1 .
.. K? . .
.. 2,121.27!

 

583%, .

7”...
Wm.“

. .3 1
:VvP; »

a. #3.;
1.542.“? ,
‘ .5... .33:
.. 2
ma .. . .2
a. has». Raw;
(Lannie! 5m
.. , $21.53?
”griffin ... .

LAzm: . “an“... .. .
. ; A :. .3:th r
5.52”. supine
L :I in.“

a
; . 1 :
iv... r.

:7

 

.
x 33. .
..
.. In:
M»???

, I 6.)...
O:

r . vPitfii. .

:
. unified. :23 an“. d

:3! 2» In.s.0f.9v,:.

,. f . .4

.2.“ 1.5.2....” . 7.94%; m ‘ ‘ ..,..W.q.,.,..m,_:..r.g.mufiz . . . unm._§..®a¢wfi ‘ $53

 

7115813

lllJHHIUHIIHIIHHillllllllllflUUHIIUIJHIUHIl

31293 01779 1959

 

LIBRARY

Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

EFFECT OF FORCING ON THE VORTICITY
FIELD IN A CONFINED WAKE

presented by

RICHARD K . COHN

has been accepted towards fulfillment
of the requirements for

PhD degree in Mechanical Engineering

 

 

 

Date March 15, 1999

 

MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1M G/CIRCIDmDuopGS-p. 14

EFFECT OF FORCING ON THE VORTICITY FIELD IN A
CONFINED WAKE

By

Richard K. Cohn

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1999

ABSTRACT

EFFECT OF FORCING ON THE VORTICITY FIELD IN A CONFINED
WAKE

By

Richard K. Cohn

Several recent studies have found that when a low Reynolds number, plane wake is
forced with sufficient amplitude, the normalized mixing product, measured as the amount
of mixed fluid per unit width of the wake, can be increased to levels larger than those seen
in high Reynolds number mixing layers. However, no studies examining the velocity and
vorticity fields of this flow have been conducted. The present study examines the velocity
and vorticity field of a low Reynolds number plane wake within a confining channel in order
to better understand the vortex-vortex and vortex-wall interactions in order to shed light on
the mechanisms which lead to increases in the amount of mixed fluid within the wake.

Molecular Tagging Velocimetry (MTV) is used to measure the velocity field in both
the streamwise (u, v velocities in x, y plane) and cross-stream (v, w velocities in y, z plane)
measurement planes. The spanwise and streamwise vorticity components are then computed
from their respective velocity fields. The experimental results represent the first time MTV
has been used to make whole-field velocity measurements in a plane where the mean flow
is moving directly out of the measurement plane. Increases were also made in the number
of measurement points per image plane which has been increased by more than the 50% over
previous MTV studies. Advances were also made in several post-processing aspects of MTV

including the determination of the best method to remap the velocity data onto a regular grid

and methods to compute vorticity from this data.

Measurements in the streamwise plane have found that a distinct spatial periodicity
exists in the um, field that is not found in either the unforced case or in unconfined forced
flows such as the wake of an oscillating airfoil. A model was develbped which relates this
spatial periodicity to the phase difference between the forcing input and the rolling up of the
vorticity shed from the splitter plate. From these data, it was also determined that the phase
at which vorticity is shed is dependent upon the forcing amplitude.

The forced wake flow is dominated by the shedding of concentrated, spanwise vortex
core rollers. As these cores develop downstream, the levels of peak vorticity within the core
decrease. This decrease has been shown to be dominated by negative vortex stretching,
rather than diffusion. A very small amount of ”Va is sufficient to generate a very large
decrease in peak vorticity levels. This same quantity has also been found to be a good
predictor of the spatial location where mixing enhancement will occur in the forced wake.

Mixing enhancement is accomplished by the generation of regions of streamwise
vorticity from the reorientation of the primary spanwise vortex cores. In flows where mixing
enhancement occurs, multiple regions of streamwise vorticity quickly convect towards the
center of the test section. A model was developed which describes how these cores develop.
The multiple regions of streamwise vorticity are the result of the passage and reorientation
of multiple spanwise rollers. As each roller is reoriented into the streamwise direction, it
experiences a large amount of stretching which increases the peak vorticity levels. These
reoriented “legs” of streamwise vorticity interact with the regions of streamwise vorticity
resulting from the passage of previous spanwise vortex rollers to generate the additional

surface area necessary for mixing enhancement.

 

Copyright by
Richard K. Cohn
1999

ACKNOWLEDGMENTS

It would not have been possible to complete this work without the help of many
people. I would like to thank all of my lab-mates over these past years, for the many hours
of conversation and assistance needed to complete this projects. I would like to specifically
thank, Chuck Gendrich, Colin MacKinnon, Doug Bohl, Hooman Rezaei, Greg Katch, and
Suman Chakrabarti for their assistance in completing this work. I would also like to thank
my advisor, Dr. M.M. Koochesfahani, for his years of assistance. This work would also not
have been possible without the financial support of the AFOSR, the Center for Sensor
Materials at Michigan State University, a National Science Foundation Graduate Research
Fellowship, and the Air Force Research Laboratory Palace Knight program.

Finally, I would like to thank my wife Misty, who stuck through all of these years of
late nights and odd hours necessary for the completion of my degree. Your support was

essential to the completion of my program.

TABLE OF CONTENTS

LIST OF TABLES ..................................................... viii
LIST OF FIGURES ..................................................... ix
LIST OF SYMBOLS AND ABBREVIATIONS ............................. xvii
Chapter 1
Introduction ............................................................ 1
1.1 Motivation ....................................................... l
1.2 Previous work .................................................... 4
1.3 Potential mechanisms for increased mixing ............................ 10
Chapter 2
Experimental Procedure .................................................. 13
2.1 Mixing layer facility ............................................... 13
2.2 Forcing mechanism and conditions ................................... 16
2.3 Velocity measurement method (Molecular Tagging Velocimetry) ........... 19
2.4 Implementation of MTV in the mixing layer facility ...................... 23
2.5 Post-processing of MTV data ....................................... 34
2.6 Post-processing of higher density data ................................. 40
Chapter 3
Streamwise Measurements of a Wake Forced Near its Natural Frequency ........... 42
3.1 Phase-averaging of streamwise measurement plane data .................. 42
3.2 Mean and RMS flow properties at center span .......................... 46
3-3 Phase-averaged (“instantaneous”) flow properties at center span ............ 63
................................................................... 73
3-4 Flow properties away from the center span ............................. 74
3-5 Higher Density Measurements ....................................... 82
Chapter 4
Cross~Stream Measurements .............................................. 86
4°] Effect of out-of-plane motion on in-plane velocity measurements ........... 86
4-2 Phase-averaging of cross-stream plane data ............................ 9O
4-3 Mean flow properties of the cross-stream plane ......................... 93
4-4 Downstream development of the phase-averaged streamwise vorticity ....... 99
4-5 Variation of the streamwise vorticity field with phase ................... 105
4.5 Relationship of streamwise vorticity field to mixing results ............... 119

4-7 Model of the development of the vorticity field in the highly forced wake . . . . 121
4-8 Evidence of axial flow along the cores of the spanwise vortices in the forced wake

' . ................................................................ 124
4-9 Higher density measurements ...................................... 125

vi

Chapter 5

Measurements at the 4 Hz and 8 Hz Forcing Frequencies ....................... 129
5.1 Mean streamwise measurement plane flow properties at center span ........ 129
5.2 Phase-averaged streamwise measurement plane results at center-span ....... 141
5.3 Mean streamwise vorticity and cross—stream plane velocity results ......... 147
5.4 Phase averaged streamwise vorticity and cross-stream plane Velocity results . . 152

Appendices ........................................................... 160

Appendix A

The Placement of Irregularly Spaced Velocity Measurements on a Regular Grid and the

Calculation of Out-of—Plane Vorticity ...................................... 161
A] Introduction .................................................... 161
A2 Comparison Method ............................................. 166
A3 Interpolation Results ............................................. 173
A4 Vorticity Calculation Results ...................................... 178
AS Conclusions .................................................... 187

Appendix B

The Computation of Mean Quantities of Phase-Locked Signals Using Sparsely Sampled Data

. .................................................................... 190

Appendix C

Velocity Forcing Amplitude Calibration Curves for the Two-stream Mixing Layer Facility

. .................................................................... 194

BIBLIOGRAPHY ..................................................... 197

vii

 

9r- ‘1

in”. ._.l

x
J
4,.)

mm

LIST OF TABLES

Table 2.1: Forcing conditions for forced wake experiments. The forcing amplitudes indicate
the percentage of the free stream velocity of the root mean square fluctuations of the
streamwise velocity. Superscripts 1, 2, and 3 are the cases which correspond to the
low, middle, and high amplitude forcing cases (respectively) of Nelson (1996),
Koochesfahani and Nelson (1997) and MacKinnon and Koochesfahani ( 1997). 17

Table 2.2: Streamwise measurement planes. Full and Reduced refer to the number of
measurement cases. ............................................... 24

Table 2.3: Spanwise measurement planes. Full and Reduced refer to the number of
measurement cases. ............................................... 25

viii

LIST OF FIGURES

Figure 1.1: Schematic of the wake flow. Note that the wake is confined both on the top and
bottom walls as well as by the sidewalls of the facility . . . . ................. 1

Figure 1.2: Laser induced fluorescence visualization of the unforced (left) and forced (right)
wake from MacKinnon and Koochesfahani (1997). (a) Streamwise view for 1 cm <
x < 12 cm downstream of the splitter plate tip. The flow direction is from left to
right. (b) Spanwise views of the test section cross-section at x = 8 cm at three
representative times in the forcing cycle. ................................ 3

Figure 1.3: Streamwise variation of total mixing product (a) and normalized product
thickness (b) at the mid-span location for different forcing amplitudes. Data is from

the mixing studies of Koochesfahani and Nelson (1997). ................... 7
Figure 2.1: Schematic of the mixing layer facility. ............................. 14
Figure 2.2: Schematic of test section. ....................................... 16
Figure 2.3: Sample MTV measurement grid. ................................. 21
Figure 2.4: Sample beam block used in MTV experiments. ...................... 22

Figure 2.5: Sample measurement planes. (a) Streamwise planes. (b) Spanwise (end view)
planes. Note that the origin of the coordinate system is at the center of the test
section at the tip of the splitter plate. .................................. 24

Figure 2.6: Optical arrangement for MTV measurements. (a) Streamwise plane. (b)

Spanwise plane. .................................................. 27
Figure 2.7: Timing Schematic for MTV experiments ............................ 30
Figure 2.8: Block diagram of the MTV processing procedure. .................... 35
Figure 2.9: Conditions for selecting a regular grid point. ........................ 37
Figure 2.10: Movement of beam blocks to increase grid measurement density. ....... 39

Figure 3.1: Phase ordered free-stream velocity fields for two downstream locations in the
forced wake. (a) x = 5 cm. (b) x = 18 cm. The small undulation in the v-component
of velocity at x = 18 cm is the signature of the vortex cores passing this downstream
location at the y location of the measurement. ........................... 43

Figure 3.2: Phase—averaged vorticity field (flooded contour) along with vorticity contour

ix

 

“JV
.1»;u

1(.

I- A Q
v”: t. \'

14ml. -»- -
.

".14
.»u

.‘L

 

1 o
r-VIO‘ ‘ ‘.
“nit\ . .

.. .“
'. ht“!
.‘.-J~ 1"
.

0..
‘r‘
Lbs
—
..,’;‘-
ho5jv
P
"v .
.,J
uh.‘\
I "v-‘
.
~"‘.’a‘.‘i
\u'
\.
_'\.,,_
0"
.
...(.'“
t
T u‘ I!
.‘4
.""w‘ H
. - p.
2'.)
Mg‘
.
‘1‘;: I.
-~.1
l
l
I
.113
.
’1. .
~ .
~«.: ‘\
. .~
7."
w.
s
V".
s
-- 1 ‘~
‘u. 1
. ~~ ‘
|.
I
n. ,
\y: V t
.W.’
\.
‘ k.\
\
.-
,.. A
\ ‘ 'o
0“! ‘ \
“.
‘ I
1‘. ,
7‘

lines of 6 instantaneous realizations overlaid on top. Dashed contour lines indicate
negative values of vorticity and contour levels are i4 3", :8 s", :24 s". (a) 3 <
x<7cm. (b) 15.75<x<19.75cm. .................................. 44

Figure 3.3: Streamwise velocity profiles at z = 0 cm for several forcing amplitudes at 3
streamwiselocations. ................ 47

Figure 3.4: Momentum thickness as a function of (a) forcing amplitude and (b) downstream
location. ........................................................ 48

Figure 3.5: RMS of streamwise (u) and spanwise (v) velocity for the unforced and l 1.6%
forcing amplitude cases. (a) um. for unforced cases. (b) um. for l 1.6% forcing case.
(c) vm for unforced case. ((1) vm for 11.6% forcing case. Contour levels are 0.3
cm/s, 0.6 cm/s, ..., 1.5 cm/s. ......................................... 51

Figure 3.6: Effect of the phase relationship between flow forcing and vortex shedding on the
streamwise velocity fluctuations in the forced wake. ..................... 53

Figure 3.7: Dependence of phase time at which shedding occurs on forcing amplitude. 55
Figure 3.8: Line plot of um. for three forcing amplitudes ......................... 56

Figure 3.9: Reynolds stress distribution in unforced and forced wake. The contour lines
indicate the 1:0.6, :tI.2,...,:t:3.0 levels. ................................. 57

Figure 3.10: Mean vorticity field for unforced wake and wake forced at 6 Hz. Contour lines
are spaced at :3 s", :6 s", :15 s". Dashes indicate negative contours. . . . . 58

Figure 3.11: Streamwise location of phase-averaged vortex cores versus phase. A linear fit
has been subtracted from the locations. ................................ 59
Figure 3.12: Mean %::— for the unforced wake and four perturbation amplitudes. The solid
and dashed contours separate regions of positive and negative value. ........ 61

Figure 3.13: Example of recirculatory flow pattern. ............................ 62

. — 6
Flgure 3.14: Comparison of 35:— in 11.6% forcing case and fifrom Koochesfahani and
Nelson, 1997. .................................................... 62

Figure 3.15: Vorticity field in forced and unforced wake. The unforced wake data are
instantaneous realizations and the forced wake data are phase-averaged results.
Contour lines are spaced at 15 s", :10 s”, :25 s". Dashes indicate negative
contours. ........................................................ 64

Figure 3.16: Effect of forcing on peak vorticity magnitude. (a) Effect of forcing on vorticity
magnitude at several downstream locations. (b) Variation of vorticity magnitude for

3 forcing amplitudes with respect to streamwise location. ................. 65

Figure 3.17: Comparison of the decay of the peak spanwise vorticity with the decay due to
solely stretching and solely diffusion. (a) 2.1% forcing amplitude. (b) 1 1.6% forcing
amplitude ........................................................ 68

Figure 3.18: Vortex core circulation. (a) Effect of forcing on circulation at several different
downstream locations. (b) Effect of downstream location for several forcing
amplitudes. (c) Effect of downstream location on vortex core radius. ........ 69

Figure 3.19: Effect of forcing on vortex spacing parameters. (a) Definition of spacing
parameters. (b) Effect of forcing on spacing ratio at several different downstream
locations. (c) Effect of downstream location on spacing ratio for three forcing
amplitudes. ...................................................... 71

Figure 3.20: Flooded contour plots of spanwise vorticity in wake forced at 11.6% of free-
stream velocity for 7 different spanwise locations. Contour lines are spaced at :5 s",
21:10 s", :25 s". Dashes indicate negative contours ..................... 75

Figure 3.21: Flooded contour plots of spanwise vorticity in wake forced at 7.3% of free-
stream velocity for 7different spanwise locations. Contour lines are spaced at :1-5 s",
:10 s", ..., :I:25 s". Dashes indicate negative contours ..................... 76

Figure 3.22: Flooded contour plots of spanwise vorticity in wake forced at 2.1% of free-
stream velocity for 7 different spanwise locations. Contour lines are spaced at :5 s”,
:10 s”, :25 s". Dashes indicate negative contours. . . . . . . . . . . . . . ....... 77

Figure 3.23: Variation of streamwise (a) and transverse (b) vortex spacing with spanwise
location. ........................................................ 78

Figure 3.24: Effect of spanwise location on peak vorticity and spacing ratio at x = 4.5 cm.
(a) spacing ratio. (b) peak vorticity. .................................. 79

Figure 3.25: Momentum thickness of the forced wake (11.6% forcing amplitude) for the
center span and two off-center span locations. .......................... 81

Figure 3.26: Comparison of high density and standard density measurements at three
different spanwise locations. Contour lines are spaced at :5 s", :10 s", :25 s".
Dashes indicate negative contours. ................................... 83

Figure 4.1: Diagram of the effect of out-of-plane motion on in-plane velocity estimates. .
............................................................... 87

Figure 4.2: Comparison of the phase-averaged streamwise vorticity field with several

instantaneous realizations of the streamwise vorticity field for four downstream
measurement planes. Five instantaneous realizations are displayed in each plot and

xi

'.

N

[5
.rir; “
Lu.s"-..

r- y;

l'
.;_._ y)

..
a-v'b ‘5
..l‘ ‘4

‘ ~.

 

\
s t
11
o".
D.
I“
I
-.‘;‘.
if; 1.
- '§ ‘ i
‘5
R...
u”;
.w

contour lines indicate the :3 s", :6 s",..., :15 s" with the dashed lines indicating the
negative contour levels. (a) x = 3 cm. (b) x = 10 cm. (c) x = 16 cm. (d) x = 24 cm.
............................................................... 91

Figure 4.3: Mean streamwise vorticity field for the unforced, 7.3%, and 1 1.6% forcing cases
at four different downstream locations. Contour lines indicate the :3 s", :6 s”,..., :15
s" with the dashed lines indicating the negative contour levels. ............. 95

Fi gure 4.4: Circulation of the average vorticity field of the right half measurement plane for
the 7.3% and 11.6% forcing conditions. ............................... 96

Figure 4.5: Mean velocity field over the cross—stream plane for the unforced, 7.3%, and
11.6% forcing cases at four different downstream locations. ............... 98

Figure 4.6: Downstream development of the phase-averaged streamwise vorticity for the
l 1.6% forcing case at two arbitrarily selected phases within the forcing cycle. Note
that the facility sidewalls are located at z = -4 cm. The axis displayed at z = 0 is the
center of the test section. Contour lines indicate the :6 s", :9 s",...,:15 s" with the
dashed lines indicating the negative contour levels. (a) (1) = 0.21. (b) (1) = 0.62. . .

.......................................................... 100-101

Figure 4.7: Variation of the magnitude and position of the peak value of <wx> field for the
7.3% and 11.6% forcing cases. (a) Variation of peak magnitude of <(o.,> with
downstream distance. (b) Change in location of peak <u)x> with downstream
distance. The farthest upstream location (x = 3.25 cm) is marked with an “F” and the
farthest downstream location (x = 23.75 cm) is marked with an “L”. ........ 103

Figure 4.8: Downstream development of the phase-averaged streamwise vorticity for the
7.3% forcing condition for (1) = 0.21. Note that the facility sidewalls are located at
z = -4 cm. The axis displayed at z = 0 is the center of the test section. Contour lines
indicate the :6 s", :9 s",...,:15 s" with the dashed lines indicating the negative
contour levels. .................................................. 106

Figure 4.9: Variation of the streamwise vorticity with phase at x = 3 cm for the 11.6%
forcing amplitude. Contour lines indicate the :6 s", :9 s",..., :15 s" with the dashed
lines indicating the negative contour levels. ........................... 107

Figure 4.10: Variation of the streamwise vorticity with phase at x = 5.25 cm for the 11.6%
forcing amplitude. Contour lines indicate the :6 s", :9 s",..., :15 s” with the dashed
lines indicating the negative contour levels. ........................... 109

Figure 4.11: Spanwise vorticity field in the range -3.5 cm < z < -2.5 cm. The phase

corresponds to phase 6 of Figure 4.10. Contour lines indicate the :5 s", :10 s'
’,...,:25 s" with the dashed lines indicating the negative contour levels. ..... l 12

xii

 

Figure 4.12: Variation of the streamwise vorticity with phase at x = 8.5 cm for the l 1.6%
forcing amplitude. The displayed phases are separated by approximately A4) = 0.08
and phase runs from right to left and then top to bottom. Contour lines indicate the
:3 s", :6 s",...,:15 s" with the dashed lines indicating the negative contour levels.

.............................................................. 114

Figure 4.13: Variation of the streamwise vorticity with phase at x = 16 cm for the 1 1.6%
forcing amplitude. The displayed phases are separated by approximately At) = 0.08
and phase runs from right to left and then top to bottom. Contour lines indicate the
:3 s”, :6 s",...,:15 s" with the dashed lines indicating the negative contour levels.

.............................................................. 115

Figure 4.14: Variation of the streamwise vorticity with phase at x = 3 cm for the 7.3%
forcing amplitude. Contour lines indicate the :6 s", :9 s",..., :15 s'1 with the dashed
lines indicating the negative contour levels. ........................... 1 17

Figure 4.15: Variation of the streamwise vorticity with phase at x = 5.25 cm for the 7.3%
forcing amplitude. Contour lines indicate the :6 s", :9 s”,..., :15 s" with the dashed
lines indicating the negative contour levels. ........................... 1 18

Figure 4.16: Variation of the streamwise vorticity with phase at x = 8.5 cm for the 7.3%
forcing amplitude. Contour lines indicate the :6 s", :9 s”,..., :15 s" with the dashed
lines indicating the negative contour levels. ........................... 120

Figure 4.17: Diagram of the development of the vorticity field in the forced wake. . . . 122

Figure 4.18: Axial flow in theA = 11.6% wake at x = 5.25 cm. .................. 125

Figure 4.19: Comparison of the standard and high density measurements of the streamwise
vorticity at x = 3 cm. Contour lines indicate the :5 s", :10 s",...,fl5 s" with the
dashed lines indicating the negative contour levels. ..................... 126

Fi gure 5.1: Mean velocity profiles for the wake forced at 4 Hz at three streamwise locations.
.............................................................. 130

Figure 5.2: Mean velocity profiles for the wake forced at 8 Hz at three streamwise locations.
.............................................................. 130

Figure 5.3: Effect of forcing on the momentum thickness across the 4 Hz and 8 Hz forcing
frequencies. (a) Effect of forcing amplitude on 0 at constant streamwise location.
(b) Effect of streamwise location on 0 at constant forcing conditions. ....... 132

Flglll‘e 5.4: RMS of streamwise (u) velocity and transverse (v) velocity for the highest
amplitude cases of the 4 Hz and 8 Hz forcing. The contour lines start at 0.2 cm/s and
spaced every 0.2 cm/s. (a) um for F = 4 Hz, A = 11.4%. (b) am for F = 8 Hz, A =
13.8%. (c) vm for F = 4 Hz, A =11.4%. (c) vm for F = 8 Hz, A =13.8%. ..134

xiii

- I
1.. x
1.;olb .
.
.i
If
.4
r’PJ‘
l.-4b .
4
it
.
71‘
k.
I
""2 \
.1.‘.§ .
u
..
11
- I
. v '_ ‘
Ԥ-..
' .
‘
I.
u.
I
' 7': \
r“ .
it
u
n
o; ‘
“<‘ ‘
h
1.
§
a.“ .7 .
- ‘
1“
~ .
..
ll
K
’§_ 0
.v‘y
:4 ,1
7:
L1
.
n.
n
n“ h .
’5. ‘.
~ *‘~
u“:
\l
.H
~
'~
. n
a; F N
. ‘~ ~
1r
it}
9‘ -
~."\ '
x "
V,

Figure 5.5: RMS of streamwise velocity vs y location at x = 4 cm. (a) 4 Hz forcing. (b) 8
Hz forcing. ..................................................... 135

Figure 5.6: Mean streamwise vorticity for unforced case and three different amplitudes for
the 4 Hz forcing. The contour lines indicate the :3, :6, :15 s'l contour lines with
the dashed lines indicating the negative values. ........ . ................ 136

Figure 5.7: Mean streamwise vorticity for unforced case and three different amplitudes for
the 8 Hz forcing. The contour lines indicate the :3, :6, :15 s"contour lines with
the dashed lines indicating the negative values. ........................ 137

Figure 5.8: Mean % field for 4 Hz forcing at three perturbation amplitudes. The contour
lines indicate regions of positive and negative sign. ..................... 139

Figure 5.9: Mean % field for 8 Hz forcing at three perturbation amplitudes. The contour
lines indicate regions of positive and negative sign. ..................... 140

Figure 5.10: Phase-averaged streamwise vorticity for unforced case and three different
amplitudes for the 4 Hz forcing. The contour lines indicate the :5, :10, ..., :25 s'
1contour lines with the dashed lines indicating the negative values. ......... 142

Figure 5.11: Phase-averaged streamwise vorticity for unforced case and three different
amplitudes for the 8 Hz forcing. The contour lines indicate the :5, :10, :25 s'
1contour lines with the dashed lines indicating the negative values. ......... 144

Figure 5.12: Effect of downstream location on peak <wz> and spacing ratio. (a) Peak
vorticity for 8 Hz forcing. (b) Peak vorticity for 4 Hz forcing. (c) b/a for 8 Hz
forcing. (d) b/a for 4 Hz forcing. Note that the scale in (c) and (d) are not the same.

.............................................................. 145

Figure 5.13: Mean (a)x at 4 streamwise locations for the 4 Hz forcing frequency cases. The
contour lines indicate the :3, :6, :15 s"contour lines with the dashed lines
indicating the negative values. ...................................... 148

Figure 5.14: Mean (1)" at 4 streamwise locations for the 8 Hz forcing frequency cases. The
contour lines indicate the fl, :6, :15 s"contour lines with the dashed lines
indicating the negative values. ...................................... 150

Figure 5.15: Mean v-w velocities in the y-z plane at 4 streamwise locations for the 8 Hz
forcing frequency cases. ........................................... 151

Figul‘e 5.16: Variation of phase-averaged (1)x with phase for F = 8 Hz, A = 13.8% at x = 3.25

cm. The contour lines indicate the :6, :9, :15 s"contour lines with the dashed
lines indicating the negative values. ................................. 153

xiv

 

Figure 5.17: Variation of <mx> with phase for F = 8 Hz, A = 13.8% at x = 20.25 cm. The
displayed phases are separated by approximately M) = 0. 08 and phase runs from right
to left and then top to bottom. Contour lines indicate the :3, :6, :15 s"contour
lines with the dashed lines indicating the negative values. ................ 154

Figure A. 1: Sample MTV measurement grid. ................ . ................ 163

Figure A.2: Sample velocity and vorticity field of Gaussian core vortex. (a) Original velocity
vector field. (b) Velocity vector field placed on a regular grid. (c) Flooded contour
plot of the vorticity field. .......................................... 167

Figure A.3: Schematic of velocity measurement locations used in the estimation of the
spatial derivatives. (a) lSt order finite difference. (b) 2nd order finite difference. (c)
8-point circulation method. ........................................ 170

Figure A.4: Profiles of the mean velocity bias error for 2““, 3’“, and 4‘h order polynomial fits
for grid densities ranging from [/8 = 2.5 to U8 = 10.5. .................. 174

Figure A.5: Comparison of the error in the velocity field resulting from the 2"“, 3m, and 4"1
order polynomial fits for the 0% and 6% noise added cases for a normalized grid
density of 3.5. (a) um. (b) um, ..................................... 176
Figure A.6: Effect of decreasing the radius from which irregular points are drawn in the
interpolation process on the velocity mean bias and random error. (a) um for 0%
noise added case. (b) am 0% noise added case. (c) uh,“ for 6% noise added. ((1) um,
for 6% noise added. ....................................... ' ....... 177

Figure A.7: Comparison of the vorticity mean bias error profile resulting from the use of
differentiating the 2"“, 3’“, and 4‘h order polynomial fit for the calculation of the out-
of-plane vorticity for four different grid densities. Results from simulations with 0%
added and a maximum of 6% uniform noise are shown. .................. 180

Figure A.8: Comparison of the random vorticity error profile resulting from the use of
differentiating the 2'”, 3'“, and 4‘h order polynomial fit for the calculation of the out-
of—plane vorticity for four different grid densities. Results from simulations with 0%
added and a maximum of 6% noise are shown. ......................... 181

Figure A.9: Comparison of the mean vorticity bias error profile resulting from the use of the
circulation method, lSt and 2nd order finite difference methods, and differentiating the
3rd order polynomial fit for the calculation of the out-of—plane vorticity for four
different grid densities. Results from simulations with 0% and 6% noise added are
shown. ........................................................ 183

Fi gul‘e A. 10: Comparison of the vorticity random error profile resulting from the use of the
circulation method, 1St and 2nd order finite difference methods, and differentiating the

3rd order polynomial fit for the calculation of the out-of—plane vorticity for four

XV

different grid densities. Results from simulations with 0% and 6% noise added are
shown. ........................................................ 185

Fi gure A.11: Effect of varying the radius from which points are selected in the interpolation
process on the mean bias and random error for the vorticity computed by means of
differentiating the 3rd order polynomial and the use of the 2nd order finite difference
method ......................................................... 186

Figure B.1: Comparison of the estimation of the mean using direct computation versus
calculating the phase average and the computing the mean. (a) Average value of
sinusoidal signals with two different starting phases. (b) Root mean square of 16
starting phases. .................................................. 191

Figure C. 1: Velocity perturbation calibration curves for the forced wake. .......... 195

xvi

AD

MTV

RMS

<X>

>“I

S zmbols

n

random

R

LIST OF SYMBOLS AND ABBREVIATIONS

Abbreviations and Notations

Analog to Digital Converter

Molecular Tagging Velocimetry

Root Mean Square

Phase average of quantity x

Mean of quantity x

Bias error in measurement of quantity x
Maximum value of quantity x

Value of x at peak vorticity

Root mean square of quantity x

Forcing amplitude measured as ums/u0 in percent

Forcing frequency in cycles per second

Generic value for frequency in cycles per second

Spacing between points on the regular grid

Characteristic flow length scale — equal to the vortex core radius
Molarity

Maximum percentage of noise added

Actual percentage of noise added to a velocity component

Maximum radius used in the least squares fit

xvii

COTC

min

T*

cam

At

Radial location
Vortex core radius

Minimum distance from regular grid location to velocity
measurement location

Total record length

Time

Velocity magnitude at point A at phase 0
Instantaneous velocity magnitude in free stream region
Streamwise velocity

Velocity in x component direction at node i, j
time-averaged free stream velocity

Azimuthal velocity about center of concentrated vortex core
Transverse velocity

Velocity in y component direction at node i, j
Spanwise velocity

Streamwise coordinate direction

Transverse (y) coordinate direction

Spanwise coordinate direction

Distance from measurement plane to camera

Time delay between undelayed and delayed images
Mean spacing between velocity measurements
Wake product thickness

Wake visual 1% thickness

xviii

 

 

O

(pmax

¢min

An integer number
Phase of measurement data — 0 < (1) < 1

Required spatial distribution of velocity measurements used in least
squares fit '

Required spatial distribution of velocity measurements within rmin
region

Circulation

Wavelength between vortex cores
Wavelength of laser light used in experiments
Phosphorescence lifetime

Wake momentum thickness

Maximum vorticity

Streamwise vorticity

Spanwise vorticity

Peak level of forcing amplitude

xix

Chapter 1

Introduction

1 - 1 Motivation

Several recent studies have examined the mixing properties of forced, low Reynolds
number, confined plane wakes. These studies have shown that when forced with sufficient
amplitude, both the total amount of molecularly mixed fluid in the wake region as well as
normalized mixing product, the amount of mixed fluid per unit width of the wake, can be
increased to values larger than those seen in unforced, high Reynolds number mixing layers.
Figure 1.1 shows of two views this flow. Two streams of fluid which are initially separated
by a thin splitter plate are allowed to interact once the plate ends. Unlike in a shear layer, the
speeds of the two streams are equal which presents several important differences in the

structure of the flow. The flow to be examined also has characteristics of both an external

 

 

Splitter u° }
Plate — '
”0 ,

 

 

 

 

F1?“ re 1 - l : Schematic of the wake flow. Note that the wake is confined both on the top and
tom Walls as well as by the sidewalls of the facility.

1

O G
"331:“
Rea—us»...-

‘ far.) 1"

“mung ,

1 rs

Vtr;1+‘
‘-‘¥h l...

. J
t.

 

5‘ ‘
rl .,.
. wk

and an internal flow. In the initial stages of the development of the flow field, the center
region can be considered to be a free region and it behaves as if it is not confined. However,
as the wake continues to develop, the confining walls of the facility, particularly the side
walls, play an important role in the flow structure and development Since this flow has
characteristics of both confined and unconfined flows, it will be referred to as “semi-
confined”.

A recent study by Koochesfahani and Nelson (1997) has shown that the normalized
mixed fluid fraction in this low Reynolds number forced wake is three times larger than that
seen in high Reynolds number liquid mixing layers and 60% larger than that in gas-phase
turbulent shear layers. However, no studies of this velocity and vorticity fields of this flow
have been conducted. The research presented in this work examines the three-dimensional
vortex-vortex and vortex-wall interactions of a confined plane wake as well as many other
properties of this flow field in order to examine the behavior of the underlying vorticity field
in this flow and to try to connect the mixing increase to its behavior. In this study, the term
vortex is used to denote a concentrated region of vorticity with the same rotational sense.

Figure 1.2 shows sample streamwise flow visualizations of an unforced and forced
plane wake from MacKinnon and Koochesfahani (1997). The measurement range for the
streamwise flow visualization results are 1 cm < x < 12 cm, which corresponds to the first
8 wavelengths of the vorticity field. This study was conducted in the same facility with the
same forcing conditions as will be examined in this research. The forcing in this flow is
applied by means of oscillating the speed of one of the free streams. In the streamwise view
of the unforced wake, the flow is starting to become unstable and undulations are beginning

to become apparent in the interface separating the two streams as they move downstream.

Unforced Forced

 

(b) Spanwise View

Figure 1.2: Laser induced fluorescence visualization of the unforced (left) and forced (right)
wake from MacKinnon and Koochesfahani (1997). (a) Streamwise view for 1 cm < x < 12
cm downstream of the splitter plate tip. The flow direction is from left to right. (b)
Spanwise views of the test section cross-section at x = 8 cm at three representative times in
the forcing cycle.

 

n-fn-‘v ’

.. ....1.u.

'0 i

15.5. n

t}; Lav
Lsih. tax

I“ rQ-o

A..’—.L_“

in (Liz:

.-.l

t"" 1.3'
..A ~

4

 

ll

:1 ‘~l-.

 

However, there is little evidence of vortical structures. In contrast, the forced case is
dominated by a concentrated Karman vortex street of spanwise vortices. Several studies also
refer to these regions of spanwise vorticity as the spanwise “rollers”. Although not shown
here, further flow visualization results presented in MacKinnon and Koochesfahani (1997)
show that the strength and spacing of these vortices vary depending upon the amplitude of
the imposed forcing in the flow. When examined in the cross-stream (y-z) plane, it is
apparent from these visualizations that the flow near the sidewalls of the test section is not
two-dimensional and the existence of streamwise vorticity is apparent. As in the streamwise
view, very little motion is seen in the unforced case. Further flow visualization images of
the forced case, also not shown here, suggest that the region of three-dimensional motion
moves closer to the center of the test section as the downstream distance increases.

Wake flows such as the one being studied are qualitatively different from the often
studied shear layer flows. Most notably, the forced wake contains vorticity of both signs
whereas shear layers contain only one sign of time-averaged vorticity. Breidenthal (1980)
examined a wake and a shear layer forced by various configurations of wedges placed on the
trailing edge of the splitter plate. With this method of forcing, it was shown that shear layers
rapidly relax to the characteristic two-dimensional large vortex structures, whereas the forced
wake remains three-dimensional. Furthermore, the forced wakes exhibited large spanwise

variations in mixing products far downstream which was not seen in the forced shear layers.

1.2 Previous work

Several research groups have measured the mixing field characteristics of a confined,

forced wake. These studies have shown that forcing can result in a significant increase in the

4

 

 

 

 

.
_.
‘."'-"
I
i
U.
“-1.
‘ D
i'; r»
-.l '
"h‘li

 

\
1"
_. a
...;
i

 

'9-
5. , ‘F
'v .
"H.
‘L.
‘. 1",

 

V.'
I
T in
‘t i '
‘l,r *1
a ll
3..

,_ .rv.‘ .
,_‘ ~_
'1

.,l
-
~.
~\‘£u “
.
vs
a
I ‘
\ . t
-_ u.
N ,~‘
1, ‘
FM
I

 

amount of mixed fluid. Breidenthal (1980), whose results are mentioned above, used the
reaction of phenolphtalein and a base to measure the visible mixing product in a wake forced
with a three-dimensional perturbation. Roberts (1985) applied two-dimensional forcing,
similar to that applied in the present study, to a wake flow by varying the pressure drop in
one of the streams. This study made use of both an absorption based technique similar to
Breidenthal as well as passive scalar measurements to examine the effect of varying the
forcing frequency on the amount of molecular mixing. This study observed a significant
increase in mixing when the flow was forced at twice the natural shedding frequency.
Examination of the cross-stream (y-z) plane at different x locations showed that in the
upstream region, regions of concentrated streamwise vorticity exist near the sidewalls. At

locations father downstream, these regions of (r)x move rapidly inwards towards the center

of the facility.

Robert’s study also proposed the following model to partially explain the increase in
mixing. Close to the sidewalls of the facility, the boundary layer will cause the
predominately spanwise vortex tubes to be reoriented into the streamwise direction. If the
streamwise vortices, which were generated by reorientation, are modeled as two point
vortices close to a wall, the induction of the vortices and their images in the side walls of the
facility will cause these original vortices to propagate away from the sidewall towards the
center of the test section. It was noted that forcing was found to cause the vertical separation
between the vortices in the wake to change and when the wake was forced at twice the
natural frequency, vertical distance between the positive and negative signs of vorticity
reached a minimum. In the proposed model, as the vertical distance between the counter-

rotating vortices decreases, the propagation speed of the pair away from the wall will

increase. This increased motion of the streamwise vortices in the cross-stream plane created
the additional surface area necessary for the two reactants to combine.

MacKinnon and Koochesfahani (1997) studied the effect of varying the forcing
amplitude on the mixing field of this flow. The forcing frequency was fixed near the natural
shedding frequency of the wake. Using a passive scalar technique, this study found that high
amplitude forcing caused a significant increase in the amount of mixing product generated
over the unforced case. Flow visualization results showing the evolution of the cross-stream
plane of this flow suggest the existence of additional streamwise vortices and more complex
interactions than can be described by Robert’s model of the reorientation of the spanwise
vorticity and its propagation towards the center of the test section.

Nelson (1996) and Koochesfahani and Nelson (1997) used a chemical reaction
technique to directly measure the amount of chemical product under the same conditions as
MacKinnon and Koochesfahani (1997). Data from Koochesfahani and Nelson (1997)
presented in Figure 1.3a, show the extent of the increase in the chemical product as the
forcing amplitude is increased. When compared to the unforced layer, forcing at low (2%)
and moderate (5%) amplitudes results in a small increase in the amount of mixed fluid. The
high (9%) amplitude forcing case, however, results in a much more significant increase in
the amount of mixed fluid. The measured product thickness of the highly forced wake is 40
times larger than that in the unforced layer. The leveling out of this profile at approximately
20 cm is likely due to the vortical structures impinging upon top and bottom walls of the test
section. If the test section had a greater height, the amount of mixed fluid would likely
continue to increase. Figure 1.3b shows the amount of mixed fluid normalized by the local

wake width. This normalized mixing product represents the percentage of the wake width

 

 

 

 

1'5 _ '5 0'5 ' - Untamed
“ v I ~ v Low Forcing
: ‘5 : 0 Middle Forcing
1.2 r- . a 0.4 1. 0 High Forcing "’
A .. _
g 1 ° 3 .
- , r 0 Q .0
g 0.9 - ,' g’ 0.3 - ,
a ’ a ; -
O
a. L o E . o
O 0—6 " . o 0.2 - 0‘
.E ' I' u r '0
.5 Z a F e W
. i-
0.3 .. 3 0'1
0M Obxtxi it ltltlllll—r—lgi xiii]
0 5 10 15 20 25 30 0 5 10 15 20 25 30
(a) X (cm) (W x (cm)

Figure 1.3: Streamwise variation of total mixing product (a) and normalized product
thickness (b) at the mid-span location for different forcing amplitudes. Data is from the
mixing studies of Koochesfahani and Nelson (1997).

Occupied by mixed fluid. Although the amount of the increase is less than that seen in the
tOtal product, the normalized mixing product shows a significant increase in the high
amplitude forcing case over the other forcing cases. Thus, the increase is not solely in the
Width of the mixing layer. Rather, the amount of mixed fluid within the mixing layer has
increased as well.

The studies of this flow field have so far been limited to flow visualization and scalar
Concentration levels. No studies of the fluid dynamical properties that generate this enhanced
mixing field are known. However, based on the flow visualization information, the
Streamwise vorticity field is believed to play a major role in the mixing enhancement. The
cuI‘l‘ent study, using the same facility as the works of Koochesfahani, Nelson, and
IVIEICKinnon will quantitatively examine the fluid dynamical properties of this flow field in
Order to better understand its mixing properties.

Although there have been no quantitative studies of a forced, semi-confined two-

stream wake such as those in which the mixing studies have been conducted, several detailed
quantitative studies of the free plane wake of a flat plate have been conducted. Sato and
Kuriki (1961) made hot-wire velocity measurements in the wake behind a thin flat plate in
order to examine the laminar-turbulent transition. Both the natural wake and wakes
perturbed with low-level acoustical forcing were studied. Three distinct regions were found
in the wake. A “linear” region in which sinusoidal velocity fluctuations were exponentially
amplified, a “non-linear” region where harmonics of the input fluctuations were amplified,
and finally, a “three-dimensional” region where the velocity fluctuations were believed to be
Perpendicular to the measurement plane. Sato (1970) expanded this study to include forcing
W ith multiple frequencies.

Mattingly and Criminale (1972) examined the near wake region both experimentally
and using a theoretical analysis based on inviscid stability theory. This study suggested that
the development of forced wakes, even those forced with very small amounts of excitation,
Could be different from the development of natural wakes. Forcing near the natural
frequency eliminates the approximately 10% drift observed in the natural frequency of the
Shedding which is perhaps an inherent part of the flow. Further, forcing generates an

Oscillation in the free-stream region which will interact in an undetermined way with the shed
Vorticity. Wygnanski et al. (1986) studied a variety of different wake flows, including
cyiinders, flat plates, airfoils, and screens, in order to determine the universality of their self-
Preserving states. Although the wake of each vortex generator was self—preserving, the
Characteristic velocity and length scales, when suitably scaled by the momentum thickness
and free-stream velocity, as well as the turbulence intensity distribution, when suitably

“Ormalized, were found to depend upon the geometry of the wake generator, which indicates

Niliiilll
_ 1
1'11 $75.11.

.\1

37.51761 1

“ail... '-
1'“. Ad:

‘9-
i- .
“-1” “ET“ .
1’.“

--‘~l
‘7‘ “a 4'5
94;
\uLt‘. W
‘V gli

' J.

a sensitivity to the initial conditions of the wake. The mean velocity profiles, however, were
universal.

Meiburg and Lasheras (1988) and Lasheras and Meiburg (1990) examined the wake
of a flat plate with periodic spanwise and streamwise perturbations. Depending upon
whether the initial perturbations were in the spanwise or streamwise direction, the
streamwise vorticity developed either a “symmetric” or a “non-symmetric” pattern. These
Studies suggested a mechanism for the self-amplifying increase in the amount of streamwise
Vorticity. Once the primary spanwise vorticity develops a kink, the induced velocity due to
the spanwise vorticity causes the kink to lift up. Then, the induced velocity of this portion
Causes the development of additional kinks and additional streamwise vorticity. Thus, the
development of the wake is dominated by the reorientation and stretching of vorticity rather
than by diffusion.

Weygandt and Mehta (1995) studied the three—dimensional evolution of plane and
Curved wakes. In comparison with shear layers, it was found that the wake is slow to recover

from input perturbations. Shear layers are constantly energized by the velocity difference

between the two layers. The resulting turbulent mixing can, in some cases, “wash-out” the
input disturbance. The streamwise vorticity was also found to be extremely sensitive to the
initial conditions. When the initial boundary layer was tripped, no stationary streamwise
vOrtical structures were observed.

LeBoeuf and Mehta (1996) conducted detailed measurements of a wake forced with

Very low amplitude acoustic waves. The purpose of the forcing in this study was to ensure
adequate coherence to allow the velocity measurements to be phase averaged, so the forcing

16Vel was set to the smallest value allowed by their amplifier. Velocity measurements were

made with a cross—wire hot-wire anemometer probe that was traversed through the
measurement volume. The probe was rotated in order to measure all three components of
velocity. As the flow is phase-averaged, the entire velocity field at a given phase can be
recovered and the vorticity components calculated. It was found that the phase-averaged
streamwise vorticity levels were 40% of the spanwise vorticity levels. The peak streamwise
Vorticity values were also found to coincide with the location of the tubes of spanwise
Vorticity. The streamwise circulation was found to be 20% of the levels found in the
Spanwise vortical tubes. It is interesting to note that the average circulation of the streamwise
Vortices remained constant as they travel downstream. This study, along with the studies by
Lasheras and Meiburg, concentrated on the central regions of test sections where the effect
of walls and other boundary conditions are minimal. In these cases, it is believed that the
Streamwise vorticity is generated by a local distortion in the spanwise vorticity field which
i s then reoriented into the streamwise direction by the induced flow generated by the vorticity
in the legs of the distortion as suggested by Lasheras and Meiburg. The origin of the initial

kinks in the roller span is believed to be a small disturbance in the flow field.

1.3 Potential mechanisms for increased mixing

Several possible mechanisms for the increase in mixing will be explored in this work.
RObert’s model that the initial spanwise vorticity is reoriented in the streamwise direction
Will result in the development of more surface area for constituent chemicals to mix.
However, MacKinnon and Koochesfahani (1997), have shown that streamwise vorticity field
iS more complicated and cannot be described solely by the reorientation of a single spanwise

VOrtex roller. The recent study of the transition of a plane wall jet by Visbal et al. (1998)

10

may have particular relevance to the current in work. This study has shown that in a forced
wall jet, vortex tubes can split into discrete concentrations of vorticity. The splitting process
was found to begin near the wall and to propagate towards the center of symmetry. The
spiral branches formed are wrapped around the original vortex core twisting in the opposite
direction of the vortex swirl.

As suggested by Lasheras and Mehta, once a spanwise distortion of the spanwise
vortex roller develops, it will generate more distortions resulting in more streamwise
vorticity. Near a wall, as in the present study, a distortion deve10ps as the boundary layer
causes the spanwise vorticity to be reoriented. Additionally, vortex stretching will have a

Significant impact on the streamwise vorticity for the portion of the vorticity near the
sidewall of the facility slows due to the boundary layer. The resulting stretching will result
in the increase in the magnitude of the streamwise vorticity.

A second possible mechanism for the increased mixing can be the existence of an
axial flow within the spanwise vortex cores. In studies of concentrated vortex cores in the

Wake of oscillating airfoils, it has been shown that an axial flow is generated along the vortex
core whenever a vortex encounters a solid boundary (Cohn and Koochesfahani, 1992,
Koochesfahani, 1989). Axial flow along the vortex core can develop if the core passes over

a reg ion where the no-slip condition is applied over a length of the order of the vortex core

di ameter. Cohn and Koochesfahani (1992) have further shown that the no-slip condition is

n 0t necessary for the generation of axial flow. This type of cross-stream fluid transport has
a] so been found within vortices in a forced shear layer and in the wake of a cylinder
(K11 I.()S€I1<a et al., 1988) and within hairpin vortices in boundary layers (Hagen and Kurosaka,

1 99
3 ) r In the forced wake flow currently being studied, the concentrated spanwise vortices

ll

(1.

:19 01

...,.
.035.

 

"" t.
.. \‘PHI"
.5 Jay.“

 

 

»
ill"
a.“

31‘

, F‘l-s

'. hi

pass over a solid wall, so it seems likely that an axial flow might exist within the vortex
cores.

Although it is not known whether core-wise axial flow exists in this flow, strong
cross-stream transport could act as a pump, bringing unmixed fluid into areas where mixing
is taking place, and moving mixed fluid into regions normally occupied by pure fluid. It
should be noted that this mechanism may be acting in concert with the previously mentioned
mechanisms. The study by Visbal et al. (1998) found that in terms of the induced velocity,

the streamwise vorticity arrangement generated by the splitting process found the plane jet
is consistent with axial flow towards the plane of symmetry.

In the research presented, the velocity and vorticity fields in a forced confined wake
are mapped in order to gain a better understanding of the vortex-vortex and vortex-wall
interactions that generate the three-dimensional motions. These interactions are then

examined in order to shed light on the mechanisms which could result in the significant

increases in the amount of mixed fluid found in the forced wake.

l2

Chapter 2

Experimental Procedure

2.1 Mixing layer facility

The experiments were conducted in the gravity driven, liquid mixing layer facility
located in the Turbulent Mixing and Unsteady Aerodynamics Laboratory at Michigan State
University. This is the same facility used in the mixing studies conducted by MacKinnon
and Koochesfahani (1991 and 1997), Nelson (1996), and Koochesfahani and Nelson (1997).
In this facility, shown schematically in Figure 2.1, fluid (water mixed with the chemicals
necessary for the molecular tagging diagnostic technique utilized in these experiments) from
two reservoir tanks is pumped into two constant head reservoirs approximately 3.5 meters

above the test section. From the constant head section, the two separate streams of fluid pass
through their own set of control valves and flow meters.

The flow meters and valves allow the mass flow rate through each side of the facility
to be controlled to within :1 .5% during the course of the experiments. That is, the mass flow
rate Of each of the streams of fluid may vary by a maximum of :1 .5% during the course of

11 1 0f the experimental runs presented. During a typical experimental set, the mass flow rate
Vari ed by less than :0.75%. For the experiments conducted for this work, the mass flow rate
was Set such that the free stream velocities in the two sides were U 1 = U2 = 9.4 cm/s. This

a1 1 . . . . . .
OWS the measurements made ln thls study to be directly compared With the mlxmg results

Dre ‘ .
v 1 OHS 1y mentioned.

13

r, - overhead tanks

 
        
  

     

 

 

 

 

 

 

 

 

upstream flow control valves
H bellows/shaker assembly
° quartz
test section
recirculationL [ll—g] I II' 1 .\d°wn5"°am
line viewport
mervoirtanks ...................
' flow control
valve

 

 

 

 

 

 

 

Fig!" e 2. l : Schematic of the mixing layer facility.

14

“3, no}
54.5 Ll. \\

‘t‘tn'
‘1'” H

After passing through the valves and flow meters, the fluid travels through a flow
management unit consisting of foam, honeycomb, and several layers of perforated plates in
order to eliminate large scale motions. Within the flow management unit, a flat splitter plate
separates the two streams. This splitter plate ends 1.5 cm upstream of the test section and
the two streams are allowed to interact. The origin of the streamwise coordinate, x, is the end
of the splitter plate.

The return section of this facility is unique in that it contains a viewport that allows

the cross-stream flow to be examined. This is accomplished by the presence of a straight
section and turning vanes in the downstream return section of the facility which direct the
flow to either side of the test section before it travels into the dump tank. The downstream
unit contains a Plexiglas window which allows direct viewing of the flow in this plane.

The test section of this facility is 30 cm in the streamwise dimension and has a cross-

section that has a span of 8 cm and transverse dimension of 4 cm. The test section of this
facility is normally constructed of Plexiglas. However, the molecular tagging diagnostics
utilized in these studies require that ultraviolet light be passed into the test section and
Plexiglas absorbs these wavelengths. A new test section made of quartz was designed for
these experiments. It was necessary to glue Aluminum bracing on the edge of the test section
in Order for the section to be able to withstand the pressure of the water. A schematic of the
[es t Section is shown in Figure 2.2. This test section is connected to the flow management
uni t and the return sections by means of plexiglass flanges. Due to these modifications, the

fartheSt upstream location where measurements can be made is approximately x = 3 cm .
This flow facility is designed to run as a blow—down facility when scalar mixing is

b -
elng Studied with the fluid being dumped after passing through the test section. However,

15

 

30cm
. 91

Flow Direction

 

 

 

+

 

 

 

AluminumChanne\ ‘ ,

 

 

 

 

1cmthick ¢4cm

 

 

 

Figure 2.2: Schematic of test section.

for this purely velocimetry study, the facility is operated in a closed 100p mode with the fluid
being recirculated from the dump tank back into the reservoir tanks. This allows the facility

to be continuously operated during the course of the experiments.
Previous visualization studies conducted in this facility determined that the natural
frequency of the shed vortices is approximately 6 Hz when the free-stream velocities are set
to 9.4 cm/s. In the present study, this was measured to be 6.5 Hz. However, measurements
wi l 1 still be made at F = 6 Hz in order to compare with the previous mixing studies. Velocity
measurements made at the splitter plate tip have also found the momentum thickness across

\ the unforced wake, 0 = 0.1 cm. This value is consistent with the values of 0 measured at a

Variety of downstream locations in the current study. Thus, Ree for this flow is about 100.

2-2 Forcing mechanism and conditions

Two-dimensional perturbations are introduced into the upper free—stream of this flow

b
y mean 3 of an oscillating bellows mechanism in the upper supply line. The diameter of the

m

16

Table 2.1: Forcing conditions for forced wake experiments. The forcing amplitudes indicate
the percentage of the free stream velocity of the root mean square fluctuations of the
streamwise velocity. Superscripts 1, 2, and 3 are the cases which correspond to the low,
middle, and high amplitude forcing cases (respectively) of Nelson ( 1996), Koochesfahani and

 

 

 

 

Nelson (1997) and MacKinnon and Koochesfahani (1997).
Frequency 4 Hz 6 Hz ' 8 Hz
2.7 2.11 2.4
l 2.8
‘1 Forcing 4.3
, Amplitudes 5.3
(% RMS) 7.7 7.32 9.8
8.5
“ 11.4 10.2 13.8
i 11.63
13.3

 

 

 

 

 

 

bellows is 5 cm. This mechanism is driven by an electromagnetic shaker whose command

 

< signal originates from a Hewlett-Packard 3314a function generator. With this setup, it is

possible to input an arbitrary forcing perturbation into the flow. However, only the effect of

purely sinusoidal perturbations will be examined in this study. A velocity and a position

transducer are attached to the bellows mechanism to track the actual motion of the bellows.

These signals, along with the command signal, are digitized by a 12 bit Analog to Digital

W (AD) Converter simultaneous with the velocity measurements in order to facilitate phase
aver aging of the measured data.

Table 2.1 shows a list of the forcing conditions used in the study. The majority of the

S tUdieS are conducted at the 6 Hz sinusoidal forcing perturbations, which is very close to the

natural Karman shedding frequency of this flow. A larger frequency (8 Hz) and a smaller

f
re"allency (4 Hz) are also examined, although not in as great detail as the 6 Hz case.

17

 

l A

In order to ascertain the effect of increasing the forcing amplitude, a range of
perturbation levels will be examined. The forcing amplitude is defined by dividing the
measured free stream RMS fluctuations measured in the flow by the mean free stream
velocity. For the 6 Hz case, nine different perturbation amplitudes are studied ranging from

' the unforced ease, up to the maximum fluctuation level being in excess of 10% of the free

 

11 stream speed. It should be noted that due to the size of the test section, no “free stream”
region exists in the 4 Hz case. Therefore, the RMS magnitudes listed for that case are only
used in order to provide a label for these cases. This will be described in more detail in
Chapter 5.

t In order to directly compare the results of the present velocimetry study with the
i mixing results, conducted in the same facility, of Nelson (1996), Koochesfahani and Nelson

( l 997) and MacKinnon and Koochesfahani (1997), several of the forcing conditions were

selected to match those used in these previous mixing studies. This was accomplished by
matching both the input perturbation waveform, as well as the peak-peak amplitude and the
RMS of the measured bellows position. The cases in the present study which match those
1' n the mixing studies are denoted by superscripts l, 2, and 3 in Table 2.1. These will be

referred to as the low, middle, and high amplitude forcing cases.
1 A numerical difference is seen between the quoted values of the perturbation levels
i n the mixing and velocimetry studies. A portion of this difference is due to the random error
i l'll'lel‘ent to the measurement technique which will tend to increase the measured fluctuation
levels in the present study. Visualization results indicated that the structure of the low,
middle and high amplitude mixing results is very similar to the 2.1%, 7.3%, and 11.6%

f -
Oren n g ]evels in the present study.

18

‘

2-3 Velocity measurement method (Molecular Tagging Velocimetry)

Measurements are made using Molecular Tagging Velocimetry (MTV). MTV is a
ful l-field optical diagnostic which allows for the non-intrusive measurement of the velocity

field in a flowing medium. This technique has previously been used by several authors such

 

as Gendrich et al. (1994), Stier (1994), Cohn et al. (1995), Hill and Klewicki (1996),
Koochesfahani et al. (1996), Cohn and Koochesfahani (1997), and Gendrich, Bohl, and
Koochesfahani (1997), Gendrich, Koochesfahani, and Nocera ( 1997) to make measurements

in a wide variety of flows.
The molecular tagging technique takes advantage of molecules having a long-lived

’ excited states which can be viewed after tagging by a photon source. The displacement of

the luminescence of these molecules is tracked over the luminescence lifetime in order to

determine an estimate of the velocity field. The tracer used in the present study is a recently

\ engineered molecular assembly containing l-Bromonapthalene, GB-Cyclodextrin and
C yclohexanol dissolved in water. This combination has a phosphorescent lifetime, 1: E 5 ms

When excited by an ultraviolet laser light source. It should be noted that lifetime is measured

as the time when the intensity drops to e‘1 of the original intensity. Measurements can be

made using time delays much longer than T with the use of proper detectors. Details about

the use of this molecular assembly in MTV can be found in Gendrich, Koochesfahani, and

x N OCCI‘a, 1997. Additional details about the chemical properties of this molecular assembly

can be found in Ponce et al., 1993 and Hartmann et al., 1996.
The concentration of the GB-Cyclodextrin in the flowing solution is driven by two

CO - . . . .
mpetmg factors. The larger the concentration, the higher the intensny of the

19

¥

phosphorescence. However, the penetration of the laser beam decreases because of increased
absorption by the phosphorescent complex. Since the working distance in the mixing layer

facility is relatively small, concentrations of GB-Cyclodextrin between 2 x 10" M and 3 x I 0“1

M were used. The concentration of the alcohol was kept at 0.06 M. The study of Gendrich,
Koochesfahani, and Nocera (1997) shows that there is no increase in intensity for alcohol
concentrations larger that 0.055 M. The additional alcohol used in the present study was to
make certain the intensity was as large as possible. Only a limited amount of
Bromonaphthalene will dissolve in a water solution. Enough Bromonaphthalene was placed
in the solution so that it was saturated and a small amount of undissolved Bromonaphthalene
could be seen. The presence of these compounds in solution has the effect of decreasing the

kinematic viscosity of water by approximately 5%.

""'—"*\_—~—- _ .

This technique can be thought of as the molecular equivalent of Particle Image
Velocimetry (PIV). Rather than tracking particles placed in the flowing medium, the
luminescence lifetime of the tracer molecules is tracked. An abbreviated description of the
molecular tagging technique will be given here. A more complete description the
implementation of the molecular tagging technique and the parameters necessary for an
0Ptimal experiment can be found in Gendrich and Koochesfahani (1996) and Gendrich
(1998)

Two velocity components can be measured in a flow when there is a spatial gradient
in the luminescence intensity field of the tagged region in two (preferably) orthogonal
directions. An intensity field of this type can be generated by crossing a pair of laser beams
in the flowing medium containing luminescent markers. The displacement of the tagged

reglon can then be measured by means of a spatial cross-correlation between two images

20

IIIIIII-.__

 

 

 

acquired at different times within the luminescence lifetime of the marker. In order to gain
sub-pixel accuracy, the cross-correlation field is then fit to a high order polynomial.
Gendrich and Koochesfahani (1996) shows the accuracy of the MTV technique is dependent
upon a number of conditions including the signal to noise ratio in the images, contrast, the
laser line widths and line profile. Based on the quality of the images and conditions in the
Current experiments, it is believed that the 95% confidence interval of the results presented
here is less than 0.1 pixel. This means that 95% of the measurements are more accurate than
0- 1 pixel. Thus, a displacement of 10 pixels will yield a dynamic range of 100. In the
Cul‘rent measurements, 0.] pixels corresponds to 0.15 cm/s and 0.2 cm/s for the for the
Streamwise and cross-stream measurement planes respectively.

In order to measure the velocity field throughout a region, many laser beam crossings
muSt be generated. This can be accomplished by generating a grid of intersecting laser lines
ShOWn in Figure 2.3. Each intersection is a “location” where a velocity measurements is

made. In the experiments reported here, these grids are created by passing a laser sheet

21

 

 

\ Thick Line Thickess:
0.625-O.71 1 mm

 

Spacing:
0.275-0.5 m

t
T

Figure 2.4: Sample beam block used in MTV experiments.

 

Thin Line Thickess:

/ 0.4-0.5 mm

 

 

 

/

 

 

 

 

through a brass beam block as shown in Figure 2.4. The beam blocks used have both thin
and thick slots. The range of dimensions of these slots is shown in the figure. The use of
1 thin and thick lines allows the maximum displacement (and thus the dynamic range) of the
experiment to be increased. The reason is that the correlation technique cannot distinguish
between intersections of lines with the same thickness, it is necessary to limit, the maximum
allowable displacement to one half of the spacing between like intersections. The use of two
line thicknesses allows the maximum displacement to be doubled since an intersection
bfitWeen two thin lines will not correlate well with an intersection between two thick lines.
Using beam blocks does result in the loss of a large percentage of the laser beam
enel’gy. This does not pose a problem in the current work as the intensity levels are sufficient
fol” the detectors used in the present study. Several previous works such as Hilbert and Falco,
1991 and Stier, 1994, have used an arrangement of offset mirrors in order to generate the grid
of intersecting laser lines. This arrangement will cause the loss of a much smaller amount

of the laser beam energy. Beam blocks, however, have the advantage of being very easy to

¥

 

 

manufacture so many different combinations of line spacing may be tried easily. This allows
for the line thickness and spacing to be optimized for the particular experimental setup and
field of view being examined. This optimization allowed between 600 and 800 velocity
vectors to be measured over each individual measurement plane in the current work. This
measurement density is more than 50% larger than previous MTV work that has been
conducted in the Turbulent Mixing and Unsteady Aerodynamics Laboratory. Even with the
loss of energy due to the use of the beam blocks, the energy provided by the laser was more

than sufficient for the experiments.

2.4 Implementation of MTV in the mixing layer facility

M'TV velocity measurements of the wake flow will be made in two planes as shown
in Figure 2.5. The streamwise (x, y) plane allows the measurement of the (u, v) components
of velocity along with the spanwise component of vorticity, (1),, at a particular spanwise (z)
location. The cross-stream (v, 2) plane allows the measurement of the (v, w) velocity
Components along with the streamwise vorticity, (it)x at a particular streamwise (x) location.
In the cross-stream plane, the mean flow is moving directly towards the camera. This work
is the first time this type of measurement has been conducted using MTV. Figure 2.5 also
Shows the Coordinate system used in this work. The origin of the coordinate system is at the
Center Of the test section at the splitter plate tip.

Table 2.2 lists the locations of the streamwise wake measurements. Measurements
are made in the center span (z = 0 cm) for streamwise locations 3 cm < x < 20 cm
downstream of the splitter plate tip for all of the cases listed in Table 1. This is listed as

fun” in the listing of forcing cases in Table 2.2. In order to study the effect of three-

23

~

 

 

 

 

 

 

 

 

 

 

(a) (b)

Figure 2. 5: Sample measurement planes. (a) Streamwise planes. (b) Spanwise (end
view) planes. Note that the origin of the coordinate system is at the center of the test
section at the tip of the splitter plate.

Table 2.2: Streamwise measurement planes. Full and Reduced refer to the number of
measurement cases.

 

 

 

 

 

 

 

 

 

 

 

 

_ Z (cm) Streamwise Measurement Range(cm) Forcing Cases
\ O 3 < x < 20 Full
§ ‘1 .5 3 < x < 20 Reduced
~2 3 < x < 7 Reduced
-2.5 3 < x < 20 Reduced
‘3 3 < x < 7 Reduced

 

 

 

 

 

 

 

-3.5 3 < x < 7 Reduced
, ‘3.7 3 < x < 7 Reduced
l

 

24

 

1A

 

'-~ .
...
nuijh

‘I
4",

I‘M“

Table 2.3: Spanwise measurement planes. Full and Reduced refer to the number of
measurement cases.

 

 

 

 

 

 

 

 

 

 

X (cm) Forcing Cases X (cm) Forcing Cases

3 Reduced 1 1 . Full

3 .25 Full 1 2 Reduced

4.25 Reduced 14 Reduced
5.25 Reduced 15.75 Full

6.5 Full 16.25 Reduced

, 7 Reduced 1 8.5 Reduced
1 8.5 Reduced 20.25 Full

10 Reduced 23.75 Reduced

 

 

 

 

 

 

 

dimensionality on (0,, two additional streamwise planes, 2: = -l.5 cm, -2.5 cm are measured

with a reduced number of forcing conditions. This reduced set consisted only of the 6 Hz,
2.1%. 7.3 %, 11.6%, and 13.3% forcing cases. Additionally, in order to gain more detailed

information on the spanwise development of the interactions between the vortex tubes, the

 

particular range of range 3 cm < x < 7 cm, a total of 7 spanwise locations were measured
with the reduced number of forcing cases.

In the cross-stream (y, 2:) plane, data will be collected in one half of the test section
(‘4 cm < z < 0 cm). Symmetry in the locations of both the instantaneous and mean
Streamwise structures result in the other half of the test section being identical to the portion

measured. In order to determine the downstream evolution of the streamwise vorticity,

 

measurements are made at several downstream locations as shown in Table 2.3. Five of
those Planes will be measured with the full set of forcing conditions and the remaining planes

WI“ be measured using the reduced set described above. In addition, several measurements

25

y

all] .1

"357.1117

37’
zi-

51
m 1'"
L“
l k
.‘l
”19".
ii"; '1"\
”‘31
,“
5
...
‘11-: P.
. n".

 

will also be made in the range -3 cm < z < 1 cm in order to spot check the symmetry

assumption.

gnaw! Experimental configuration

The optical setup used to generate the phosphorescent laser grid for the streamwise
and spanwise measurement planes are shown in Figure 2.6. This arrangement is similar to
that used in Gendrich, Koochesfahani, and Nocera, 1997. The beam from a pulsed
u l traviolet laser (Lambda-Physik LPX 220i) passes through a pair of cylindrical lenses which

are used to thin the rectangular shaped beam generated by the laser into a thin sheet.
Optically, the pair of lenses is equivalent to one cylindrical lens with a much longer focal
length. This sheet is then split into two portions with a nominally 50% transmittance, 50%
I‘eflectance beam splitter. These beams and are then reflected through focusing lenses to
iTlcrease the breadth of the beam, and finally through the brass beam dividers to create a
SE-‘rl‘ies of laser lines. The laser lines pass through the walls of the quartz test section and into
the flow. The beams are carefully aligned so the lines from the two sides are in the same
pl ane and form the necessary intersecting grid pattern. A sample grid experimental grid has
previously been shown in Figure 2.4.
A two-camera measurement system which will be discussed later in this section is
uSed to track the luminescence of the tagged line crossings. Two Pulnix 9701 charge
C()Upled device (CCD) cameras are synchronized using a common synchronization generator.
Although these cameras output their images in RS-170, interlaced format which simplifies

the recording on image processing systems, the two fields are acquired simultaneously.

26

 

 

Top View

      

Pulsed UV Side View

Laser

Beam
Mirror Splitter

  
 
   
    

\Mirror

Focusing
Top Lens
View

Focusing

 

Pulsed UV
Laser

(b)

Figure 2.6: Optical arrangement for MTV measurements. (a) Streamwise plane. (b)

Spanwise plane.

27

'1’,

xi. .

A.» L11

 

Cl

 

These cameras have a resolution of 768 horizontal pixels by 484 vertical pixels. However,
they were digitized to a resolution of 512 pixels horizontally and 512 pixels vertically. The
difference between the 484 vertical pixels on the camera and the 512 pixels recorded by the

digitizer are filled in with zeros. The remainder of the 768 horizontal pixels are not digitized
by the image processing system.

For the streamwise plane measurements, the cameras view the flow through a Nikkor

50 mm f/1.2 lens. With the working distances between the measurement plane and the

camera a field of view of approximately 4 cm x 4 cm to be examined. Using this lens and

the cameras with a 0.5 ms shutter time, it was found that a delay time of At = 5.4653 ms

produced delayed images with a sufficiently high signal to noise ratio to produce high quality
correlati on results. The maximum displacements found in the streamwise plane
measurements was approximately 9 pixels. This corresponds to a velocity of approximately
13.5 cm/s.

The spanwise plane images were recorded from the downstream viewport of the flow
facility- As the distance from the downstream viewport to the image plane was larger than
the distance from the camera to the image plane in the streamwise measurements, a lens with
a larger focal length and consequently smaller aperture was needed. A Nikkor 105 mm focal
length, f/ 1 .8 lens was used. With the working distance in the experimental setup and this
lens, a 4 Cm x 4 cm field of view was examined. In order to maintain high signal to noise

ratio images, the delay time between the cameras was limited to At = 4.00365 ms. The

Shutter titne remained at 0.5 ms. The maximum displacement measured in this plane is

aPPIOXimately 3 pixels, which corresponds to a velocity of about 5.5 cm/s.

28

 

 

. w.
F“
5 iii

 

r.) n7

~.i

ll.)

 

 

(a .,
l\‘\: ,I

 

r, 4.
gin-"w

I.“

'\
(.4
'...’O
'1.

S nchroni 'on 0 ex erimental a aratus

In the two-camera MTV setup, the timing between the components is absolutely
crucial . An error in the delay time between the two cameras will cause a bias error in all of
the vel ocity measurements. It is also possible that improper timing could result in the digital
image processor recording camera fields in the wrong order which could result in correlation
fields being computed using intersections containing information occurring at different times
which would have a significant impact on the measurement accuracy. It is also necessary for

the pulsed laser to fire at the proper time in order for the molecules to be luminescent while
the shutter of the cameras is open. Further, the phase averaging procedure used in these
experiments, which will be described later in this section, require that the bellows control,
position , and velocity signals are recorded at the proper time. Figure 2.7 shows a schematic

of the setup used to synchronize the cameras, the pulsed ultraviolet laser, and the A/D

 

converter.

A single synchronization generator is used to create the timing signals for both of the
cameras. This ensures that the two cameras are recording images that are delayed an exact
amount in relation to each other. The horizontal drive signal for the two cameras is common.
The vertical drive signal is used to delay the cameras relative to each other. A Stanford
Research Systems DDGS35 digital delay generator is used to create the two vertical drives
1301363 With a fixed delay between them. These signals are then input into the cameras. It
Should be noted that when recording images in RS-170 (standard video) format, it is
necessary to delay the vertical drive signal in units of the horizontal drive timing signal

(63-55 113). Ifthe delay is not in units of this timing signal, it is possible to cause the digitizer

29

‘

    

 
     
  

Pulnix
Camera
#1

VD
In

 

HD
in

   

      
 

   

  
   
 
  
 

  

VD
In

 

Pulnix
Camera
#2

HD

  
  

in

To Image

 

 

 

 

Acquisition System

 

 

 

       
   

Synch
Generator

 

    

m

  
     

Trigger Trigger

To AD Trigger

 
  
 

To Laser Trigger

 

 

OUI

SRS D0535
Digital Delay
Generator #1

 

ALIB CUD

    

 

  
     

  
     

 

Delay Generator #1 Settings

Trigger:
Trigger Threshold:
Trigger Slope:

Trigger Termination:

All output loads:

All output voltages:

Delay A:
Delay B:
Delay C:
Delay D:

External
-l.5 V

high Z
high Z
TI'UNormal
T0

TO + 0.57 ms
A+n*63.55 us
C + 0.57 ms

Used for camera vertical drives.

 

AD
Enable
Laser
Enable
Trigger Trigger .
in out A d B C U D
SRS DG535
Digital Delay
Generator #2
Delay Generator #2 Settings
Trigger: External
Trigger Threshold: 1.5 V
Trigger Slope: +
Trigger Termination: high Z
All output loads: high Z
All output voltages: TIT/Normal
Delay A: T0+l6.721 ms
Delay B: A + 2 ms
Delay C: TO + 16.721 ms
Delay D: C + 2 ms

Used to trigger laser at 30 Hz.

Figure 2.7: Timing Schematic for MTV experiments.

30

to reverse the order of the fields in one of the cameras. Delaying in units of the horizontal
drive signal does not have a significant impact on the current experiments as the period of
this signal is small relative to the typical delay times desired in these experiments (3-5 ms).

The second DDG535 delay generator displayed in Figure 2.7 is also triggered by the
vertical drive control signal of the synchronization generator. This unit is set to provide a
30 Hz control signal to fire either a 90 mJ pulse (streamwise plane measurements) or a 110
m] pulse (cross-stream plane measurements) from the Lamda-Physik LPX 220 laser and the
trigger the analog to digital converter. The timing of the second delay generator is set such
that the laser fires and the AD converter triggers just prior to the beginning of the image from

the undelayed camera. Thus, the undelayed camera records a phosphorescent image.

Two camera measurement system

The experiments reported in this work will use the camera measurement system that
has been used in Cohn et al., 1995, Gendrich, Bohl, and Koochesfahani, 1997, and Gendrich,
Koochesfahani, and Nocera, 1997. In this system, the two cameras are placed at a 90 degree
angle from each other. Both are looking directly into a cube beam splitter. In this manner,

bOth of the cameras receive half of the intensity of the grid crossings. Using the two camera
S)fstem, it is extremely important for the relative alignment between the two cameras to be
kflown in order for it to be eliminated from the flow measurements. In the current
e)erriments, the relative displacement between the two cameras is known and steps were
tElken in order to minimize this alignment displacement field.

In order to align the two cameras relative to each other, the undelayed cameras is

31

 

mounted onto a three-axis rotational stage system and the delayed camera is mounted to a
three-axis translational stage system. The initial displacement field between the two cameras
is measured by acquiring images of the laser grid with zero delay between the two cameras.
Since both images are recorded shortly after the laser fires, the signal to noise ratio of the
images is very high and the displacement field between the two images can be measured with
accuracy that exceeds 0.] pixel subpixel accuracy previously quoted for the present
experiments. For the results presented here, 40 images of the undelayed field are measured
and the resulting displacement field is averaged. Since averaging reduces the uncertainty in
a measurement by a factor of (number of points)”, the initial camera displacement field
should be accurate to better than 0.016 pixels.
Before each experimental run is started, the initial displacement field is measured and
adjusted such that the maximum displacement between the grid intersections of the two
cameras is less than 1 pixel. Typically, all of the measurement locations have a displacement
less than 0.9 pixel and over half have a displacement of less than 0.5 pixel. This
measurement is repeated at the end of each experimental run to confirm that the cameras

have not shifted during the course of the measurements.

I‘mage acquisition and processing

One thousand realizations of the velocity field are measured for each measurement
C30ndition. As each realization is separated by 1/30’" of a second, this corresponds to 200
fOrcing cycles of the 6 Hz case. Between 600 and 800 velocity vector measurement are

Ihade in each realization over the 4 cm x 4 cm field of view. This corresponds to a mean

32

SPJQIl

Spain

 

.‘xl; Ii;

"NICE

5? '- -.
talk?)

{V
{Y7

spacing of 1.5 mm between standard resolution velocity measurements. A set of higher
spatial resolution measurements was also made and will be discussed in section 2.6.
Multiple experiments were conducted at the same forcing conditions at different downstream
locations because the total streamwise measurement range is significantly larger than the
field of view that can be measured in a single experimental realization,. These data sets can
be linked together because the phase time reference is known from the forcing signals
recorded by the A/D converter.

The images from the two cameras were recorded by a dual Trapix-Plus acquisition

system manufactured by Recognition Concepts Inc. This system contains two image

processing systems which are controlled by a single imbedded controller. Each image

processing system contains its own disk array that is capable of recording over 9 minutes of

RS-17O video data in digital format. This system is capable of recording up to 10 bits of
video data, however, only 8 bits of resolution were recorded as that is the limit of the
Cameras being used in these experiments. The use of the beam splitter to record the two
images causes one of the images, the undelayed one in these experiments, to be flipped
horizontally during image acquisition. Thus, it was necessary to horizontally flip the
undelayed images within the image processing system in order to orient both images
Properly.

After recording, the images are stored in “tar” format on 8 mm backup tapes using
an Exabyte 8505 tape drive. These tapes can store up to 16 GB of data. The unprocessed
iImages from each forcing condition at each downstream location occupy 0.5 GB of space,
So many forcing conditions were placed on each tape. The data were processed directly

from these tapes. Processing of the experimental data is performed using an in-house code

33

___.__—A -

developed over the past several years. A four-processor Silicon Graphics Origin 200 is used
as the primary data processing device. Each processor of this machine is capable of
processing 40 MTV velocity vectors/second. Thus, each data set takes approximately 5

hours to process.

2.5 Post-processing of MTV data

A block diagram of the procedure used to process the measured velocity data can be
found in Figure 2.8. Once the velocity components are measured using the correlation
procedure and the zero delay reference grids are subtracted as described above, it is necessary
to rescale the data from the original pixel units of the acquired images into physical units.
This was done using the top and bottom edges of the test section as reference points in order
to determine the distance scale factors. The distance between these two edges is 4 cm and

they were visible in all of the recorded images.

After the images are re-scaled into physical units, it is necessary to place the
measured points onto a regular grid. This is necessary so that flow variables such as vorticity
and strain rate can be computed via a finite difference technique. Unlike data acquired using
PIV, the original measurement field is not regularly spaced. Furthermore, the best estimate
for the velocity field acquired by any tracking-based measurement technique is actually at
a location midway between the initial and final location of the feature being tracked. Thus,
Clata from many whole-field techniques (including PIV) must be re-mapped onto a regular
grid. (Spedding and Rignot, 1993).

Several authors including Agui and Jimenez (1987), Spedding and Rignot (1993),

Abrahamson and Lonnes (1995), Fouras and Soria (1998), and Luff, et al. (1999) have

34

 

 

Image Correlation

 

) Subtract Two Camera
Displacement Field

 

 

 

 

 

 

+

Scale into Physical Units

+

Place Velocity Vectors on Regular Grid

i

Compute Finite Difference Derivatives

+

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Phase-Average ‘—— Phase Order

 

 

 

 

 

 

Averaged Resuts

 

 

 

 

1 +

 

Filter Phase Ordered Data Based on
Mean and RMS Phase-Averaged Fields

 

 

 

v

 

 

 

Phase-Average K——— Phase Order

 

 

 

 

+ i

 

 

Filter Phase Ordered Data Based on
Mean and RMS Phase-Averaged Fields

 

 

 

v

 

Final Phase- ‘—

 

 

Phase-Average +—— Phase Order

 

 

 

 

 

+

 

Final mean and ‘
RMS Resuts

 

 

 

 

Compute overall average
and RMS Fields

 

 

Figure 2.8: Block diagram of the MTV processing procedure.

35

examined different methods for the interpolation of velocity measurements and the
computation of vorticity. In the current work, the remapping onto a regular grid of
measurements is made by performing a local least squares fit of the irregularly spaced
velocity measurements to a 2nd order polynomial. A minimum of 9 points are used in the
least square fit. For a 2nd order polynomial, 6 is the minimum number of points needed to
perform the analysis. When the measurements are interpolated onto a regular grid, the
spacing between points on the regular grid, h, is kept at approximately the mean spacing

between points in the irregular grid, 5. Thus, no information on the spatial structure of the

flow is lost, and no new information is unintentionally generated by the interpolation
procedure. The spatial derivatives are then computed using a 2nd order central finite
difference method (4lh order accurate).

Appendix A contains more complete information on the use of a polynomial least-
squares fit to remap data onto a regular grid and on several methods which can be used to
calculate vorticity. These results show that a mean bias error, caused by spatial filtering of
the data, exists in the vorticity measurements, which will cause the peak vorticity values to
be underestimated. The bias error is extremely sensitive to the normalized measurement

density, the ratio of a length scale of the flow, L to 5. Studies have shown that if [/5 is not

sufficient, the peak values of vorticity can be underestimated by over 40%. In the present
study, these effects were examined by making an additional set of measurements at twice the
standard resolution which will be discussed in section 2.6. It should be noted that
simulations show that a small amount of overshoot of vorticity is also found in the range
from between one and two core radii from the center of the vortex.

Extrapolation into regions where there are no measurement points can greatly

36

Points within the region r < R

31'? “56d m the fit- These Three velocity measurements
13011.13 must havea 31331131 must be within the region r <
(1133113111103 covering an arc rmin for a measurement point to
larger than dam be considered valid. These

‘\ points must have a spatial
(bmin distribution covering an are
larger than (pm

¢m rm

R\
./

 

Figure 2.9: Conditions for selecting a regular grid point.

increase the error in the process of placing points onto a regular grid. In order to prevent this,
additional criteria were placed on each regular grid point before a valid velocity vector was
placed at that point. Figure 2.9 shows these conditions graphically. First, all points on the
regular grid are required to have at least 3 actual measurement points within a radius of rm.

In order to optimize the number of valid points, 1.675 < r

m < 25. A factor greater than
unity is necessary to account for variations in the local spacing of points. Furthermore, it was
required that these points were spatially distributed such that at least these three points were
distributed over an arc of at least 90 degrees. In addition, it was required that the points used
in the fit are distributed in an arc of at least 180 degrees around the regular grid point.

The results presented in Appendix A also show that the bias error is also affected by

the size of the region, R, from which velocity measurements are used in the local fit. If the

37

L11

Iii:

1*.

1'
h

region is too small, there will not be sufficient numbers of measurement points to perform
the least squares fit. If the region is too large, the bias error is increased. In order to optimize

the results, the size of this region varied in the range, 2.335 < R< 2.85.

The data rate of MTV velocity measurements is limited by the framing rate of the
cameras. For the cameras used in these experiments, this spacing is approximately 33.3 ms.
The period of the forcing signal is 167 ms. This corresponds to only 5 measurements per
period. This flow, however, is extremely periodic in nature and lends itself very well to
phase averaging. The repeatability of the flow will be shown in Chapters 3 and 4. The phase
averaging is accomplished using the command input signal to the electromagnetic shaker
recorded by the AD converter. This signal is first phase ordered by minimizing the RMS
between measurements that are close in phase after the appropriate reference frequency is
applied to the signal. Zero phase is arbitrarily selected as the positive zero crossing of the
forcing signal. Since each AD measurement of the command signal corresponds to a velocity
field measurement, the MTV velocity information can be phase ordered based on the same
reference frequency. After phase ordering the velocity field information, the data set is
divided into 64 bins with a width of 1.6% of the period. All data that fall within a bin are

averaged to form the velocity field at that phase, 4). The value of (1) will be normalized such
that 0 < 4) < I. If the phase-averaged velocity field is representative of the instantaneous

field at a particular point in time, this procedure yields a set of data in which consecutive
phases can be thought to be separated by 2.6 ms. In Chapters 3 and 4, it will be shown that
the phase-averaged field is representative of the instantaneous results for a large portion of
the collected data.

In addition to increasing the data rate of the measurements, phase averaging also

38

 

Line intersections created when beam blocks are in there initial location.

V created when beam block L is moved from position 1 to 2.
I created when beam block R is moved from position 1 to 2.
0 created when beam block L is moved from position 2 to 1.

Figure 2.10: Movement of beam blocks to increase grid measurement density.

reduces the measurement uncertainty. Since there are, on average, nominally 16
instantaneous realizations in each phase bin, the measurement error will be reduced by a
factor of 4. This corresponds to 95 % confidence interval of 10.038 cm/s for the streamwise
plane (u, 'v velocities in the x, y plane) and 10.05 cm/s for the cross-stream plane (v, w
velocities in the y, 2: plane). A standard uncertainty analysis for a 2"d order finite difference
technique reveals that the uncertainty in the derivative quantities, such as vorticity, is equal

1.34
to —h—5u , where h is the grid spacing used for the differentiation and 5u is uncertainty in

the velocity measurement. Thus, the 95% confidence interval for the phase-averaged spatial
derivatives can be found to be $0.4 s" for the streamwise plane and 120.45 5" for the cross-
stream plane. Ifthe error distribution is Gaussian, it can be shown that the width of the 95%
confidence interval is twice that of the standard deviation. Thus, the standard deviation of

these errors is 10.019 cm/s for the streamwise plane velocities, :0. 025 cm/s for the cross-

39

stream plane velocities. For the spatial derivative quantities, the standard deviations for the
streamwise and cross-stream plane are :02 s" and :0.23 s" respectively.

The phase averaging process is also used as a means of filtering the velocity data of
spurious bad vectors. As the phase average for each velocity point is calculated, the root
mean square of the velocity values in that bin is also computed. All velocity values that are
more than 3 times the RMS value away from the mean are replaced by the mean value for
that location. Then, the averaging process is repeated. For all of the measurements
conducted, no more than 1% of the vectors at any measurement time were replaced. After

the substitution, the phase averaging process is repeated.

2.6 Post-processing of higher density data

Higher spatial density measurements for the 6 Hz, 1 1.6% forcing case were made for
several measurement locations in order to determine the effect of the spatial filtering inherent
in the measurements. These experiments were conducted by making four separate
measurements of this forcing condition and adjusting the location of the beam blocks as
shown in Figure 2.10. This causes individual velocity measurements to be made at slightly
different locations. The displacement of the beam block is approximately 1/2 of the spacing
between the initial measurement locations. During the course of these measurements, the
bellows forcing is not stopped. This technique effectively halves in each direction the mean
spacing between velocity measurements.

To take advantage of the increased measurement density, the procedure used in the
standard density measurements cannot be used. If this procedure were used, the increased

spatial data density information would not be used in the regularization procedure. In order

40

to guarantee that this information is used, the data from all four sets are first phase ordered.
Then, the measurement planes from all four data sets that are within each phase bin are
combined so that they are considered one super—measurement plane. This plane consists of
the approximately16 realizations from each of the four data sets that occur within the phase
bin being examined.

After all of the irregularly spaced velocity measurements at a particular phase are
placed in their appropriate phase bin, the data is then placed on a regular grid using the 2nd
order polynomial described previously. However, a minimum of 120 points are used in the
fit, rather than 9 used in the standard measurements. With the large increase in the number
of measurement points inherent in this technique, the maximum radius from which velocity

measurements are used for the fit, R, is decreased to 2.335 which should further increase the

measurement accuracy. It will be shown in the following chapters that the high density
measurements result in a significantly larger value of the peak vorticity. These increases are

consistent with the increases in peak vorticity levels due to an increase in [/5 as shown in

Appendix A.

41

Chapter 3

Streamwise Measurements of 3 Wake Forced Near its Natural
Frequency

As stated in the introduction, few studies of the effect of forcing amplitude on the
vorticity distribution and other flow characteristics have been conducted in a confined wake
flow. In this chapter, the streamwise flow characteristics of the wake forced at 6 Hz will be
examined and discussed. First, the phase-averaging procedure will be examined and results
will be presented showing the rationale for considering the phase-averaged data to be
equivalent to the “instantaneous” data. The second section will then describe the mean flow
properties at center span and the third section will examine the instantaneous (phase-
averaged) results at this spanwise location. The fourth section will examine the flow
properties of several different spanwise locations, and the final section will compare data
acquired at the standard density measurement results with measurements made at a higher

spatial density in order to determine the effect of spatial filtering on the results.

3.1 Phase-averaging of streamwise measurement plane data

As has been described in Chapter 2, the measured data have been phase-averaged in
order to both increase the effective temporal sampling rate and reduce the measurement error.
Figure 3.1 shows the phase ordered (u, v) velocities in the free-stream region of the flow at

two representative downstream locations in the flow for the 11.6% forcing case. Each plot

42

 

 

 

 

 

Figure 3.1: Phase ordered free-stream velocity fields for two downstream locations in the
forced wake. (a) x = 5 cm. (b) x = 18 cm. The small undulation in the v-component of
velocity at x = 18 cm is the signature of the vortex cores passing this downstream location
at the y location of the measurement.

is composed of approximately 1000 measurement points. However, in order to make the
plots more readable, only every 10‘h point is shown. Figure 3.1a shows a measurement
station 5 cm downstream of the splitter plate tip while Figure 3. lb shows a measurement
station 18 cm downstream of the splitter plate. The u-component of velocity in the upstream
example (Figure 3.1a) has a sinusoidal appearance, which is as expected for the free-stream
region of this flow forced with a sinusoidal waveform. The v-component shows very small
velocities. The collapse of both velocity fields to the expected instantaneous values at these
locations is excellent. The downstream example shows similar features as the upstream
example. In fact, the collapse of the u-component of velocity to the expected velocity at this
location appears to be even better than that of the upstream example. The small undulation
in the v-component of velocity is the signature of the vortex cores passing this downstream

location at the y location of the measurement.

43

 

(”1(8 ) 25-20-1540 -5 0 5 10 15 20 25

 

 

 

 

 

IIJAAIJAAAIA AllAlll'

 

I A I I I J I l A I l I l A A I
4 5 6 7 16 17 1
(a) x (cm) (b) x (cm?

Figure 3.2: Phase-averaged vorticity field (flooded contour) along with vorticity contour lines
of 6 instantaneous realizations overlaid on top. Dashed contour lines indicate negative values
of vorticity and contour levels are 21:4 s", 1-8 s", ..., 1-24 3". (a) 3 < x < 7 cm. (b) 15.75 < x
< 19.75 cm.

The collapse of the instantaneous realizations applies as well to the vorticity field.
Figure 3.2 shows flooded contour plots of the phase-averaged vorticity field at one
representative phase for the region 3 < x < 7 cm and 15. 75 < x < 19. 75 cm. Overlaid on top
of the flooded contours are vorticity contour lines of several of the instantaneous realizations
used in computing the phase-averaged vorticity. In Figure 3.2a, containing the upstream
results, the spatial overlap between the instantaneous and phase-averaged is nearly perfect
in the central region of the test section. On the bottom and top of the section, there are some
variations due to the top and bottom wall boundary layers. Figure 3.2b shows the
instantaneous vorticity contour lines overlaid on top of the phase-averaged field for the
farthest downstream regions examined in this study. It can be seen that the instantaneous

vorticity field does not line up with the phase-averaged field as well as the upstream

example. This is caused by a small amount spatial wandering of the vortex core from
realization to realization. It is also noted that the spatial distribution of the vorticity field
(i.e., the elongated shape) is very similar in all of the realizations.

Two quantities will be used to quantify the numerical differences between the

vorticity values among the various realizations within each phase bin; namely (<mz>mrs)pc'uk

and (<(oz>m)m will be used for this purpose. The former is defined as the RMS among the
vorticity values in the instantaneous realizations at the location of peak vorticity, which is
typically the center of the vortex core in this study. The latter is defined as the peak value
of the RMS among the vorticity values in the instantaneous realizations at any point in the
measurement field. These same two quantities will be used in Chapter 4 to examine the
numerical difference between the realizations for the streamwise vorticity measurements.
For the 11.6% forcing case that is shown, the maximum value of the spanwise

vorticity is 40 3". Within the phase bin shown, («1) )pwk < I s". This value is

z>rms

approximately equal to the uncertainty in the vorticity measurement caused by the random

error inherent to the measurement technique. Examining all of the phase bins, («02>

"715)!er

= 2.5 s". This is likely due to the small amount of spatial drift in the location of the vortex
cores which was previously mentioned. This can result in large deviations in regions of high
gradient. Thus, it is felt that the phase-averaged field is an accurate representation of the
instantaneous result.

In this downstream region, the peak vorticity levels have decreased to approximately
10 3". However, the values of both ( ((1)2) )IM and (<(oz>

remain nearly identical

"713 "713 )max

to the upstream case with values equal to 1 s"and 2.5 s‘1 respectively. Again, the spatial

location where (<(oz>,,m.)m is near the edges of the vortical structures. The variation from

45

realization to realization, however, is more noticeable since the peak vorticity values have
decreased by a factor of four. Even with the increased spatial wandering, it is believed that
the phase-averaged results provide a good estimate of both the spatial location and numerical
value of the instantaneous field measurements. This allows the phase-averaged results to be

used as if they are the instantaneous results.

3.2 Mean and RMS flow properties at center span

Mean free stream velocity

Flow characterization experiments were conducted in the center span of the test
section in order to determine the basic streamwise structure of the flow. First, the mean
quantities of this flow will be examined. Figure 3.3 shows a plot of the mean streamwise
velocity of this flow versus y location for the unforced case as well as for several of the 6 Hz
forcing cases at x = 4, I I , and 17 cm. The free-stream velocity for all cases is approximately
9.4 cm/s. In general, at upstream locations, the effect of forcing is to reduce the wake deficit,
as measured by the difference between the free stream velocity and the smallest velocity in
the wake. At x = 4 cm, the wake velocity deficit for the unforced case is more than twice the
deficit of the forced cases. However, at locations farther downstream, this is not necessarily
the case. At x = I 1 cm, the magnitude of the deficit for the unforced case is approximately
the same as the forced cases, and at x = I 7 cm, the forced deficit is larger than the unforced
deficit. The 13.3% amplitude forcing case is interesting because for all downstream
locations, the deficit appears to remain nearly constant. The value of the deficit is also much

smaller than the deficit of any of the other forcing cases.

46

2 an)

 

   

 

 

D Untamed 1.5 mm OM19“,
A f=6Hz,A=2.1% ,
<> f=6Hz,A=7.3% A 05
+ f=6HZ,A=11.6% ‘E, '0 [5]
O f: 6 Hz, A: 13.3% V
> -o.5
-1
-1.5 0 W O,“
'20 é ‘37 i é ‘ ‘ ‘
x = 4 cm H (cm/8)
2 m) 00 2 one
1.5 1.5
1 1
A 0.5 A 0.5
5 o E o
>no.5 >~ -o.5
-1 -1
- _ -1.5
1 5 A_L LLI A a n #1 + a 4 42m; fQ Ami94 A 4 l J_L r . I A A A I l A a . x l x A n A I . . L I
‘20 2 4 6 a 10 ‘20 2 4 6 8 10
x =11 cm 0 (cm/8) x :17 cm H (cm/8)

Figure 3.3: Streamwise velocity profiles at z = 0 cm for several forcing amplitudes at 3
streamwise locations.

Momentary Thickness

The momentum thickness across the layer is also used to quantify the total

momentum deficit in the flow. The momentum thickness is computed according to:

9m: 1. m(,_m,d

akc u0(x) u0(x)
The value of no is chosen to be the average of the free stream velocities in the upper and
lower stream at the x location where 9 is computed. The value of the free stream velocity of
the two streams typically differ by less than 2%. As defined, 9 will almost always be positive

for wake type flows, however negative values of 9 have been noted in the wake of an

oscillating airfoil by Koochesfahani (1989). The momentum thickness can be thought of as

47

 

 

 

 

0.3:- g x=4cm 03: ' Unforced
» A x=7cm - Cl 6H2,2.1%Forcing
:v x=11 cm :A 6Hz,7.3% Forcing
0.25 -<> x-14 cm 025— v 6Hz,11.6% Forcing
:O x=17cm ; O 611213.396 Forclng
02} 02:
. o .
A __ O A .-
5 015*- g 015-
V . v t
G " Q _
01- 9 O O O O o O 01
3i go 5% g ‘22 ' :
- D V r
. D a A v -
0.05: D A o 0.05:
* 0 § :
. L
i . i r 1 a . i i l . r J3 r I i i
00 5 1o 15 0o
(a) forcing (% RMS) (b)

 

Figure 3.4: Momentum thickness as a function of (a) forcing amplitude and (b) downstream
location.

one component of the drag on a hypothetical flat plate upstream of the region being
measured. A second important term will be related to the non-uniform pressure distribution
generated by the lower pressure present in the cores of the streamwise vortices. Since the
drag on this hypothetical plate must be constant, an increase in the momentum thickness at
one spanwise location must be accompanied either by a decrease at others or by changes in

the pressure conditions. Note that negative values of 0 have been previously seen in studies

of the wake of oscillating airfoils (Koochesfahani, 1989) where they were inferred to be
related to a thrusting condition on the airfoil.

Figure 3.4a shows a plot of the effect of forcing amplitude on the momentum
thickness at several different downstream locations. For the farthest upstream location (x =
4 cm), the momentum thickness generally increases as forcing amplitude is increased. The
13.3% forcing amplitude is an exception. As noted in the mean profiles, the wake deficit for

the 13.3% amplitude is very small at all downstream locations, which results in a small value

48

for 6. For locations farther downstream, such as x > 14 cm, this trend reverses and increases
in the forcing amplitude result in a decrease in 9.

Figure 3.4b shows the behavior of the momentum thickness for the unforced and
three forcing cases with respect to downstream location in more detail. For the unforced

case and the 2.1% and 7.3% forcing amplitudes, 9 remains relatively constant with

downstream location and has a value that is approximately 0.] cm. The small decrease in the
unforced case is likely the result of the slight increase in free stream velocity caused by the
growth of the sidewall boundary layers. For the 1 1.6% forcing case, the momentum
thickness increases dramatically with downstream distance. Initially, its value is 0.03 cm.
By I 7 cm downstream, 6 has increased to nearly 0.2 cm. Since the momentum thickness is
proportional to the drag of the plate, which for each forcing case is constant in the mean, the

value of 6 must either decrease at other spanwise locations or the pressure conditions must

change in order for the drag to remain constant. In section 3.4, it will be shown that the
former condition is at least partially satisfied. It is also likely that the latter condition will
hold as well since the size of the vortex cores is expanding, which will decrease the absolute
value of the pressure within the core.

The 13.3% forcing case maintains a value that is always less than the momentum
thickness for all other cases. In the upstream region, this value is almost zero. It increases
slightly as the downstream distance increases, however, by I 7 cm downstream, it is still less

than the 0.] value found in the unforced flow.

49

Veloc ’ RM elds

Figure 3.5 shows a contour plot of the root mean square of um and vm, for the
unforced and the l 1.6% forcing amplitude cases. It is interesting to note that in the forced
case shown in Figure 3.5b, the um, field is neither uniform nor monotonically decreasing in
the streamwise direction. Rather, there is a distinct periodicity between regions of large and
small RMS in both the streamwise (x) and the cross-stream (y) direction. This periodicity
is seen in the urms for all of the forced cases, however, decreased perturbation amplitude
decreases the magnitude of the effect. This periodicity is not seen in the um field of the
unforced case in Figure 3.5a. At the same transverse (y) location, the streamwise spacing
between locations of maximum (or minimum) values of um, is the wavelength of the forcing
perturbation. Locations of local minima in the RMS field on the one side of the layer are
vertically aligned with regions of local maxima of the RMS field on the opposite side. This
pattern is very different from what is seen in unconfined forced wake flows, such as in the
wake of an oscillating airfoil studied by Koochesfahani (1989).

The vm field for the unforced and l 1.6% forcing cases is shown in Figure 3.5c and
3.5d. In the unforced case, no 12",”. is seen in the region x < 8 cm. At locations farther
downstream, a weak region of vmu. is found in the center of the test section. For the forced
case, the strongest vmu is seen in the center part of the test section in the region x < 10 cm.
Examining locations closer the top and bottom walls of the test section causes a decrease in
vm. Farther downstream, vm weakens and begins to spread towards the sidewalls. This

pattern is created by an increase in the vertical spacing between the regions of <wz> which

will be shown in section 3.3.

50

v (cm)

11 (cm)

V (cm)

V (cm)

0 0.25 0.5 0.75 1 1.25 1.5

 

 

 

‘7f5““1o“"12.5”“15' ' “171.5“ 20

m»-

‘25 ‘ U

o.

 

 

 

l
t

“50

(b)

 

1 1.25 1.5

 

 

 

6 “2.5“ ‘5 “75 ‘ ‘ ‘10‘ ‘ 7121.5 ‘15“ 171.5“ ‘20

 

 

 

 

 

‘75” 1o““12'.5‘ ‘15““17‘.5““2o

O
9’
0'1
(J11-

Figure 3.5: RMS of streamwise (u) and spanwise (v) velocity for the unforced and 11.6%
forcing amplitude cases. (a) um for unforced cases. (b) am for 1 1.6% forcing case. (c) vmu.

for unforced case. ((1) vm for 1 1.6% forcing case. Contour levels are 0.3 cm/s, 0.6 cm/s,
1.5 cm/s.

51

This pattern in the um, field can be explained by the phase difference between the
forcing and the resulting vortex shedding response. In the model, shown pictorially in Figure

3.6, the vortices are assumed to be inviscid point vortices and it is assumed that the free
stream velocity can be described by U .1- = no + Cu0c0s(21rft)iwhere no is the mean velocity,
f is the frequency of the perturbation, C is the forcing level and U s is the instantaneous free
stream velocity. It is further assumed that vortices are shed at a distinct phase, 0 < (l) < 1,
within the forcing cycle and convect downstream at nearly a constant velocity. It should be
noted that it is not necessary for the convection speed to be constant for this model. For the
following argument, a negatively signed vortex is shed at a phase of ft = d) and a positively
signed vortex is shed 1/2 cycle later at a phase of ft = q) + 0. 5. Only the effect of the induced

velocity due to the closest vortical structure will be considered and the effect of the image
vortices in the top and bottom walls are ignored in this model. In the explanation of the
model, it is assumed that the vortex convection speed is constant. This requirement is not
necessary and is only used to simplify the explanation.

A negatively signed vortex will pass the streamwise location of x = n?» - (1))», where
it is the forcing wavelength and n is any integer, at a phase time of ft = 1]. At this phase, the
flow free stream velocity is a maximum, U s = uo + Cuo. Similarly, when the positively
signed vortex passes this point, the phase time is ft = (n + 1/2) and the free stream velocity
is at its minimum value, U s = uo - Qua. Examining point A from Figure 3.6, which is located
above the vortex street, it can be found that when the negatively signed vortex passes this

point, the velocity at this point is u“, = U], + Uhmcd = 14,, + Cuo + 172m, where U -,,,,m.,‘, =

I

F/Zrtr, is the induced velocity due to the vortex, F is the circulation of the vortex, and r is

52

j‘qu-fA/Zrt4>“/A
—— O Q m 0

1:05 —-——— m 63m 0

Point A: Peak-Peak Variation:
At t = O; 11A.0 = Ufs + Uinduced = uO + (no + I‘l21tr 2(Cu0 + P/21tr)
At t = 0.5; um”, = U,s + Uinduwd = 110 - Cuo - I‘l21tr

Point B: Peak-Peak Variation:
At t = 0; “3‘0 = Ufs + Uinduced = 110 + C110 ' 1721'” 2((“0 47/27”)

At t = 0.5; “5.0.5 = Ufs + Uinduced = u0 - Cuo + I‘l2nr

Figure 3.6: Effect of the phase relationship between flow forcing and vortex shedding on the
streamwise velocity fluctuations in the forced wake.

53

the distance between the vortex and the point A. Similarly, when the positively signed

vortex passes 1/2 a cycle later, the velocity at this point is um; = Ufl. + U -,,,,W,,, = no - Qua -

l

l"/21tr. Thus, at point A, the velocity will vary in the peak to peak range of 2( C140 + 172m).
Examining point B located below the vortex street, when the negatively signed vortex

passes, the velocity is “3.0 = Ufl. + UWWJ = no + Cuo- F/Zrtr, whereas when the positively
signed vortex passes, the velocity is 11305 = U], + U,,,,,m.,,d = uo - Qua + 172m. Thus, the peak
to peak velocity variation at point B is 2( C110 — F/21tr), which is less than the peak to peak

variation at point A. Consequently, the RMS value at point B will be less than that at point
A. Similarly, at a point 71/2 either upstream or downstream, the situation will be reversed and

the “large” deviation will be below the vortex array while the “small” deviation will be above
the array.

Examining the um. data from the various forcing amplitudes, it is apparent that the
locations of the minima occur at different spatial locations for the different forcing
amplitudes. This location is an indication of an amplitude dependance in the phase when the
spanwise vortex cores shed from the splitter plate tip. From this spatial information, the
difference in the phases at which shedding occurs can be determined. This is shown in
Figure 3.7. Increases in the forcing amplitude cause the shedding to be delayed to a later
phase. Between the highest and lowest forcing amplitude, the phase difference is
approximately V2 of the forcing cycle. This dependence of amplitude on the phase at which
vorticity is shed highlights the non-linear nature of the process by which the vorticity align
into a vortex street.

For the 6 Hz forcing frequency, the spatial periodicity does not effect the
measurement of the free stream RMS values. As seen in Figure 3.6, on the top and bottom

54

0.5 »-
0.4 L
1

Q3-

4‘1)

02-

Q1~

 

 

 

l l 1 I l l I l l l 1 l
00 5 10 15

forcing (%RMS)

Figure 3.7: Dependence of phase time at which shedding occurs on forcing amplitude.

portions of the test section, a region of uniform um. can be seen. Figure 3.8 shows a line plot
of the um, profile for several forcing amplitudes. From these data, the value of am can be

determined and compared with no.

Reynolds Stress

Figure 3.9 shows the Reynolds stress in the unforced wake as well as the wake forced
with 4 different forcing amplitudes. The grey scale level in the flooded contour plot is used
to signify the intensity of the variable being examined. Overlaid on top of the flooded
contours are coarsely spaced contour lines to differentiate between the positive and negative
signs of the variable. The dashed lines indicate the negatively signed data. This format will
be used in many of the successive plots. In the unforced and the 2.1% forcing cases, the
value of Reynolds stress is small. However, the higher forcing amplitudes show a spatial

periodicity in their value. For the 1 1.6% and 13.3% forcing cases, the positive and negative

55

N
I

 

:13 o O F=6Hz
:51 0° 0 a A=2.1%
153?, 8 g o A=7.3%
f5] (9 o o A=11.6%
1-0 0 0
:DO 0
Zoom O
059 a 8
A :00 O
. a o o
E .0 oo
o 0- a 5 o
>‘ I 1:1 0 O
0.5- a o o
: 0 oo
. a 50
-1'-0D 00
'D O
:1: o
.D
1.5-u g
'1]
:e
n>.i..li.iil.i l llllllll|l
‘ 0.5 2.5

 

1 "1.5' ' ' '2'
um (cm/s)
Figure 3.8: Line plot of u,,m for three forcing amplitudes.
signs of Reynolds stress are nearly aligned vertically. However, the 7.3% forcing shows an
offset in the vertical spacing. This is quite similar to the patterns seen in the vortex spacing

that will be shown in Section 3.3.

Mean Vorticifl Field

Using the MTV technique, it is also possible to examine other quantities based on the
ISI (and higher) spatial derivatives. These quantities, such as vorticity and strain are very
useful in determining the nature of the flow field. The accuracy of the vorticity calculation
has been discussed in Appendix A.

Figure 3.10 shows the mean vorticity field for an unforced wake and for the wake
forced with four different forcing amplitudes. It is interesting to note that magnitude of the
mean vorticity field is larger in the unforced case than in the forced case. This is the result

of the vortex alignment in the forced cases which causes the positively and negatively signed

56

LI:
Unforced

Reynolds Stress: -3-2-1 0 1 2
2

-2

f=6Hz,A=2.1°/o

E1
0 0
V
>‘-1
-2

F=6Hz,A=7.3°/e

 

V (cm)

I
u-L

   

 

 

 

 

 

“ .1, 1, \1/ \'TS\’/> I\\/."ii ‘.\
iv‘ ' ‘ I.“

' .1- 1 ‘1 1 1'
\’ ‘4 “a, ‘ ’\_/ '

 

 

o.
1"
0'1

(,1_

‘715"“10““1é.5““15 "17‘5”“

Figure 3.9: Reynolds stress distribution in unforced and forced wake. The contour lines
indicate the 320.6, :tI.2,...,:1:3.0 levels.

57

 

.548")

45-105 0 5 1o 15

 

 

 

 

 

,
.
.
-2»-

 

2:

p

 

 

-zt

 

 

 

 

 

 

 

 

 

2 r
5 ‘3"
o 0:-
V .
>. _1 :_
.2i
0 g‘ ‘25‘ 5 ‘75 ‘ ‘10 12.5 15 17.5 20
x(cm)

Figure 3.10: Mean vorticity field for unforced wake and wake forced at 6 Hz. Contour lines
are spaced at :3 s", :6 s", :15 s". Dashes indicate negative contours.

58

 

 

0.05 -
_ . -<mz>
+<wz>
I
‘3' 1.":
0 025 - 2: ..° .
no . ° -
‘< - 0 0 ‘?o ’ % q, t
A _ a ‘0 .' '11. ~. 0
l " o . .:° 9 °'0
j 0 - q, . a .0 ° ° 0
~ - s " .‘z'
I " 3' Q q, .. . . .
5 » ‘L‘. g "’1‘”
h n a", 3 °
p.025 L '13:
l- . .g
+_ I
O 050 1 ¢ 2 3

Figure 3.11: Streamwise location of phase-averaged vortex cores versus phase. A linear fit
has been subtracted from the locations.

vorticity to cancel out in the mean. The vortex spacing parameter will be discussed in more
detail in the following section. Between 2.1% and 7.3% forcing, the mean vorticity increases
slightly, however it decreases again before the 1 1.6% forcing case. In the highest amplitude
case (13.3%) the positively and negatively signed vortices are nearly perfectly aligned
resulting in near zero mean vorticity throughout the measurement range.

In the 7.3% and 11.6% forcing case, compact regions of (3; are apparent. These

regions are generated because the convection speed of the vortices is not constant. This was
determined by plotting the x location of the positively (and negatively signed) vortex cores
over several forcing cycles. A linear fit of the x location with respect to phase is then
performed. The value of the linear fit is then subtracted from the actual location at each
phase. The result of this process is shown in Figure 3.11. Note the oscillatory appearance
which indicates that the mean convection speed is not constant. The peak difference amounts

to about 2.5% of the mean vortex convection speed. The compact regions of (T)Z are also

59

seen in the 13.3% forcing amplitude. However, the near perfect alignment between the

positively and negatively signed (7); tends to reduce the impact of the non-constant

convection speed.

Effect of forcing on 93%

Using the incompressible continuity equation, it is also possible to derive the
spanwise derivative of the cross-stream velocity, that is, i“— . This term is important in the
vorticity transport equation in the term for the stretching of the spanwise component of
vorticity. A negative value indicates that the magnitude of the spanwise vorticity will be
decreasing. As seen in Figure 3.12, the mean value of this quantity, % is very small for the
unforced and low amplitude forcing cases. Even though the magnitude is quite small, its

effect on (Dz is significant as will be discussed later. However, for the larger forcing

amplitudes, negative values of % can be seen in a wedge shaped region in the center of the
facility. On the top and bottom surfaces of the facility, % is positive. The sign of this
quantity is consistent with a mean recirculatory flow in the cross-stream plane, as seen in
Figure 3.13, which transports fluid from the sidewalls of the facility towards the center, and
then back out towards the sidewalls in the region near the top and bottom walls of the
facility. In Chapter 4, the cross-stream plane will be examined and it will be shown that this
pattern is seen in this flow field.

It is also interesting to note that the increase in {a corresponds quite closely with the
location where the amount of mixing product was found to increase in the studies by

MacKinnon, Koochesfahani, and Nelson. Figure 3.14 shows a plot of g for the 11.6%

60

  

.1 v.’
I

32(3 )-3-2-10 1 2" 3

 

 

2 .-

 

.
.
p
-2 -

-2
F=6
,2?

 

-zi
F=6
2:

 

Figure 3

 

 

 

 

 
 
 
 
 

 

Hz, A =11.6°/o

 

 

 

Hz, A = 13.3%

 

o-
N
a:
ml-
'4
o:
—l
o
A.
[0.
o:
—L
a:
d
V
or
N
o

.12: Mean %- for the unforced wake and four perturbation amplitudes. The solid

and dashed contours separate regions of positive and negative value.

61

 

 

 

 

Figure 3.13: Example of recirculatory flow pattern.

forcing case overlaid on top of the Z—P curve from the high amplitude forcing case
i

Koochesfahani and Nelson, 1997. The increase ingll slightly precedes in streamwise
location the increase in 3:511. The value of? even exhibits a slight decrease just before the
l

Z—"begins to level out. Note that general presence of g2.- alone does not generate mixing
l

enhancement. Rather, it is indicative of other characteristics which generate the mixing

 

 

enhancement.
0 37,,
o ”1,, (From Koochesfahani and Nelson 1997)
0.5 ' D D -1 .75
Z Ct)
” rm [:1 -1.5
0.4 - afiffimfi or»
* ED 5315 4.25
*‘ [I] D
- $3“ @@ -1
0.3 :' D U .
"2a _ DD ‘3; fl " o7s|,\
0.2 L- C] C1? '
5: _ U '05
l at “ "
D- D E .
~ :13? do -o.25
0.1 '- (8 c]
.. D 0
"titlinll.il..li4I1111.1111..
o 5 " 10 15 20 25 30025

— 5
Figure 3.14: Comparison of 95%— in 11.6% forcing case and a—Tfrom Koochesfahani and
Nelson, 1997.

62

3.3 Phase-averaged (“instantaneous”) flow preperties at center span

At this point, it is useful to examine the properties of the phase-averaged <C°z> field.
In section 3.1, the phase-averaged field was found to very closely match the instantaneous

realization. Figure 3.15 shows an instantaneous (Dz field for the unforced case and a
representative phase bin of the <wz> field for 5 forcing cases. In the unforced case, it is

difficult to discern organized vortical structures in the far upstream region. Structures begin
to become apparent at approximately z = 6 cm. However, the horizontal spacing between
the structures is evolving until approximately 10 cm downstream. By x = I 7 cm, the
magnitude of vorticity within the spanwise rollers has decreased to a very small magnitude.

When forcing is applied, identifiable vortex core structures are apparent at the earliest
downstream location measured. With increasing forcing amplitude, it is apparent that the
strength of the vortical structures is increased in upstream locations. However, at locations

farther downstream, the peak values of <(oz> are lower in the high amplitude forcing cases

than they are in the low amplitude forcing cases.

These effects of perturbation amplitude on the peak magnitude of the spanwise
vorticity field are shown more clearly in Figure 3.16. Figure 3.16a shows the effect of
forcing amplitude on peak vorticity levels for several different downstream locations. At the
farthest upstream location, x = 3.5 cm (shown by the squares), as forcing amplitude
increases, the peak vorticity also increase from about 15 s" to nearly 40 s". In the range from
unforced to about 6% forcing, the rate of increase is relatively small. For forcing amplitudes
larger than 6%, the peak vorticity levels increase rather quickly. As the distance from the

splitter plate increases, the trend of peak <wz> increasing with forcing amplitude

63

 

 

 

 

 

 

 

 

 

 

 

p
p
p
-2 _

 

 

 

 

 

 

Figure 3.15: Vorticity field in forced and unforced wake. The unforced wake data are
instantaneous realizations and the forced wake data are phase-averaged results. Contour
lines are spaced at :I-5 s", :10 s", :25 s". Dashes indicate negative contours.

64

Th

ex;

 

 

 

40 1:! 3.75 cm ~11), 40 F D 2.1% forcing «n,
A 6.75 cm «1), D L ‘ A 7.3% forcing «o,
0 10.25 cm «1, ' » o 11.6% forcing «o,
35 0 13.75 cm «1), D - (é - 2.1% forclng +1.),
l> 16.5 cm «1), . ' ~ A 7.3% forcing +11),
30 I 3.75 cm +11), A 30 _ O 11.6% forcing +11)l
A 6.75 cm +11)' 0 A ~
0 10.25 cm +11)z I ‘ ‘ t a
p 25 O 13.75 cm M)' g A g“ L
'0) p 16.5 cm +0), . 3 _ g
V A L
A. 2° 0 a g 8 3 O O ’3‘" 20 L A %
‘3, 15 I g l I 8 . V t a i
60 9 9 8 . O O O “
9 D F
10 g? , i p . 5 ’ 9 10 r a
5 ’ b i
O 1 1 1 1 l 1 1 1 1 I 1 4 1 4 J 0 1 1 1 1 l 1 1 1 1 I 1 14 1 l 1 1 1 1 I
0 5 10 15 0 5 10 1 5 20
(a) forcing (%rms) (b) x (cm)

Figure 3.16: Effect of forcing on peak vorticity magnitude. (a) Effect of forcing on vorticity
magnitude at several downstream locations. (b) Variation of vorticity magnitude for 3
forcing amplitudes with respect to streamwise location.

changes. At a downstream location of 10.25 cm (diamonds), there is little change in the
magnitude of the peak vorticity with increased forcing. At x = 13.75 cm (circles) and x =
16.5 cm (right pointed triangle), the magnitude of the vorticity peak decreases slightly as
forcing amplitude is increase.

According to the results presented in Appendix A, the spatial filtering resulting from
the spatially under-resolved velocity measurements in the data set will result in the peak

vorticity levels being underestimated. The current data set has a U5 (feature size measured

by the vortex core radius to mean data spacing) ratio of approximately 2.5. This will result
in the peak vorticity values being underestimated by about 18%. However, this does not
affect the reported trends. Only the magnitudes of the peaks would be increased by 18%.
This results in an actual peak vorticity level of approximately 47 s". This will be further

examined in section 3.5.

65

Figure 3.16b shows the peak vorticity value as the vortices convect downstream for
three representative forcing amplitudes. It is apparent that the vorticity decreases much faster
for the higher amplitude forcing cases than for the other cases. Although the vorticity
magnitude of the 11.6% forcing case is initially over double that of the 2.1% forcing case,
by 15 cm downstream, the magnitude of the vorticity for the higher amplitude forcing is less
than that of the lower amplitude forcing.

One might think that the decrease in <(oz> is due to viscous diffusion. However, it
will be shown that the rate of decay of <(1)z> due to diffusion is much too slow to describe

the decrease. Rather, the decay in the peak vorticity level is best described by negative
vortex stretching, which is connected to 957‘: . The vorticity transport equation will be used

to assess the drop in the peak value of <mz> from viscous diffusion as well as stretching. The
vorticity transport equation which describes the change in phase-averaged spanwise vorticity
is

D<a)z> 8w aw 8w 32
-—-—-=((o —+o) —+co —)+vé—,<mz>.
Z

Recall that it has been previously shown that the <(1)z> is an excellent representation of (1)z
and is used in place of the instantaneous value. Assuming constant vortex convection speed,
and assuming that at least initially, <0)? and <co)> are small, the above equation takes the

following form.

8x —u0 ' 82 822 z '

Note that the two terms describing vorticity reorientation have been eliminated and the only

remaining terms are related to diffusion and stretching. Further note that the free stream

66

velocity uo is used for the vortex convection speed. The vortex convection speed is actually
slightly smaller than the free stream velocity.

Figure 3.17 compares the decay of the peak vorticity observed in the 2.1% and 1 l .6%
forcing cases with the decay that would be observed due. solely to stretching and solely to
diffusion. For both the stretching and the diffusion results, the vortex strength and core
radius at the initial location is used as a boundary condition. For computing the stretching

results for both forcing amplitudes, 35:; is set to a constant value. For the 2.1% forcing case,

8w
‘3? = -0.35 s" and for the 11.6% forcing case -a—“’— = -l s". These values are reasonable

81
estimates of the actual value of 132‘”— found in the two forcing cases. However, these small
values are near the detection threshold of the current experiments. The value of gall also
tends to increase with the downstream development of the structures. This was not modeled.

The decay of the peak vorticity for the low amplitude forcing case is shown in Figure

3.17a. Within the measurement range, the measured peak <(1)z> decays approximately 45%.

HoWever, the decay estimated solely to diffusion (solid line) is only approximately 5 %. The
estimated decay of peak vorticity caused by the small amount of vortex stretching agrees
quite well with the decay seen in the actual results. This agreement is much more apparent
in the 11.6% forcing case examined in Figure 3.15b. The dashed line indicating the decay
due solely to the stretching parameter agrees quite well with the actual decay whereas the
solid line indicating the decay due to diffusion does not yield the decay seen in the actual
experiments. It is of course recognized that the actual decay of the peak vorticity is caused
by the combination of the two effects. However, the stretching term dominates in this flow.

Although the peak vorticity magnitude is decreasing rather quickly in the high

67

F=6Hz,A=2.1% mm“,l1 F=6Hz,A=11.6%

 

 

 

 

 

 

4o — ————— Stretchlng 40 -
30- 301.
1'.” 20- l". 20-
N N
3 ~- 3
1o_- ‘W‘l‘r- 10:
F r
L.
0L1111l1111l1111l1111l 011111114111111l1111]
o 5 1o 15 20 o 5 1o 15 20
(a) x (cm) (b) X (cm)

Figure 3.17: Comparison of the decay of the peak spanwise vorticity with the decay due to
solely stretching and solely diffusion. (a) 2.1% forcing amplitude. (b) 11.6% forcing
amplitude.

amplitude forcing cases, the circulation of the cores in this case decreases at a much slower

rate. Figure 3.18a shows the effect of increasing the forcing amplitude on the circulation, F2.
Circulation is computed by integrating the <(oz> above the +1 s" for the positively signed
core and below the -I s" for the negatively signed core threshold in the region within $0.51»

of the center of the core. For all downstream locations, increased forcing results in an
increase in the circulation. This is in contrast to the peak vorticity plot of Figure 3.16a in
which the peak vorticity decreased with forcing amplitude at x > 10 cm for forcing
amplitudes greater than 10%. Figure 3.18b shows the variationof the circulation in the
vortex cores for three of the forcing amplitudes. In the high amplitude forcing, the
circulation decreases by approximately a factor of two over the range being examined,
compared to the factor of nearly four seen in the peak vorticity. In the low amplitude cases,
the circulation decreases by 25%, however, this is still a smaller decrease than is seen in the

peak vorticity for this forcing amplitude. This smaller level of decrease is likely a result of

68

3.75 cm +r, [:1

3.75 cm -r

 

 

 

10 I z 10 D 2.1% forcing -F,
A 6.75 cm +1“, A 6.75 cm 4‘, A 730/, {mug -r
9 0 10.25 cm +r, 0 10.25 cm -r, 9 o 1 1.5% forcing 5;
O 13.75 cm +1“, 0 13.75 cm -l", . 2_1% forcing +1“,
8 r 16.5 cm +1“, t> 16.5 cm -I‘, 8 A 7.3% forcing +I‘,
g @ O 11.6% forcing +1“,
7
7 [:1 D E i O. a @‘
’0? 6 c1 - ' 8 o O 70‘ 6 45; c@
a? I . 01‘
5 5 9 g 5 9 5 5 9‘1 A3
N 4 A D N 4 ‘ A
“ $1 is . » ‘— 3‘“ 11 .
3 3 " $5 59‘
2 2
1 1
0 1 1 1 4 I L L 1 1 1 1 I o 1 1 1 I 1 1 1 I 1 g 1 I 1 1 1 1 I
O 5 10 15 0 5 10 15 20
(a) forcing (%rms) (b) x (cm)
1 I 2.1% forcing rm
11.6% forcing rm
0.9
0.8
5 1
0.7 ‘
E 0.6 I .
8. 0.5 ' ' f I I-
§ Q
h 0.4
0.3
0.2
0.1
0 1 1 1 I 1 I 1 1 I 1 1 I
0 5 10 15 20
(C) x (cm)

Figure 3.18: Vortex core circulation. (a) Effect of forcing on circulation at several different
downstream locations. (b) Effect of downstream location for several forcing amplitudes. (c)
Effect of downstream location on vortex core radius.

69

the core radius increasing due to the stretching term.

As the vortex cores convect downstream, the effect of stretching causes the originally
Gaussian shaped vorticity profiles to flatten out and to stretch in the vertical direction. This
makes it difficult to determine the core radius in terms 'of the location at which vorticity
drops to the e'1 point. In order to determine an effective radius for the vortex cores, a “radius
of gyration” is computed. The radius of gyration is defined as the square root of the second

polar moment of inertia of vorticity divided by the circulation. That is,

1......

When the Gaussian vorticity profile is used with this formulation, that is = e

 

the expected value core radius is recovered. Figure 3.18c shows the vortex core radius for
two different forcing amplitudes. Note that in the 2.1% forcing condition (squares), the core
radius remains nearly fixed at a value of 0.5 cm. For the l 1.6% forcing case, the core radius .
nearly doubles between x = 5 cm and x = I 8 cm, increasing from 0.45 cm to 0.8 cm. This
results in a threefold increase in the area of the core. Thus, the decrease in vorticity
magnitude is tempered by the increase in core radius.

Figure 3.1% defines the vortex spacing parameters a and b. Re-examining Figure
3.15 and looking at the locations of the vortices, qualitatively, it can be seen that the
horizontal spacing between vortices, a, remains fixed for all of the forcing cases. This is
indicative of the nearly fixed vortex convection speed and the constant forcing frequency.

The vertical spacing, b, appears to remain approximately constant between the 2.1% and

70

 

1 1__

 

 

 

o 3.75 cm : o 2.1% forcing
0.9 A 3.75 cm 0,9 ; A 7.3% forcing
O 10.25 cm D E v 11.6% forcing V
0.8 O 13.75 cm D 0.8 E.
D 16.5 cm 5 W
0-7 I unforced 0.7 .- V
0.6 D O 0.6 L
a a V
0.5 0 5 :-
3 D a I .1 :1
0.4 0.4 L 410 g
o 9 > Q I o g E m 151
0. 0 o 8 D 0.3E- @ ‘3 U
o g :
A .
0.2 O 0.2 E- g
0.1 a 0.1 :- ‘9 W
t
0 . 1 1 1 1 I 1 1 1 1 I L 1 1B 1 I 0 " 1 _1 L I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I
0 5 10 15 0 5 10 15 20
(b) forcing (% ms) (6) x (cm)

Figure 3.19: Effect of forcing on vortex spacing parameters. (a) Definition of spacing
parameters. (b) Effect of forcing on spacing ratio at several different downstream locations.
(0) Effect of downstream location on spacing ratio for three forcing amplitudes.

7.3% forcing cases. Then, it decreases until the vortices are nearly aligned in the 13.3% case.

Figure 3.1% plots the ratio of b to a, as determined from the locations of the peak <(0,> and

shows the effect of forcing on vortex spacing more quantitatively. The spacing parameters
are measured from the location of peak vorticity within each core. For the unforced case, b/a
= 0.31. This value is slightly larger than the theoretical Karman spacing ratio of 0.281 . The
measured value of the unforced value of b/a in this study is significantly different from that
found in the work of Roberts (1985), where the unforced b/a = 0.52. It should be noted,
however, that the Roberts study measured the vortex spacing by visually locating the center
of the vortex cores. Using a data set in which flow visualization and velocity/vorticity

measurements were acquired simultaneously, it was found that the value of b/a estimated by

71

visual methods tends to be larger than that measured by locating the center of vorticity.
However, the visual estimation is highly dependent upon the individual making the
measurement.

Atx = 3. 75 cm (squares), b/a remains relatively constant, possibly increasing slightly
for forcing levels less than about 7.5%. The value is slightly larger than that of the unforced
case. For larger forcing levels, the spacing ratio decreases and at 13.3% forcing, the value
is nearly zero. It is believed that if the forcing amplitude is further increased, this ratio would
become negative, which could indicate a thrusting type condition. However, this was not
examined in this work. At locations farther downstream, such as x = 16.5 cm, the trend is
reversed and, except for the 13.3% forcing condition, the vertical spacing increases with
increasing forcing amplitude. In the 13.3% forcing case, the spacing remains nearly constant.
This is likely due to the initial alignment in this case in which the positive and negative
signed vortices are at the same vertical location.

Figure 3.19c shows the effect of downstream location on the spacing ratio for the
several forcing cases. For the two low amplitude forcing cases, the spacing ratio remains
nearly constant between 0.3 and 0.4. However, for the l 1.6% forcing case, the spacing ratio
is initially a very low value, 0.1. In the range greater than 10 cm downstream, the spacing

ratio increases dramatically to a value of 0.9. By this downstream location however, <(1),>

is very stretched out and no longer has a Gaussian character. In the region x > 10 cm it has
also corresponds to the range of locations where 37"“ values increased and where
Koochesfahani and Nelson (1997) found large increases in mixing product.

The effect of variation of the forcing frequency on b/a presented in Roberts (1985)

is quite different from what is seen in this study when the forcing amplitude is varied. At

72

low forcing frequencies, Roberts found that b/a is approximately 0.52. As the forcing
frequency is increased to about twice the natural shedding frequency, b/a decreases to its
minimum value which is slightly less than theoretical Karman spacing ratio of 0.281. As
previously mentioned, it was at this frequency where Roberts saw the largest increase in the
amount of mixed fluid. At higher forcing frequencies, the spacing increased again. Plotting
these spacing results versus forcing frequency would result in a u-shaped curve with a
minimum located where the forcing frequency is twice the twice the natural frequency. It is
interesting to note that the high mixing levels found by MacKinnon and Koochesfahani
(1997) and Koochesfahani and Nelson (1997) are also for cases where the forced wake
generates a small vertical spacing between vortices in the upstream locations. However, it
will be shown in Chapter 5 that decreased b/a is most likely not an indicator of a forcing
condition which will contain significant mixing enhancement.

In the current work, it has been shown that the b/a is influenced by the forcing
amplitude as well as the downstream location at which the measurements have been made.
Neither of these two characteristics were examined in Roberts (1985). The flow is going to
respond differently to input perturbations at different forcing frequency. Thus, measurements
need to be made to determine whether the forcing levels can indeed be compared. As the
forcing frequency changes, each downstream location is a different number of forcing
wavelengths downstream. Thus, at the same downstream location the vortex array may be
at a different point in its evolution. Both of these effects will generate changes in the spacing

which may alter the u-shaped spacing curve found by Roberts.

73

3.4 Flow properties away from the center span

As the spanwise vortex cores encounters the sidewalls of the test section, the
boundary layers on the sides of the test section are going to affect the core. Very few studies
have examined these types of flows close to sidewalls. Nelson (1996) examined mixing
results at the center span location as well as two locations closer to the sidewalls of the
facility. As previously mentioned, this study was conducted in the same experimental facility
used in the present work. At the first off center span location, 2 = -0.8 cm, the results of both
the flow visualization and mixing levels show effects very similar to those seen at the center
span. However, it appears that the spanwise vortex rollers become three-dimensional at a
location farther upstream than at the center span.

At the second off center span measurement plane, z = -2 cm (halfway between the
centerline and the wall), the mixing and flow visualization results were markedly different
from those at the center span. In the highest amplitude forcing case, three-dimensional
motion was found very far upstream. Furthermore, the vortex pattern changed dramatically
as it convected downstream. It was also found that the flow is well mixed at a significantly

earlier downstream location than at center span. At x = 11 cm, both the visual thickness, 8'
and the product thickness, 5,, reach their maxima. Farther downstream, 51 remains constant,
while 5, actually decreases.

Figures 3.20, 3.21, and 3.22 show the vorticity field of the 11.6%, 7.3% and 2.1%
forcing cases for a common range of downstream locations at seven different spanwise
locations in the flow. All plots are for a common phase which was selected such that a

negatively signed vortex core is located at a common location in the z = 0 cm spanwise

74

F = 6 Hz, A=11.6%

<0),> (s'1

 

-25-20-15-10-5 0 5 1015 20 25

 

 

 

 

 

 

 

Figure 3.20: Flooded contour plots of spanwise vorticity in wake forced at 11.6% of free—
stream velocity for 7 different spanwise locations. Contour lines are spaced at :5 s", :10
s", :25 s". Dashes indicate negative contours.

75

F = 6 Hz, A=7.3%

<0),> (3")

 

-25-20-15-1o-5 0 5 101520 25

 

 

 

 

 

 

 

Figure 3.21: Flooded contour plots of spanwise vorticity in wake forced at 7.3% of free-
stream velocity for 7 different spanwise locations. Contour lines are spaced at :1-5 s", :10
s", £5 s". Dashes indicate negative contours.

76

F = 6 Hz, A=2.1°/o

<0),> (s'1

 

-25-20-15-10 -5 O 5 10 15 20 25

 

 

 

 

 

 

 

Figure 3.22: Flooded contour plots of spanwise vorticity in wake forced at 2.1% of free-
stream velocity for 7 different spanwise locations. Contour lines are spaced at -1_-5 s", :10
S", :25 s”. Dashes indicate negative contours.

77

 

 

 

1.5-
: ——B—- a:F=6Hz,A=2.1%
L —9—— a:F=6Hz,A-7.3%
1.25; —e— a:F-6Hz,A=11.6%
: —-—'— b:F=6Hz,AI2.1%
- —-—O— b:F=6HZ,A=7.3°/o
1; + b:F=6Hz,A=11.6%
A .
$0.75-
v .-
G :
0.5:-
0.25:-
"L 1 1 1 11
Q4 0

z (cam)

Figure 3.23: Variation of streamwise (a) and transverse (b) vortex spacing with spanwise
location.

location for all three forcing amplitudes. A line has been drawn at x = 4.3 cm to allow for
a comparison of the spatial location of the vortices. With the exception of the z = -2.5 cm
measurement which will be described in more detail later in this section, as the measurement
plane moves closer to the sidewalls, the location of the vortex. core generally moves
upstream. This indicates slowing of the core due to the boundary layer on the sidewall of the
facility. Note that the effect is the same for the all three forcing amplitudes.

The slowing of the vortex cores can be examined quantitatively be examining the
spacing between the cores. Figure 3.23 shows the variation of both the b and a parameters
with spanwise location. Since the forcing frequency is fixed, a decrease in the streamwise
separation between the cores indicates a decrease in the vortex convection speed. For the
A = 11.6% case, between z = 0 cm and z = -3 cm, a decreases from 1.49 to 1.20. This
indicates a 20% decrease in the convection speed of the spanwise vortex rollers. A similar

effect is noted in the two lower forcing amplitudes with a reduced magnitude. It is noted that

78

-——e— 2.1% Forcing
—A— 7.3% Forcing
——a—— 11.6% Forcing

. .° .0
CD CD
II'I'IIU'TT‘I

 

 

 

-
b

14 0W1277‘3HH4
(b) 2191'“)

 

 

—
O
VII
1-

 

Figure 3.24: Effect of spanwise location on peak vorticity and spacing ratio at x = 4.5 cm.
(a) spacing ratio. (b) peak vorticity.

in the high amplitude forcing case, the value of a is seen to increase again at z = -3.5 cm.
This effect is not seen in the two lower forcing amplitudes. The transverse spacing
parameter, b, also shows a variation with spanwise location. In the central region of the test
section, the value remains relatively constant. However, for z < -2 cm, b increases for the
11.6% forcing case. For the two lower forcing amplitudes, the increase in b is delayed by
0.5 cm.

Figure 3.24a shows this variation of b/a for three forcing cases shown above. A
slight increasing trend is seen in b/a for z > -2.5 cm. This is caused by the slight increase in
b previously seen to exist in this region. At z = -3 cm, the value of b/a increases very rapidly
in the A = 1 1.6% forcing case. The two lower forcing amplitudes do not exhibit this feature.
At 2 = -3.5 cm, b/a decreases again in the 11.6% forcing case, however, it continues to
increase in the results for the other two amplitudes. It will be shown in Chapter 4 that in the
11.6% forcing case, 2 = -3 cm is approximately the spanwise location where the spanwise

vorticity is reoriented into the streamwise direction. However, for the lower amplitude cases,

79

the location where the reorientation occurs is closer to the sidewalls. It is believed that a
peak in b/a exists for these forcing amplitudes, but at a location closer to the sidewalls than
could be measured.

Figure 3.24b shows the spanwise variation of the peak <(1),> at x = 4.5 cm for the

three forcing amplitudes shown in Figures 3.20-22. In the high amplitude forcing case, the
peak vorticity value initially remains relatively constant as the measurement plane moves
closer to the wall. From the center of the test section to z = —2.5 cm, the peak vorticity level
increases slightly. For z < -2.5 cm, the peak vorticity levels begins a dramatic decrease. This
decrease in vorticity magnitude is the result of the bending of the spanwise vortex roller. As

the roller bends, the vorticity is reoriented from (1), into 0),. A similar effect is noted in the
7.3% forcing case, however, the increase in peak (1)x at z = -2.5 cm and the subsequent

decrease at larger absolute values of z is not as substantial as that seen in the higher forcing
amplitude. For the 2.1% forcing case, the peak vorticity remains nearly constant over the
entire span.

In the previous discussion on peak vorticity levels, it was noted that at z = -2.5 cm,

(1), is being reoriented into 0), for the 1 1.6% forcing case. Further, when the locations of the

spanwise vortex rollers was discussed, it was noted that as the measurement plane moved
closer to the wall, the location of the spanwise vortex roller generally appears to have moved
upstream. However, at this same z = -2.5 cm spanwise location where the vorticity
experienced a dramatic decrease, the roller had moved downstream. At z = -3 cm, the roller

has moved downstream again. In order for the core to display these characteristics, it must

80

ll.

Ill!- “7151‘

   

 

 

0.25" A z=0cm
r C] z=2.5un
1' O z=3.5cm
0.2-
t
A 0.15:
E
0 1
v
a 1-
0.1-
0.05- M
o_1111I114LI4111I1111fl
O 5 1O 15 20

x (cm)

Figure 3.25: Momentum thickness of the forced wake (11.6% forcing amplitude) for the
center span and two off-center span locations.

be experiencing a large amount of bending. This bending will result in the reorientation of

a significant amount of (1), into (1),.

It is noted that it a second possibility for this arrangement is that the core closest to
the line drawn at x = 4.3 cm is not a part of the same core as those seen in the locations closer
to center span. That is, the core centered near x = 4.3 cm is part of the same vortex tube as
the vortex seen atx = 5. 75 cm at center span. This seems unlikely as it would require a very
large gradient in the mean velocity of the vortex core This would likely have been seen in the
plots of the spacing parameters seen in Figures 3.23 and 3.24.

Figure 3.25 shows the variation of the momentum thickness with downstream
distance for the 1 1.6% forcing case at center-span and at two additional spanwise locations.
As has been previously seen in Figure 3.4, the momentum thickness for the 11.6% forcing
case increased with downstream location. Figure 3.21 shows the streamwise development

of 9 for the center span and two off center span locations. For both the z = -2.5 cm and z =

81

.UI—I Il- ’fl‘fl

-3.5 cm spanwise locations, 0 is initially nearly 0.1 cm. This is the momentum thickness of
the unforced wake at the center span. In contrast, at center span the value of 9 for the A =
11.6% forced wake is very close to zero. As the flow moves downstream, the momentum
thickness increases for the spanwise location 2 = -2.5 cm. This increase is not as large as that
seen at z = 0 cm. For z = -3.5 cm, the trend is decreasing. In section 3.2, is was noted that
the span-averaged momentum thickness, in addition to changes in the pressure field are
related to the drag on the splitter plate, which must remain constant at all downstream

locations. In order to determine if the span-averaged 9 is constant, it would be necessary to
integrate 9 across the entire span. However, not enough measurement locations exist in the
current set of experiments to accurately perform this integration. From these three locations,
however, it does not seem likely the span-average of 9 will be constant. Rather, it is likely
that span-averaged 9 will increase with downstream location, which would indicate

increasing drag. Thus, a decrease in the mean pressure terms must account for this increase.

3.5 Higher Density Measurements

Measurements were made at twice the standard density in order to assess the effect
of spatial filtering due to spatial under-resolution in the data. Only the 11.6% case was
measured with the higher resolution. Figure 3.26 shows a comparison of the low density and
high density measurements for the far upstream location at three different spanwise locations.
In all three cases, the spacing between vortices and the general structure of the flow remain
the same. However, the vorticity magnitudes are larger in the higher density measurements

and there are several features that are more easily distinguishable in the higher density

82

F = 6 Hz, A=11.6%

 

<0), >(s'1) ..
-25-2o-15-1o-5 o 5 1015 20 25

Low Density High Density

z=0cm

 

 

 

 

 

 

 

Figure 3.26: Comparison of high density and standard density measurements at three
different spanwise locations. Contour lines are spaced at 15 s", :10 s", :25 s“. Dashes
indicate negative contours.

83

results.

Examining the center span (z = 0 cm) results, it can be seen that the vorticity
magnitudes are increased in the high density measurements as expected. The peak
magnitude of the vortex core located at x = 3.5 cm is slightly more than 35 s" in the low

density results where [/8 = 2. This is increased to nearly 46 s" in the higher measurement
density results where [/5 = 4. Even with this large increase in vorticity magnitude, the

structure of the vortex array and the locations of the peak magnitudes of the results of the two
different densities looks very similar. Similarly, the high and normal density results at z =
-3.5 cm are very similar. The higher density results have larger vorticity magnitudes and
structures appear slightly more elongated in the streamwise direction.

This increase in vorticity magnitude is slightly larger than would be predicted by the
results presented in Appendix A on the effect of grid spacing on the peak vorticity
magnitude. An examination of the location of the measurement grid reveals that in this
measurement set, the center of the vortex core is between two grid points in the low density
data and very close to a grid point in the high density data. This effect was not examined
in Appendix A, however, it is believed that it will result in the vorticity levels being further
underestimated. Using the estimated error values for the 2nd order finite difference method
and the high density data presented in Appendix A and the approximate number of vectors
per core radius, it is estimated that the actual peak vorticity should be 47 .9".

Comparing the high and standard density results at the z = -3 cm spanwise location,
in the high density results, it appears as if the vortex cores at this location have undergone
some type of splitting process. Looking at the large positive and negative cores located at

(x, y) = (4.5, -0.5) and (x, y) = (5, 0.4), respectively, it appears as if regions of vorticity are

84

splitting away from the primary core. The magnitude of the peak vorticity in the secondary
vortex is approximately 1/3 of the peak vorticity in the primary vortex core from which it
splits. Although this effect is present in the standard measurement density data, it is much
more difficult to distinguish. An examination of an animation of the data further reveal that
the small positively signed core located at (4. 75, 0. 5) has split off from the positive core that
is located (4.5, -0.5). It has convected into the upper stream as a result of the induced
velocity of the negatively signed core at (5, 0.4). Similarly, the negatively signed vorticity
breaking off from the core located at ( 5, 0.4) will eventually move into the bottom stream.
The splitting process in this flow seems similar to that found in the study of a wall jet by

Visbal, Gaitonde, and Gogineni (1998).

85

Chapter 4

Cross-stream Measurements

4.1 Effect of out-of-plane motion on in-plane velocity measurements

In optical based velocimetry methods which track the motion of a tracer, whether it
be particles or some other features, such as a phosphorescent laser grid, within a flow, out-of-
plane motion can cause an apparent in-plane motion to be seen in velocity field
measurements as seen in figure 4.1a. This is an issue in all flows, however, in flows where
the out-of—plane component of the velocity field is of the same order, or larger than the in-
plane velocities, the error generated in the velocity field measurements can be considerable
and proper experimental design is very important in order to minimize its effect. The (v, w)
velocity measurements discussed in this chapter require that the detector is placed parallel
to the (y, 2) plane such that the free-stream velocity is moving directly towards the detector.
In fact, the out-of-plane velocities in this measurement plane are nearly twice that of the in—
plane velocities.

The general mechanism causing this in-plane motion caused by the out-of-plane
displacement is shown in Figure 4.1b. An object in the image plane (e. g. a particle, an MTV
grid crossing, or some other tracer) initially located at point A is imaged onto point B on the
detector array. The out-of—plane motion in the fiowfield causes the tracer to move a distance

Az towards the camera to a location A’. Assuming that the tracer remains in the depth of

field of the imaging system, this new point will be imaged at a location B’ on the

86

 

' \\\\ \\
Sam
E\\\\\\\\
'5'.\\\\\\\:

R\\‘\\\\ \
L\\\\\\ \ \
kakxxx \ \
kkkkk‘s V». x

_
4+«ki—Q— ‘. Q.

Trrrrrrr
fry/r1111
r////////
r////////
f////////
V ////

 

 

 

I I I IIJJJJJ—o

\k‘\\\\\\\\“"”,,III”/vl
I

i—

-kkkx\\\\\ \ \1

I ‘0 ’JJaaw

#fifi‘fi“ \ s I 1 4

. ‘-.—.—.—.—._+—.—>—

x 1M‘—‘_.—¢—.— o: c .

\“‘—“~’w

_w«««.—.— I r t l u

\
We’rrrrx/z l I 1 \

p-

\ \ \\\““*‘

”(III/IIIIIIO\\\\\\\\\\\F

I I /////V/'/
I /////'/'/'/
I I’ll/7"),
I [I’ll/7’2;
.0 .0..—'—'-—'—-.—-.—

1
i
1
J
1

x x \\\“‘i‘
\ \ \\\\\\\
\ \ \\\\\\\
\ \ \\\\\\\

Lens

 

87

Detector
array

 

Figure 4.1: Diagram of the effect of out-of—plane motion on in-plane velocity estimates.

detector array. Using geometric optics, it can be shown that the point B’ on the detector
array is the image of the point A” in the image plane. Therefore, the forward motion of the
tracer would be registered as an apparent upward motion on the detector.

In this arrangement, the in-plane displacement, Ax and Ay, caused by the out-of—plane

motion, A2, is related to the distance between the imaging system and the image plane, 2

(ram ’

and the distance from the center of the image plane, x, y. To first order, the relationship can

 

be described by:
Z
Ay = —-y
2mm
Al
Ax = x,

where it is assumed that the out-of—plane motion Az is small relative to anm- Similarly, the

error in the velocity field measurements resulting from the out-of plane velocity for MTV (or

PIV) type measurements can be found to be:

 

 

A A2
u:
z,,mAt
A
Av: Z y,
2 At

where At represents the time delay between the successive image pairs and Au, Av represent

the errors in the two in-plane velocity components. As seen in Figure 4.1a, the apparent
velocity caused by this effect is zero in the center of the image plane and is a maximum at
the edges of the image if the optical axes are exactly lined up. If the axes are not lined up,
there will be an offset in the profile. The experiments reported in this chapter have been
designed to attempt to minimize this effect. From the free-stream velocity and the time delay

between image pairs, the mean out-of-plane motion in this experiment, Az can be estimated

88

to be 0.0375 cm. The distance from the imaging plane to the camera, 2 is 75 cm, and the

maximum distance from the center of the measurement field to the edge (x and y) is 2 cm.
Thus, the maximum apparent displacement in either the x or y direction is 0.001 cm, which
corresponds to a velocity of 0.25 cm/s. This is approximately 5% of the peak velocities
measured in this flow. Note that the 5% error is only at the edges of the measurement. Other
locations will have an error that is a smaller percentage of the peak velocity. This velocity
error can be further minimized by the use of lens with a larger focal length. This would

allow 2 to be increased further. However, the 105 mm lens used was the largest available

for use in this study.

If the out-of-plane motion is uniform, the vorticity value calculated from the in-plane
velocity measurements is not affected by this error. The measured velocity can be defined
as u

umm, + Au and v",,,,,_,,,,,,, = vmm, + Av where Au and Av are the apparent 1n-plane

measured '-

motions caused by the out-of—plane motion as have been defined above. Substituting the
above expressions for “M.M.: and vmmwd into the definition of vorticity:

.. _2v__a_u
l—ax ay'

It is easily shown that vorticity is not affected by Au and Av. Returning to the expressions
for Au and Av, it is clear that Au is not dependant upon x, and Av is not dependant upon y.

Thus, the computed value of the vorticity field will be correct. A calculation of the vorticity
field from the velocity field in Figure 4. la confirms that the vorticity due to the out-of—plane
motion is zero. Thus, even though the velocity vectors near the edges of the measurement
field of view have an error which exceeds those specified in Chapter 3, the error in the

vorticity field is not affected by this source of error.

89

4.2 Phase-averaging of cross-stream plane data

The data collected in the cross-stream measurement plane is phase-averaged in the
same manner as the streamwise measurement plane data. Figure 4.2 shows a flooded contour

plot of the phase-averaged streamwise vorticity, <0),> for four different downstream

locations at a given phase. Overlaid on top of the flooded contours, contour lines of five
instantaneous realizations from which the phase-averaged field is created are drawn. Note
that only the right half of the test section is measured in these experiments. Both flow
visualization, and a limited set of velocity and vorticity field measurements have shown that
the locations of the vortical structures are nearly symmetric about 2 = 0 in this flowfield.
However, the sign of the vortical structures is opposite in the two half-planes.

Figure 4.2a and 4.2b show the phase-averaged and instantaneous realizations for a
typical phase at x = 3 cm and 10 cm respectively. As can be seen, the instantaneous and the
phase-averaged results overlay on top of each other very closely. 'At x = 3 cm, the phase-

averaged peak vorticity magnitude at this phase is 22 3". At this location (<0) > )M, = 1

X nm

S". Discounting several points near the edges of the measurement field where some of the
measurements are suspect due to the grid crossing impacting the wall of the facility,

(<(1) > = 1.25 s". The region where (<(1)_,>

X "715) max

occurs at this, and all of the other

,...)...“
streamwise locations, is in the shoulder regions of the <(o,> distribution where a small
amount of spatial wandering in this region of high gradient will result in large RMS values.
At x = 10 cm, structure of the streamwise vorticity field has changed rather dramatically.
Several distinct structures of both positive and negative sign can be seen. The phase-

averaged peak <(1),> has dropped significantly to approximately 11 5". At the location of

90

rl‘r‘rrfl

if'rw

 

 

 

 

Irivilrrivlv

 

 

[rrrrlv

 

 

 

Figure 4.2: Comparison of the phase-averaged streamwise vorticity field with several
instantaneous realizations of the streamwise vorticity field for four downstream measurement
planes. Five instantaneous realizations are displayed in each plot and contour lines indicate
the :13 s", :6 s",...,:15 s'1 with the dashed lines indicating the negative contour levels. (a)
x=3cm. (b)x= 10cm. (c)x= 16cm. (d)x=24cm.

91

= 0.5 3", however (<0) > )

I I771.“ mar

the (<(1),>,m_,),,,,,,, has increased slightly to approximately 1.5

s". These results are comparable to that seen in the streamwise measurement results and are
very near the expected uncertainty values inherent in the measurement.

Figure 4.20 compares the instantaneous and phase-averaged streamwise vorticity field
at x = 16 cm. It is apparent that the cycle to cycle variation in the location of the regions of

<(1),> has increased from the earlier streamwise locations. The magnitude of the peak
vorticity in this phase is approximately the same as that seen at x = 10 cm results, tom“, =

10.5 s"; however, (<(1),> )pwk has increased dramatically to 1.5 s". An increase is also seen

"713'

in (<o),> = 2.25 s", nearly 25% of the peak vorticity levels. At x = 24 cm, shown in

4.2d, the streamwise vorticity field shows further increases in the variation between the

instantaneous realizations. The phase-averaged peak vorticity has been reduced to 7.5 s" and

(<(1),>m)pwk has increased to 1.75 3" while (<(1),>,.,,,,),,,,,, has increased to 2.75 s" at this
downstream location. The increase in the spatial wandering of the <0),> field likely results

in the peak vorticity values being reduced compared to the instantaneous results due to
averaging out of the peak vorticity values. However, even with the considerable amount of
deviation from realization to realization in this far downstream location, it is apparent that
much commonality exists among these instantaneous realizations in terms of the general
structure of the vorticity field. For example, examining at the vortical structure located at
(y, z) = (0.5, -2. 75), it is apparent that all of the instantaneous realizations contain a
positively signed vortex at this location. However, the exact location of the center of this
vortex has wandered from realization to realization.

At the x = 3 cm and x = 10 cm measurement stations, it is felt that the phase—

averaged results provide an accurate representation of both the spatial location and the

92

magnitude of the instantaneous streamwise vorticity field. At these stations, the variation
of the individual realizations of each phase is comparable to those in the streamwise
measurements and nearly identical to the expected measurement uncertainty. As the fluid
travels downstream, small scale motions and spatial wandering result in the smearing of the

<(1),> field in the phase-averaged results. This will result in the peak values of (1)x being

reduced from the instantaneous realizations and the apparent size of the regions of
streamwise vorticity being slightly increased. However, the phase-averaged results do

provide an accurate description of the spatial location of the a), in this flow.

4.3 Mean flow properties of the cross-stream plane

Figure 4.3 shows the overall mean of the streamwise Vorticity field for the unforced,
A = 7.3%, and A = 11.6% cases at four different downstream locations. These two forcing
cases were selected as they are representative of the qualitative features seen in the other
forced cases. The lowest amplitude (2.1% forcing) case will not be shown in this section as
the small levels of the streamwise vorticity are very difficult to discern without the extensive
use of color to accentuate the vorticity field, and the qualitative features found in this forcing
amplitude are also seen in the 7.3% forcing case. As has been previously described, these
cases are the counterparts to the medium and high forcing amplitude cases in the mixing
study of Koochesfahani and Nelson (1997).

The unforced flow shown in the first column does not show the presence of any 6, .
This absence is not the result of positively and negatively signed vorticity averaging to a

small value. Rather, the instantaneous levels of 0), are nearly zero at all (y, z) locations. In

93

contrast, 61 is seen at all of the downstream locations studied for all of the forcing cases.
At the far upstream measurement location, the presence of a counter-rotating vortex pair can
be seen close to the sidewall of the facility. Generally, as the downstream location increases,
the strength of the mean vorticity decreases. In the 7.3% forcing case, the peak value of '(T)"
decreases from 10 s” at x = 3 cm, to 5 s" at x = 8.5 cm and remains relatively constant at this
level through the last measurement location, x = 24 cm.

Examining the streamwise vorticity field for the 11.6% forcing case, it can be seen
that at x = 3 cm, the structure of Ex looks very similar to that seen in the 7.3% forcing case.
However, the magnitude of the peak Ex is about 50% larger than in the 7.3% case, "()3x =
15 s" . At x = 8.5 cm, the mean streamwise vorticity field shows significant differences from
the lower amplitude cases. Three distinct regions of mean ‘0')", are seen; two in the upper
half-plane and one in the lower half-plane. The peak vorticity among these three cores is 11
5". At x = 16 cm, there are at least 4 distinct regions of concentrated vorticity. The peak
mean 6', = 7 3" among the four regions. At x = 24 cm, most of the. regions of vorticity have
moved near the centerline of the test section. These additional structures will have the effect
of increasing the amount of mixed fluid in the wake as their presence causes more fluid from
the top and bottom streams of the test section to interact.

In addition to the appearance of the additional cores, the spatial location of these
cores is very different from that observed in the 7.3% forcing case. At x = 8.5 cm, the
streamwise vorticity field for the 11.6% forcing case occupies a region of the test section
ranging from -3.5 cm < z < -1 cm. In contrast, the mean fix in the 7.3% case remains less
than 1 cm from the sidewall of the facility. By x = 16 cm, this region of mean vorticity

encompasses nearly the entire measured half of the test section in the high amplitude forcing

94

0),(s ) -15 -12 -9 5 -3 0 3
Unforced A = 1 1 .6%

       

.1

= 3 cm
1! (cm)

X

6.5 cm
1! (cm)

x 1
‘2
2
1
E
°§
cov°
F>~
11
x
1
2
2
1
E
0?
«3°
N>-
II
x

 

 

Figure 4.3: Mean streamwise vorticity field for the unforced, 7.3%, and l 1.6% forcing cases
at four different downstream locations. Contour lines indicate the :3 s", :6 s",...,:15 s"
with the dashed lines indicating the negative contour levels.

95

><2><i

—e— Negatively signed 7.3%
—I— Positively signed 7.3%

1 —e— Negatively signed 11 .696
+ Positively signed 11.6%

I“, (cm’ls)
N OJ «h 01 05 \l 00 (O

[LIIILIPI

I I I l I I I I l
10 20 25

x (cm)15

0

Figure 4.4: Circulation of the average vorticity field of the right half measurement plane for
the 7.3% and 11.6% forcing conditions.

case, whereas these locations have changed very little in the lower amplitude case. Since the
positioning of both the mean and instantaneous vorticity field is symmetric about z = 0 cm,
the 11.6% forcing case results in the entire test section containing regions of mean
streamwise vorticity. These large scale motions generate additional surface area for the
unmixed fluids to come into contact and likely result in an increase in the amount of mixed
fluid. By x = 24 cm downstream, the regions of Eb: in the 11.6% case have moved even
closer to the centerline of the facility.

The additional cores found in the high amplitude forcing case afterx = 8.5 cm are not
the result of the primary vortices splitting. Rather, there is an increase in the amount of
vortical fluid. This can be seen in the Figure 4.4 showing the mean circulation of both the

positively signed and negatively signed 5, for the 11.6% and 7.3% forcing cases. 1", is

computed by integrating all positive (or negative) vorticity values above the i1 5‘1 threshold
over the measured half of the test section. The boundary layer regions on the top and bottom

surfaces were excluded from the integration, however the boundary layer region on the

96

sidewall was included. Small positive and negative background values of 6)., found within
the integration region will cause the value of I“, to overestimate the circulation values of the

vortex cores. However, as these background values should be similar at the different

downstream locations, it is fair to compare F, with one another. For the l 1.6% forcing case,

there is a significant increase in the overall circulation between x = 3 cm and x = 16.5 cm.
In contrast, the circulation of the 7.3% forcing case is relatively constant. It will be shown
in section 4.8 that these additional structures are the result of the reorientation of the

spanwise vorticity into the streamwise direction. It is noted that the value of I“, are not the

same for the positively and negatively signed vorticity distributions. It is believed that this
is caused by small differences in the velocity of the two streams. However, because of the
symmetry of the flow, the Ex field in the left half-plane will have the same magnitude and
location of those measured in the right half-plane, but with an opposite sign. This will result
in a net zero circulation if the entire test section is considered.

In Figure 3. 12, the % field for the wake forced wake at several different perturbation
amplitudes was shown. For the 11.6% and larger forcing cases, the arrangement of the %
field was consistent with the presence of a mean recirculating flow pattern in the cross-
stream plane. The lower forcing amplitudes showed no indication of this type of motion.
The velocity measurements over the cross-stream plane shown in Figure 4.5 confirm the
presence of this flow field for the l 1.6% forcing amplitude results. The mean recirculating
flow pattern is most obvious in the x = 8.5 cm measurement plane, however, this pattern
continues to be seen at x = 16 cm and x = 24 cm. Note that this pattern is not observed in
the unforced or the 7.3% forcing results.

Based on the mean velocity and vorticity results, it can be reasoned that this

97

 

      

 

 

     

 

 

 

 

 

      

 

 

 

 

 

 

 

  

2-
1 -rr...7 -
. ... -~1 .
i III/111‘ I
I I/IIIIIa 1
' II/;/-.- .
1_ . III/z--. .
I If fl! -
- if fi»\
1 if l—)
~ \\\\ \* i 1 --------
E A\ . .‘\\ I :-
. ..< . 1
0 E o- 7 ..---\\ . I-
\a , ...... . . ..
m > , ,1, ......
II - ”...,”. .
- .\.<./;/. ..
x - \‘\<-/I/r ..
. ---///r ..
-1- .............. - ----,,. ..
......-. ._.. \‘<---;/1 ..
111......~- ------‘1 .1
-—.. ........... . ,....-....-------.,,.. .\
m.1~\\7:. - W/1\\\\lit\\‘\\\-... WI
.2-
2-
If! - \\\\\\\\1Irun. ,\\\\\‘H,,,,....
, ......,,.----,,-.,,-- 1;,-\-,’----_,,’..*.‘
. . ........-’,’...‘.. .;,_<---_ --..a....~‘
, ............;z; Ivv- \
. ; 1........71;/// \ \11.;
1 ‘¢. ; ......1/1/11/1 \ \11!
r ‘V ; .. ~11rltlI/III”/I~‘\\\ 1111
‘ V... I .. .1II1111114+\\\ 1111
‘.. . » 1 \l\\\~‘\ \-1\\ 1111
E [ .-. . . “1‘-“ \~11\ \11
A‘ *u. .. . ‘.<--.-\\\*-11 \1\
O E -.. ... .... . ....»..—‘\\‘. \\\\
m 00 ........---- . -1 ........\\\. \\\
I v ....--——-—-.-’4‘ 1 - ........... .11; \ \\\
> ..-.-.-.-------- 1 -....... ....... ... 1 ..-
O [ ...-.......---.--I,,. -. . ,.... . ....” ...
II ,.-.-..__-..------ ,. ........... -~‘-IH' ...
,...-“.--. ----- - I. . ....-..\-W/// 111
x ....--.----.--------,,. ....----W,-//I \1\
—1- .----.~.;, I ....-.------<.....q41/1 \\\
.---..m..,,; ...------------~o—~c’;/1 \\\
.----.¢;,,-. \---4 _------_ 1 \\\
.------,, ,. -...-. -_-----..**.;; \ \\\
--- - ... - ._---------.~.a.-..\ \.\
Q-
2-
,, .—.-111 It - -- ~ \\ \i
<.41y[//;;;4¢— —-—11/;1;,/ a,» ,...-\
. ..2/11..;;;; .....,/// tiff/1”), ,-
-. ......,.... ,...-,..,,zzz x, - -
-. ,..)..;.,r . o~n;plllflflllll£‘w+.-\i
1_ ,............, . r....;;,,;,,z 1; ~\\
,.......,.,... . ,.....1;7;I.....,“\\‘\1
(.....;-I;-rlp | I-tllllIlllt‘nilfI—\\\\i
r........-.... . ,1.;7.11\\\<-~~1\1..1-.1
’............. f..1-1\‘\~-~...\\\\....
g A \....... . /...............\\\<...
E _ ...... .... \\\1.1.
g o- . .... .....\...1111«
‘9 v ........ ....,\.,,;;;.
'— >. . ...... ....\\-;/ 1
. . ...\ 4111/}.
|| . ... ,1].
X f "7
-1-
.2-
2r
‘\
1- \
E .- :
° 50. .
1f ~a ,
N > .
|| -. .
-. .
X 1 -. .
.2-

 

 

 

 

 

 

Figure 4.5: Mean velocity field over the cross—strearn plane for the unforced, 7.3%, and
11.6% forcing cases at four different downstream locations.

98

recirculating pattern will result in more mixed fluid in the 11.6% forcing case as compared
to the lower amplitude cases. The mean “pumping” action resulting from the vortex
arrangement causes fluid located in the top and bottom regions of the facility which is
typically unmixed, to be pumped towards the central region where they can combine. This
action will result in an increase in the amount of mixed fluid. The fluid in the central region
of the test section, which is typically mixed, is also pumped back into the typically unmixed
top and bottom regions. This results in an increase in the thickness of the mixed fluid layer.
Thus, the mean recirculatory flow pattern will result in an increase in the amount of
molecularly mixed fluid, and in the mixed fluid taking up a larger width within the test

section.

4.4 Downstream development of the phase-averaged streamwise vorticity

Figure 4.6 shows a perspective view of the phase-averaged streamwise vorticity field
for the 11.6% forcing results at two different phases within the forcing cycle. The two
phases are separated by approximately 1/2 of the forcing cycle. Results from 11 different
measurement planes are displayed on the same plot. Note that the side-walls of the flow
facility are located at z = -4 and no wall exists at mid-plane (z = 0). At both phases, the
farthest upstream plane shows evidence of two distinct vortex cores located close to the side-
walls of the facility. In the first phase, shown in 4.6a, the negatively signed vortex located
just above the center-line has a peak <mx> = -26 s". This is twice as large as that seen in
the positively signed vortex core (13 3") below the center-line. This situation is nearly
reversed in the second phase shown in Figure 4.6b where the positively signed vortex core
below the center-line has a magnitude (24 s”) that is 50% larger than the negatively signed

99

Ned n 6 3V .36 n 9 AS .232 59:8 “3:wa: 05 wan-BEE 8:: 354% 2: 5:5 1. wiri-h on” .1. bfi 2: 886-: mom: 59:00
donoom :8 05 b8 SE8 2: 2 Q n N am tuba—mm:- wca 2C. .59 V- n N E @882 2a Enact-m bio-fl 05 35 Bo Z .298 mEB£ 2: 52:3
mow-Em @0828 35:33 95 :- 8.8 mic-8 use; - 2: co.- bfi-tg 85:88; Bwfia>uomwnm 05 we Eon-@292- Emobmcaon— “9v beam-“—

100

  

05.92:-

m— Nr m w m 0 m- m. m- NTmF.
AvaN-BV

 

$w.wpu< .5..me

mr NF m o m o m- o- m- NTmT

 

 

Army A-8v

o3.:n< firm-u"-

.UoscncoQ 6.? Bum-m

101

core (-16 s") above the center-line. The variation of the strength of the streamwise vorticity
with phase will be discussed further in section 4.5.

An interesting effect is seen in the next two downstream planes (x = 4.25 cm and x
= 5.25 cm) of Figure 4.6. In both the x = 5.25 cm plane in Figure 4.63 and the x = 4.25 cm
plane in Figure 4.6b, the negatively signed vortex core appears to have split into two
portions. However, the x = 4.25 cm plane of Figure 4.6a and the x = 5.25 cm plane of Figure
4.6b do not show signs of these multiple cores. The multiple structures are not
distinguishable at these downstream locations due to the averaging process. More vortical
structures are seen as the distance downstream of the splitter plate increases. However, the
magnitude of the peak vorticity of the structures is decreasing with the increasing x.

When examining Figure 4.6, it is important to not visually connect the streamwise
vorticity among the various planes in order to create a volumetric image of the underlying
vortex structure. The vortical structures seen in the various planes are not necessarily part
of the same vortex tube. Rather, the measurement planes show the streamwise projection of
the vorticity vector for several different vortex tubes. Within each measurement plane, the

spanwise slice of the vorticity field will intersect with the projection of <wx> for different

vortex tubes. The multiple structures that are seen in several planes are the result of the
measurement plane intersecting multiple vortex tubes Each measurement phase plane will
cut a different number of tubes. A detailed description of the topology of the vorticity field
in this flow will be discussed in section 4.8.

Figure 4.7a shows the decay of the peak levels of <0)x> with downstream distance.

This value was determined by searching for the peak level of <mx> within all of the phases

at a given x location. For the 1 1.6% forcing case the absolute value of the peak streamwise

102

 

 

 

 

 

35 r O m, 7.3% Forcing 2 :' -~--~ - o m, 7.3% Forcing
: Ci +m,11.6% Forcing I ——9—— +(D' 11.6% Forcing
~ 0 «n, 7.3% Forcing 15 L ~ -- ~ 0 «0' 7.3% Forcing
30 r u «n, 11.6% Forcing : ——0— «1511.696 Forcing
: 1 L
A 25 :- 9 D D _
v- : - D D n
.m _ 0.5
v 20 - I D E
8 -
i : o 0 ' I D 3 o —
. I Z
N 15 - ' > ~
3 I 0 ' D -o 5 L
V - I - .
_ O O . . C] .
10 - 0 a ' I
Z O O O O 8 . a '1 '_'
_ O r I
5 -_ O 1 5 F x increasmg
E
O 1 1 l 1 I 1 14 4_l 1 A 1 A l 1 1 1 1 l L i 1 1 I -2 '- L J 1 1 l 1 I i l l A 1 l l l 1 i 1 l I
0 5 10 15 20 25 O -1 -2 -3 —4
(a) x(W1) (b) 2 (cm)

Figure 4.7: Variation of the magnitude and position of the peak value of <mx> field for the
7.3% and 11.6% forcing cases. (a) Variation of peak magnitude of <(ox> with downstream
distance. (b) Change in location of peak <o)x> with downstream distance. The farthest

upstream location (x = 3.25 cm) is marked with an “F” and the farthest downstream location
(x = 23. 75 cm) is marked with an “L”.

vorticity is <mx>m = 25 s" at x = 3 cm. This value decreases to 11 s" by x = 24 cm. Note
that this value may be underestimated by the smearing of the phase-averaged vorticity due
to the spatial wandering of the streamwise vortices. For the 7.3% forcing case, the initial

peak streamwise vorticity level is <mx>max = 18 s" at x = 3 cm and decreases to about 9 s"

at x = 24 cm. This is similar to decay of the spanwise vorticity where it was found that the

peak vorticity decreased with streamwise location. As is also seen in the <(oz> results, the
rate at which <(ox> decays for 1 1.6% forcing is faster than the rate at which <(ox> decreases

in the 7.3% forcing and lower forcing levels.
Comparing the peak streamwise vorticity levels with the peak spanwise levels at the

same downstream location,fl—>—'fl‘— = 0.6 for the 11.6% forcing case. For the 7.3% forcing

z>max

. . . <0) > ,
Case, this value 18 found to be even larger With —<—‘—"”—"- = O .8 . These values are conSIderably

I max

larger than those found in the free wake studies conducted by LeBoeuf and Mehta (1996),

103

who found the ratio of the peak streamwise vorticity to spanwise vorticity values to be 40%,
and the study by Weygandt and Mehta (1995), who found this ratio to be 20%. The
Reynolds number for these studies was Re9 z 350, which is slightly higher than the present
work

It is believed that the true value of the ratio of <0) > to <0) > in the forced

X max Z max

confined wake flow depends on downstream location and has a peak value of at least I . In
Appendix A, the effect of spatial filtering on the peak vorticity levels is examined. It was
found that as the measurement density decreased, the peak vorticity levels are underestimated
by an increasing amount. As the streamwise vortex cores are generally seen to be more

compact, the normalized measurement density, U8, is smaller than that found for the

spanwise vortex cores and the resulting bias error in the measurement will be larger. Thus

the peak values of <wx>m are expected to be underestimated by a larger fraction than that
for <wz>m. This matter will be examined further in section 4.9 where the standard density

measurements will be compared with a set of higher density measurements.

The spatial location where the peak streamwise vorticity occurs within the cross-
stream plane, seen in Figure 4.7b, reveals an interesting difference between the 1 1.6% and
the 7.3% forcing cases. This plot displays the (y, z) location where the peak vorticity is
found. The first and last streamwise locations for each vortex is specially marked. In
general, x increases towards the left. For both forcing cases, the peak positive vorticity is
always located in the bottom half of the test section while the peak negative vorticity is
located in the top half of the test section. For the l 1.6% forcing case, the spanwise location
of the peak streamwise vorticity moves in from near the sidewall of the test section located

at z = -4 cm towards the center. However, for the 7.3% forcing case, the location of the peak

104

vorticity remains confined in the region -3. 75cm < z < -2. 75 cm. This effect was also seen
in the results for the overall mean vorticity

The location of the phase-averaged streamwise vortical field with respect to
downstream location for the 7.3% forcing case can be'seen in the perspective plot found in
Figure 4.8. At the farthest upstream planes, there is evidence of only one negatively signed
region of vorticity. As will be seen in section 4.5, this is a result of the particular phase
selected. Other phases show the presence of a positively signed vortex core in the bottom
half-plane. As the flow develops downstream, the existence of additional streamwise
vortical structures closer to the center-plane is noted, however, their strength is considerably

weaker than that of the regions of vorticity close to the wall. In contrast, the regions of (1),,

that move towards the center in the 11.6% forcing case are stronger than those close to the

wall.

4.5 Variation of the streamwise vorticity field with phase

A = 11.6%

As with the variation seen with downstream location, the streamwise vorticity field
also shows a large variation with phase at each downstream location. The variation of the
strength of the <(ox> field at x = 3 cm for eight equally spaced phases, (1), within the forcing
cycle can be seen quite clearly in Figure 4.9. Recall that o varies in the range 0 < (b < 1 in

this study. In order to more easily reference the phases shown in the figure, they have been

labeled [-8. The vorticity magnitude of the two structures is clearly changing between a peak

105

.m-o-fi- 59:8 gnome: 05 mas-SEE 3:: team-6 2: £3» 1. w???
QH .1. on 2: 236:- mo-E 59:00 .853 :2 2: do 5E8 2: m- c n w a cox-“Bio £5- 2? .Eo v-

u N a @282 2a Ema-02m bio-d
05 :2: 802 .36 n 9 c8 cos-950 mac-88 com.» 2: 5a- bsEo> “cm-3E3? cowwcoébécm 2: ho EoEmoEZ-o Enema-son “we Emmi

106

  

95.92:-

mF NF m m m o m- o- m-NTmT
. :3 say

 

«.2: axon“-

    
     

<0» (s") .. --
" -15-12 -9 -6 -3 0 3 6 91215

 

Figure 4.9: Variation of the streamwise vorticity with phase at x = 3 cm for the 11.6%
forcing amplitude. Contour lines indicate the :6 s", :i:9 s"...:15 s" with the dashed lines

indicating the negative contour levels.
107

of the negatively signed streamwise vortex tube, located in phase 2, and a peak in the
ositively signed streamwise vortex tube located in phase 6. Both peaks are located at a
spanwise location of z = -3.5 cm. In between these vorticity peaks, such as in phase 4 and
8, there are phases in which the streamwise vorticity extends out from the wall towards the
center region of the test section. Note that these levels are less than the smallest contour line
displayed, however, they can still be seen in the flooded contour.

Figure 4.10 shows the variation with respect to phase for the downstream location x
= 5.25 cm. In this plane, the spanwise location of the region of peak vorticity has moved
inwards to z = 3 cm. Some noteworthy dynamics are occurring in this measurement plane.
Starting from phase 1, two distinct vortex cores are noted. A positively signed core is
located in the lower half-plane and a negatively signed core is located in the upper half-plane.
At the second phase, two nearly distinct negatively signed vortex cores are apparent in the
upper half-plane. In this plane, the outermost core has a larger radius and peak vorticity. It
is hypothesized that these two cores are the result of the reorientation of the primary
spanwise vorticity from two of the negatively signed <(ox> cores located one wavelength
apart.

Although the splitting mechanism described in Visbal, Gaitonde, and Gogeneni
(1998) was seen in the streamwise measurements presented in section 3.5, it is believed that
this mechanism is not responsible for the increase in the number of regions of streamwise
vorticity and the enhanced mixing field found in this flow. In section 3.5, it was seen that

the regions of negative <(oz> that split off of the “primary” negative spanwise vortex roller

moved from the bottom half of the test section to the top half of the test section. These

“secondary” cores became aligned horizontally with the positively signed primary roller in

108

 
 

-303691215

.1
(0? (s ) -15 -12 -9 -6

 

Figure 4.10: Variation of the streamwise vorticity with phase at x = 5.25 cm for the 11.6%
forcing amplitude. Contour lines indicate the 16 s", :9 s",...,:i:15 s" with the dashed lines

indicating the negative contour levels.
109

the top half of the test section. Similarly, the secondary cores split from the positively signed
spanwise roller moved into the bottom half of the test section. If these secondary cores,
which were generated by splitting from the primary core, are reoriented into the streamwise

direction, regions of both positive and negative <0); would be seen in both the top and

bottom halves of the test section. This is not the case. It is believed that if these secondary
cores are reoriented into the streamwise direction, they are quickly canceled by reoriented

primary roller. Furthermore, in Figure 4.4, it was found that I“x increased with x. If the
multiple cores were the result of the splitting of one core, I“x would remain constant.

Continuing on to phase 3 of Figure 4.10, two regions of negatively signed vorticity
are still present, however, they are not as distinct as seen in phase 2. The lack of distinctness
is due to the lack of sufficient spatial resolution to properly distinguish the two regions. It
is also apparent that in this phase, the innermost (farthest away from the wall) core has a
larger radius and vorticity value. In phase 4, the weakening outermost region of negatively
signed vorticity is drawn around the positively signed region, which is at its maximum value.
This region of negatively signed vorticity is further weakened by vortex cancellation as it
merges with the positively signed vortex, or will be absorbed into the boundary layer region.

The second half of the cycle is nearly the reverse of the first half. From phase 4 to
5, the region of positively signed vorticity shrinks in core radius. This continues into phase
6, however, a second positively signed core can be just seen to the inside of the core
weakening outer (closer to the sidewall) region of positively signed vorticity. As this core

becomes smaller it is continues to move around the strengthening <wx>. In phase 7, the
“new” region of positively signed <mx> is clearly apparent while the “old” region has nearly

disappeared. In phase 8, two oppositely signed regions of vorticity are again noted much as

110

in phase I . In this phase, the region of positively signed streamwise vorticity is a maximum.
The appearance of this new region of streamwise vorticity in phases 2 and 6 can be
connected to the passage of the spanwise vorticity at this x location. This effect will be

shown by examining the causes for the development of the new region of +<mx> at

approximately z = -3 cm in phase 6 of Figure 4.10. Figure 4.1 1 shows the spanwise vorticity
field for the z = -2.5 cm, -3 cm, -3.5 cm measurement planes at the phase corresponding to
phase 6 of Figure 4.10. In 2 = - 3 cm plane of Figure 4.11, a positively signed spanwise
vortex is positioned at the streamwise location of the measurements seen in Figure 4.10, x
= 5.25 cm. This location is marked with a line in the three streamwise plane measurements.
Examining the streamwise locations of the spanwise rollers in Figure 4.1 1, the x location of
the spanwise roller at z = -2.5 cm is seen to be located nearly V27. downstream of the location
of the roller streamwise position at z = -3 cm. Between z = - 3 cm and z = -3.5 cm, spanwise
location of the tube remains nearly constant, however the magnitude of the peak vorticity
within the core has decreased dramatically.

The new region of <wx> formation in phase 6 of Figure 4.10 is the result of the
reorientation of the positively signed spanwise vortex tube as it passes this location. From
the measurements of the spanwise vorticity, it can be seen that the spanwise roller is
experiencing a large amount of bending in the streamwise region where the new <(i),> is
formed. This bending is the reorientation of the vortex tube. As the phase continues to
increase, the magnitude of the peak vorticity of the streamwise vorticity increases as the
central part of the roller convects downstream and the streamwise “legs” are stretched. The

maximum magnitude of streamwise vorticity is achieved 1/4 of the forcing cycle later in

phase 8.

ll]

F = 6 Hz, A=11.6%

 

_1 . _,
<wz> (s ) ._ » .
-25-2o-15-1o-5 o 5 1015 20 25

 

 

 

 

Figure 4.11: Spanwise vorticity field in the range -3.5 cm < z < -2.5 cm. The phase
corresponds to phase 6 of Figure 4.10. Contour lines indicate the :5 s", :10 s",...,fl5 s"
with the dashed lines indicating the negative contour levels.

112

In the study of the free-wake conducted by LeBoeuf and Mehta (1996), it was found
that the location of the peak streamwise vorticity values were coincident with the locations
of the peak spanwise vorticity values. In the semi-confined wake of the current work, the
peak magnitude of the streamwise vorticity at a particular location in the flow occurs after
the passage of the spanwise vorticity at that point.

Figure 4.12 shows the streamwise vorticity distribution for the 1 1.6% forcing cases
for several phases at x = 8.5cm. This plot contains the same type of information displayed
in Figures 4.9 and 4.10; however, the inward movement of the streamwise vorticity makes
it difficult to discern important features in the three-dimensional perspective plot. As seen
at x = 5. 25 cm, multiple vortical structures are apparent at all phases within the forcing cycle.
These structures, occupy a considerably larger region of the test section than the regions of
<(ox> at x = 5.25 cm. The variation in the streamwise vorticity field with respect to phase
is caused by the passage of the most recent spanwise vortex tube, which generates more

streamwise structures the reorientation, and the interaction of the new <cox> with the
“legacy”<(i)x> created earlier by the passage of spanwise vortices several forcing cycles

prior. Examination of an animation of these data reveal that the “newest” regions of
streamwise vorticity are generated closer to the center of the test section. This process will
be described further in section 4.8. As the flow continues downstream, more of these
structures are generated. Figure 4.13 shows the streamwise vorticity distribution for this
forcing case 16 cm downstream of the splitter plate. At this location, even more structures
are apparent than in the x = 8.5 cm case and the vast majority of the test section is occupied

by vortical fluid.

113

   

-15-12 .‘9 -6 -3 o 3 6 9'1215

«a.» (8")

_.
I

1..

..
' l

 

 

 

l .1 I .I

2 (gm) ‘4 o z (cam)
Figure 4.12: Variation of the streamwise vorticity with phase at x = 8.5 cm for the l 1.6%
forcing amplitude. The displayed phases are separated by approximately A4) = 0.08 and
phase runs from right to left and then top to bottom. Contour lines indicate the :3 s", :6
s",...,:i:15 s" with the dashed lines indicating the negative contour levels.

114

.1 ..'
«1) s --
?( )-15 -12 -9 '6 '3 0 3 6 9 12 15

V (cm)

V (cm)

 

l 1 1. ..l

 

.2 ‘
1 (cm)

.2 ‘
1 (cm)

Figure 4.13: Variation of the streamwise vorticity with phase at x = 16 cm for the l 1.6%
forcing amplitude. The displayed phases are separated by approximately AC1) = 0.08 and phase

runs from right to left and then top to bottom. Contour lines indicate the :3 3”, 1:6 s",..., :i:]5
s" with the dashed lines indicating the negative contour levels.

115

A = 7.3%

As with the 11.6% forcing case, the 7.3% forcing case also shows a large variation
with phase. Figure 4.14 shows a perspective plot of 8 different phases in the forcing cycle
for the streamwise location of x = 3 cm. As with the measurements of this plane for the
l l .6% forcing case, two regions of streamwise vorticity are noted; a negatively signed region
in the upper half plane, and a positively signed region in the lower half plane. Unlike in the

l 1.6% case, there are many phases where one of the signs of <(ox> is not present. Note that

the region of negative <(i),> in phases 3-8 is located very close to the sidewall and is difficult

to resolve.

The 7.3% forcing amplitude case develops additional streamwise vortices as the flow
develops. Due to the decreased magnitude of the spanwise vorticity, from which the
streamwise vorticity is generated through the reorientation, the additional vortices have a
significantly reduced magnitude. Figure 4.15 shows the streamwise vorticity field at x = 5.25

cm. Similar to the results of the l 1.6% forcing, the locations of the new regions of <(ox> are

farther apart in the transverse (y) direction. This is similar to the effect noted in the 7.3%
forcing case in Chapter 3 where it was found that the transverse spacing among regions of

<mz> increased. Phase 1 shows the presence of two regions of (1),. As phase increases, a

second region of streamwise vorticity begins to develop starting in phase 3. It is located
farther away from the sidewall of the facility and is closer to the top wall of the facility.
Continuing on in phase development, the negatively signed streamwise vortex increases in
intensity as it is stretched while the positively signed streamwise vortex decays. In phases

7 and 8, the positively signed region streamwise vorticity field is regenerated by the passage

116

   

-1
< s
w">( ) -15-12 -9 -6

o
3

-(3 (3 (3 (5 S) 1 22 1 55

Jun:
II II II

3
6H:
73%

 

Figure 4.14: Variation of the streamwise vorticity with phase at x = 3 cm for the 7.3%
forcing amplitude. Contour lines indicate the :6 s", :9 s",...,:t15 s" with the dashed lines
indicating the negative contour levels.

117

 
 

<oox> (5'1)

-15-12i-9" -6 -3 o 3 6 9 1215

 

Figure 4.15: Variation of the streamwise vorticity with phase at x = 5.25 cm for the 7.3%
forcing amplitude. Contour lines indicate the 3:6 3", 21:9 s",...,:tI5 s" with the dashed lines
indicating the negative contour levels.

118

of the positively signed spanwise vortex which is reoriented into the streamwise direction
at a location farther from the sidewall and closer to the bottom wall of the test section.
The process of the creation of streamwise vorticity from the reorientation of spanwise
vorticity continues at locations farther downstream.~ Figure 4.16 shows the streamwise
vorticity field at x = 8.5 cm. At this downstream location, the presence of multiple vortical
structures of both sign is apparent. Note that the perspective view makes some of these
structures difficult to discern. Unlike the 11.6% forcing results, these structures have not
moved towards z = 0 cm. Rather, they are staying close to the sidewall of the facility.
Although the peak vorticity levels are decreasing, the qualitative characteristics remain the
same; multiple regions streamwise vorticity is noted, however, these structures remain close
to the sidewall. As seen earlier in Figure 4.7b, although the peak vorticity levels are

decreasing, the mean circulation value remains constant through x = 24 cm.

4.6 Relationship of streamwise vorticity field to mixing results

Several interesting comparisons can be made between the streamwise vorticity results

and the mixing results of Nelson (1996). ForA < 7.3%, the regions of both 5x and <wx>
remain close to the sidewalls of the facility and no 55x or <mx> is present in the central

region of the test section. This is consistent with the mixing results of Nelson (1996) where
it is shown that when the low and middle amplitude forcing are applied, the 2 location of the
peak amount of mixed fluid is initially located near the sidewall of the facility. As x
increases, the region of peak mixed fluid also moves very slowly towards the center of the
test section. In contrast, in the high amplitude forcing results, the 2 location of peak mixing

moves towards the center of the test section very quickly. This is the same as the behavior

119

  
 

«1)? (5'1)

-15-12-9 -6 -3 o 3 6 91215 A=7.

 

Figure 4.16: Variation of the streamwise vorticity with phase at x = 8.5 cm for the 7.3%
forcing amplitude. Contour lines indicate the 16 s", :9 s",...,:tI5 s'1 with the dashed lines
indicating the negative contour levels.

120

of the regions of 6x and «01>.

Generally, increases in the amount of mixed fluid in this type of flow have been
attributed to increases in the amount streamwise vorticity in a flow. At center span, the
Nelson found that the middle amplitude forcing case was a better mixer than the low
amplitude forcing case which was a better mixer than the unforced case. This was seen in
Figure 1.3. However, the present study shows that no 55,, or <wx> are found at this spanwise
location for any of these forcing conditions. Thus, the mixing increase for these cases must

be a result of the increase in a),

4.7 Model of the development of the vorticity field in the highly forced wake

The data presented point to a model of the development of the vorticity field in the
forced, confined wake flow. Figure 4.17 shows a simplified schematic of this model. The
spanwise vorticity at the splitter plate is shed into the free stream. This vorticity organizes
itself into a predominately spanwise vortical field. Very close to the sidewall, the boundary
layer slows the spanwise vortical tube and it begins to reorient itself into the streamwise
direction. As the rollers of the spanwise vortical field convect downstream, the streamwise
vorticity is stretched, causing the core radius to decrease and the peak vorticity to increase.
It is believed that some of these small radius, high peak vorticity regions were not observed
in the present measurement set as the size of the vortical region was reduced beyond the
limits that could be measured in the current experiment. It is recognized that as the core
continues to decrease in size, the spatial gradients will increase which will eventually

increase the effect of diffusion.

121

M id-span

 

 

 

 

 

 

Figure 4.17: Diagram of the development of the vorticity field in the forced wake.

122

The streamwise vorticity field at each measurement location is the result of the
reorientation of all of the spanwise rollers which have passed by that particular point in
space. As will be described later, not all of these regions are separately distinguishable in
the current data. As the measurement plane moves farther downstream more of these “legs”
are observed. In the high amplitude forcing case, the location of these legs moves towards
the center of the test section while in the lower amplitude cases, this motion is much less

pronounced. With the passage of each spanwise vortex roller, more (1)" is added at each

downstream location by the reorientation of the spanwise vorticity. The newly formed
vortical regions are located slightly farther away from the sidewall than the regions resulting
from the passage of previous rollers. In some of the lower amplitude forcing cases, the new

region of (1)x is also located farther from the transverse center (y = 0) than the regions. Note

that the end of the streamwise “legs” of the vortex tube remains fixed at the splitter plate tip.
This description is consistent with both the measurement results and the schematic of the
model shown in Figure 4.17.

The variation and disappearance of the structures at certain phases is caused by three
broad effects. The first is the stretching of the vortex core. As the spanwise vortex rollers
convect downstream, the streamwise legs are continuously stretched. This results in the core
size decreasing and the peak vorticity increasing. The second reason for the variation in the
magnitude of the peak vorticity is the lack of spatial resolution in the experimental
measurements. Since the data density remains constant during the measurements, as the core

radius shrinks, the effective [/8 decreases. The results presented in Appendix A show that
as [/5 decreases, the bias error increases. This will result in a decrease in the measured

value of the vorticity. As rcore continues to decrease because of the stretching, a point will

123

be reached where the core radius is smaller than size which can be measured in the current
experiment. Section 4.9 will also show that in upstream regions where the spacing between
the regions of streamwise is small, the lack of spatial resolution results in multiple vortex

cores having the appearance of a single core. The third reason for the variation in the <(ox>
structure is diffusion. As the regions of <mx> are stretched, the core radius will decrease and

the peak vorticity will increase. This increase in spatial gradients will result in an increase

in the effect of diffusion. However, it should be noted that diffusion is a slow acting process.

4.8 Evidence of axial flow along the cores of the spanwise vortices in the forced wake

As described in the introduction, core-wise axial flow has been found to exist along
vortex cores in a large number of vortex dominated flows. One of the questions this study
set out to answer was whether axial flow exists along the cores of the vortices in the forced
wake. Figure 4.18 shows vector plot of the (v, w) velocities in the (y, z) plane at x = 5.25 cm
for the phase corresponding to the passage of the negatively signed spanwise vortex core.
When this core passes the x = 5.25 cm plane, a well-defined, jet-like flow can be seen
moving from right to left at y = 0 cm. The width of the core-wise jet is approximately 0.6
cm, which is approximately 60% of the spanwise vortex core diameter, and it is located near
the vertical center of the test section where the spanwise cores are located for this forcing
amplitude. The peak velocity within the core-wise jet is 15% of the free stream velocity. It
is also approximately 1/3 of the peak velocity of the jet of fluid resulting from the pair of
counter-rotating streamwise vortex cores that develops in the cross-stream plane.

Due to its small size and velocity, it is believed that the axial flow does not play a

primary role in the mixing increase in this flow. The amount of fluid transported by the

124

 

-\\W\so—tt a c I t | i
o 4 \ f I - a I I I I I - 0-0\\‘.o\
- I] u - s \ ooooooo ’-'\\V\

I [’4‘\\~.\

O I

 

I---‘\\-

-~**&’/IIV\

\
\\
\\
\\
\\\\\\\w’/f
\\\\\\\\~s-o—ol I
\\\\\\\\~.-ot a c
\\\\\\\\\\~-on
\\\\\\\\\\\atc

,-I’l
III/aforpo
(rerovoooo
(Ira-rt-o'\
I’llo’l’vt‘

 

 

[fl I T I

 

—
-

lllllll

I
d
N
A
V .-

Figure 4.18: Axial flow in the A = 11.6% wake atx = 5.25 cm.

streamwise vortices is significantly more than that transported by the axial velocity.
Furthermore, the fluid transported by this core-wise velocity is located within the vortex
cores. Visualization and mixing results indicate that the fluid within each of the vortex cores
is from only one of the two streams. Thus, the fluid transported by the core-wise flow will

not come into contact with fluid from the other stream, and therefore, cannot mix..

4.9 Higher density measurements

The effect of spatial filtering on the <wx> field due to the measurement density was

examined by means of a limited set of measurements conducted at twice the standard
measurement density in each direction. Due to vortex stretching, the size of the streamwise
vortical structures has been found to vary with the phase of the measurement. This results

in a distribution of the non-dimensional measurement density, [/8 ranging from 2.5 to less

than I in the standard density results. As the actual rc decreases to smaller than the mean

spacing between measurements, the vorticity will appear to be spread over a larger region

125

.1 '. ..-
<0)x >(S ) 5:5
-25 -20 ~15-10 -5 0 5 10 15 20 25

Standard Density High Density

2. nqunmmmm‘mmmzmM-wulm-mauam-ran-nygmr-m-ttmlaw‘: 2., simummmumwamam.waanmnmmm ~

 
   

  

...

#J‘LJAJAlll

V (cm)

ii_i

 

 

-2- n- nmlum-«wfin-m‘fi
r

o..1,--1..1.-2..1_3..r._.4 OHI'-'1""-2""-3""-'4

 

Figure 4.19: Comparison of the standard and high density measurements of the streamwise
vorticity atx = 3 cm. Contour lines indicate the fl s", :10 s",..., £5 s" with the dashed lines
indicating the negative contour levels.

than it actually covers. If multiple cores are located within this region, they will appear as
a single structure. If rm, is much smaller than the measurement density, the structure might
not be detected at all. Doubling the resolution allows some of these features to be
distinguished.

Figure 4. 19 compares the high and standard measurement density data for the 1 1 .6%
forcing case at x = 3 cm. This pair of images displays the effect of the increased spatial
resolution. In the lower density measurements, a single negatively signed structure can be
seen in the upper half-plane. However, the higher density measurement clearly shows two
distinct negatively signed vortical structures. In the lower density measurement, the limited
resolution resulted in the two structures appearing to be one. The appearance of the

additional structures lends further credence to the model described in section 4.8 as it shows

126

further evidence of multiple vortical structures decreasing in radius due to stretching.

As expected, the magnitude of the measured vorticity has also greatly increased with
the increase in spatial measurement density. In the standard density results, U8 = 1.5. The
peak measured vorticity level at this phase of the positively signed vortical structure is 20 s".

In the higher density measurements with [/5 = 3, this value has doubled to 40 s". From the

results presented in Appendix A, the mean bias error for the high density results is
approximately 5 %. Using this value, the vorticity values can be corrected to yield an actual

estimated peak <0),> = 42 s'1 for this phase. This value will be called the “corrected” peak
vorticity. [/8 = 1.5 was not examined in Appendix A, however, a bias error of a factor of

two is not viewed as unreasonable given the very low measurement density.

When comparing the ratio of the peak levels of streamwise vorticity to the peak levels
of spanwise vorticity, different normalized measurement densities can result in this quantity
being reported incorrectly. As the physical size of the regions of streamwise vorticity is

. . . . <09"... . .
typically smaller than the spanWise vortiCity, —— is likely to be under reported.

I max

Examining the downstream location x = 4.25 cm, it is found that the corrected peak <wx> =

. . . . <0) > .
37 s" . At this location, the corrected peak <mx> = 44 3", resulting in ———"——"‘i = 0.85 . Usmg

I max

the uncorrected values from the experimental results, this ratio is calculated to be only 0.65.
As the streamwise vortex core stretches, its peak vorticity will increase and its core radius
will decrease. As the core radius decreases however, the peak vorticity levels will be
underestimated by a larger fraction. It is believed that this effect results in the peak levels
of streamwise vorticity reported in this series of experiments to be underestimated even after
they have been corrected because as the core radius shrinks to less than the mean spacing

between points, the structures will not be detected. It is at these small radii where the peak

127

levels of <wx> will be found.

The true value of 3%:423 for this flow could be greater than or equal to I . As was
shown in the previous sections, the streamwise vorticity is primarily formed by the
reorientation of the spanwise vorticity. Since the total circulation cannot change within the
vortex tube, the streamwise legs must have the same circulation as the spanwise vortex roller.

Thus, in the absence of effects which tend to reduce vorticity levels, such as diffusion which

has a relatively long time scale, as the “leg” is stretched, its peak levels <0)x> could increase

to levels larger than the peak levels of <mz>.

128

Chapter 5

Measurements at the 4 Hz and 8 Hz Forcing Frequencies

This section will discuss the results of measurements made for the 4 Hz and 8 Hz
forcing frequencies. These frequencies have been selected as flow visualization results
indicate that the general structure of the flow observed at these frequencies is similar to that
seen at 6 Hz. Note that the range of measurements in this study is significantly smaller than
that of Roberts (1985). Although there are many similarities between these sets of
measurements and the 6 Hz results presented in the previous two chapters, some interesting
differences are found. Unfortunately, as of present, no molecular mixing studies have been
conducted in the experimental facility at these frequencies, so the velocity and vorticity fields

cannot be compared to mixing field data.

5.1 Mean streamwise measurement plane flow properties at center span

Figures 5.1 and 5 .2 show the mean velocity fields at three downstream locations for
the 4 Hz and 8 Hz forcing conditions respectively. These are quite similar to the patterns
seen in the 6 Hz case. In the 4 Hz case, however, the free stream region, especially on the
top side of the test section, is very thin in comparison to the unforced results. This makes
it to determine the value to be used for the free stream velocity as there is no region where

the streamwise velocity is constant.

129

 

 

 

a Unforced 1.5
A 1: 4 Hz, A: 2.7% 1
o f: 4 Hz, A: 7.7% A 0.5
o f=4Hz,A=11.4°/o E
3 o
> -o.5
-1
-1.5
’20 ‘ 2
x=4cm
2 2
1.5 1.5
1 1
A 0.5 A 0.5
5, o 5 o
>. -o_5 >‘ -0.5
-1 -1
-1.5 -1.5
'20 * 2 20 ‘ ‘ ‘ 2
x =11 cm x :17 cm

 

Figure 5.1: Mean velocity profiles for the wake forced at 4 Hz at three streamwise locations.

 

 

 

 

2
D Unforced 1.5 as om @
A f: 8 ”Z, A: 2.4% 1
o f: 8 Hz, A: 9.8% ... 0.5
O f: 8 Hz, A: 13.8% E
3 o
>no.5
-1
-1.5
’20‘ 7 2
x=4cm
2 to O 2
1.5 w: 0 '3" O 1.5
1 1
A 0.5 A 0.5
5 o E o
>no.5 >no.5
-1 -1
-1.5 -1.5
Do
_20 .hé....a....é. _20. é
x =11 cm U (cm/8) x :17 cm

 

Figure 5.2: Mean velocity profiles for the wake forced at 8 Hz at three streamwise locations.

130

In the 4 Hz forcing cases, the peak velocity deficit of the wake, measured by the difference
between the free-stream velocity and the minimum velocity of the wake, always decreases
when forcing is applied. An increase in the forcing amplitude usually corresponds to a
decrease in the velocity deficit. However, at x = 11 cm and x = 17 cm, the deficit is constant
for all three forcing amplitudes.

Similar trends are seen in the 8 Hz forcing results. As would be expected, at x = 4
cm, the minimum deficit occurs for A = 13.8%, the largest of the three forcing amplitudes.
However, the largest deficit is found for the middle 9.8% forcing level, not in the smallest
amplitude results as was found in the 4 Hz and 6 Hz results. Forx = I 1 cm, the wake deficit
of all of the forced cases as well as the unforced are nearly identical. At x = I 7 cm, the deficit
of the three forcing cases remains approximately the same, however, the deficit of the
unforced case is lower than that of the forced cases. Between these two streamwise
locations, the width of the wake, measured as the distance from the free-stream region on the
bottom and top surface of the wake differs dramatically. Since the wake deficit for the
different cases is similar, this measure of the wake width can be easily seen in Figures 5.2.
At x = I I cm, the width of the wakes for the different forcing amplitudes and the unforced
case are nearly identical. By x = 17 cm, it can be seen the velocity deficits the three forcing
cases varies dramatically. This is especially noticeable in the high amplitude case where the
wake occupies nearly the entire width of the test section. A similar effect was seen in the 6
Hz forcing results for the A = 11.6% forcing amplitude. The 2. 7% forcing amplitude has the
narrowest wake and the 13.8% is in between.

Figure 5.3 shows the effect of the 4 Hz and 8 Hz forcing on the momentum thickness

at center span. In Chapter 3, it was discussed how this quantity could be related to the drag.

131

 

 

 

 

03:0 F=4Hz,x=4cm 03: - Unforced
. A F=4Hz,x=7cm - C1 4H1,2.7%Fordng
:v F=4Hz,x=11cm :0 4H2,7.7%Forclng
0.25-0 F=4Hz,x=14cm ’ 0.25-0 4H2,11.4%Fo¢cing
:O F=4Hz,x=17om : I 8H2,2.4%Forcing
. - F=8Hz,x=4cm . - o st,9.a%Forcing $
- A F=8Hz,x=7cm - o 8H2,13.8%Fon:lng ‘
0.2:, F=8Hz,x=11cm 02."
A ~ 0 F=8Hz,x=14cm A
E :0 F=8H2,x=17cm . E
0 0151- 0 0.15-
v . v r
o . 0 :
V ..
E x 0.1:
6 I I
V 9 ‘ 0.05;
I .-
0 8 t
1 or i l l L 1 L j o 1 1
00 10 1S 0
forcing (°/o RMS) (b)

 

Figure 5.3: Effect of forcing on the momentum thickness across the 4 Hz and 8 Hz forcing
frequencies. (a) Effect of forcing amplitude on 9 at constant streamwise location. (b) Effect
of streamwise location on 6 at constant forcing conditions.

The dependence of 8 on the forcing amplitude is shown in Figure 5.3a. For the 4 Hz forcing
results, 9 decreases slightly from the unforced value of 0.1 cm with increased application of
forcing. For the 8 Hz results, as in the 6 Hz, at x = 4 cm and x = 7 cm, the momentum
thickness decreases as forcing increases. For the locations farther downstream, 9 increases
with forcing amplitude.

Figure 5.3b shows the downstream development of 9 for the 4 Hz and 8 Hz
frequencies. For the 4 Hz cases, 6 decreases with downstream distance. This is likely an

artifact of the value used for the free-stream velocity. As in Chapter 3, the value of uo used
in the calculation of the momentum thickness is estimated from the average of the upper and
lower free stream velocities at the x location where 6 is computed. As the measurement
moves downstream into regions where no free stream value can be found, such as at x = I 7
cm for the 4 Hz cases, the value of uo tends to be overestimated due to the growth of the top
and bottom boundary layers.

132

For the 8 Hz forcing cases, an increase in 9 is apparent at all forcing amplitudes. The
A = 2.7% forcing shows a very slight increase. ForA = 9.8%, 9 triples as x increases from
4 cm to 20 cm. The momentum thickness for the 13.8% forcing case starts out very small
and then rapidly increases by more than factor of I 0 (from 0.02 cm at x = 3 cm to 0.28 at x
= 15 cm). This is very similar to the F = 6 Hz, A = 11.6% case. However, for x > 16 cm,

9 begins to decrease in the F = 8 Hz, A = 13.8% results, which was not seen in the 9 results
for F = 6 Hz, A = 11.6%. However, at the location of the maximum value of 9 in the F =

8 Hz, A = 13.8%, x = 15 cm, the 8 Hz results have traveled through I 3 forcing cycles
whereas the 6 Hz results have only traveled through 9.5 forcing cycles at this downstream

location. Thus, a maximum on 9 for the 6 Hz results may exist at x = 25 cm, which is I 3

forcing cycles downstream for this forcing frequency.

In the results for the 6 Hz forcing described in Chapter 3, a spatial periodicity was
noted in the umu field. This same periodicity is seen in the example of the um“ of the 4 Hz
and 8 Hz cases in Figure 5.4a and 5.4b respectively. For both forcing frequencies, a spatial
periodicity with a wavelength equal to the forcing wavelength is seen. Although only the
highest measured amplitude results are shown, the effect is also seen in the lower amplitude
forcing results for these frequencies however with a reduced magnitude. The v velocity RMS
for these same two forcing cases is shown in Figure SAC and 5.4d. Generally, a pattern with
the largest RMS values in the middle of the layer and the values decrease moving towards
the top and bottom of the facility. However, in the F = 8 Hz, A = 13.8% the pattern begins

to split apart similar to what will be seen for the 62 later in this section and for <(oz> in

section 5.2.
Examining the 4 Hz results, it can be seen that the region of spatial periodicity

133

u (cm/s) :2:

'0 0.5 1 1.512 2.5 3

F=4Hz,A=11.4% m
2.-

 

 

'rvvv'vvvv vav vav

 

F = 8 Hz, A =13.8% (‘0
2 .-

 

 

 

vm(cmls) .5: (b)

o 0.51 1.5 22.5 3
2:

 

 

 

 

 

 

 

 

2:
e ‘2'
o O?
V :
>43
;
-2L

6 2:5 ‘ é ‘ ‘ 7‘5 1'0 7712.5‘ ‘ ‘ '1‘5‘ ‘ ”1715‘ ‘ ‘ ‘20

x(cm) (d)

Figure 5.4: RMS of streamwise (u) velocity and transverse (v) velocity for the highest
amplitude cases of the 4 Hz and 8 Hz forcing. The contour lines start at 0.2 cm/s and spaced
every 0.2 cm/s. (a) umu for F = 4 Hz, A = 11.4%. (b) um, for F = 8 Hz, A = 13.8%. (c) vmu.
for F = 4 Hz, A =11.4%. (c) vm for F = 8 Hz, A =13.8%.

134

 

 

 

 

25 C] F=4H2 2? D O O F=8Hz
£0080 a A=2.7% : DD 0° 00 a A-2.4%
15? DD O o A=7.7% 15:3 0 O o A-9.8%
E g 0% o A=11.4% : g o A-13.8%
1.— D 00 1- U
D O " D
i a 1 a o
. O - D O O
0.5- 03 0.5- a o o
. o a . a o o
is: 0" 33a) g 0L3 0<> 0 O
I : OCO ; DC] 000
I C] C]
-o.5- a 08 -o.5- a 08 O
: D O O C] o
. D O O E]
.13 O O U
-1-a <> o -1-0
:0 O O C]
e 8 00 a
15:3 0 0o -15-Cul
1:1 8 O _ 1:1
-2L1111111111L1114L11llllllLLJl] -2pllllllillllllllllllll111111!ll
o 0.5 1 1.5 2 2.5 3 o 0.5 1 1.5 2 2.5
(11) um (cm/s) (b) um (cm/s)

Figure 5.5: RMS of streamwise velocity vs y location at x = 4 cm. (a) 4 Hz forcing. (b) 8
Hz forcing.

reaches nearly to the wall of the test section. For this forcing case, no region of a uniform
value for the um, can be seen at any downstream location. This is also apparent in line plots
of the umu , an example of which is seen in Figure 5.5. For the two higher forcing amplitude
of the 4 Hz results in Figure 5.5a, no uniform region of um. is found with transverse
location. It is therefore not possible to measure an exact, constant value for the free-stream
fluctuation levels as the value of the um. is always changing. For this reason, it was noted
in Chapter 2 that the forcing levels listed for the 4 Hz forcing frequency are used only for
labeling the different cases. For the 8 Hz results displayed in Figure 5.5b, a steady region
of um, is seen. This was also seen in the 6 Hz results discussed in Chapter 3.

Figures 5.6 and 5.7 show the mean vorticity field for three measured forcing
amplitudes for the 4 Hz and 8 Hz forcing frequencies as well as the unforced measurement
results. For the 4 Hz forcing results, a region of nonzero 52 is seen in the lowest forcing

amplitude at early x locations. As will be seen in section 5.3, this is the region where the

135

as") I 7

-15-10 -5 O 5 1015

 

 

.
.
-2-

 

 

2:

 

 

..
.
.

 

 

2 r

 

 

I
.
.
-2 -

 

 

2 .
p
p
y.

 

 

 

Figure 5.6: Mean streamwise vorticity for unforced case and three different amplitudes for
the 4 Hz forcing. The contour lines indicate the 1:3, 1:6, :15 s'l contour lines with the
dashed lines indicating the negative values.

136

.1. x.“
,n: f!

-15-1o-5 o 5 1o 15

   

 

.
p-
h

-2»-

 

 

 

 

 

2."

 

 

-2;

 

 

2."

 

 

 

 

0"
N
01
U]-
\1
01
d
O
..1
gm
0'!
—.L
U!
c—A
N
u:
N
0

Figure 5.7: Mean streamwise vorticity for unforced case and three different amplitudes for
the 8 Hz forcing. The contour lines indicate the 13, :6, :1115 s"contour lines with the
dashed lines indicating the negative values.

137

vorticity shed from the splitter plate is forming into the gaussian shaped spanwise vortex
tubes. In the higher forcing amplitudes, 652 is nearly zero in the wake region. This is very
different from what was seen in the 6 Hz forcing amplitude results in which regions of mean
vorticity were seen in the all but the highest forcing amplitude.

The 52 field for the 8 Hz forcing results is very similar to the mean field for the 6
Hz forcing. Two regions of non-zero 52 are seen for the 2.4% and 9.8% forcing amplitudes,
a negative region in the top half-plane and a positive region in the bottom half-plane. For
the 13.8% forcing amplitude, two regions of mean vorticity are seen again. However, these
regions split apart with the negative mean vorticity moving towards the top wall of the test
section while the positive region moves towards the bottom of the test section. A similar
effect was previously seen in the vm field. This splitting apart of the mean vorticity field
was not seen in the 6 Hz forcing results. In the 6 Hz forcing frequency results, isolated
regions of mean vorticity were seen. In Chapter 3, this was explained by the variation of the
convection speed of the spanwise vortex rollers. This same effect is also seen in the 8 Hz
forcing results.

In Chapter 3, the mean field of g at z = 0 cm was seen to be a good predictor of the
x location where molecular mixing field begins the increase dramatically for the 6 Hz
forcing. Figures 5.8 and 5.9 display the % field for the 4 Hz and 8 Hz forcing frequencies.
No region of increased % is seen in the 4 Hz forcing results. However, a wedge-shaped
region of strong negative % is seen in the highest amplitude 8 Hz forcing results. In section
5.3, it will be shown that the strong recirculatory flow pattern found in the F = 6 Hz, A =
11.6% case is also found for this forcing case. It is believed that this will indicate a rapid

increase in the amount of mixed fluid. Without molecular mixing results, however,

138

 

a337,413")

F=4Hz,A=2.7% ~3Q2-1o123

 

 

 

 

 

 

 

 

Figure 5.8: Mean 93% field for 4 Hz forcing at three perturbation amplitudes. The contour
lines indicate regions of positive and negative sign.

139

Walk-1) I

-3-2-1012 3

F=8HLA=2A%

 

 

 

 

-2

 

 

 

 

 

Figure 5.9: Mean “21% field for 8 Hz forcing at three perturbation amplitudes. The contour
lines indicate regions of positive and negative sign.

140

this cannot be confirmed.

5.2 Phase-averaged streamwise measurement plane results at center-span

In Figure 5.10, a sample of the instantaneous vorticity field for the unforced case as
well as the phase-averaged vorticity field for the three 4 Hz forcing frequencies cases. In the
upstream region of the F = 4 Hz, A = 2. 7% case, the process of the vortices forming into
well-defined, circular cores can be seen. Initially, the vorticity is continuously shed from the

splitter plate. As the vorticity convects downstream, the elongated, isolated regions of <mz>

form. In addition to convecting downstream, it is apparent from animations of the data that
the elongated regions are also rotating about their center. As these regions rotate and convect
downstream, the cores become circular in shape and take on the Gaussian vorticity profile.
This is close to the classic picture of the formation of vortex cores.

In the F = 4 Hz, A = 7. 7% results, the vortex street is nearly vertically aligned. This
alignment was seen in the F = 6 Hz, A = 11.6% forcing case which was one of the high
mixing cases. This case, however, does not exhibit any of the other characteristics found in
the high mixing case such as a region of negative 7%.— ,a mean recirculatory pattern in the
cross-stream plane velocity field and regions of streamwise vorticity moving towards the
spanwise center of the test—section. As previously stated, no mixing results exist for this
forcing condition so it is not possible to confirm that this case will not result in increased
molecular mixing.

In the highest amplitude forcing case for the 4 Hz forcing case, the vortex cores shed

from the splitter plate exhibit a splitting type process. At each downstream location where

141

   

1

15 20 25

 

y
p
p

-2-

 

2 ’
p
b

 

 

b
r
b

-2-

 

 

 

 

 

 

 

Figure 5.10: Phase-averaged streamwise vorticity for unforced case and three different
amplitudes for the 4 Hz forcing. The contour lines indicate the :5, 110, :1:25 s"contour
lines with the dashed lines indicating the negative values.

142

<(oz> is present, two vertically aligned vortex cores of the same sign are seen. For x < 8 cm,
these regions are connected. As the cores convect downstream, the splitting process
completes and two separate cores can be.

Along the top wall of the facility, a second region of vorticity is being shed from the
top wall boundary layer. Although this second shedding region is most apparent in the
11.4% forcing, it can be seen in the 7.7% forcing results as well. A similar effect is not seen
on the bottom boundary layer. These shed vortices are not observed in the 6 Hz or 8 Hz
forcing cases. It is believed that the 4 Hz forcing is exciting a disturbance in the surface of
the top wall which is then shedding vorticity. This disturbance is not excited in the other two
forcing frequencies studied.

Figure 5.11 displays the phase—averaged 8 Hz forcing frequency results. Although
the peak vorticity magnitudes of the 2.1% and 9.8% forcing amplitudes are differ
significantly, the spacing between these two cases is nearly identical for all streamwise

locations. In the 13.8% forcing results, the peak values of <(oz> in the vortex array are larger

yet. In this case, however, the transverse separation between the vortex cores increases

dramatically. For x > 12.5 cm, the negatively signed <(oz> moves towards the top wall while
the positively signed <wz> moves towards the bottom wall. This pattern is similar to what

was seen for the F = 6 Hz, A = 11.6% case, however the effect is greatly enhanced in the
high amplitude 8 Hz forcing case.

Figure 5.12a shows the effect of the 8 Hz forcing amplitude on the peak levels of (i)z
as the vortex tube convects downstream. For all three forcing amplitudes, the vorticity levels
are the largest at the locations farthest upstream, but then decreases. At x = 4 cm, the peak

levels of (02 range from 20 s" for to 2.4% forcing amplitude to 40 s" for the 13.8% forcing

143

 

<coz>(s")

 

 

 

 

 ~ Stream-.5

 

p
-

 

 

 

 

 

 

6‘ ‘2.5“"é ‘ 7i5““1o““12.5““15‘ ‘17.5“ ‘20

Figure 5.11: Phase-averaged streamwise vorticity for unforced case and three different
amplitudes for the 8 Hz forcing. The contour lines indicate the :5, 1:10, :i:25 s"contour
lines with the dashed lines indicating the negative values.

144

 

 

 

 

 

 

 

 

F = 8 H2 F = 4 H2
4 4O
0 II. 1:] 13.3% forcing «o, _ o 11.4% forcing «a,
A 9.8% forcing «n, A 7.7% forcing «0,
Q o 2.4% forcing «n, o 2.7% forcing «a,
. U I 13.8% forcing +03. _ I 11.4% forcing +0),
30 A 9.8% forcing +631 30 A 7.7% forcing +6)z
- % (Rm 0 2.4% forcing +10, . O 2.7% forcing +111,
.A ' 0 :~ i
'0) _ i .m .
V 20 7f 20 I
A Ii-
3" ' Q a" ’ 9 a-
D
W 8 a '.
. A .
10 ‘ 10 e 2 % %
obi l 111111111 1' ohiaiiliLgliixilJnxxl
0 5 10 15 20 0 5 10 15 20
(a) X (cm) (b) x (cm)
2 - 0.5 -
, a 13.8% forcing «n, . a 11.4% forcing «n,
_ A 9.8% forchg ~03, A 7.7% forchg «o,
o 2.4% forchg «a, [@3 _ o 2.7% forchg «n,
- 0.4 -
1.5 - ’
_ 1'
D : CD CD
:19 0.3 - (53 ‘9
a _ 1:1 ‘5
z 1 L A 3 _
- A 0.2 :- Q)
0 5 P % é CD 0]
.. . A
' . as 6% -
. m 0.1 . Cb E
_ O - 243 AA AA AA
0 . .fi 1 . . i i . l i I 0 i i l i d] i . i i
0 5 1O 15 20 0 5 1O 15 20
(c) X (cm) (d) x (cm)

Figure 5.12: Effect of downstream location on peak <a)z> and spacing ratio. (a) Peak
vorticity for 8 Hz forcing. (b) Peak vorticity for 4 Hz forcing. (c) b/a for 8 Hz forcing. (d)
b/a for 4 Hz forcing. Note that the scale in (c) and (d) are not the same.

145

amplitude. The decay rate of the vortices is significantly different and at x = 17.5 cm, (02 =
6 s" for all three forcing amplitudes. This rate of decay of the peak <(nz> is slightly faster
than that observed with the 6 Hz forcing amplitude. This is likely the result of the larger
values of % found in the 8 Hz forcing frequency results.

The behavior of the peak <wz> for the 4 Hz cases, seen in Figure 5.11d, is
significantly different than that of the 6 Hz or 8 Hz forcing. Unlike the 6 Hz and 8 Hz cases,
the magnitude of the peak <wz> only increases by 50% with increases in the forcing
amplitude. In the 6 Hz and 8 Hz cases, increasing the forcing amplitude results in the peak
<(uz> increasing by over a factor of two. The vorticity levels of the 4 Hz forcing cases are
also significantly lower than in the other cases. The highest forcing amplitude for the 4 Hz
cases has a maximum <(oz> = 20 s'1 . This value is approximately the same as the peak
values seen for A = 2.4% for the 8 Hz case and A = 2.1% for the 6 Hz forcing. As the
vortices convect downstream, the decay of the peak <0)z> for the 4 Hz forcing amplitude is
significantly less than that of the other two forcing frequencies. In the range 3 cm < x < 20
cm, the peak <(oz> only drops by a factor of two for all forcing amplitudes. This is again
similar to the low amplitude results of the other two forcing frequencies. It is worthwhile
to noted that the value of g which causes the majority of the decrease in peak <wz> values,
measured for the 4 Hz forcing frequency is also of the same order as that of the low
amplitude forcing results for the 6 Hz and 8 Hz cases.

The non-dimensional spacing between the vortices, b/a, for the 8 Hz and 4 Hz forcing

is found in Figures 5.2c and 5.2d respectively. Note that the scale on the two plots is

different because of the large differences in b/a between these two cases. For the 8 Hz

146

forcing, the highest amplitude forcing initially has a value very close to zero. This indicates
the near alignment seen in Figure 5.1 1. As the flow convects downstream, it increases to a
value of nearly two. At locations far downstream, the b/a quantity may be misleading since
the two-dimensional rollers likely do not exist at these downstream location. This is
approximately double the maximum value seen in the 6 Hz forcing results. This dramatic
increase in spacing is clearly seen in Figure 5.11 where the positive and negative vortices
split apart with the negatively signed rollers moving towards the top of the test section and
the positively signed rollers moving towards the bottom of the section. The two lower
forcing amplitude cases for 8 Hz forcing also show an increase, however it is not nearly as
large as seen in the A =13.8% results.

In both the 6 Hz and 8 Hz forcing data, the high amplitude forcing results show a
large increase in spacing with x. For the 4 Hz case shown in Figure 5.11d, however, the
spacing ratio remains relatively constant for the higher amplitude results. An increase in b/a
is only observed in the lowest amplitude forcing results where b/a increases from 0.2 to 0.32
between x = 4 cm and x = I 1 cm. The ratio remains constant for x > I 1 cm. It is believed
that the small values seen for x < 1 I cm is due to the formation process of the vortex core.
From Figure 5.10, it appears that the vortex cores are still in the process of evolving into the
tightly wound core. Since the regions of vorticity are initially shed from the splitter plate,

they are closer together then in the fully formed vortex street.

5.3 Mean streamwise vorticity and cross-stream plane velocity results

Figure 5.13 shows the development of the (17x field for three forcing amplitudes of

the 4 Hz forcing frequency. For the lowest amplitude, no streamwise vorticity can be seen

147

.x

F=4HZ
A=11.4°/o

-15-12 .945 -3 o a 6 91215

3.25 cm
V (cm)

x
E A
0 E
in 8.
.5 >.
11
X

15.75 cm
7 (an)

X:

X = 20.25 cm
V (cm)

 

 

 

Figure 5.13: Mean 0),, at 4 streamwise locations for the 4 Hz forcing frequency cases. The
contour lines indicate the 1-3, 1:6, :15 s"contour lines with the dashed lines indicating the
negative values.

148

at the farthest upstream location. As the flow develops downstream, regions of 5): can be
seen at x = 6.5 cm. As the measurement plane moves further downstream, both the peak
magnitude of the 61—): as well as the size of the region where vortical structures are found
decreases. ForA = 7. 7% andA = 11.4% two counter-rotating regions of a: are seen at the
farthest upstream location. As the flow develops downstream, these regions show a small
increase in the value of peak vorticity, reaching a maximum at x = 6.5 cm. Further
downstream, the peak magnitude of these regions begins to decrease. Multiple regions of 61—)—x

are seen at the farthest downstream locations. However, these regions remain close to the
sidewall.

The development of the 5): field for the 8 Hz forcing seen in Figure 5.14 is very
similar to the 6 Hz forcing results. For A = 2.4% and A = 9.8%, the counter-rotating
streamwise vortex pair is present. The downstream development of these cases is similar to
the F = 6 Hz, A = 7.3% results. As the measurement plane moves downstream, these
regions weaken and they remain relatively close to the wall. For A = 13.8%, the counter-
rotating pair of mean 5; splits into multiple regions at farther downstream locations. As
the measurement plane moves farther downstream, the regions of mean streamwise vorticity
moves towards the center of the test section. This is similar to what was seen in the high
mixing case ofF = 6 Hz, A = 11.6%.

The cross-stream plane velocity field (v-w velocities in the y-z plane), for the 8 Hz
case, shown in Figure 5.15, also displays characteristics similar to the 6 Hz forcing. For the
A = 13.8% forcing, a mean recirculatory pattern is seen at x > 6.5 cm. This is again
reminiscent of the results seen in the cases which have shown a large amount of mixed fluid.

The lower forcing amplitude results do not show the presence of the mean recirculatory

149

'“3 " F=8HZ
-15 ~12 ‘9 ’5 '3 0 3 6 9 12 15

<col> (3")

A = 2.4% A = 9.8% A = 13.8%

EA
35.
(ah
ll
X
EA
",5.
d>
ll
X

x = 15.75 cm
Han)

20.25 cm
Y (CM)

x:

 

 

 

Figure 5.14: Mean (1)" at 4 streamwise locations for the 8 Hz forcing frequency cases. The
contour lines indicate the :1-3, :6, :15 s"contour lines with the dashed lines indicating the
negative values.

150

 

\\\\‘
\iii‘
\\\\\
\\\\\

‘1‘

 

 

/€::

5 A . ‘3
..r .

a 5. f);

- >. \~/

'3 :15;

x ‘41}

E E

IQ 8

o >-

II

>1

 

 

 

 

 

x = 15.75 cm
y(cm)

 

 

 

 

 

x = 20.25 cm
1! (cm)

 

 

 

 

 

 

 

Figure 5.15: Mean v-w velocities in the y-z plane at 4 streamwise locations for the 8 Hz
forcing frequency cases.

151

pattern. The mean cross—stream plane velocity field for the 4 Hz forcing (not shown here)

does not display the recirculatory pattern at any of the forcing amplitudes.

5.4 Phase averaged streamwise vorticity and croSs-stream plane velocity results

As with the 6 Hz forcing results, the streamwise vorticity field for the 4 Hz and 8 Hz
forcing cases display a variation with phase. Figure 5.16 shows the variation of the 8 Hz, A
= 13.8% forcing case with phase at x = 3.25 cm. As the flow moves through the forcing
cycle, a variation between the peak of the positively signed streamwise vortex tube and the
negatively signed vortex tube is seen as the flow convects past the measurement location.
At x = 20.25 cm, shown in Figure 5.17, the regions of streamwise vorticity have moved
towards the center of the test section and nearly the entire test section is occupied with
regions of streamwise vorticity. For the low amplitude, 8 Hz forcing cases and all of the 4
Hz cases, this inward movement is not seen.

Based on the data presented, it is believed that the F = 8 Hz, A = 13.8% case will
exhibit enhanced mixing seen in the F = 6 Hz, A = 11.6% case. It displays the characteristics
seen in this known, high mixing case such as region of negative % , a recirculatory flow

pattern in the cross-stream plane, and regions of <mx> moving quickly towards center span.

None of the other 4 Hz and 8 Hz measurement exhibit these features. It will be interesting

to see if mixing studies will confirm this hypothesis.

152

  
 

<oox> (5")

-15-12 -9 -6 -3 0 3 6 9 1215 A=13.8%

 

Figure 5.16: Variation of phase-averaged (i)x with phase for F = 8 Hz, A = 13.8% at x = 3.25
cm. The contour lines indicate the 1:6, :9, 1:15 s“contour lines with the dashed lines
indicating the negative values.

153

<(n‘> (8")

-15-12-9—6-3 O 3 6 91215

2-

Man) 2

 

 

 

.‘2 .‘2
1 (cm) 2 (cm)

.‘2
2 (cm)

Figure 5.17: Variation of <wx> with phase for F = 8 Hz, A = 13.8% at x = 20.25 cm. The
displayed phases are separated by approximately Act) = 0.08 and phase runs from right to left

and then top to bottom. Contour lines indicate the fl, i6, :15 s‘lcontour lines with the
dashed lines indicating the negative values.

154

Chapter 6

Conclusions

Velocity and vorticity field measurements were made in a forced, confined wake in
order to both better understand the vorticity interactions in this flow and to examine the
mechanisms which may lead to the mixing enhancement that has been found in previous
studies. Molecular Tagging Velocimetry (MTV) was used to measure two components of
the velocity field over the streamwise plane (u-v velocity components in the x-y plane) as
well as over the cross-stream plane (v-w velocity components in the y—z plane). The spanwise
and streamwise vorticity is computed from their respective velocity fields.

The experimental results represent several advances in the MTV measurement
technique. Between 600 and 800 velocity vectors were measured per image plane. This
represents an increase of more than 50% over previous MTV studies. The present study also
represents the first time that this technique has been used to make whole-field measurements
in a plane where the mean flow is moving directly out of the measurement plane. Advances
were also made in the post-processing of data and the determination of the best methods to
remap velocity data onto a regular grid and to compute the vorticity.

The results over the streamwise plane measurement yielded new information about
the overall flow properties. The um, field shows a distinct spatial periodicity that is not
found in the unforced case or in unconfined forced flows such as in the wake of an oscillating

airfoil. A model was developed which relates this spatial periodicity, which is a consequence

155

 

of the confinement, to the phase difference between the forcing input into the flow, and the
shedding of vorticity. From these data, it was determined that the phase at which vorticity
is shed from the splitter plate is dependant upon the forcing amplitude, highlighting the non-
linear nature of the shedding process.

The forced wake flow is dominated by the shedding of concentrated, spanwise vortex
cores. The behavior of these cores is highly amplitude dependant. As forcing amplitude
increases, the vortex spacing ratio, b/a, initially decreases. However, as the flow develops
sufficiently far downstream, b/a increases with increased forcing amplitude. It is recognized,

however, that at some point downstream, concentrated regions of (1)z no longer exist and the
meaning of b/a at those locations is unclear. As the regions of (i)z convect downstream, the

levels of peak vorticity within the cores decrease. This decrease has been shown to be
dominated by vortex stretching rather than diffusion. A very small negative value of 93%;; has
been found to be sufficient to generate a very large decrease in peak vorticity levels.

In addition to causing the decay of peak vorticity levels in this flow, the enhanced
mixing in the center span was found to be well correlated with increases in jail.“— at center
span. A large region of mean -2177 is found in the central region of the test section and a large
region of +% is found near the top and bottom walls of the test section for the forcing cases
which resulted in large amounts of mixed fluid. This pattern is consistent with a mean
recirculatory flow-field found in the cross-stream plane. It is of course recognized that in
general, the presence of _% alone does not result in mixing enhancement. However, in this
confined flow, it is indicative of patterns generated by the streamwise vorticity which
generates the increased mixing.

An examination of the structure of the spanwise vortex rollers close to the sidewalls

156

 

 

of the facility was also conducted. For the high amplitude forcing cases (for 6 Hz forcing,
those cases with A > 11.6%) the level of peak spanwise vorticity decreases at locations
within the tube that are closer to the sidewall. This is likely connected to the vortex tube

reorienting into the streamwise direction. However the peak (1),, remains relatively constant

for the low and moderate amplitude results (A < 8.5%). Further, it was found that the
spacing ratio between the vortex cores increases only slightly as the tube moves closer to the
wall for these lower amplitude results, whereas it shows a large increase in the ratio is seen
for the higher amplitude results.

Although core-wise axial flow was found to be present within the spanwise rollers,
its likely does not result in a large increase in the amount of molecularly mixed fluid. The
velocity generated by this motion is small and it is only present over a very small spatial
extent. Further, the axial motion is located in the center of the vortex core where mixing
results indicate the presence of fluid from only one of the two streams. Rather, it is the
streamwise vorticity generated by the reorientation of the primary spanwise vorticity which
generates the increased molecular mixing. In the high amplitude forcing case, multiple
regions of streamwise vorticity are present. These regions generate a recirculating type flow
pattern which will result in large quantities of unmixed fluid to be pumped from the free-
stream regions in the top and bottom of the test section into the middle where they can
interact. These regions also generate small scale motions which will increase the surface
area over which the unmixed fluid can interact and mix.

Results of the measurements in the cross-stream plane show that for the high
amplitude forcing cases a mean recirculatory pattern is present. This feature is absent in the

lower forcing amplitude cases. This flow pattern is consistent with the measurements of "“782

157

 

 

previously discussed. In the high amplitude results, multiple regions of (17x can be found

close to center of the facility sufficiently far downstream. Mixing is of course generated by

the instantaneous (1)" (or <wx>) which is also found in the central region. In contrast, the

regions of both a: and <mx> remain close to the wall in the lower amplitude cases.

The development of <(ox> differs dramatically between the high and lower forcing
amplitude cases. Initially, a counter—rotating pair of streamwise vortices is found close to the
side wall of the facility for all forcing amplitudes. As the flow develops downstream, these
regions begin to move closer to the center of the test section. However, the rate at which
these vortices move inward differs greatly between the low and high forcing cases. For the
6 Hz forcing frequency, at 1 1.6% forcing, the regions of streamwise vorticity quickly occupy
the entire test section. For amplitudes of 7.3% and lower, they remain close to the sidewalls
of the facility. Furthermore, multiple regions of <(l)x> are found in the high amplitude cases.
Although present in the lower amplitude results, the number as well as the strength of these
structures is greatly reduced.

A model was developed to account for the development of streamwise vortical
structures in this flowfield. As the spanwise rollers convect downstream, the spanwise
vorticity is reoriented into the streamwise direction. The ends of these streamwise legs are

fixed near the tip of the splitter plate. The multiple regions of <(ox> in each cross-stream

measurement plane are the result of the passage of multiple reorientated spanwise rollers.

As a <wz> roller convects past a particular downstream location, a region will be reoriented
into the streamwise direction. The newly reoriented <wx> located closest to the center plane.

The “legacy” <wx> from the passage of the roller the previous cycle moves closer to the side

158

 

wall under the action of the newest region.

Similar effects were seen in the development of the velocity and vorticity field for
frequencies both larger (8 Hz) and smaller (4 Hz) than 6 Hz case. No mixing studies are
available at those frequencies to compare with the present results. From the velocimetry data
at these frequencies, it is believed that enhanced mixing will be present in the high amplitude
8 Hz forcing. This case had many of the characteristics, such as a mean recirculatory region
and regions of <wx> moving quickly towards the center of the facility that were present in
the cases where it is known that a large amount of mixing occurs at the 6 Hz forcing
frequency. It is believed that little to no mixing enhancement will be found in any of the 4
Hz forcing cases studied. However, mixing studies need to be conducted to confirm this

hypothesis.

159

 

 

Appendices

 

160

Appendix A

The Placement of Irregularly Spaced Velocity Measurements on a
Regular Grid and the Calculation of Out-of-Plane Vorticity

A.l Introduction

In recent years, many authors have made use of full-field, two-component optical
velocity measurement techniques, such as Particle Image Velocimetry (PIV) to derive other
flow quantities such as the out-of—plane vorticity. The data gathered from PIV are normally
thought to be gathered on a uniformly spaced grid which allows for a variety of methods to
be utilized to calculate these quantities. The development of Molecular Tagging Velocimetry
(MTV) has placed an additional complication on the calculation iof these quantities in that
the data are not normally collected on a uniformly spaced grid. This section deals with the
questions related to remapping the MTV data onto a regularly spaced grid and the method
used to compute the out-of—plane vorticity component from these remapped data sets.

Molecular Tagging Velocimetry is a full-field optical diagnostic which allows for the
non-intrusive measurement of a velocity field in a flowing medium. This technique takes
advantage of molecules which have long-lived excited states when tagged by a photon
source. The evolution of the luminescence of these molecules is tracked over the
luminescence lifetime in order to determine an estimate of the velocity field. MTV has been

used by several authors such as Gendrich et al (1994), Stier (1994), Cohn et al. (1995), Hill

161

 

 

“Y
I II .. w

 

 

 

 

and Klewicki (1996), Koochesfahani et al. (1996), Cohn and Koochesfahani (1997),
Gendrich, Bohl, and Nocera (1997), and Gendrich, Koochesfahani, and Nocera (1997) to
make measurements in a wide variety of flows.

This technique can be thought of as the molecular equivalent of Particle Image
Velocimetry. Rather than tracking particles placed in the flowing medium, the luminescence
lifetime of the tracer molecules is tracked. A more complete description the implementation
of the molecular tagging technique and the parameters necessary for an optimal experiment
can be found in Gendrich and Koochesfahani (1996) and Gendrich (1998). The accuracy of
velocity measurements made using Molecular Tagging Velocimetry is equivalent to that of
PIV. Gendrich and Koochesfahani (1996) report that the 95 % confidence interval for the
accuracy of this technique is 0.] pixel. This means that 95 % of the measurements are more
accurate than 0.1 pixel. Thus, a displacement of 10 pixels will yield a dynamic range of 100.

In the implementation of MTV, a series of laser-lines is used to generate a two-
dimensional spatial distribution in the intensity field within the flowing medium. Figure A. 1
shows a sample grid used in MTV images. As seen in this figure, these measurement points
are typically not regularly spaced. Thus, it is necessary to place data on a regular grid before
flow variables can be computed via standard finite difference techniques. It should be noted
that it is possible to generate the lines on a regular grid. However, both a study at MSU and
Spedding and Rignot (1993) have reported that the best estimate of the velocity value
acquired through the use of a measurement technique which tracks a tracer in a flow is
located at a point midway between the initial and final location of the feature being tracked.
Thus, even if the laser-lines shown in Figure A.1 were uniformly spaced, it would still be

necessary to remap the data onto a regular grid. Further, unless special care is taken in the

162

 

 

  

1 l “ R , , 3
Figure A. 1: Sample MTV measurement grid.
selection of the measurement windows, data collected from PIV measurements is not actually
on a uniformly spaced grid and should be re-mapped onto a regular grid to achieve the most
accurate results.

Agui and Jimenez (1987) examined several different means of interpolating velocity
data acquired through the use of particle tracking onto a regularly spaced grid. Comparing
the root mean square of the difference between the interpolated velocity and the actual
velocity in a simulated flow, it was found that the best results were obtained using certain
polynomial interpolaters and a “k-rigging” technique. However, the advantage was small
with respect to others methods, so interpolation was performed using a simple convolution
with an “adaptive Gaussian window”. No additional comparisons between the techniques
were included.

Spedding and Rignot ( 1993) compared the adaptive Gaussian window technique with
a “global basis function” in both the accuracy of the interpolation of velocity information and

the calculation of out-of—plane vorticity information. Using the root mean square error of the

163

 

entire flow field, comparisons were made of the results of both the simulated velocity field
of an Oseen vortex and the “bootstrap” error between several computed and one real flow
field. This error calculation method attempts to place together into one measurement both
the mean bias component and the random component of the error found in the entire
remapped velocity (or vorticity) field. It will be shown in the following sections that this
may lead to misleading results. This study found that the global basis function produced
results that were generally superior to the adaptive Gaussian window. It was reported that
using the global basis function, the velocity field could be reconstructed to an overall
accuracy of 2.5% and the vorticity field could be reconstructed to an overall accuracy of 5%
given a suitable choise of grid density.

Several studies have examined the accuracy of vorticity calculations from data
already on a regular grid. Abrahamson and Lonnes ( 1995) compared calculating the vorticity
from the local circulation of the velocity field as well as using a local least-squares method
to calculate the vorticity field. It should be noted that computing the vorticity on by the use
of the circulation method is identical to the use of a finite difference calculation on a
specially filtered version of the velocity field. This will be shown in the next section. The
study by Abrahamson and Lonnes found that both methods spatially filter the flow field and
had the largest error in the regions of local maxima and minima in the flow field. However,
the circulation method generally out-performed the least-squares technique in this study.

Luff, et al. (1999) used the simulated Oseen vortex field to compare the effect of
noise on 1St and 2“d order finite difference techniques as well as the 8-point circulation
method for the calculation of vorticity. The 8-point circulation method will also be used in

the present study and will be described in the next section. Luff, et al. studied the

164

 

 

propagation of uncertainties into the vorticity field from several sources including the
experimental velocity uncertainty, smoothing of the velocity field in order to eliminate spikes
in the vorticity data, and spurious or “drop-out” velocity vectors generated in the velocity
field. It was found that the circulation and 1St order finite difference methods produced
smaller experimental uncertainties than the 2nd order finite difference technique. However,
the average uncertainty was found to be :3 7% of the average value of vorticity (16.5% of
the peak vorticity) with a uncertainty of the order of 550% of the average value of vorticity
(97% of the peak vorticity). The use of an “PS-smoothing” procedure resulted in a factor of
10 improvement in both the average and peak uncertainties.

Fouras and Soria (1998) separated the error in the vorticity computation into two
portions, the mean bias error and the random error due to the propagation of the error in the
velocity field measurements to the vorticity field. The mean bias error is a result of the
spatial filtering of the original data and is the cause of the underestimation of the peak
vorticity levels. The random error is the result of the errors in the original velocity
measurement technique used to acquire the velocity information. These two errors cannot
be minimized simultaneously as they are both effected by the spacing between velocity
measurements. As the distance between the velocity measurements decrease, the value of
the computed vorticity measurement will more closely match the exact value. However,
decreasing this spacing makes the measurement more sensitive to the random error found in
the initial measurements. In many cases, the mean bias error is significantly larger than the
random error propagated into the vorticity field from the velocity field. Simulations of an
Oseen Vortex showed that for data already on a uniformly spaced grid this study found that,

differentiating a local 2nd order polynomial least squares fit produced more accurate results

165

 

 

then estimating the vorticity using a 1"t order finite difference technique.

The present study makes use of a simulation of an Oseen vortex in order to study the
effect of remapping an irregularly spaced velocity field onto a regular grid and on the
calculation of the out-of—plane vorticity component. The velocity field will be remapped by
fitting the irregularly spaced data to a 2'”, 3’“, or 4‘h order polynomial. The vorticity field will
then be calculated through the use of one of four methods: differentiating the polynomial fit,

1St and 2nd order finite difference techniques, and an 8-point circulation method.

A.2 Comparison Method

In order to determine the accuracy of the various remapping and vorticity calculation
techniques, a simulation with the often used Oseen vortex was conducted. This velocity field
has been used in the majority of the previously mentioned studies to determine the accuracy
of various interpolaters. This flow has an out-of-plane vorticity, 0),, and an azimuthal

velocity, ue profile described by:

—(1‘2 I rczmc)

wz :wmaxe
2
a) I' _ 2/2
UQ=M(1_C (r ram).
2r

In order to simulate the irregular spacing found in the MTV velocity field caused by
the laser grid generation and the placement of the velocity vectors in between the initial and
final location of the feature being tracked, the initial velocity measurements are irregularly
spaced as shown in Figure A.2a. The irregular spacing was generated by sub-dividing the
measurement field into 8 x 5 sized regions, where 8 is the mean spacing between velocity

measurement points. A random number generator is then used to place each velocity

166

 

 

I/rzrrrAkkx\\\\
//////«<—<AVA\\\\ \
//////M\\\\\\
I /////M‘\\\\\ \

/
/
ré
\
\
\
\
2r\\\\\\\»~w//////

i\\\\\\\~>—.Av’/////
i\\\\\\~.-—u—v’/////

-3 -2 -1 o 1 2
(b) x

..t
1I

/ //rA-—<—<—x\\
\ \\\W//'/'
\ \\\~>—>al//

uyru-li-r711ivr

 

0) (s“)

0 5 1o 152025303540

 

Figure A.2: Sample velocity and vorticity field of Gaussian core vortex. (3) Original velocity
vector field. (b) Velocity vector field placed on a regular grid. (c) Flooded contour plot of

the vorticity field.

 

measurement point at a random location within the 5 x 5 sized regions. In this manner, the

mean spacing between measurement points will remain a constant value of 5, however the

actual location of the measurements will vary.

In Figure A.2b the irregular velocity field shown in Figure A.2a is remapped onto a
regular grid by one of the polynomial interpolaters. It is interesting that even though the
irregularly and the regularly spaced measurements have the same mean densities, the Oseen
vortex seems more distinguishable in Figure A.2b. From these regularly spaced
measurements, the vorticity field can be computed through the use of many methods. Figure
A.20 shows a flooded contour plot of the vorticity field computed from the regular grid data
is seen in Figure A.2c.

The data will be remapped onto a regular grid by means of a local two-dimensional
least squares fit to a 2““, 3'“, or 4th order polynomial. The u and v velocity fields are fit
separately. After the fits for the two velocity components are generated, each local fit is
evaluated at the regular grid point location to determine the velocity at that particular grid
point. In all cases, the fit is over-determined. That is, for any order fit, there is a minimum
number of points necessary in order to perform the fit. It can be found that the minimum
number of points necessary for a fit to be properly determined is (order+ I ) *(order+2)/2. So,
for the 2"d order polynomial fit, a minimum of 6 points is necessary for the fit to be properly
determined. For all of the measurements reported more than the minimum number of points
are used in the fit. This will tend to reduce the random errors found in the original
measurements.

The computation of the out-of-plane vorticity component will be computed in four

different ways. The first three methods differ in the manner in which they estimate the

168

 

spatial derivatives of the velocity field. The out-of—plane vorticity, 002 can be computed from

the spatial derivatives of the velocity field using the relation

In the first method, the velocity derivatives will be estimated by directly differentiating the
local polynomial fit. As the fit is a polynomial, the calculation of the derivative from the fit
is straight-forward and does not require any added computation after placement onto a
regular grid. This method has the advantage that it can also be used to compute the vorticity
on the original irregular grid.

Once the velocity is on a regularly spaced grid, it can be numerically differentiated
using finite difference techniques. Two different finite difference methods will be examined:
a first order central (2"d order accurate) and second order central (4th order accurate). In the

first order central method, the two spatial derivatives used in the vorticity calculation are

defined by

g1 _ “1.141 ‘um
ay U.” 2h

(91) _vi+l.j_vi-l.j
axu— 2h ’

where (i,j) subscript indicates the relative location of the point in the regular grid. These
locations are indicated in Figure A.3a. This is the standard algorithm used in several
commercial packages for the computation of 0),. The 2nd order central (4th order accurate)
utilizes 8 points in the estimation of the spatial derivatives to provide a higher order
estimation of the spatial derivatives. The location of the measurement points used can be

seen in Figure A.3b. The spatial velocity derivatives for the 2nd order method are defined by

169

 

 

 

[fl] —ui.j+2 +8ui,j+l ” 8u1.j—l + “1.1—2
i.j

 

ay 12h
(av) —vi+2.j+8vi+1,j_8vi-l,j+vi-2.j
ax 1.1 _ 12h

will be examined.

The final vorticity calculation method computes the circulation around a rectangular
circuit that extends one regular grid point in each direction around the point to be examined
as shown in Figure A.3c. This circulation value is then divided by the area in order to

determine the vorticity. Using this method, the vorticity can be calculated using the relation:

1
_.____ L .L L
wz;i.j — 4h2 {2M4 ui—l,j+l + 2u1.j+1+ 4 ui+l.j+l) +
L L .L
2h(4 Vi+l.j—l '1' 2 vi+l.j + 4 Vi+l.j+l) +
l 1 l
2h('4'ui—i,j—i +Tui,j—i +Tui+i.j—i) +

.L L. L
2M4 vi+l.j—l + 2 V+li.j + 4 Vi+l,j+1)}’
where h is the separation between neighboring grids and the integrals are estimated using the
trapezoidal rule. It can be shown that this method is identical to a filtered version of the 1‘“

order central finite difference technique where the velocity at each point (i,j) is replaced with

its “3-point average” defined as :

C C C C 0
555355 ©
2 3%: I @

(b) (c)

@656
C
“@9393:

 

Figure A.3: Schematic of velocity measurement locations used in the estimation of the spatial
derivatives. (a) 1St order finite difference. (b) 2nd order finite difference. (c) 8-point
circulation method.

170

 

1 1

“1,1: ‘2-“1.j+Z(“1+i,j+ L‘s—1.1“)
1 l

vi,j — 2 vi.j + 4 (vhf-1 + vi.j-l)'

The use of the 15‘ order central difference relation on the above filtered field will recover the
previous expression for vorticity.
There are several parameters which are important in the determination of the accuracy

of both the remapping scheme and the calculation of 0),. In this study, we will examine the
effect of the mean spacing of the original velocity measurements, 5, and the radius R, from

which points are drawn from for use in the fitting procedure. The mean spacing between the
original velocity measurements yields information about the smallest structure size which

can be resolved by the measurement technique. This size must be larger than 5. In order to
examine this effect, the ratio of the structure size in the simulation, L to 5 will be examined.
The characteristic flow scale used in this study is the vortex core radius, rm, defined as the
distance from the peak vorticity to the location where the vorticity has dropped by a factor
of e". Simulations will be conducted for values of U5 ranging from 2.5 to 10.5.

Using the interpolation techniques, it is possible to create a regularized velocity field
that has a mean spacing between points, h, that is larger or smaller than that of the original
field. For all of the studies described here, h = 5. If the interpolater is used to generate data
with h < 5, it may appear that information about small-size structures can be determined,
however, no new information can be generated during the interpolation process. Information
is simply re-used. The information a. If h > 5, information about small-size structures could

be lost.

171

 

As previously described, the least—squares fitting process was over-determined for all
of the simulations performed in this study. However, it was found that the radius from which
points are drawn, R, for use in the fit can play a significant role in the estimation of both the

velocity and vorticity field. R will be normalized by 5' for the results presented in this study.

Decreasing the size of this region has the of decreasing the number of points available for
use in the fitting procedure and will provide a more local estimate of the velocity and
vorticity field. This typically results in an increase in the accuracy of the measurement,
however it also tends to increase the propagation of error from the velocity measurement.

Since higher order polynomials require more points for the fit to be properly
determined, when the results of the remapping and vorticity calculations are compared
between different fit orders, different R values are used for the different order polynomial
fits. For each fit order, there will be a minimum radius, Rm," necessary for the fit to be
properly determined. For the studies comparing the different fit orders, R/Rmm is kept at a
constant value of 4.5. This results in R values for the 2"“, 3'“, and 4‘h order polynomial fits

of 2.95, 3.85, and 4.65 respectively.

As in the results of Fouras and Soria (1998), the accuracy of the three different
polynomial least-squares fit orders and the different vorticity calculation methods are
measured by the mean bias error caused by spatial filtering and a random error caused by
both the propagation of the error in the original measurements to the remapped field and the
placement of the randomly spaced points onto the regular grid. Within each sample set, 100
random simulations are conducted. The mean bias error will be denoted by the subscript bias
and refers to difference between the mean value of these 100 measurements and the exact

value at a point in the flowfield. The random error is quantified by the root mean square of

172

 

 

difference between the interpolated velocity or computed vorticity value and the actual value
at each point in the 100 measurement samples of the simulation. This will be denoted by the
subscript rms.

In order to more closely simulate actual velocity measurements which will contain
a random error inherent in the measurement technique, noise was added to the simulated data
field. The method used is similar to that in Luff, et al. (1999). A random number generator
is used to add a random percentage of noise, with a maximum value of in% to each
component of the velocity field. With this formulation, the velocity at each point in the

simulation had a value of:

u =uw(l+n

random)

V : vac! (1 + nrandum)

where nmm is a random number with a value -n < n < +n. of the actual velocity value

random

to each component of the velocity.

A.3 Interpolation Results

Figure A.4 shows a profile of the velocity bias error for the 2‘“, 3'“, and 4‘h order fits
The error is normalized by the peak velocity found in the flow. A radial cross-section of the
vortex core ranging from the center of vorticity to a radius where r/r = 3 is examined for

care

all plots presented in this work. Grid densities ranging from [/5 = 2.5 to [/5 = 10.5 is

examined in this work. In all cases, the maximum error occurs at approximately r/rcm = 0.5,
which is in a region of large velocity gradient. Increasing the grid density greatly decreases

the error for all fit orders. In the 2“d order fit case, increasing the grid density from 2.5 to 3.5

decreases the maximum error from 5% of the maximum velocity to less than 2% of the

173

 

 

0.02 :-

11.011...

-0.06

-0.08 -

 

 

2nd order fit 0.02

-
b
p-

r

 

 

3ml order fit

 

U5=3.5

 

 

h
i-
i-
p-
b
-

    

[15:45
U5=5.5
U5=6.5
U5=7.5
U5=8.5
U5=9.5
U5=1 0.5

3

Figure A.4: Profiles of the mean velocity bias error for 2“", 3rd, and 4‘h order polynomial fits
for grid densities ranging from [/5 = 2.5 to U5 = 10.5.

174

maximum velocity. For [/5 larger than 4.5, the interpolation error is less than 0.5% for all

locations. In addition to the large underestimation of the velocity error which is a maximum

at r/r

C0”!

= 0.5, the interpolation overestimates the velocity field in the region r/rwn, > 1.5.
It should be noted that the presence of this overshoot} in this region can lead to misleading
results if the mean error is integrated over the entire region to determine one value. Several
previous studies have attempted to quote a single number for the overall accuracy of the
interpolation procedure. Since the area over of the region of the overshoot is about three
times larger than that of the undershoot region, the mean velocity error averages to nearly

zero. This is not indicative of the error found in this flow. If an RMS is taken, the

undershoot near r/r

CUR?

= 0.5 will dominate the error value and the fact that interpolater
produces values with little error at many points in the flowfield will be lost.

It is also apparent that the mean velocity error for the 2"d and 4‘h order polynomial fits
is significantly smaller than that found using the 3rd order polynomial interpolater. This is

seen more clearly in Figure A.5. A.5a shows the mean error for [/5 = 3.5 for the three fit

orders. Results with a maximum added of 0% and :t6% of the velocity added to the original
irregularly spaced field are presented. The effect of adding additional noise is very similar
to the 16% value. No discemable impact can be seen in the mean results as a result of the
added noise. The 2“d and 4‘h order polynomial fits do a nearly equivalent job interpolating
the irregularly spaced data onto a regular grid. The magnitude of the um associated with
the 4‘h order fit is less than 0.5% of the peak velocity less than that of the 2nd order fit. It
should be noted, however, that different R values used for the different order polynomial fits.
Decreasing the radius will result in more accurate results for all fit orders.

Figure A.5b show the profile of the root mean square of the error between the actual

175

 

- -I—— 2"“orderflt-O96noioo
+ 3:0rerfI-096nobe
—Q— 4 orderflt-O%noico
U5 = 3.5 ----- a ----- 2"“ «dorm-6% noise
..... Am» 3"ordor1n-6%noiu
»~Q»-~- 4"orderflt-6%nobo

0.02 . 0.01

“bin/um“

 

 

 

 

 

(a) "'m (b)

Figure A.5: Comparison of the error in the velocity field resulting from the 2““, 3rd, and 4‘h
order polynomial fits for the 0% and 6% noise added cases for a normalized grid density of
3.5. (a) uhm. (b) um.

velocity and the interpolated velocity for the 100 samples. The addition of noise into the
original measurements results in a small increase in the um, for all of the fit orders.
However, the least squares fitting process tended to minimize the effect of this randomly
distributed nose error. Note that even the noise-free data contains a random error. This is
due to the remapping process itself. The um, of the error for the 2nd order polynomial fit
results is slightly larger than the error for the other two cases, but this difference is fairly
small. The difference is likely caused by more points being used in the fit for the higher
order polynomial fits. For all cases, the um. error is less than approximately 0.8% of the
maximum velocity.

Figure A.6a and A.6c shows the effect of reducing the maximum radius from which

176

 

 

 
   

 

 

 

 

 

 

0.02 0.02- —J— R18=1.5
* —e— 318:2.0
—e~— 315:2.5
———v—— RI5=30
-o.02_ 0.015 —*’— W5=3-5
-0.04’-
‘- I
g as :
in ‘5 '006?
\o = I
o P
c -0.08:
-0.15-
-0.12:-
c
E
I.l.l
o\°
co

 

 

 

Figure A.6: Effect of decreasing the radius from which irregular points are drawn in the
interpolation process on the velocity mean bias and random error. (a) um, for 0% noise
added case. (b) um, 0% noise added case. (c) um for 6% noise added. ((1) um for 6% noise
added.

177

the irregularly spaced points are drawn for use in the fit. As all of the cases show similar

effects, only the [/5 = 3.5 results are examined. The mean velocity error results for both no
noise added and the 6% noise added are very similar. Small values of R/5 generates the
smallest mean error. The addition of noise has the most significant effect on the error for R/5
= 1.5 causing a slight increase in the maximum mean error from 0.2% of the maximum

velocity to 0.6% of the maximum velocity when noise is added. When noise is added, the

maximum mean velocity error is smaller in the R/5 = 2.0 case. For R/5 values greater than
3.0, ubm continues to increase and at R/5 = 6.0, the maximum value of ubm = 12% of the

peak velocity.
The random component of the error is shown in Figure A.6b and A.6d. For the 0%

noise added results, u is minimized at small values of R/5. However, increases in R/5

r1713

cause only a small change in u and the maximum value of um. for the 0% noise added

results is less than 0.8% of the maximum velocity. Figure A.5d shows the effect of the

addition of the random 6% noise to the original measurement data on um. When noise is

added, increasing R/5 results in the minimum values of um. However, the decrease in um.

with increasing R/5 is small in comparison to the increases seen in um. for increasing R/5.

A.4 Vorticity Calculation Results

In this section, the error generated by the four methods for calculating the out-of-
plane vorticity field will be examined. First, the results of differentiating the various
polynomial fit orders to determine the vorticity value will be discussed. Then, the results

from this method will compared with those from the finite difference and circulation

178

 

 

methods.
Figure A.7 compares the mean accuracy of the vorticity calculation using the fit

differentiation method for four values of U5. In all cases, the result of differentiating the

third order polynomial results in a smaller error than'the result of differentiating the 2nd or
4‘h order polynomials. For all cases, the maximum mm...- is located at the center of the vortex
core. For U5 = 2.5, the third order fit results in the vorticity being underestimated by nearly
25 % while the differentiation of the 2“d and 4‘h order fit results in an error of 45% and 35%
of the maximum vorticity respectively. As the grid density is increased, (um, decreases to

nearly zero for all cases.

The random error associated with the vorticity field calculations is shown in Figure
A8. The same four grid densities shown in Figure A.7 are examined in this plot. For the
0% noise added case, as the grid density is increased, the random error decreases. The fact
that fluctuations are found when no noise is present in the data is the result of the random

placement of the velocity information. Note that decrease in com with increasing grid density

is not contradictory to the findings of Fouras and Soria (1998) that increasing the separation
between velocity measurements results in a decrease in the random error. As the grid density
is increased, additional information is being added to the velocity field. If the mean spacing
in the initial, irregularly spaced grid, 5, is kept constant and the spacing in the regular grid
is decreased, the random noise error will increase as found by Fouras and Soria (1998).
When noise is added, the random error increases as the grid density increases. However, the
maximum random error is less than 3% of the maximum vorticity. This is small compared
to the 25 % bias error.

Since the results of differentiating the 3rd order polynomial fit clearly outperform the

179

 

 

Differentiate 2"“ order fit - 0% noise
Differentiate 3'“ order fit - 0% noise
Differentiate 4" order fit - 0% noise
Differentiate 2"" order fit - 6% noise
Differentiate 3" order fit - 6% noise
Differentiate 4" order fit - 6% noise

 

   

 

 

 

 

 

 

 

 

 

=
E
§
-0.3:-
-0.4 :-
1 r l l I
'0'50 1 2 3
mm
-0.1 :- -0 1 '-
32 " 83
\. -0 2 _- \. -0 2 :-
3! . 2% .
: 3 t
'0'3 T -0 3 f
- L
-0.4;- -0.4 :-
4 I I I - ’- 1 4 I 4 L J .L I I
'0 50 1 2 3 0'50 1 2 3
f/I'm rlfcon

Figure A.7: Comparison of the vorticity mean bias error profile resulting from the use of
differentiating the 2"", 3'“, and 4‘h order polynomial fit for the calculation of the out-of—plane
vorticity for four different grid densities. Results from simulations with 0% added and a
maximum of 6% uniform noise are shown.

180

——-I——— Differentiate 2"" order fit - 0% noise
——+— Differentiate 3" order fit - 0% noise
——0— Differentiate 4" order fit - 0% noise
— - - 13 - - - Differentiate 2"" order fit - 6% noise
— — — A - - - Differentiate 3" orderfit- 6% noise
- - — G— - - Differentiate 4"I order fit- 6% noise

 

0.04 - ”5 = 2-5 0.04 .. U5 = 4.5

 

 

 

 

 

 

0.03 -

.0

O

00
l

. 3
3 0.02- 3 0.02
\E ' AAA 25 - M AA AA

 

 

 

Figure A.8: Comparison of the random vorticity error profile resulting from the use of
differentiating the 2"“, 3'“, and 4‘“ order polynomial fit for the calculation of the out-of—plane
vorticity for four different grid densities. Results from simulations with 0% added and a

maximum of 6% noise are shown.

181

 

results from the other two fit orders, only the 3rd order fit results will be compared with the

(i)z values calculated using the finite difference and circulation methods. For the two finite

difference methods and the circulation methods, only the results from the 2nd order
polynomial least squares fit shown in the previous section will be utilized. In the previous
section, it was shown that the vorticity estimates calculated through the use of these methods
rely solely on the velocity information on the regularly spaced grid. The interpolation results
for the 2nd and 4th order polynomial fits were very similar. As expected, this results in very
similar results for the vorticity calculation. Since the 2"d order fit is less computationally
intensive and allows for a smaller number of surrounding points to be used in the
interpolation process it was used instead of the 4th order fit. Both of these interpolation
techniques produced results significantly better than the 3rd order least squares fit. As
expected, the decreased accuracy in the remapped 3rd order fit velocity information results
in decreased accuracy in the vorticity calculation.

Figure A.9 compares 00““, found by differentiating the 3rd order polynomial least

squares fit with the finite difference and circulation methods. Once again, the effect of
adding noise to the initial velocity field is negligible. The qualitative features of the four

methods are very similar. The maximum (rim is at r/r = 0 which is the location of the

C(H‘C‘

peak vorticity. At r/rcm =1.5, there is a small overshoot where the vorticity value is
overestimated. Note that although the numerical amount of the overshoot is small relative
to the undershoot, the area occupied by the region of overshoot is roughly three times larger
than the region of undershoot. Thus, the overall averaged random error will be significantly

smaller than the peak. The estimate of the overall circulation of the vortex computed by

integrating the vorticity field will yield a result accurate to better than 0.1% even for U5 =

182

 

 

——I—— Circulation method - 0% noise
+ 1" order finite difference - 0% noise
—0— 2'” order finite difference - 0% noise
———A—— Differentiate 3" order fit - 0% noise

 

-——13—-— Circulation method-6% noise

--—-—<> 1‘orderfinitedifference-6%nobe

—--G-—- 2““orderfinitedifference-6%noise

”-23-" Differentiate3"orderfit-6°/onoise L
ti

0.1 :- U5 = 2'5 01 L15 = 4.5

 

 

 

 

 

 

 

 

 

1 2 3
rlrm
U5 = 10.5
1 l I r L J
1 2 3
rlrm

Figure A.9: Comparison of the mean vorticity bias error profile resulting from the use of the
circulation method, 1St and 2nd order finite difference methods, and differentiating the 3rd
order polynomial fit for the calculation of the out-of—plane vorticity for four different grid
densities. Results from simulations with 0% and 6% noise added are shown.

183

2.5 which underestimated the peak vorticity by nearly 20%.

Quantitatively, (om decreases as the grid density increases. The use of the 2nd order
finite difference method outperforms all other methods for low grid densities. As grid
density increases, the difference between the methods becomes smaller and smaller as the
all methods do a very good job. For the U5 = 2.5 case, the maximum error for the 2nd order
finite difference method is 18%. In comparison, differentiating the 3rd order polynomial
results in a maximum bias error of 22%, the 1"t order finite difference method results in a
maximum bias error of 24%, and the circulation method results in a maximum error of 28%.
For the U5 = 10.5 case the maximum bias error is less than 1% of the maximum vorticity
for all calculation methods.

The random error in the vorticity computation is plotted in Figure A. 10. As with the
results found in comparing the results from differentiating different order polynomials, (om,
decreases with increasing grid density in the no noise added results, and increases with
increasing grid density for cases where noise is present in the initial data. The maximum
random error, however, is small compared to the maximum bias errors. It should be noted
that the circulation method is the least sensitive to the addition of noise.

The effect of varying the maximum radius from which irregular grid points are drawn
from in the least squares fit on the vorticity calculation is shown in Figure A.11. Only the
results from the two methods showing the smallest (0m, differentiating the 3rd order fit and
the 2“d order finite difference method, are shown. Note that in Figure A.6 which examined
the variation of um and um. with R/5, the smallest value examined was R/5 = 1.5. This
value of R/5 is not present in Figure A.11 as both mm and 00m, found using this radius are

very large. As with the results for velocity, decreasing the R decreases the mean bias error.

184

 

—I— Circulation Method - 0% Noise

2 7+7 2“d order finite difference - 0% Noise
rk— Differentiate 3"1 order fit - 0% Noise

~~~~~ B—~-~ Circulation Method - 6% Noise

----- G» - — ~ 2“‘1 order finite difference - 6% Noise
----- A ~ - -- Differentiate 3'“ order fit - 6% noise

U5 = 4.5

 

 

O
/
—® /
ru— ‘1“ ’
1—0\ 5‘31? /@
/,O
4
3
U5: 10 5
O

AA‘wQ :ibo

 

 

Figure A. 10: Comparison of the vorticity random error profile resulting from the use of the
circulation method, 1St and 2nd order finite difference methods, and differentiating the 3rd
order polynomial fit for the calculation of the out-of—plane vorticity for four different grid
densities. Results from simulations with 0% and 6% noise added are shown.

185

 

 

U5 = 3.5

0% Noise 6% Noise
—I— RI8=2.0 - - - 13 - - - RI5=2.0
+ Rl5=2.5 - - - 1A - - - RI5=2.5
——v—— RI8=3.0 - - - v - - - R15=3.0
—>— RI5=3.5 - - - -l> - - - 315:3.5
——<—— RI5=4.0 - - - <1 — - - RI5=4.0
—o— R16=5.0 - - - <> - - - RI8=5.0
—~O— 815:6.0 - - - G - - - Rl5=6.0

Differentiate 3"I order fit

 

2"" order finite difference

 

 

 

 

 

 

 

Figure A.1 1: Effect of varying the radius from which points are selected in the interpolation
process on the mean bias and random error for the vorticity computed by means of
differentiating the 3rd order polynomial and the use of the 2nd order finite difference method.

186

The effect of adding noise to the mean error is negligible except for the smallest radius for
the differentiation method results. For small values of R/5, the differentiation method results
in a mean bias error of less than 0.5%. For the same R/5, the 2nd order finite difference
method results in a 03m- of 2% of the maximum vorticity. Both methods are very sensitive
to increases in R/5. For R/5 = 6, the (ohm. for the 2nd order finite difference method has
increased to 35 % compared to 25 % for differentiating the 3rd order polynomial.

As in the study of the effect of R/5 on the remapped velocity field, when no noise is
added to the measurements, increasing R results in an increase in (om. However, when noise
is added to the velocity field, increasing R results in a decrease in (pm. In the 0% noise
measurements, (um for the differentiating the 3rd order fit is less than that for the 2nd order

finite difference technique. However, the differentiation method is more sensitive to the
addition of noise in the velocity measurements. Again, these errors are generally small in
comparison to the 00km errors present. For the differentiation results, the maximum 00m is
1% and 3.5% of the peak vorticity value for the 0% and 6% noise added results respectively.

For the 2"d order finite difference technique, the maximum mm. for both the 0% and 6% noise

results is 2% of the peak vorticity value.

A.5 Conclusions

The effect of remapping an irregularly spaced velocity measurements onto a
uniformly spaced grid and the accuracy of the out-of—plane vorticity computed from this
information has been studied through the use of a Gaussian core vortex simulation.

Remapping onto a regular grid was performed by the use of a 2"“, 3m, and 4‘h order least

187

 

 

squares fit. The vorticity is computed by means of calculating the derivatives necessary for
the computation of vorticity by means of directly differentiating the least squares fit,
performing a 1“ or 2nd order finite difference calculation on the regularly spaced data, and by
computing the local circulation of the region around'a point and dividing by the area. The
effect of varying the normalized grid density (ratio of the flow characteristic length to the
mean spacing in the initial velocity measurement) and the maximum radius from which
points are used in the remapping process was examined. For all of the studies conducted,
the density of the remapped, uniformly spaced grid remained the same as the initial
irregularly spaced measurement grid.

Two types of error are present when remapping data from an irregularly spaced grid
to a regularly spaced grid and in the computation of the out-of—plane vorticity: a mean bias
error due to spatial filtering and a random error due to the remapping process itself and the
propagation of the error in the original data. The mean bias error is not effected by noise in
the original measurements and can be decreased by increasing the grid density. The random
error can be affected by the presence of noise in the original measurements, which causes an
increase in the random error. However, the filtering inherent in the fitting process tends to
decrease the magnitude of the random error in the regular grid as compared to that which
would be expected. Generally, the errors resulting from um are significantly larger than
those in ubm.

When remapping data from an irregular grid onto a regularly spaced grid, 3 least
squares fit to an even order polynomial does a better job then an odd order least square fit.
It is also apparent that it is necessary to have a sufficient grid density in order to accurately

reproduce the flow features. Ifthe ratio of the flow characteristic length to the mean spacing

188

 

 

 

between measurement points, U5, is greater than 3.5, the maximum ubm is less than 1.5%

of the maximum velocity. Even in the presence of noise, the random error is less than 1%.

If U5 = 2.5, the um, is less than 5% of the maximum velocity. It was also found that

decreasing the radius from which irregular points used in the fit are drawn from can
significantly decrease the values of um.

Using the 2nd order finite difference technique and differentiating the 3rd order
polynomial fit provide the most accurate estimates of the out-of—plane vorticity. It is
interesting that the differentiating the 3rd order fit yields more accurate vorticity calculations
than differentiating the 2nd or 4Lh order fits since the even order fits provide a better estimate
of the velocity field. Using the 2'”d order finite difference technique to estimate the vorticity
provided the best representation of the actual vorticity field with a maximum bias error of

2% of the peak vorticity can be achieved for U5 = 4.5, and for U5 = 2.5, the maximum error

is less than 20% of the peak vorticity. Differentiating the 3rd order fit produced bias errors
that were only slightly larger than the 2“d order finite difference method. However, it was
found that decreasing R significantly improved the results of the 3rd order fit. For both
methods, um, is relatively constant across the vortex core and is less than 2.5% of the peak
vorticity. It should be noted that although a mean bias exists in the lower density results, a
vortex core will still be found in the interpolated results and the peak vorticity will be at the

same location as is found in the higher density results.

189

 

Appendix B

The Computation of Mean Quantities of Phase-Locked Signals Using
Sparsely Sampled Data

When a phase locked signal is sparsely sampled in time, it is possible that simply
summing the signal and dividing by the number of samples may not yield the best estimate
of the true mean value of the signal. This scenario occurs in many experiments utilizing
whole field velocity measurement techniques based on the recording of images from video
cameras. For example, in the Molecular Tagging Velocimetry Experiments conducted in this
thesis, the characteristic frequency of the flow is 6 Hz, while the measurement sampling rate
is approximately 30 Hz. Thus, only five measurements are made per forcing cycle.

In cases where the signal being measured is periodic, it is possible to use the
frequency information to construct a more accurate time average with a shorter record length.

To show this, a sinusoidal signal of the form:

y = sin(2nft + (p),

where f is the signal frequency and (1) is an arbitrarily selected starting phase was examined.
In the results presented here, the frequency of the input si gnal, f = 6.01 Hz and the sampling
rate is 30 Hz. Given this choice of parameters, approximately 5 measurements of the
sinusoid per cycle were used in the construction of the average. These numbers were

selected to best match the data to be collected in the forced wake MTV experiments. In this

190

 

  
   

 

 

 

 

0.25 —— 8V9 0.15 P —— avg
- avg of phase avg j ----- avg of phase avg
0.2 _.___._..__ avg _
0.15 0 avg of phase avg 0'12 __
0.1
O o -
O 0.05 o 0.09 -
1- 1- —
C
r: 0 :0
3 0 05 5 0 06 i
E . a: . _
-0.1 i
_ 1
-0.15 0.03 - El
- t ' ' .
0'2 L l WHMM
-0 25 g1_1 i I i 1 #1 I i i r 1 I i i 1 l I o P i i i 1 I i I A. i ‘1‘?“‘1‘5‘0‘1’ '1 l i I
' 0 250 500 750 1 000 0 250 500 750 1 000
I t
(a) T (b) T

Figure B.1: Comparison of the estimation of the mean using direct computation versus
calculating the phase average and the computing the mean. (a) Average value of sinusoidal
signals with two different starting phases. (b) Root mean square of 16 starting phases.

study the non-dimensional record length, T* defined as the total time of the data record
divided by the period of the signal, is varied. It should be noted that for the technique
described to be successful, the sampling rate must not be an exact measurement multiple of
the frequency of the signal to be measured.

Using the known signal frequency, the sparsely sampled data can be first phase-
averaged. The mean value of the signal is then computed by the average of the phase-
averaged signal. Results have shown that for a wide range of T*, this method can result in
a more accurate estimation of the mean quantity. Figure B. 1 a highlights this finding for two

different values of the starting phase, (ii. The dashed lines indicate the mean computed by

averaging the phase-averaged data. This estimate of the true mean consistently provides an
estimate very close to the exact value of zero for T* values larger than 100. In contrast, the

mean computed by the classic procedure of summing all of the values and dividing by the

191

 

 

number of samples yields a result significantly poorer estimate of the actual mean. As T*
increases, the mean computed in this manner generally improves in accuracy as expected
from the increased number of samples in the record. However, the error does not decay
uniformly and oscillations are noted in the estimated value of the mean. Zero error occurs
when the total number of samples occupy an exact number of cycles.

Since varying (b can result in differences in the estimated value of the average

obtained using either method, it was deemed necessary to find a quantity which assesses the
sensitivity to this effect. The mean value was computed for 16 different values of the starting
phase and the RMS of these 16 samples gives an indication of the variation expected in the

estimation of the mean caused by changes in (1). Figure B.1b shows similar trends as the

previous figure. For the majority of the range examined, computing the average of the phase-
averaged data yields more accurate results than computing the average via standard methods.
It is noted that for a record length of exactly 500 periods yields zero error for the standard
method. This location of T* = 500 is an artifact of the particular signal and sampling
frequencies chosen. Different combinations of these two parameters will yield a different
T* value where zero error is found..

The computation of the mean using the direct method yields less accurate results than
averaging the phase averaged data because the record length may contain a non-integral
number of cycles. The effect is similar to sampling the signal with a sufficient sampling
frequency, however, not sampling the entire waveform. When sampling a waveform with
only a few measurements per cycle, the signal will not be sampled uniformly over its period.
Thus, certain phases of the signal will be disproportionately represented in the mean. This

effect gets reduced as the record length increases. Phase-averagin g the signal first eliminates

192

 

 

 

this bias. The zero value at T* = 500 is a good example of this effect. At this non-
dimensional sampling time, the sampling frequency and function frequency are such that all
phases of the signal are equally represented. Thus, the standard procedure for the
computation of the mean generates a very good estimate of the average for all starting phases.
The exact number of samples necessary for this nodal point to occur will vary with the exact
selection of sampling and signal frequency.

When the frequency of a sparsely sampled signal is known, that information can be
used to improve the estimate of the calculation of the mean. The additional information
provided by the signal frequency allows for this increase in accuracy. It should be noted
however that in absolute terms, the difference in accuracy between the two methods is small.
The error resulting from the direct computation of the mean at T*=100 is only 0.2% of the

signal amplitude.

193

 

 

Appendix C

Velocity Forcing Amplitude Calibration Curves for the Two-stream
Mixing Layer Facility

In the experiments conducted in this dissertation, the velocity perturbations were
input into the through an oscillating bellows mechanism. This bellows is driven by an
electromagnetic shaker which derives its command forcing signal from a function generator.
Typically, the perturbation is input by setting the peak to forcing frequency and peak-to-peak
amplitude of the command signal. The bellows is driven in open-loop mode, so the
amplitude of the bellows motion is dependent upon the frequency. The amplitude or the
bellows motion is measured by means of linear position transducer. The velocity
perturbation in this, as well as previous experiments, is reported as the RMS streamwise
velocity fluctuations, urms normalized by the free-stream velocity, uo. As the fluctuation
levels decrease with downstream location, this measurement is made as far upstream as
possible.

Figure C. 1 shows the calibration of the free-stream velocity disturbance in the mixing
layer facility with the displacement of the bellows/shaker mechanism and with the input
peak-to-peak levels of the command signal. All of the data was collected at x = 3.75 cm and
for a wake case with 110 = 9.4 cm/s. The vast majority of the data is for the 6 Hz forcing
frequency, however, a smaller amount of data for the 4 Hz and 8 Hz forcing frequencies are

also shown. As described in the main portion of the thesis, the u velocity perturbations do

194

 

 

 

 

 

0.5
Shaker Displacement (rms, mm)

_ 13
A10" 0 " A
a; E;
a 5; . :3
__ I
I 6H!
0 ' 0 4H:
q 0 8H:
1 1 1 1 l 1 #1 l J 1 L 1 l
00 1.5

 

 

l l L L
100

l l l
200

. I
300

Function Gen (peak-peak, mV)

Figure C. 1: Velocity perturbation calibration curves for the forced wake.

not reach a steady level for the 4 Hz forcing cases. Thus, the perturbation level is dependent

both upon the spanwise and streamwise position of the measurement. For all three forcing

frequencies, the relationship between the input forcing and the velocity disturbance is linear.

However, the SIOpe is not constant between the different forcing frequencies. This indicates,

‘ as expected, that the flow responds differently to the different input forcing frequencies.

195

 

 

 

 

BIBLIOGRAPHY

196

 

BIBLIOGRAPHY

Abrahamson, S. and Lonnes, S. [1995] “Uncertainty in Calculating Vorticity from 2D
Velocity Fields using Circulation and Least Square Approaches.” Experiments in
Fluids, 20, pp. 10-20.

Agui, J. and Jimenez, J. [1987] “On the Performance of Particle Tracking.” Journal of Fluid
Mechanics, 186, pp. 447-468.

Breidenthal, R. [1980] “Response of Plane Shear Layers and Wakes to Strong Three-
Dimensional Disturbances.” Physics of Fluids, 23(10), pp. 1929-1933.

Cohn, R.K., Gendrich, C.P., MacKinnon, CG, and Koochesfahani, M.M. [1995]
“Crossflow Velocimetry Measurements in a Wake Flow.” Bulletin of the American
Physical Society, 40(12), p. 1962.

Cohn, R.K. and Koochesfahani, M.M. [1993] “Effect of Boundary Conditions on Axial Flow
in a Concentrated Vortex Core.” Physics of Fluids A, 5(1), pp. 280-282.

Cohn, R.K. and Koochesfahani, M.M. [1997] “Structure of the Velocity and Vorticity Field
in a Confined Wake.” Bulletin of the American Physical Society, 42(11), p. 2197.

Fouras, A. and Soria, J. [1998] “Accuracy of Out—of—Plane Vorticity Measurements from In-
Plane Velocity Field data.” Experiments in Fluids, 25(5/6), pp. 409-430.

Gendrich, C.P, Bohl, D.G., and Koochesfahani, M.M. [1997] “Whole-Filed Measurements
of Unsteady Separation in a Vortex Ring/W all Interaction. AIAA Paper #97—1780.

Gendrich, CF. and Koochesfahani, M.M. [1996] “A Spatial Correlation Technique for
Estimating Velocity Fields Using Molecular Tagging Velocimetry.” Experiments in
Fluids, 22, pp. 67-77.

Gendrich, C.P., Koochesfahani, M.M., and Nocera, DC. [1997] “Molecular Tagging
Velocimetry and Other Novel Applications of a New Phosphorescent
Supramolecule. ” Experiments in Fluids, 23, pp. 361-372.

Hartmann, W.K., Gray, M.H.B., Ponce, A., Docera, D.G., and Wong, RA. [1996] “Substrate
Induced Phosphorescence from Cyclodextrin-Lumophore Host-Guest Complexes.”
Inorganica Chimica Acta, 243, pp. 239-248.

Hagen, JP. and Kurosaka, M. [1993] “Corewise Cross-Flow Transport in Hairpin Vortices--
the ‘Tornado Effect’.” Physics of Fluids A, 5(12), pp. 3167-3174.

197

A /_ ?

 

 

Hilbert, HS. and Falco, PE. [1991] “Measurements of Flows during Scavenging in a Two-
Stroke Engine.” SAE 910671.

Hill, RB. and Klewicki, J .C. [1991] “Data Reduction Methods for Flow Tagging Velocity
Measurements.” Experiments in Fluids, 20(3), pp. 142-152.

M. M. Koochesfahani, R. K. Cohn, C. P. Gendrich, and D. G. Nocera [1996] “Molecular
Tagging Diagnostics for the Study of Kinematics and Mixing in Liquid-Phase
Flows.” Plenary presentation at the Eighth International Symposium on Applications
of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8-1 1, 1996: pp. 1.2.1
— 1.2.12.

Koochesfahani, M.M. [1989] “Vortical Patterns in the Wake of an Oscillating Airfoil.” AIAA
Journal, 27, pp. 1200-1208.

Koochesfahani, M.M. and Nelson, KB. [1997] “Molecular Mixing Enhancement in the
Confined Wake of a Splitter Plate.” AIAA paper #97-2007.

Kurosaka, M., Christiansen, W.H., Goodman, J .R., Tirres, L., and Wohlman, RA. [1988]
“Crossflow Transport Induced by Vortices.” AIAA Journal, 26(1 1), pp. 1403- 1405.

Lasheras, J .C. and Meiburg, E. [1990] "Three-Dimensional Vorticity Modes in the Wake of
a Flat Plate." Physics of Fluids A, 2(3), pp. 371-380.

LeBoeuf, R.L. and Mehta, RD. [1996] "Topology of the Near-Field Vortical Structures in
a Three-Dimensional Wake." Experimental Thermal and Fluid Science, 13, pp. 223-
238.

Luff, J.D., Drouillard, T., Rompage, A.M., Linne, M.A., and Hertberg, JR. [1999]
“Experimental Uncertainties Associated with Particle Image Velocimetry (PIV)
Based Vorticity Algorithms.” Experiments in Fluids, 26, pp. 36-54.

MacKinnon, CG. and Koochesfahani, M.M. [1997] “Flow Structure and Mixing in a Low
Reynolds Number Forced Wake inside a Confined Channel.” Physics of Fluids,
9(10), pp. 3099-3101.

Mattingly, GE. and Criminale, W.O. [1972] “The Stability of an Incompressible Two-
Dimensional Wake.” Journal of Fluid Mechanics, 51(2) pp. 233-272.

Nelson, KB. [1996] Structure and Mixing in a Low Reynolds Number Forced Wake in a
Confining Channel. MS. Thesis, Michigan State University.

Meiburg, E. and Lasheras, J .C. [1988] "Experimental and Numerical Investigation of the

Three-Dimensional Transition in Plane Wakes." Journal of Fluid Mechanics, 190,
pp. 1-37.

198

 

 

Ponce, A., Wong, P.A., Way, J .J ., and Nocera, DC. [1993] “Intense Phosphorescence
Triggered by Alcohols upon Formation of a Cyclodextrin Ternary Complex.”
Journal of Physical Chemistry, 97, pp. 1137-1142.

Visbal, M.R., Gaitonde, D.V., and Gogineni, SP. [1998] “Direct Numerical Simulation of
a Forced Transitional Plane Wall Jet.” ALAA paper #98-2643.

Roberts, FA. [1985] Efiects of a Periodic Disturbance on Mixing in Turbulent Shear
Layers and Wakes. Ph.D. thesis, Caltech.

Sato, H. [1970] “An Experimental Study of the Non-Linear Interaction of Velocity
Fluctuations in the Transition Region of a Two-Dimensional Wake.” Journal of
Fluid Mechanics, 44(4), pp. 741-765.

Sato, H. and Kuriki, K. [1961] “The Mechanism of Transition in the Wake of a Thin Flat
Plate Placed Parallel to a Uniform Flow.” Journal of Fluid Mechanics, 11, pp. 321-
352.

Spedding, GR. and Rignot, E.J.M. [1993] “Performance Analysis and Application of Grid
Interpolation Techniques for Fluid Flows.” Experiments in Fluids, 15, pp. 417-430.

Stier, B. [1994] An Investigation of Fluid Flow during Induction Stroke of a Water Analog
Model of an IC Engine Using an Innovative Optical Velocimetry Concept -LIPA.
PhD. Thesis, Michigan State University.

Weygandt, J.H. and Mehta, RD. [1995] "Three-Dimensional Structure of Straight and
Curved Plane Wakes." Journal of Fluid Mechanics, 282, pp. 279-311.

Wygnanski, 1., Champagne, F., and Marasli, B. [1986] “On the Large-Scale Structures in

Two-Dimensional, Small-Deficit, Turbulent Wakes.” Journal of Fluid Mechanics,
168, pp. 31-71.

199

 

 

 

fl:

1.: 1.1.1.1.?!
. 1.. vaafi: (1:01.31

1:12... 0:1:
.. t...) .
I ...
. I
t liMhai:
.1. 1.1.1.... . “its:
... ~ I .y
.1. .... if

«xuflflfi...uu.

1.1 .11. t0.

1. ..1 Hm... «I

n 11. E1:
.31.... .31
istfivu.

.r... .....ufl. ... .
inhibiialhrtl
1:184:171.
1:01!!!- .11..»
1.2.3.3}... .

..ns .
.Tkflwhrflrifirut

. 5.3.153 its

r . .
..z .. H91”.

$31,115:? ..Nsth. 3:5
corner/Iii 1.3;" a

I lit“;

.. ...1

a ...
.....r
3.931....

12.131. :1
$3.... .hs:

.0
1 n. u.
5211 s . c
, .... 1..
n.1-F-naii ....
.
3%.-..19.
3:3.
.1 ..

.9“.