be 
/1,,BD/ K/ _ [2 - 2 COS Kb.t] 2 
Kb.t 
54 
(5 .11) 
The phase angle ( for the vector "'BD/ K can be shown to be 
( = arctan(cot!:it - csc !:it). (5.12) 
The ratios Kev/ K and /1,,BD/ 1,, / are shown in Fig. 5. 7 (a) and ( is shown in Fig. 
5. 7 (b) as functions of the frequency f for the range used in this experiment. In Fig. 
5.7 (a), both the C-D and B-D schemes have low attenuation for low frequencies and 
significant attenuation for high frequencies. 
At half the sampling frequency, the C-D scheme attenuates all the differentia ted 
signal intensity, but the B-D scheme retains about 63% of the signal. This high value 
Is erroneous and comes from the complex component , which also indi cates a phase 
shift. T his phase lag (negative values) is shown in Fig. 5. 7 (b) to vary linearly with 
f . It is practically zero for low frequency and is 90° at half the sampling frequency. 
At the vortex shedding frequency (about 168 Hz) , the C- D scheme, whi ch does not 
introduce any phase shift , attenuates only about 5% of the signal. At this frequency, 
the B-D scheme attenuates only 1 % of the signal but introduces a 16° phase lag. If 
a higher sampling frequency had been possible, the phase lag would be smaller. In 
the present res ults the C- D scheme has been used to calcula te all time deri vatives . 
' 
5.3 Vorticity Spectra 
The spectra for three components of the vorti city vector are presented and di scussed. 
The spectra are for 6 wake locations covering both the lower and upper layers, 
and they are normalized by the local variance of the indi vidual component. The 
discuss ion is representative of all t he 21 wake locations where data were taken. 
Fig. 5.8 shows that the spectra for W x are similar in the region /17/ < 1.65. The 
distributions are all broad band , and there are no di stinct peaks at the shedding 
frequency. Since the Karman vorti ces are aligned only in the z-direction (if they are 
not obliquely shed) , there should not be any evidence of the vortex shedding in the 
X-directio ·t· ·t TI· r·ef'ore the W x spectra provide anoth er indication th at the n VO) ,JC) ,y. 1e , 
55 
12-sensor vorticity probe accurately resolves the components of the vorticity field. 
All these spectra have relatively flat distributions up to about the shedding frequency 
and then drop off for higher frequencies. 
Fig. 5.9 shows that the spectra for wy are also similar in the region /77/ < 1.65. 
The distributions are also all broad band with no distinct peaks at the shedding 
frequency, reflecting the parallel nature of the Karman vortices. The wy spectra are 
quite close to the wx spectra, in both magnitude and in the distribution. However, 
the Wy spectra have very low energy content in the high frequency range, undoubtedly 
due to the attenuation effect of central differencing. At the highest frequency, the 
Wy spectral value is about one half the W x value. 
The spectra for W z in Fig. 5.10 show a strong intensity narrow band spike at 
the vortex shedding frequency, f s = 168 Hz. This strong peak is a signature of the 
Kannan vortex. At locations 17 = -0.4 7 and 0. 71, the peaks seem to be highest 
indicating that these locations are somewhere near the centers of the vortices shed 
on either side of the wake. Near the centerline (ry = 0.12), the peak is a little lower. 
The effect of the vortices can be seen, from these peaks, to extend beyond 77 = 1.29 
but less than 77 = 1.65. The cross-stream distribution of the spectral peak can be 
predicted by the linear stability theory [38]. For the same reason as in the wy spectra, 
the W z spectra have very low energy content in the high frequency range. All three 
spectra span about one decade in magnitude indicating that the sampling frequency 
Was really two low. However, the vortex shedding frequency is not in the attenuated 
high f t d thus the filtered W z signal can still serve 
requency range of the spec ra, an 
as a phase f f 1·t·on·• I analysis of the measured turbulent flow field , as 
re erence or cone 1 1 <• 
described in Chapter 6. 
56 
5.4 Probability Density Functions of the Vorticity 
Components 
The probability density functions (PDFs) for all three components of th fl . 
· · e uctuatmg 
vorticity are presented in Fig 5 11 for the freestream and centerline 1 · 
· · ·eg1ons. Each 
vorticity component is normalized by the scale uo/ Lo. The narrow width f s o the 
d" . . . istnbut1ons for the freestream PDFs compared to those for the centerline PDFs 
demonstrate the high signal-to-noise ratio (up to two orders of magnitude) of the 
vorticity measurements. Figs. 5.12 to 5.14 show the PDFs of all three vo t· · r 1city 
components for six wake locations. For each location, a Gaussian distribution with 
the same rms value is plotted for comparison with the PDFs. These six locations 
are the same as those in the spectra plots for ease of comparison. 
In Figs. 5.12 and 5.13 , the PDFs for the Wx and Wy components have similar 
patterns. Both have symmetric shapes which reflects the near-zero skewness values 
in this region. The differences from Gaussian distribut ions can be quantified with 
the flatness factor, which is 3 for a. Gaussian. The differences are smallest near the 
centerline region where Wx and Wy have flatness values of approximately 5 (see Fig. 
5·1(d)). Near the lower wake edge (TJ == -1.GS), the W x and wy PDFs have much 
lower probabi lity away from the mean vorticity than the Gaussian distributions, 
which indicates high intermittency. Note that these vorticity components a.re not 
as close to the Gaussian distributions as the velocity components are in Figs. 4.9 
to 4.11, especially for the centerline region. For both components, the PDF at 
'IJ ===- -1 65 h l · l k . nd the mean vorticity. This means that tl1e w 
· as a very 11g 1 pea.' a.rou · x 
and w fl . . . . . . t pr·oba.bly quite small. The highfluct t· 
Y uctuat1ons m this region a.re mos ' · ua mg 
values, approximately 15 times the sea.le uo/ Lo, occur very rarely. Near the wake 
centerline ti- k 
1 
. 11 lower with a wider distribution a.round the mean , 1e pea va. ue 1s rnuc , · · 
This patte fl t tl . . t . vor·t i· ca.l activity taking place in the center regi011 
rn re ec ·s 1e 111 ense · 
In Fig. 5.1 4, the PDFs for W z show the asymmetry expected for a skewed variable. 
57 
.. 
-· ..,. .... . 
These PDF , . · ·fi l d'ff f i G · d' ·b · s are sigrn cant y 1 erent rom t11e auss1an istn utions even at the 
centerline region. For the lower wake, the PDFs have - wz peaks but have longer +wz 
tails, giving positive skewness values. For the upper wake, the opposite pattern is 
seen, and this yields negative skewness values . Near the lower wake edge (77 = -1.65), 
the tails are long relative to the width of most of the distribution, which is why the 
flatness val · h' h · h ' · I th t 1· · ( 0 12) tl PDF ue is so ig m t is region. n e cen er me region 77 = . , 1e 
is quite symmetric so that t he skewness value is approximately zero. Near the mean 
spanwise flu ctuating vorticity value (wz = 0), the PDF in the centerline region is 
broader than those in the outer wake, indicating more intense vortical act ivity. 
5.5 Joint Probability Analysis of the Vorticity Components 
The contours of the joint probability distribution fun ctions (JPDFs) P(wx,wy), and 
the covariance integrand w w P(w w ) where X y Xl Y l 
(5 .1 3) 
a re shown in Fig. 5_15 for positions 77 = -1.65 to 1.29. The contours of the JPDFs 
are inclined about the horizontal axis by approximately -45° for the lower wake, and 
about + 45° for the upper wake. The contours of the covariance integrand offer the 
additional information that there is a greater contribution to the covariance, wxwy, 
frorn Q2 and Q4 than from Ql and Q3 for the lower wake, and vice versa for the 
upper Wake. Fig. 5.1 6 quantifi es the quadrant contributions. These plots show that 
there is a strongly preferred orientation in the direction of the mean shear of the 
projection of the vorticity vector on the x - y plane. This quantitatively illustrates 
the Presence of vortical structures that are most probably inclined at approximately 
±45° from tlie 1 t 1. w111·ch is consistent with the computational results wa-:e cen er me, 
of M:eiburg & Lasheras (391 in Fig. 1.5. This preferred orientation is along the 
diverging 1 . ( b 'd ) h · F'igs ·1 3 and 1.4, which is parallel to the axis of 
egs or ra.1 s s own rn · · 
111axirnurn positive strain rate. Approximately perpendicul ar to thi s is the preferred 
58 
I 
,: 
orientation £ 
or the uv correlation: which is along the converging legs. The most 
Probable incl· t· 
ma 10n angles of ±4-5° show that the vortica1 structures called braids 
tend to ma . . . 
ximize the covariance wxwy, as shown in Section 4.4. 
In 
th
e quadrant analysis of wxwy in Fig. 5.16, the Ql curve is almost identical to 
th
e Q3 curve, and Q2 curve is almost identical to the Q4 curve. This is consistent 
With the al 
most perfect symmetry observed in the JPDF and covariance integrand 
contours. Tl . . 1
e asymmetry of the total correlat10n from one side of the wake to the 
other (hi h . . . . 
g er magmtude m the lower wake) 1s probably due to experimental accuracy, 
Which also · · . . . 
is evident m the asymmetry of -uv ll1 Fig. 4.4. 
Meiburg & Lasheras [39] calcula ted the dynamics of the laminar wake of a flat 
Plate with . . . . 
a corrugated tradJJ1g edge. They showed that flow structures with stream-
Wise Vorti · d" · 1 
ces mteract with the Karman vortices to produce the three- 1mens10na 
closed v . . 
oitex loops illustra ted in Fig. 5.17. The two legs of these closed vortex loops 
have the orientation and rota tion similar to the "double-roller" model of the vortical 
structure d" . / . fl 
suggested by Grant [21] in Fig. 1.2, and also the 1vergmg convergJJ1g ow 
st
ructures · . l [9] · F" l 3 aiound the saddle from the results of Browne et a.. ll1 • 1g. · · 
In the current measurements, the correlation wxwy is positive for the upper wake, 
Which . d" · · J · .· . 
1U icates tha t the diverging leg is the preferred onentat10n oft 1e p10.1ect10n on 
the X - . h .. 
Y plane and has a larger contribution than the convergmg one. T e positive 
Value of w- · . · f "h · · " ·t x structures xWy is mterpreted by Ong [ 44] as an indicat10n o airprn vor e 
in a turbulent boundary-layer. Perhaps the closed vortex loop in plane wake flow is 
a tw · · 1 · d 
o-sided "hairpin" vortex. Since the current measurements are srng e-pornt an 
th
e Meibti . & L . . 1 · . k -' . 11 of these speculations ig asheras calculat10n is for a amrnar wa e, a 
Point to the need for a direct simulation or a large-eddy simulation of the turbulent 
Plane Wak 
e. 
'I'he contours of the JPDF P(wx· , Wz) and the covari ance integrand Wx-WzP(w,c, wz) 
are shown in Fig. 5.18. There is symmetry in the di stributions of both w.1,· and Wz 
59 
which is reflected in the JPDF plots. The covariance integrand conto . h urs s ow som 
r e 
s ight preference in certain quadrants which is likely due to experiment· I . a uncertamty 
The quadrant analysis in Fig. 5.19 indicates that the total correlatio · 1. . 
. n l S S 1ghtly 
positive in the centerline region and slightly negative in the outer reg1'0 . ( . ns roughly 
for 77 > 0. 7) , but this is within the accuracy of the data. 
s 1own m For P(wy,wz), contours of the JPDFs and the covariance integrand are l . 
· a most Fig. 5.20. These contours and the quadrant plots in Fig. 5.21 clearly show 1 
re a 10n across the wake. zero co1· 1 t . 
5.6 JPDF Analysis of Vorticity and Velocity Gradients 
The correlations between the vorticity and the velocity gradient components make up 
the stret h' / · tl c mg compress ion and reorientation term in 
1
e mean vorticity transport 
equation 
u
1
~ aui awi - -ax - Wj- - Uj-O + f)_jSi j + -~· 1 fJx x R e u.r:j uXj 
J • j 
1 (5.14) 
This term, the first on the right hand side, consists of nine correlations: wxau/ ox, 
~) WzOu,/fJz, WxfJv/ox , wyfJv/oy , -:;;zav/8z, wJJw/ fJx, WyOw/fJy, Wz8w/ 8z. 
JPDF analysis of the individual correlations can shed some light on the question: how 
do the reorientation, stretching and compression processes of 
th
e velocity gradients 
se or ecrease the vort icity? 1ncrea d 
Schematic models in Fig. 5.22 of the different physical processes will aid in the 
discussion of the individual correlations. As illustrated in the top sketch , the wx 
component of either sign is stret ched by +du/dx, and conversely, compressed by 
- du/dx . In the bottom sketch, the velocity gradient du/ dy will shear and reorient 
Wy about th . . r1~1 . l tter J)rocess represents a transfer of Wy vorti city 
e z -ax1s mto Wx· n1s a . 
into w s· .
1 
l . 
1 1 
t 1 the dw/dx gradient will reorient wx about the 
x · 1m1 ar y, m t 1e same s <e c 1, 
Y-axis into w 
z· 
60 
Figs. 5.23 to 5.40 show the JPDF contours, covariance integrand contours and 
th
e quadrant analysis of the above nine correlations . The quadrant analysis plots 
indicate that two correlations which involve the gradients of the spanwise velocity, 
~
wxow / Bx and WyOW / 8y, have the largest magnitudes ( due to limitations of the graph-
ics software, the partial derivative symbol is represented as regular derivative symbol 
in all fi . ) Th . 
· gures . e other correlat10ns are almost zero across the wake. Quadrant 
analysis of the individual correlations can also explain how the velocity gradients 
Increase or d th . . ecrease e vort1c1ty components. 
Fig. 5.23 shows that P(wx, ou/ 8x) is symmetric with respect to the vertical axis 
(wx == 0) . Even though it is not so obvious in the covariance integrand contours, 
th
e vortex stretching ( QI & Q2) contribute more to the covariance than the vortex 
compression (Q3 & Q4) as seen in Fig. 5.24, yielding a net gain in Wx. Q2 and 
Q4 have higher magnitudes than QI and Q3 in the lower wake resulting in a small 
negative correlation there but this is due to measurement inaccuracy. 
' 
For P(wy, 8u/ By), Fig. 5.25 shows that the four quadrants have almost the 
same level of contributions as seen in both the JPDFs and the covariance integrand 
' 
contours. There seems to be preference in certain quadrants even though the contour 
levels for such cases are quite small. The quadrant plots in Fig. 5.26 confirm that , 
for the lower wake, the Q3 and Q4 contributions are higher than QI and Q2, and 
vice Versa in the upper wake. The term represents the shearing of Wy vortex lines 
by the velocity gradient 8u/8y into the streamwise direction. The gain of +wx by 
+au I oy shearing on +wy is balanced by the gain of -Wx by +au I oy shearing on 
-wy . Similar conclusion can be drawn about the shearing due to -8u/ 8y. The total 
correlation is almost zero across the wake, which indicates that there is no net gain 
of w b ~ 
X y uU/ Oy Shearing. 
For P(wz, f.Ju / oz), Fig. S.27 shows the symmetry with the horizontal axis (8u/8z = 
O). In th . . th . ns to be 1neference in certain e covariance mtegrand contours, e1e seer 
61 
quadrants which distorts the symmetry picture but the contour levels for such cases 
are quite small, and this is very likely just due to measurement error. The quadrant 
plots in Fig. 5.28, which show that the magnitude of Ql is almost the same as Q4, 
a
nd Q2 is almost the same as Q3, confirm this observation. These plots also show 
th
at, for the lower wake, the Ql contribution is higher than Q3, and Q4 is higher 
th
an Q2, and vice versa in the upper wake, although all four quadrants are more 
nearly equal there. This small asymmetry is probably due to measurement limita-
tions s· ·1 -=--,-,,.- . . . 
· nru ar to wyl}u / By, discussed above, the total correlation here is about zero 
across the wake, which indicates that there is no net gain of Wx by Bu/ Bz shearing. 
For P(wx, Bv/Bx), Fig. 5.29 shows that the four quadrants have almost the same 
level of contribution as seen in both the JPDFs and the covariance integrand con-
tours Th · · · b h 1 1 
· ere seems to be preference m certam quadrants ut t e contour eves 
for such d·ir · · h' 1 t cases are again quite small, and the werence 1s wit m t 1e measuremen 
accuracy. This quadrant equality is confirmed in Fig. 5.30. The correlation repre-
sents the shearing action of the velocity gradient Bv / Bx on the streamwise vortex 
lines reo · t· h d ' · Tl 1 t 1 t · nen mg t em into the transverse !fect10n. 1e near y zero ne corre a 10n 
value indicates that there is no net gain or Joss of wx to wy due to Bv / fJx shearing. 
The d' · f 11 d · t . gra ient Bv / Bx is one component of wz, and the patterns o a qua ran s m 
~/. 
x v Bx m Fig. 5.30 are almost the same as those of WxWz in Fig. 5.19 except for 
th
e magnitudes. 
For P( Wy, Bv /By), the correlation represents a stretching/ compression term and 
Fig. S.31 shows that the four quadrants have almost the same level of activity as seen 
in b 1 · 0th
· the JPDFs and the covariance integrand contours . The quadrant Pots m 
F' . 
ig. 5·32 confirm this with the total correlation about zero across the wake. For part 
of the upper wake, the Ql contribution is slightly higher than Q3, and Q2 is slightly 
higher than Q4. But these differences are certainly due to measurement error. In any 
case, Fig. 5.32 clearly shows that the stretching and compression processes balance 
62 
out so no net gain or loss of wy occurs. 
For P(wz, av/oz), Fig. 5.33 shows general symmetry about the horizontal axis 
(av /az = 0). In the covariance integrand contours, there seems to be some small 
preference in certain quadrants which distorts the symmetry picture, but the contour 
levels for such cases are again quite small, and these differences are within the noise 
of the data. The quadrant plots in Fig. 5.34 confirm the symmetry showing that 
the magnitudes for all quadrants are about the same. The total correlation is almost 
zero across the wake indicating that the transverse vorticity component has no net 
gain or loss from av/ az shearing on the span wise vorticity component. 
For P(wx, aw/ ax) , Fig. 5.35 shows that the JPDF contours are ti lted approx-
imately +25° from the wx axis at 'T/ = -1.65 and - 25° at 17 = 1.29. All other 
correlations between vorticity components and velocity grad ients discussed so far 
have been symmetric around either the horizontal, or vertical axis. Part of the rea-
son for the non-zero correlation here is because aw/ox is one component of wy . The 
JPDF contours for wx and wy have been shown above to be inclined at approxi-
mately ± 45°. The quadrant plots in Fig. 5.36 show that the JPDF contours are 
almost perfectly symmetric. The Ql curve lies on top of the Q3 curve, and the Q2 
curve is on top of the Q4 curve. The correlation wxBw / Bx is antisymmetric across 
the wake; its value is positive for the lower wake and negative for the upper wake. 
This means that ow/ ox predominantly shears W x of like sign to increase positive W z 
in the lower wake, and predominantly shears wx of unlike sign to increase negative 
W z in the upper wake. 
For P(wy, 8w/8y), Fig . . 5.37 shows that Ql contribut ion is about the same as that 
for Q3, and Q2 is the same as Q4, as seen in both the JPDFs and the covari ance 
integrand contours. This pa ttern is confirmed in the quadrant plots in F ig . 5.38 
in which the Ql curve is almost on top of the Q3 curve, and likewise for Q2 and 
Q4. Across the wake, Ql and Q3 generates +wz; Q2 and Q4 generates -w
2
• The 
63 
equal contributions from Ql and Q3 means that the generation of +wz by +fJw/fJy 
shearing of +wy is the same as that by -8w / 8y shearing of -wy. Similar deduction 
can be made about the equal contributions from Q2 and Q4 across the wake. The Q2 
and Q4 contributions are dominant in the lower wake so that the total correlation 
wy8w / 8y is negative there. The opposite is true in the upper wake. This means 
the correlation reduces the strength of f22 which is positive in the lower wake and 
negative f22 in the upper wake. These reductions are approximately balanced by 
the enhancements of n2 due to the correlation wxfJw / 8x discussed in the previous 
paragraph and seen in Fig. 5.36. Therefore, one can deduce for plane wake flows 
that 
-8w 8w 
Wx- + W - Q 8x Y 8y ~ · (5.15) 
Physically, this approximation means that for plane wake flows, whatever inten-
sity Wz gains from wx is mainly taken by wy, and whatever intensity Wz gains from Wy 
Is, in turn, taken by wx. In this regard, the Wz just serves as the vehicle for vorticity 
exchange between wx and wy. 
Now we come to the last stretching/compression term, wz8w/8z. This corre-
lation· 1 t · t· ·t r.'o · IS a so the last among the three that can genera e span wise vor ·1c1 y. r , r 
P(wz,8w/ o z ), the JPDF contours in Fig. 5.39 are almost symmetric about the hor-
izontal axis (ow/ oz = O). But, the covariance integrand contours indicate a weak 
Preference. for Ql in the lower wake and for Q2 in the upper wake. This pattern 
is confirmed in the quadrant plots in Fig. 5.40. These plots clearly show that the 
magnitudes of Ql and Q2 curves are larger than those for Q3 and Q4 in the lower 
and upper wake wa.ke regions, respectively. This means vortex stretching (Ql & Q2) 
of either sign Wz dominates vortex compression (Q3 & Q4) of either sign Wz· The 
total correlation is thus approximately zero across the wake indicating no net gain 
of w Z • 
64 
5.7 Summary 
The discussion in the above sections has demonstrated the following: 
1. The miniature 12-sensor probe can measure simultaneously velocity and vorti c-
ity with good accuracy in the near-wake of a circular cylinder for x / d = 20- 30. 
2. Central-differencing, which results in proper phase alignment of the stream-
wise velocity gradients with the directly measured variables , should be used in 
applying Taylor's hypothesis. 
3. The measured mean spanwisc vorticity component agrees well with the deri va-
tive of the mean streamwise velocity. The other mean vorticity components 
are about zero. This is a good test of the accuracy of the measurements. 
4. At x/ d = 30 , the rms of the streamwise vorti city is highest, followed by the 
transverse and the spanwise components. The nns values for the transverse 
and spanwise components are somewhat attenuated clue to the low sampling 
rate. 
5. The skewness and flatness values for the vorticity components have the proper 
characteristics for a plane wake. These values are high at the wake edges and 
have nearly Gaussian values in the centerline region. 
6. Only the spectra of the spanwise vorticity component show intensity peaks at 
the shedding frequency, indicating parallel shedding. 
7. The extremely narrow PDFs of all vorticity cornponents in the frees trearn com-
pared with the centerline region indi cate a high signal-to-noise ratio of up to 2 
orders of magnitude. 
8 . The JPDF contours of w.T:wy show a preferred orientation that is a long the 
direction of the mean shear. Therefore, W x and w 11 are negatively correlated 
65 
in the lower wake and positively correlated in the upper wake. The pattern is 
consistent with the computational results of Meiburg & Lasheras [39]. Other 
vorticity correlations are about zero across the wake. 
9. The correlations of vorticity and velocity gradients are about zero across the 
wake except for those that involve gradients of the spanwise velocity. 
1 O. Quadrant analysis of the correlation WxOu,/ ox and WzOW / oz show more stretch-
ing than compression activity across the wake. 
11. The Wz component serves as the vehicle for vorticity exchange between the w.1: 
and W y components. 
66 
6 Conditional Analysis 
6.1 Motivation 
The goal of this chapter is to characterize the "coherence" of the turbulent fluctuating 
velocity and vorticity fields with the shedding vortices. "Coherence" here means the 
phase locking of the fluctuating velocity (or vorticity) field with a certain phase of 
the periodic shedding vortices. In the near-wake of a bluff body, the vortex shedding 
is regular and the shedding frequency is governed by the Strouhal-Reynolds number 
relationship. This has been illustrated in the spectra of the variables u,, v, and W z in 
C hapters 4 & 5. 
Using a phase of the vortex shedding as a conditioning detection criterion is just 
one of many different possibilities of conditional averaging. Browne et al. [9] condi-
tionally sampled on temperature to detect large-scale coherent structures in the far 
wake region ( x / d = 420) of a circular cylinder. They decomposed the fluctuating 
velocity components into the incoherent, and coherent parts. For these structures 
detected by Browne et al. , the contribution to t he Reynolds shear stress from the 
incoherent motion is much larger than that from t he coherent motion . As described 
in Section 1.3, Cantwell & Coles (10] measured in the region within 8 diameters 
downst ream of a circular cylinder using two X-wire probes mounted on a pair of 
whirling arms. T hey deduced a pattern of moving vortex centers and saddles (these 
flow regions a re illustrated in F ig. 1.4) by analyzing the decomposed signals and 
found that a substantial part of the turbulence production is concentrated near the 
saddles . Their data showed that t he Reynolds normal stresses due to the incoherent 
motion have maxima near each vortex center , and the magnitude of the incoherent 
shear stress has a maximum near the saddle between the vorti ces. In the near-wake 
of a circular cylinder, Hussain & Hayakawa (26] measured the large-scale spanwise 
67 
.... -., ... 
vo rticity using a ra.ke of X-probes positioned in the plane of the mean shear. They 
d educed the passage of the spanwise vorti ces using the measured coherent vorti city 
as t he trigger signa l. For the measurement station of x /d = 30, they found that the 
sp a nwise vortex center was slightly less than one cylinder diameter from the center-
line . They observed similar patterns for the Reynolds shear stresses and turbulent 
kineti c energy as had Cantwell & Coles. No observations about enstrophy was made , 
b ecause only the spanwise component of vorticity was measured. These investiga-
tions point t o a need for simultaneous measurements of all the velocity and vorti city 
vector components to elucidat e some ha.s ic characteristi cs of the vortical structures . 
T herefore , in this chapter , the simultaneous measurements of velocity and vorti city 
will be a nalyzed using the passage of the shed vortex for phase referencing. The 
following section will describe the detection scheme. 
6.2 Description of the Conditioning Detection Scheme 
The det ect ed event used as the conditioning criterion was chosen to be the passage 
of the shed vortex. This was achieved by: (1) band-pass filtering the time series of 
the spanwise vorticity component in a narrow band around the shedding frequency, 
and (2) usin g a phase of this periodic time series as the phase reference for ensemble 
averaging all other variables considered. 
First, the w 2 spectrum was narrow band-pass filtered around the sheddin g fre-
quency f s, and an inverse Fourier t ran sform ation was performed to produce a time 
series of the phase reference signal. A square band-pass filter was used over 3 fre-
quency bins ( ~ 4 Hz bandwidth) with the shedding frequency centered in the middle 
one . The two additional bins are for smoothing the spectrum of the background 
turbulence signal which is the difference between the total fluctuating signal and the 
phase reference signal. The square filter has negligible leakage effect s on neighbor-
ing frequency bins. Because the inverse Fourier transform was perforrned on blocks 
68 
containin 512 . . 
g samples out ot approximately 180000 samples to obtain a time series, 
a moving · d 
Wm ow technique was used to smooth out the difference between blocks 
of data · h 
m t e transformation. This technique simply overlaps consecutive blocks 
resultin · 
g m an averaging effect for the overlapped regions. 
The result of this process is illustrated in Fig. 6.1 which shows the spectra of 
the total fl t . 
uc uatmg Wz signal, the phase reference signal, and the background tur-
bulence signal. The reference signal spectrum has a peak at the shedding frequency, 
and negli 'bl l gi e energy at all other frequencies. The background turbu ence spectrum 
is smooth · . . . . . 
m the filtered frequency band srnce it was designed to filter only the ad-
ditional c t 'b . 1 . 
on l'1 ution of the shed vortex. For each wake position, t 1is process was 
repeated t · · " 1 · · £ 0 produce separate and local detect10n signals. There1ore, P iase m orma-
t' ion hetwee · · l ·1 bl n Vanous wake locations is not dll'ect y avai a e. 
An instantaneous turbulent signal can be decomposed into 3 parts : mean, coher-
ent, and incoherent . This technique was first introduced by Reynolds & Hussain [52] 
to stud th . . fl r,, · bl F( t) y e dynamics of travelling waves ma turbulent ow. i•or a vana e 1 Xi, 
one ca . 
n Wnte 
F(x· t) - -i, == F(x ·) J( ) '( ) i + Xi,t +f Xi,t, (6.1) 
Where F( - · 1 d f '( · t) 
. xi) is the mean or time average, J( xi, t) is the coherent signa , an . Xi, 
is the incoherent signal. The last two parts make up the total fluctuating signal 
f(x. t) -i, === !(xi, t) + J'(xi, t). (6.2) 
'I'he coherent signal J can be obtained from the phase average of F which is defined 
as 
".F( X · ) 1 N - -
i,t >:::: )~N~F(xi,t+jT) = F+f, 
J=l 
Where T is th . . d d N ,....., 14700 for this experimental data 
e vortex sheddmg peno an ,..., 
(6.3) 
set. The h . . omes from the reference signal P ase mformation for the phase average c 
69 
obt · 
ained with th fi . 
and e ltermg process described earlier. It follows that <F> is periodic 
<I>- -
- f, and <!'> = O. (6.4) 
Th 
e coherent and . 
mcoherent fields can be shown to be uncorrelated, i.e., 
<.ff'> -=---
= f !' = 0. (6.5) 
Therefo . 
re, it follows that 
<(] + !')2 -
> = <!2> + </'2> . (6.6) 
Fig. 6 2 
· shows th t t 1 fl · · e o a uctuatrng, coherent, and mcoherent parts of Wz , The 
C:1z sign I . 
a is quite . d' 
. per10 ic except for some occasional small jitter. The period T 
IS app . 
roximatel 6 . . . [ Y ms, because the sheddmg frequency 1s about 168 Hz. Hussam 
251 
a
nd 
Hussain & H k [26] I · d' d 1 tl · · · h Id b aya awa rnve m icate t mt 1e spanwise vorticity s ou 
e Used to det . 
ect coherent structures m turbulent wake flows. Furthermore, they 
ernphasized th . 
at usmg the local signal for phase-referencing smears the structure less 
than . 
Using an t J [ 
'I' ex erna phase-reference signal as was done by Cantwell & Coles IO]. 
he Vortex cente· · d · h 1 l · 1 passage, they assumed would be associate wit a oca maximum 
~- ' 
wz in the lower wake, and a local minimum of Wz in the upper wake. They described 
a saddI 
. e as the region between two consecutive spanwise vortices that is associated 
With a local . . . f - . h 
mmimum of w in the lower wake and a local maximum o Wz m t e z 
Upper Wak . 
. e. Therefore, in the current work an event detected by a local maximum 
in0 ' 
z Was Presumed to be near the vortex center in the lower wake and to be near the 
saddle in th . . . 
e upper wake. To describe the saddle and vortex center locat10ns m terms 
of <n .. 
z> it is important to recall that the mean vorti city Dz is positive in the lower 
Wake 
a
nd 
negative in the upper wake. One can visualize a vortex shedding pattern, 
detected b . . - . . . . ; . 
Y maxima m Wz on either side of the wake, as depicted m Fig. 6.3. This 
lll0 de1 · . _ 18 derived from a combined analysis of the velocity vector plots and W z contours 
70 
of Hussain & H k 
aya awa [26] , and of the present database. The streamwise and lateral 
dim · 
ensions of the vortices are estimated to give a rough idea where the vortex center 
and sadd! . 
. e regions are located. The vortex shedding period and mean streamwise 
Velocity at th h . . 
e alf-deficit point are used to estimate the streamwise extent of the 
vortices Th 
· e vortex center locations and their lateral extent are estimated from the 
Wz contour t /d 
s a x = 30 of Hussain & Hayakawa. The vortices are shed alternately 
so that th 
e vortex center on one side is ahead ( or behind) a vortex on the other 
side by a . . . . 
pproximately half the distance between consecutive vortices. The arrows 
between th . . . . . . . 
e shed vortices illustrate the convergmg and d1vergmg flow regions which 
drive fl · d 
ui toward or away from the saddle region, respectively, creating a highly 
strained fl 
ow there. The streamwise location of the detected event can be visualized 
as 1· 
a lOe ap · · · · th h th ·t PIOximately normal to the wake centerlme, runnrng roug e vor ex 
center in th 1 . d' d p· 
e ower wake, and the saddle region in the upper wake as m 1cate on ig. 
6
·
3
· Inter · t · Id k · t t th pre ation of the following conditional analyses shou ta em O accoun e 
fact that . . . . the current database was obtarned by smgle-pornt measurements. Also, the 
locatio . . . · · 
ns of the vortex centers and the saddles m Fig. 6.3 are rough eS t1mates smce 
the R . h £ 
eynolds number in Hussain & Hayakawa is 13400, almost seven tunes t at or 
the 
Present data. For all analyses in this chapter, measurements at x/d = 30 were 
Used. 
6.3 Conditional Analysis of the Velocity Vector Field 
'I'he <::u> (- -) . h · F'g 6 4 Here, and in 
= 'll values for 6 cross-stream locat10ns ares own 111 1 · · · 
subsequent l k d fi ·t . nd time is normalized P ots, velocity is normalized by the wa e e lCJ u0 a 
by th · · --s;- 0 For 
e period T of the vortex shedding. Note that, by de:firntwn, <u = · 
17 ::::: -1 06 . t· t' which is associated 
· , this component has a peak near the detec 10n ime 
With th · k p :t've time corresponds 
e Passage of the vortex center in the lower wa e. osi 1 
to Upst t Therefore, for the 
ream and negative to downstream of the vortex cen ;er. 
71 
lower wake t / T O . . 
, = corresponds to the passmg of the vortex center, and t/T = ±0.5 
corresponds to the passing of the saddle regions. At r; = -1.06, the peak near 
t / T - 0 I . d ' 
- · m 1cates that u has strongest coherence at a location slightly upstream of 
the Vortex center. 
For 'f/ = 0. 71 and 1.29 in Fig. 6.4, in the upper wake, the detected event cor-
responds to the saddle region so the velocity at t / T = 0 is negative, with positive 
values of < > . h . u on e1t er side due to the passage of the vortex centers. The degree 
of phase coherence varies depending on the wake positions; it is low near the wake 
edges and around the wake centerline, and high in the region in between. The wake 
centerline region (r; = 0.12) has almost zero phase coherence, probably because it 
is outside the domains of influence of the lower vortex center and the upper saddle 
regions (see the schematic model in Fig. 6.3). 
Fig. 6.5 depicts the <u 2> values . From Eq. (6 .6), 
(6 .7) 
hence both the intensity of the coherent part and the phase alignment of the in-
coherent part are represented in this quantity. Unli ke <u>, <u 2> is nonzero and 
represents the conditional variance of the total streamwise velocity fluct uations. The 
lficoherent part, < u'2 >, is almost the same as the total, and thus is not shown in 
th
e figure. The values of <u2> are everywhere less than 0.007. One striking feature 
IS th t . . f ' t} . 
a the vortex center has a maximum and the sadd le has a mm1mum or 11s 
Reynolds normal stress . In the absence of vortex shedding <u'2> would have been 
a constant equal to the variance u'2 . The fact that <u12> oscillates around u'2 is an 
indication that a significant fract ion of the incoherent fluctuations are phase locked 
to the Vortex shedding. In Section 6.5, an attempt to quantify the degree of locking 
Will be presented. 
The schematic mode] in Fig. 6.3 can help to explain the patterns of <v> in Fig. 
6
,6. For TJ = - 0.47 and -1.06 (the lower wake) where the vortex center passage is at 
72 
_._,_. _____ - - - - - -
t/T === 0, the d . . 
ownstream region should have an upward motion, as seen for t/T < O, 
and the upstr . 
earn region should have a downward motion, as seen for t/T > 0. For 
t/T:::: ±0.5 i e in th ddl · L'k · h · 
' · · e sa e reg10n, <v> returns to zero. 1 ew1se, t e combmed 
effects of th . 
e convergmg and diverging legs of the saddle region in the upper wake 
('r/:::: O. 71 and 1.29) produce positive v motion for t/T < 0 and negative for t/T > 0. 
Note furth h 
er t at the coherent v-fluctuations are strongest near the wake centerline 
and are Weaker t d h . . h 1 . . 
owar t e wake edges. This JS because t e ower spanw1se vortices 
cornbine th . . . . 
eu effects rn the centerline region with the upper saddle reg10ns for the 
Vertical m . 
obon. The schematic model in Fig. 6.3 shows that, for y / Lo > 0 near the 
Wake cent 1' 
er me, the downstream region (x/L 0 > O) of the saddle has upward flow. 
'I'h. Js is comb· d . . 
me with the upward motion of the lower vortex at the same streamw1se 
region s· . . 
· 
1milar reasoning can be made to show that the downward motion near the 
Wake cent -1· · 1 ·· Th 
er me receives contributions from both the vortex and sadd e regwns. e 
syrnrnetr f · · · tl Y O the <v> plots indicates that the vortex mot10ns are symmetnc 111 1e 
'(v> field. 
Fig. 6 7 h · · 
· s ows the <v2> and <v'2> terms 111 the equat1011 
'(v2> - -2 
- '(v > + <v12> . (6.8) 
Dnlike -2 . d·a b t 
'(u > 1Il Eq. (6.7), <i?> is quite significant and is the werence e ween 
"v2 > 
a d tl · o 47 < v'2 > peaks at n 1e mcoherent contribution < v'2 >. For T/ = - · ' 
.t/ T:::: 0, whereas <v2> peaks at t/.T = 0.2. This is due to ii, which has been shown 
In p· · 
Ig. 6 6 t h /T ±0 25 I the upper wake, for 
· 
0 ave maximum values around t = · · 11 
"I ::::: 0. 71 and 1 29 h . . t t/1' = ±0.5 and the saddle is at · w ere the vortex center is a 
t/T - 1 
- 0 the· l . . . t/T- 0?.5 For TJ = 0.71, t1e 
' mco 1erent part has a mm1mum near - .~ · 
total p ·t · · · f - } ere Since ii2 
. ar Is minimum near t/T = 0 due to zero contribution rom v 1 . 
Is largest . 1 ~ . l t zero v2 is larger than u 2 
In t 1e wake centerline region where u 1s a Jou' ' 
here even though v'2 is not. For T/ = -l.06, and -0.4 7, <v'z> is higher in the region 
Upstrea.zn f . . . · 1 . of the saddle) than in 0 a vortex center (which 1s m the convergmg eg 
73 
the region downstream. One can also see a similar pattern in the upper wake ( where 
upstream of the saddle is equivalent to downstream of the vortex center) except near 
the centerline where there is almost symmetry. Therefore, this phenomenon reveals 
the significance of the converging leg of the saddle, which is also discussed in Section 
4.4. 
The <w> values in Fig. 6.8 show weak coherence with the vortex motion. This is 
not surprising since the vortex shedding is parallel. The maximum <w> is about 3% 
of the wake deficit u 0 , and only slightly higher than the 1 % measurement uncertainty 
reported in Section 3.5. 
Fig. 6.9 shows the <w2> values in the equation 
2 . 
<w > = <ui> + <w'2> . (6.9) 
The incoherent part <w'2> is not shown since it is practically the same as the total 
' ' 
due to the extremely small values of <u?>. In other words, <w 2> is completely due 
to the phase locking of the incoherent w' :fluctuations. This phase locking is small 
near the wake centerline and the wake edges, and is maximum near the centers of 
the upper and lower vortices (t/T = 0 at 'f/ = -0.47 and t/T = ±0.5 at 'f/ = 0.71). 
The values of <u'2> and <v'2> show similar behavior as seen in Figs. 6.5 and 6.7. 
Therefore, the incoherent Reynolds normal stresses have maxima near the vortex 
center. This is consistent with observations of Cantwell & Coles [10]. 
Fig. 6.10 presents the phase averages of the total and the incoherent turbulent 
kinetic energy k which is defined as 
. ' ' 
<k> = <k> + <k'>' (6.10) 
where 
k l( 2 2 2 
- 2 U + V + W ), 
(6.11) 
(6.12) 
74 
k' l 
== -(u'2 + v'2 + 12) 2 'W . (6.13) 
Across the w k th 1 . . . . a e, e tota 1s only shghtly higher than the mcoherent contribution. 
For both th . h 
e mco erent and total fluctuating parts, Fig. 6.10 shows a maximum 
coherence th near e vortex center, at t/T = 0 for T/ = -0.47, and a minimum near the 
saddle reg· t 11., . 10n, a t = 0 for T/ = 0. 71. The coherence 1s lower near the wake center-
line and . t th . . 
. a e wake edges. The coherence shows near symmetry m the streamw1se 
direction a . d h . mun t e vortex center and saddle reg10ns. These patterns are consistent 
with earli b . er o servat10ns for the component variances. 
Fig. 6.11 depicts the phase averages of the stress uv and its incoherent part , u'v' , 
which are rela ted by 
<uv> :::: <uv> + <u' v'> . (6.14) 
Across th k · · · · h 'b · e wa -e, the total 1s only slightly different from the mco erent contn ut10n. 
T . 
he coherent contribution is very small mainly because < u > is small. Fig. 6.11 
reveals a strikingly clear pattern that at the vortex center, < uv > has the lowest 
magnitude (~ 0.01) , and at the saddle, it has the highest(~ 0.08). The left column 
is for th l · · · t · · I t/T - 0 for e ower wake, and < 1w > has a pos1t1ve sign excep arounc -
17 :::: - 0.47; the right column is for the upper wake, and <uv> has a negative sign 
everywhere except near the centerline (r; = 0.12). The high < uv > values in the 
saddle region (o.og for T/ = -0.47 and - 0.08 for 'T/ = 0.71) means that turbulence 
Production is predominantly occuring there, and the vortex center just collects the 
turbulent kinetic energy produced in the saddle regions. The wake centerline region 
(17 :::: 0.12) also has significant < uv > values, but this high correlation does not 
result in high d t · . e the average over the period is almost zero energy pro uc ·10n smc 
here. These observations are consistent with the findings of Hussain & Hayakawa 
[26] and Cantwell & Coles [10]. Hussain & Hayakawa reported a peak <u'v'> value of 
approximately 0.006 U! at x /d = 30. The peak <u'v'> at 'T/ = - 0.47 in Fig. 6.11 is 
7.5 
,,·: 
.. ; 
::; 
~:: 
,, 
,,, 
i,, 
'/ 
. ; 
J.: 
about 0.0025 u2 'Th 
oo· e value from Hussain & Hayakawa is higher probably because 
their accepted t . t h "l . . . s I uc ures must ave arge-scale vorticity concentratwn", whereas 
th
e structures in the current measurements are the periodic Karman vortices. 
An interesting feature revealed in Fig. 6.11 is that u'v' is a little higher than uv 
for 'rJ == -0.47, and a little lower than uv for T/ = 0.71. Therefore, Eq. (6.14) shows 
th t = 
a uv has the opposite sign compared to uv. This corresponds to negative coherent 
energy production (i .e., the vortex is decaying) as presented in the next chapter. 
6.4 Conditional Analysis of the Vorticity Vector Field 
The vorticity values are normalized with the wake deficit U o and the wake half-width 
Lo for tl1e l t · · . Po s 111 this sectwn. Across the wake, the <wx> values are about zero as 
seen in p· 6 12 . . cl cl t . f' ig. · . There is very low level of coherence upstream an owns ream o 
th
e detected events but this is within the error of the data. 
Fig. 6.13 presents the phase averages <w;>. From Eq. (6.6), 
<w2> - <- 2 2 
X - W.> + <w' > X X ) (6.15) 
but the incoherent part (not shown) is practically the same as the total due to the 
extremely small values of <w;>. Therefore, <w;> is completely due to the phase 
locking of the incoherent w~ fiuctuations. The <w;> values show strong coherence 
With th . · F' . 6 13 Th n - - 1 06 plot e passage of the shed vortices as seen m 1g. · · e ·, - · 
has a p k · f h t t · (t/T > 0) This ea 111 <w; > for the region upstream o t e vor ex cen er · · 
Phenomenon is directly observed here for the first time due to the availability of these 
Vorticity data. It can be explained with the aid of the vortex model in Fig. 6.3. With 
regard to lateral position' the T/ = - 1.06 point is slightly outside of the vortex center 
and is about the level of the saddle region. At this lateral position, upstream of 
the detection event is the diverging leg, and downstream is the converging leg of the 
saddle region. At T/ = - 1.06, the high <w; > in the upstream region compared to 
the downstream region means that the diverging leg is dominant over the converging 
76 
.. 
,, 
J •'I 
' ' ::: ,: 
-:: , 
· ,, j 
,., I 
:! I 
f I 
,, ' ,.  
leg (the diverging leg is associated with stretching of W x , which increases <w_;> ; by 
contrast , the con verging leg is associated with compression). 
This phenomenon can also be seen in the upper wake, a t r; = 1.29, where the 
detected event is the saddle region. The downstream region is the diverging leg of 
the saddle (t/T < 0) and has higher <w;> than the upstream converging leg. The 
schema ti c vortex model in Fig. 6.3 illustrates these flow regions as arrows in the 
plane of the mean shear. This di verging leg, commonly called "braid ", could extend 
to the wake edges and join with the converging leg of the downstream saddle at a 
different spanwise location. This is a possible mechanism for the generation of three-
dimensional flow stru ct ures. Hussain & Hayakawa {26] speculated tha t these braids 
with predomin antly streamwise vorticity interact with the spanwise vortices to pro-
duce "sm all-scale, three- dimensional vorticity flu ctutations" . Mei burg & Lasheras 
{39] showed from inviscid vortex calculation tha t the braids containing streamwise 
vortices interact with the Karm,in vortices to produ ce three-dimensional closed vor-
tex loops illustrated in Fig. 5.17. 
The <wv> values also show very low coherence witl1 the vortex shedding across 
the wake as seen in Fig. 6.1 4. For the correlation w;, Eq. (6.6) shows tha t 
(6.16) 
but the incoherent part is practically the same as the tota l due to t11 e extremely 
sm all <w;>. Therefore, <w;>, shown in Fig. 6.1 5, is completely due to th e ph ase 
locking of the incoherent <w~2 >. The coherence is quite strong, although less than 
<w;
2
>. Near the vortex centers, r; = -0.47 for the lower wake and r; = 0.71 for 
the upper wake, the coherence is stronger than a t other wake locations. The high 
coherence lateral positions for <w; > are relati vely closer to the wake edges. No te 
that even though the rms values of the flu ct uating w2 . and w11 are approximately the 
same across the wake, as seen in Fig. 5.l (b) , the streamwise vorti city variance has 
77 
-# ~->-•·· 
significant} h · 1 . 
Y iguer coherence with the passage of the shed vortices than the lateral 
component. 
The <wz > values obviously show high coherence at all wake positions in Fig. 
6
-16 sine ·t · h 
e 
1 is t e phase reference variable. The peaks at t/T = 0 characterize the 
detection 
events and the phase averages are symmetric around these points. The 
large pe k 1 . . . 
a Va ue of <w2 > at T/ = -0.47 compared to other locat10ns mdicates that 
th
e vortex center is near this point. Also, the symmetry around the detection time 
t/T:::: O for all 6 locations show that the educed vortex center is probably half-way 
between t 
wo saddle regions. This observation is consistent with Hussain & Hayakawa 
[26). 
For the .. 1 · 2 
cone at10n w2 , Eq. (6.6) shows that 
(6.17) 
aud Fig 6 17 · d · · · · · · 1 t tl tl 
· · m 1cates that the mcoherent contribut10ns are a mos 1e same as 1e 
total ex t .c f O 47 B th l 
' cep ior small cliff erences at the detected event o T/ = - · · · 0 s 1ow 
signific· t h k · · O cl d an co erence with the vortex passage at all wa e pos1t10ns. ne can e uce 
th
at the vortex center lies somewhere in the vicinity of T/ = -0.47 ( or T/ = O. 71 for 
th
e Upper wake) since the coherence is strongest there. At T/ = -0.47, the peak of 
,w'2> A . t t' 2 at t/T == 0 can be considered a signature of the vortex center. 11 111 eres mg 
Point is that the minimum level of <w;> is about 2 sample intervals upstream of 
th
e detected event in the upper wake. Recall that for <w;> in Fig. 6.1 3, the region 
upstream of the saddle also has lower coherence than the downstream region. These 
observations indicate that the region upstream of the saddle is characterized by lower 
Vorticit fl . . . · Y uctuat10ns, relative to the reg10n downstream. 
Enstrophy, c, is defined as one-half the sum of the vorticity variances, and the 
coherence of the enstrophy reflects a combination of the 3 variance phase averages. 
For the total c, Eq. (6.6) shows that 
78 
<s> = <f> + <c'> , (6.18) 
w h e re c , f and c:' are defined similarly to their kinetic energy analogs (Eqs. (6.11) 
to (6.13)) . Fig. 6.18 indicates that the incoherent contributions are almost the 
sam e as the total across the wake. Only near the detected events, vortex center 
o r saddle, is the difference significant. Fig. 6.18 shows that the peak coherence 
occurs at T/ = -0.47 at the detection event (t/T = 0) which is near the vortex 
center . At T/ = 0.71 , t he coherence is minimum at t/T = 0.25 or 3 sample intervals 
upstream of the saddle. This means that even though the turbulent vorticity field is 
predominantly incoherent , the magnitude of the vorticity Huctuations are in tensified 
s ignificantly when a vortex center passes by. The opposite is true when the saddle 
region passes by. Near the wake edges, the T/ = - 1.65 location still exhibits some 
coherence, which indicates that the field of influence of the spa.nwise vortex probably 
extends greater than l.6L0 • 
The phase averages of the dissipation rate, E, are shown in Fig. 6. 19. Eq. (6 .G) 
shows that 
(6.19) 
but t he incoherent contribution is practically the same as the total which is the 
only quantity shown in Fig. 6.19. The phase averages of the dissipation rate follow 
closely the pattern of the ens trophy ( they should be identical if the fluctuating field 
was isotrophic) . 
6.5 Phase-Locking with the Vortex Shedding 
In th is section, the degree of coherence or phase locking of incoherent, turbulent 
fluctuat ing quantit ies with the vortex shedding will be quantified . Only locking of 
second order correlations will be presented. 
The phase locking parameter , L , for the incoherent cornponents of two variables 
Ji and h is defined as 
79 
(6.20) 
where r1 and r2 are the rrns values of the two va1·1·ables. Tl ·t d f 
ie magrn u e o the phase 
locking, r1-2 , is defin ed as 
r1-2 
(6.21) 
where NP is the number of points per period of the vortex shedding (12 for the 
present data). For !1 = f 2 = u', the normalization factor r1r2 becomes simply the 
local variance u'
2
. £ would be zero in this case if <u'2> does not vary with respect 
to u'2 over th e period of the shedding. 
The phase locking values for the incoherent Reynolds normal stresses are shown 
in Fig. 6.20. For u'2 , the value of£, reaches a maximum of 15% of u'2 at approxi-
mately r; = 0. 7. The centerline region has a lower phase locking value of 5%, a.bout 
the same as those at the wake edges . The phase locking values for v'2 are slightly 
higher than those for u'2 and peak around 17 = 1.0. The value near r; = 0 is also a 
minimum. Interes tingly, although tu is about zero, w'2 has the highes t phase locking 
among the normal stresses . This ilJustrates strong ph ase-locking of spanwise veloc-
ity fluctuation s to the vortex passage. The maximum locking values are also near 
r; = ±0. 7, locations determined in the previous sections to be near the vortex centers. 
The values of £, for the varian ces of the incoherent vorti city components are 
shown in Fig. 6.21. For w.? , the value of,[, reaches a maximum of 18% of w2 at 
approximately r; = l .O. The locking for w/ are mu ch lower at approximately 5% 
of w~2 across the wake. The Jocking values for w/ are substanti al higher than the 
other two values and reach maxima of approximately 26% of w? at the vortex center 
regions (r; = ±0.7). Like the Reynolds normal stresses, the vorticity variances also 
have minimal locking around the centerline region. 
The phase locking values for the incoherent Reynolds shear stress u'v' are shown 
in Fig. 6.22 to reach a maximum of about 20% of u1v' at the wake centerline. Fig. 
80 
--·- _:__-.. ..:.. --- - ._; -_ - :- - - - ·_.:. -· - . .::_:: _-
6.11 shows that the m agnitude of phase locking for < u'v' > is almost the same 
as the time average for most wake locations. This means practically all of u'v' 
generation is phase-locked with the vortex passage. The above 20% value is due 
to the normalization factor. In surrunary, a significant percentage of the incoherent 
fluctuations a re phase locked with the Karman vortex shedding. This is yet further 
evidence that the large-scale coherent structures affect t he entire t urbulent field. 
6.6 Summary 
The discussion in the above sections has demonstrated the following: 
l. Detection a t the peak of the periodi c w 2 signal due to vortex shedding corre-
sponds to passage of the vortex center in the lower wake and passage of t he 
saddle in the upper wake. 
2. The transverse velocity is more coherent with the shed vortex than t he stream-
wise velocity, especially in the wake centerline region, which results in higher v2 
there. The spanwise velocity is incoherent since the vortex shedding is parallel. 
3. The incoherent Reynolds normal stresses and turbulent kinetic energy are max-
imum near the vortex center and minimum near the saddle region. The incoher-
ent Reynolds shear stress u,'v' is maximum near t he saddle region and minimum 
near the vortex center. These observations are consistent wi th the findings of 
Cantwell & Coles [10] and Hussain & Hayakawa [26] . 
4. Although Wx a nd w11 are negligible, <w';> and <w~
2> are not. T he variance of 
the streamwise vorticity has the strongest coherence in the diverging braids of 
the saddles, which suggests a three-dimensional vortex line orientation of these 
flow regions. 
5. <w~2> has m aximum coherence near the vortex center and minimum coherence 
near the saddle region. 
81 
' 
·~ 
.,, 
·' 
ti. The coherence of the ens trophy is maximum near the vortex center and mini-
mum slightly upstream of the saddle region. The dissipation rate follows the 
same pattern. 
7. The incoherent shear stress u'v' is almost entirely phase-locked to the Karman 
vortex shedding . The variance w'2 has the highest phase-locking compared 
with other incoherent normal stresses. The variance w? has the highest phase-
locking compared with other vorticity components. 
82 
·-. -......0:o:-
7 
Enstrophy Balances 
7.1 Motivation 
Ens trophy . 
. ' €' 
18 
a scalar quantity defined as one half the trace of the tensor w;wj, 1.e., 
e 2: 1 
-w·w 2 1 i, 
and· · 
(7.1) 
it 18 analo · . 
gous to kmet1c energy k, since the latter is one half the trace of the 
Reynolds stress tensor u Th t· · f 1 · 1 · · f 1 
iUj. ere ore,€ 1s a measure o t 1e vort1ca act1v1ty, or o t 1e 
rotational int . . . 
ens1ty of turbulent flows. Although the govermng transport equat10ns 
for€ are 1 
We l known [59], the use of these equations in practical calculations has not 
been Wid 
espread . One of the reasons is the Jack of vorticity data in various turbulent 
flows t 
o qu t"f 
an 1 Y the terms in the equations. 
Recent1 . . . 
Y enstrophy has been used in turbulence modelmg. A mean vorticity and 
enstro 1 . 
p 1Y transport model has been developed to calculate time averaged turbulent 
flows . h 
Wit some success [6]. The simultaneous measurements of velocity and vorticity 
avai1abJ . 
e in th 1 d 1· · · e current database can contribute to turbu ence mo e mg 111 two 1m-
Portant w 
ays: (1) providing benchmark data including many previously unavailable 
Velo 't 
CJ Y-vort" ·t 1 d 1 ·tl 
. ici Y correlations, and (2) helping to improve turbu ence mo es w1 1 
insight 8 
derived from the data analysis. To guide theoretical modeling, the current 
datab 
ase contributes to the development of a physical model, which should help to 
quantitat· 
IVely characterize the various enstrophy transport processes. 
Balint t . · 
e al. [4] determined the balances of terms 111 the transport equat10ns 
of 
lilean a d fl . . . 1 . . . . 1t. -
n uctuatrng ens trophies for a turbulent bounda1y- ayei, usrng s1m u a 
neou8 . . 
measurements of velocity and vorticity. With a miniature 9-sensor vortic1ty 
Probe, th . - . . 
ey measured all the terms in these equat10ns except those that mvolve the 
83 
I 
·i :; 
:,1 .f 
. ., I 
:/ 'I 
·• I I I 
I ;/ 
'i J 
I I 
I 
'": 'C"":'-: --:---.:-.....!- - -~-
Instantaneous t. . 
vor 1c1ty gradients. For the present project, similar measurements 
Were obtained . th . . . . 
lil e nea1-wake of a CJrcular cylmder. The rest of this chapter will 
Present the t b 1 
ur u ent energy production due to the coherent and incoherent fields, 
the derivat. 
. ion of the enstrophy transport equations, and the balances of the terms 
in these equations. 
7.2 
Distributions of Kinetic Energy Production 
Before d' . . 
iscusslllg the various terms in the enstrophy transport equations, it is help-
ful to look b .· 
IJefly at the energy equations. When the instantaneous quantities are 
decomposed i . . 
nto mean, coherent, and incoherent parts (see Sect10n 6.2), the time 
averaged to . . . . . 
tal k1Uetic energy must be equal to the sum of the time averaged kinetic 
energ f 
y O the three fi ld · 
I~ 
2uiuj _ 
As d. 
e s, 1.e., 
(7.2) 
iscussed i·n C . fl hapter 6, the vortex shedding results in coherent velocity uctu-
ations wh· 1 
Jc 1 generate the coherent kinetic energy, ½uiui. The transport equation 
for th 
e Illea k' f th h n llletic energy, ½U;U;, has two production terms coming rorn e co-
erent and . . . . ,. . h 
lllcoherent fields. tor the mean kmetic energy equat10n, F 1g. 7.1 s ows 
that the - - . . . 
. coherent production term, uvdU / dy, has small positive values whereas the 
incoherent production term u'v'dU/dy has large negative values across the wake. 
p . . ' ' 
OS] t1 Ve J . . . . . . . f T va ue Is a gam, and negative value 1s a loss m the mean energy equa ion. 
herefore tl . . . . . 1(,. , ·t· b t 
. ' le mean flow 1s gammg energy from the decaymg · aunan vor ices, u 
Is losin 
g a great d 1 h · l . t fi Id 51·ffil·1ar energ·y transfer ea more energy to t e rnco 1e1en e · 
111echan· 
isrns between 
[25]. d 
. 1 t fi 1d i·e discussed by Hussain the coherent an mco 1eren e s a 
Thus, inspite of the presence of the shed coherent spanwise vortices, almost all 
the total e . . . . ·1 t t 
nergy is lil the mcoherent field. This chapter will study sirni ar ranspor 
Phenoni · · £ · th 3 ·t 
ena for enstrophy. To achieve this, the governmg equatwns 01 e par s 
of the tot I . . 
a enstrophy will be denved next. 
84 
• 
,' 
: ;: 
! ;. 
cl ,1, 
, ,, 
. ,· 
! :, 
i .:· 
,, 
.. , 
,· 
' 
.: :/ 
::I 
! r 
'I 
: ' 
J ;J; 
! ·(1 
' 
7 .3 Derivation of the Enstrophy Equations 
In the presence of coherent structures, one can decompose any instantaneous vari-
ab le into mean, coherent, and incoherent parts, as discussed in Section 6.2. The 
transport equations for each decomposed component of the turbulent kinetic energy 
are given by Hussain [25). Here the corresponding enstrophy transport equations 
w ill be presented. The derivation of these equations involves the following steps : 
1. Substitute the triple-decomposition expression for each variable into the mo-
mentum equations for in compressible flows. 
2. Phase average the resulting equations from step 1. 
3. Time average the equations from step 2 to obtain the momentum equations for 
the mean field. 
4. Subtract the time averaged equations from the phase averaged equations 111 
step 2 to obtain the momentum equations for the coherent field. 
5. Subtract the equations for the mean and coherent fields from the instantaneous 
equations in step 1 to obtain the momentum equations for the incoherent field. 
6. Cross the momenturn equa.t ions for each field with the gradient operator to 
obtain the vorti city equations. 
7. Multiply the vorticity equations for each field by the corresponding mean, 
coherent or incoherent vorticity vector to obtain the enstrophy transport equa-
tions. 
8. Time average the coherent and incoherent enstrophy equations. 
As the intermediate momentum equations have been documented by Hussain 
(251, only the enstrophy balances for each field are presented here . All quantities 
have b een nondirnensionalized by the velocity deficit 'U. a and the wake half-width L0 • 
85 
" ii 
,, 
.. , 
The mean t h . . 
ens rop y equat10n JS: 
(I) (2a) (2b) (3a) (3b) v. a 1 __ 
---,-, an j 
-z----z-an i a -- a --J a;;( 2Di Di) - 'U·W ·- + 'UjWj8 8 (Di u;w:) -(D·u ·w-) J I Bx · f] I J I x· x· x · J J J J 
(4) (5a) (5b) 
- a -- n-,-, + D1w/§ii + Di8 ( Ui Di) + iWjSij x· J 
(6) (7) 
1 a2 1 _ _ 1 ani ani 
+ -=--- (-D;Di) - -- -
Re BxJ8.rci 2 R e Bxi 8.1:i · (7 .3) 
The cohere t . . 
n enstrophy equa tion is: 
(1) (2a) (2b) (3a) (3b) ~ a 1 
-z----z-B Di I 8 (- _ _ ) f] (- I I ) u j --:-- ( - ~) I I OW; 8xj 2W1Wi - -u ·w·- + <ujwi> ox· 2~ W1UJ·Wi -w·<U·W·> a I J I J 1 8x · X· J J XJ J 
( 4a) ( 4b) (5a) (5b) 
~IJU; n ~ + W1·WJ·S1·J· + w· <w' s~ ·> + WjWJ·- + HjWiSij 
l J I) IJXj 
(6) (7) 
1 I) 2 ( 1 ~) 
+ Re IJx ·ox. 2WjWj 
J J 
1 !Jwi !Jwi 
---
Re IJxj IJxj (7.4) 
The incoh 
erent enstrophy equation is: 
(I) (2a) (2b) (3a) (3b) [J .j!__ 1 __ 
I~(w'u'w') l f] c- / I ) -,-,!JDi I I OWi J a (-w'w~) - - u·<w-w> Xj 2 I I - - u -w.- <ujwi> OXj ? f] . ' J ' ? I) . J I I J I IJx · ~ XJ ~ .7:J J 
( 4a) (4b) ( 4c) (5a) (5b) 
--8Ui n -,-, 
+w'w'·S' · + W·<W~S' ·> + <w'w'>s·· + WiWj~ + HjWiSij I J IJ ) I IJ I J I) UX) 
86 
(6) 
1 82 1-+ ----(-w'w') Re ox ·ox. 2 i i 
. J J 
(7) 
l ow' ow' I I 
---
Re OXj OXj · (7.5) 
Tennekes & L 
umley [59] derived the transport equations for the mean and the 
fluctu t· 
a ing enst h' 
rop 1es and presented interpretations of the resulting terms. The 
above t . 
nple-deco · · mpos1t1011 equations have additional interaction terms; therefore, 
all term . 
s are mterpreted here for ease of reference. 
In the me 
an enstrophy equation: 
• Term (l) 
represents the advection of the mean enstrophy. 
• Ter ( 
m 2a) describes the loss of mean enstrophy to the incoherent field by 
gradient production. 
• Term (2b) fi corresponds to the gain of mean enstrophy from the coherent eld 
by gradient production. 
• Term (3 ) d · h · l · t 1 · t a escnbes the transport of mean enstrophy by t e rnco 1eien ve oci y-
Vorti ·t · 
ci Y interaction. 
• Term (3b) represents the transport of mean enstrophy by the coherent velocity-
Vorticity interaction. 
• Term (4) d . 1 ·f t . tching/com1nession mo els the gam or loss of mean enstrop 1y rom s re 
of the . . . h · - · The reorientation 
mean vorticity by the mean stram rate w en z - J · 
of mean vorticity ( also in terms (5a) and (5b )) when i =/ j results in no net 
change . . S . 5 5 in mean enstrophy, as discussed m ectwn · · 
• Term (5a) is for the amplification/ attenuation of the mean enstrophy through 
stretching/ compress ion of the incoherent vorticity by the incoherent strain rate. 
87 
- --•Ill 
• Term (5b) represents the amplification/ attenuation of mean ens trophy 
re c mg compression of the coherent vorticity by the coherent strain through st t h' / · · · 
rate. 
represents the viscous d1ffus10n of mean enstrophy. • Term (6) . . . . 
• Term (7) describes the viscous dissipation of mean enstrophy. 
In the coher ent enstrophy equation: 
corresponds to the advect10n of coherent enstrophy. • Term (1) . . 
• Term (2a) describes the gain of coherent enstrophy from the mean field by gra-
dient prod t· T t h uc ion. his term appears with opposite sign in the mean ens rop Y 
equation. 
• Term (2b) indicates the loss of coherent enstrophy to the incoherent field by 
gradient production. 
m a represents transport of coherent enstrophy by coherent velocity-
• Ter (3 ) 
vorticity interaction. 
• Term (3b) corresponds to the transport of coherent enstrophy by incoherent 
velocity- t · · . . vor 1c1ty 111teract10n. 
• Term ( 4a) models the amp I ifi cation/ attenuation of coherent ens trophy 
th
rough 
stretch ' / . · · b th h ·ent strain rate 
mg compress10n of the coherent vort1c1ty Y e co er 
When i ~ j. The reorientation of coherent vorticity (also in terms (4b), (5a) 
and ( 5 b)) when i -=f. j results in no net change in coherent enS
t
rophy. 
• Term ( 4b) is for the amplification/ attenuation of coherent enstrophy through 
stretching/ compression of the incoherent vorticity by the incoherent strain rate. 
88 
.:, 
! 
' 1, 
•:I 
,' 
I 
·'' 
,I 
I 
,' 
- ~--·"!~ - -----·...:.· 
• T 
erm (Sa) models the amplification/attenuation of coherent enstrophy through 
stretching/ · 
compress10n of the coherent vorticity by the mean strain rate. 
• Term (5b) · · is mterpreted by Tennekes & Lumley [59] as a mixed production 
term and it appears in the mean balance with the same sign. According to 
Tennekes & Lumley, this could mean that the stretching/compression of coher-
ent vorticity by the coherent strain rate produces mean enstrophy and coherent 
enstrophy at the same rate. 
• Term ( 6) describes the viscous diffusion of coherent enstrophy. 
• Term (7) corresponds to the viscous dissipation of coherent enstrophy. 
In th · e incoh 
erent enstrophy equation: 
• Term (1) represents the advection of the incoherent enstrophy. 
• Term (2a) indicates the gain of incoherent enstrophy from the mean field by 
gradient production. This term appears in the mean enstrophy balance with 
0 PPosite sign. 
• Term (2b) describes the gain of incoherent ens trophy from the coherent field by 
gradient production. It has the opposite sign as the same term in the coherent 
enstrophy equation. 
• Term (3a) represents the transport of incoherent ens trophy by incoherent velocity-
Vorticity interaction. 
• Ter (3b) f · h t t ·ophy by the inter-m corresponds to the transport o mco eren ens 1 . 
action of incoherent vorticity with coherent velocity. 
• Ter (4 ) / t· of incoherent enstrophy m a represents the amplification attenua wn 
through st t h" / . f the incoherent vorticity by the incoherent re c mg compresswn o 
89 
·-=--~------.--. -----..,.--,-- .· .. · .. · 
strain rate when i ::::: j . The reorientation of incoherent vorticity ( also in terms 
(4b), (4c), (5a) and (5b)) when i =/:- j results in no net change in incoherent 
ens trophy. 
• Term ( 4b) appears with the same sign as term ( 4b) in the coherent ens trophy 
balance. 
• Term (5a) models the amplification / attenuation of incoherent enstrophy through 
stretching/ compress ion of incoherent vorticity by the mean strain rate. 
• Term (5b) is for the a mplification/ attenuation of incoherent enstrophy through 
stre t ching/ compression of the incoherent vo rticity by the incoherent strain rate. 
It appears as t errn (5a) in the mean enstrophy equation which means that the 
stre t ching / compress ion of incoherent vorti city by the incoherent strain rate 
produces mean enstrophy and incoherent enstrophy at the same rate. 
• Term (6) describes the vi scous diffusion of incoherent enstrophy. 
• Term (7) re presents the vi scous dissipa tion of in coherent enstrophy. 
For a plane wake flm~, the above equations simplify signifi cantly. For example, the 
production t erm (2a) in the mean enstrophy equation includes only th e correlation 
v'w: and the gradient clDz/ cly instead of the full correlation and vorti city gradient 
t ensors . The mean enstrophy equation becomes : 
(1) (2a) (2b) (3a) (3b) 
u ~( ~n2 ) __ fjD, z =-z-onz fJ - - fJ - -) v'w' -- + VWza a(nz v'w~) -(Dz VWz ox 2 z z ay y y fJy 
(4) (5a) (5b) 
- fj -- n, - ,-,- Dzwzs33 + Dxa(U Dy) + zW_jS3j + y 
90 
-=~"!:'".....,,,_ ___ ~--=- " .-.- .. ------- - ·-. ~-
(6) (7) 
(7 .6) 
The coher t . 
en ens trophy equation becomes: 
(1) (2a) (2b) (3a) (3b) 
-- 8 I 
ua(-w2) =---anz awz 1 a C __ ) !._(w <v'w' >) - -vw- + <v'w'>- -8 WzVWz X 2 z z 8y 8 z z z 8y 2 y y 
(4a) ( 4b) (5a) (5b) 
(6) (7) 
(7. 7) 
The incol . 1e1ent enstrophy equation becomes: 
(1) (2a) (2b) (3a) (3b) 
va 1- 1 a l O ( - / / ) - ,- , anz Bwz a(-w'2) 
2 oy (w:v'w:) -- v <ww> X 2 I - -vw- <v'w'>- 2 0y I I z av z av 
( 4a) ( 4b) ( 4c) (Sa) (Sb) 
__ f}U D-,-, 
+ w'w's' . - I I + I I - + W
1 W1 - + zWi8 3i 
I J IJ + Wz <wis3i> <wiwj>Sij X y 0y 
(6) (7) 
1 82 1- 1 aw: aw: 
+ -R .!:) 2(-w:2) - -R ~~· (7.8) 
e uy 2 e uXj u,'E j 
91 
' 
' ., 
7 .4 The lVlean Enstrophy Balance 
Before discussing the various terms in the enstrophy transport equations, it helps 
to study the time mean distributions of the 3 parts of the total enstrophy. Fig. 
7.2 shows these distributions for the mean, coherent, and incoherent enstrophies 
across the wake at x / d = 30. The incoherent part is the largest and peaks near the 
wake centerline. Note that it is already divided by 10 to show some features of the 
other two parts. The mean and coherent enstrophies have roughly the same order 
of magnitude, and they are smaller than the incoherent part by a factor of 40 to 
50. The mean enstrophy has a minimum at zero near t he wake centerline since the 
mean vorticity is zero there. The coherent enstrophy value at the wake centerline is 
a little lower than the values at rJ = ±0.5 due to t he presence of the vortex centers. 
In summary, the incoherent field contains much more enstrophy than the mean or 
the coherent fields. 
The mean enstrophy balance is presented in Fig. 7.3. The various terms in Eq. 
(7.6) a re grouped into 3 plots for clarity. Fig. 7.3 (a) is for terms (1) to (3b) . Fig. 
7.3(b) is for terms (5a) an<l (5b) which are called the "vortex stretching/compression" 
terms. Fig. 7.3(c) shows the viscous diffusion, viscous dissipation, and residual 
terms. 
The plane wake does not have a continuous source of enstrophy generation as 
does the turbulent boundary-layer; therefore, the mean enstrophy must decay in the 
strearnwise direction. This is illustrated in Fig. 7.3(a) where term (1) is shown 
to be negative across the wake, demonstrating the loss of mean enstrophy by fluid 
particles as they advect downstream . Term (1) has two minima near 17 = ± 0. 5, 
and is approximately zero near the wake centerline since the mean enstrophy is zero 
all along it . Term (2a) is negative across the wake, represent ing the loss of mean 
enstrophy to the incoherent field , with a large minimum at the centerline. Term 
(2b) has very small positive values across the wake; it represents the gain of mean 
92 
enstrophy from the decaying Karman vortices. Term (3a) is positive around the 
wake centerline region within 17 = ±0.5, and is negative in the outer region. This 
means the transport of mean enstrophy is from the outer wake to feed the centerline 
region. This is how the mean field has sufficient enstrophy to lose to the incoherent 
field (term (2a)) in the centerline region. The other transport term, (3b), is nearly 
zero across the wake. This term is due to the interaction of the coherent velocity 
and vorticity, and like other "coherent" terms, is quite small. 
Term (5a), which represents enstrophy amplification/attenuation due to the in-
coherent strain rate s~j, is shown in Fig. 7.3(b) to be positive in the centerline region 
(roughly within 17 = ±0.7), and negative in the outer wake. It seems that amplifi-
cation is occurring in the centerline region and attenuation in the outer wake, but 
the magnitudes are near the limit of the experimental error. The mean spanwise 
vorticity nz in term (5a) has the same sign on a given side of the wake so the sign 
change is entirely <lue to the change in sign of the correlation w5 s;_1. 
In Fig. 7.3(c), both the diffusion and dissipation terms are seen to be negligibly 
small across the wake. The residual term is the remainder after summing all the terms 
in Eq. (7.6). It therefore contains the cummulative measurernent un certainties of all 
t erms in the balance. The res idual varies in the range ±0.03 across the wake. It is of 
the same order of magnitude as the larges t term in the equation, i. e. about one- third 
of the maximum of term (2a) . This fact should be kept in mind while interpreting 
the various terms in the mean enstrophy balance. 
7 .5 The Coherent Enstrophy Balance 
In the coherent enstrophy balance from Eq. (7.7), all terms are practically within 
the noise level since the magnitudes are even smaller than the residual for the mean 
enstrophy. To analyze this balance properly, the amplitude should be one order of 
magnitude larger since the uncertainty level cannot be decreased much smaller. The 
93 
structure detection scheme of Hussain & Hayakawa [26] gave higher magnitudes for 
these coherent terms since they only detected structures with high magnitude of Wz , 
Section 6.3 shows that their peak coherent Reynolds shear stress is more than twice 
that found from the current measurements for the same streamwise position. 
Besides using a different structure detection scheme to increase the coherent 
signal-to-noise rati~, one can: (1) measure at higher speeds, (2) measure at wake 
locations closer than x/d = 30, and (3) measure with a higher data sampling rate to 
minimize data attenuation. 
7.6 The Incoherent Enstrophy Balance 
The incoherent enstrophy balance is presented in Fig. 7.4. The various terms in 
the balance are grouped in 3 plots as was done for the mean balance in Fig. 7.3. 
Similar to Fig. 7. 3, all terms are plotted with their signs as presented in Eq. (7.8) . 
These terms have much larger magnitudes than those in the mean and coherent 
enstrophy balances. Recall from Fig. 7.2 that the incoherent enstrophy is about 40 
times larger than the mean enstrophy. All terms in this balance are, in principle, 
symmetric around the wake centerline, as for the mean enstrophy balance. 
In Fig. 7.4(a), term (1) is positive across the wake, and represents the advection 
of the incoherent ens trophy by the mean strearnwise velocity. Therefore, it means 
the advecting fluid particles gain incoherent enstrophy because the incoherent en-
strophy increases in the downstream direction. This term has a. minimum in the 
centerline region and maxima a t roughly 77 = ±1 .0, implying that the streamwise 
gradient of incoherent enstrophy is largest there. Term (2a) represents the gain of 
incoherent enstrophy from the mean field; it is the negative of term (2a) in the mean 
enstrophy equation. Term (2b) represents the interaction of the incoheren t fi eld with 
the coherent vorticity gradient ; it is negligibly small. Term (3a) is the transport due 
to the incoherent field. It is negative within the region 17 = ± l.5, and positive in the 
94 
... 
'·;: 
' ~i 
''J 
• l' ' 
'·11 , • . 
·/':.· ,,· 
i:' , ', . 
'; ':'· •. I ' ,' ' 
, ,J' ', ,:· II ' , 
l ·,• ., ,• 
Jl , ' 
. I 
o ute r wake. This means incoherent enstrophy is drawn from the centerline region 
toward the wake edges , which is consistent with the wake spreading. Term (3b) is 
t h e transport due to the coherent lateral velocity, and it has similar trend as term 
(3a) . The difference is that term (3b) is smaller by roughly a factor of 5, and it 
ch a nges sign roughly at 77 = ±1.0. 
Fig. 7.4(b) shows the "vortex stretching/ compression" terms. Term ( 4a) corre-
sponds to the incoherent enstrophy amplification due to the incoherent action; it is 
b y far the largest term in the balance. This term is much larger than all t he other 
terms and is scaled down by a factor of 10 in the plot. Vortex stretching increases 
enstrophy by straining of vorti ces to increase their component vorticity. The very 
la rge positive value across the wake indicates that, on balance, incoherent vortex 
stretching is much greater t han vortex compression. T his phenomenon is observed 
experimentally here for the first time for cylinder wake flows. 
The source term involving coherent vort icity, ( 4b ), is negligble compared to ( 4a). 
Term ( 4c) is stretching / compression of t he incoherent vorticity by the coherent 
s tra in rate tensor. It is much sm aller than term ( 4a), but is still about twice as 
large as the largest t erm in the mean balance. Term ( 5a) represents vortex stretch-
ing / compression due to t he mean velocity gradient. It is the second largest source 
term, with peaks about one order of magnitude smaller than the peak of term (4a) . 
This term is zero at the wake centerline, because the velocity gradient is zero t here . 
The t erm comes from the mean velocity gradient dU / cly, a nd the incoherent vortic-
ity correla tion w~w~ , which are shown in Fig. 5.1 and F ig. 5.16 to have maxima 
around 77 = ±0.7, respectively. Furthermore, the correlation w~w;, is due t o high 
stre t ching activity in the braid regions, as discussed in Section 5.5. Therefore, t he 
two maxima of term (5a) at roughly 17 = ±0 .7 are probably due to vort icity aligned 
with the mean strain field being stretched in the braid regions. 'I'errn (5b), which is 
the production of incoherent enstrophy involving the mean vorticity, and appears in 
95 
the mea 
n enstrophy balance as term (5a) with the same sign, is negligible compared 
tooth 
er source terms. 
p· 
ig: 7·4(c) shows the diffusion term (6) and the residual term. Since the instan-
taneou · . 
. s vorticity gradients cannot be measured, the viscous dissipation is included 
lil the resid 1 h 
ua; t us, the measurement uncertainties from all other terms are lumped 
Wi th it. The diff · · 1 h k d · · · "fi d us10n 1s near y zero across t e wa e, an 1s ms1grn cant compare to 
the dissi t · . . . . 
pa ion (plus measurement uncertainty). Note that the d1ss1pat10n (residual) 
term· 
is scaled down by a factor of 10 in the plot; it is just slightly smaller than the 
" Vortex t. t h" " 
s Ie c mg term (4a) in the Fig. 7.4(b). This means the enstrophy gener-
ated by the. . . . 
lllcoherent vortex stretching appears to be largely dissipated m the same 
flow 1. · eg1on. 
In 1938, Taylor [58] used dimensional analysis to show that turbulent vortex 
stret h. . 
c ing is approximately balanced by viscous dissipation for a high Reynolds num-
ber flo B . . 
w. asically, he showed that the other terms are of higher order and can be 
neglected Th . . Th . h . t 
· e current measurements support Taylor's findmgs. e mco e1en 
Vortex st . 1 
retching/ compression term ( 4a) is larger than all other terms by at east an 
order of . . d. . t" . 
magnitude, and it is approximately balanced by the viscous issipa 10n, 1.e, 
l ow' ow' 
- I I 
Re "1h; oxj · (7.9) 
7.7 Summary 
The discu · · d th £ 11 · g for x/d = 30· ssion m the above sections has demonstrate e O owm · · 
1
· The mean fl fi ld . f decaying Karman vortices, but loses a ow e gams energy rom 
great deal more energy to the incoherent field. 
2. The incoherent field has much more enstrophy than the mean or the coherent 
fields. 
3. The mean enstrophy decreases with downstream direction. 
96 
___ .. _ ... ____ ,_ ·------- --.-- -- - ---
4. The mean fi eld loses enstrophy to the incoherent field in the centerline region 
and the transport terms balance it by taking enstrophy from the outer wake. 
5. The terms in the coherent enstrophy balance are within the noise level. All 
coherent terms in the mean and incoherent equations are small compared with 
the other terms in these equa tions. 
6. The incoherent enstrophy increases with the downstream direction. 
7. Incoherent ens trophy generation by incoherent vortex stretching/ compression 
is larger than any other terms by at leas t an order of magnitude. This term is 
a pproximately ba lanced by viscous di ssipation. 
97 
', 
:J ,,, 
,·, 
', . 
. 
, J 
I 
1 
' 
' '.
' 
8 Conclusions and Recommendations 
The following conclusions a re for the calibration and data reduction techniques. 
l. The present calibration and data reduction method is independent of the probe 
geometry and sensor orientation. For vorticity measurements, the only geo-
metrical information needed is the distance from the center of each wire to the 
centroid of the frontal sensing area of the probe. 
2. The error minimization method , when used with more sensors than the number 
of unknowns, can produce unique solutions of the measured velocity for angles 
of attack within ±20° . 
3. For calibration angles wit hin ±20°, neglecting the sensor with the lowest cool-
ing velocity from the plus probe results in divergence during data reduction. 
Conversely, neglecting the sensor with the highest cooling velocity results in 
convergence to practically the sarne solution as when all sensors are used. 
4. The zonal calibration scheme can improve measurement accuracy. 
The following conclusions are for the characteristics of the velocity field . 
l. At x /d = 20 and 30, t he maximum rms of the transverse velocity component is 
higher than the streamwise, followed by the spanwise component. T his trend 
is consist ent with X- probe measurements of Yamada et al. [70] at :r / d = 30. 
2. T he small prong spacing of the 4-sensor and 12-sensor probes results in more 
accurate measurements. 
3. T he skewness and flatness values of the velocity components show t he proper 
characteristics of a t urbulent p lane wake. 
98 
," . 
,, 
l 
) 
•: 
4. The 12-sensor probe accounts for velocity gradients in the measuring volume 
and can measure the Reynolds shear stresses more accurately than the 4-sensor 
probe. 
5. The velocity spectra show peaks at the vortex shedding frequency ls for the 
streamwise and transverse components. The spanwise velocity spectra donot 
have any peak at l s, indicating parallel shedding. spanwise component. 
6. The JPDFs of the streamwise and transverse velocity components show pre-
ferred orientations at +45° from the centerline for the lower wake and at -45° 
for the upper wake. The other correlations are about zero across the wake due 
to flow symmetry. 
7. The converging legs of the saddle region are the dominant uv generators. These 
flow regions are a long the direction normal to the mean shear. 
The following conclusions are for the characteristics of the vort icity field. 
1. The miniature 12-sensor probe can measure simultaneously velocity and vortic-
ity with good accuracy in the near-wake of a circular cylinder for x / d = 20-30. 
2. Central-differencing, which res ults in proper phase alignment of the stream-
wise velocity gradients with the directly rneasured variables, should be used in 
applying Taylor's hypothesis. 
3. The measured mean spanwise vort icity component agrees well with the deriva-
tive of the mean strearnwise velocity. The other mean vorticity components 
are about zero. This is a good test of the accuracy of the measurements. 
4. At x/ d = 30, the rms of the streamwise vorticity is highest , followed by the 
transverse and the spanwise components. The rms values for the transverse 
and spanwise components are somewhat attenuated due to the low sampling 
rate. 
99 
", J 
,3 ~ .. 
,• I 
5. The skewness and flatness values for the vorticity components have the proper 
characteristics for a plane wake. These values are high at the wake edges and 
have nearly Gaussian values in the centerline region. 
6. Only the spectra of the spanwise vorticity component show intensity peaks at 
the shedding frequency, indicating parallel shedding. 
7. The extremely narrow PDFs of all vorticity components in the freestream com-
pared with the centerline region indicate a high signal-to-noise ratio of up to 2 
orders of magnitude. 
8. The JPDF contours of WxWy show a preferred orientation that is along the 
direction of the mean shear. Therefore, wx and wy are negati vely correlated 
in the lower wake and positively correlated in the upper wake. The pattern is 
consistent with the computational results of Meiburg & Lasheras [39]. Other 
vorticity correlations are about zero across the wake. 
9. T'he correlations of vorticity and velocity gradients are about zero across the 
wake except for those that involve gradients of the spanwise velocity. 
10. Quadrant analysis of the correlation wxfhl/ox and wzow/oz show more stretch-
ing than compression activity across the wake. 
11. The W z component serves as the vehicle for vorticity exchange between the w.1: 
and wy components. 
The following conclusions a re from the analysis of the turbulent wake conditioned 
with the passage of the shedding vortex . 
l. Detection at the peak of the periodic W z signal due to vortex shedding corre-
sponds to passage of the vortex center in the lower wake and passage of the 
saddle in the upper wake. 
100 
· .. '
-': : ' I ,(: 
' . 
1 .. ; ·:/: 
: .. ' ·-, ,.,,, . 
' ~ : . 
'; : 
i 
2. The transverse velocity is more coherent with the shed vortex than the stream-
wise velocity, especially in the wake centerline region, which results in higher v 2 
there. The spanwise velocity is incoherent since the vortex shedding is parallel. 
3. The incoherent Reynolds normal stresses and turbulent kinetic energy are max-
imum near the vortex center and minimum near the saddle region. The incoher-
ent Reynolds shear stress u'v' is maximum near the saddle region and minimum 
near the vortex center. These observations are consistent with the findin gs of 
Cantwell & Coles [10] and Hussain & Hayakawa [26]. 
4. Although wx and wy are negligible, <w:2> and <w~2> are not. The variance of 
the streamwise vorticity has the strongest coherence in the diverging braids of 
the saddles , which suggests a three-dimensional vortex line orientation of these 
flow regions. 
5. <w?> has maximum coherence near the vortex center and minimum coherence 
near the saddle region . 
6. The coherence of t he enstrophy is maximum near the vortex center and mini-
mum slightly upstream of the saddle region . The d issipation rate fo llows the 
same pattern. 
7. T he in coherent shear stress u'v' is almost entirely phase-locked to t he Karman 
vortex shedding. The variance w'2 has the highest phase-locking compared 
with other incoherent normal stresses. The variance w? has the highest phase-
locking compared with other vorticity components. 
The following conclusions a.re for the enstrophy balances at x / cl = 30 for the 3 parts 
of the measured flow field: mean , coherent, incoherent. 
l. The mean flow field gains energy from decaying Karman vortices, but loses a 
great dea l more energy to the incoherent field. 
101 
,., 
., 
... 
j. 
' ~ ' 
. :·' 
,i ;_. 
2. The incoherent field has much more enstrophy than the mean or the coherent 
fi elds . 
3. The mean enstrophy decreases with downstream direction. 
4. The mean field loses enstrophy to the incoherent field in the centerline region 
and the transport terms balance it by taking enstrophy from the outer wake. 
5. The t erms in the coherent enstrophy balance are within the noise level. All 
coherent t erms in the mean and incoherent equations are small compared with 
the other t erms in these equations. 
6. The incoherent enstrophy increases with the downstream direction. 
7. Incoherent ens t rophy generation by incoherent vortex stretching/ compression 
is larger than any other terms by at least an order of magnitude. This terrn is 
approximately balanced by viscous di ssipation. 
The following recommendations can be drawn from the discussion of the results 
in the previous chapters. These suggestions can be useful in either further analysis 
of the current database or further experiments in turbulent wakes . 
1. Develop techniques to make miniature probes more robust , and able to function 
properly in other fluids such as water . 
2. Repeat the measurements in this project with higher sampling frequency. Also, 
perform parametri c studies by varying the Reynolds nurnber. 
3. Build a vortex generator to calibrate the vorti city probe with known vortex 
strength. 
4. Use the probes , and the data reduction technique to measure in the far-wake 
region (x/d > 200) where accuracy is critical , and in the very-near wake (x/d < 
20) where the angles of attack are extremely high. 
102 
.. --
5. Perform two-point measurements using vorticity probes to investigate the three-
dimensionality of the vortical structures. 
6. Analyze the velocity gradients as thoroughly as the velocity and the vorticity 
components. 
7. Improve the data reduction method to make use of the fact that the error 
equation is parabolic in all the velocity gradients. 
8 . Perform conditional analysis with trigger condition such that the incoherent 
vorticity has high magnitude with the same sign as the coherent vorticity. 
9. Analyze the kinetic energy balances of the mean, coherent and incoherent flow 
fields. 
10. Analyze the enstrophy balances for x/d = 20 to confirm findings at x/d = 30. 
103 
,.. 
,,, 
" ,. 
'" ,, ,, 
,,I 
'\ 
?, ·,: 
'.~ . 
y 
V 
w, 
z 
F" 
'9· 1-1. Definition of the coordinate axes. 
104 
" ,,
,1 
,. 
: 
ii 
ii 
r' . ,' 
: , 1.,. 
I ~. ( 
.,:)t 
·,I . 
... :·,: 
' ,, ,, 
Pig. 1 
~ 
~ 
~ y 
z 
/ 
I 
X 
' \ 
\ 
\ 
\ 
I 
\ 
\ 
·
2
· Sketch of the double- roller model for the vortical structure 
of Grant [21 J. 
105 
,. 
,, 
,, 
I' 
/' 
I~ 
. ; P! 
_: ii 
··"' 
: ·/.i 
I·' 
: .,, 
. ,·J 
I 
,. 
,, 
•1 
Fig. 
___ Flo" Direction 
1 
-2 -4 
1 
·
3
· Velocity vector plot in the x-y plane from Browne et al. [9J. 
The saddle region is between two consecutive vortices. The 
con~erging and diverging separatrices are regions of flow 
moving toward or away from the saddle. 
106 
-6 
Fig . 1 .4. Sketch of the model of the wake vortical structure from 
Hussain & Hayakawa [261. The braids transfer fluid away 
from the sadd les, which are marked by symbol X. 
107 
·, : ; 
. ' 
.• 
Fig . 
I 
z 
0 - iµJ,UJ~llfl+I.LJ..lJJ-JJJJJlLUlJJJ~-12 x- II +l O -) O 
-r 
+lr-___ _____ _ 
I . i«,tRe1111J£<f {:,e111111111<f£.t'1!l(IJ//, 
Y I\\\\"'~" •\\\\\\\~~ •\\\\\~ 
-) ot--___ 6 ____ .--.--......J 12 x- II 
M . ex mes at various section cuts in the woke calculated by 1 
·
5
· Vort 1· fl~iburg & Losheros [391. The flow is o laminar _woke of a tapered 
1 
t plate. The upper woke is represented by solid lines, and the 
oter woke by dashed lines. The bottom plot in the x-y plane 
s ows vortex lines inclined in th e direction of the mean shear. 
108 
:1 1\1tlt- -? 
......... 
0 
0:.0 
IB 
1 -. N 
In 
2.91 1~, 5.04 
0.08 0.18 
~t=fil -
Blo\Jer 
A. 
- -
IA '<:!" 
~ 
.... 
•I • I 
8.81 3.15 
Fig. 2.1. Scale drawing of the wind tunnel. 
Al l dim ensions ( m) 
~r:r I c=) -t 
t--1 
A-A 
C"? 
(0 
0 
l-1 
~u ]}63 
l 4.4 
n-n 
' ,, 
,__. 
,__. 
0 
DC MOTOR 
OFAN 
41 .91 
'~ r 
22.136 
I 
I 
.... ,~ .... ,~ 
SCREEN 
40.65 
~~ 
Fig. 2.2. Scale drawing of the calibration jet. 
All dimensions (cm) 
75 
1 
25 
..... ~ ,,,., 
I --J 
w 
z 
yaw 
angle 
y 
pitch 
angle 
-------c::::=~-----probe axis 
X 
Fig. 2.3. Definition of the pitch and yaw angles. 
111 
' ' 1'• 
:·: .·'. ' :• 
', '. · ,,' 
y ty ,. I ~. 
4 
L 2 
J_ 3 j 
. 
z ·~L_j 
L = 0.51 mm, O! = 42 ° 
· · Schematic sketch of the Kovasznay-type 4-sensor probe. Fig. 2 4 
112 
z 
. .. ,·,,· 
. .,, . ~ 
.: · · .,·: 
. '. 
4 
• • 
4 
2 
• • 
3 
1
.,.. 0.51 mm . ~, 
2 
X 
0 
45 
z 
· ·
5
· Schematic sketch of the plus-shape 4-sensor probe. Fig 2 
113 
. , .. '-' .,-.;; 
Dimensions in mm 
End view 
Fig. 2.6. Schematic sketch of the 9-sensor vorticity probe. 
114 
/--\--"' / "\ 
I \ 
I !\------------~ 
I \ I 
\"\ I \ I 
'-I ~/ /--pJ '- - \-
/ I "\ / \ 
/ 66 ~--1- 66 l 
I I \ _ I f" 4 4 _\ ) __ \_9_9~ 
"' / "' 
"' / "' 
--- -- ......_ 
--
I 
/ 
/ 
·---- 132 ---- --- 53 -
Dimensions in 1/100 mm 
/ 44 I 
1-42-
Fig. 2.7. Schematic sketch of the 12-sensor vorticity probe. 
115 
114 
F. 2 8 Smoke f\ow visualization of tne Karm6n vortex lines snowing 19. . . parallel shedding. 
UG 
wire 
UN-----
prongs 
Fig . 3.1. Schematic sketch showing the three components of the 
coo ling velocity for a hot-wire sensor. 
117 
50 
40 
N' 30 C/) 
~ 
E 
'-' 
C"l 
:t:: 
::) G> 20 
10 
0 
-2 -1 0 
e (volts) 
F' ig. 3.2 C l'b t· . · a I ra 10n curve of one sensor 1n 
!he line is a 4th-order polynomial 
18 measurement. 
118 
2 
the 12-sensor probe. 
curve fit; the symbol 
.,---· 
\.. .. -~~ -
Q) 
CJ) 
C 
0 
.c 
() 
;'='. 
20 
15 
10 
0 
o_ -5 
-10 
-15 
-20 
-20 
4 
I 
- - - - - -t 
8 
I 
5 1 
--- - ------+-----
9 6 
-r - - - - - - - -- ---,-----
3 7 2 
-15 -10 -5 0 5 10 15 
yaw angle (deg) 
Fig. 3.3. Division of calibration ang le space into calibration zones. 
119 
• . • ~ ! 
~,.. ' . 
20 
(b) 
1.50 1.50 
-
(/) 
'-
.S 1 .45 
VJ 
> 
--.s 1.45 
> 
1.40 
4 .00 
1.40 
4.05 4.10 4.00 
4.05 4.10 
U(m/s) U(m/s) 
I ... ,
Fig. 3 .4. (Comparison of solution techniques for the plus probe: 
a) 4 sensors used, and (b) 3 sensors used. 
120 
Fig. 
50 
40 
,-.... I v 
U) 30 ~ I 
E 
'--' 20 
LL \ 
10 
' (a) \ .-
'- _,. 
0 
50 
40 
v,-.... 
U) 30 ~ 
E 
'--' 20 
LL 
10 
(b) 
0 
50 
40 
,-.... 
v 
Cl) 30 ~ 
E 
'--' 20 
LL 
10 
(c) 
0 
-10 - 5 0 5 
10 
velocity (m/s) 
3·5· Solution characteristics of the 12-sensor probe velocity 
c,om~o~ents: (o) U, (b) V, and (c) W. The vertical dashed 
line 1nd1cates the solution. For each plot, all other flow 
variables are held at the converged values. 
121 
:Jo 
" 1S I 
:)8 
'--' 
Cl) 
Cl) 
Q) 
c:: 
3: 
Q) 
.::{_ 
Cl) 
Fig. 
1.2 
(a) 
1.0 
0.8 
0.6 
0.4 
0.2 
o.o 
-0.2 
5 
4 (c) 
3 o
0 
2 
0 
-1 
-2 
-3 
-4 
-5 
-5 
8 
0 
0 
0 
0 
0 
Tl 
5 
Cl) 
E 
I-
Cl) 
en 
Q) 
C 
-+-' 
0 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
0.0 
50 
40 
30 
..::: 20 
10 
0 
-5 
(b) 
~ 
(d) 
D 
0 
t,. 
D 
8 
t,. 
§ 
t,. 
~ 
D 
0 
~DO 
D 
~t,.t,.D.§ 
0 
Tl 
t,. 
§ 
t,. 
~ 
@ 
~ 
9 
0 0 0 
5 
4.1. Mean I . a\ x/ d ve o~i\y and moment~ of the fluctuating. velocity . components ( b) - 30. ( o) meon veloc 1\y w1\h \he curve t,t as solid line, 
Therms, _(c) skewness, ond (d) flatness. o, u; o, v; A, w. 
and velocity component rms values are normalized by u0 in this 
subsequent figures; ri ==y/L
0 
in this and subsequent figures. 
122 
, .· 
F'ig. 4 
0 .7 
0.6 
0.5 t:,.t:,.1:,,./:;./),. 
* 
0.4 ."IJN. 
:::i 
I:,. 
Ab. 
0.3 ... 
I:,. • A 
0.2 ... 
0.1 
0.0 
0.7 
0.6 (b) 1:,,./:;.t, I:,. A 
0.5 
0.4 I:,.. 
~. 
* 
I:,. 
> 
... 
0.3 I:,. ... A/:;. 
0.2 ... 
I:,. ... I:,. 
0.1 • 
0.0 
0.7 
0.6 (c) 
0.5 
• 
0.4 
1:,,.•~ ~ JP 
0.3 A 
I:,. A A 
0.2 A 
0.1 
0.0 
-6 -4 -2 0 2 
4 6 
11 
· t mparison of the velocity component rms values at two 
.2 Co · 
s reamwise locations: (a) u .. (b) /, and (c) w·. The open 
symbols are for x/d==20, and the closed symbols are 
for x/ d= 30. 
123 
·· ' 
Fig. 4 
* ::J 
* > 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
0.0 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
0.0 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
0.0 
(b) /:t:, !::,. !::,. 
t::,. t, 
oD 
t::,.~Dt::,. 
L:IJ! A~ 
~ AtrJ 
6 
~ 
lj6 
~ 
-6 -4 -2 0 
Tl 
& 
q 
2 
i 
iJ 
12 
mpanson af the velocity component rms values from 
.3. Co · 
-sensor probe measurements to V-probe and X-probe 
~alues at x/d=30: (a) u', (b) v', and (c) w'. 
v' V- probe; 6, X- probe; and •, 12-sensor probe. The 
Th- probe comes from one array of the 12-sensor probe. 
e X-probe data are from Yamada et al. [701. 
124 
' • 
Fig. 
I?, 
I 
I~ 
I 
I~ 
I 
0.06 
0 .03 
0.00 
-0.03 
-0.06 
0.06 
0 .03 
0.00 
-0.03 
-0.06 
0.06 
0.03 
0.00 
-0.03 
-0.06 
(a) 
• 
. '
A 
• A AAAAA AAAA 
• i 
i 6i 
A 
(b) 
AAAAAAt•~tAAAAAA 6 N:f:l:A 
(c) 
- 6 -4 - 2 0 
1l 
2 4 6 
4·4· Comparison between the 4-sensor and 12-sensor probe 
(
measurements of the Reynolds shear stresses at x/d==30: 
a) -uv, (b) -uw, and (c) -vw. The open symbols are for 
the 4-sensor probe, and the closed symbol .for th~ 2 
12-sensor probe. These stresses are normalized with u
0 
, 
125 
Fig . 
0.10 
0.05 
I~ I 0 .00 
-0.05 
-0.10 
0.10 
(b) 
0.05 
I~ 
I 0.00 ~~~~ ~ ~~~-- ~~~~~~~ 
-0.05 
-0.10 
0.1 O 
(c) 
0 .05 
I~ OD 
I 0 .00 
~~~~~~~~~~~~~~~~ 
-0.05 
-0.10 
-6 -4 -2 0 
Tl 
2 4 6 
4.5 . Com · Id panson of 12-sensor probe measurements of the Reyno s (~r)esses to V-probe, and X-probe values at x/d==30: (a) -uv, 
Th -uw, (c) -vw. o , V-probe; a, X-probe; .,, 12-sensor. 
The V-probe dota are tram one array of the 12-sensor probe. 
e X- probe data are from Yamada et al. [701. These stresses 
are normalized with u 
2 
0 • 
126 
10 _, 
10 - 2 
~ 10 - 3 
10 - 4 
10 - 5 
10 -, 
10 - 2 
:, 
-e- 10 - 3 
10 - 4 
10 -5 
10 -, 
1 O -2 
:, 
-e- 1 O -3 
1 O -4 
1 O -5 
Fig. 
10 ° 10, 10 2 
10 3 10 o 
101 102 
f(Hz) 
f(Hz) 
4·6· Power spectra of the streamwise velocity. All spectra are norm_alized by the local variance and integrate to unity 
in this and subsequent figures. 
127 
10 3 
10 - 1 
10 - 2 
-& 10 - 3 
10 - 4 
10 - 5 
10 - 1 
10 - 2 
-& 10 - 3 
10 - 4 
10 - 5 
10 - 1 
10 - 2 
-& 10 - 3 
10 - 4 
10 - 5 
10 o 10 1 10 2 
f(H z) 
101 102 
f(H z) 
Fig . 4. 7. Power spectra of the transverse velocity. 
128 
.. ~~ .. 
.'.., ·· ' .' 
. - ......... . 
10 3 
10 _, 
10 - 2 
~ 10 - 3 
10 - 4 
10 - 5 
10 _, 
10 - 2 
~ 10 - 3 
10 - 4 
10 - 5 
10 -, 
10 -2 
3: 
-B- 10 - 3 
1 O -4 
10 - 5 
10 ° 10 1 10 
2 
10 1 10
2 
f(Hz) 
f(Hz) 
ig. 4.8. Power spectra of the spanwise velocity. F' 
129 
10 3 
_.., ..... ...... 
2.0 
1.5 
'? 1.0 
'--' Q. 
0.5 
o.o 
-0.5 
2.0 
1.5 
~ 1.0 
Q. 
0.5 
o.o 
-0.5 
2.0 
1.5 
1.0 
0.5 
o.o 
-0.5 
I 
11 = -0 .47 
ri= 1.29 
I 
I 
H, 
I 
I 
I 
-2.0 
- 1.0 0 .0 
u 
1.0 2.0-2.0 -1.0 
o.o 
LI 
Fig. Go 
0
• 
1 
,ty density function (PDF) of the streomwise velocity. 4.9. Prob b'I' co uss,on distributions ore shown os dashed lines. The velocity 
no mpo~ents in this and subsequent PDF and JPDF figures ore 
rmalized by Lio· 
130 
· . .. :;·;,. 
, . ··.· 
. · .. , 
2.0 
~ ,;! ..... •• 
1.5 
-;- 1.0 
'--' Q. 
0.5 
o.o 
-0.5 
2.0 
1.5 
-;- 1.0 
'--' Q. 
0.5 
o.o ~ 
-0.5 
2.0 
TJ = -0.47 
1.5 
-;- 1.0 
~ 
Q. 
0,5 
1/ 
o.o 
-0.5 
-2.0 
-1.0 
I, 
I 
I 
/ I 
0 .0 
V 
\ 
\ 
1.0 2.0-2.0 
I 
'I 
-1 .0 o.o 
V 
1.0 
Pig. 4.10. PDF sh of the transverse velocity. Gaussian distributions ore 
own as dashed lines. 
131 
.... ~ .·,,,. 
2.0 
~ 1:-"' .. ~,. -- •• 
2 .0 
1.5 
..-.... 1.0 ~ 
'-' Cl.. 
0 .5 
0 .0 
-0. 5 
1.5 
..-.... 1.0 ~ 
'-' Cl.. 
0.5 
o.o 
- 0 .5 
2.0 
1.5 
'i' 1.0 
.....__, 
Cl.. 
0.5 
o.o 
-0.5 
- 2.0 
/, 
- 1.0 0.0 1.0 
2.0-2.0 
w 
I. 
/2 
-1.0 
I. 
0.0 
w 
~ 
Fig. 4 · 11 · PDF of the spanwise velocity. Gaussian distributions are 
shown as dashed Jines. 
132 
I!-,..,. ..... 
5 
4 
3 
2 
02 
I 
I 
Q1 
0 
_ _ ____ ___ ___ I ___________ _ 
-1 
-2 
03 
-3 
-4 
-5 
-5 - 4 - 3 -2 -1 
F' 19,412 Df''t' · · e 1ni 10n of the quadrants. 
133 
I 
I 
I 
2 
Q4 
3 4 5 
........ .. ,-···- . 
- -- -- --
1.0 ,,. -1.65 1.0 
,,. -1.06 
0.5 0.5 
> o.o 0 .0 
0.0 
-0.5 
-0.5 
-0.5 
:::,. 
-1.0 
:i -1.0 -0.5 
-1 .0 
-1.0 
0.0 0.5 1.0 
'-
-1.0 -0.5 0.0 0.5 1.0 
-1.0 -0.5 0.0 0.5 1.0 
Q. 
11= 1.29 
1.0 
1.0 
0.5 
0.5 
> o.o 0.0 
o.o 
-0 .5 
-0.5 
-0.5 
-1.0 
-1.0 
-0 .5 0.0 0.5 1 .0 -1.0 -0.5 0.0 
0.5 1.0 -1.0 
-0.5 0.0 0.5 1.0 
- -----
------ -- -- ----- --------------- - ----- --- ---
11'" -0.47 
1.0 ,,. -1.6 5 
-0.5 
-0.5 
-1.0 c,._;:,,=.J..d,..L.J.~~~.;..J -1.0 w,,.~~:Ll.J..~ca::...,,!........J 
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 o.o 0.5 
1.0 
0.5 
-0.5 
-0.5 
-0.5 
-1 .0 
· 1.0 -0.5 0.0 0.5 1.0 
-1.0 -1.0 
.1.0 -0.5 o.o 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 
u 
u 
u 
Fig. 4. 13 J · ) ( ) d . · . o,nt Probobility density function (JPDF top on covariance 
integrand (bottom) contours for the streomwise and transverse 
velocity components. 
134 
. : : 
!' , 
~ ... ·._ ; 
~ 
0.05 
0.04 
0.03 
0.02 
0 .01 
' 
'.I 
0 .00 
- 0 .01 
-0.02 
-0.03 
-0.04 
:::::o,,_ I • 
0,_ --- I \ ,' 
',,_ --,0.. \ , 
---o -,,___ \ ,,' 
--,, ---o__ , ,, 
\ ~- ,' 
',, ..... \ ,,',,' 
-,_, ', ,' ,',' 
'-o, -, \ ,,' ,,',,' 
, I 0. ' ,' •',,' 
', ', \ , _,,., 
', I ', \ ,'_.,.,, .. G 
' ' \ , ,.. , 
', l ',, g,,,- __ ,, 
' ' , ,.. 
' , ' ~ , I ~------- , 
- - - -
-------
_t_ 
' ,.. 
-------
-0.05 
- 2.0 
- 1.5 
-1.0 
-0.5 0.0 
T) 
Fig. 
0.5 1.0 1.5 
4. 14. Quadrant analysis for the correlation of the streamwise 
and transverse velocity components: D, 01; 0, 02; 
t!,., 03; 0, 04; and Y, total (sum of quadrant values 
in this and subsequent figures). 
135 
2.0 
,. 
;: 
:j 
-a.. 
1.0 
0.5 
s: 0.0 
-0.5 
-1.0 
·1.0 
s: 0.0 
·0.5 
T\= -1 .65 
-1.0 
-1.0 
·0 .5 0 .0 0 .5 1.0 -1 .0 -0.5 0.0 0.5 
1.0 -1.0 -0.5 0.0 ,0.5 
1.0 
T\= 0.71 1.0 
11= 1.29 
1.0 
o.o 
0.0 
-0.5 
-0 .5 
-0 .5 
-1.0 
-1.0 
0.0 0 .5 1.0 -1.0 -0.5 0.0 
0.5 1.0 -1.0 
. -0.5 0.0 0.5 1.0 
-- -- ------- ---- ----- ------ --- --- -------- -- ---- -
0.5 
o.o 
0.0 
-0.5 
-0.5 
-1.0 
-1.0 
1.0 -1.0 -0.5 0.0 
0.5 1.0 -1.0 
-0.5 0.0 0.5 1.0 
-1.0 W=<~;:...,.....L-l.....,_...1.A-~ -1 ,0 
-0.5 0 .0 0.5 1.0 -1.0 -0.5 o.o 0 .5 1 .0 -1.0 -0.5 0.0 0.5 1.0 u 
u 
u 
Fig. 4. 15. J PDF (top) ond covariance integrand {bottom) contours for 
the streamwise and spanwise velocity components. 
136 
. . ,>.:/' ·, 
; . . . ~ 
. . .· ~ ' . 
_'l•,I' 
1. :: .. ' , 
I ' •••• 
~. -. f; ,. ,· 
,~ 
0.05 
0.04 
0.03 
0.02 
0.01 
o.oo 
-0.01 
-0.02 
-0.03 
-0.04 
-0.05 
I 
I 
,ti.-- --:~~~--W=::====r~,--~~~~:ft-._ 
,,' 0---- --- ----
,, ,' I -•• ·-D. 
r: , / I ••• • •• 
,, = , I 1.l •• ·-••• 
- 'LJ ·-.. ·-. I •• _ 
1 ·o 
-:..: -.- --=-- :..:-:.: ~--=--- -~ _1_::Y-- - -- -- --~- ----- - --•---- ---- -
- ....... -- -= I - - - - - - - - - - -
.. .. ...~ .. ,, I ,,0 
,' 
'B ·••. I o,,,,-_,,,'' 
• •, I ,' ,, 
', ,' r:v'' 
' ~ , >:.I 
-. 'v• I ,, ,, 
'•o ::::::: ... ,Q ••• --::t· _:@:-:::,-,,..,_A',,,::,,,' 
-~- ---- ~ 
I 
I 
I 
I 
I 
0.5 1.0 1.5 
.2.0 
-2.0 -1 .5 -1.0 -0.5 o.o 
Tl 
Fig. 4 · 16· Quadrant analysis for the correlation of the streamwise a,,nd sponwise velocity components: D, 01; 0, 02; 
0 , 03; 0, Q4; and Y, total. 
137 
... : ,J': 
,'- ',•. 
. ·.· ,; 
- ,. , ' ? 
·. :· ,/ . 
. . . , : 
.:.(} ·:\ 
,. 
1.0 11= · 1 .65 1.0 11= -1.06 
0 .5 0.5 
s: 0 .0 0 .0 
· 0.5 
-0 .5 
· 1 .0 
-10 - -1.0 s: 
·1 .0 
·0.5 0.0 0.5 1.0 
-1 .0 -0.5 0 .0 0.5 1.0 -1.0 -0.5 0.0 ,0.5 > 
-
11= 1.29 
a.. 1 .0 11- 0.12 Tl= 0.7 1 10 1.0 
0.5 0.5 0.5 . 
s: 0.0 0.0 
-0.5 
-0.5 
-0.5 
! -1 .0 
-1 .0 
-1.0 
·1 .0 
-0.5 0.0 0.5 1.0 
-1.0 -0 .5 0.0 0.5 1.0 -1 .0 -0.5 0.0 0.5 
--- ----------
------- ----- -- - --- - -- - - - -- -- --- ---
1 .0 ri= -1 .65 1 .0 ri= -1.06 
0.5 0 .5 
s: 0 .0 0 .0 
-0.5 
·0 .5 
s: -1.0 
·1.0 
-1.0 . 
·1 .0 
·0.5 
-1.0 ·0.5 0.0 0.5 
> 0.0 0 .5 1 .0 
-1.0 
·0.5 0.0 0 .5 1.0 
-a.. 
s: 1.0 
1. 0 > 
0 .5 
0.5 
s: 0.0 0 .0 0.0 
· 0.5 
·0.5 
-0.5 
·1.0 
-1.0 
-1 .0 
-1.0 
-0.5 0.0 0 .5 1 .0 
-1.0 
-0 .5 0 .0 0.5 1 .0 -1.0 -0 .5 0.0 0.5 V V V 
Fig. 4.17. JPDF (top) and covariance integ ra nd (bottom) contours for 
the transverse and spanwise velocity components. 
138 
1.0 
1.0 
.. 
. ' 
' ·.· 
' 
1.0 
1.0 
.... . ,t ···. 
,~ 
0.05 
0 .04 
0 .03 
0 .02 
0 .01 
o.oo 
- 0.01 
-0.02 
-0.03 
- --=- -'Y- --=--=-=--=-=.Jf':.:-__ - - - - _ I_ - - - - - - - - - -:.--=--
----~-------!-.---------..lf'-- ---- ---.. -- ---
- , , '¢, , I _ _,,.8 
', ' ,. -· .0 
'0 . ,,0' ,' 
' ' -,. ',, / 
. '<>, ... ,• -·· 
',,_ · -. ,,' _()' 
··o , ,, ,---
----.::::~:::::::{:S-~~,-~,~=~:--· 
- 0.04 
-0.05 0.5 1.0 1.5 
2.0 
- 2.0 - 1.5 -1.0 -0.5 0.0 
Tl 
Fig. 4 · 18· Quadrant analysis for the correlation of the transverse 
'X'd sganwise velocity components: 0, Q1; 0, QZ; 
, 0 ; ¢, Q4; and Y, total . 
139 
~ ·1 .; . 
... / .. ' . . .: 
. . ~ ~'. ' .' 
... ·. ·:,, . 
1 
C 
0 
(l) 
E 
(/) 
(/) 
(l) 
C 
3: 
(l) 
.:x 
(/) 
Fig. 
- ·------- - - -
1.0 
3.0 (a) 
(b) 
2.5 D 0 .5 oooo 
6~0 
2.0 0 6° 0 
6 0 
60 
(I) 
0 
o.o 
E 1.5 
I... 
6 
0 0 
6 1.0 0 0 -0.5 
!& 
0.5 ~ 0 
~ 9 i:j -1.0 0.0 
2.0 
50 (c) (d) 1 .5 
6 
1.0 40 
6 0.5 6 
6 30 (I) 
D (/) 0.0 oogEfB:Baao Q) 
0 C ...., Do 6 0 0 6 .._ 
-0.5 20 6 6 
0 6 6 
6 -1.0 
0 10 
~a~S 
0 
-1 .5 
-2.0 
0 
-5 0 5 
-5 0 5 T] 
T] 
5. 1. Mean vorticity and moments of the fluctuating vorticity components 
at x/d=30: (.a) mean vorticity with solid line as -dU/ dy, (b) rms, 
(c) skewness, and (d) flatness. o , O ; o, 0 ; 6, 0
2
• The mean 
and rms values are normalized by u ;L in t~is and subsequent 
figures; T]=y /L0 in this and subsequ;nt 0figures. 
140 
··, · , 
' ..
~ _ ... 
Fig. 
* 
• 3>. 
* N 
3 
3 
2 
0 
3 
2 
0 
3 
2 
0 
(a) 
6. 
• 
6.A 
(b) 
4. 
(c) 
~ 
6.A 
-6 -4 
........ 
It, 
MMt1.A 
6. 
A 
6. 
6. 
• 
• 
6. 
• 
~ 
A..t,.6.6.6, 
6.& 
k 6. 
• 
6 
• 
6 
A 
6 
• 
~ 
_.6 6. 
• 
JI>. ~ 
A 6 
• 
-2 0 2 4 
6 
Tl 
5.2 c stream . n ° the vorticity component rms values at two 
· ompariso f 
symbol;ise locations. (a) w,', (b) w ', and (c) w,'- The open 
ore for x/d~2D; the closed symbols ore for x/d~3D. 
141 
·: ,- ' 
0.30 
0.20 
0 .10 
0.00 
I 
I 
I 
I 
I 
I 
I 
I 
~ 
I 
I 
I 
I 
I 
I 
I 
I 
"··1-.. 
,' I 
I I '. 
,' I 
I \ 
I \ 
I \ 
I \ 
I \ 
I I 
I 
' 
' .. 
' 
' 
' 
' 
' 
' 
' 
' I 
' 
' 
' 
' 
' 
' 
-,- - - - - 1- - - - 'rtu-.,..,,._ ,...., _ ...... ,e-----=--=-=-: -
- --·-EJ-- -- ---- - -G- I 
- - - - - • - ~'- - - - - - Q 
' 
' 
' 
' I 
' ' I 
' I 
'' 'al 
' , 
, ' , , 
,,' ' ,,' 
, .. , 
',, l(;j ---- ..................... ,' 
\ I 
\ I 
' , 
-0.10 
'1:3_ I / 
· 1-0' 
I 
-0.20 
Fig. 
- 2.0 - 1.5 -1 .0 -0.5 0.0 
i) 
0.5 1.0 1.5 2.0 
5.3. Comparison between the central-difference scheme (C-D) (open 
symbols) and the backward-difference (B-D) (closed symbols) 
scheme. The affected term represents production in the mean 
enstrophy balance . 
142 
· ·, : .: · . . ,
. ': : .". 
.. 
>... 
-0 
"-!cf 
-0 
If 
Fig . 
0. 8 
0 .6 
0.4 
0 .2 
0.0 
- 0.2 
- 0.4 
-0.6 
- 0.8 
- 1.5 
---- •-, 
_t:3-___ __ - -
. - -- ____ _ ,-£1._ 
' --\ ' \ .... 
' \ 
' \ 
' \ \ \ 
\ \ 
\ \ 
\ \ 
\ \ 
\ \ 
\ \ 
- - - - ',~ - - - - - - - - - - - -
-1.0 
- 0.5 
,, I 
'~11.i, 
,, 
I \\ 
\ ' 
\ \ 
\ ' 
\ \ 
\ \ 
\ \ 
\ \ 
\ \ 
I \ 
I \ 
I \ 
\ \ 
._ __ \ 
' ' 19.-:. :---------0 
---. 
0.0 0.5 1.0 
time delay (ms) 
1.5 
5.4. Correlation vw~ with t ime delay using central - difference (C- D) 
(open symbols) and backward-difference (B-D) (closed symbols) 
schemes. The location is r, = 0.12. The 8-D curve has been 
corrected by s.b.i_fting to the left by half a time step or 0.25 ms. 
The gradient dOJ dy is included to compare with Fig. 5 .3. 
143 
10 
8 
• 
' ,, ,, 
6 I 0 I 
' I I
I I 
• I 
' 
4 
,, I 
,, I 
,, I 
I I I 
I 
I 
' 2 ,. 
N 0 3 
-2 
-4 
- 6 
-8 
-10 
0 2 3 4 5 6 7 8 9 10 11 
12 13 14 
time (ms) 
Fig . 5.5. Compari son of th e w time series from the central-difference 
(open symbols) and backward-difference (closed symbols) 
sc hemes. Th e locati on is 11= 0.12 . 
144 
15 
10 3 
10 2 
10 1 
10 o 
10 ° 10 
1 10 
2 
f (Hz) 
- . ... 
.. 
,, 
'" 
,, 
' 
' 
Fig . 5 ( mpansan of the w spectra from the central-difference .6. Co · T~ol,d Ii ne) and back~ard-diff erence ( dashed line) schemes. 
e location is Tl=0.12 . 
145 
\ 
'I, 
~. 
I 
' 1 
., 
10 3 
~ 
" 
1 .0 
0.9 
0.8 
0 .7 
0 
;: 0.6 
ci 0.5 
~ 0.4 
0 
0 
~ 
CJ) 
0 
0.3 
0.2 
0. 1 
0.0 
0 
-10 
-20 
-30 
-40 
v -50 
(f) 
0 
..c 
o. -60 
- 70 
- 80 
-9 0 
--
--
--
-
-
-
-
-
-
Kco/K 
(a) 
(b) 
0 100 200 300 400 500 600 700 800 900 1000 
f (Hz) 
Fig. 5. 7. Signal atten uation and phase shift of the two differencing 
sc hemes: (a) Kco/K or IKBD/KI, (b) phase shift due to the 
B-D sc heme. So lid lin e is for the C-D scheme, dashed line 
is fo r th e 8 - D sc heme. 
146 
II 
10 -2 
3)( 
-e- 10 -3 
10 -4 
10 -2 
3)( 
-e- 1 O -3 
10 -4 
10 -2 
3)( 
-e- 10 -3 
10 -4 
10 1 10 2 
10 1 10 2 
f(Hz) 
f(Hz) 
Fig, • 
0
,i;,e~lro of. the slreomwise vorticity component. All spectra 5.8 s in th" ormol1zed by the local variance and integrate to unity 
is and subsequent figures. 
147 
,' •';:' 
. , 
~' ...  
10 3 
10 - 2 
T)=-1.65 
~>, 10 - 3 
10 - 4 
1 0 - 2 r_ -.--,-,-rn"TTr"--r----r..,.,.,-,,---..-.-.rrrn 
10 - 4 
10 - 2 
10 - 4 
10 ° 10 1 10 2 
f(Hz) 
ri=0.71 
10 1 10 2 
f(Hz) 
Fig. 5.9. Spectra of the transverse vorticity component. 
148 
10 3 
10 - 2 
10 - 4 
10 - 2 
10 -4 
10 - 2 
10 - 4 
'fl= -1 .06 
10 ° 10 10 
2 
f(Hz) 
n==0.12 
101 102 
f(Hz) 
Fi g, 5· 1 O. Spectra of the sponwise vorticity component. 
149 
:··:·'' 
' .·' 
: :·~. '. ~ .. 
\'t ;• ,' 
10 3 
0.4 
0 .3 
0.2 
0 .1 
0.0 
-0.1 
0.4 
0.3 
i>' 0.2 
'--' Q 
0.1 
0.0 
-0.1 
0.2 
0 . 1 
o.o 
-0.1 
(a) 
--------
(b) 
__________ .... 
-- ------- -
-20 -10 
. 
I • 
I \ 
I \ 
I \ 
,' \ 
I \ 
I • 
I \ 
,' I \, 
' 
, ' 
I 
,' 
,, 
I • 
I • 
I 
I 
I • 
I \ 
\ 
\ 
I• 
I \ 
I \ 
I • 
I \ 
I \ 
I • 
I \ 
I • 
I ' 
I 
0 
' 
' 
-----------------------
___ .. ___ ____ __ _ 
--------- --- ---· 
10 
Fig. w k of the vorticity components in the freestreom ond the 5. 11. PDF iso / centerline: (o) w , (b) w , and (c) w . The solid line 
v i°'· the freestreom: 'the dos'hed line is for ~~0.12. The 
r°r ,c,ty components in this ond subsequent PDF ond JPDF 
igures are normalized by u/L0 , 
150 
' ,• .. ,~ 
'. • ••• ' 1 
... ,, .. 
. : 
'. ~' 
", !• 
20 
0.4 
0.3 
.......... 0.2 
3"" 
'-..J 
Q_ 
0.1 
0.0 
-0.1 
.......... 0.2 
3"" 
'-..J 
Q_ 
0.1 
\ 
I, 
o.o 
-0.1 
.......... 0.2 
3"" 
'-..:., 
Q_ 
0.1 I 
I, 
0.0 
-0.1 
- 20 -10 0 10 
20 -20 -10 
0 10 
20 
Cl.\ 
wx 
Fig. 5 
· 1
2
· PDF of the strearnw·1se rt· ·t G · d' t 'but·1ons are 
vo 1c1 y. auss1on 1s n 
shown as dashed lines. 
151 
0.4 
0.3 
......._ 0.2 3>, 
'-' Q. 
0.1 
o.o 
-0.1 
0.4 
0.3 
......._ 0.2 3>, 
'-' Q. 
0. 1 
0.0 
-o. 1 
11=-0.47 
0.2 
/2 
0. 1 
o.o 
-0.1 
- 20 -10 
-10 
0 10 
20 -20 
· of the tronsverse vorticity. Gaussian distributions ore 
Fig. 5 .13 PDF 
shown as dashed lines. 
152 
'"' 0 .2 3"' 
'--" Q_ 
0.1 
o.o 
-0.1 
? 0.2 
'--" Q_ 
0.1 
o.o 
-0.1 
? 0.2 
'--" Q_ 
0.1 
o.o 
-0.1 
I I I 
I 
ri==0.71 
-20 
-10 0 10 
20 -20 -10 
sh of the spanwise vorticity. Gaussian distributions are Fig. 5.14. PDF 
own as dashed lines. 
153 
11= -0.47 
10 TJ= -1 .65 10 
11= -1.06 10 
5 
5 5 
8~ . o ................ ~ 
~ -·····----------
.5 
0 C,) 0 
. 5 
.5 
-10 
--,::,>-. Q ·10 -1 0 
-5 0 5 10 
-10 .5 0 5 10 
~ 8)( ·10 .5 0 5 10 -10 11" 1.29 
'--Q. 10 11-0.12 10 
TJa 0 ,71 10 
5 
5 
8~ 
-,----
5 
0 
.............. (j)---------- -- --
0 
0 
.5 
. . 
.5 
.5 
· 10 
-10 
-5 
-10 
-10 
0 5 10 -10 .5 0 
5 10 -10 
.5 0 5 10 
------- ------- -- --------- ----- --------------
11"' -1 ,06 
5 5 
8~ 0 0 
-5 
--
-5 .5 
8>-. 
8)( • 10 
'--
-10 
-10 
0 5 10 -10 -5 
0 5 10 
-10 .5 0 
5 10 
11" 1.29 
Q. 
8-,.., 10 
8)( 5 
-5 
·5 .5 
-5 
-10 
-10 
0 5 10 -10 .5 
0 5 10 
-10 -5 0 
5 10 
co, 
ro, 
ro, 
Fig . 5.15 J . . · · oint probability density function (JPDF) (top) and covariance 
~~erand (bottom) contours fo r the streamwise and transverse 
rt icity components. 
154 
'' 
. t,. 
2.0 
1.5 
1.0 
0.5 
I I ... .. ... , ...... ,, 
00,,-- ' ...... '" L,c.:i ,' ',, ,,,'fij 
,,r I ... .. 
,, I "'.., ' 
, ,, I ,' • 'f ',, 
rm' ' ', ... , 
,.. ....... ·= I / ••• '-
,, ,,,g,,,•'"''"' I / •• •• -l!l / ','f 
' I ,' 
- - - --,- - - - - - - - - - -
I , 
,~ 
- - -
-0.5 
-1.0 
-1.5 
-2.0 Q.5 1.0 1.5 
-2.0 -1.5 -1.0 -0.5 a.a 
Tl 
Fig. 5.16 0 . · uodrant analysis for the correlation of the streomw,se 
~
nd 
transverse vorticity components: 0, 01; 0, OZ; 
in • 03; <>, 04; ond 'f, total (sum of quadrant values 
th1 s and subsequent figures). 
155 
.. . , . ~-· . ' . 
\ \ 
z 
\ 
' ' :;· \ 
' • I 
' - , 
Fig . 5.17 S h · Mc_ emotic sketches of the vortex loops calculated by 
eiburg & Lasheras (391 for the laminar wake of a flat 
plate with a corrugated trailing edge. Shaded region_s 
represent the spanwise shed K6rrncfo vortices. 
156 
~':: ~ .. - --·--. 
11"' -1.65 
: 
I 
···· ·······--@: .... ............ . 
-5 ! 
a" o 
-5 
' 
-10 ! 
-10 ~c...................i~.....,_,__~_J 
-5 0 5 10 -10 -5 O 
5 10 
-10 ......... ~__,_,_._._...._,__._._~~ 
-10 .5 0 , 5 10 
11= 0 .12 10 
1'1"' 0.71 
5 
5 
a" 0 0 
-5 
-5 
-10 
-10 
-5 0 
-10 
5 10 -10 -5 0 
_:- , 
-1 0 W-'--......,__JW----'_...J....,.,_~..,__._.~ 
5 10 -10 .5 0 
5 10 
---- ----------- ---- ---- -- ------------------
5 
0 
a" 0 
.5 
-
·5 
~f 
8)( · 10 
....__ 
• 10 
0.. 
-5 0 
SN 
10 8><. 
-5 
. 10 
-10 
5 10 - 10 -5 
0 5 10 
-10 .5 0 
5 10 
n= o.71 
n= 1.2 9 
10 
-5 
-5 
-10 
-1 0 
-5 0 5 
5 10 
.10 .5 0 
5 10 
10 -10 
.5 0 
ro. 
(J), 
(J) 
K 
· F (top) o nd covorio nee integrond (bottom con ours for F"ig. 5.18 JPD ) t 
th
e streamwise and spanwise vorticity components. 
157 
-
2.0 
1.5 
1.0 
0.5 
-0.5 
-1.0 
-1.5 
-2.0 
-2.0 - 1.5 -1.0 -0.5 0.0 
T) 
0.5 1.0 1.5 
Fig. 5 .19. Quadrant analysis for the correlation of the streamwise 
and spanwise vorticity components: D, Q1; 0, Q2; 
6. Q3· o Q4· and ~, total. 
' , ' , 
158 
! I ,• 
:·, 
2.0 
~ - ·~-· ·--
llg -1 .65 
s" 0 
········· @ ..... ; ... ............ 
-5 
-10 I 
-10 
-5 0 5 
10 11 2 0.12 
5 
···· ·~ 
s" 0 
··  i .···· ·· 
-5 
-10 
-10 
a" o 
·5 
·10 
·10 
-5 
-5 
·5 
0 5 
0 5 
0 5 
1 0 
10 
10 
10 
11=-0.47 
11~ -1 ,06 10 
• 
5 
0 
0 
-5 
-5 ! 
l 
-10 
-10 
-10 -5 0 5 
10 -10 
-5 0 ' 5 10 
11= 1.29 
11= 0.71 
10 
i 5 
5 
0 : ·············~ ·············· 
-5 
-10 
-10 -5 0 
-1 0 r_......,_,__._J~~...,__._.~ ........ ...., 
5 10 -10 -5 0 
5 10 
-- --- -- -------- --- ----- ---- ----- ---
11" -0.47 
-5 
-5 
-10 
-10 
-10 -5 0 
5 10 
-10 -5 0 
5 10 
11" 1.29 
11= 0 .71 10 
5 
-5 
-5 
-10 
.10 10 
-10 -5 0 
5 
-10 -5 0 
5 10 (f)y 
(f)y 
Fig, 5 
.20. JPDF ( . d (b ttom) contours 
top) and covariance integron ° 
th
e transverse and spanwise vorticity components. 
for 
159 
2.0 
1.5 
I 
1.0 
0 .5 
1:: 0.0 
-0.5 
___ .G------~-~ 
__ .o--··:_.---6-------r :::..._·------~--
.... · \:r::: ·.·_· ... ,t,, • • • • I ----------------fr 
• • . . • I • --.:·:::::.ti 
--- _.,._ -- - - - ~ - - - - - - - - • - - - - - - l . - - - - - --' ,4---~-31'-~--~-=-'f·-----
<::3-.. 1 -·B 
.. ::_ ···o -s··:>·· . 
• . • '0·.· .. _·_-_ --~ · o--..... Le, _____ ,:: :';J ;:'.·. <- .. 
·o- -- ----r -er 
-1.0 
-1.5 
-2.0 1.0 1.5 2.0 
-2.0 
- 1,5 -1.0 -0.5 o.o ,, 
0.5 
Fig, 5
.21. Q uadra t . . and n ?nalysrs for the correlation of the transverse 
6. 
0
spanw1se vorticity components: 0 , Q1; 0, Q2; 
' 3 ; O, 04; and 'f, total. 
160 
z 
y 
~ 
... C) 
shear 
reorientation 
z 
y 
+dU!d.x () nx 
stretching 
X 
-dU!d.x 
Ox ) ... 
compression 
) 
i 
I 
I 
I 
I 
'----/ 
I 
/ dU!dy 
I 
• I 
~ I 
1 ,/ 
: / 
: / 
: / 
: / 
X 
... 
() Ox ) 
·<~z7··········· ··7 ....................  
dW!d.:x ',, 
' · 
' 
Fig. 5.22. Schematic sketches of vortex stretching (top) and reorientation 
due to shearing (bottom) by velocity gradients. 
161 
;·-
3 T\• -1.65 
3 Tja ·1.06 
2 
··············~ ·············· 
2 
ct 
>< 
"O 0 
0 :l "O 
-1 
-1 
...--... -2 
-2 X 
u 
-3 
-3 
---- -10 -5 0 5 ::J 10 
-10 
-5 0 5 10 u X 
T\= 0 .12 8 3 
3 T\= 0.71 
..__, 
o._ 
2 
2 
3 
2 
0 
- 1 
-2 
-3 
-10 
-5 0 ,5 10 
2 
X 
"O 0 
0 :l 
"O 
-1 
- 1 
- 2 
-2 
-3 
-3 
-10 
-5 0 5 10 
-10 
-5 0 5 10 
-~ • 
-2 ! 
' -3 '-'-'-~ .,_.____._~~~~~ 
-10 ·5 0 5 10 
---------- - ----- --- - ----- - -- -- - - - - --- - - - - - -- - - - ----- -- - - -- - - ---. 
3 T\= -1. 65 
3 11= -1.06 
3 11-= -0 .47 2 
2 
2 
X 
"O 0 
0 
0 
:l 
----
"O 
X 
-1 
-1 
- 1 
"O 
---
-2 
-2 
-2 
:J 
"O 
-3 
-3 
-3 
x 
8 -10 -5 0 5 10 
-10 
-5 0 5 10 
-10 
-5 0 5 10 
..__, 
o._ 
>< 
3 
3 T\= 0.71 3 11= 1.29 
"O 
T~ ---:J 2 2 2 "O X 8 
X 
-0 
0 
0 
0 
:l 
-0 
-1 
-1 
-1 
-2 
-2 
-2 
-3 
-3 
-3 -10 -5 0 5 10 
-1 0 
-5 0 5 10 
-10 -5 0 5 10 (()x (() ((), X 
Fig. 5.23 . JPOF (top) and covariance integrand (bottom) contours for CJ\ and du/ dx . 
162 
X 
3 
1.0 
0.5 
0.0 
- 0.5 
-1.0 
I 
I 
-E3- - - - --- --G--- - ----G--
- - - - - - - I -----
_.O- A----------b.---------A --G-. 
--- ----= I a----- -.. 
-· A.---- ----.A •. D -- --a o.· • • • • • f;1 :::::li-------- I ··-··n 
- - - - - - - - - - - - I_ -_::_:-:.-.,--=----=--.=-7-,a:.---
- - - - . ..,_ - - - - - - - - .. - - - - - - - - - ..,. - - - - - - - t _ ..,_ - - ••• -~ 
---- <>- I 0-··· .. -8 
----:o::·-- ··-o. .__ __ I ------(}-------~~;-----· 
-- --0--- ------0--- -· 
·0- . .. __ I 0--------0· 
---0--- ----,--
-2.0 -1.5 -1.0 -0.5 0.0 
,, 
0.5 1.0 1.5 2.0 
Fig. 5.24. Quadrant analysis for the correlation of w and du/dx. 
D, Q1; 0, Q2; 6, Q.3; <>, 04; .., , total. x 
163 
10 T)=-0.47 
5 
5 
~ i: T. 
""O 
-
::, 
""O 
>, 
8 
-10 
-10 -5 
10 
0 5 
T)=0.12 
10 
0 ............... . 
-5 
-1 0 ~ ............ ........._....._,._..........._........._.-.L..L........_.._._, 
-10 -5 0 5 10 
5 
0 
-5 
-10 
-10 
-5 0 ' 5 10 
0.... 
5 
5 l 
>. 
:!2 0 ~ 
-0 
-5 
-10 
-10 
-5 0 5 10 
.: • 
- 1 0 ~~~ .......... _.__,_._......__.-'--L.........._L.....J 
-10 -5 0 5 10 
: ~} 
l 
-10 u..._...._,_..........._.........._.........._........_ ........... ~ 
-10 -5 0 5 10 
-- --- - -- ----- - ------- -- ------ --- - ------- - ----------- ------ -
5 
~{:--~----
- 0 ::, 
-0 
>, 
-1 0 .....__....,_,_...,__._,..,_.____~~~~~ 
5 
0 
-5 
8 -10 -5 0 5 10 
-10 
-5 0 5 
-1 0 w..,........__...,__,_..........,......,_........_L.W..L.W..~ 
10 
-10 
-5 0 5 10 
5 
0 
-5 
- 1 0 l.J...L.w..,...L...w..,__,_L.W............_......__.~~ -1 0 ~ ............ ---W...LL.L-'-u...,_._,_._,_LL.L'-'..J -1 0 '-'-'---'----'-'..L.c..,--'-'-'......,_........_L.W..L.W..~ 
-10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 
Fig. 5.25. JPDF (top) and covariance integrand (bottom) contours for 
wy and du/ dy. 
164 
i • 
t'- ·~- . .•.• 
1.0 
0.5 
o.o 
-0.5 
-1.0 
1 
I 
I 
, , ,fr - -- -. - -i- A- •• 
/1', , , G' , , -, --r E3- --·~,._:: :{p .• 
,,,' ,,,,' u •• ·-. 
,,, r:,.---- I •• ·-. 
, ~ . . / _,,,- I ··-.:ts-. 
, ,l5. , , •A • , , c;. • / _,B' · •. :-, .. I ',, '8 
I ···t,. 
-----T-·-- I - - - -:..:·:.:"!:-·.:...·- ---~-·---------•-------- --Y.-- --- ----- .:-:~- ,- - - - - - - - - - - - -
' ' 
' ' 
',, '0. 
'~ ',, 
' ' . 
' 
' 
' 
' . 
' 
' . 
0. 
·o. 
I 
I ,~ 
, ,,',O 
,, ,' 
, ,' ,,' 
t:!>' , , , 
, , 
'O' 
I 
, , ,, , 
, , 
, , 
, , 
•• I ,,,, 
··o---- ,,£:!~,-
. -· 1-0---,-,-,-,~ ::e.i 
• I , ' ''0- ________ <}' 
I 
2.0 
-2.0 - 1.s -1.0 - 0.5 o.o 
'Tl 
Fig. 5,26 Q · 
0 
uadran\ analysis far the correlation of ", and du/ dy. 
' 01; 0, Q2; 6, Q3; <>, Q4; Y, total. 
165 
,., 
. .. 
,•' 
\.' 
T)=-1.65 10 T)=-1 .06 10 Tl=-0.47 10 
i : i 
5 i 5 
·· ········ • ···· ···  
5 
--·- ~}-N -0 0 0 0 ~ 
-0 
-5 I 
-5 ! 
-5 
---N : -10 
"l:J -10 -10 
-
-10 -5 0 5 10 -10 -5 0 5 10 -10 
-5 0 
'5 
::::, 
-0 
N Tl"0.12 11~ 0.71 T]= 1.29 8 10 10 
.___... 10 
{~ 
CL 
5 5 
5 
N 0 
-0 0 0 ~ 
-0 
\ 
-5 -5 -5 I 
\ 
-10 -10 -~ 0 
-10 -5 0 5 10 . 1'() -5 0 5 ~o -10 -5 0 s 
-- ---- --- ----=---~---- ------ ·------· -- --' --
10 T)= -1.65 10 '11= -1 .06 10 
5 5 5 
N 
-0 
~ 0 0 0 
-0 
-N 
-5 "O -5 -5 
-
::::, 
"O 
-10 -10 -10 
-;:,. 
;.....L, 
8 -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 
..__.. 
Q_ 
N 
"O 10 
T)= 0.12 
10 
T)= 0.71 
10 
-
::::, 
"O 5 5 5 N 
8 N 
-0 
~ 
-0 
-5 -5 -5 
-10 -10 -10 
-10 -5 0 5 10 -10 -5 0 5 10 · 10 -5 0 5 
co, co, co, 
Fig. 5.27. JPDF (top) and covariance integrand (bottom) contours for 
u.1z and du/ dz. 
166 
10 
Hl 
10 
10 
~· . 
N 
3 
1.0 
0.5 
0 .0 
-0.5 
- 1.0 
I 
,,B-------1..0 
,,,' I ··- .• 
,,, ,/J,-------i·A------·.,.··A 
0' ,,"' t!t: .. .. 
,,' ,,,,,"' I ............. ... 
, , ... ... 
, Iv, .. ... ... 
,,' ,,Ll ....... 'b 
,,' ,,,' '0.' ......... 
,0 ,' ',, ........ 
,',, .e;,, . ·-. ·-.fj_ 
, ,,' I .. B 
I 
---------. -.-----.,_ - - - - - - - - -
--=-,it-------~~ - - - - - - ,:..ir--------1l---------Y---------
I 
,0 
'•, '0 ,,,'_,0 
--~::·- '• ,,0-:,----
............. '0 ,,,',,,,0 
.... .... , ,' 
, I , , , 
', ........ ,' ,,' 
O • • , 0- - - I -8-, 
···-... -----L-G:_-::-----"'· 
-~------- -<} 
I 
-2.0 -1.5 -1 .0 -0.5 0.0 
ii 
0.5 1.0 1.5 2.0 
Fig. 5 .28. Quadrant analysis for the correlation of wz and du/dz. 
D, Q1; 0, 02; 6, 03; 0, 04; T, total. 
167 
,i: -- · 
3 11- -1.65 3 
11= - 1.06 3 
2 2 
--· 
ct) 
2 
>< 
"O 
...._ 0 > 0 
"O 
0 
-1 -1 
- 1 
..--... -2 
>< i 
-2 i -2 
l:J 
-3 
-
:::,. 
-10 
-3 
-3 
l:J -5 0 5 
10 -10 -5 0 5 
10 -10 -5 0 
, 5 10 
)( 
8 
.......... 3 
11= 0.71 
11= 1.29 
Q 
3 
3 
2 2 
2 
<f> >< 1 "O ...._ 0 > 0 0 "O 
- 1 -1 
-1 
-2 -2 
-2 i 
-3 -3 
-3 
-10 -5 0 5 10 - 10 
.5 0 5 10 -1 0 
-5 0 5 10 
-- - - - - - ----- - -------- -- ---- - - - -- -- - - - - - - ----- --- - - ----
3 
11= -1.65 3 
2 
>< 
"O 
...._ 0 > 
..-.... "O 
>< -1 
- 1 
l:J 
-1 
-2 
'> -2 -2 l:J 
)( -3 -3 
-3 
8 
....__ -10 -5 0 5 10 -10 
-5 0 5 10 
-10 .5 0 5 10 
Q 
>< 3 
11= 1.29 
l:J 
-
:::,. 
l:J 2 2 
2 
«i~ >< 8 
>< 
"O 
...._ 0 > 0 
0 
·--~ · "O 
-1 - 1 
-1 
-2 
-2 -2 
-3 -3 
-3 
-10 -5 0 5 10 -10 
-5 0 5 10 
-10 -5 0 5 10 
w, 
w, 
(I), 
Fig. 5.29. JPDF (top) and covariance integrand (bottom) contours for 
wx and dv/dx. 
168 
. \':" ,.- · 
)( 
3 
1.0 
0.5 I 
- -~ - :.-e.- .. =: t t- -- ------A,. B.--_-_:·:.-- -- I ---------s:·-,,, 
--- ~--- I ·-,, -- •• 
-. ,:::--- '• --~-. ---
:::: :~--- ·----::::'8 
0.0 __ -:-:_ _; _~_ -~ _ ~.; _ -:. _-:. _-:-~- --=-= ~ -· ..:.- -__._ ~- ~-::: _-:-:_ -~- ;_ -~- ~-~ 'Y 
-. ··-o I .-----:2 
···o:·-., .e-· --
. -.. _- - - . ~ I - - -- - - ~ 0· - -- - -
···-o·:_-_- ... __ I _____ .--G~~------
.. ---:s-:: = =:·- ... , ~ --- -----~ 
-0.5 
-1.0 
-2.0 -1.5 -1.0 -0.5 0.0 
,, 
0.5 1.0 1.5 2.0 
Fig. 5.30. Quadrant analysis for the correlation of wx and dv/dx. 
D, 01; 0, 02; t::., 03; <>, 04; 'Y, total. 
169 
• 
. r:• 
!:···············~··············· 
~ l 
'D 
·10 
-> 
'D 
";... 
8 
----Q_ 
>-. 
"O 
--> 
"O 
·10 ·5 0 5 
10 Tl"' 0.12 
5 
0 
·5 
·10 
·10 .5 0 5 
10 
.5 
· 10 L..,.~.L...,.~~~w...._ ........... ..., 
·10 ·5 
5 
0 
· 5 
0 5 10 
5 
0 
. 5 
·10w..... ............ ......._..........._......._....._._....._,_............, 
·10 
·5 0 '5 10 
10 ~~,.,...,~~1ln"'~1~.2~9~~ 
:+ 
l · 10 LI..L~..C.....,. ........... ......._.~ ....................... ~ • 10 .................... ,......._..........._....._,_..........._,......._~ 
10 · 10 . 5 0 5 10 ·10 -5 0 5 10 
------ ---- --- - ------ - - -- - - - --- - - ----
-- - --- - - - ------- - ---- - -- - --
1l'=·1.65 
10 1l'=·1.06 10 T]'=·0.47 
10 
5 5 5 >-. 
"O 0 
0 0 
> 
"O 
.--.... 
~ 
.5 
-5 
·5 
'D 
-> 
'D 
· 10 
·10 
·10 
";... 
·10 
·5 0 5 10 
· 10 .5 0 5 10 
· 10 
-5 0 5 
8 
..___.. 
Q_ 
Tl"' 0 .12 
11"' 1.29 
~ 10 
10 10 
'D 
,,,,,,..,.....,,,.,_ 
-> 
'D 5 5 5 
>-
8 
>-. 
"O 0 
0 0 
> 
"O 
-5 
-5 
·5 
-10 
·10 
-10 -10 ·5 0 5 10 
·10 .5 0 5 10 
· 10 
-5 0 5 WY 
WY 
(>)y 
Fig. 5.31. JPOF (top) and covariance integrand (bottom) contours for 
wy and dv / dy. 
170 
10 
10 
1.0 
0.5 
0.0 
>, 
3 
-0.5 
-1.0 
-2 .0 -1 .5 - 1.0 - 0.5 0.0 
T) 
0.5 1.0 1.5 
Fig. 5 .32. Quadrant analysis for the correlation of wy and dv/dy. 
D , Q 1; 0 , 02; 6. , Q3; <> , 04; ,.. , total. 
171 
2 .0 
10 TJ=- 1.0 6 10 TJ= · 0.47 
5 
~ 0 ·················@>················· 
l 
-5 
..--... 
N 
-10 "O 
- -10 -5 > 
"O 
N 
0 5 10 
5 5 
0 
·· ··· ··· ··, ....... 
0 
-5 
-5 
-10 
-10 
- 10 
-5 0 5 10 
-10 
-5 0 , 5 10 
8 10 
'--' 
TJ= 0.12 Tl= 0.71 
0... 
5 
N 
"O 0 > 
"O 
-5 
j 
: ........ ....... ~················ 
-5 i 
-10 
-1 0 
-5 0 5 
-10 ~~~~~~~~ 
10 -10 -5 0 5 
! 
-10 ............................. ~~~~........., 
10 -10 -5 0 5 10 
---- --- - - - - ------ -- ------------------- - ------------ -
10 T1= · 1.65 10 T1= · 1.06 10 ~.47 
5 5 5 
N 
"O 
0 0 > "O ..--... 
N 
-5 
-5 
-5 
"O 
-> 
"O 
-1 0 
-10 
-10 ~ 
;., 
8 - 10 -5 0 5 10 
-10 
-5 0 5 10 
-10 -5 0 5 10 
'--' 
0... 
T)= 0.12 T)= 0.71 T)=1 .29 
N 
10 10 10 
"O 
-> 
"O 5 5 5 
N 
8 
N 
TI 
0 0 0 
.._ 
> 
TI 
-5 
-5 
-5 
-10 
-10 
-10 
-10 -5 0 5 10 
-10 
-5 0 5 10 
- 10 -5 0 5 10 w, 
O)z 
oo, 
Fig. 5.33. JPOF (top) and covariance integrand (bottom) contours for 
w2 and dv/ dz. 
172 
1.0 
0.5 
0.0 
N 
3 
-0.5 
-1.0 
-2.0 - 1.5 -1.0 - 0.5 0.0 
Tl 
0.5 1.0 1.5 
Fig. 5.34. Quadrant analysis for the correlation of wz and dv/dz. 
D, Q 1; 0, 02; 6., Q3; <>, Q4; ~, total. 
173 
2.0 
3 3 
'f1= • 1.06 3 
2 i 2 
--(j) 2 X 
------ --,----------
"O 
~ 0 0 
0 
"O 
·1 ·1 
·1 
--
·2 ·2 
·2 
X 
"O 
· 3 
i 
~ 
.3 .3 
·10 . 5 0 5 10 · 10 .5 0 5 10 
·10 -5 0 ,5 10 
"O 
x 
8 3 'f1= 0.12 
'f1= 0.71 Tj= 1.29 
.___,. 
3 3 
0.. 2 2 
2 
• 
X 
"O 
~ 0 0 
0 ·•········· · . . ... .. ...... 
"O 
· 1 ·1 
-1 
-2 -2 
-2 ' ! 
-3 .3 
-3 
·10 .5 0 5 10 -10 .5 0 
5 10 -10 -5 0 5 10 
----- - -------- - --------- - - -- -- - --------------------------
3 3 
3 
2 2 
2 
X 
"O 
--~ 
0 0 
0 
X 
· 1 -1 
-1 
"O 
-3: -2 -2 
-2 
"O 
x .3 
.3 -3 
8 ·10 .5 0 5 10 -10 -5 0 
5 10 ·10 -5 0 5 
10 
.___,. 
0.. 
X 
"O 3 3 
3 
-3: 2 2 
2 
"O 
>< 
8 X 
"O 
i 0 0 
0 
·1 ·1 
-1 
-2 
· 2 ·2 
·3 .3 
.3 
·10 . 5 0 5 10 -10 -5 
0 5 10 -10 -5 
0 5 10 
ro, 
ro, 
w, 
Fig. 5.35. JPDF (top) and covariance integrand (bottom) contours for 
wx and dw/dx. 
174 
1.0 
0.5 I !'---~-,,_.,_.,_.$, 
,~~-:.-:.-------• ,,,, I 
J', \ ........ ,,~,, , , ... ~ Ihf/' \ I ............... ... 
_.,_.,_~~ \ I ........ ,&, ... , ... 
•:-' , ........ ,en \ ~... ,, \, I ............... fi 
- - - - - - - - - - 'l, - - - - - - - - - - -
I' 
• ···~ I ~',, ,---~ 
•••• ••• -~ .© -- \ ,, .. 
-----e - - ',,,,_ ,,-!';::, ,,• 
•• ' , ,J?J 
""J , , , 1·0 ', ,, ,,' I • - •••••• • g-:,,, 
-0.5 
-1.0 1.0 1.5 2.0 
-2.0 -1.5 -1.0 -0.5 0.0 ,, 
0.5 
Fig. 5.36. / Quadrant analysis for the correlation of wx and dw dx. 
D, 01; 0, Q2; 6, Q3; 0, Q4; T, total. 
175 
: 
t: ~--
::,... 
u 
...... 
i 
-5 ! i 
: 
-1 0 ~ ....... L.,._~Lo.: ~..J..,......_......l 
- 10 -5 0 5 10 
5 
0 
-5 
-10 
-10 -5 0 5 10 
10 
5 I i 
0 
- ~-
-5 
: 
-10 
-10 -5 0 5 10 
11= 0.71 
10 
5 
0 
~) 
-5 
: 
-10 LI-L-~__,_._........_......,._._._J..~..<.J 
- 10 -5 0 5 
10 
10 
11=-0.47 
5 
0 
-5 
-10 
-10 -5 0 , 5 10 
11= 1.29 
10 
5 
0 .............. ~··············· 
: 
-5 ! 
! 
-10 w.,......,..;.JW....._....J_~......L~~ 
-10 .5 0 5 10 
---- ---------- --- -- -------------------- --------------- --
11= -0.47 
10 
11=-1.65 
10 
5 
5 5 
-s 
...... 0 ~ 
0 
.--...u -5 
-s 
-5 
-
-5 
~ 
"'O 
>, -10 
- 10 
-10 
8 -10 -5 0 5 10 
-10 -5 0 
5 10 -10 
-5 0 5 
'--" 
Q_ 11= 0.71 
11= 1.29 
10 
>.. 
"'O 10 
~ 
"'O 5 
5 
>, 
8 ::,... 
u 0 
0 
...... 0 i -5 
-5 
-5 
-10 
-10 
-10 
- 10 -5 0 5 10 
-10 -5 0 5 
10 - 10 -5 
0 5 
(J)y 
(J)y 
(J)y 
Fig· 5.3 7. J PDF (top) and covorionce integrand (bottom) contours for 
wy and dw/ dy. 
176 
10 
10 
1.0 
0.5 
0.0 
-0.5 
-1.0 
-2.0 -1.5 -1.0 -0.5 0.0 
Tl 
0.5 1.0 1.5 
Fig. 5.38. Quadrant analysis for the correlation of w and dw/dy. 
D, Q1; 0, Q2; 6., Q3; <>, Q4; ~. total. Y 
177 
2.0 
10 
llc-1.65 10 11=-1.06 10 i 
5 5 
t 
5 
N 
·················@················· -0 0 0 ~ 
-5 -5 -5 
---.. 
N 
-0 
-10 -10 -10 ~ -10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 ,5 10 
-0 
-,::. 
11= 0 .1 2 1.29 8 10 11= 0.71 11= ,,____. 10 10 
Q_ ! 
5 5 5 
......... , ............... N -0 0 0 ~ 
-0 
-5 -5 -5 
- 10 - 10 
l 
-10 
-10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10 
---- - - -- ---- - - -- ---- ------ - - ------ - - --- - - --- --
- ------ - - - - - --
10 
11= -1 .65 10 11= -1 .06 10 
T)=-0.47 
: 
5 5 5 
N 
• 
-0 0 0 ~ 
-0 
.... .. . . ~ 
---... 
N 
I . -5 -0 -5 -5 
-s: 
-0 
- 1 0 - 10 -10 
-,::. 
-10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 8 
..._ 
Q_ T)= 0 .1 2 T)= 0.7 1 T)= 1.2 9 N 10 10 10 
--0 
-s: 5 5 5 
--0 
N 
8 N 
-0 0 0 0 ~ 
-0 
-5 -5 -5 
-rn ·10 -10 
-10 -5 0 5 10 - 1 0 -5 0 5 10 - 10 -5 0 5 
w, w, w, 
Fig. 5.39 . JPDF (top) and covariance integrand (bottom) contours for 
wz and dw/dz. 
178 
10 
10 
N 
\J 
1.0 
0.5 
~ 
-u 0.0 
-0.5 
-1.0 
,G'' 
,' 
I 
I 
__ ,,E3-------Ls 
,,,' ,1i.··---
,[3' ,f:::r · - - - -· · r ·o· - - - • -----8 
. ,.-· ,' '~ 
_..e;' ~ ~ 
I 
__ _. ___ -.-= - - ,-.,..-:- - _I_ - - - - - - - - - - -
----- - ------- -----. __ I_ __ ••• ---•---------
1 ~---------Y' .. '> 
····~@ I ,,...,.----~~--8 
:::::. ,V • 
·::::::. I _ •• -·· G' .. -· 
---~--- I ¢ •• · ,,-
• --:::: - -~ - . - - - - -~ -0- - - - - • • • • ' ••• 
··0-... I .. ·· 
·----./':\. __ , .. --G 
Io -
I 
1.5 2.0 
-2.0 -1.5 -1.0 -0.5 0.0 
,, 
0.5 1.0 
Fig . 5.40. Quadrant analysis for the correlation of wz and dw/dz. 
D, Q1; 0, Q2; 6, Q3; 0, Q4; T, total. 
179 
Fig. 
' N 3 
-e-
N 
13 
10 3 
10 2 
N 
3 
-e-
10 
10 o 
10 3 
10 2 
-e-
10 
10 o 
10 3 
10 2 
10 
10 o 
10 ° 10 10 2 10 
3 
f(Hz) 
6. 1. Spectra of w signals at 77=0. 12: (a) total fluctua ting (w
2
), 
(b) phase reference (wi;), and (c) incoherent (w;). The phase 
reference signal is use a for conditional analysis of all 
measured variables. 
180 
6 
4 
IJI 
2 
N 0 3 
-2 
-4 
• 
- 6 
0 2 3 4 5 6 7 8 9 
time (ms) 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
10 11 12 13 14 15 
Fig . 6.2. Time series for the three parts of the nondimensional spanw,se 
vorticity. o, tota l fluctuating (w
2
); t., coherent (0\); 
•, incoherent (w:). 
181 
3 
2 
0 
- 1 
- 2 
- 3 
-3 
- 2 
I I 
: reference 1 
' station - - > 
: I 
I 
I 
-1 0 2 3 4 5 
Fig. 6.3 . Schematic sketch of th e vortex shedding and the saddle regions. 
Th e dimensions are estimated from current database and from 
Hussain & Hayakawa [26] at x/ d= 30. The reference station at 
x/L0 = O indicates the detected events across the wake. 
182 
vortex center 
0.4 
11 = -1 .65 
0 .2 
Is n-t:::,-6-A-&--6. -8 -6 -6-0.0 
I 
-0.2 
downstream upstrea m 
-0.4 
0 .4 
11 =- 1.0 6 
0 .2 
6 & 6 6 
' 6 0 .0 - - - - - - -6 - - - ~ - - - - - - - - -13. - -
6 6 6 : 6 
- 0 .2 
-0.4 
0.4 
11= -0.4 7 
0.2 
0 .0 - -n-8-6 _4_~_ AA!J -15.--
6 : 6 
-0.2 
-0.4 
- 0.5 0 .0 
t/T 
I\ 
::) 
V 
I\ 
::) 
V 
I\ 
::J 
V 
0 .5 
saddle 
0.4 
11= 1.29 
0.2 
666~ I 0.0 ------ - ·15. · , - -- -- --- -15,. -~ 666 
I 
I 
I 
-0.2 
downstream upstream 
-0.4 
0.4 
11=0.71 
0 .2 
66 ' 6 
---. A ti --- ~ - -----Li -6- -0.0 
6 6 66 
-0.2 
-0.4 
0.4 
11=0.12 
0 .2 
' 
' 0.0 b b. h_ ~ -A -l 15, -6_- 6 -6 -t:s. 
-0.2 
-0.4 
- 0.5 0.0 
t/T 
Fig . 6 .4. Conditional ave ra ges of u fluctuations . 
183 
0.5 
t;.-
vortex center 
0.2 
ri=-1.65 
I 
0.1 6 b,-6 -6 -6. ~ D.-~-t:,. 
downstream upstream 
0.0 
0.3 
ri=-1.06 
0.2 
0.1 
0.3 
ri=-0.47 
0.2 
0.1 
-0.5 0.0 
t/T 
0.5 
0.2 
I\ 
N 0.1 :) 
V 
0 .0 
0.3 
I\ 
N:J 0.2 
V 
0.1 
0 .3 
0.1 
saddle 
ri= 1.29 
I 
6 6 : 
- -t::.7::, 6 EE b,-b,-6...t. 
downstream upstream 
ri=0.71 
ri=0.12 
-0.5 0.0 
t/T 
0.5 
Fig. 6.5. Conditional averages of the total u2 fluctuations. The incoherent 
contribution is almost the same as the total therefore not 
shown. The long dashed lines indicate the time averages. 
184 
0.4 
0.2 
0.0 
-0.2 
-0.4 
0.4 
0.2 
0.0 
-0.2 
-0.4 
0.4 
0.2 
0.0 
-0.2 
-0.4 
vortex center 
ri=-1.65 
6 ' b b:, j)_ - - .D,_t;:,. ti - '/\ - - - - - - 7'' 
' D 6 6 Ll 
' 
downstream upstream 
6666 
6 & 
----- --- --- -,- -- ---- -----
' 
' 
------------4- -----------
-0.5 0 .0 
t/T 
0.5 
0.4 
0.2 
I\ 
> 0.0 
V 
-0.2 
-0.4 
0.4 
0.2 
I\ 
> 0.0 
V 
I\ 
> 
V 
-0.2 
-0.4 
0.4 
0.2 
0.0 
-0.2 
-0.4 
saddle 
' 6666: 
6. _ - - - - - -- --f- zs- --------
: 6 666 
downstream 
66 6 
6 6 
' 
' I 
' 
' 
upstream 
---- - ---- ---/;!.. -----------
' 
' 
------- -----&-----------
' 
' 
-0.5 0.0 
t/T 
Fig. 6.6. Conditional averages of v fluctuations. 
185 
0.5 
0.2 
vortex center sadd le 0.2 
ri=-1.65 ri= 1.29 
I\ ,ii,l N 
> ._ ___ ~ ... ---
0.1 
V 
A A A..A 0.1 : 6 iii 
•-..A- a .i"._ - - " A , I\ N 
> 
V 
downstream upstream 
downstream upstream 
0.0 0.0 
0.3 0.3 
ri=-1 .06 
ri=0.71 
I\ 6 6 N 
> 
V 6 6 
0 .2 66 " 0.2 • 6 (\ •• .._6: 666 
,•••i N > - - - A t f6- - -& 
,-6 -6 6 6 + ---- V I ••• ' I 
• 
•: 
I 
' 
0.1 0.1 
0.3 0.3 ri=0.12 
ri=-0.47 
I 66 I I\ 66 
:6 N !:,. 6 66 
I 6 > !:,. 
666 •• V 
!:,. ' 6 
0.2 6 I 0.2 A 4
6 
6- • : • 6 " (\ -·- - • +- ._ - - .! 
- - - +- - -•--i. _... N A A : A A A 
A • I > 
•• 
V 
0.1 
-0.5 0 .0 
t/T 
0.5 
0.1 
-0.5 0 .0 
t/T 
0.5 
Fig . 6. 7. Conditional averag es of the total and incoherent v2 fluctuations . 
The closed symbols are for the incoherent fluctuating field. 
The open symbols are for the tota l fluctuating field. The long 
dashed lines indicate the time averages of the incoherent field. 
186 
1' 
-
vortex center saddle 
0.4 0.4 
11=-1.65 11= 1.29 
0.2 0 .2 
I /\ 
0.0 !:s f:i.-A-/J-6-4s ~-6.-f::s -6- 3: 0.0 ts n-6-ts.-6-ts b.~ ~-8.-
V I 
-0.2 
-0.2 
downstream ups\reom downstream upstream 
-0.4 
-0.4 
0.4 0.4 
ri =- 1.06 ri=0.71 
0.2 0.2 
6._ 6. -A. f':I -6 -1 li -6 - ts. -6, - /\ -----s f::s -8.-~ 6 _6.. 6. -D.-t,; 0.0 3: 0.0 
V 6. 6. : 
- 0 .2 -0.2 
-0.4 -0.4 
0 .4 0.4 
ri = -0.47 ri=0.12 
0 .2 0.2 
I /\ I 
0.0 t:,,. D.-b.- f::s -6.-i ZS -6- 6 ~ - 3: 0.0 6- -t1 -6,- f::s -6-i$- 6-8.-~ ~ -
V 
-0.2 -0.2 
-0.4 
-0.4 
-0.5 0 .0 0.5 -0.5 0.0 0.5 
t/T t/T 
Fig. 6.8. Conditiona l averages of w fluctuations. 
187 
0.2 
0.1 
0.0 
0 .2 
0.1 
0.0 
0.2 
0.1 
0.0 
- 0.5 
vortex center 
11=-1.65 
downstream 
0.0 
t/T 
upstream 
0.5 
0.2 
I\ 
('\:: 0.1 
V 
0.0 
0.2 
I\ 
"'3: 0. 1 
V 
0.0 
0.2 
/\ 
N 
3: 0. 1 
V 
0.0 
- 0.5 
saddle 
11= 1.29 
downstream upstream 
-cc: 0.716 
' 
- - - - -:- - - --n-
6 I 
6 ' 6 /2 /'1 6 
' I 
' 
0.0 
t/T 
11=0.12 
0.5 
Fig. 6 .9. Conditional averages of the tota l w2 fluctuations. The incoherent 
contribution is almost the same as the tota l therefore not 
shown. The long dashed lines indicate the time averages. 
188 
vortex center 
0.4 
11 = - 1.65 
0.3 
0.2 
I 
0.1 
... - ... ~"'A ..... J..-..._.__...-A 
I 
I 
downstream I upstream I 
0.0 
0.4 
11=-1.06 
0.3 
I 
- ~''i 0.2 A j i 1' : - - _j-a 
I 
0.1 
0.0 
0.4 
0.3 
0.2 
0.1 
./\ 
:y 
V 
-(\ 
.:::t.. 
V 
~ 
V 
-(\ 
.:::t.. 
V 
-(\ 
.:::t.. 
V 
saddle 
0.4 
11= 1.29 
0.3 
I 
0.2 
·-·~ ... : ~ .-..-,-.... 
I 
0.1 
downstream upstream 
0.0 
0.4 
11=0.71 
0.3 it. : At. : 6 i 
- ~i- +- - -t,~-
0.2 •+•i• I 
I 
0.1 
0.0 
0.4 
0.3 
0.2 
0.1 
o.o 
-0.5 0.0 
t/T 
0.5 
0.0 
-0.5 0.0 
t/T 
0.5 
Fig . 6.10. Conditiona l averages of the total and incoherent k fluctuations. 
The closed symbols are for the incoherent fluctuating field. 
The open symbols are for the total fluctuating field. The long 
dashed lines indicate the time averages of the incoherent field. 
189 
0.10 
0.05 
0.00 
-0.05 
-0.10 
0.15 
0.10 
0.05 
0.00 
-0.05 
0.15 
0 .10 
0.05 
0.00 
-0.05 
vortex center 
Tj=-1.65 
------------ .... ------------
downstreom upstream 
Tj=-1.06 
------- -- ----1------ ------
Tj = -0.47 
A 
1 
I 
- - - - .L- - - - -j : A 
----------.-1- -_ _A_ -----
+A 
I 
I 
. 
I\ 
> 
:J 
V 
0.10 
0.05 
0.00 
-0.05 
-0.10 
0.05 
0.00 
-0.05 
-0.10 
-0.15 
0.10 
0.05 
0.00 
-0.05 
-0.10 
saddle 
Tj= 1.29 
------------~------------
.. : . 
- -'-. - -}- - __._._ 
···' downstream 
I 
' 
I 
I 
upstream 
: A 
' : 
-------- -- --~--------•--
- - . - -:- - - - -
: A 
I 
A : A 
AA 
I 
I 
I 
0.5 
-0.5 0.0 
t/T 
0.5 -0.5 0.0 
t/T 
Fig. 6.11 . Conditional averages of the total and incoherent uv fluctuations. 
The closed symbols are for the incoherent fluctuating field. The 
open symbols are for the total fluctuating field. The long dashed 
lines indicate the time averages of the incoherent field. 
190 
vortex center 
1.0 
11=-1.65 
0.5 
' 0.0 1:r ~-6-6-6-& n-t,.- tJ.-6-A 
' 
-0.5 
downstream upstreom 
-1.0 
1.0 
11=-1.06 
0.5 
0.0 
66 I ~ b,._ - - - _ .b,_~ zs-s 75, -,,s-li 
I 
-0.5 
- 1.0 
1.0 
0.5 
0.0 b 6A6.b.-~ -6-s 2:;-;j-li 
' 
-0.5 
-1 .0 
-0.5 0.0 
t/T 
0.5 
I\ 
X 
3 
V 
I\ 
X 
3 
V 
I\ 
X 
3 
V 
saddle 
1.0 
11= 1.29 
0.5 
0.0 
-0.5 
downstream upstream 
-1.0 
1.0 
11=0. 71 
0.5 
I 
0.0 1::,--b,./],..l::,, ..6.-,t fj-8.-1:s.-6-!i. 
-0.5 
-1.0 
1.0 
0.5 
I 
0.0 ts-b,..t:i- 6 .D.-,t b.-R:,.-/:s.-11-li 
-0.5 
-1.0 
-0.5 0.0 
t/T 
Fig. 6.12. Conditional averages of wx fluctuations. 
191 
0.5 
6 
5 
4 
3 
2 
8 
7 
6 
5 
4 
3 
8 
7 
6 
5 
4 
vortex center 
11=-1.65 
I 
6-A- _ ..,.,_ : A 6 6 6 
Ll 6 6 Ll "IS ts:{.j-' - -
downstream 
I 
I 
:6 
I 
upstream 
___ .l- ___ _ 
~ 
6 I 6• 
66 : 
11 = -0.47 
3 
-0.5 0.0 
t/T 
0 .5 
7 
6 
/\ 5 
N 
X 
3 
V 4 
3 
2 
8 
7 
saddle 
downstream upstream 
11=0.71 
I\ 6 
66666 
N 
X 
3 
V 
I\ 
N 
X 
3 
V 
----4'-ts: ___ 
: 6 
5 I 6 I 66 I 
4 
3 
8 
11=0.12 
7 
I 
6 
6 I 6 
~.../:-,,-8 ti 4r - - 6 -
5 
4 
3 
-0.5 
,666 
0.0 
t/T 
Fig. 6.13. Conditional averages of the tota l w 
2 
fluctuations. The 
incoherent contribution is almost th~ same as the total 
therefore not shown. The long dashed lines indicate the 
time averages . 
192 
0.5 
).,- -
vortex center 
1.0 
ri=-1.65 
0.5 
I 
0.0 ts i:s. ·8-6-6-A-6.-6-6-6-6 I 
-0.5 
downst ream upstream 
-1 .0 
1.0 
ri=-1.06 
0.5 
0.0 6- ts.-t:,.-ts, ·t,_-.6,- 6-~-6-6-
-0.5 
-1.0 
1.0 
ri=-0.47 
0.5 
I 
0.0 b 6 -t,-l:s-6-C:r 6 -A-6-1:::, -ty, I 
-0.5 
-1.0 
-0.5 0.0 
t/T 
I\ 
>. 
3 
V 
I\ 
>. 
3 
V 
I\ 
>. 
3 
V 
0.5 
saddle 
1.0 
ri= 1.29 
0.5 
I 
0.0 A -Di.-~-t:,. -D--4r n ·s ts.-6-~ 
-0.5 
downstream upstream 
-1.0 
1.0 
ri=0.71 
0.5 
0.0 6- .ti.A~ h .~ ~ -t,_· 75. ·El:,. 
I 
-0.5 
-1.0 
1.0 
ri=O. 12 
0.5 
I 
0.0 A D.-A-t':,.-6-f;; t:,. ·&· tr!::,.· ti 
- 0.5 
-1.0 
-0.5 0.0 
t/T 
Fig. 6.14. Conditional averages of wy fluctuations. 
193 
0.5 
, ... -~ 
vortex center 
5 
ri=-1.65 
4 
3 
' 
ts"-6---6 7:, A ~ 6-6~ -n 
' 2 
0 
7 
6 
5 
4 
3 
2 
8 
7 
6 
5 
4 
downstream 
ri=-1.06 
ri=-0.47 
3 
-0.5 0.0 
t/T 
upstream 
0.5 
/\ 
('\j 
>, 
3 
V 
(\ 
('\j 
>, 
3 
V 
/\ 
('\j 
>, 
3 
V 
saddle 
5 
ri= 1.29 
4 
3 
~6 6 ?i t b.-!:r,6.-.6. _l:::,. 
2 
0 
7 
6 
5 
4 
3 
2 
8 
7 
6 
5 
4 
3 
-0.5 
downstream 
0.0 
t/T 
upstream 
ri==0.71 
Fig. 6. 15. Conditional overages of the total w/ fluctuations. The 
incoherent contribution is almost the same as the tota l 
therefore not shown . The long dashed lines indicate the 
time averages. 
194 
0.5 
1.0 
0.5 
0 .0 
-0.5 
-1.0 
1.0 
0.5 
0.0 
-0.5 
-1.0 
1.0 
0.5 
0.0 
-0.5 
-1.0 
vortex center 
11=-1.65 
6 In 6 6 : 6 . 
- - - - -6- - - - - - , - - - - - -ts - - - -
66 : 66 
downstream 
11=-1.06 
I 
I 
I 
I 
I 
' 
upstream 
6~6 
6 6 
-----s -----~ ------n ----
6 6 
6 6 
' 
' 
- - -- -1:::,- - - - - -~- - --- -?5.- ---
-0.5 0.0 
t/T 
0.5 
1.0 
0.5 
I\ 
N 3 0.0 
V 
I\ 
N 
3 
V 
I\ 
N 
3 
V 
-0.5 
-1.0 
1.0 
0.5 
0.0 
-0.5 
-1.0 
1.0 
0.5 
0.0 
-0.5 
-1.0 
saddle 
6_ In A 6 : Ll 6 
-----A-----~------f:..----
6. 6. 
6. 6. 
downstream upstream 
11=0.71 
6.Lp 6 
6. : 6. 
' 
' 6 : 6. 
------------~------------
' I 
11=0.12 
6. ~ 6 
' 6. 6 
- ___ _6,__ ---- ~- - --- _6, ___ _ 
-0.5 0.0 0.5 
t/T 
Fig. 6.16. Conditional averages of w2 fluctuations. 
195 
vortex center 
5 
ri=-1.65 
4 
saddle 
5 
ri= 1.29 
I\ 4 
N 
.._ N 
3 3 3 
~ - -;t A ~ ~A.-A-al -A 
, .... V 
~ 
__ _., _ ___ _.. 
I\ ; A A A A 2 
.... 1 
1 
1 
1 
1 
1 
downstream upstream 
2 N 
N 
3 
I 
V 1 
downstream upstream 
0 0 
6 
ri=-1 .06 
5 
4 
~,. 
t: .. 
1 
7 
11=0.71 
I\ 6 !:::,, 
N .. 
-.. N 
3 5 
' 
V 
~ .. 
3 - - - - .;... - - -..-
' : j 
1 
I\ 4 - - ~~ t - - ~-N 
N 
3 ¼i, .. 2 , ... V 3 I 
I 
2 
7 
11=0.12 
I\ 6 
N !:::,, !:::,, 
' N .. 
7 
6 
5 
4 
3 
2 
3 
V 5 
_._.~iii---&-ti,• 
- 0 .5 0.0 
t/T 
0.5 
~ 
I\ 
N 
N 
3 
V 
4 
3 
2 
-0.5 
I 
1 
0.0 
t/T 
Fig. 6.17. Conditional averages of the total and incoherent w/ 
fluctuations. The closed symbols are for the incoherent 
fluctuating field. The open symbols are for the total 
fluctuating field. The long dashed lines indicate the 
time averages of the incoherent field. 
196 
0.5 
5 
vortex center 
11=-1.65 
' 
' 
4 .- ___ A A_ 4-_A _A _j. --A 
...... : 
3 
downstream 
2 
8 
11 = -1.06 
7 
' I 
I 
I 
I 
I 
upstream 
- - -,- - - - -
6 
• ;. 
5 
10 
11=-0.4 7 
9 
8 
7 
-0.5 
,: 
I 
0.0 
t/T 
0 .5 
saddle 
6 
: 11= 1.29 
•• .& I A• : ,,
.j\ 5 
Lu 
V 
ft 
(\ 
v 4 
, I\. 
w 
V 
ft 
(\ 
w 
V 
,I\. 
w 
V 
ft 
(\ 
w 
V 
3 
9 
8 
7 
6 
10 
9 
8 
7 
-0.5 
----·----
: ;. ;. 
A_._. 
downstream upstream 
ri=0.71 
ri=0.12 
6. 
A 
• I 
- c..A°i ZS i 5' - - -
... ,. 
, A 
I 
I 
0.0 
t/T 
0.5 
Fig. 6.18. Conditional averages of the total and incoherent enstrophy 
fluctuations. The closed symbols are for the incoherent 
fluctuating field. The open symbols are for the total 
fluctuating field. The long dashed lines indicate the 
time averages of the incoherent field. 
197 
0.040 
0.035 
0.030 
0.025 
0 .020 
0.050 
0 .045 
0.040 
0.035 
0 .030 
0.060 
0.055 
0.050 
0.045 
vortex center 
-ri=-1.65 
downstream upstream 
-ri = -0.4 7 
I 
I\ !::, 6 
Ll : 6 
: 6 
- - -6 - .!.- - ---t:::, -
6 
6 
66 
saddle 
0.040 
T]= 1 .29 
0.035 
I\ 
UJ 0.030 
V 
0.025 
downstream upstream 
0.020 
0.060 
T]=0.71 
0.055 
I\ 
UJ 
V 
0.050 6 6 6 
- -- -..L.-----
6: 
f 6 
0.045 
~6 6 
: 6 
0.040 
0.060 
T] = 0.12 
0.055 
I\ 
UJ 0.050 
V 
0.045 
0.040 
-0.5 0.0 
t/T 
0.5 
0 .0 40 
-0.5 0.0 
t/T 
0.5 
Fig . 6. 19 . Conditional averages of the total dissipation rate fluctuations . 
The incoherent contribution is almost the sam e as the total 
therefore not shown. The long dashed lines indicate the 
time averages. 
198 
30 
25 
01 20 
C 
.;y_ ,,... 
u 15 0 ; \ 
~ 10 /J( --.. ..._ 
I \ 
I \ 
5 I 
\ 
...._.I A 
' 0 
30 
25 
(b) 
CJ) 20 
C 
.;y_ 
u 15 ,-A'-
,,..- A 
0 I '\ 
/ 
~ 10 I ~ 
f 
1 '\ 
I 
..... JI!' 
5 
0 
30 
25 
(c) 
I\ 
20 x" 
~ \ 
01 I ~ I C I 
'.52 I \ I 
u 15 
I 
0 I \ I A 
~ 10 
\ I 
I 
,, 
5 1 
0 
-2 -1 0 
2 
T] 
Fig. 6.20. Phase locking of the incoherent velocity flow field with the 
vortex shedding: (a) <u12>, (b) <v'2>, and (c) <w'
2
>. 
199 
30 
25 
(a) 
O'l 20 
C 
'.Y /""). 
0 15 \ 0 I \ / ...... 
~ 10 I \ 
,A 
I \ 
I 
I 
5 1 ~ ),. 
0 
30 
(b) 
25 
O'l 20 
C 
'.Y 
0 15 0 
~ 10 
..... 
5 " 
/JI;' 
, ...... A 
0 
30 I 
(c) r\ 
25 
,. 
,J. ..... I \ 
I 
\ 
O'l 20 ~ I 
C I i 
'.Y \ 
I 
0 15 I I 0 \ ,, 
~ 10 
5 
0 
-2 -1 0 
2 
Tl 
Fig. 6.21. Phase locking of the incoherent vorticity flow field with the 
vortex shedding: (a) <w/>, (b) <w/>, and (c) <w~
2
>. 
200 
30 
25 
CJ) 20 / .... 
C /I( ' 
.Y. / ~ 
u 15 x '\. 0 I '\. 
~ 10 I ~ 
I 
5 I 
0 
-2 -1 0 2 
Tl 
Fig. 6.22. Phase locking of the incoherent Reynolds shear stress 
field <tfv'> with the vortex shedding. 
201 
...,... 
0.01 
0 .00 z 
0 \ 
I- \ u 
:::) \ 
0 \ 
0 
O:'.'. \ 
o._ \ -0.01 
>- I (.'.) 
O:'.'. I w 
z I 
w I 
u I 
ti I 
z -0.02 I 
::.:'.: I 
LL I 
0 I 
w \ I 
I 
~ l' O:'.'. 
-0.03 
- 0.04 
-4 -3 -2 -1 
' 
' 
'~ 
/. I 
,: I 
I I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
f 
0 
T) 
I 
I 
I 
I 
I 
1 
I 
I 
I 
I 
\ I 
\ I 
i 
/ 
/ 
' I I 
I 
I 
I 
I 
f 
I 
I 
I 
I 
2 3 4 
Fig. 7.1. Distributions of the rate of mean kinetic energy production 
by the mean velocity gradient. The open symbols are for 
production due to interaction with the coherent field, 
and the closed symbols are for the incoherent field. 
202 
1.0 
0.9 
0.8 
I 0.7 j I 
0. 6 I \ >-I I Q_ 
I \ 0 0 .5 Q'.'. f- I CJ) z I I w 0.4 \ 
0.3 I \ I 
0 .2 } I \ 
0. 1 
' 
0.0 
-4 -3 -2 -1 0 2 3 4 
Tl 
Fig. 7 .2. Enstrophy distributions at x/ d=30. 
o, mean; 6., coh erent; D, incoherent/1 O. 
203 
0.10 
0.07 
0.03 
0.00 
-0.03 
- 0.07 
-0.10 
0.10 
0.07 
0.03 
0 .00 
-0.03 
-0.07 
-0.10 
0.10 
0.07 
0.03 
0.00 
-0.03 
-0.07 
-0.10 
Fig. 
-
-4 
(a) 
• 
(b) 
I ~, 
I ' ' 0, : o, 
- - -e- - • - • • C = ~: :: =~- • -... -,'• .•.• •-.. ,. --';-e- - - -•- - -,·,~.--·- - • -----,' Ofd ', 
0 -. I \ o--
""-.,. ,' ""'"'_ ...... 
'0 I u 
I 
(c) 
,.,,~ I / \ ,. __ __ 
__..-- \ ~~ I "' 
--a--~- --G- -- ·S-. - -E>- - •• :l'D- rt'.'\" '-Q--.§-y-..e:,----Er----~-=-•----
1 '.,. 
-3 -2 -1 
I 
0 
Tl 
2 3 
7.3. Balance of the mean enstrophy at x/d=30. 
1
al o, 11 ); o, (2a); •, (2b); b. , (3a); • , 
b o, 5a); e, (5b). 
(3b). 
c o , 6); o, (7); •, residual. 
204 
4 
1.0 
0.5 
0.0 
-0.5 
1.0 
0.5 
0.0 
-0.5 
0.5 
0.0 
-0.5 
-1.0 
-4 
(c) 
I 
-~~ -..:_&..:: - ~ ---El- - - - D - - - -13-EJ- rEl--G-EJ- -- --!3- - --G- -~-~ ·;: §-- -...:-{j-
' I 0"' ~ / 
-3 -2 
" ? 
'0 I / 
" ef 
" 
-1 
G-.~c-cf 
0 
Tl 
2 3 
Fig. 7.4. Balance of the incoherent enstrophy at x/d=30. 
(
al O , 11 ); o, (2a); •, (2b); 6 , (3a); A , (3b) . b o, 4a)/10; o, (4b); •, (4c); 6, (5a); A, (5b). 
c o, 6); o, (dissipation rate and residual)/10. 
205 
4 
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