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This is to certify that the

thesis entitled

EXPERIMENTAL STUDY OF THE
2-D AND 3-D STRUCTURE OF A
CONCENTRATED LINE VORTEX ARRAY

presented by
Douglas Gordon Bohl

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_£h..D_.__degree in Mechanical Engineering

 

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Experimental Study of the 2-D and 3-D Structure of a
Concentrated Line Vortex Array

By

Douglas G. Bohl

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering
2002

ABSTRACT

EXPERIMENTAL STUDY OF THE 2-D AND 3-D STRUCTURE OF A
CONCENTRATED LINE VORTEX ARRAY

By

Douglas G. Bohl

It is well know from previous work that when a two dimensional vortex interacts
with a no-slip boundary flow along the axis of the vortex core, axial core flow is
generated. To date, work on the generation and structure of axial core flow has been
limited to mainly analytic investigation and qualitative experiments. In this work,
Molecular Tagging Velocimetry (MTV) is used to provide quantitative measurements of
the structure of a developing vortex array.

A NACA-0012 airfoil is pitched about the quarter chord point at varying
amplitudes and frequencies to produce a semi-infinite vortex array with vortices of
varying characteristics (i.e. circulation, peak vorticity, core size, etc.). Two-component
MTV is used to detail the initial formation of the vortex array for the first chord length
downstream of the airfoil trailing edge. MTV results combined with flow visualization
images indicate that the unsteady Kutta condition is not valid for high reduced frequencies
in this flow field. Mean drag results show that the mean force switches from drag to thrust
as the reduced frequency is increased. The condition of zero mean drag does not occur
when the vortices are aligned, as predicted by thrust calculations from the mean flow field
along the tunnel midspan, but rather at a reduced frequency that is 30% higher. This
difference is due to the velocity fluctuations and lateral pressure distribution p(y) in the
flow field which are not accounted for when the mean profile only is used.

Cutting walls are placed into the flow field at half a chord downstream of the

trailing edge to initiate axial flow within the vortex cores. A combination of line tagging

(single component velocity perpendicular to the tagged line) and stereoscopic (three
component velocity) MTV measurements are used to detail the interaction of the vortex
array with the cutting walls. As expected the results show that axial flow is generated
within the vortex core and that fluid moves away from the cutting walls. The maximum
axial flow speeds are nominally equal to the maximum swirl speeds. Evidence for the
presence of area varying waves is given by a local change in the vortex core area that
propagates away from the cutting walls. The spatial location of the area varying waves
correlates to local spikes in the peak vorticity and the axial core flow. Two wave speeds,
the speed of the axial flow front and the speed of area varying waves on the vortex core,
are calculated and found to increase with increasing reduced frequency. These two speeds
are however different with the area varying wave increasingly lagging behind the axial
flow front.

Reverse spanwise flow (i.e., flow opposite to the direction of the axial core flow) is
observed outside of the vortex cores. The reverse flow indicates a large scale recirculation
is set-up to return fluid towards the wall, satisfying the conservation of mass. The vortex
array structure changes in character as it convects downstream. The peak vorticity level is
reduced and the core radius increases. The vorticity profile changes from Gaussian to non-
Gaussian. At some span locations and reduced frequencies “hollow core” vortices are
observed. The vortices eventually loose coherence and the spanwise flow, initially limited
to the vortex core regions (i.e. the axial core flow), is found throughout the flow field. This
break—up of the vortical structures begins near the wall and propagates towards the center

of the tunnel.

Copyright by
Douglas G Bohl
2002

This book is dedicated, with love, to Coreen and Bailey who somehow managed to
live and grow with me through this long strange journey. And to Noah who joined us for

the final sprint.

ACKNOWLEDGEMENTS

I have found that the idea of a Ph.D. representing individual research and work to
be false. No work of this magnitude could ever be accomplished without the help and
support of many people. I would like to acknowledge the following people for their
contributions to my “individual” work.

My local family Coreen and Bailey who kept me grounded and helped me keep my
priorities straight during the maddening processes. My extended family, including but not
limited to, my mom (Marilyn), dad (Gordon), brother (Dave), and mother-in-law (Sue),
who gave me places to go far away from East Lansing to get away. My happy place is the
family farm in Wisconsin. Thanks To Grandma and Grandpa Bohl for letting us visit and
Uncle Kenny who let me clean cow stalls while on “vacation”. My local friends: Amy
who I could be bitter with, Darryl and Shela who let me sit in their woods during the fall,
and Chuck and Elaine who have been great friends to Coreen and myself.

From “the office” I would like to thank Mike who built (and rebuilt) much of my
experiment, and made my days brighter with friendly conversation. Ann and Darla from
MRSEC, without whom ordering (and returning) equipment would have been a
nightmare. Special thanks to Ann for keeping me involved with kids and science! Jill, who
was always helpful with departmental issues. And Hu Hui and Chee Lum who provided a
lot of support (technical and moral) in the lab.

Much of the experimental technique used in this work is a result of long hours of
work by Chuck Gendrich and Richard Cohn. I have added to the technique, but much of
the difficult work developing MTV is theirs.

Finally, thanks to Manooch. A student advisor relationship is a lot like a parent

Child relationship during the teen years. Thanks for being a good “dad”.

vi

TABLE OF CONTENTS

LIST OF FIGURES ......................................................................................................... IX

LIST OF SYMBOLS ...................................................................................................... XV
CHAPTER 1

INTRODUCTION .............................................................................................................. 1
CHAPTER 2

EXPERIMENTAL METHODS ....................................................................................... 14

2.1 Facility ............................................................................................................ 14

2.2 Two Component Planar Molecular Tagging Velocimetry ............................. 17

2.2.1 Technique Description ........................................................................... 17

2.2.2 Optical Arrangement .............................................................................. 19

2.2.3 Imaging Details ...................................................................................... 20

2.2.4 Instantaneous Planar MTV Error Estimation ......................................... 25

2.3 Stereoscopic MTV .......................................................................................... 26

2.3.1 Angular Stereoscopic Method Description ............................................ 26

2.3.2 Generalized Undistortion Reconstruction .............................................. 28

2.3.3 sMTV Acquisition / Processing ............................................................. 31

2.3.4 sMTV Experimental Validation ............................................................. 34

2.3.5 sMTV Optical Issues ............................................................................. 36

2.3.6 sMTV Error Estimation ......................................................................... 36

2.4 Line Tagging MTV ......................................................................................... 38

2.4.1 Technique Description ........................................................................... 38

2.4.2 Error Analysis ........................................................................................ 40

2.5 Data Acquisition ............................................................................................. 42

2.6 Phase Averaging ............................................................................................. 44

2.7 Calculation of Statistical Quantities ............................................................... 45
CHAPTER 3

CENTERSPAN PLAN AR MEASUREMENTS, NO CUTTING WALLS ..................... 48

3.1 Static Airfoil Characteristics ........................................................................... 49

3.2 Oscillating Airfoil ........................................................................................... 54

3.2.1 Time Averaged Data .............................................................................. 54

3.2.2 Mean Force Calculation ......................................................................... 65

3.2.3 Trailing Edge Phase Averaged Data ...................................................... 67

3.2.4 Downstream Phase-Averaged Data ....................................................... 71

3.2.5 Vortex/Cutting Wall Initial Conditions ................................................. 89
CHAPTER 4

VORTEX / SIDE-WALL INTERACTION ..................................................................... 90

4.1 Vortex / End Wall Interaction ......................................................................... 90
CHAPTER 5

vii

VORTEX / CUTTING WALL INTERACTION ........................................................... 101

5.] Line Tagging Results .................................................................................... 103

5.2 Stereoscopic Results ..................................................................................... l 11

5.2.1 Mean Flow Field, k=5.2, 11.5 .............................................................. l 13

5.2.2 Phase Averaged Flow Field, k=5.2. ..................................................... 126

5.2.3 Phase Averaged Flow Field, k=l 1.5. ................................................... 137

5.2.4 Area Varying Waves on the Vortex Cores .......................................... 148

CONCLUSIONS ............................................................................................................ 15 1

REFERENCES ............................................................................................................... 155
APPENDIX A

MEAN FORCE EQUATION DEVELOPMENT ........................................................... 159

viii

LIST OF FIGURES

Figure 1-1: Schematic representation of flow field investigated. ....................................... 2
Figure 1-2: Top view dye visualization in the flow behind a pitching NACA0012 airfoil
from Koochesfahani (1989). Flow is right to left, the airfoil is upstream of the right

side of the image. Flow facility side walls, not shown, were above and below the dye
traces. ......................................................................................................................... 2

Figure 1-3: Schematic of rotational flow over a fixed solid boundary. Adapted from Figure

10-1 from Schlicting ( 1960). Figure courtesy of Kind Design, Inc. ......................... 5
Figure 1-4: Schematic of initial vortex array formation flow field. ................................. 10
Figure 2—1: Definition of the airfoil motion. ..................................................................... 14

Figure 2-2: Schematic of the Turbulent Mixing and Unsteady Aerodynamics Laboratory
10,000L water tunnel at Michigan State University. Cameras A and B shown posi-
tioned for stereoscopic measurements. Camera C shown positioned for single line
tagging experiments. ............................................................................................... 15

Figure 2-3: Representative example of real airfoil motion (A = 2°, k=8.6) with a compar-
ison to programmed functionality. Difference between the real and programmed in
counts. ..................................................................................................................... 17

Figure 2-4: Typical MTV image pairs and the resultant 2-component velocity vector field.
The flow shown is from a vortex ring impacting a flat wall at normal incidence. The
axis of symmetry is indicated by the dashed lines.(a) Undelayed grid imaged 1ms af-
ter the laser pulse (“undelayed” image) (b) same grid imaged 8ms later (“delayed”
image) (c) velocity field derived from (a) and (b). Gendrich and Koochesfahani

(1996). ..................................................................................................................... 18
Figure 2-5: Schematic representation of optical set-up for planar and stereoscopic MTV ex—
periments. ................................................................................................................ 19
Figure 2-6: Timing diagram for the “double shot” camera operation. Each Fsynch pulse in-
dicates the start of a new image. ............................................................................. 21
Figure 2-7: Mean image pattern from a SCO intensified camera. .................................... 23

Figure 2-8: Deviation from a uniform displacement, without (a) and with (b) background
correction. ................................................................................................................ 24

Figure 2-9: Uncertainty level in MTV planar technique for SCO intensified cameras as a
function of camera gain. .......................................................................................... 26

Figure 2-10: Schematic of angular displacement stereoscopic method. Distances and angles
are derived independently for each camera. ............................................................ 27

Figure 2-11: Sample left and right image calibration grid at z = 0. .................................. 30

Figure 2-12: Representative derivatives of mapping function, F, for the right camera, x-di-
rection. (a) F1 1,], (b) F112, (c) F113. Distances in real units (cm, mm, inches etc.).
Derivative units are (pixels) / (real distance unit). .................................................. 31

ix

Figure 2-13: Instantaneous (a) left camera, (b) right camera, and (c) combined velocity field

for flow under the rotating disk. .............................................................................. 33
Figure 2-14: Schematic of the rotating disk experiment. .................................................. 34
Figure 2-15: Comparison of V9 from stereoscopic MTV with planar reference MTV data.

Data at y = -0.43 cm ................................................................................................ 35
Figure 2-16: Stereoscopic camera set-up with liquid prism. ............................................ 37

Figure 2-17: Ratio of the out-of-plane and in-plane error to the correlation technique error
level as a function of the camera angle, j. ............................................................... 38

Figure 2-18: Example of undelayed and delayed lines for a single tagged line. Delay time:
16 msec; A=2°, k= 11.5; x=0.85c, z=0.85c. ............................................................ 39

Figure 2-19: Intensity profiles taken from Figure 2-11, in the region spanwise displace-
ment. Curves shown are best fit 2nd order polynomial to :1: 5 data points about the
peak intensity location. ............................................................................................ 40

Figure 2-20: Phase averaged spanwise flow profiles for vortex / side wall interaction. A=2°,
k=2.84; x=1.27c, z=0.85c. ....................................................................................... 41

Figure 2-21: Simulation of an Oseen vortex with a Gaussian distribution of axial flow (a)
and the expected measured axial velocity distribution considering equation (8)

(b) ............................................................................................................................. 43
Figure 2-22: Phase ordered u and v velocity components for the 2 degree, k=1 1 case at x=6
cm downstream of the trailing edge. Note 2 cycles are shown. .............................. 46

Figure 2-23: Sample map of experiment fields of view combined to create large data sets
in phase space. ......................................................................................................... 46

Figure 3-1: Mean u, v and (02 fields for the stationary airfoil at AOA=O°, tunnel midspan.

.................................................................................................................................. 50
Figure 3-2: Mean velocity vectors in the near wake of the static airfoil. (a) vector length
proportional to velocity, (b) vector length uniform. ................................................ 51

Figure 3-3: Flow visualization of recirculation bubble in the near wake of a stationary
NACA-0012 airfoil at AOA=O°. ............................................................................. 51

Figure 3-4: Profile of ii, V and tit—)2 for the stationary airfoil at x=c/2 (a) and x=c (b), midspan.
Every 3rd point shown. ........................................................................................... 53

Figure 3-5: RMS u and v fields for the stationary airfoil at AOA=O°, tunnel midspan. ..53

Figure 3-6: Mean of all phases streamwise velocity, u, field for the A = 2°, k = O, 5.2, 5.7
and 11.5. Note scale change for k=11.5 .................................................................. 55

Figure 3-7: Mean streamwise velocity, 5, for A=2° k=5.2 5.7, 1 1.5 at y=O. .................... 56

Figure 3-8: Mean streamwise velocity, 6, and vorticity, El, profiles for A=2° k=5.2 5.7,
11.5 at x=O.5c, c. ..................................................................................................... 57

Figure 3-9: Schematic of the vortex orientation for the velocity deficit cases. ................ 58

Figure 3-12: Time averaged mean v velocity field for the A = 2°, k = 0, 5.2, 5.7 and 11.5.

.................................................................................................................................. 60
Figure 3-13: Mean spanwise vorticity, 232, field for the A = 2°, k = 0, 5.2, 5.7 and 11.5.
.................................................................................................................................. 61
Figure 3-14: RMS u velocity field for the A = 2°, k = 5.2, 5.7 and l 1.5 oscillating cases.
.................................................................................................................................. 62
Figure 3-15: RMS v velocity field for the A = 2°, k = 5.2, 5.7 and l 1.5 oscillating cases.
.................................................................................................................................. 63
Figure 3-16: RMS spanwise vorticity, (02, field for the A = 2°, k = 5.2, 5.7 and 11.5 oscil-
lating cases. ............................................................................................................. 64
Figure 3-17: Control volume used for determination of mean drag equation .................. 65

Figure 3-18: Mean force coefficient, CF, as a function of airfoil oscillation amplitude and
reduced frequency. .................................................................................................. 68

Figure 3-19: MTV flow visualization (k = 8.8) and velocity vectors from the planar MTV
data (k = 8.6) near the airfoil trailing edge. Airfoil at AOA=2°, trailing edge moving
up (AOA decreasing). Dashed lines indicate initial line locations. ........................ 69

Figure 3-20: MTV flow visualization (k = 8.8) and velocity vectors from the planar MTV
data (k = 8.6) near the airfoil trailing edge. Airfoil at AOA=O°, trailing edge moving
up (AOA decreasing). Note flow visualization images flipped up/down. .............. 70

Figure 3-21: Phase averaged vorticity, <(oz>, field for A=2°, k=4. 1, 5.2, 5.7 and 1 1.5 cases.
¢=0. Dashed line indicates edge of data grid. Note: Vorticity contour levels different
for each case as shown. ........................................................................................... 72

Figure 3-22: Vorticity profiles for 2 degree, k=5.2, 11.5 case at x = 6 cm, 2 = 0 cm. Curve
is best fit to a Gaussian vortex profile. Horizontal profiles show quantity as a function
of x, and vertical profiles a function of y. Equation represents the vorticity profile for
a Gaussian vortex. ................................................................................................... 74

Figure 3-23: Azimuthal velocity profile for the 2 degree, k=l 1.0 case at x=6 cm, z=0 cm.
Fits shown for portions of the profile indicated by the line sections. Functionality in-

dicated by equation near the fit line. ....................................................................... 75
Figure 3-24: Vortex array properties as a function of the streamwise distance. ............... 78
Figure 3-25: Schematic representation of vortex spacing variables. Positive b indicates pos-

itive vorticity is above negative vorticity as shown. ............................................... 80
Figure 3-26: Vertical vortex spacing, b, as a function of downstream distance. .............. 82
Figure 3-27: Vortex streamwise spacing, a as a function of downstream distance. ......... 83
Figure 3-28: Vortex convection speed, Uc, as a function of downstream distance. ......... 84

Figure 3-29: Peak vorticity wpeak, circulation F, core radius rc, convection speed U0, and

xi

spacing (a,b) for the 2-D planar measurements at X: 6 cm, 2 = 10 cm. .................. 85

Figure 3—30: Maximum swirl velocity, sz, as a function of the reduced frequency at x=c/
2, midspan. .............................................................................................................. 88

Figure 4-1: Schematic of the vortex / side wall interaction. ............................................. 90

Figure 4-2: Sample line tagging imaged. A=2° k=1 1.5 x=38rci. Dashed white lines indicate
undisplaced line center location for reference. ........................................................ 92

Figure 4—3: Schematic representation of axial flow profiles relative to vortex location.

A=2°, k=11.5, x=38rci. ............................................................................................ 93
Figure 4-4: Local phase averaged peak spanwise flow, <w>peak’ versus the downstream
distance, x, and the reduced frequency, k. Data plotted dimensionally. ................. 94
Figure 4-5: Local peak spanwise flow, <w>peak, versus the downstream distance, x, and the
reduced frequency, k. Data plotted non-dimensionally. .......................................... 95
Figure 4-6: Local peak spanwise flow, <w>peak’ versus the downstream distance, x, and the
reduced frequency, k. Distance normalized by vortex array wavelength, a. ........... 96
Figure 4-7: Peak axial flow speed, <w>peak, versus the reduced frequency for A=2° at
2: 10cm off the centerspan. ..................................................................................... 97
Figure 4—8: Peak reverse axial flow speed, <w>rev, versus the reduced frequency for A=2°
at z=10cm off the centerspan. ................................................................................. 97
Figure 4-9: Maximum axial flow, <w>max, versus reduced frequency. ........................... 98

Figure 4-10: Spanwise flow versus vertical distance, y, and phase. A=2°, k=11.5. Down-
stream location indicated on figure. Line plots show all phases (i.e., y-profiles) simul-

taneously. ................................................................................................................. 99
Figure 4-1 1: Axial velocity profiles for A=2°, k=9.6. Multiple phases shown simultaneous-
ly. ........................................................................................................................... 100
Figure 4-12: Axial velocity profiles for A=2°, k=5.2. Multiple phases shown simultaneous-
ly. ........................................................................................................................... 100
Figure 5-1: Schematic of flow field with cutting walls. Coordinate system origin is located
at the cutting wall leading edge, as shown. ........................................................... 101
Figure 5-2: Impact speed, Vi, versus reduced frequency from the mid-span data. ......... 102

Figure 5-3: Example phase-averaged spanwise velocity profiles for A = 2°, k = 11.5, x =
13rd. Phase, ¢, with the highest spanwise flow shown at each 2 location. Distance be-

tween profiles not shown to scale. ........................................................................ 104
Figure 5-4: Peak spanwise flow speed for A = 2°, k=5.2. .............................................. 105
Figure 5-5: Peak spanwise flow speed for A = 2°, k = l 1.5. .......................................... 105
Figure 5-6: <W>peak versus spanwise location for k = 1 1.5. .......................................... 106

xii

Figure 5-7: Peak to peak magnitude of velocity spike, V08, as a function of reduced frequen-

cy. .......................................................................................................................... 106
Figure 5-8: Comparison of the front speed, Vf, the wave spanwise flow speed, Vw, and the
equivalent sound speed, ce. ................................................................................... 108
Figure 5-9: Maximum spanwise velocity, <w>max comparing no-wall and cutting wall ex—
periments. .............................................................................................................. 109
Figure 5- 10: <w>my as a function of the reduced frequency. ......................................... l 10

Figure 5-11: Spanwise velocity profiles for A = 2°, k = 1 1.5, x = 58rd all phases (i.e. y-
profiles) shown. ..................................................................................................... 1 11

Figure 5-12: Comparison of the peak spanwise velocity between the line tagging and sMTV
data. ....................................................................................................................... l 13

Figure 5-13: Comparison of the planar (midplane) and stereoscopic (z=8cm, near mid—
plane) measurements of the mean streamwise velocity, u, for k=1 1.5. ................ 114

Figure 5-14: Mean streamwise velocity, u, fields, k = 5.2, 2 = rci, 2rci, 10rd, 20rd. ....l 15
Figure 5-15: Mean streamwise velocity profiles for x = 1 1rd, k = 5.2 at different 1’s. .1 16
Figure 5-16: Mean streamwise velocity, 11, fields, k = 11.5, 2 = rci, 2rd, 10rd, 20rd. ....1 17

Figure 5-17: Mean streamwise velocity profiles for three downstream positions at z = 2rd,
k = 11.5. ................................................................................................................. l 18

Figure 5-18: Mean lateral velocity, v, fields, k = 5.2, 2 = rci, 2rd, 10rd, 20rd. .............. 1 19
Figure 5-19: Mean streamwise velocity, v, fields, k = 1 1.5, 2 = Ici» 2rd, 10rd, 20rd. ....120
Figure 5—20: Mean spanwise vorticity, (02, fields, k = 5.2, z = rci, 2rd, 10rd, 20rd. ...... 122
Figure 5-21: Mean spanwise vorticity, (Dz, fields, k = 11.5, 2 = rci, 2rd, 10rd, 20rd. 123
Figure 5-22: Mean spanwise velocity, w, fields, k = 5.2, 2 = rci, 2rd, 10rd, 20rd. ........ 124
Figure 5-23: Mean spanwise velocity, w, fields, it = l 1.5, 2 = rci, 2rd, 10rd, 20rd. ......125

Figure 5-24: Phase-averaged vorticity, «02>, for ¢=0 at multiple span locations. k=5.2. ..
127

Figure 5-25: Phase-averaged vorticity, «02>, profiles for k = 5.2. Dashed lines <w> = 0.
Spacing of profiles in figure not to scale. .............................................................. 129

Figure 5-26: Phase-averaged spanwise velocity, <w>, for (1)20 at multiple span locations.
k=5.2. ..................................................................................................................... 130

Figure 5-27: Phase-averaged spanwise flow, <w>, profiles for k = 5.2. Dashed lines <w>
= 0. Spacing of profiles in figure not to scale. ...................................................... 131

Figure 5-28: Comparison of vortex parameters (F, <ml>pcak, rc, Uc) from no-wall midspan
data and cutting wall, 2 = 20rd (near midspan) data. k = 5.2. Symbols common. 133

xiii

Figure 5-29: Vortex parameters (1", <wz>peak, rC, UC, <w>pcak, b/a) for multiple span loca-
tions. k=5.2. Symbols common. ............................................................................ 134

Figure 5-30: Relative location of the peak vorticity to the peak spanwise flow, k = 5.2.
................................................................................................................................ 136

Figure 5-31: Phase-averaged vorticity, «02>, for ¢=0 at multiple span locations. k=1 1.5.
................................................................................................................................ 138

Figure 5—32: Phase-averaged vorticity, <0)Z>, profiles for k = 1 1.5. Dashed lines <w> = 0.
Spacing of profiles in figure not to scale. .............................................................. 139

Figure 5-33: Phase-averaged spanwise velocity, <w>, for ¢=0 at multiple span locations.
k=11.5. ................................................................................................................... 140

Figure 5-34: Phase averaged spanwise flow, <w>, profiles for k = 1 1.5. Dashed lines <w>
= 0. Spacing of profiles in figure not to scale. ...................................................... 141

Figure 5-35: Comparison of vortex parameters (F, <wz>pcak, rc, UC) from no-wall midspan
data and cutting wall, 2 = 20rd (near midspan) data. k = 11.5. Symbols common.

................................................................................................................................ 143
Figure 5-36: Vortex parameters (F, <wz>peak, rc, UC, <w>pcak, b/a) for multiple span loca-
tions. k = 11.5. Symbols common. ........................................................................ 144

Figure 5-37: Peak vorticity, <wz>peak, as a function of downstream distance for k = 1 1.5.
Curves shown are vertically offset to allow simultaneous viewing. ..................... 145

Figure 5-38: Spatial correlation of initial local vorticity peak to re, Uc, <mz>peak, k=11.5.
Vertical major grids are 0.1 apart. Magnitudes for the vertical axis can be found on

Figure 5-36. ........................................................................................................... 147
Figure 5-39: Relative location of the peak vorticity to the peak spanwise flow, k=1 1.5.
................................................................................................................................ 148
Figure A-l: Airfoil mean drag formulation control volume. .......................................... 159

xiv

AOA

FOV

IV

MTV

RMS

sMTV

LIST OF SYMBOLS

Airfoil angle of attack

Field of view

Camera intensifier voltage
Molecular Tagging Velocimetry
Root mean square

Stereoscopic molecular tagging velocimetry

Maximum airfoil angle of attack
Airfoil span length
Airfoil chord length

Mean force coeffient from the mean velocity profile
Mean force coeffient including pressure terms

Center to center time between gate pulse
Mapping function between real and image planes

Partial derivative of mapping functing; i with respect to j

Airfoil ossilation frequency

delayed gate period

undelayed gate period

Distance from the lense plane to center of the measurement plane

image plane direction

real plane direction

XV

max

Reduced frequency: k = (1th)/Um

z-direction distance from lense plane to center of the measurement plane
Chord reynolds number

Vortex core radius measured at x = 6cm from the midspan data

Initial vortex core radius measured at x = 6cm from the midspan data
Vortex convection speed

Tunnel free stream approach speed

Tunnel free stream at a given downstream measurement location

Streamwise velocity component

Equivilant sound speed

Front speed

Impact velocity, Uc/sz

Peak to peak spanwise flow speed difference for area varying wave.
Radial velocity component

Peak swirl speed

Initial peak swirl velocity measured at x = 6cm from the midspan data
Wave speed

Azimuthal velocity component

Lateral velocity component
Spanwise velocity component

Highest axial flow speed measured in the spatial region investigated

xvi

wpeak

Local maximum in the axial flow speed

Streamwise direction

x-direction distance from lense plane to center of the measurement plane
Lateral direction

Spanwise direction

Time delay between undelayed and delayed images

Data phase referenced relative to airfoil motion

Circulation

Initial circulation measured at x = 6cm from the midspan data
Stereoscopic camera angle

Tagged molecule phosporescence life time

Initial maximum vorticity measured at x = 6cm from the midspan data
Maximum spanwise vorticity

Local maximum spanwise vorticity

Spanwise vorticity

xvii

CHAPTER 1
INTRODUCTION

Flows with strong concentrated regions of vorticity (here after referred to as a
“vortex”) are prevalent in many technological (e.g., wing tip vortices) and natural flow
fields (e.g., tornadoes). The interaction of vortices with other vortices, boundaries (e.g.,
no-slip, free surface), particles, etc. is a fundamental problem in fluid mechanics. In this
work the interaction of concentrated vortices with a no-slip boundary condition and the
subsequent axial core flow was investigated using Molecular Tagging Velocimetry
(MTV).

It has been shown in prior work (e.g., Laursen et al., 1997; Cohn and
Koochesfahani, 1993; Koochesfahani, 1989) that the interaction of a vortex with a solid
boundary perpendicular to the axis of the vortex core causes the flow along the axis of the
vortex core (axial flow, core flow, or secondary flow). Figure 1-1 shows schematically the
array of alternating sign vortices generated in the wake of an oscillating airfoil. When the
vortices interact with a cutting wall, flow is developed in the vortex core. Figure 1-2 is
taken from Koochesfahani (1989) and shows the top view of a dye visualization
experiment in the wake of an oscillating airfoil. In Figure 1-2 the flow direction is right to
left and the airfoil is directly upstream of the right side of the picture. The axial flow
within a vortex core is made visible by lateral displacement of the dye. Figure l-2(a)
shows the development of axial flow as a result of the interaction of the vortex array with
the side walls of the tunnel. In Figure l-2(b) a false wall is placed into the center span
location of the tunnel. Both (a) and (b) show dye being ejected away from the no-slip
boundary as expected. A similar finding was reported by Wiegand (1993) for an

axisymmetric vortex ring cut by a solid wall. In the present work the flow field of a line

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Flow

(b) with false walls

Figure 1-2: Top view dye visualization in the flow behind a pitching NACA0012
airfoil from Koochesfahani (1989). Flow is right to left, the airfoil is upstream of the
right side of the image. Flow facility side walls, not shown, were above and below the
dye traces.

vortex interacting with a no-slip boundary was investigated with the goal of understanding
the effects of the interaction on the vortex.

Interest in flow fields with concentrated vortices containing axial flow is driven by
several technical areas. For example, the passage of helicopter rotor blades creates trailing
vortices behind the rotor blades. During forward flight, vertical descent and landing
(forward based descent) following rotor blades can interact with the trailing vortex of the
leading blade. This interaction, Vortex Blade Interaction (VBI), is a major source of noise
and, as Conlisk (1997) noted, “reducing the number and intensity of the interactions is
critical for reducing rotor noise.” Much of the work with helicopter rotor fluid dynamics is
conducted computationally. Reduction of the noise is dependant upon the ability to
compute blade loads accurately. Conlisk (1997) indicates that many researchers/designers
specify the vortex core structure and do not allow it to change temporally. This approach
yields computed blade loads that are fairly accurate for a single interaction but the
solutions were found to be sensitive to the core size chosen. When the vortex core
structure was allowed to change with interaction and calculated along with the flow field
for multiple VBI encounters the results no longer agreed well with experiments. Failure to
predict the vortex structure changes is a limitation since multiple blades may interact with
the same vortex. These observations were also echoed by Hassan et al. (1992) who noted
that the accuracy of the simulations of VBI was dependant upon the user specified core
size and strength.

A second area of technical interest in the subject of vortices with axial flow is in
turbomachinery. Kotidis (1989) performed a study on the losses in a turbine. One of the

identified losses was the transport of fluid from the tip of the blade to the hub via axial

flow within the spanwise shed vortices. As a final technical example, vortex breakdown,
important in many technical areas, has been associated with sudden widening of the vortex
radius and a deceleration of the axial flow within the core (e.g., Lopez 1990). Naturally
occurring flow fields that exhibit strong regions of vorticity also show flow along the axis
of the vortex. A vivid example of axial flow within a strong vortex occurs within
tornadoes and hurricanes, and to a lesser extent in dust devils and water spouts
(Maxworthy, 1973).

Much of the past work in the area of rotational flow near a no-slip boundary,
including the start-up axial flow within a vortex core, and the resulting 3-D core structure
is analytical. First solutions in this area considered either a quiescent fluid above a rotating
infinite plate (von Karman, 1921), or solid body rotation over a infinite plate (Bodewadt,
1941). These flows are illustrative in that the source of axial flow, or secondary flow,
within the core of a vortex can be understood by examining these “simple” flows.
Consider the case of solid body rotation over a fixed plate shown in Figure 1-3. The flow
has a constant angular velocity equal to (11 Away from the wall where, the flow can be
assumed inviscid, the r-momentum equation can be simplified to the Euler-n equation
which equates the radial pressure gradient to the azimuthal velocity and radial location.
Near the plate viscosity cannot be neglected. The azimuthal velocity is reduced but the
radial pressure gradient is assumed to be the same as the far field via boundary layer
approximations. The corresponding imbalance indicates fluid particles move radially
inward (for rotating fluid, fixed wall) near the wall. Note that the Euler-n equation does
hold near the wall in the boundary layer, but the r—momentum equation can be used to re-

derive the expression with additional terms (from the viscous term in the r-momentum

 

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equation) with V,(r,z) added. The fluid particles transported towards the center of the

vortex are expelled away from the wall due to mass conservation. For example, in Figure
l-2(a) the axial flow that develops due to the interaction of the vortex with the side walls
of the tunnel is directed away from the walls. Similarly, when the cutting plate is placed
into the flow near the centerline of the tunnel, as in Figure 1-2(b), the axial flow is directed
away from the flat plate.

While the solid body rotation flow field is illustrative, it does not represent the
vortical flow field of a concentrated line vortex. Other researchers (e.g. Rott and

Lewellen, 1966; Burggraf et al., 1971) have sought similarity solutions for a generalized

vortex, defined by the azimuthal velocity field, V0 ~ r—a, over a no-slip wall. The flow

field of a generalized vortex is defined at the extremes by on = -l for a solid body rotation

and CL = +1 for a potential vortex. Rott and Lewellen (1966) were limited to 0t < 0.1217

because the solution diverged for 0t > 0.1217; however, Burggraf et al. (1971).were able to
find solutions for the entire range of or by creating composite similarity solutions. Kuo
(1970) developed analytical solutions for vortices that simulated tornadoes, or flows in
tornado chambers. The solution showed a strong upward motion of fluid within the core of
the vortex and a weaker descending or recirculating motion outside the core region.

Donaldson (1956) and Donaldson and Sullivan (1960) investigated axial flow
solutions allowable under the Navier Stokes equations. Donaldson and Sullivan (1960)
were able to show that several different axial flow core structures were possible. These
core structures included single axial flow peak in the center of the vortex (single-celled
structures) and axial flow distributions with both positive and negative axial flow
velocities (multi-cell).

Experimental work on the formation of axial flow, and the resulting core structure,
is limited. Koochesfahani (1989) studied the flow behind an oscillating NACA0012 airfoil
and showed that axial flow developed within the vortex core due to interaction with solid
boundaries (i.e., the tunnel side walls and/or false walls placed in the flow). The
magnitude of the axial flow speed was estimated by the displacement of dye and appeared
to vary linearly (for constant oscillation amplitude) with the reduced frequency. The

magnitude was significant with respect to the freestream velocity (around W/U m= 0.3-

0.65). Koochesfahani (1989) noted that the use of dye to estimate the axial flow speed
represents an integrated or average velocity during the dye transport. Cohn and
Koochesfahani (1993) looked at the boundary conditions that cause axial flow to be
initiated in a flow field similar to Koochesfahani (1989). The axial flow was initiated for

the no-slip boundary as expected. However, axial flow was also initiated for a shear

boundary condition. The core structure was visually different between the two
experiments. The solid wall boundary created a core structure with the highest axial flow
near the center of the vortex core (single-cell). In contrast, the shear boundary created a
core axial velocity profile that indicated the maximum axial velocity occurred near the
edge with a very small axial velocity near the center (multi-cell).

Laursen et al. (1997) performed Particle Image Velocimetry (PIV) measurements
and dye visualization for a 2-D vortex pair, created by ejecting fluid through a slit,
traveling in a still fluid. They noted that the vortex structure is initially 2-D, but quickly
becomes unstable and breaks up. Evidence for the “break up” indicated axial flow within
the vortex core was responsible. An estimate for the axial flow speed was determined
using dye markers. The data indicated that the axial flow speed was nearly equal to the
convection speed of the vortex pair. Hagen and Kurosaka (1993) used heated fluid as a
passive tracer in a hairpin vortex created by a ramp protruding into a flow. The results
confirmed the corewise transport of the heated fluid within the vortex core. The velocity
of the transport was found to be on the same order as the free stream velocity.

The previous experimental studies were performed for free or convecting vortices.
More work had been performed for confined vortices (e.g., vortex tubes, “tornado
chambers”, etc.). Measurements of the velocity field of a vortex in a tornado chamber

were made by Wan and Chang (1971). These experiments indicated that the axial flow

structure for tornado-like vortices was determined by a parameter ec = Qc/(Frc),

where Qc is the volume flow through the vortex core, I" is the circulation and rC is the core

radius of the vortex. Two cases were studied, a “low” cc, and a “high” ec. The low eC

vortex had a single cell axial flow structure, while the high eC had a two cell axial flow
structure.

Hirsa et al. (2000) studied the flow field of a vortex created by a pair of flaps in a
tank using PIV and flow visualization. The data indicated that there was a region of
recirculation near the wall within the vortex core with very little axial flow. This region
was characterized by a sudden increase in the radius of the vortex and was considered to
be analogous to vortex breakdown seen in vortex chambers. The structure of their vortex

indicated that maximum axial flow was at approximately r = rc, rather than r = 0. The

recirculation region was also discussed in Maxworthy (1972) and was described as a
“vortex jump”. Maxworthy indicated that this jump region allows the flow to match
boundary conditions between the natural upflow from the vortex no-slip interaction, and
the much larger outflow from the chamber. The vortex jump discussed in Hirsa et al.
(2000) and Maxworthy (1972), has also been observed in natural flow fields such as dust
devils, waterspouts, and tornadoes, Maxworthy (1973).

Vortex interaction with walls has been studied for vortices with axial flow already
present by Krishnamoorthy and Marshal (1994), Marshall and Yalamanchili (1994) and
Marshall and Krishnamoorthy (1997). In these works the results show that a vortex with
axial flow present that is cut will produce either expansion or compression waves at the
wall, depending on the direction of the axial flow with respect to the cutting wall. The area
varying waves travel along the core of the vortex away from the cutting surface. Lundgren
and Ashurst (1989) investigated the propagation of area varying waves on the vortex core
and found that the equations governing the propagation of wave on a vortex core were

analogous to gas dynamics wave transport equations.

To date, quantitative measurements of any kind are limited and whole field
measurements of the vortex core with axial flow do not exist. Rockwell (1998) reviews the
literature on vortex body interaction. In the section on normal interactions (i.e., were the
axis of the vortex is perpendicular to the cutting body) Rockwell (1998) states that the
“study of vortex distortion in the vicinity of the surface, fully accounting for viscous
effects, would be fruitful.” Therefore, well resolved measurements in the core of a vortex
interacting with a no-slip boundary are needed to provide fundamental understanding for
further work in this area. The current research is aimed at acquiring an understanding of
the development of axial flow due to normal incident interaction of a columnar vortex
with a no-slip boundary and the associated effects on the vortex structure.

In this work, measurement and analysis of the undisturbed flow field (flow in the
near wake of a pitching airfoil Figure 1-4) was required to provide reference and initial
conditions. However, the problem of the trailing edge flow field and initial development
of the vortical wake behind an oscillating airfoil is also of fundamental interest to
aerodynarnicists. Three specific areas of interest are: the validity of the unsteady Kutta
condition at the trailing edge, the formation region of the vortex array, and the mean
thrust/drag of an oscillating airfoil.

Much of airfoil theory and design is based upon inviscid theory, use of which is
reliant upon assumptions of the flow conditions at the trailing edge of the airfoil. Inviscid
approximations require that the flow leave the airfoil trailing edge “smoothly”. The Kutta
condition requires that a stagnation point exists and is fixed at the trailing edge for the

static airfoil (McCrosky, 1982). The unsteady Kutta condition is an extension of the steady

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Kutta condition to allow inviscid analysis on a oscillating airfoil and assumes a zero
pressure difference across the trailing edge (Maskell, 1972; Giesing, 1969 ).

Experimental work, used to determine how well the inviscid analysis represents
real viscous flows, has shown that the classic unsteady Kutta condition may be valid at
low reduced frequencies; however at higher reduced frequencies the validity is less certain
(Poling and Telionis, 1986). Uncertainty over the importance of trailing edge conditions
can be illustrated by the conclusions of two works. Katz and Weihs (1981) concluded that
the use of the unsteady Kutta condition, though it may be physically inaccurate, does
provide accurate estimates of airfoil forces and downstream flow patterns using inviscid
solution methods. Stanek and Visbal (1989) performed a simulation using the full Navier

Stokes equations of the flow field experimentally investigated in Koochesfahani (1989).

The results showed only qualitative agreement between the calculations and experimental
data. The differences were attributed, in part, to resolution near the trailing edge and
details of the trailing edge shape. These differences highlight importance of the trailing
edge flow conditions.

The study of the mean flow field downstream of oscillating airfoils has been
motivated by the use of oscillating and/or heaving airfoils as a thrust generation
mechanism (Anderson, 1998; Streitlien and Triantafyllou, 1998; Platzer and Neace, 1993).
The formation of the vortex array behind an oscillating airfoil has been studied mainly
with flow visualization and point measurements (Wilder et al., 1996; Koochesfahani,
1989). Koochesfahani ( 1989) showed, using dye visualization, that the vortical pattern in
the flow field generated by the oscillating airfoil can be complicated with multiple
structures for each oscillation depending up on the oscillation amplitude, frequency and
wave form. The mean thrust has been calculated using point data based on the mean
streamwise velocity field (e.g. Anderson et al., 1998; Koochesfahani, 1989). Whole field,
time resolved velocity and vorticity measurements of the formation region behind an
airfoil are limited. Whole field measurements would increase the general understanding of
the physics of the flow field and allow the thrust/drag to be estimated more precisely.

The current work can, therefore, be divided into two broad categories: the initial
formation region of the 2-D vortex array without cutting walls and the interaction of the
vortex array with the no—slip cutting walls. The flow field for the first section of this work,
shown schematically in Figure 1.4, is characterized by the airfoil parameters, the chord, c,
and reduced frequency, k. The coordinate system origin for this subsection of the study

was at the trailing edge of the airfoil when the airfoil passed through 0 degrees, as shown.

The measurements of the undisturbed flow field were needed to determine the
initial conditions for the axial flow study. The same measurements of the undisturbed flow
field simultaneously provided the secondary goals for this work. Specifically, details of
the near field development of the flow behind an airfoil oscillating at a high reduced
frequency touching on the associated issues and characterization of the development of
the wake of an oscillating airfoil over the first chord length downstream from the airfoil
trailing edge.

It is important to note that the work detailing the vortex / wall interaction has
considered the dynamics of a vortex interacting with a no-slip boundary without regard to
the generation method. The airfoil is used as a method of generating vortices because it
allows control of the vortex properties through the airfoil peak amplitude and oscillation
frequency.

The primary goal of this work was to investigate the interaction of the vortex array
with a no-slip boundary (Figure l-l ). The flow field for this portion was characterized by

the vortex strength (i.e., circulation, F, and peak vorticity, (oz) and core radius, rc (defined

as the He point for a Gaussian vortex). This flow field was shown schematically in Figure
1.1. The origin was fixed at the leading edge of the wall with 2 positive away from the
wall. The goal of this portion of the work was to detail the vortex and axial flow structure
after the vortex/cutting wall interaction and the changes in the vortex properties after the
interaction. The axial flow properties were measured and the effect of the axial flow on the
vortex peak vorticity, circulation and core size was detailed. Finally, the presence of area

varying waves on the vortices was addressed.

The remainder of this work is divided into three broad categories: experimental
methods (Chapter 2), results/discussion (Chapters 3-5) and conclusions (Chapter 6). The
results and discussion are further subdivide into the initial evolution of the vortex array
without the cutting walls (Chapter 3), the spanwise flow generated by the vortex / tunnel
side wall interaction (Chapter 4), and the vortex array properties after interaction with the

cutting walls (Chapter 5).

CHAPTER 2
EXPERIMENTAL METHODS
2.1 Facility

The interaction of a vortex core cut by a solid wall is investigated in this work. The
resulting flow field is shown schematically in Figure 1-1. A NACA-0012 airfoil (chord,
c=12cm) was oscillated in a periodic motion about the quarter chord location to generate a
sequence of alternating sign vortices. The motion was described by a sine function with a

peak amplitude or airfoil Angle of Attack (AOA), A, and a frequency, f, (Figure 2-1).

 

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Airfoil Angle of Attack

 

 

 

 

Figure 2-1: Definition of the airfoil motion.

Walls were placed into the flow field after isolated vortices were formed to cut the
vortices. The work of Cohn and Koochesfahani (1993) and Koochesfahani (1992, 1989)
indicate that shortly after the vortex/cutting wall interaction occurs a transport of fluid

within the core region of the vortex is initiated.

The research was conducted in an Engineering Laboratory Design (ELD) 10,000
liter close-loop water tunnel with a 2’x2’x8’ (61 cm x 61 cm x 243 cm) test section, see
Figure 2-2. The facility is located in the Turbulent Mixing and Unsteady Aerodynamics

Laboratory at Michigan State University.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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shown positioned for stereoscopic measurements. Camera C shown positioned for
single line tagging experiments.

The freestream velocity, U”, was adjusted to 10.5 cm/s, which corresponds to a
chord Reynolds number, ReCzIZOOO. The ReC was chosen to match the conditions
reported in Koochesfahani (1992, 1989). The flow upstream of the airfoil had a RMS level

of 0.015Um. The unsteady pitching of an airfoil can be characterized by the reduced

frequency, k, where k = (ncfl/ U00. Given c = 12 cm and free stream velocity, U00 = 10.5

cm/s, the reduced frequency was k = 3.59f, where f is the frequency of the airfoil motion.
The airfoil was held in a false wall assembly that allowed linkage between the motion
controller and airfoil. The airfoil was held with only a small gap (< 1mm) between the
airfoil and assembly walls. The aspect ratio of the airfoil/assembly was 4:1.

The motion of the airfoil was controlled by a Galil DMC-1030 motion control unit.
The resolution of the motor providing the motion was 10,000 counts per revolution or 27.7
counts per degree. The controller was programmed to provide a symmetric sinusoidal
motion to the airfoil. An example of the airfoil motion is shown in Figure 2-3. The
deviation level was found by subtracting the actual airfoil position from the best fit sine
wave for the data. The peak deviation was 2 counts (0.07 degrees). The RMS deviations
level was 0.6 counts (0.02 degrees). The deviation levels, peak and RMS, were
representative of all airfoil motions.

The airfoil zero angle of attack was found by oscillating the airfoil at several

reduced frequencies and measuring the vortex array properties, <wpcak>, F, and re. The

ratio of these properties between the positive and negative vortices was calculated and
plotted versus mean angle. The airfoil zero AOA was determined by the condition that
most caused the ratios to be one. Once the airfoil zero AOA was determined, a camera was
aligned with a physical target placed on the airfoil and fixed securely in place. The airfoil
zero AOA could then be checked as needed based on camera images of the target. This
ensured a consistent zero degree AOA throughout the experimental procedure.

Two oscillation amplitudes, A = 2° and 4°, with frequencies in the range of f =
1.14-3.20 Hz (k = 4.3 - 12.1) for A = 2° and f= l.l4-1.78 Hz (k = 4.3 - 6.7) for A = 4°

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comparison to programmed functionality. Difference between the real and
programmed in counts.

enough to ensure that isolated vortices were formed by the first chord length downstream
of the airfoil trailing edge. The upper end of the reduced frequencies was limited by the
spatial stability of the vortex array. Specifically, at higher reduced frequencies the vortex

array was found to change lateral position during and between runs.

2.2 Two Component Planar Molecular Tagging Velocimetry

2. 2. 1 Technique Description

Molecular Tagging Velocimetry (MTV) is a whole field optical technique which
relies on molecules that can be turned into long lifetime tracers upon excitation by photons
of an appropriate wavelength. Typically a pulsed laser is used to "tag" the regions of
interest and those tagged regions are interrogated (i.e. imaged) at two successive times

within the lifetime of the tracer. The displacement of the tagged region between the two

interrogation times is determined and the measured Lagrangian displacement vector

provides the estimate of the velocity vector, see Figure 24. MTV can be viewed as a

 

 

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Figure 24: Typical MTV image pairs and the resultant 2-component velocity vector
field. The flow shown is from a vortex ring impacting a flat wall at normal incidence.
The axis of symmetry is indicated by the dashed lines.(a) Undelayed grid imaged lps
after the laser pulse (“undelayed” image) (b) same grid imaged 8ms later (“delayed”
image) (c) velocity field derived from (a) and (b). Gendrich and Koochesfahani
(1996).
molecular analogue to Particle Image Velocimetry (PIV) and can be utilized in flows
where particle seeding, needed for PIV, is either difficult or problematic. MTV was better
suited to the current study because particle seeding within the core of a vortex is difficult.
The 2-component and 3-component measurements in this study utilized a grid of
intersecting laser lines, as shown in Figure 2-4, for tagging purposes. A direct spatial
correlation method was utilized to determine the displacement of the tagged regions.
Detailed discussions of the planar 2-component MTV technique, and the chemistry of the
chemical complex used, can be found in Koochesfahani et al. (1996), Gendrich and

Koochesfahani (1996), Gendrich et al. (1997), and Koochesfahani (1999).

The molecular tracer used in this study was a chemical triplex with the following

molar concentrations: 1 x 10'4 M g1 B-cyclodextrin, 0.055 M cyclohexanol, and a saturated

solution of l-bromonaphthalene (approx. 1 X 10'5 M). The lifetime for this chemical
complex was nominally 13 =3.5ms. The photon source was a Lambda Physik LPX 210i

XeCl excimer laser which generates 20 ns long pulses at a wavelength of 308 nm.

2.2.2 Optical Arrangement

The optical arrangement for the 2-component and 3-component MTV experiments

is shown in Figure 2-5. The excimer laser produced a rectangular beam that was first

      
    
 

      

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passed through a cylindrical lens combination to expand the beam along the long axis. A
second cylindrical lens combination was used to focus the beam along the short axis into a
thin sheet approximately 0.5 mm thick. The use of cylindrical lens combinations for these

two operations allowed variable focal lengths to be achieved which in turn allowed the

beam to be focused where needed (i.e., at the measurement plane). The beam next passed
through a 50-50 beam splitter to create two laser sheets. The sheets were redirected into
the tunnel using an array of mirrors. Brass plates with machined slots were used to
alternately block and pass portions of the beam, creating a series of lines. Quartz windows
were set on the top free surface of the water tunnel to allowed the beams to enter the
working fluid without distortion from surface waves. The series of lines from the two
sheets intersected in the measurement region, creating a tagged grid pattern similar to that

shown in Figure 2-4.

2. 2.3 Imaging Details

Past implementations of MTV (see e.g. Gendrich 1998, Cohn 1999) have used a
two camera approach to image the illuminated flow patterns. One camera begins to
acquire an image a short time (~ 1 us) after the laser fires to provide the initial pattern
location or the “undelayed” image. An image from the second camera is taken a known
time after the first camera to provide the distorted, or “delayed”, image. Use of this
technique requires the two cameras to be spatially aligned precisely. A different strategy is
used in this study to provide the image pairs. Specifically, a strategy was employed which
that allowed the two cameras, described above, to be replaced with a single intensified
Charge Coupled Device (CCD) camera.

Intensified cameras, analogous to “night vision” cameras, allow amplification of
low light scenes. The image intensity is controlled by setting the gain (i.e. amount of
amplification) and the gate period. The camera electronics allow the user to arbitrarily
place the gate pulse (i.e. “intensifier on” pulse) within a frame. When the pulse is high the

intensifier is activated and the camera CCD array is exposed to photons. The period of the

20

F synch
4— 1/60 sec H

 

 

 

 

 

 

 

 

 

 

—>l At=d+(gd-gu)/2 ‘41——

undelayed gate: gu 9W— _>L ‘— gd:delayed gate

‘4—d——>l

laser pulse

 

 

 

 

 

 

Figure 2-6: Timing diagram for the “double shot” camera operation. Each Fsynch
pulse indicates the start of a new image.

gate pulse is analogous to the shutter time on non-intensified video or still cameras. By
utilizing the ability to place the shutter at any point within the frame time, an image pair
can be created by gating one frame at the end of a frame (the undelayed image) and gating
the next image at the beginning of the following frame (the delayed image) for the “double
shot”. A Stanford Computer Optics (SCO) 4QuikE intensified camera was used in this
work. This camera is a full frame CCD camera with a resolution of 640 x 480 pixels and a
framing rate of 60Hz. This timing strategy reduced the effective data rate by one half,
from 60 Hz to 30Hz, as two frames were needed to provide each image pair (i.e. each
instantaneous whole field velocity measurement).

A schematic of the “double shot” camera timing diagram is shown in Figure 2-6.
The timing signals were created using a function generator and two Stanford Research

Systems (SRS) Digital Delay Generators. The laser was fired shortly (~lusec) before the

21

undelayed gate was applied. The center to center time between the gate pulses, d, was
precisely known.

The delay times used in this study were near to the lifetime value for the chemical
triplex (3.5-4.5 msec) and therefore the light intensity of the tagged regions was expected
to drop approximately 66% from the undelayed to the delayed images. The intensified
cameras used did not allow change in the camera intensifier gain between images. The
tagged lines in the delayed image would therefore become too dark compared to the
undelayed tagged lines to be used in the correlation routine. Different gate periods were
used for the undelayed and delayed images to compensate for the loss of intensity. The
delay time between images was therefore defined for this timing strategy by the time from
the center of the undelayed gate to the center of the delayed gate plus as shown in Figure
2.6. Typical gate periods were 0.2 msec and 0.4 msec for the undelayed and delayed
images respectively.

The use of intensified cameras can be complicated by a mean background pattern
on the image. Figure 2.7 shows a mean background image from a SCO camera.
Observation of this image indicates a small scale honeycomb pattern on the image caused
by the intensifier coupling. MTV data are affected by this background pattern in the
following way: There are two intensity patterns present on each data image, the tagged
lines and the background honeycomb pattern. The tagged line patterns move spatially,
however the honeycomb pattern remains fixed relative to the CCD array. There is a
relative displacement between the tagged lines and the honeycomb pattern. This relative
displacement is measurable, and causes an extra error in the measurements. The error was

significantly reduced by correcting for the honeycomb pattern. A series of images were

22

 

Figure 2-7: Mean image pattern from a SCO intensified camera.

taken of a relatively uniform image (uniformly illuminated white background), and those
images were averaged to create an average image that contains the background
honeycomb pattern as shown in Figure 2.7. All data images were then divided by that
mean background image reducing the effect of the mean background image. There was
also a large scale non-uniformity in the image intensity (i.e. the image is darker on the
edges) shown in Figure 2-7. This intensity pattern does not affect MTV measurements
because of the global, rather than local nature of the pattern, however, this non-uniformity
was also compensated for in the above strategy.

The affect of the background and the background compensation technique were
illustrated via target tests. A target was displaced uniformly in the x-direction in these
tests. The measured displacement was found and the difference between the measured

displacement and the actual uniform displacement was taken to be the error. This was

23

0.2 pixel -> (a)

 

 

k“ V: v ‘v‘ ' v “ ' ' ‘ ‘
* ‘\\‘ " - ‘ '0 .1
~ «x ‘V ‘ . C c C 0 V.” ‘ ' A , ’
‘ ‘ ’ ' . . o
F'r‘,’\ I. . . ‘ h ‘
‘ ~ ‘ C _
‘ ‘ ‘ § ‘
-100-/;\L’ ' ' "’ ' l. - . .
>- I I J 1 . _ . o , f ‘ =0 09 l
L. I I _. V y a 1“ O A a . ‘ ‘ rms . plxe
i ’ . ' m” ' . A' , , ‘
i- c. * ., " . v
“ A h.
V t
— I", \ I - 9" . ‘ ‘v ‘ . . . . . ‘ a “
“-200 I? I / I i ‘ o y ‘ - . . . . . . ‘
a bth.. ‘ O ' A - ‘ ' V ' - . . fio V‘
*__ a ' ‘
V T ‘— " '< “ . 5“.
I b o ‘ ‘ ‘
> 1 f .‘ T. 'V‘ ‘ < .
.- 1 I 1 . I ‘
i, o. ‘O ' ‘f ‘ r O ‘ . ,
.300 -"~ ‘5’ x V ' ' .5 -> 0 . . v a
k - 1‘ t . . . . . V v .v s ‘
s
\‘- \ f ‘. fl . ‘
i- , A ’ ‘1 4 a . . o . ,g
\0— ' - ' ‘ a " ‘
G ¢ ‘ ‘ ' G
V e
t \\ . “‘ V t» D ‘. ‘ ‘ .
.400 . [/h'.‘ . ‘ a I i ' 5“ . s-.' ’V .
s ‘ a g ‘ ' ' '
\ . b. p. . . ‘
l l l J L 1 1 L 1+1 1 l l l l l ‘1 l l l l l 1

 

 

\. ( .1
100 200 300 400 500 600
x (pixel)

 

0.2 pixel —> (b)

=\\|\|Il\\//1I ' P v».
r\\ "\ cs ‘. I
I
“H“... . . ' ""rms=0.03prxel
r-\..'--‘ I '."d
.100"\ I O I‘ ‘ "
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L\ O 4: “

I)

I
' \ O ‘

-200

 

 

r—‘
p— ‘ u
k. .
b. I ‘
a l-‘ . \ - ' i Q _., ,
> j..\ . \ \ ' - ‘
p ' I l . ' ‘ I ‘ *. -
.300 _. - . , a .....
.. I ‘ , . ...... ‘
l.‘ ‘ O I ‘ I - - ' .
t—- -' I ‘ \ , ‘ .1 ‘
r-' ' \ o. .0- “.
-400 L. . . -- .
L, . t . , . ‘ \ .. ' . a I . _. . ~ _ . ‘ .
L L L [AL 1 l i] I l [4 LI l l I 1 l I l l l l l

 

100 200 300 400 500 600
x (pixel)

Figure 2-8: Deviation from a uniform displacement, without (a) and with (b)
background correction.

performed on the same images with and without background compensation. Figure 2.8
shows the error for the target tests. The figure clearly shows the decreased error in the

compensated data over the majority of the image plane. Quantitatively, the error level,

24

measured by the RMS level, was reduced by a factor of three through the compensation

technique used. Typical error levels are discussed in the next section.

2. 2.4 Instantaneous Planar MTV Error Estimation

A series of images of a target with zero displacement were taken with the SCO
cameras used in this study and processed using the MTV correlation technique to provide
an estimate in the uncertainty in the planar implementation of the technique. The u and v
displacements for these images, when processed, should be zero because the target was
fixed spatially. Any non-zero displacements were a result of noise on the image and error
from the processing technique. This error estimate is not affected by the background
pattern discussed in the previous section because there was no displacement for these error
tests. The results from these tests is shown in Figure 2.9 for the intensified cameras used in
this study. An extensive discussion of general error estimation for the planar technique in
general can be found in Gendrich and Koochesfahani (1996). The camera intensifier was a
significant source of image noise so the error level in Figure 2-9 was found as a function
of camera gain. Typical intensifier voltage (IV) levels are between 1200 and 1240 in this
study which corresponds to an error level of 0.17 to 0.24 pixels. The error level quoted
was for the 95% confidence level, indicating that 95% of the measurements will be better
than the quoted error estimate. This corresponds to a 0.08 to 0.12 pixel RMS error level
for a Gaussian distribution. Typical magnifications and delay times of 0.0088 cm/pixel
and 3.5msec, respectively, were used in this study. The 95% error level for an

instantaneous measurement in dimensional units was therefore 0.4 to 0.6 cm/sec.

25

 

F’
N
O

(pigell

note: curve is an exponential fit to data

9’
or
c

.°
N
o

O

b

O
I'IIII'ITFT'IIII'IIII'ITII'IIII'I

95% configence level
a.» .
O

 

P
.5
O

 

I l L 1%] I l 1

 

 

.°
0
o

l l l l l l l
1200 300 1400

11 1100‘ 1
intensifier votage (IV)

Figure 2-9: Uncertainty level in MTV planar technique for SCO intensified cameras
as a function of camera gain.

2.3 Stereoscopic MTV
2. 3.1 Angular Stereoscopic Method Description

Standard planar optical techniques such as MTV and PIV measure velocity
components in the plane of the laser sheet (i.e. they provide two components of the
velocity vector in the plane of the laser sheet). In a flow with all three velocity components
standard planar techniques have two weaknesses: First, the accuracy of the two
dimensional velocity vectors suffers due to artificial in-plane velocities measured as a
result of the out-of-plane motion. This effect can be minimized by the appropriate choice
of optics. Second, a potentially significant portion of the flow dynamics (i.e. the out-of-
plane velocity component) is not measured.

Measurement of all three velocity components was needed for this study and

therefore a stereoscopic Molecular Tagging Velocimetry (sMTV) technique was

26

developed. Stereoscopic measurement techniques make use of two detectors that view the
measurement region from two different perspectives. Comparison of the simultaneous
flow velocities between the two cameras yields the complete velocity vector in the plane
of the laser sheet.

An angular stereoscopic measurement technique was developed and tested for use
with the base planar MTV technique for this work. A schematic of a general angular
stereoscopic system is shown in Figure 2-10. The construction of the 3 component

Measurement

object plane Region

X

   
  
  

 

‘P
lens plane \‘A

image plane
Lefi Camera Right Camera

 

 

Figure 2-10: Schematic of angular displacement stereoscopic method. Distances and
angles are derived independently for each camera.

velocity field from the individual cameras 2-component measurements can be
accomplished using two primary techniques: geometric reconstruction, and the
“generalized undistortion” method. The geometric reconstruction method uses ray tracing
and pin hole approximations, see for example Westerweel and Newstadt (1991) and Bohl
et al. (2001). The reconstruction equations are dependant upon the physical parameters of

the experimental system as shown in Figure 2-10. In addition, the geometric

27

reconstruction equations are also dependant upon the magnification ratio between the
image and object planes. This method is “easy” to implement and provides very accurate
results if there is no index of refraction change between the camera and image planes and
the magnification is uniform across the image. In the presence of an index of refraction
interface the geometric method still works well, although there is a systematic error across
the image in the measured velocity components, see Bohl et al. (2001). Geometric
relationships can also be determined for non-uniform magnification ratios, however they

are significantly more complicated to derive and are specific to a given geometry.

2.3.2 Generalized Undistortion Reconstruction

The generalized undistortion method accounts for the imaging geometry of the
system via calibration of the imaging system. The present implementation of the
generalized undistortion method follows from the work of Solof et al. (1997) which
described the method for a PIV system. The method described in that work is general in
that it is independent of the planar method (i.e. MTV, PIV) used to provide the
displacement measurements. It is also general in that methods other than the angular
displacement method (e.g. the linear translation method) can be used with it.

First consider the transform relationship for a single camera. A location in space
can be related to a location on the image array by the following relationship:

X = F(x) (1)

where X is the image plane location, x is the real space location and F is a function that
relates the two. The displacement recorded by the camera can be related to (1) by:

AX = F(x + Ax) - F(x) (7—)

A first order expansion of (2) gives the displacement as

28

leI

where

11 ill
direct?
plane
displa
solve:
(llSpi;
displ;
Smal

l'ectr

Whe
dist

dis;

AX z VF(x) . Ax (3)

where
BFI.

In (4) i represents the image plane directions (1,2) and j represents the real-plane
directions which can, in general, be three (1,2,3). If the flow is 2-dimensional the third real
plane direction drops out. In that case (3) can be solved to provide the real plane
displacements using a single camera. If there is out-of-plane motion present (3) cannot be
solved using a single camera because the number of unknowns, the real plane
displacements dimensions, 3, outnumbers the number of knowns, image plane
displacements dimensions, 2. In practice, the effect of three dimensionalities may be
small, and equation (3) cane be used with good results to determine the in-plane velocity
vector.

In a stereoscopic system (2) can be written uniquely for each camera as:

AXC = Fc(x+Ax)—Fc(x) (5)
where c=1,2 represents the left and right cameras. The 3-component version of (4) is

written as:
C
AX? z VFi,j(xj) - Ax]. (6)

where i=l,2 (each camera in-plane displacement direction) and j=1,2,3 (real plane
displacement direction). Equation (6) can be expanded to relate the image plane

displacements to the real plane 3- component displacements as follows:

29

_ , 3
- sz (7)

E. E. N? R

Equation (7) is solved for (ij) via least squares fit. The real displacements, (ij), are
converted directly to real units (i.e. inches, cm, mm, etc.) by this transform.
The solution of (7) requires the establishment of the relationship between real

space locations and image plane locations. This is done via calibration. In this work a

Left Camera Right Camera

 

 

II
1111

 

 

 

 

Figure 2-11: Sample left and right image calibration grid at z=0.

calibration target with printed intersections at known spatial locations was placed into the
flow field (Figure 2-11). The target was moved to and imaged at seven 2 planes, with the
cameras held fixed, and a polynomial fit (4th order in x and y, and 2nd order in 2) was

created for each camera. The polynomial fits were differentiated to provide FiJ for use in

(7). The firnction F, and its derivatives F, were smooth, slowly varying and well

J

described by the polynomial fits (Figure 2—12).

30

A
a:
v
A
a
v

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: T / .— \ \\\ /
0.5? i / / ~ \\\\\ “21x // /
: ,I' L 0.5 51‘7\ 1 9 ~ __J /
°’ / °§' \ / a i
.0... .3. l / .055-.. ‘\ \
E 1E 3 3 5‘; “é 2‘: E / E. E .__- \
. - in, .m ,_ .. .. . .-
‘;é, é“: l); a?
-1.5 7/ / i i .1 dark 2., \, l
.2:- .' ! g 53 § .2 '39 \ i A
E I L_ > N9 /’ ,/ , /
"'53/ i i i i l l 45;) / ”/fl is ‘3'
1 11 111.1ii 14.11.... 1. 1 11"..A/1f:;+/’1A«
.2 -1 o 1 2 1 o 1 2
x(cm) x(cm)
055- L 41 i is (c)
of- i
Ans?- 42 o -
IS. "i‘ j -7
>. E
4.5,» -
.25. / l /
E l
.. - 42
”:2: 1° 9 i‘.
1 2

X (cm)/

Figure 2-12: Representative derivatives of mapping function, F, for the right camera,
x-direction. (a) F 11,1, (b) F'm, (c) F113. Distances in real units (cm, mm, inches etc.).
Derivative units are (pixels) / (real distance unit).

As discussed, this method automatically accounts for non-uniform magnification
in the fit of the calibration data. The drawback of this method, compared to the geometric
undistortion method, is a more complicated calibration procedure. In this study the non-
uniformity in the magnification due to multiple index of refraction interfaces (see 2.3.4)

makes the use of the generalized undistortion method more desirable.

2.3.3 sMT V Acquisition / Processing

The acquisition / processing of sMTV data was as follows: The laser sheet and

grid intersections were created in the location to be studied in the same manner as the

31

planar MTV in section 2.2.2. Camera timing and use was identical to that describe in
section 2.2.3, except that two cameras were used and aligned for angular measurements as
shown in Figure 2-10. The cameras were aligned so that the middle of the measurement
domain occurred at the same location (i.e. the center of the image) on both image planes.
A computer generated target with known grid spacing was placed into the test section at
the same location as the laser grid (Figure 2-11). The was accomplished by moving the
calibration grid in the z-direction until the center intersection was in the same pixel
location for both cameras. This insured that the calibration grid for z=0, and the laser light
sheet, occurred at the same z-plane. The grid was displaced 0i0.15 cm in steps of 0.05 cm.
Maximum z-displacements were less than 0.10 cm during the delay times used. The target
grid had a 0.2 cm Ax and Ay, which corresponded to the nominal laser grid spacing.
Images were recorded by both cameras at each 2 location. The calibration grid
intersections are correlated to provide the mapping of image to real coordinates. The
calibration data are fit via least squares to determine the functions, F, from equation (1).
The calibration grid was then removed from the test area and data were acquired
for each camera simultaneously. The images from each camera were processed separately
using the standard planar MTV processing routine to provide the apparent velocity
components for that perspective. Figure 2-13a,b shows an instantaneous velocity field for
the left and right cameras in the technique validation flow field as discussed in section
2.3.4. It is instructive to note that the estimation of the out of plane velocity is primarily a
function of the x-displacement. Comparison of Figure 2-13 a and b shows that the v-
component was similar between the two cameras, while the u-component showed

differences, as expected. Since each grid intersection was unique, the real-space starting

32

(a) (b)

 

 

 

 

 

 

 

 

 

    

 

 

 

_150:_.-”l”’_...--_._..o-"-‘\\ _1503‘M‘~\\ . .--.._.-o-.----'
3'~.-.’-'l’/’—>-.-o-o-o-,,..--\ w\\\ .--o—o—.-...-..--
.p...o..l///av.....oa ..... - .-- W \\\ - . a---“.......o
200’»..,.oll//’_.g” a -\ 200M“\\\\\1_11.99--l-l
1"”1;/ .. ,4-.. .\ _- W§§§\..at ..... ’4'-
’v’ l----.‘ ».,. wk \\ . v .....
‘ZSOJ’N:I/;//:”pll| ..‘._’ 2503*‘a—\\\\\\- Ill ,,,,,
e l”*l;f;/;Ia--.vs ,-- I i t\%\\§{\:\x .to . .‘
’2' II- ,a\.\‘ ~\‘~v~\ \ .l. .
fi-aoo4I/I,,/H/n,. H..-” iaookx an‘nu'l. '
; ///,-,////r;,. ..--“" ; .xx .3. \\\\‘,\\- ‘ '1
I’l’lll,f’f"lu\.\w l\\\“\\ \\\\‘\\\\;“”
3507,,,:,,//” to t 1‘ ,--.."‘1’ 35°5\\\\x\‘\\::\\\\“\ -.‘ "‘
:lell ll\\\‘- f""'"““‘ ‘\ \\ ‘ ""
m __...,__.-.., I f t o \ \ \ H, "w »-... \\\\\::._“ ‘w-w
-—»........-... I I f \ xx ' 'w 400;- ,-..._\\\\\\\\V‘_::‘~o—«.—,
-_, ‘-r f H\\\\‘._ L , ,,,._\\ “\MN—««l
W ._- ”‘4
450-”:me x’i’.'.‘.‘l W 450~ / K K. Ah T”
100 200 400 400
x (pixel) x (pixel)
axlsof (c)
rotation V8(cmls)
0.5 " 1 ' "
0.
-0.5
A
E
3 1
> .

.1
x (cm)

Figure 2-13: Instantaneous (a) left camera, (b) right camera, and (c) combined
velocity field for flow under the rotating disk.

location for each intersection was determined and it’s match found for the other camera.
The data for the same intersection were combined as described in section 2.3.2 to provide
the three component velocity measurement in the plane of the laser sheet, see Figure 2-
130. Data post processing (i.e. phase averaging, mapping to a regular grid, determination

of vorticity, etc.) proceeded as with the planar technique.

2. 3.4 sMT V Experimental Validation

Experimental validation of the technique was made in the flow under a rotating

disk as shown in Figure 2-14. A detailed description of the sMTV validation can be

250

L6 RPM
0

 

 

 

 

 

 

 

 

 

 

 

 

 

Top View
/ DC MOtOT
Disk Y
/
r
. . Measurement
Side View Region

Figure 2-14: Schematic of the rotating disk experiment.

found in Bohl et al. (2001). The setup shown in Figure 2-14 illustrates a rotating flat disk
(radius R = 6.35 cm) inside a container of square cross section. A DC motor was used to
rotate the disk at 250 RPM (i1 RPM) for all experiments. The rotating disk was located
3.2 cm (0.5R) above the bottom fixed wall. The region of the flow imaged here was
characterized by primarily an azimuthal flow (maximum V9 about 16 cm/s) and a weaker
secondary flow (V x and Vy typically 5 times smaller). The imaged region for 3-component
measurements was in the (x-y) plane and was approximately 3.5 cm (in x) x 2.5 cm (in y),
starting about 0.15 cm below the disk (outside of the Ekman boundary layer). The baseline

reference data for this flow were obtained from the measurement of the (V9,Vx) velocity

34

components over the (x-O) plane at a given y distance away from the disk using standard 2-
component MTV with viewing through the bottom of the test section. Because the out-of-
plane velocity component (Vy in this case) was very small, accurate measurements of the
in-plane velocity components (V9,Vx) could be made and used for comparison.

The approach just described allowed the comparison of velocity data (V9,Vx)
between the stereo method and the measured reference data along the radial line defining
the intersection of the two measurement planes. Since the two measurements were not
carried out simultaneously, comparisons cannot be provided on the basis of the
instantaneous velocity data. Instead, the mean and rms fluctuation of the velocity data
computed from 7200 realizations were compared. The comparison of the mean and rms
profiles of the primary velocity component V9 is illustrated in Figure 2-15 with excellent

agreement between the profiles measured by the stereo method and the reference data.

 

 

 

 

 

 

” mmm D reference (mean)
15 "' 43 0 reference (mm)
L AB A stereo (mean)
F AD 0 stereo (me)
A
10 - Ch
2': ca
§ DA
5 -
5, : ‘55
>° Woobgoo e g g
o _
~ [21
C a
r 9
-5 _ é
'— r l 1 1 i i l g l I L l r a 1
-2 -1 0
x (cm)

Figure 2-15: Comparison of V9 from stereoscopic MTV with planar reference MTV
data. Data at y = -0.43 cm

35

2. 3.5 sMT V Optical Issues

Implementation of the angular stereoscopic method in the current work was
complicated by several index of refraction surfaces (i.e. air / acrylic and water / acrylic) .
The most significant refraction occurred for the air/acrylic interface on the outside of the
water tunnel because the measurement plane was viewed at an angle with respect to the
tunnel sidewalls. All other interfaces were parallel refractions which had much less
significance to the imaging characteristics of the system. A liquid prism was employed to
compensate for the refraction due to the air / acrylic interface (Figure 2-16). The walls of
the liquid prism were made of acrylic and the working fluid was water. The liquid prism
allows light rays to make the air-acrylic transition at nearly 90° which eliminated the

refraction at interface and increased the image quality.

2. 3. 6 SM T V Error Estimation

The error level in a stereoscopic measurement is dependant upon the error in the
base planar technique and the set-up of stereo imaging system. The error in the angular
stereoscopic method is dependent specifically upon the angle between the cameras.
Analysis of the error as a function of the camera angle, (p, (as defined in Figure 2-10)
using the geometric reconstruction equations, is shown in Figure 2-17. The ratios of the
in-plane component and out-of-plane component errors to the measurement correlation
error (i.e. the planar technique error) are plotted as a function of the camera angle, (p. The
camera angle used in this work was nominally 17° due to optical limitations. The error
ratio of out-of-plane to correlation error for this camera angle was approximately 3:1. The
corresponding error ratio for the in-plane component to the correlation error is 1:1. Given

the correlation error estimate discussed in section 2.2.2 of 0.17-0.24 pixels (0.4 - 0.6 cm/

36

 

    

a... iv . .«t-

Figure 2-16: Stereoscopic camera set-up with liquid prism.

37

 

.. _ — - in-plane component
—— out-of-plane component

 

 

uncertainty ratio
A

 

(a)
Il'jITrTIT—I—IITIITIIllll’rTll'lI

 

 

o l 1 1 J l l l J l l l 1 1 L #1 l_l l l

10 20 30 40
camera viewing angle, q>, (deg)

 

Figure 2-17: Ratio of the out-of-plane and in-plane error to the correlation technique
error level as a function of the camera angle, (p.

sec), the corresponding instantaneous out-of-plane error was 0.5 - 0.7 pixels (1.2 - 1.8 cm/
sec), respectively. For reference, out-of-plane velocities of up to 15 cm/sec were

measured in this study.

2.4 Line Tagging MTV
2.4.1 Technique Description

A single tagged line written into the flow field provides a measurement of the
velocity component perpendicular to the line. Initial measurements of the axial flow speed
in this study were made using a single tagged line viewed from the downstream end of the
test section. Lines were written into the flow by using a single laser sheet (rather than
splitting the sheet with a 50/50 beam splitter as shown in Figure 2-5) and beam blockers.

The lines were passed vertically into test section (i.e. normal to the bottom surface of the

38

tunnel). The tagged lines were imaged from the downstream end of the tunnel, camera c in
Figure2-2. Undelayed and delayed images of the single tagged line were acquired with a
single intensified camera using the timing described in Figure 2-6 with delay times of 16
ms and gate periods of 0. l4 and 2.3 ms. Displacement normal to the tagged line indicates

spanwise, z—component, flow in this orientation, see Figure 2—18.

Undelayed Delayed

 

Figure 2-18: Example of undelayed and delayed lines for a single tagged line. Delay
time: 16 msec; A=2°, k= 11.5; x=0.85c, z=0.85c.

The data were processed to provide the spanwise velocity component (w) via the
following method: The location of highest intensity was found to the nearest pixel for
each row in an image. The intensity profile was fit using a polynomial, typically 2nd order,
using the intensity from the 5 pixels on each side of the maximum intensity location
(Figure 2-19). The center of the lines in the undelayed and delayed images were
determined by finding the maximum of the fit polynomials. The displacement was taken
to be the difference between the peak intensity locations in the delayed and undelayed
images. The resulting displacement was scaled, using the magnification ratio and delay
time, to provide the dimensional velocity. The procedure was carried out for each row in

the image.

 

260 —E—— undelayedline
240 —e— delayed line

 

    

O I l I I l l l l I I l l l l l l l l l l 1

1o 20 3o 40 50
location (pixel)
Figure 2-19: Intensity profiles taken from Figure 2-11, in the region spanwise
displacement . Curves shown are best fit 2“d
about the peak intensity location.

order polynomial to :h 5 data points

2.4.2 Error Analysis

There were two sources of error in the line tagging measurements. They were
measurement error, and error due to flow velocities in the direction parallel to the beam.
The measurement error level could be characterized by investigating regions in the flow
where there was no spanwise displacement measured and determining the noise in the
measurements in those locations. The phases that corresponded to the two vortices
passing through the measurement location, and the resulting spanwise axial core flow, are
shown in Figure 2-20. The two vortices are differentiated by the line colors. Figure 2-20
highlights the localized axial core flow and the near zero spanwise flow outside of the
vortex cores. Estimates for the measurement error taken from the near zero spanwise flow

region were 0.35 cm/s for the phase averaged data.

.3
N

 

_h —A
CA
I'YWTTTTTTI

y(cm)
mwhmmwooco

YII'TITFIITTT'YIUI'IIll'lllITTTFT‘ITT'IIIII'III

—A

 

 

 

0 ” 1 m1 1 I 1 1 1 1 l 1 1 1 1J 1 1 1 1
-20 -15 -10 -5 0
W (cm/s)

Figure 2-20: Phase averaged spanwise flow profiles for vortex / side wall interaction.
A=2°, k=2.84; x=l.27c, z=0.85c.

The second source of error in the line tagging measurements was due to flow
velocities in the direction of the tagged line. Molecular tagging techniques rely upon a
difference in contrast to provide the spatial location of the tagged region. A single tagged
line provides contrast only in the direction perpendicular to the line. Along the direction
parallel to the line there will be no change in contrast, neglecting beam attenuation issues,
and therefore no spatial location information can be determined in this direction. The
intersecting lines used in the planar and stereoscopic techniques provide a 2-dimensional
contrast field in a region where two lines intersect, thus limiting the possible measurement
locations to those regions. Hill and Klewicki (1994) discusses this issue and provide an
estimate for this error. An equation for the error estimate was adapted from Hill and

Klewicki (1994) and is given by the equation:

41

Aw _ aw)
? — tan0($/At (8)

In (8) w is the estimated velocity component perpendicular to the tagged line, Aw is the
error in the estimated velocity, 0 is the local flow angle (tan0 is therefore v/w if v is the
velocity component parallel to the tagged line) and At is the delay time for the
measurement. The error is clearly a function of the flow investigated, and knowledge of
the flow field is required to determine the error level.

Figure 2-23 shows a simulation of the effect of this error on a vortical flow field
similar to those measured in this work for illustration. In F igure2-23(a) the real flow field
of an Oseen vortex with a Gaussian distribution of axial flow is shown. Figure 2-23(b)
shows the flow field that would be measured given equation (8). In this work, the phase

averaged peak spanwise velocity, <w>pcak, will primarily be reported for the line tagging

data. The spatial location of this velocity should be relatively coincident to the center of
the vortex and should, therefore, have the least error due to the vertical velocity. Lateral
velocities at other phases (i.e. locations different to the vortex center) will, however, be

affected.

2.5 Data Acquisition

Each MTV data run was composed of 1000 measurements (i.e image pairs) at a
data rate of 30Hz. The SCO cameras used in this study provided an image that was
640x480 pixels in size. Typical FOV’s were 4cm x 3cm in extent. Between 350
(stereoscopic) and 500 (2-D planar) grid intersections were placed into this spatial region.
Multiple FOV’s were taken and later combined in phase space to increase the data

spacing, as recommended in Cohn and Koochesfahani (2001), to limit the systematic error

42

 

bution of axial flow

istri

ian d

th a Gauss

f an Oseen vortex w

ion 0

lat

S
(a) and the expected measured axial velocity distribution consider

(b).

Figure 2-21

(8)

ion

mg equat

43

in the calculation of the vorticity. Different FOV’s were combined and the data were re-
mapped into regularly spaced data points. Other flow properties, e.g. vorticity and strain,

were calculated at this point.

2.6 Phase Averaging

While MTV is a whole field measurement technique, it was necessary at times to
investigate multiple fields of view, either to increase the spatial extent of the data or to
increase the spatial resolution. One method that allows multiple data sets to be combined
into a single data set is phase ordering/averaging the data sets. In this study the data were
sufficiently repeatable from cycle to cycle that measurements from different airfoil
oscillation cycles could be combined relative to the reference motion of the airfoil, or the
airfoil phase, (1) and discussed as representative of the instantaneous flow field. Phase
averaging the data has several desirable effects on the data processing for this problem. By
phase averaging the temporal resolution of the data could be increased. The data were
ordered and divided into 64 phase bins. With a nominally 30 Hz data rate, the number of
points per cycle ranged from 42 (f=0.7l Hz) to 8 (f=3.55 Hz) for the frequency range
investigated. None of the frequencies investigated was an integer multiple of the camera
framing rate so the camera frames would move relative to the airfoil motion populating
the entire cycle. The cycle was broken down into finer phase (temporal) spacing than
allowed by the camera framing rate.

Phase averaging allows the individual measurement error to be reduced by
averaging the measurements within the phase bin to determine the phase averaged value
for that portion of the phase. There were typically 16 realizations per phase bin which

reduced the measurement error level by a factor of 4. Phase averaged noise levels were

44

therefore be 0.1-0.15 cm/sec for the planar measurements as well as the in-plane portion
of the stereoscopic measurements while the corresponding out-of-plane measurement
error was 0.3-0.45 crn/sec.

The airfoil trajectory shown earlier in Figure 2-3 is the phase ordered airfoil AOA
for the time series data and indicates the repeatability of the airfoil motion. The phase, 4),
was normalized to run from 0 to 1 rather than 0 to 21:. Figure 2-22 shows the phase
ordered velocity for the u and v components at an arbitrary spatial location for the A = 2°,
k = 11.0 case from a planar MTV measurement. The width of the velocity band and the bin
RMS levels (i.e. the RMS of the points that were contained in a bin l/64th of a cycle wide
at a given spatial location) were nominally 1.5 cm/s and 0.4 cm/s respectively. These data

showed both the cycle to cycle variation, the effect of averaging over a finite portion ( 1/

64th) of the phase, as well as experimental noise. The phase, ¢=0, will be defined as an
airfoil angle of attack (AOA) of zero degrees with the AOA decreasing (i.e. airfoil
pitching down) for this work.

Figure 2-23 shows typical measurement FOV boundaries from the individual
experiments. The use of phase averaging allowed the creation of a spatially large, phase
resolved (i.e. pseudo instantaneous) data field over the entire spatial region shown in
Figure 2-23. The increased spatial data extent allows the development of the flow field to

be examined continuously in space.

2.7 Calculation of Statistical Quantities

In this work the mean and RMS levels for the oscillating cases were calculated
from the phase averaged data. Specifically, the data were re-ordered with the phase

information then phase averaged as described above. The large FOV data sets were

45

20

 

o u (cm/s)
v (cm/s)

 

A
OI
nflrjrfi I7

 

.5
O

 

velocity (cm/s)
0|
""'I""lfil‘

 

 

o
-5
.101111I111111111111LL111I111111111I111
o 0.25 0.5 0.75 1 1.25 1.5 1.75
phase,¢

Figure 2-22: Phase ordered u and v velocity components for the 2 degree, k=1] case
at x=6 cm downstream of the trailing edge. Note 2 cycles are shown.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

45' | J
-. .
..
’E‘ 2 E‘- -. -.....- 1' """""""""" ‘ 1--.... ~
>. ~ -I. ____________
-2 E’
-4H1 1 1 1 1— 1 1 . 1 1 1 1 1
10 20
x(cm)

Figure 2-23: Sample map of experiment fields of view combined to create large data
sets in phase space.

46

created by combining the smaller F OV phase averaged data sets and the mean and RMS
level were computed by averaging the data in phase space. These statistical quantities
were therefore different from the long time average and RMS, however, because the
phases were nearly equally populated the differences were negligible. The statistical
quantities for the static airfoil case was computed using the complete time series. The
large FOV data for the static case was created by combining the small FOV statistical

data.

47

CHAPTER 3
CENTERSPAN PLANAR MEASUREMENTS, NO CUTTING WALLS

Standard 2-D planar MTV measurements have been made at the center span of the
tunnel over a distance of -1 cm < x < 13 cm (-0.08c < x < 1.08c) downstream of the
trailing edge of the airfoil without the cutting walls present, see Figure 1-4 with the free

stream speed, U oc= 10.5 cm/sec. Data from multiple FOV’s were combined to generate a

data set over the range -4 cm < y < 4 cm (-0.33c < y < 0.33c) for -1 cm < x < 13 cm (-0.lc
< x < 1.080) with two exceptions: first, for the range 10 cm < x < 13 cm, additional data
were taken to increase the lateral range of y to 21:8 cm ($0.67c). This allowed free stream
information to be gathered. Second, near the trailing edge of the airfoil, portions of the
laser lines were blocked by the airfoil, and therefore data were not available in that region.
The areas where data were not available are marked by dashed lines in the figures of this
chapter.

The data in this chapter are subdivided into two sections, static and oscillating
airfoil cases. The static airfoil data were acquired at A=0°, the mean airfoil angle of attack
for the oscillating cases. These data provide contrast to the oscillating cases, emphasizing
the dynamic nature of the flow field. The mean flow fields of the oscillating cases are first
discussed followed by the “instantaneous” phase averaged flow fields. Three reduced
frequencies, k=5.2, 5.7, and 11.5 are considered in this section of the work. The k=5.2 and
11.5 cases are the cases that will be investigated in detail with the cutting walls. The k =
5.7 case is included because the vortex array is aligned vertically giving a nominal zero
mean thrust. An additional case, k = 4.1, is included in the discussion of the phase
averaged flow field because the formation process was significantly different at this low

reduced frequency. The spatial evolution of two reduced frequencies, k = 5.2 and 11.5 for

48

A=2°, will be discussed in detail. Vortex array properties (peak vorticity, circulation, core
radius, convection speed, etc.) will be discussed for the vortex arrays at x = 6cm

downstream, where the cutting walls will be placed, for all cases.

3.1 Static Airfoil Characteristics

Figure 3-1 shows the mean u, v and (02 fields, along the centerspan of the facility,
in the wake of an airfoil held fixed at an AOA=O°. The vorticity shown in Figure 3-1 was
calculated from the mean velocity field. The mean streamwise velocity was slightly

greater than the freestream speed (5 z 1.04U ,0) above the airfoil and outside of the airfoil

boundary layer. The average streamwise velocity then relaxed to the freestream level for x
> 0.50. This was expected given the decreased tunnel area caused by the airfoil section and
the wake deficit. The velocity in the wake directly behind the trailing edge of the airfoil ( x

< 0.1c) was nearly zero (5 == -0.04U.,,). The velocity deficit was clearly indicated by the

vector field, Figure 3-2(a). The weak reverse flow in the wake was observed, when the
vectors were scaled to a uniform length regardless of their magnitude. That is a
recirculation region was observed, Figure 3-2(b). Note that the data in Figure 3-2 are the
irregularly spaced data before they were re-mapped onto a regular spaced field. These data
are used in the vicinity of the airfoil surface to minimize re-mapping artifacts due to lack
of resolution in the airfoil boundary layer. Figure 3-3 is a flow visualization image taken
using the MTV chemical triplex. The recirculation region was indicated in this image by
the relatively bright region below and in the wake of the airfoil. The separation point was
marked by the boundary of the luminescent fluid/dark fluid at the airfoil surface. In this
region it was believed that the concentration of free bromonaphthalene was increased, due

to absorbtion and release from the airfoil, allowing it to be visualized. The transverse, a7,

49

 
 
  

 

static

 

 

 

 

 

0.5 '
x/c

Figure 3-1: Mean u, v and col fields for the stationary airfoil at AOA=O°, tunnel
midspan.

 
 

50

 

 

 

     
  

 

 

-0114 * fr * 1*i"*'<**rr+re"1r'
' -0.05 0 0.05 0.1
x/c
0.04 (b)
0.02 _, a “a
2 o a/ r / a/
> ,I / x / / ,/ \
/' \ \ “a —.—-> ->—> —>—>
'002 a»... ->—"—.>—. 4" —>—> ..—>—-> —->—>
.004 *7 TT TT T" 77 r? T” ** 1*?
-o 0 0 0.05 01
x/c

Figure 3—2: Mean velocity vectors in the near wake of the static airfoil. (a) vector
length proportional to velocity, (b) vector length uniform.

.—/‘

 

Figure 3-3: Flow visualization of recirculation bubble in the near wake of a
stationary NACA—0012 airfoil at AOA=O°.

velocity showed the flow being directed first towards the centerline of the shear layer
downstream of the airfoil trailing edge then becoming predominantly streamwise, V 2 0,
for x > c (Figure 3-1). Near the airfoil surface the recirculation region was shown by the 3
directed away from the surface of the airfoil (Figures 3-1 and 3-2).

The mean vorticity, 60.2, was initially high in the boundary layer, and reducing in

magnitude with increasing downstream distance (Figure 3-1). Note that the velocity at the
airfoil surface was not used as a boundary condition in the velocity re-mapping (from
irregularly spaced to regularly spaced data), therefore the actual vorticity in the boundary
layer near the airfoil surface was underestimated.

At x z 0.60 the structure of the G and 002 fields changed with the wake showing a

decreasing width from x = 0 to x = 0.60. The minimum wake width was found at x z 0.60.
The width of the wake region then increased for x > 0.60 with a marked decrease in the
sharpness of the velocity deficit profile. The 62 field also showed this change in the
structure, i.e., a lateral spreading and a decrease in magnitude. Profiles of the same data
are shown in Figure 3-4 for x = 0.50 and 1.00 to clarify the current discussion by
illustrating the average vorticity decrease, and wake width increase.

The RMS fields are shown in Figure 3-5 for the same measurement area. The
wake directly downstream of the airfoil trailing edge showed low RMS levels ( z 0.05Um).
Peak RMS of nominally uRMS = 0.16U,° and VRMS = 0.18Uw occurred at x = 0.60
downstream. The RMS streamwise, uRMS, velocity showed two local regions of high
fluctuations symmetrically placed about the y = 0 line. A single RMS peak was shown for
vRMS. The downstream locations of the RMS peaks correlated with the change in the

mean field structure noted above.

52

 

 

 

 

"0 15 10 5 _'C6IU_ 5 10 15 20
oe‘_ ........ ,....,........., .
: —a— _ :
05:- +VIU_ ’
: —e'*‘w¢’U ':
0.4:- _3
0.3:- _1
0.2:- :_
2 : 00‘2:
>1 . 3. .
0.17 0.1;
0:- oi
-0.1:- -o.15-
-0.2[- .02.:—
t ':
p
-0.3 “ .r11111111 .11..
.| -0.o.' 1.5

 

 

 

 

 

-0.5

 

anneal”?

fi/u_, V/u_
Figure 3-4: Profile of E, P and (T), for the stationary airfoil at x=c/2 (a) and x=c (b),
midspan. Every 3"d point shown.

 

-0.3

Figure 3-5: RMS u and v fields for the stationary airfoil at AOA=O°, tunnel midspan.

53

3.2 Oscillating Airfoil
3.2.1 Time Averaged Data

Mean streamwise velocity, 5, fields for the A = 2°, k = 5.2, 5.7 and 11.5 the
oscillating cases are shown in Figure 3-6. The airfoil shown in the following figures is
shown at an AOA of 0°, which for the oscillating airfoil represents the mean location of
the airfoil. The ii field for the A = 2°, k = 5.2 case showed a velocity deficit downstream

of the airfoil with a nominally constant (in x) minimum mean velocity, 6 z 0.85U o, , in the

wake for x > 0.40 and extending past the measurement range (Figure 3-7). The y-profiles
revealed an overshoot in the streamwise velocity above and below the wake before the
velocity relaxed to the freestream speed (Figure 3-8). The mean u field for the A = 2°, k =
5.7 case showed a small velocity deficit for 0.20 < x < 0.60 (Figure 3—7). For x > 0.50 the
velocity profile in the y-direction was nearly uniform (Figure 3—8). A large velocity excess
was observed in the wake for the A = 2°, k = 11.5 case in contrast to the velocity deficit
observed for k=5.2. The switch from a mean streamwise velocity deficit to a mean
velocity excess is an indicator of a switch from an airfoil with mean drag to one with mean
thrust (Koochesfahani, 1989). The k=5.7 case, which showed a nominally uniform mean
streamwise velocity profile, was indicative of a net zero drag case, if only mean
streamwise velocity was considered. A complete discussion of the mean drag
characteristics of the measured airfoil is found in Section 3.2.2.

The qualitative aspects of the mean streamwise velocity, 5, field can be described
by the orientation of the vortices with respect to each other. The following discussion is
not unique. Similar discussions can be found, for example, in Lighthill (1975) and

Koochesfahani (1989). Consider a vortex and the two closest vortices of the opposite sign.

54

am,-
1.10
0.99
0.88
0.77
0.66
0.55
- 0.44
0.33
0.22
0.11
0.00

 

 

1.07
1.03
0.99
._; 0.95
0.91
0.87
0.83
0.79
0.75

1 .07
1 .03
0.99
0.95
0.91
0.87
0.83
0.79
0.75

 

 

2.10
1.89
1.68
1.47
, 1.26
1.05
0.84
0.63
0.42
0.21
0.00

     

0.5
x/c

Figure 3-6: Mean of all phases streamwise velocity, 3, field for the A = 2°, k = 0, 5.2,
5.7 and 11.5. Note scale change for k=11.5

55

 

 

 

 

- —a— 2°,k=5.2
1.4 —1e1— 2°,k=5.7
s - ——9— 2°, 11:11.5

 

 

 

 

 

 

 

 

 

 

 

D
.312
1 _
0.8
0.6
l I l J L l I I l l I
0 0.5 1
xlc

Figure 3-7: Mean streamwise velocity, E, for A=2° k=5.2 5.7, 11.5 at y=0.

In Figure 3-9 the negative vortices are above the positive vortex. Between the vortices
the induced streamwise velocity from both vortices are directed in the direction opposite
of the direction of convection resulting in a velocity less than the freestream velocity (i.e.,
a velocity deficit) between the vortices when viewed from a stationary coordinate system.
The overshoot in the velocity with respect to the freestream velocity is also predicted by
this geometric argument. Specifically, above and below the vortex array the induced
velocity from the vortices will sum with the freestream velocity. The induced velocity
from the vortices follow a l/r reduction away from the vortex and so the flow will return
to the freestream level far away from the vortices.

When the vortices are exactly aligned, as shown in Figure 3-10, the induced
velocities due to each vortex cancel, and the mean streamwise velocity profile is uniform

across the vortex array. Figure 3-11 shows the case where the vortex orientation has

56

0.4

x=0.50

O

20 -10 0 "

10 20 30

 

-30
0.3
0.2
0.1

‘D

30

V'TrTTYV'V'YjTT—TYYT'fi

-0.1

1'1

-0.2

-0.3

I‘VIIIV‘rrTTYY

 

c
'YY'YIIYIYfT‘V'Y‘YrT

k=5.2

  

—-8— uIU_
-9— WW.

 

 

 

 

-30

-20

1.5

 

0.4 .
0.3
0.2
0.1

g. 0

-0.1
-0.2

-0.3

IVY'I'TIY'IITYTYYTY'IIT‘TFVYI'IYYI'IIVY

 

VVY'II

k=5.7

\.

 

 

 

 

-0.4

1.5

 

V'Y'VY'IIIY'TUVI'II'Y'IYVI'YYIIIYIIIIFU

 

 

 

 

 

 

 

-30

0.4

0.3

0.2

-30

0.4

0.3

ylc

x=c

-20 -10 0 "

10 20 30

 

k=5.2

VVIIIfi

1"

YrYYlYIIlrfi'TYYI'IV‘lY7T‘IYYT

 

Fjfi—TVIYIVYIYYY‘IT'TI

—a— W.
—6— WW.

 

 

 

 

-20

1.5

10 20 30

 

YYTYIVTY

k=5.7

V'IIV'I'IrT1YYlI'YYYYIVY'Y

 

YYTY'I'YYIT1Y

—B— W,
---9-- WW.

 

 

 

 

1.5

 

  

IIIIVT'Y

I I

I'I

r

it

\4

 

l

L I I —B— W,
E 11:11.5 -—e—-— cow-
E

C

E

  
 

1“

 

 

.l l I I

 

WU

1.5 1!

 

Figure 3-8: Mean streamwise velocity, 5, and vorticity, 52, profiles for A=2° k=5.2
5.7, 11.5 at x=0.5c, c.

u<U°° +

u>U°° >

 

u=Uoo fi

Figure 3-9: Schematic of the vortex orientation for the velocity deficit cases.

®®®

u=U°° q

Figure 3-10: Schematic of the vortex orientation for uniform mean streamwise flow
case.

 

U>U°° +
u<U°° —>

u:U°° q

Figure 3-11: Schematic of the vortex orientation for the velocity excess cases.

58

1'0
1’01
1‘01
8110
1111::
10111

0175131

3110“,]
“115 a

trailing

switched. In this orientation the induced streamwise velocity between the vortices adds to
the freestream speed, and a velocity excess, or jet-like profile is observed. An undershoot
above and below the vortex array is also predicted, in contrast to the velocity deficit case.
The mean lateral velocity, V, is shown in 3-11.These data also showed that the mean
velocity was directed towards the y=0 plane as was also observed for the static airfoil. The
maximum 3 levels increased with increasing k. The downstream distance over which non—
zero ; was found decreased with increasing k. The lack of mean V downstream of the
airfoil was expected for this flow field regardless of the orientation of the vortices with
respect to each other. Specifically, the induced v velocity fluctuations from each vortex
will sum to zero, causing a net zero 3 if the orientation of the array pattern is fixed.

The mean spanwise vorticity, 052, field for the A = 2°, k = 5.2, 57 and 11.5 the

oscillating cases are shown in Figure 3-13. The k = 5.2 case showed the mean spanwise

vorticity, 00-2, in the wake to have the same orientation as 52 in the boundary layer, i.e. the

vorticity from the top (y positive) boundary layer of the airfoil remained above the
vorticity from the bottom boundary layer of the airfoil in the wake. The k = 5.7 case
showed very little 52 for x > c/2. This result indicated that the vortices were aligned as
inferred from the mean streamwise velocity profile. The k = 11.5 case shows the expected
vorticity orientation change which was also consistent with the jet like profile in 6
observed.

Fluctuating u and v velocity and spanwise vorticity fields for the above cases are
shown in Figures 3-14, 3-15, and 3-16, respectively. The RMS streamwise velocity, “RMS,
was a maximum upstream of the trailing edge of the airfoil. Downstream of the airfoil

trailing edge the uRMS levels in the vortex sheet ranged from 0.2Um to 0.6Uoo and

59

\7/ U“
11— 0.07
' 0.06

0.04
0.03

1 .. 0.01
0.00
-0.01

*= -0.03
-0.04
-0.06
-0.07

static

 

 

0.20
0.1 6
0.1 2
0.08
0.04
0.00
-0.04
"-5 -0.08
-0.1 2
-0.1 6
-0.20

0.20
0.16
0.12
0.08
0.04
0.00
-0.04
-0.08
-0.12
-0.1 6
-0.20

0.50
0.40
0.30
0.20
0.10
0.00

-0.1 0

' . -0.20

-0.30

-0.40

0 0.5 1 -0.50

xlc

Figure 3-12: Time averaged mean v velocity field for the A = 2°, k = 0, 5.2, 5.7 and
11.5.

 

 

-0.2
-0.3

 

60

 

xlc

F lgure 3-13: Mean spanwise vorticity, 5,, field for the A = 2°, k = 0, 5.2, 5.7 and 11 5

61

ll/n

“1msl use
0.40
0.36
0.32
0.28
0.24

; 0.20

3 0.16

0.12

0.08

0.04

0.00

 

1.1m,ls I Um
0.40

0.36
0.32
0.28
0.24
~ 0.20
‘-‘ 0.16
0.12
0.08
0.04
0.00

 

1 uRMS I U”
0.60

0.54
0.48
0.42
0.36
0.30

i 0.24

' 0.18
0.12
0.06
0.00

   

0 0.5 1
xlc

Figure 3-14: RMS u velocity field for the A = 2°, k = 5.2, 5.7 and 11.5 oscillating cases.

62

Vnus I U“
0.50
0.45
0.40
0.35
0.30

; 0.25

1 g. 0.20

0.15

0.10

0.05

0.00

   

 

0.5
xlc

Figure 3-15: RMS v velocity field for the A = 2°, k = 5.2, 5.7 and 11.5 oscillating cases.

63

II/r‘

FiEur
050111;

 

 

"0'3 0 0.5 1 °

xlc

Figure 3-16: RMS spanwise vorticity, (1)2, field for the A = 2°, k = 5.2, 5.7 and 11.5
oscillating cases.

64

\‘C

ch.

1116

ind

Ill-'1':

Figun

increased with increasing k. The RMS lateral velocity, VRMS, levels showed magnitudes
that ranged from 0.3Um to 0.9Uoo and also increased with increasing k. RMS spanwise
vorticity, (02, levels were 2 to 3 times larger than the mean levels in the wake. The RMS

levels indicated a highly fluctuating flow field; However it is instructive to note that this

was a result of the unsteady, not turbulent, nature of the flow field.

3.2.2 Mean F orce Calculation

Force on an immersed object can be calculated using a combination of control
volume and differential analysis on the flow field. Initial analysis of the drag
characteristics on the current data was reported in Khalid (2001). The development of the

mean thrust, CF, equations used in this work can be found in Appendix A and were derived

independently of Khalid (2001). Figure 3-17 shows a schematic of the control volume

3"3

(2):, ._.

 

My), F(y)

 

 

lllll ' "l"

 

 

Force —>

Figure 3-17: Control volume used for determination of mean drag equation

65

Whe
V9111:
app”
U0 3
down;

”0011‘

used. It was recognized that P0 and P00 were non-uniform vertically due to hydrostatic

variations, however this variation was the same for both surfaces and therefore had no
effect on the force calculation. Past researchers (e.g. Anderson et al., 1998;
Koochesfahani, 1989) have used a mean streamwise velocity profile formulation to
calculate the thrust/drag characteristics of this flow field. The thrust from the mean

streamwise velocity, CFM can be computed using:

GIN

3’
0.41 -—>4

00

Equation (9) was used when only a single velocity component could be measured in the
wake. This equation neglects the effects of the pressure variation in y due to the
fluctuations in the flow velocity as the vortices pass the control surface. If the flow field is
2—D and the complete velocity vector is known, the mean force equation can be re-derived

as shown in Appendix A to be:

y
2 ~2 -2
cUoo -V

Where CFM is defined by equation (9). The limits of integration were, in this case, the

vertical measurement extent. The mean streamwise velocity did not return to the upstream

approach velocity level within the vertical measurement extent of this problem, i.e. U0, at
U0 at y = 0.60. For this case, the pressure difference between the upstream and

downstream lateral control surfaces had to be taken into account. Equation (10) was

modified to reflect this as

66

F1

11111

The
11:5
mea

shor

abo

freq

W218

11110

FF+——2— (21de (11)

Where AP was found via the Bernoulli Equation by

_ _ _Q 2_ 2
P0 Poo—AP—2(UDo U0) (12)

The upstream velocity, U”, was determined using the conservation of mass for the A=2°,
k=5.7 case for which the vortices are lined up. This calculation of U0, agreed with the

measured free stream speed for the static airfoil case. Investigation of the velocity profiles
showed that the freestream speed above and below the centerline were slightly different

(~0.5%). The value of U0 used in (12) was found by averaging the freestream speeds

above and below the centerline.
Figure 3-18 shows the mean force coefficient as a function of the reduced
frequency, and oscillation amplitude of the airfoil calculated from the mean profile only,

CFM , and the mean profile plus the RMS and pressure profiles, Cpp. From these data it

was clear that the use of the mean profile only over estimates the thrust, and
underestimates the drag on the airfoil consistent with, Streitlien and Triantafyllou (1998).
When the full force equation was used the zero thrust case for A = 2° was found to be k z

8 rather than, k z 6 which was the case predicted by the mean field.

3.2.3 Trailing Edge Phase Averaged Data

Figure 3-19 shows flow molecular tagging flow visualization and MTV data with
the airfoil passing through AOA=-2° (minimum AOA) and beginning to pitch down

(trailing edge moving up). The undelayed image indicated the initial location of the tagged

67

 

 

 

 

 

 

 

 

0.45 _
I 0 C ,static
0'40 :- 0 C2, static
2 El ,2°
0.35 E— C.) :21: O
0.30 E- . 02.4»
0.25 E'
u- E
0 0.20 E
0.15 :— 0
0 10 E O . D
. ;- D
0 05 E O. . D
. E O O D I I I
0.00 ’ O—ga-fi—I
E El
_ .-
'0-05 1- , 4 1 fi1 1I I 1 1 1 1 l 1 1 1
0 10

Figure 3-18: Mean force coefficient, CF as a function of airfoil oscillation amplitude
and reduced frequency.

line. The flow visualization data were taken at a reduced frequency of 8.8 (the highest
flow visualization reduced frequency), and are shown in Figure 3-19 for qualitative
comparison to the MTV velocity vector data taken at k = 8.6. The unsteady Kutta
condition requires the flow to leave the airfoil trailing edge tangent to one of the surfaces
(McCrosky, 1982). These data highlight the reverse flow along the surface of the airfoil,
shown by the tagged line displacement compared to the initial line location marked by the
dashed lines. Reverse flow was also indicated by the MTV velocity vector data near the
airfoil surface. The reverse flow image was strikingly similar to the “possible” flow
pattern drawn in Figure 5 of McCrosky (1982) and contradicts the validity of the unsteady

Kutta condition for this high reduced frequency case. The reverse flow indicated that fluid

68

undelayed ¢z0.75

      

—’2U,, <002>*c/U_,

125

M 100
75
, 50
25

-25
.~. -50
-75
-100
-125

   

-0.1 0 0.1
xlc

Figure 3-19: MTV fiow visualization (k = 8.8) and velocity vectors from the planar
MTV data (k = 8.6) near the airfoil trailing edge. Airfoil at AOA=2°, trailing edge
moving up (AOA decreasing). Dashed lines indicate initial line locations.

from the top surface of the airfoil convected to the bottom of the airfoil which showed
that the flow cannot leave tangent to either airfoil surface.

Flow visualization near the trailing edge is shown in Figure 3-20 for the case of
AOA=0° with the airfoil beginning to pitch up (trailing edge moving down). The flow

visualization images are flipped up/down in this figure to allow comparison with the MTV

69

undelayed

     

   

p, . \ ‘
.WWVI‘nyfflw

 

—>2U_, <(oz>*clUm
M 125

 

-0.1 0 0.1
xlc

Figure 3-20: MTV flow visualization (k = 8.8) and velocity vectors from the planar
MTV data (k = 8.6) near the airfoil trailing edge. Airfoil at AOA=O°, trailing edge
moving up (AOA decreasing). Note flow visualization images flipped up/down.

data of fluid from the top surface boundary layer rolling up below the trailing edge of the
airfoil. Recall that data were limited above the airfoil because the beams were blocked.
Results in this region can be determined by assuming vertical symmetry in the flow field
and moving half a cycle (phase). The delayed flow visualization image showed that the
vortex formation occurred very near to, possibly upstream, of the trailing edge. The tagged

line on the upper surface of the airfoil nearest to the trailing edge highlights this

70

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observation (Figure 3-20, “delayed”, third line from the left). This line was displaced
below the bottom surface of the airfoil indicating that fluid from the upper surface of the
airfoil was transported below the bottom surface of the airfoil and formed the vortex about
to be shed. The sharp gradient marking the initial shear layer illustrated by the “kinks” in
the first three tagged lines downstream of the trailing edge. MTV data for a similar phase
is also shown in Figure 3-20. These data highlight the initial vortex formation near the
trailing edge. The sharp gradient present in the delayed images of Figure 3-20 was not
shown by the MTV data due to the data point spacing which was too coarse to capture the
feature. The velocity vectors in Figures 3-19 and 3—20, which show streamwise velocities

ranging from near zero to +2U°° , illustrated the large velocity fluctutations in this flow

field.

3.2.4 Downstream Phase-Averaged Data

Figure 3-21 shows the phase averaged spanwise vorticity, «02>, for four cases, A
= 2°, k = 4.1, 5.2, 5.7 and 11.5 at 0 = 0. The k=4.l case is included in this section since the
flow field characteristics were very different from the higher reduced frequency cases. For
k = 4.1 the boundary layer left the airfoil trailing edge forming a shear layer then rolled—up
downstream. The coherent vortical structures remained connected to the boundary layer
vorticity until approximately 6cm (0.50) downstream. Two distinct regions of the same
sign vorticity were generated for a single cycle. The orientation of the primary regions of
vorticity was suggestive of a thrusting airfoil, however, net drag was calculated based on
the mean flow. This was because of the secondary regions of vorticity which were aligned

in a “drag” orientation, and were strong enough to result in the mean drag.

71

 

F1.
Ca
dil

 

0.5
x/c

Figure 3-21: Phase averaged vorticity, «01>, field for A=2°, k=4.l, 5.2, 5.7 and 11.5
cases. ¢=0. Dashed line indicates edge of data grid. Note: Vorticity contour levels
different for each case as shown.

72

The next highest reduced frequency, k=5.2, no longer showed the multiple
structures that were in evidence for the k=4.l case. A thin “tail” of vorticity was observed
and represented the remnants of the connecting shear layer. For k=5.7 the connecting
vorticity was no longer in evidence by x>0.50 downstream. The vortices were nearly
aligned vertically as expected from the mean data. The k = 11.5 case showed the boundary
layer rolling up into distinct vortices very near to the trailing edge.

The formation of the vortex array was dependant on the reduced frequency.
Specifically, when the reduced frequency was low enough (k = 4.1 case) the boundary
layers left the airfoil and began to form a shear layer. The shear layer then appeared to roll
up into isolated vortices. The higher reduced frequency cases (k > 4.1) showed the
boundary layer vorticity rolling up into insolated vortices near to the trailing edge.

The results of this work are consistent with the observations of Wilder et al. (1996)
for lower reduced frequencies. Wilder et al. (1996) investigated the flow behind a flapping
NACA0012 airfoil and observed that the wake rolls up forming well defined isolated
structures at x z c for reduced frequencies of k = 2.05 and k = 2.8. Wilder et a1. (1996)
commented that this formation occurred well upstream of that observed at lower reduced
frequencies, and credited this with the relatively high reduced frequencies studied. The
current work shows the formation of isolated structures for x < c at higher reduced
frequencies. Wilder et a1. (1996) marked the boundary layer of the airfoil with dye. The
dye was observed to concentrate in the downstream vortical structures. They inferred,
from the lack of dye between the marked vortices, that there was little vorticity between
the isolated vortical structures. The current whole field measurements confirm

quantitatively that inference.

73

€1.11

Sm

Representative vorticity profiles are plotted in Figure3-22 for A = 2°, k = 5.2, 11.5
along with the best fit to a Gaussian vortex profile. The vortex profile showed an excellent
fit to the Gaussian model for Idl < 0.5 cm (nominally the core size). For Idl > 0.5 cm the
vortex deviated from the Gaussian profile with slightly higher measured vorticity levels.
There was near symmetry shown in the vorticity profile for k=1 1.5 while the k=5.2 case
was asymmetric. Specifically, the vertical (y-direction) profile was thinner and showed a

second peak at r z 1.5 caused by the remnants of the connecting vortex sheet.

 

 

 

 

60 ,
k=5.2 D mm” 150 k=11.5 . I: 21:20:11»
0 vertical
— - - — vm

 

 

 

 

00 (1/8)

 

 

 

 

 

1 l J L 1 J

0 1 1
distance, <1, (cm)

 

 

0 l 1 1
distance. d, (cm)

Figure 3-22: Vorticity profiles for 2 degree, k=5.2, 11.5 case at x = 6 cm, 2 = 0 cm.
Curve is best fit to a Gaussian vortex profile. Horizontal profiles show quantity as a
function of x, and vertical profiles a function of y. Equation represents the vorticity
profile for a Gaussian vortex.

Figure 3-23 shows the azimuthal velocity, V9, profile for the same vortices shown

in Figure 3-22. The velocity profiles were also fit for a Gaussian vortex. Agreement was
good for the k = 5.2 case over the entire range, while the k = 11.5 case agreed only over a

small range near the vortex center. This was consistent with the agreement of the vorticity

74

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15 -
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-15 - El k=11.5
_ —— k=11.5, fit
1 1 1 1 1 1 1 1 1 1 l 1 1 1 1
'20 -'1 0 1 2

distance, d, (cm)

Figure 3-23: Azimuthal velocity profile for the 2 degree, k=11.0 case at x=6 cm, z=0
cm. Fits shown for portions of the profile indicated by the line sections. Functionality
indicated by equation near the fit line.

profile fits already discussed. These velocity profiles were similar to those reported in
other works (see e.g., Wilder et al. (1995), Koochesfahani (1992)).

The downstream development of the vortical flow field can be illustrated by
tracking the motion of the vortex and characterizing the peak spanwise vorticity, <m>peak,
circulation, F, and core radius, rc, as a function of downstream distance. The location of a
vortex was defined as the location of the centroid of the vorticity. The centroid was used to
define the vortex location as the vortex becomes non-Gaussian downstream and the
centroid better represents the vortex location (compared to the location of the peak
vorticity) for these distributions. To find the peak vorticity a second order polynomial was

used for the fitting the data and the spatial extent of the fit was limited to 0.5rC away from

75

the nominal peak. This process was repeated for each phase and was carried out until the
vortex left the measurement field.

The circulation of the vortex is determined by

F = 2((Dz)AxAy (13)

Where Ax and Ay were the data grid spacing. When using (13) only the vorticity from the
vortex should be included while noise in the vorticity calculation should not. To achieve

this two threshold levels are used, a vorticity cut-off limit and an included spatial radius.

A cut-off limit of 2 s'1 was placed on the vorticity to reduce the effects of experimental
noise on the circulation calculation. Only the vorticity with in spatial radius of 1 cm from
the peak vorticity location was used. The 1 cm radius was justified by the vorticity profiles
in Figure 3-22 because it was small enough to not include the “tail” of the vortex for the k
= 5.2 case (making a localized measure), but big enough to be within 1% of the infinity

value for the k = 11.5 case. A conservative estimate of the uncertainty level in the

circulation calculation is 2 cmz/s based upon the effects of threshold level and radius of

included data.

The vortex core radius, rc, can be defined by the e'1 point of the vorticity field for a
Gaussian shaped vortex. The vortex initially fit a Gaussian profile as shown in Figure 3-
21, however, the vorticity distribution becomes non-Gaussian downstream. To account for
this, the “radius of gyration” of the vorticity field was used to define the vortex core

radius. The radius of gyration is defined as the square root of the second polar moment of

inertial of the vorticity divided by the circulation, or:

76

dc‘

usi

2
r = M (14)

6 ”(MA

The core radius, rc, defined by equation (14) reduces to the e”I point value for a Gaussian
vorticity profile.

Figure 3-24 shows the development of the vortex properties as a function of the
downstream distance for A = 2°, = 4.1, 5.2 and 11.5. Only the positive sign vortex
properties are shown in Figure 3-24 since vortex property data at x = 6cm (Figure 3-29,
discussion to follow) showed the vortex properties for the positive and negative sign
vortices agree well. Recall that the k = 4.] case had multiple regions of same sign vorticity
for each phase. For that case the vortex with the strongest vorticity was tracked. The
maximum of the vorticity was initially constant for the k = 5.2 and 11.5 cases. This
constant region reached nominally to x=0.4c downstream. After this point there was a
decrease in the local maximum vorticity as a function of x. The k = 4.] case showed a
nominally uniform peak vorticity until x = 0.75c. The expected decrease in vorticity based
on diffusion of vorticity is shown for the k = 11.5 case by a dashed line in Figure 3-24. The
decrease in vorticity was larger than expected if diffusion alone acted on the vortex to
decrease the vorticity magnitude, so a second mechanism must be investigated.

Vortex stretching / re-orientation was also investigated to describe the difference
using arguments similar to those found in Cohn (1999). The vorticity transport equation

can be written as

0(0))

z _ 8w 8w 8w 2
——Dt _ («0387 + ((095; +(mz)5-z-)+1)V mz (15)

If the convection speed is constant, and (ox, (oy are small then equation (15) simplifies to:

77

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A=2°, k=5.2
A=2°. k=11.5

 

0.08

0.07

0.06

0.05

0.03

0.02

0.01

Figure 3-24: Vortex array properties as a function of the streamwise distance.

78

2
§;(wz) = -U‘—[<w >3—‘Z"+n[—a—<w > +1.37“. >D (16>

a_w

82.

Using equation (16) with a constant axial flow gradient ( ) ( term I ) and viscosity (term

11) neglected allows qualitative investigation of the effects of stretching on the vortex. The
solid line in Figure 3-24 is the result of applying term I of equation (16) (term 11 is the

viscous diffusion term) with a constant axial flow gradient, found by best matching the

stretching term to the data, of -O.6 s". The existence of this axial flow gradient was
attributed to the development of the axial flow generated by the vortex interaction with the

side walls of the tunnel. This axial flow gradient was below the current measurable axial

flow gradient of 1 s]. From this analysis though, the decrease in peak vorticity appears to
be a function of both viscous diffusion, and vortex stretching.

The circulation as a function of the streamwise distance is also shown in Figure 3-
24. The data in Figure 3-24 show that the circulation was constant, to within the
uncertainty limit, over the spatial extent investigated. The small oscillations observed for
the k = 11.5 case are due to vorticity moving in (as the vortex forms) and out (as the vortex
expands) of the included range.

The vortex core radius, rc, is shown in Figure 3-24. The vortex formation

dynamics were further illustrated by these data. All three cases have nearly the same core

radius, rc=0.43 cm, at x = c downstream. The k = 11.5 case showed a linear increase in rC

from 0.4 cm to 0.43 cm over the extent investigated. The k = 5.2 case showed the same

linear increase in rC for x > 0.5c. For x < 0.5c the core radius decreased from a high level

79

of 0.55 cm near the trailing edge to a minimum of rC = 0.40 cm at x z 0.5rc. The k = 4.1

case showed the same trends as the k = 5.2 case, however the linear growth range began at
x z 0.7c. The decreasing core radius shown initially for the k = 4.1 and 5.2 cases can be
explained by the formation process. The lower reduced frequency cases formed into
isolated structures further down stream as was shown in Figure 3-15. The vorticity was
more distributed, therefore rC was larger, while the isolated structures were forming.

The vortex spacing, and orientation are also important parameters in the

description of the dynamics of this flow field. A schematic of the vortex spacing

nomenclature is shown in Figure 3-25. The sign of b is defined as the lateral distance from

 

 

Z

l
|
|
x I
l
|

 

   

Figure 3-25: Schematic representation of vortex spacing variables. Positive b
indicates positive vorticity is above negative vorticity as shown.

the negative vortex to the positive vortex, where the vortex location was again the location
of the centroid of the vorticity. The vertical was significant in this flow field because it

was known from the mean flow data that the orientation of the vortex array, and

80

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for
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11011-1

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corresponding net force, switched as the reduced frequency increased. The streamwise
spacing, a, is a measure of the vortex sheet wavelength (and frequency).

The vortex spacing as a function of downstream distance is shown in Figures 3-26
and 3-27. Figure 3-26 shows the vertical spacing between the positive and negative
vorticity for k = 5.2 and 11.5 cases. The vertical spacing, b, was calculated by determining
the vertical locations of the centroid of the vorticity at a given downstream location for the
negative and positive sign vortices; therefore, this measure is not an instantaneous
measurement. Both cases showed an initial motion of the vortex away from the y=O line.
After that initial region, the vortices moved back towards the center. The k = 5.2 case
appeared to be approaching a constant vertical spacing distance for x > c. The high
reduced frequency case, k = 11.5, showed that for x > 0.5c the distance between the
vortices remains relatively constant, though they were moving away from the y = 0 line in
the +y-direction.

The vortex streamwise spacing, a, is shown in Figure 3-27. The cases showed an
initial period of increasing a. This was explained by observing that the vortices formed
near to the trailing edge in a nearly fixed location, then convected downstream. The
downstream vortex of the same sign was already in the flow field and therefore convecting
away from the forming vortex causing the initial increase in a. Once the vortex was
formed and started to convect downstream the streamwise spacing remained relatively
constant. The k=5.2 case showed decreasing streamwise distance, 3, indicating the
downstream vortex was slowing slightly. The streamwise distance, a, increased with
downstream distance for the k=1 1.5 case indicating the convection velocity was

increasing. The vortex convection speed, UC, as a function of downstream distance is

81

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..... .'.'o‘
I'l‘i‘l'flt'fi" ‘

 

 

v
""""""
‘ A

 

 

 

 

 

 

6 12
xc (cm)

Figure 3-26: Vertical vortex spacing, b, as a function of downstream distance.

82

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1 1
o . 1

 

 

 

 

xc (cm)
Figure 3-27: Vortex streamwise spacing, a as a function of downstream distance.

shown in Figure 3-28. These data were found by numerically differentiating the vortex
position versus the airfoil phase and multiplying by the frequency. The differentiation (i.e.,
convection speed) was very sensitive to uncertainty in the spatial location, so the spatial
location data were first smoothed using a moving window averaging scheme, then
differentiated. The initial acceleration period of the vortex was clearly observed for the
k=5.2 and 11.5 cases. .The k = 11.5 case showed a trend of nominally linear increase in UC
as a function of increasing downstream distance. There was a brief, UC = constant period
for 0.67c < x < 0.75c, which was followed by a return to the linearly increasing convection
speed. The k = 5.2 case showed a nominally uniform convection speed after the initial
acceleration, then a small decrease in the convection speed for x > 9 cm (0.75c).

The vortex properties as a function of the reduced frequency of the airfoil are

presented in Figure 3-29 for x = 6 cm (0.5c). The vortex properties for the both positive

83

18 i ' 1

 

 

16

14

 

 

 

 

 

 

 

 

 

 

- O A=2°, k=5.2
4 A A=2°, k=11 .5

 

 

 

 

 

 

 

 

 

 

 

6 12
xc (cm)

Figure 3-28: Vortex convection speed, Uc, as a function of downstream distance.

and negative sign vortices are shown in Figure 3-29. These data indicated the vortices
differ by sign of vorticity (and circulation), but otherwise similar. Note that the cutting
walls placed into the flow to initiate the vortex/no-slip interaction were placed so that the
leading edge was at x = 6cm (0.5c). This discussion, therefore, serves as an indication of
the initial conditions for the vortex array before interaction with the wall.

The peak vorticity, <wz>peak, and circulation, 1", increased with increasing reduced
frequency as expected. The peak vorticity magnitude increased linearly by over 6 times
for the reduced frequency range investigated. The circulation showed a smaller increase of
3 times. The circulation data were fit to a second order polynomial. No curve was drawn
for the 4° due to the limited number of data points.

The vortex core radius was rc z 0.4 cm for the 2° (k > 5) case and rc z 0.46 cm for

the 4° cases. The core radius initially decreased as a function of k for the 2° case. The

84

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 3-29: Peak vorticity wpeak, circulation F, core radius rc, convection speed Uc,
and spacing (a,b) for the 2-D planar measurements at X: 6 cm, z = 10 cm.

85

reduction of the core radius at x = 6 cm for A = 2° was consistent with difference in the
formation process of the vortices for this oscillation amplitude. Specifically, for low k’s,
isolated and concentrated vortices form farther downstream which resulted in a more

diffuse spatial distribution of vorticity and greater re. The A = 2° rC data do indicate a

slight increase in the vortex core size for the higher reduced frequencies. This may be due
to three dimensionalities (i.e., axial core flow) from the vortex / side wall interactions
beginning to affect to vortex array. The 4° case showed higher values for the core radius,

rC z 0.47 cm. The lower rc value for A = 4°, k = 4.1 was due to the formation of multiple (2

rather than 1) isolated vortices of the same sign for each phase which result in smaller

vortices. The quoted rC value was for the strongest vortex of each sign.

A comparison of the vortex core radius can be made between the radius of gyration
measure (16) described above and the fit to a Gaussian profile. The core radius for the
vortex profile of k=11.5 case shown in Figure 3-21 was found to be 0.31 via fit
parameters. The reason the fit underestimated the vortex core size was due to the deviation
from the Gaussian profile for Idl > 0.5. The actual vorticity was higher than the fit vorticity
for Idl > 0.5 which resulted in a larger core radius using (16). The same comparison for the
k=5.2 case shows good agreement with the vortex core radius from the Gaussian fit (from

the horizontal profile with no tail) being equal to rC = 0.41 , compared to 0.40 from ( 16).
Vortex convection speed (UL) and spacing data are provided in Figure 3-29. The

vortex convection speed at this fixed spatial location was found via best linear fit to the
position of the vortex core as a function of phase for the 20 points around the x = 6cm
location. Phase was converted to time using the airfoil oscillation frequency, f. The vortex

convection speed, UC, was larger, by up to 35%, than the freestream speed, U00. The low

86

fieq

drug

((11
vom
span
was
oscfl
spat
Th6“

Vekx

Vekx

V

S 11'

“liar.

V

S W

frequency k = 4.1 cases show UC less than Ugo. For the 2°, k = 5.2, 5.7, 6.4, the near zero
drag cases, UC z U 00.

The streamwise spacing, a, and the lateral spacing, b, between <wpcak,p> and
<mpeakm> were found directly from the spatial location for each vortex when the negative

vortex was located at 6cm downstream of the trailing edge of the airfoil. The streamwise
spacing, a, of the vortex cores decreased with increasing reduced frequency. This result
was expected since the vortex spacing, or wavelength, was dependent upon the airfoil
oscillation frequency. Increasing f reduced the wavelength (and therefore streamwise
spacing). The lateral spacing showed a switch in the orientation for the A = 2°, k > 5.7.
The vortex pattern was nearly aligned for the A = 2°, k = 5.7 case consistent with the mean
velocity profile shown in Figure 3-6.

The peak azimuthal or swirl velocity, sz, is shown in Figure 3-30. The swirl

velocity was calculated via the following formula:

—/
F(1_err“) (17)

SW _ 21v
c

 

where sz occurs at r = 1.12rc. Substituting the value for r into ( 17 ) yields:

_ 0.6382F
SW - 21trc

 

(18)

and is shown in Figure 3-30 by the symbols with solid lines. The swirl velocity was also
found, for comparison, from the measured velocities when the vortices were at x = 6cm.
Profiles of the velocity were taken along the streamwise, v(x), and lateral, u(y), directions
through the vortex core. The swirl velocity was taken to be the peak azimuthal velocity in

the profile. The vortices were slightly asymmetric due the induced velocity of the

87

Figi
X:

neig
difle
Close
used
the d;
3166p
W35 8

reduo

 

 

 

 

14: ’I I' 1'4
_ I . x -
E 0 IA ,- . 1.3
12 L ,A’ ' ' 71'2
: ‘ , .f '1.1
_ , -
_ , o -1
10 '_' '1 ’A -‘ O 9
70‘ t ’ é — '
F 3 L " ' '1 ”:8
3 : a ’43 €01}
>5 6 -_ ; ~ o.5>
— A .- a
: , v.‘ - 0.5
4; 3’ €04
; , —a— 2. _ 0.3
- -—G— 4 ~
2 L' ---A--- fromv(x) profile 10-2
I ----<>---- from u(y) profile - 0.1
o l_L l l l l l l l l I l l l 14 l l l l l l l l I l l 1‘ o
0 2 4 6 k 8 10 12

Figure 3-30: Maximum swirl velocity, sz, as a function of the reduced frequency at
x=c/2, midspan.

neighboring vortices which caused the sz from the streamwise profile to be higher. The
difference increased with increasing k. The was because the vortices were stronger and
closer together increasing the induced velocity affect. The calculated swirl velocity was
used for reference in this work given the good agreement with the measured values and
the difficulty of obtaining velocity profiles through the center of the vortex because of the
steep nature of the velocity profile. The swirl velocity showed the peak rotational velocity
was significant for this study, exceeding the freestream and convection speeds for high

reduced frequencies.

88

pH

18C

3.2.5 Vortex/Cutting Wall Initial Conditions

The initial conditions for the axial flow study were taken from the vortex
prOperties at x = 6cm. The results are summarized in Table l. The initial vortex core

radius, rci = 0.40 cm was used for the 2° cases, and rci = 0.47 for the A = 4° cases.

Table 1: Vortex Properties at x=6cm downstream of Airfoil Trailing Edge

 

 

 

 

 

 

 

 

 

 

 

 

 

A k m, (3") 1“i (cm2/s) szi (cm/s)
2° 4.1 25 17.6 3.9
2° 5.2 45 21.1 5.3
2° 5.7 52 23.0 5.8
2° 6.4 61 24.6 6.4
2° 8.2 94 34.1 8.8
2° 9.6 118 43.7 11.3
2° 10.5 139 48.0 12.3
2° 11.5 156 54.1 13.5
4° 4.1 60 31.6 7.0
4° 5.2 91 46.9 10.2
4° 5.7 108 52.8 11.1
4° 6.4 129 62.7 13.2

 

 

 

 

 

89

CHAPTER 4
VORTEX / SIDE-WALL INTERACTION

4.1 Vortex / End Wall Interaction

Axial flow develops within the vortex cores because of the interaction of the

vortices with the side walls of the facility (Figure 4-1). Measurements of the axial flow

0000

Side View

\_/\_/

NACA-OO12Airfoil 9 g 5 g

8 Tunnel Side-Wall

 

 

 

 

Top View

Figure 4-1: Schematic of the vortex / side wall interaction.

due to this interaction provided a comparison for the cutting wall interaction
measurements, which had different boundary conditions. Additionally, these results
allowed the choice of placement of the cutting walls to minimize the effects of the vortex /
side wall interaction. Single line tagging, as described in Section 2.4, was used to provide
a measurement of the transverse velocity for this experiment. Images of a single tagged
line were acquired at a spanwise location of z = 10 cm off of the centerline, nominally
halfway between the tunnel centerspan and the airfoil assembly side walls, over a

streamwise extent of 13 S x/rci _<_ 113 (5-45 cm). Note that the discussion now shifts to the

90

ton

and

fieq
opU
expl
flou

111111

the

case
prol
(1181;

Was

lmai

“5H

the

COnt
1011.
51:11

39115

vortex axial flow development. The normalization variables switch from airfoil based (c

and U00) to vortex based (rd and szi)-

Flow conditioners were placed at the end of the tunnel test section to dampen low
frequency oscillations in the tunnel freestream speed. These flow conditioners blocked
optical view from the end of the tunnel and were removed, for the line tagging
experiments only, to provide access for imaging during the line tagging experiments. The
flow facility motor setting was adjusted to account for the nominally 30% increase in the
tunnel freestream speed without these flow conditioners.

Figure 4—2 shows several representative images, at x=15 cm (38rd) downstream of

the trailing edge and span location 10 cm from the center span, for the A = 2°, = 11.5
case. These images show clear evidence of axial flow within the vortex core. Velocity
profiles that are “Gaussian” were observed along with velocity profiles that were
distinctly non-Gaussian, with multiple peaks. The line displacement method utilized here
was a single component measurement technique and these data were therefore influenced
by the strong lateral (v) velocity with some of the profile shapes due to that fact. These
images also indicated that there was weak flow directed towards rather than away from the
wall (i.e. “reverse” spanwise flow) present.

Figure 4-3 serves to orient the reader to the progression of the axial flow profile as
the vortex convects past the measurement (i.e., laser line) location. White lines in the
contour plot of Figure 4-3 represent the laser line (or line tag position) relative to the
vortex position. It is important to remember that in Figure 4-3 the tagged line was fixed
spatially and the vortex convected past which is represented in the figure by time on the x-

axis. As the vortex entered the measurement region, Figure 4-3(a), the axial flow was low.

91

Fig
inc
In I

110‘.

IECIC

 

       

1 1
undelayed “Gaussian” “Multi-Lobed” “Reverse”

Figure 4-2: Sample line tagging imaged. A=2° lk=11.5 x=38rci. Dashed white lines
indicate undisplaced line center location for reference.

In this portion of the airfoil phase the reverse profiles in Figure 4-2 were found. The axial
flow speed increased, Figure 4-3(b),(c), in magnitude until the center of the vortex was
reached, Figure 4-3(d), when the peak axial flow was measured. The “Gaussian” profiles
(Figure 4-2) were most typically found in the ((1) portion (Figure 4—3) of the phase The
axial flow level decreased as the vortex left the measurement location, Figures 4-3 (e)-(g).
Typically the “multi-lobed” profiles in Figure 4-2 were found near the outer portion of the
vortex cores (b)—(c) and (e)-(f) (Figure 4-3) though farther downstream the profiles
became less Gaussian and more irregular for all phases. Figure 4-3 (f),(g) showed a
reverse axial flow over a portion of the core. The spatial extent of the displaced fluid in
Figure 4-2 was similar in size to the vortex core size in Figure 4-3 indicating that the

spatial region of the spanwise flow was similar to the spatial extent for the vortex array.

92

-150 -90 -3o 30 90 150
~20 cm/s C02

\\\\\\\ \ \ \ \
‘\\\\\\\\\\
\\\\\\\\\\ \
\\\\\\\\\\

\\\\\\\\\\'
\\\\\\\\\\\
\\\\\\\\\\\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

H
10 - -
9 . .
3 - .
7 .
5° ' '
>5 - ' '
4 - W - -
3 _ _ .
2 : - .
1 .1 -
I | l 11l111111 nil 111111
0 l 0 2 4 ll 5 10 5 1O 15 l 5
w(cmls)
g f e d c b a

Figure 4-3: Schematic representation of axial flow profiles relative to vortex location.
=2°, k=11.5, x=38rci.

The relationship, spatially and temporally, of the axial flow to the vortex core can only be
surmised using these data. Stereoscopic data in Chapter 5 will show that the reverse flow
actually occurred outside of the vortex cores not within them and the strong spanwise flow
away from the wall did occur within the vortex core.

The line tagging data were processed to provide the axial velocity profiles versus
y. The peak axial flow level, both positive and negative, for each measurement location

were recorded and plotted. Figure 4-4 shows <Wpeak> versus the downstream distance and

93

F igr
do“

the r
the I

"16:15

<w>peak (cm/s) A- o

012 3 4 5 6 77691011121314

    

m
rrr'rrrvlrr[11rrrrlrrrrlrrl’rlrrrlllr

 

A=4°

20

x (cm)
Figure 4-4: Local phase averaged peak spanwise flow, <w>peak, versus the
downstream distance, x, and the reduced frequency, k. Data plotted dimensionally.

the reduced frequency plotted dimensionally. The grid shown on the contour map marks
the measurement locations. The axial flow within the vortex core reached the z = 10cm
measurement plane close to the trailing edge (i.e. for x = 13rd), for k > 9.6. The
dimensional values show <w>peak < 2 cm/s at this location. The same data are shown non-
dimensionalized by the vortex initial conditions in Figure 4-5. Axial flow levels were
nominally equal to the swirl velocity for the highest reduced frequency (A = 2°, k = 11.5)
and approached the swirl velocity for all other cases. The data in Figure 4-4 and 4-5 show

that the peak axial core flow first increases, then decreases in magnitude. There were

94

A=2°

   
    
 
    

.5
.5
l

on

N

lllllllerlll'lllllllll

an
I r

 

rivllrrlr

 

 

5
A: °

6 7
35.5

5

4.5 .

o 25 so 75 100
xlrea

Figure 4-5: Local peak spanwise flow, <w>peak, versus the downstream distance, x,
and the reduced frequency, k. Data plotted non-dimensionally.

indications from the k = 11.5 case that <w>peak may approach a constant value or be
slowly decreasing far downstream. An alternate spatial normalization, the vortex
wavelength “a”, was tried and the results are plotted in Figure 4-6. The result of this
spatial normalization was to make the downstream location of maximum spanwise flow
occur nearer to the same location ( between 2.5 and 4) indicating that the wavelength may
be an important spatial length in this problem.

Figure 4-7 shows the peak spanwise velocity versus the x location for selected

reduced frequencies. These data showed that if the reduced frequency is high enough (k >

95

1101
ilfi
The.

data

A=2°

 

 

A=°

 

 

 

5 1o
xla

Figure 4-6: Local peak spanwise flow, <w>peak, versus the downstream distance, x,
and the reduced frequency, k. Distance normalized by vortex array wavelength, a.

6.7) the peak axial velocity level rapidly increased as a function of downstream distance.
After the maximum was reached <W>peak first decreased with a high deceleration, similar
in magnitude to the initial acceleration, followed by a slower deceleration. The
downstream location of the maximum axial flow decreased with increasing reduced
frequency. For k S 6.7 the peak axial flow velocity did not decrease within the
measurement extent, but remained nominally constant. The reverse axial flow, <wrev>,
data show that reverse flow had levels less than O'1<wpeak> for the highest reduced

frequency tested (Figure 4-8).

96

Fig]
1:1-

11,

d
e
d

 

d

 

9
co

0
b
O Tlll TTTTITIIUIIIlllIlTTI—rllltlTTTITTIIIIIIIIIlllllllllllll

 

vswi
s> s> s>
mum

I

<w>peak
s> s> s> s>
d N (a) GI

 

 

O

 

 

l l l l l 1 1 11' l l l 1|L14 1411 l

 

I
P
u-l

25 so 75 100
)1/18

Figure 4-7: Peak axial flow speed, <w>peak, versus the reduced frequency for A=2° at
z=10cm off the centerspan. '

 

 

 

 

 

ops
_ -—+§-—-k=52
+ k=9.6
" ———8——— k=11.5
- o l
3
0 p
>' L
\ \
> a,
A2 _ \\...
3 L 8’
v \,
'0-05 " \ ,.£§}~-—~€1
\ m/
l a» /
,- _7\\ ’ l
15
- .1 l l L 1 l 1 1 l l l l l 1 l I l 1 1 l l 1 l 1
o o 25 50 75 100
xlrg

Figure 4-8: Peak reverse axial flow speed, <w>,“, versus the reduced frequency for
A=2° at z=10cm off the centerspan.

97

The maximum spanwise flow velocity in the measurement domain for each k (i.e.
search through all measurements locations and find the highest magnitude spanwise flow
for each k) was found using the data in Figure 4-4. The result is plotted in Figure 4-9 for

the A = 2° cases. When the data were dimensional there was a nominally 700% increase in

 

 

 

 

 

1.1

14- 11
12_ 10.9
~ ~06

A _ -

E 1° -0.7 E
3 8- d0,6\
E * -0.5 g
S 6' :04;
v - . ' V
4- 40.3
’ -0.2

2,... _
-0.1

41111111lllllllllll|1141_lll 111.11
00 2 4 6k 8 10 12 0

Figure 4-9: Maximum axial flow, <w>max, versus reduced frequency.

<wmax> from the lowest reduced frequency to the highest (2.3 cm/s to 14 cm/s). When the

data were normalized with the swirl velocity, the difference was reduced to less than a
100% increase (0.6VSW to 1.0VSW). The curves plotted are 2nd order polynomial fits to the
data and show good agreement over the reduced frequency range investigated.

The data from the line tagging experiments can be shown in contour form with y
and phase being the axis variables (i.e., profiles as a function of phase). Figure 4-10 shows
the data presented in this way for the A = 2° , k = 11.5 case at two downstream distances

(x = 38rd, 88rd). The spanwise flow show was confined to two isolated structures at the

upstream location, that corresponded to the two vortices, for each phase at x = 38rd. The

98

y (CM)

V (cm)

ligr
Dov.
Sim]
Skev
fine

Shar

1ru<

 

..
.5

 

 

 

 

      

 

 

 

10-
’—
9 3.
1.
8 A

5"
A7 :5.
>. 3'
5 2.
4 "
3 .
1‘.
9 10-
8 9.
g.
7 7'
56 :6-
35- :5.
> ‘-
4 3.
3 2'
1.

2 0.25 0.75 3.1-111-11111111I

0.5
phase 01)
Figure 4-10: Spanwise flow versus vertical distance, y, and phase. A=2°, k=11.5.

Downstream location indicated on figure. Line plots show all phases (i.e., y-profiles)
simultaneously.

Skewed shape of these structures was a result of the flow velocity parallel to the tagged
line as discussed in Chapter 2. The velocity profiles in Figure 4—10 show Gaussian like
Shapes that corresponded isolated vortex cores. The two peaks correspond to the isolated
Structures in the vortex cores. Further downstream at x = 88rd, the axial flow was in a
Single diffuse band that spanned the entire phase rather than two isolated structures. The
magnitude dropped significantly compared to the upstream case. The profile was also
Wider, and less steep than was observed at x = 38rd. The current discussion is intended to
Show that there was a change in the structure of the axial flow that implied a simultaneous

change in the vortex structure. There were no vorticity data that accompany these

99

srm

<w> (cm/s) ‘

0.01.1 2.2 3.3 4.4 5.5

    
     

v(cm)
uhmmsrcoo

  

0.25 0.75

0.5
phase

Figure 4-11: Axial velocity profiles for A=2°, k=9.6. Multiple phases shown
simultaneously.

<w> (cm/s)

0.0 0.8 1.6 2.4 3.3

   

New
to) or 01 N on to

    
   

(L25 0575

05
phase
Figure 4-12: Axial velocity profiles for A=2°, k=5.2. Multiple phases shown
simultaneously.
measurements, however the stereoscopic data in Chapter 5 do support the conjecture.
Figure 4-11 shows the contour map of the transverse velocity versus y-location and phase
for k=9.6. These data are from the furthest downstream measurement location and showed
the transition from isolated structures. The same data for the k=5.2 case, Figure 4-12, did

not show the transition.

(IQ

l(

it

1h

CHAPTER 5

VORTEX / CUTTING WALL INTERACTION
The data in Chapter 3 and 4 (i.e. the “no-wall” experiments) presented the

boundary conditions for the vortex/cutting wall interactions (i.e. the “cutting wall”

experiments). The cutting walls were placed into the flow at x = 6cm (c/2, 15rd), with a

gap of i9.5 cm (i24rci) about the tunnel centerline (Figure 5-1). The coordinate system

 

 

 

 

 

 

 

 

 

 

 

Side View
6:1,? 9 C ’3' K‘rx’D O f)
V,
Fwy” " 7‘ 3 «"\_,. f" V4]
i Cutting Wall
T23; ., W ‘ ’
: 2,1 3331 F :1 l
‘ 3 [C3 19cm
5 l l“ .. 3. .51 F”; l
NACA-0012 l j a l #13 1 “:3 ‘ <3, 5
Airfoil F l 1
Rf. .3 .1 -1

 

 

 

 

Top View

Figure 5-1: Schematic of flow field with cutting walls. Coordinate system origin is
located at the cutting wall leading edge, as shown.

for this portion of the study is shown in Figure 5-1 and had an origin at the leading edge of
the cutting walls. The wall location represents a compromise location. This was the
minimum downstream location where the vortices were formed into isolated structures for

the low reduced frequency (k = 5.2) case. The initial axial flow from the vortex/side wall

101

Fig

interaction has however already reached this span location for the highest reduced
frequency cases (A = 2°, k = 9.6, 10.5, 11.5) with speeds of nominally 2 cm/s.

Marshal and Yalamanchili (1994) report that a cutting wall thickness greater than
the vortex core radius results in distortion and bending of the vortex as it interacts with the

wall. A wall thickness of rci or smaller results in no significant bending of the vortex. The
cutting walls are made of 0.3 cm (0-751'ci) thick clear acrylic to minimize effects from the

cutting. The leading edge of the cutting walls were rounded to prevent separation around

sharp edges. Marshall and Krishnamoorthy (1997) define the impact velocity, Vi = UC/
szi’ and determined analytically that if Vi > 0.25 no separation on the cutting surface

would occur before the vortex is cut. The impact velocity calculated from the midplane

data is shown in Figure 5-2. All cases studied show V, > 0.25 in this study.

 

 

 

 

 

 

3
L. D 2°
0 4°
2.5” D
2_ U
.. U
i F
> U
\15-
a O
3 _ El
00 n
0.5-
0.25
oriiiliiirlirrilririlrriiliitilii
0 2 4 6k 8 10 12

Figure 5-2: Impact speed, Vi, versus reduced frequency from the mid-span data.

102

The discussion in this chapter first centers on line tagging results with the cutting
walls in place. These data were available for all cases and were used to determine which
cases would be investigated in greater detail with stereoscopic MTV. Following those data
the stereoscopic data for the k = 5.2, 9.6 and 11.5 (A = 2°) are presented. These data are
organized around a discussion of the mean flow fields for all cases followed by the phase
averaged flow fields with each case discussed individually. Finally a discussion of the data

as it related to waves on vortex cores will be presented.

5.1 Line Tagging Results

Line tagging MTV was used initially to map out the extent and approximate
magnitude of the spanwise flow for all cases. Measurements were taken at multiple span
locations from 0.6rci S 2 S 25rd (0.24 - 10 cm) with an approximate spacing of 1.6rci (0.65
cm) and from -l3rci S x S 68rci (-5 - 27 cm) with steps of 2.5rci (1 cm). These data were
processed in the same way as the data in Chapter 4 to provide the local peak spanwise
flow and global maximum spanwise flow. Example velocity profiles for A = 2°, k = 11.5
at x = 13rd for multiple span locations are shown in Figure 5-3. The phase of the peak
axial flow (i.e. the velocity profile through the center of the vortex) is shown. Strong
spanwise flow was measured, with significant reverse spanwise flow also observed near
the wall. The peak spanwise flow increased in magnitude as 2 increased until the

maximum was reached at z 2 4rd. The spanwise axial flow then decreased as 2 increased.

The reverse spanwise flow decreased away from the wall. The axial flow front had not

reached the center of the cutting wall assembly at this downstream location.

103

   

 

. i f.- l i
: l i i I
A; g i g .
é; . > S i

O 5
w (cm/s)

O 2
Z=O.6I'Ci 2.3I'Ci 4rd 9rd 14rd 19rd

Figure 5-3: Example phase-averaged spanwise velocity profiles for A = 2°, k = 11.5, x
= 13rd. Phase, (1), with the highest spanwise flow shown at each 1 location. Distance

between profiles not shown to scale.

Contour maps of <w> ak over the entire spatial extent for k = 5.2 and 11.5 are

pe
shown in Figure 5-4 and 5-5. The reverse spanwise flow, <w>rev, is also shown in Figures
5-4 and 5-5 and will be discussed later. Both cases show that the spanwise core flow
begins near the wall and moves away towards the center of the tunnel with increasing
downstream distance. No steady or constant axial flow condition was found after the axial
flow was initiated. The axial flow “front” (i.e. the boundary between the spanwise flow /
no-spanwise flow) advanced at a higher rate for the higher reduced frequency case.

When the axial flow versus the z-location was plotted as a function of downstream
distance a local overshoot / undershoot or “spike” in the axial flow velocity with a peak to
peak magnitude of VOS = 019sz was observed near the axial flow front boundary (Figure
5-6). The velocity spike followed the front boundary as it progressed towards the center of
the cutting wall. Spikes at the axial flow front boundary were observed for all k’s (except
k = 4.], A 2 2° ) though the disturbance was weaker for lower k values (Figure 5-7).

Stereopscopic data, presented later in this chapter, will show that this axial flow spike was

104

k=5.2 <W >90“, V."

    
     

1
1

zlrd

   

5

0

5

00 5 10 15 20 25 30,35 40 45 50 55 60 65
X I'

d

Figure 5-4: Peak spanwise flow speed for A = 2°, k=5.2.

k=11.5 <W >m’ Vm

0 5 10 15 20 25 30,35 40 45 50 55 60 65
X l’

cl

Figure 5-5: Peak spanwise flow speed for A = 2°, k = 11.5.

l05

 

  
 
     
     

 

 

 

24 ; x-location
. —a— 2.5rd
22: ——q§——-5%
20 r ~—9—- 7.5I'd
18 P
16 r
5'8 14 -
\ -
N 123-
10b
F.
8 —
6 .—
4 -
2 _
° 9 0.25 0.5 _ 0.75 ‘ 1
<w>...» I V",

Figure 5-6: <w>peak versus spanwise location for k = 11.5.

 

 

 

 

0.25
E o
0.2 -
; n
: O
E 0.15 L D
> : [j
\ _
3 ~ D
> 01 - D
I u
_ O n
0.05 -
~ D 2°
” o
E O 4
l l l 1 l l l l l I L
00 5 k 10

Figure 5-7: Peak to peak magnitude of velocity spike, V03, as a function of reduced
frequency.

106

associated with a local spike in the vorticity and for high k, a change in the vortex core
area. These data are therefore initial evidence that area varying waves were present on the
vortex. The 2 = 12.5rci profile also showed evidence of other velocity spikes behind the
axial flow front.

Two speeds were defined using the line tagging data. The front location was

defined by the span location <w ak> = 0.5 cm/s. The threshold level was picked as it was

pe
relatively low, but above the measurement noise level. The front speed was determined by
dividing the spanwise distance travelled, 2, by the convection time (the midspan

convection speed, UC, multiplied by the downstream distance traveled, x). The speed of

the spike was estimated by differentiating the span position of the peak level of the

spanwise velocity spike by the convection time, as was done for Vf. Lundgren and Ashurst

(1989) use the gas dynamics equations for an analogous description of the behavior of

area varying waves on a vortex tube and define an equivalent “speed of sound” as:

C = -— (19)

The two wave speeds and the calculated values of ce are also shown for the data in Figure

5—8. Both wave speeds increased as the reduced frequency increased. The speed of the

Spike, Vw, was less than the front speed, Vf, for all cases (Figure 5-8) which indicated that

the axial flow front led the axial flow spike with an increasing distance. The sound speed,

Ce, was higher than the wave speed, Vw, for low reduced frequencies. Equation (19), used
to calculated cc, was derived for a potential vortex with flat-topped axial flow. The

stereoscopic data will show that the axial flow profile for the current work did not fit a

107

Th

{Fl

Cm‘

dist;
“We

Th...

 

 

 

 

20_
3 O v,,2°
1.85 E] v,,2° A 0
16E- 0 V,,4° V D
t V V..4°
14:- c, <\ O 8
7a? : -
E12:- {3' r'
u .—
.., -_ 0
131°: V
o : n
3. 8r P
en : V’ _‘ ;_,
6;- v (iii
4%-
ZE-
:¥ 1 1 l l J_ I l g i 1 l
00 10
k

Figure 5-8: Comparison of the front speed, Vf, the wave spanwise flow speed, Vw, and
the equivalent sound speed, cc.

flat-top profile. The qualitative agreement was consistent with a small amplitude traveling
wave on the vortex core.

The use of cutting walls placed symmetrically about the tunnel midspan means
that the axial flow from the two sides meet at the midspan at some downstream distance.
The axial flow generated from the symmetrically placed cutting wall was observed
(Figures 5-4 and 5-5) when the axial flow front reached the cutting wall midspan plane, x

z 45rd for k = 5.2 and x z 17rd for k = 11.5. The symmetry point was near 2 z 23rd (9.5

Cm) as expected.
The location of the peak axial flow moved away from the wall as the downstream
distance increased. The global maximum spanwise flow observed over the spatial domain

investigated, <wmax>, as a function of the reduced frequency is shown in Figure 5-9.

These data showed good agreement with the no-wall cases for high reduced frequency

108

 

 

 

 

14 -
12 E- O a
” A
1310 E . 9
a : e
o - O
v 8 _'
'3 E O I:
S 5 '
L [1 O
v t 00
4 L O a El
[J 0 2°, with walls
2 * El El 2°, no walls
F 0 4°, with walls
_ A 4°, no walls
0 _ 1 1 1 1 J 1 1 1 1 l 1 1
O 10

k

Figure 5-9: Maximum spanwise velocity, <w>max comparing no-wall and cutting wall
experiments.

while at lower reduced frequencies <wmax> for the no-wall experiments was lower. Figure

5-4, the low k case, indicated that while the maximum spanwise velocity occurred within

2.5rci of the wall. The maximum spanwise flow moved significantly farther away from the
wall (10rd) for k = 11.5 (Figure 5-5). It was, therefore, not surprising that <wmax> for the

lower k no-wall experiments was smaller than the cutting wall cases since only a single z

location, far away ("40ch from the side walls, was measured for the no-wall experiments.

The measurements revealed a significant reverse spanwise flow (Figure 5-4 and 5-
5) with magnitudes of nominally half those for the axial core flow. The span location of

the peak reverse, <wrev>, flow for the k = l 1.5 case was observed to move away from the
cutting wall as was noted for <wpeak>. Maximum <wrcv> levels are shown in Figure 5-10.

The highest <wrev> measured for the A = 2° cases occurred for k=9.6 then decreased for

109

 

 

Figur
k=10.;

Sallsfi.

Phases
Observ
legion:
increaé
aPPFOa
”1th i
flow it
OCCUFFI';

downSt:

 

 

 

 

14;" O 2°,wlthwalls
: <> 4°,withwalls
12:-
A E
£10:
9, E
:3; o
s. z o
v 45 o <><> 0
: 000
I O
2r
F l 4 l l J l 1 l l I l J
00 5 10
k

Figure 5-10: <w>rev as a function of the reduced frequency.

k=10.5 and 11.5. The presence of reverse spanwise flow is required so that continuity is
satisfied (i.e. to balance fluid transported away from the cutting walls).
Figure 5-ll shows the spanwise velocity profiles for A = 2° k = 11.5, with all

phases plotted, at x = 58rd. This figure provides a comparison with Figure 4-9. It was

observed that for the no—wall cases the axial flow structure changed from two isolated
regions per phase to a single band of spanwise flow as the downstream distance was
increased. This was also observed with the cutting walls. Near the wall the axial flow

approached zero (2 = 0.6rci, Figure 5-5 and 5-11) though at an equivalent z-location it was
much higher upstream (e.g. at z = 0.6rci, x = 20rd, Figure 5-5). A single band of spanwise
flow for all phases was observed for z < 4rd with a transition to two isolated structures
occurring between 4rd and 9rd. Multiple structures were observed for z > 9rd at this

downstream location. The less pronounced spanwise flow peaks in Figure 5-ll were a

110

 

Sp
no
str

im

'1!
3.)

Th.—

We]

-5;
-55.

 

WM
-2024

 

mm
-2 0 2
w (cm/s)
z=0.6rci 4rd 9rd 14rd 19rd

Figure 5-11: Spanwise velocity profiles for A = 2°, k = 11.5, x = 58rd all phases (i.e. y-
profiles) shown.

result of the increased downstream distance of the data in Figure 5-11 compared to the
multi-structure plot in Figure 4-9. These data were instructive in that the additional
spanwise measurement locations indicated that the change in structure initiated near the
no-slip boundary and propagated away from the wall. No transition from two to one
structure was observed for the low reduced frequency case over the downstream range

investigated.

5.2 Stereoscopic Results

The data in Section 5.1 were used to define the experimental parameters for the
more detailed stereoscopic measurements. The cases chosen were A = 2° k = 5.2, 11.5.
The k=5.2 and 11.5 cases were chosen because the represent the lowest vorticity level with
well defined vortices at x = 6cm downstream (k = 5.2), and the highest vorticity value in

the parameter range (k = [1.5). Four spanwise planes 2 = rci, 2rd, 10rd, 20rd (0.4cm,

lll

6C
l(
3P
bi.

dis

Up

Up

W3
est;
the
W3;
iéls‘e
”105

For

\Vhe

degc

EMT-

0.8cm, 4cm, 8cm) were investigate. The z = rci, 2rd cases started upstream of the leading
edge of the cutting wall and extended to approximately SOrCi (20 cm) downstream. The 2 =
lOrCi, 20rd cases began downstream of the cutting wall leading edge and also extended to
approximately 50rd (20 cm) downstream. The initial portion of the data sets were affected

by optical issues resulting from the leading edge of the cutting walls. The optical
distortion resulted in erroneous data in those regions which was not used or shown in the
analysis below with a single exception. The phase-averaged vorticity plots do include the
upstream vorticity. The vorticity fields did not appear to be qualitatively affected by the
optical issues, and so are shown for visual affect.

The stereoscopic data sets were taken with the laser sheet parallel to the cutting
walls. There was a spatial uncertainty in the spanwise location of the laser sheet. It was

estimated that at the measurement location the laser sheet was 0.13rci (0.05 cm) thick. For
the z = rci (0.4 cm) measurement plane the spanwise gradients were large, and the data set

was affected by the uncertainty in the laser sheet placement. This was because of finite
laser sheet thickness ( ~ 0.5 mm) and the method used for beam placement. This error was
most apparent in the mean data which did not show good agreement between the small
FOV’s. The data from this measurement plane were included here because they provide
important information about the flow field near to the cutting wall but care must be taken
when analyzing the data given the additional source of measurement uncertainty
described. The remaining planes did not show the effect of the laser sheet placement
uncertainty.

A comparison of the spanwise velocity component between the line tagging and

sMTV data sets is shown in Figure 5-12. The sMTV data (lines in Figure 5-12) show

112

I: A
II'
.'

 

clearly the initial local spike in the spanwise flow already discussed. In addition these data
also show multiple secondary spikes in the peak spanwise flow. The line tagging data do
not show the secondary spikes. This may be due in part to the data density in the
streamwise distance for the line tagging data which was coarse compared to the

stereoscopic data. Figure 5-12 is instructive because these data show excellent agreement

 

 

 

 

 

 

 

 

12
11
1o
:1
a 9 D
E 8 1:1
8. 7 u
8
16 °
:1
4 '31::
3
2 El “musslns
1 D m
1 I 1 1 1 4 1
° 10 20
x(cmls)

Figure 5-12: Comparison of the peak spanwise velocity between the line tagging and
sMTV data.

between the data sets which allows later discussion using the information from the

different the data sets.

5.2.1 Mean Flow Field, k=5.2, 11.5

A comparison of the average streamwise velocity fields near the tunnel midspan is
show in Figure 5-13 for k = 11.5. These data show the repeatability of the mean
streamwise flow field was to within 2% near the centerline (y=0), and better than 0.5% for
the freestream between the no—wall and cutting wall experiments. Good agreement

between the no-wall experiments and cutting wall experiments (near the midspan) was

113

Fig
mi

als

“’3

no
am
he
inc
\‘e
lhc

are

 

 

0.8
0.6

 

0'4 30 stereoscopic

- — - - 20 planar

 

 

0.2

 

—L
llIlllrlllllllllllIIIIIIIIT—IlrrTll'llllllIIIIIIII

 

 

o l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 I 1 1 1 1 l 1 1 1
0 0.5 1 1 .5 2
xlc

Figure 5-13: Comparison of the planar (midplane) and stereoscopic (z=8cm, near
midplane) measurements of the mean streamwise velocity, u, for k=11.5.

 

also observed (but not shown here) in the mean flow data. These data indicate the cutting
walls did not change the base flow field significantly from the no-wall experiment.

The freestream showed an acceleration for the cutting wall experiment that was
nominally uniform for all cases and span locations considered. The speed increased
approximately 4% over the 20cm streamwise experimental fetch. Laminar flat plate
boundary layer analysis on the facility walls (cutting and side) predicted a 2% velocity
increase over the same range because of boundary layer growth. The remainder of the
velocity increase was accounted for by the non-planar shape of the thin acrylic walls (i.e.
there was a bulge in the cutting walls) near the middle of the plate which decreased the
area and increased the mean flow speed.

The mean streamwise velocity fields for k = 5.2 are shown in Figure 5-14. The low

reduced frequency case showed an decreasing wake thickness as a function of the distance

114

 

 

I. II

F1g

   
  

IIIIIIIIIIIIIIIIII

 

1o , . , . . . u I U...
’ 1.25
1.20
1.15
1.10
1.05
1.00
0.95
0.90
0.85
0.80
0.75

Figure 5-14: Mean streamwise velocity, 3, fields, k = 5.2, 2 = rci, 2rd, 10rd, 20rd.

115

V1

81

be

16‘

from the wall. Mean velocity profiles at x = llrci (Figure 5-15) show that the velocity

deficit increased and formed an increased overshoot while the wake widened from 2 =

20rd to z = 10rd. From 2 = 10rd to z = 2rd the wake profile changed character with the

mean velocity becoming flat across the wake.

 

 

 

 

 

 

 

10 Z
5 2r¢I
“ 10rd
5_ ----- 20rd
5'5 ’ , ...........
\ o_ - .........
> L ......
.5-
-1—lllllllllllllllJtlllllllllIE‘llllllll
‘1.5 1.6 1.7 1.8_ 1.9 2 2.1 2.2
ulVsw

Figure 5-15: Mean streamwise velocity profiles for x = 11rd, k = 5.2 at different z’s.

Figure 5-16 shows the mean velocity fields for k = 11.5. The mean flow field

Velocity excess decreased and the wake thickening near the centerline (z = 20rd). More
Striking in this figure, closer to the cutting wall, 2 = 10rd, the wake showed the same
behavior until x z 45rd when the wake changed from a velocity excess split into two

regions with a velocity deficit between. While the split of the wake was more evident at z

116

ITIIIIII'IIII'IIII

 

 

Figure 5-16: Mean streamwise velocity, 3, fields, k = 11.5, z = rci, 2rd, 10rd, 20rd.

117

= rci and 2rd, these locations continue to develop and show the wake profile transitions
from a velocity excess (x = 10rd) to a velocity deficit with overshoot (x = 39rd) to a

velocity deficit (z = 50rd). Figure 5-17 shows the velocity profiles for z = 2rd at three

10

 

 

 

 

 

 

 

Figure 5-17: Mean streamwise velocity profiles for three downstream positions at z =
2rd, k = 11.5.
downstream locations to highlight this transition.

The mean lateral velocity fields, v, are shown in Figure 5-18 and Figure 5-19.
These data indicate significant lateral velocities develop near the wall with increasing
downstream distance. There was also significant V found away from the wall. This was a
significant departure from V = 0 measured at the midspan in the no-cutting wall

experiment.

118

 

 

 

 

 

10_
I =20I‘ci

5L

1.":

\0:

>:

.5:

i.

1.
_1o"111I1111[1111I1111I1111I11

 

z=10rc,

Y c
O
TTII'ITII'IIII'IIII

-10 llllLlLLlLLLlLLJ—LllllILJLJ

 

 

 

 

 

 

  

5 _:__ z=2r¢i
.. C
5" :
\ 0 _-
>~ :
-5 :—
_1O : 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 l 1 1
‘05 le
E z=rc1 0.20
5 _- 0.16
'5 3 ”W 0.12
: 0 :- 0.08
> E 0.04
_ 0.00
-5 3 -0.04
-10 _ 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 -0'08

 

 

10 20 30 40 50 * H '0-12

-0.16
x, r°i -0.20

Figure 5-18: Mean lateral velocity, 3, fields, k = 5.2, z = Feb 2rd, 10rd, 20rd.

119

‘ ‘il

00

 

2:201‘ci

y/rci
o
llllllllllllllllll

 

 

-5
_1011111111111111141111441111
10
z=10rci

y/rci
O
lllllllFTlTIlllllll

 

 

 

 

 

   

 

 

 

-5
_1o 4 141 I l I 1 I l 1 1 l I l l 1 l l 1 l l 1 I 1 l
10_
5E_2==218
L? E
\ 0 _-
> :
-5 :_
_10 : I 1 l I 1 l 1 1 11 1 1 1 1 I 1 1 1 1 I 1 1 l 1 J 1 1
10_ V/Um
E Z=rci 0.30
5 E- 0.24
" t 0.13
:0 O '- 0.12
> : 0.06
-5 :_ 0.00
: -0.05
-10 : 1 l 1 I 1 | l 1 l l L l 1 l J 1 1 I I I I I l I 14L ‘0-12
10 20 30 40 50 333-342
x/ r“ a -0130

Figure 5-19: Mean streamwise velocity, 3, fields, k = 11.5, 2 = rci, 2rd, 10rd, 20rd.

120

The mean vorticity field for the low reduced frequency case, k=5.2, is shown in
Figure 5-20. The mean vorticity field near the centerline, z = 20rd, was nearly uniform in
thickness and magnitude for x < 35rd after which the lateral separation between the two
signs of vorticity increase slightly. At 2 = 10rd the‘vorticity field was noticeably thicker,
though still of nominally uniform thickness as a function of x. The vorticity field at z =
2rd was qualitatively similar to that observed at z = 10rd for x < 15rd. The two signs of
vorticity separated laterally for x > 15rd, with no mean vorticity between the layers
consistent with the flat region in the mean velocity profile showing (Figure 5-14). For x >
45rd the layers thickened and the lateral spacing decreased. The mean vorticity field at z =
rci showed multiple regions of non-zero mean vorticity indicating that multiple vortical
structures must be present at this span location.

The mean spanwise vorticity, normalized by the no-wall peak vorticity (Table 4- l ),
for the high reduced frequency case, k = 11.5, is shown in Figure 5-21. The orientation of
the mean spanwise vorticity field reversed orientation for this reduced frequency case at
all span locations except 2 = 20rd. Close to the wall, 2 = rd and 2rd, the orientation of the
vorticity changes between 25 and 30rd, consistent with the mean velocity field (Figure 5-
16). The 2 = 10rd case also shows the reversal farther downstream for x > 45rd. At 2 =
20rd the vorticity layer was beginning to switch orientation at the end of the measurement
extent.

The mean spanwise velocity fields for the low and high reduced frequency cases
are shown in Figure 5—22 and Figure 5-23. The spanwise flow from the vortex cores

entered into the flow field for x > 0 and 10rd (z 2 rd and 2rd respectively). At 2 = 10rd the

121

5

i’sin' '
.

' PM? ‘5 ifihfmwsfiifigd

     

, 2 .. 1 ’ . .. 3 .. i - ' ' -0_12
10 20 30 40 50 -0.16
X] rci -0.20

Figure 5-20: Mean spanwise vorticity, 6, fields, k = 5.2, 2 = rci, 2rd, 10rd, 20rd.

122

z=2rci

‘ WMfiWMM '

0.20
0.16
0.12
0.08
0.04
: . 0.00
’ ’ -0.04
-0.08

 

   
     

-10~

 

. .1 . . ., 2. . . .. -0.12
10 20 30 40 50 -o.16
x/ rci -0.20

Figure 5-21: Mean spanwise vorticity, 52 fields, k = 11.5, z = rci, 2rd, 10rd, 20rci.

123

    
    

llll'llllllllllllll

 

1 0 -..-_ - .. . 1.13:... 3..., 7.3;.” 3 .. . w I Um
._ . _ "‘F-‘fi. . . - - ' . ' ' i 0.20
0.1 6
0.1 2
\ '2- .1 0.08
‘ - ' 0.00
-0.04
-0.08
30 -0.1 2
y/ re. -0.16
-0.20

Figure 5-22: Mean spanwise velocity, w, fields, k = 5.2, 2 = rd, 2rd, 10rd, 20rd.

20

124

Il—llllllllll'llll

 

30
y l rci
Figure 5-23: Mean spanwise velocity, v—v, fields, k = 11.5, 2 = rd, 2rd, 10rd, 20rd.

125

 

axial flow was observed for x > 40rd. The magnitudes were about 30% of those measured
at z = rci. Both signs of spanwise flow observed at all three span locations, consistent with
the line tagging results. The 2 = 20rd plane indicated no mean spanwise velocity.

The high reduced frequency case, k = 11.5, showed axial flow near the wall, 2 = rd
and 2rd, for x z 0 (Figure 523). The mean positive spanwise flow increased in magnitude
for increasing downstream distance until the peak level which occurred near x = 35rd. The

mean positive spanwise flow then decreased as a function of increasing downstream
distance. Reverse spanwise flow was found to exist above and below the positive

spanwise flow. The spanwise flow entered into the z = 10rd plane at x 2 6rd and increased

in magnitude as was observed closer to the wall. The peak positive spanwise flow

occurred at x z 46rd. Reverse spanwise flow was also observed above and below the

positive spanwise flow for this lateral location. A comparison of the maximum mean

spanwise velocities for the z = 2rd and 10rd show that the magnitude was largest at z =
10rd (0-35szi compared to 0-26szi) which indicated the peak axial flow speeds
occurred well away from the wall for this case. At 2 = 20rd minimal spanwise flow was
observed until x z 30rd. No significant reverse spanwise flow was observed at this

measurement location.

5.2.2 Phase Averaged F 10w Field, k=5.2.

The phase averaged vorticity field, <w>, for (11:0 and multiple span locations is
shown in Figure 5-24. Though data for all phases are available, a single phase, (11:0, is
shown as being representative of the “instantaneous” vorticity and velocity fields. These

data show the downstream and spanwise development of the vorticity field. Close to the

126

”all”?

L:

   

   
    

-1.0 -0.7 -0.4 -0. 101.111.10.11 0.7 1.0
<002>I01l

10_
59

a? E
\ 0 _-
>1 :
-5 :—
-10‘

Illllllllllllllllll

 

 

  

   

“20" "§0m"”140"1
x/r¢i

10

 

50

Figure 5-24: Phase-averaged vorticity, <w>, for ¢=0 at multiple span locations.
k=5.2.

127

cutting walls (2 = rci) the vorticity quickly looses its initial shape (x z 0) with distributed
multiple, rather than a single, peaks (x z 10rd). The vortices continue to deform as the

downstream distance increased with skewed, highly stretched rather than round structures
formed. The initial single structure was observed to break-up into multiple structures as
was inferred from the mean vorticity field. Vorticity profiles, shown in Figure 5-25,
confirm these observations. The profiles shown in Figure 5-25 (and profiles for
subsequent sections) were vertical ( y ) profiles taken at the x-location nearest to the peak
vorticity location.

Farther away from the wall at z = 2rd the vortices also significantly distorted from
their initial shape. The initially Gaussian profile vortices (x z 10rd) develop into hollow
core profile vortices (observable at x z 48rd). However, the development into a hollow

vortex was not as a result of the decrease in the central peak vorticity of the Gaussian
profile, but a distortion and re-distribution of the vorticity. Evidence for this is clear when
the data are viewed as function of phase dynamically, however by observing the regions of

vorticity in Figure 5-24 at z = 2rd progressively, the break-up and redistribution of the

vortex can be inferred. The profiles provided in Figure 5-25 show the initially Gaussian
profile with the vorticity tail progress into the hollow vortex profile downstream. The z =

10rd and 20rd plots and profiles show that the vortices remain essentially Gaussian

shaped with the peak level dropping with increased downstream distance.

The axial flow fields and corresponding profiles are shown in Figures 5-26 and 5-
27 respectively. Note the profiles shown in Figure 5-27 (and subsequent profile figures)
correspond to the same spatial location of the vorticity profiles shown in Figure 5-25. At 2

= rci the axial flow profiles do not have a Gaussian distribution or even round shape,

128

y/_rcl

F1;

Z = I‘ci 2rd 10rci 201'ci
<w> I (1) X = 10rd

% e
a e

%
1 %
. 1%

5
“'8

\0
>0

-5

 

T

- 1

20rd

T

29rd

i»
2

Figure 5-25: Phase-averaged vorticity, <w>, profiles for k = 5.2. Dashed lines <w> =
0. Spacing of profiles in figure not to scale.

38rd

48rd

129

-1.1 -0.8 -0.5 02 01.5.1105 0.8 1.1
<w > I szi

    

 

    

L5 o
>1 ,
.5 __
-10 20 . 30 , 40 .

x/rci

1Figure 5-26: Phase-averaged spanwise velocity, <w>, for (11:11 at multiple span

locations. k=5.2.

130

z= rci 2r 10rd 20rd

 

 

 

5% a? i i
$55 57> % é
a: a; % t
<7: % t f?

1‘Figure 5-27: Phase-averaged spanwise fiow, <w>, profiles for k = 5.2. Dashed lines
<W> = 0. Spacing of profiles in figure not to scale.

131

rather a local maximum and minimum spanwise flow level is observed. The positive and
negative spanwise flows were nearly equal in magnitude. As the downstream distance
increased the spanwise flow profiles were similar, though the peak levels changed. The
contour data of Figure 5-26 indicate that the spanwise flow field also skewed into oblong
structures shaped qualitatively similar to the vorticity structures at the same downstream

distance. At 2 = 2rci the spanwise velocity was near zero initially and developed a profile
similar to the z = rci profiles. Peak levels were higher at this span location compared to the
z = rci location.

The spanwise flow fields for z = 10rd and 20rd were significantly different from
those observed for z = rd and 2rd. At 2 = 10rd the spanwise flow was near zero for x <
lSrci after which a single peak, Gaussian like profile was observed. The peak level
increase until x z 40rd after which it decreased. No significant reverse spanwise flow was
observed. For 2 = 20rd the same trend was observed, but the development was further

downstream.
The vortex parameters were determined by tracking the vortex through its
downstream evolution as discussed in section 3.2.4. Figure 5-28 shows a comparison of

several vortex parameters between the no-wall midspan data and the cutting wall 2 = 20rd

(the span location closest to the midspan) data. These data show good agreement between

the two data sets. The circulation was lower, by nominally 2 cmz/s, for the cutting wall
data. This difference was near to the previously discussed uncertainty level in that

measure.

132

In! 1*

II't'
1 _
.

I._

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

50:
E
155°"
\ .
1:40
‘1‘.
A30.
3 :
20:
,. l-
5 r --—-B-—-—- k=5.2, z=20rcl :
1 —e— mldplane,k=5.2 ‘0.
obllllllllllIIIIIIL‘IIIIIILILLLIIIII1 OZLJILIIIIIII-llllllllllllllllllll'lll
-10 0 10 20 30 40 50 -10 0 10 20 30 40 50
x/r,j xlrc,
0.7: 14:
0.5: 12:
i .5“ 2mm; «1: “‘5 *1
0.5 _ mi. 5.1:“ ’ A 10 _ " '
" We“ ‘ ° 2
504_ ”*5 E 0_
v : 3 :
h"0.3: 3° 5:
0.2_ 4-
0.1, 2,
02114111111AILVIIIAiALLALAIlllLlAIJLI OEIIIIIIllIlLlllLllIIlllllllllllllll
.10 0 10 20 30 40 50 -10 0 10 20 30 40 50
x/rc, xlrd

Figure 5-28: Comparison of vortex parameters (F, <w>peak, rc, Uc) from no-wall
midspan data and cutting wall, 2 = 20rd (near midspan) data. k = 5.2. Symbols
common.

The vortex parameters for the k = 5.2 case are shown in Figure 5-28 for multiple 2-

locations. The local vortex parameters, F, <w>peak, rC, U0, and <wpeak> are shown
normalized by the initial vortex parameters, 1",, (oi, rci, szi from the no-wall data

respectively. There was a nominally 5% difference in the circulation values comparing the
midspan and off center values. This was accounted for by day to day variability in the

water tunnel freestream speed which showed a 4% peak to peak variation. The effect of

133

 

 

 

1.2
1.1

0.9

0.8
[—1. 0.7
\ 0.6
0.5
0.4
0.3
0.2
0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.5
L.
.2; 3 “:33
3 E
h >
‘~ \
a.” 0
D ..
1.5
11111 1111 1'1111I1111I111114111I11
.10 0 10 20 30 40 50
x/ rd
0.5 1 g
0.25 0'8;
E0.13 :
0 . .. .. , :
.. 1’ 14:13:51; 377.: "1 awn-m: $22.4? _
_ ”"777“ iii. . " > 0.4 _
a : ‘2 »
\ 2
£0.25 £02 -
-0.5 F v ° :
-0.75 i '0'2 ; k=5.2, z=10rd ”‘
t .04: —v-—— #52, z=2rd
r- . —-‘
.1r-ILIlllllllllllllllllJlllIIIIIIIIL :IILIIIIIIIlllllljilllillllllli:lllllll
-10 0 10 20 30 40 50 -10 0 10 20 30 40 50
x I rei xl re,
Figure 5-29: Vortex parameters (F, <w>peak, rc, Uc, <w>peak9 b/a) for multiple span

locations. k=5.2. Symbols common.

134

 

free stream speed was to change the actual reduced frequency. Figure 3-29 shows the
effect of reduced frequency on the circulation for the midspan data and indicated that the
circulation will be lower for lower reduced frequencies. The circulation was nearly

constant as a function of downstream distance for z = 10rd and 20rd. For 2 = 2rd the
circulation was initially constant, then decreased for x > 20rd. This decrease was caused

by the expansion of the vortex, and vorticity leaving the circulation calculation radius.
The peak vorticity showed significantly different downstream characteristics

depending upon the span location. At z = 2rci the initial rate of decrease in <m>peak was

higher than the no-wall decrease. Several local maximums in <w> ak were observed at x

p6

= 12r 201 27rd, and 34rd. There are two possible explanations for this phenomena.

ci’ ci’
First, a variation in the vortex area (i.e. a vortex stretching effect) would increase or
decrease the vorticity levels. There was no measurable change in the vortex core radius at
the spatial locations of the local maximums of minimums which suggested this was not
the cause. A second possible explanation was that the deformation and re-organization of

the vortices created increases and decreases in the peak vorticity level. Given the dynamic

break-up of the vortex observed at z = 2rd (Figure 5-23) this seams to be a more plausible
explanation. The z = 10rd data indicated that peak vorticity initially followed the same
decreasing rate of the midspan data. However for x > 10rd the rate of decease was again
higher until x z 25rd after which <w>peak was nearly constant. At z = 20rd <w>peak
followed the midspan trends though the rate of decrease was higher for x > 30rd. Core

radius measurements show that the core radius increases at all span locations with the rate

of increase being higher closer to the wall. The vortex spacing ratio b/a was nominally

135

 

 

constant at z = 20rd and slightly increasing at z = 10rd. Recall that the vortex location was

defined by the location of the centroid of the vorticity, which better accounts for vortices

like those observed in Figure 5—24 at z = 2rd. At 2 = 2rd the spacing first decreases then

was nominally constant as the axial flow developed in the vortex core. The peak spanwise

flow initially increased from zero. Small overshoots in <wpeak> were observed after the

axial flow was initiated at all three measurement locations. The highest spanwise flow was

measured at z = 2rd.
The relative location of the centroid of the vorticity, xc, and spanwise flow, xw, is

shown in Figure 5-30. This figure shows that the relative locations were within a core

 

 

 

 

 

 

 

 

 

   

 

 

 

 
 
   
   

k=5.2, 2820!;

k=5.2, 2310!“c

k=5.2,z=2rc . . .
IILALILJL .2llllllllLllllLlllll-llLLLlll

o 10 20 so ‘ 40 so 0 1o 20 so 40 so
x I r¢i x I re,

 

 

 

 

 

 

 

 

 

Figure 5-30: Relative location of the peak vorticity to the peak spanwise flow, k = 5.2.

radius with a maximum difference observed at z = 2rd. The relative locations at z = lOrC
were nearly zero for x < 30rd after which the relative locations were no longer coincident,
though to a lesser degree than was observed for z = 2rd. The relative locations were nearly

zero for z = 20rd.

I36

 

 

 

5.2.3 Phase Averaged Flow Field, k=11.5.

Contour plots of the phase averaged peak vorticity, <w>, and corresponding line

profiles are shown in Figure 5-31 and Figure 5-32 respectively. At 2 = re, and 2rd the

vortices changed from Gaussian (x < 0), to non-Gaussian ( x > O) and then break-up into

multiple regions of vorticity (x > 15rd). Further away from the wall at z = 10rd and 20rd

the vortices also change from Gaussian to non-Gaussian profiles with the break-up of the

vortices was observed for the z = re, and 2rd, also observed further downstream for z =
10rd. The peak levels at z = 20rd decreased with increasing downstream distance and the

size of the vortices appeared to expand.
The spanwise flow field, <w>, is shown in contour form in Figure 5-33 with lines

profiles in Figure 5-34. The structure change from isolated to “single band” for z = rci, 2rd
and 10rd. Of particular note is that the spanwise flow was nearly zero for x > 48rd at z =
rci. At z = 20rd the spanwise flow enters the flow field with Gaussian like profiles, but
change to multi-lobed profiles for x > 30rd. This may be a result of influence of the axial

flow from the far cutting wall and the proximity to the midspan of the plane. Reverse

spanwise flow was measured at all span locations except 2 = 20rd with the highest levels

close to the wall. The spanwise velocity profiles in Figure 5-34 showed good qualitative
agreement with profiles from the line tagging experiments. An example is the velocity

profile at x = 19rd and z = rci of Figure 5-34 compared to the near wall profile (2 = 0.6rci)

of Figure 5-3.

137

 

 

 

   
    

 

Illllllllllll‘lllll

 

 

IllllllllllllllllII

x/rci

Figure 5-31: Phase-averaged vorticity, <w>, for ¢=0 at multiple span locations.
k=11.5.

138

1‘: rci

<w> I (1) x = 9rCi

-Q2 0 0:2 0:4 0:6
% l9rci
30rd
I
l

381‘ci

N
2"!
p—t
c
27
N
G
9"!

 

 

“‘17
“‘3’

50rd

War—v—

Figure 5-32: Phase-averaged vorticity, <w>, profiles for k = 11.5. Dashed lines <w> =
0. Spacing of profiles in figure not to scale.

[39

-1.1 -0.8 -o.5 -o.2 0.5 V0.5 0.8 1.1
<w > / sz,

 

    

5
s.“ o
>
.5_
'10 1o ' 20 7' 3o ”so
xlrc,

Figure 5-33: Phase-averaged spanwise velocity, <w>, for ¢=0 at multiple span
locations. k=11.5.

140

 

z: rci 2rd 10rd 20rd

 

 

I V ._
43v; o.2"6'.4 x " 9rd
L 19rd
I E F %
I 30rd
' ? % F
I
I I I I
38rd
I I
I I I
I
I I I I
I 50rd
I I I I
I I I
I I
I I
I I
I I I
I I I

Figure 5-34: Phase averaged spanwise flow, <w>, profiles for k = 11.5. Dashed lines
<w> = 0. Spacing of profiles in figure not to scale.

I41

 

 

 

II

31

A comparison of the vortex parameters between the midspan no-wall experiments

and near center span (2 = 20rd) cutting wall experiments are shown in Figure 5-35. The

data show good agreement between the data sets. The peak vorticity levels differ by about
5%. It was expected that the peak vorticity level would be smaller for the cutting wall data
because the data density was slightly smaller in the cutting wall data sets. However, from
Cohn and Koochesfahani (2000) this difference was expected to be less than 3%. It was
shown in section 3.2.4 that vortex stretching due to a small spanwise flow gradient did
affect the peak vorticity level significantly and it is possible that for this high reduced
frequency case, the effects of the axial core flow were already apparent in the peak
vorticity at this location.

The vortex parameters are shown in Figure 5-36. The normalization parameters
are the same as used in the k = 5.2 discussion. The circulation data showed lower values
of the circulation in the initial region (over-lapping with the midspan data) as was
observed for the k = 5.2 case for the same reasons. The circulation decreased over the

measurement extent for z = 2rd. The rapid decrease in circulation for x > 20rd was caused

by the break-up of the vortices into multiple structures, and the subsequent motion of the
smaller structures away from the tracked largest structure. The break-up of the vortices at

z = lOrC was indicated by the decrease in circulation for x > 35 rd.

Peak phase averaged streamwise vorticity decreased with increasing downstream
distance at all span locations. The peak vorticity showed local maximums, or spikes, for z

= 2rd as was observed for the k = 5.2 case, however, for this reduced frequency there were
also local peaks for z = 10rd and 20rd span locations. Discussion on this point to follows

in the next paragraph. The local vortex core radius, rc, increased with increasing

I42

 

 

 

 

 

 

 

 

 

60
’ Wm 54mm ' ”When
50
40
a C
a?“ E
5 so
v .
L‘ .-
20
» -—-—+:—3—-— Io-11.5.1820rd
C . —9— midplano.kn11.5
10
0’111.1111...111141441.1..111....1H

 

 

 

-1 0 0 10 30 40 50

 

20
X] 'a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

’ 14’ a _.-.-. ,
~ : 1m": ”awn.
. - a:
0.6 4,5? T is. 12 _
’ .. a" .«;' C
b J.“ .-.v,:.sv D
:E' 10
- tar-3* A :
A '- ' u .
g 0.4 E 8 _
V ' 3 :
0 0 _
h I.
3 8 _,
r :
0.2 4 -
2
ollllllllllllllllllllJllllll‘lillllj oillllllllljllllllllllllllllllllllll
-1o 0 1o 20 so 40 so -10 o 10 20 so 40 so
x l rd x I rd

Figure 5-35: Comparison of vortex parameters (F, <m>peak, rc, Uc) from no-wall
midspan data and cutting wall, 2 = 20rd (near midspan) data. k = 11.5. Symbols
common.

downstream distance with the rate of increase higher closer to the wall. The drop in the

core radius for z = 2rd at x = 15rd was caused by the break-up of the vortex into multiple

structures as noted for the circulation recalling the core radius was calculated using (16).
The spacing ratio b/a showed an initial roughly constant region for all three span

locations. At x = 15rd b/a decreased and eventually changed sign for the z = 2rd

measurement plane. This sign change indicated a change in origin of the vortex array,

143

1.2
1.1

0.9
0.8
0.7
\ 0.3
(- 0.5
0.4
0.3
0.2
0.1

     

 

 

 

VVTY IIVrV

 

 

 

 

Til! IIIY IVTY TYIr'I‘l.

 

llPlllllJllllllllllllllJlL

10 so so 40 so
xlrd

 

 

 

 

 

 

YU‘I 7111' III!

   
 

 

 

 

 

   

    

 

 

 

k=11.5, z=20rd
k=11.5,z=10rd :
k=11.5,z=2rd :
k=11 : .. .
"l1 llllllilllllepllllll
0 1o 20 so 40 so
x/g xlg

Figure 5-36: Vortex parameters (I‘, <w>
locations. k = 11.5. Symbols common.

peak, rc, Uc, <w>mk, b/a) for multiple span

144

switching from a velocity excess to a velocity deficit consistent with the Ii and (7) data in
section 5.2.1. Careful observation of the phase resolved data in Figure 5-31 also show this

trend. Further from the wall for z = 10rd the vortex array orientation switch was initiated
at x = 35rd indicating that further downstream the array would also change orientation.
The peak spanwise velocity showed the characteristic overshoot in <w>peak as the
spanwise flow was initiated. The highest level was measured at z = 2rd, though the levels
were still increasing for z = 10rd.

The local spikes in <m>peak are show in Figure 5-37 for z = 2rd and 10rd. Two

k=11.5, z=10rcl
——-V— k=11.5, z=2rc,

 

0 1o 20
xlrd

Figure 5-37: Peak vorticity, <w>Peak, as a function of downstream distance for k =
11.5. Curves shown are vertically offset to allow simultaneous viewing.

local spikes (indicated by the arrows pointing to solid lines solid lines) were observed at
both span locations. A weaker spike (indicated by the arrows pointing to the dashed lines)

was also observed. These data indicated the local vorticity spikes were not isolated

145

 

incidents but events that convected away from the wall through multiple measurement
planes.

Figure 5-38 shows re, <w>peak and Uc plotted on the same figure along with
<w>peak for z = 2rd, 10rd and 20rd. The vertical axis of the curves are offset to allow the
curves to be displayed on the same figure without significant overlapping. At 2 = 2rd the

first local minimum in the peak vorticity (solid line) corresponds to an increased vortex
area, the characteristic spike in the spanwise flow velocity, and a decrease in the
convection speed. The following spike in the vorticity (dashed line) correlated spatially
with the undershoot in the spanwise flow, a decrease in the vortex area and a positive spike
in the convection speed. The correlation for the subsequent vorticity spikes was less clear,
however, at z = lOrci the spatial correlation was evident for the first three vorticity spikes.
The vortex core radius was least likely to show correlated local maximums and
minimums. The core radius variation for the first local peak was relatively small, 4% or
0.02 cm. This level was significant, but near to the noise level in the radius estimation.
The subsequent vorticity peaks were less intense, and the corresponding area change was
also less making it more difficult to observe. At 2 = 20rd the correlation between the
vorticity and spanwise velocity was clear for the first two vorticity spikes. A surge in the
convection speed appeared to lag spatially behind the initial vorticity and spanwise flow
spikes. No change in the vortex core radius could be reliably detected at this span location.

The relative spatial locations of the vortex core (defined by the centroid of the

vorticity) and <w>peak is shown in Figure 5-39. The locations were initially coincident,

but separated as the downstream distance was increased. There were small oscillations in

146

 

z = 20rd

 

z = 10rd

 

Z = 2rd

 

xlr

cl

Figure 5-38: Spatial correlation of initial local vorticity peak to re, Uc, <w>peak,

k=11.5. Vertical major grids are 0.1 apart. Magnitudes for the vertical axis can be
found on Figure 5-36.

147

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

" E u k=1 1 .s, 2:205, " _
I- “ - = r-
4.5, , k—11.5,z 10rcl 4.5
, v k=11.5, z=2r‘=l Z . .
_211111111111111]114L 111111; .211111111L1114_1_111_11l14_LLlL1
II 10 20 30 4O 50 0 10 20 30 40 50
xlrd xlr,I

Figure 5-39: Relative location of the peak vorticity to the peak spanwise flow, k=11.5.

the spatial location observed which may indicate small motions of the axial flow within

the vortex core though the locations remained within a core radius of each other.

5.2.4 Area Varying Waves 0n the Vortex Cores

Evidence for area varying waves on the vortex cores has been presented
throughout this chapter. This evidence was first given in the line tagging data with the
spanwise velocity spikes that followed the axial flow front boundary. These data showed
waves were present for all reduced frequency cases except the lowest (A = 2°, k = 4.1).
The trends in the wave speed were consistent with the calculated equivalent speed of
sound. The more detailed stereoscopic data showed the spanwise velocity spikes were
correlated to spikes in the peak vorticity and a change in the vortex core radius of 5% for
the high k case. The low reduced frequency case did not show evidence of area variation

or spikes in the vorticity to within the measurement resolution.

148

 

As already discussed, Lundgren and Ashurst (1989) presented equations for area
varying waves on a vortex core that were analogous to the gas dynamics equations. In that

work, the wave propagation speed, Vw, was defined as:

A2
V =w1+c 1n——-(1+A /A
e A 1

w 1 (20)

2)

where w] is the velocity in front of the wave relative to the velocity behind the wave (i.e.
the velocity change across the wave). A1 and A2 are the vortex areas downstream and

upstream of the wave. For small disturbances A1 = A2 and equation (20) simplifies to:

Vw = w1+ce (21)

where C6 is the previously defined “equivalent speed of sound”. In the limit of no axial
flow upstream of the wave w] = O and VW = ce. The low reduced frequency case was
consistent with this equation (i.e. there was no change in the vortex area observed and the
wave speed was approximately equal to the equivalent sound speed).

The high reduced frequency case, k = 1 1.5, showed an area variation of nominally
5%. The wave speed was calculated using (20) by substituting experimental values for the
area ratio, speed of sound, and relative upstream axial flow velocity. For the high reduced
frequency case cc = 15.1 cm/s, A1/A2 = 0.95 and w] = 0.65 cm/s. Using (20) VW = 16.16
which compares well to the VW = 15.95 (from the line tagging data) and VW = 16.4 (from
the stereoscopic data). The analytic work does, therefore, appear to describe the
experimental data to first order.

The high reduced frequency data also showed a correlation between the area

varying waves and a change in the vortex convection speed, Uc. In this work UC was

149

calculated by differentiating the downstream vortex core location versus time. A local
maximum (or minimum) in the convection speed is also the same as a local surge (or lag)
in the downstream location of the vortex core. Maxworthy et al. (1983) indicates that an
area varying wave with area ratio greater than 20% will cause an unstable helical or spiral
disturbance behind the wave to be generated. For smaller area ratios a series of decreasing
amplitude waves will exist. For k = 11.5 (area ratio of 5% rather than 20%), the first
primary wave was followed by a series of smaller amplitude disturbances consistent with
the results of Maxworthy et al. (1983). The line tagging data, for cases with waves, and
stereoscopic data, high k case, also appeared to have smaller axial flow spikes (i.e. waves)
behind the initial wave consistent with the Maxworthy et a1. (1983) conclusions.
Additionally, a small amplitude displacement, evidenced by the local maximums and
minimums in the convection speed, was generated. This kink in the vortex is consistent
with schematics of axisymmetric waves in Maxworthy et a1. (1983) and may be the

precursor to the unstable waves discussed.

150

"“l
.

lla-
I

r:
1

Chapter 6

CONCLUSIONS
In this work the velocity and vorticity fields in the wake of an oscillating NACA-

0012 airfoil were measured with the goal of better understanding the development of a
vortex array with and without axial flow, as well as the characteristics of the axial flow if
present. Molecular tagging techniques were the primary experimental tool for this
purpose.

Planar MTV was used to detail the flow field (from which the vorticity field was
calculated) at the tunnel midspan, from the trailing edge of the airfoil through the first
chord of development. Cutting walls were placed into the flow field half a chord
downstream of the trailing edge of the airfoil to initiate spanwise flow within the core of
the vortices (axial core fiow). Single and multiple line tagging experiments were
performed to detail the spanwise flow with and without cutting walls. Stereoscopic MTV
was developed and used for the first time in this work. This technique allowed the
complete velocity vector, in the plane of the laser sheet, to be measured which in turn
allowed the axial core flow and vorticity field to be measured simultaneously.

Results from the planar midspan data without cutting walls showed several
interesting features. The unsteady Kutta condition was found to not be valid for high
reduced frequencies in this flow field. Reverse flow along the surface of the airfoil was
observed using molecularly based flow visualization techniques. The vortex array
switched vertical spatial orientation with increasing reduced frequency indicating a switch
from mean drag to mean thrust for the oscillating airfoil. Mean force calculations from the
data show that the use of the mean streamwise velocity field overestimated the mean

thrust force. The vortices in the array were aligned for k z 6, which would signify a zero

151

 

net thrust/drag using the mean profile only. The zero net thrust case was found to exist for
k z 8 using a more complete calculation of the thrust force developed from the Navier-
Stokes equations.

The formation and development of the vortex array was governed, in part, by the
reduced frequency. For k = 4.2 the boundary layer vorticity left the trailing edge of the
airfoil and then formed isolated vortices downstream. For k > 5 the vortices were formed
at, and possibly upstream of the airfoil trailing edge. Fluid from one surface was
transported to the other surface during the formation process. The peak vorticity,
circulation, and convection speed were functions of the reduced frequency and airfoil peak
oscillation amplitude while the vortex core radius was nominally a function of angle of
attack only.

Results of the line tagging data without cutting walls showed that spanwise flow
was generated because of the vortex / tunnel side-wall interaction. The magnitude of the
spanwise velocity was on the order of the swirl velocity. The spanwise flow was initially
distributed in two isolated structures for each phase. The structure changed into a single
band of spanwise flow through the entire phase for moderate to high reduced frequencies.
No transition from multiple to single structures was observed for the lower reduced
frequency cases.

Line tagging data were also taken for the cutting wall case. These data showed that
the axial flow from the vortex cutting wall interaction propagates away from the cutting
walls. Area varying waves were also present near the axial flow front and were also found
to propagate away from the cutting walls. The speed of the area varying waves was found

to be slower than the front speed indicating the area varying waves increasingly lag the

152

 

front boundary. The span location of the highest spanwise flow at given downstream
distance was found to move away from the wall with increasing downstream distance. The
peak spanwise velocity showed an overshoot after the front passed. Reverse spanwise
flow was measured for all reduced frequency cases investigated. The structure change
observed for the no-wall case was also observed for the cutting wall case.

Two reduced frequency cases, k = 5.2 and 11.5, were investigated using
stereoscopic MTV. These data revealed many details of the flow field with the cutting
walls. The mean streamwise flow field was found to change from a velocity excess to a
velocity deficit for increasing downstream distance in the wake for the high reduced
frequency case. The structure of the vortices near the wall broke up into multiple regions
of vorticity with the profile becoming non-Gaussian and asymmetric. Hollow vorticity
profiles were found for several conditions, though the evolution from Gaussian to hollow
was not a result of decreasing peak vorticity, but a distortion and re-distribution of the
vorticity. Further away from the wall the vortices remained somewhat Gaussian with a
decreasing peak level.

Spanwise flow in the core of the vortices was observed with reverse spanwise flow
measured outside of the vortex core. The distribution of the spanwise flow within the
vortex core was significantly different depending upon the span location and varied from
Gaussian like, to multi-lobed, to non-specific. The transition of the axial flow structure
from isolated to band like, as was noted in the line tagging data, was also observed in the
sMTV data for the high reduced frequency cases. The sMTV data showed that this change
in structure occurred over the same region as the break-up of the vortices was observed.

Spikes in the peak vorticity were observed to move away from the cutting walls as the

153

 

vortices convected downstream. These spikes appeared to be well correlated to spikes in
the peak spanwise core flow and the convection speed which also moved away from the
cutting walls. A change in the vortex core radius accompanied the first spike in the peak
vorticity for the highest reduced frequency cases indicating that an area varying wave was
initiated along with the axial core flow. A characteristic signature of the initiation of axial
flow within the vortex core was found to be a local decrease, followed by increase, in the

vortex core convection speed.

154

Chapter 7

REFERENCES

Anderson, J.M., Streitlien, K., Barrett, D.S., Triantafyllou, M.S., “Oscillating Foils of
High Propulsive Efficiency”, Journal of Fluid Mechanics, vol 360, pp. 41-72, 1998.

Bodewadt, U.T., Die Drehstromung Uber Festem Grund, ZAMM, Vol. 20, 1941

Burggraf, O.R., Stewartson, K., Belcher, R., “Boundary Layer Induced by a Potential
Vortex”, Physics of F luids, Vol. 14, No. 9, pp. 1821-1833, 1971.

Cohn, R,K, Efi‘ects 0f Forcing on the Vorticity Field in a Confined Wake, Ph.D.
Dissertation, Michigan State University, 1999.

Cohn, R., Koochesfahani, M., “Effect of Boundary Conditions on Axial Flow in a
Concentrated Vortex Core”, Physics of Fluids A, Vol.5 No.1, pp.280—282, January 1993.

Cohn, R., Koochesfahani, M., “The Accuracy of Mapping Irregular Spaced Velocity Data
onto a Regular Grid and the Calculation of Vorticity”, Experiments in Fluids, 861-869,
2000.

Conlisk, A.T., “Modern Helicopter Aerodynamics”, Annual Review of Fluid Mechanics,
Vol.29, pp. 515—567, 1997.

Donaldson, C., Solutions of the Navier-Stokes Equations for Two and Three Dimensional
Vortices, Ph.D. Dissertation, 1956.

Donaldson, C., Sullivan, R., “Examination of the Solutions of the Navier-Stoke Equations
for a Class of Three-Dimensional Vortices Part 1: Velocity Distributions for Steady
Motion”, AF OSR TN 60-1227, 1960.

Gendrich, C.P., Dynamic Stall of Rapidly Pitching Airfoils: MTV Experiments and Navier-
Stokes Simulations, Ph.D. Dissertation, Michigan State University, 1999.

Gendrich, GR, and Koochesfahani, M.M., “A Spatial Correlation Technique for
Estimating Velocity Fields Using Molecular Tagging Velocimetry (MTV)”, Exp. Fluids,
vol. 22, 1996, pp. 67-77.

Gendrich, C.P., Koochesfahani, M.M., and Nocera, D.G, “Molecular Tagging Velocimetry
and Other Novel Applications of a New Phosphorescent Supramolecule”, Exp. Fluids,
vol. 23, 1997, pp. 361-372.

Giesing, J. P., "Vorticity and Kutta Condition for Unsteady Multi-energy Flow,"
Transactions of ASME, Journal of Applied Mechanics, Vol. 36, pp. 608-613, 1969.

155

i‘“"—'-“"I

Hagen, J ., Kirosaka, M., “Corewise Transport in Hairpin Vortices- The ‘Tornado Effect’”,
Physics of Fluids A, Vol.5 No.12, pp.3167-3174, 1993.

Hassan, A.A, Tung, C., Sankar, L.N., “Euler Solution for Self-Generated Rotor Blade-
Vortex Interactions”, International Journal for Numerical Methods in Fluids, Vol. 15, pp.
427-452, 1992.

Hill, R.B., Klewicki, J.C., "Data Reduction Methods for Flow Tagging Velocity
Measurements", Experiments in Fluids, Vol. 20, pp. 142-152, 1996.

Hirsa, A., Lopez, J.M., Kim, 8., “Evolution of Initially Columnar Vortex Terminating
Normal to a No-Slip Wall”, Experiments in Fluids, Vol.29, pp.309-321, 2000.

Karman, T. von, “Uber Laminare und Turbulente Reibung, ZAMM, Vol. 1, pp. 233-252,
1921.

Katz, J ., Weihs, D., “Wake Rollup and the Kutta Condition for Airfoils Oscillating at High
Frequency”, AIAA Journal, Vol. 19, No. 12, pp. 1604-1606, 1981.

Khalid, A., “Pressure and Drag Force Determination on an Oscillating Airfoil Submerged
in Fluid”, Masters Project Research Report, Michigan State University, 2001 .

Koochesfahani M.M., “Molecular Tagging Velocimetry (MTV): Progress and
Applications”, 30th AIAA Fluid Dynamics Conference AIAA99-3786, June 28 - July 1,
Norfolk, Virginia, 1999.

Koochesfahani, M.M., “Vortical Flow in the Wake of a Pitching Airfoil”, International
Conference on Engineering Applications of Mechanics, June 1992.

Koochesfahani, M.M., “Vortical Patterns in the Wake of an Oscillating Airfoil”, AIAA
Journal, Vol.27, No.9, pp. 1200-1205, Sept. 1989.

Koochesfahani M.M., Cohn, R.K., Gendrich, C.P., Nocera, D.G “Molecular Tagging
Diagnostics for the Study of Kinematics and Mixing in Liquid Phase Flows”, Proceeding

of the Eighth International Symposium on Applications of Laser Techniques in Fluid
Mechanics, July 8-11, 1996, Lisbon, Portugal

Kotidis, P.A., “Unsteady Radial Transport in a Transonic Compressor Stage”, MIT Gas
Turbine Lab Report #199, September, 1989.

Kuo, H.L., “Axisymmetric Flows in the Boundary Layer of a Maintained Vortex”, Journal
of the Atmospheric Sciences, Vol. 28, pp.20-41, January, 1971.

Krishnamoorthy, S., and Marshall, J., “An Experimental Investigation of ‘Vortex
Shocks’”, Physics of Fluids, Vol. 6, No. 11, pp. 3737-3741, 1994

156

 

 

Laursen, T.S., Rasmussen, J.J., Stenum, B., Snezhkin, E.N., “Formation of a 2D Vortex
Pair and its 3D Breakup: an Experimental Study”, Experiments in Fluids, Vol.23, pp.29-
37, 1997.

Lighthill, J ., Mathematical Biofluiddynamics, SIAM, Philadelphia, 1975.

Lopez, J.M., “Axisymmetric Vortex Breakdown Part 1. Confined Swirling Flows”,
Journal of Fluid Mechanics, Vol. 22], pp.553-576, 1990.

Lundgren, T.S., Ashurst, W.T., “Area-varying Waves on Curved Vortex Tubes with
Application to Vortex Breakdown”, Journal of Fluid Mechanics, Vol. 200, pp. 283-307,
1989.

Marshall, J., Krishnamoorthy, 8., “On the Instantaneous Cutting of a Columnar Vortex
With Non-Zero Axial Flow”, Journal of Fluid Mechanics, 1997.

Marshall, J., Yalamanchili, R., “Vortex Cutting by a Blade, Part H: Computations of
Vortex Response”, AIAA Journal, Vol. 32, No. 7, pp. 1428-1436, 1994.

Maskell, E. C., "On the Kutta-Joukowski Condition in Two-Dimensional Unsteady Flow,"
unpublished note, Royal Aircraft Establishment, Famborough, England, 1972.

Maxworthy, T., “On the Structure of Concentrated, Columnar Vortices”, Astronautica
Acta, Vol. 17, pp. 363-374, 1972.

Maxworthy, T., “A Vorticity Source for Large-Scale Dust Devils and Other Comments on
Naturally Occurring Columnar Vortices”, Journal of Atmospheric Sciences, Vol.30,

pp.1717-1722, 1973.

Maxworthy, T., Mory, M., Hopfinger, 13.1., “Waves on Vortex Cores and Their relation to
Vortex Breakdown”, AGARD CPP-342, paper 29, 1983.

McCrosky, W. J., “Unsteady airfoils," Annual Review of Fluid Mechanics, Vol. 14, pp.
285-311, 1982.

Platzer, M., Neace, K., “Aerodynamic Analysis of Flapping Wing Propulsion”, AIAA
Paper 93-0484, 1993.

Poling, D.R., Telionis, D.P., “The Response of Airfoils to Periodic Disturbances- The
Unsteady Kutta Condition”, AIAA Journal, Vol. 24, No. 2, pp. 193-199, 1986.

Rockwell, D., “Vortex Body Interactions”, Annual Review of Fluid Mechanics, 1998.

Rott, N ., Lewellen, W.S., “Boundary Layers and Their Interactions in Rotating Flows”,
Progress in Aeronautical Sciences, Pergamon Press, Vol. 7, pp. 1 1 1, 1966.

157

 

 

Schlichting, H., Boundary Layer Theory, McGraw-Hill Book Company, 1960.

Solof, S.M., Adrian, R.J., and Liu, Z-C, “Distortion Compensation for Generalized Stereo-

scopic Particle Image Velocimetry”, Measurement and Science Technology, 8, 1997,
pp. 1441-1454.

Stanek, M. J. and Visbal, M. R., "Study of the Vortical Wake Patterns of an Oscillating
Airfoil," AIAA Paper No. AIAA-890554, 1989.

Streitlien, K., Triantafyllou GS, “On Thrust Estimates For Flapping Airfoils”, Journal of
Fluids and Structures, Vol. 12, pp. 47-55, 1998.

Wan, C.A., Chang, C.C., “Measurement of the Velocity Field in a Simulated Tornado Like
Vortex Using a Three Dimensional Velocity Probe”, Journal of the Atmospheric Sciences,
Vol. 29, pp.1 16-127, January 1972.

Westerweel J., and Nieuwstadt F.T., “Performance Tests on 3-Dimensional Velocity Mea-

surements with a Two Camera Digital Particle Image Velocimeter”, Laser Anemometry,
Vol. 1, 1991. [313349-355

Wiegand, A., The Response of a Vortex Ring to a Transient Cut, Ph.D. Dissertation,
University of California Sandiego, 1993.

Wilder, M.C., Mathioulakis, D.S., Poling, D.R., Telionis, D.P., “The Formation and

Internal Structure of Coherent Vortices in the Wake of a Pitching Airfoil”, Journal of
Fluids and Structures, Vol. 10, 1996, pp.3-20.

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APPENDIX A
MEAN FORCE EQUATION DEVELOPMENT

Given a submerged body in a flow field, determine an equation that can be used to
calculate the mean force from the velocity field. For this analysis the flow is assumed to be

incompressible, 2-D, and steady in the mean. Figure A.1 shows the control volume used

 

 

 

m3v3
/ U0, Pm
....................... 1
(2): -._.
1100,1309 I
I—°-
I—'-

 

 

Figure A-l: Airfoil mean drag formulation control volume.

for the following formulation. The integral linear momentum equation can be written as:

$1 VpdV+ ijW-mdA = 2F (Al)
CV CS

 

For this problem the x-component is considered and the forces are composed of the drag

159

force and the pressure forces on the control surfaces:

2F, = D+ I Pds (A2)

CS
Consider first the left hand side of (A1). If the flow is steady (i.e. the mean flow is not
changing with time), the first term in (A1) is zero. The second term can be expanded to for
the four control surfaces indicated as:
A 2 2

IVp(V-n)dA = juuppdAl—ju (y)pdA,—ju3pv3dA3—jU4pv4dA4 (A3)
CS 1 2 3 4

Assume that U U3, V3, U4, V4 are constants and that V3 = -V4 , U3 = U4. The mass flux

Up’
across the top and bottom surfaces is defined as

n13 = pV3A3 (A4)

P V4A4 (A5)

”’4

The mass flux across the upstream and downstream surfaces is defined as:

"i. = Ipr..,.d.v (A6)
1

vi. = Ipbuowy (A7)
2

Since the flow is assumed to be 2-D, the span length may be removed from the areas and

differential areas. The conservation of mass for this control volume indicates that
m1 = mz+m3+m4 (A8)

Combining (A4)-(A8) and (A3) and simplifying yields:

160

 

 

I Vp(V-n)dA = pb[IUiI,dy-JU2(y)dy+[— jU,,,,dy—ju(y)dyju3] (A9)
CS 1 2

l 2

It is further assumed that U3 2 U therefore

up,

I vp(v.h)dA = pr Uuptt(_v)d_v— I 112mm,] = DTM (A10)
CS 2 2

Equation (A10) is important in that if Uup = U00 and the contribution from the fluctuating

velocities are not important (A10) is the force as noted. The force coefficient is

Cm = ZKL-lmXM—ljdy (All)

C Uup Uup

With the current data set, the 2-D flow field is know. The mean and RMS
quantities for U and V can be calculated and used to formulate a more complete thrust
coefficient. Let the velocity be broken into a mean and fluctuating portion as:
u(1)=('fi+u’) (A12)
The fluctuating quantities enter into the drag formula via (A10) (uRMS) and the pressure in

(A2) (vRMS). Consider first the effects of uRMS. Substituting (A12) into the RHS of (A10)

and taking the average yields

[WV-mm = pb[IUu,,u(y)dy—Iu2(y)dy—Iu)2ms(y)dy] (A13)
CS 2 2 2

The vertical pressure distribution, P(y) is composed of the hydrostatic pressure
distribution plus the dynamic pressure. Since the hydrostatic pressure distribution is the
same for surfaces (1) and (2) it is neglected in the following analysis. Using the y-
component Navier-Stokes (N-S) equations a formula the y-pressure distribution can be

found. The y component N-S can be written as:

161

 

 

(A14)

 

2 2 2
mamas—-195 .dv dv dv
at 8x By dz — pay dxz CIVZ dZZ

It was shown in Khalid (2001) that contributions from the x derivatives are small. The flow

is assumed to be 2-D and steady which simplifies (A14) to:

2
av ldP d v
_ -_- --_ +v — A15

Let v(t)=(V+v’) and substituting into (A15) yields:

2 2
—a—(1‘22+ 29v’+ v,2) = —1E+v izv+—a—2v' (A16)
3y Pay ay ay
The mean of (A16) is
8P _ 82- a -2 2
Since V 2 0, the (A17) reduces to
8P 3
8_y = —P§;(V12QMS) (A18)

Integration of (A18) gives the pressure distribution in the y-direction on surfaces (1) and

(2) of the control volume. For surface (1) VRMS is small so that P = const. Integration of

(A18) from the freestream to an arbitrary y location on surface (2) yields:

P(y)—P... = woman—vim...) (A19)

Assuming that the freestream RMS level is small the pressure distribution is:

P(Y) = Poo—DV12QM5(Y) (A20)

Equation (A20) can be substituted into (A2) and (A10) to get

162

 

DFP = pb[lv,.,,u())d.v- Ju2(_v)d.v - Juimow— [(122,456) — was] (A21)
2 2 2 2

where AP is the pressure difference between the freestream pressure for surface (1) and
(2). For this work, that pressure difference is significant since the measurements do not
extend to where the freestream returns to the upstream speed. AP is found using the

Bernoulli Equation by:

AP = 123w 2—U,,,2) (A22)

up

Note that AP is a constant. Finally the force coefficient is:

 

2 ‘ 2 , _
CFP = 2[I(%v_))(l;j(—V) — 0"." + .IfiR—Mgfldy _ .IWRMSO) Ade] (A23)

- 2
L up up U U

up up

163