, JPL PUBLICATION 85-72-

•

Effect of FloW Oscillation^ on".

Cavity Drag and ;a Teehniq;U6 for

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|OSCILLATIONS ON CAVIT Y DfiAG AND A TECHNIQUE

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JPL PUBLICATION 85-72

Effect of Flow Oscillations on

Cavity Drag and a Technique for

Their Control

M. Gharib

A. Roshko

V. Sarohia

September 30, 1985

Prepared for

U.S. Army Research Office

Through an agreement with

National Aeronautics and Space Administration

by

Jet Propulsion Laboratory

California Institute of Technology

Pasadena, California
The research described in this paper was carried out by the Jet

Propulsion Laboratory, California Institute of Technology, and

was sponsored by the U.S. Army Research Office and the National

Aeronautics and Space Administration.

Reference herein to any specific commercial product, process, or

service by trade name, trade mark, manufacturer, or otherwise,

does not constitute or imply its endorsement by the United States

Government, or the Jet Propulsion Laboratory, California

Institute of Technology.

The view, opinions, and/or findings contained in this report are

those of the authors and should not be construed as an official

Department of the Army position, policy, or decision, unless so

designated by other documentation.
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2. GOVT ACCESSION NO, 3. RECIPIENT'S CATALOG NUMBER

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4. TITLE (end Subtitle) 5 TYPE OF REPORT. & PERIOD COVERED

EFFECT OF FLOW OSCILLATIONS ON CAVITY DRAG

AND A TECHNIQUE FOR THEIR.CONTROL -

6. PERFORMING ORG. REPORT NUMBER

7. AUTHORO)

M. GHARIB, A. ROSHKO, AND V. SAROHIA

B. CONTRACT OR GRANT NUMBERfs.)

9. PERFORMING ORGANIZATION NAME AND ADDRESS

JET PROPULSION LABORATORY

CALIFORNIA INSTITUTE OF TECHNOLOGY

4800 OAK GROVE DRIVE, PASADENA. CA 91109

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AREA ft WORK UNIT NUMBERS

U.S. ARMY RESEARCH OFFICE

CONTRACT No. 17134E

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U. S. Army Research Office

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12. REPORT DATE

SEPTEMBER 30, 1985

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The view, opinions, and/or findings contained in this report are

those of the author(s) and should not be construed as an official

Department of the Army position, policy, or decision, unless so

Ri cmafpfi hv rvl-ViC'r HnriiTnpnf'flf i nn .

19. KEY WORDS (Continue on revert* aid* It necesemry and Identify by block number)

Aerodynamics, Cavity Flows, Active Fluid Control

20. ABSTRACT fCooOaum ma, r*-nrmm Identify by block number;

Experiments to relate the state of the shear layer to cavity drag have

been performed in a water channel using a 4" ax i symmetric cavity model. Detailed flow measurements in various cavity flow oscillation phases, amplitude

amplification along the flow direction, distribution of shear stress, and other

momentum flux obtained by laser Doppler velocimeter are presented. Measurements show exponential dependence of cavity drag on the length of the cavity.

A jump in the cavity drag coefficient is observed as the cavity flow shows a

» JAN 73

EDITION OF » NOV 65 IS OBSOLETE

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bluff body wake type behavior. Natural and forced oscillations are introduced

by a sinusoidally heated thin-film strip which excites the Tollmein-Schlichting

waves in the boundary layer upstream of the gap. For a large gap, self-sustained periodic oscillations are observed,while for smaller gaps, which do not

oscillate naturally, periodic oscillations can be obtained by external forcing

through the strip heater. The drag of the cavity can be increased by one order

of magnitude in the non-oscillating case through external forcing. Also, it is

possible to completely eliminate mode switching by external forcing. For the

first time, it is demonstrated that amplitude of cavity flow Kelvin-Helmholtz

wave is dampened or cancelled by introduction of external perterbation of

natural flow frequency but different phase.

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-mACKNOWLEDGMENTS

This effort presents the results of one phase of research carried out at

the Jet Propulsion Laboratory, California Institute of Technology, Contract

NAS7-918 and sponsored by the U.S. Army Research Office, Contract No. 17134E.

We are indepted to Drs. Luis Bernal, for his assistance during the difficult LOV measurement phase, and Dan Nosenchuck, for his assistance during the

early strip-heater experiments. Thanks are also due to the technical staff of

the Fluid Dynamics Group at JPL, especially to chief technician, Mr. Wayne Bixler, and Messrs. Barney Green, Stan Kikkert, and Robert Smither for their excellent workmanship and assistance in construction of the experimental apparatus.
-ivABSTEACT

The phenomenon of cavity flow oscillation is investigated to determine the

conditions for onset of periodic oscillations and to understand the relationship

between the state of the shear layer and the cavity drag. Experiments have

been performed in a water tunnel using a 4" axisymmetric cavity model instrumented with a strip heater on the nose cone and pressure taps in and around

the cavity. A complete set of measurements of oscillation phase, amplitude

amplification along the flow direction, distribution of shear stress and other

momentum flux is obtained by means of a laser Doppler velocimeter. Drag

measurements were made by integrating the mean pressure over the solid surfaces of the cavity. Results indicated exponential cavity drag dependence on the

length of the cavity. A jump in the cavity drag coefficient is observed as the cavity flow shows a bluff body wake type behavior. An independent estimate of the

drag, which is obtained by integration of shear and mean momentum transfer

terms over the peripheral area of the cavity, confirms the exponential dependence of drag on the length of the cavity. Results, also reveal that the drag of

the cavity in the non-oscillating mode is less than the case if the cavity were

replaced by a solid surface. Natural and forced oscillations of the cavity shear

layer spanning the gap are studied. The forced oscillations are introduced by a

sinusoidally heated thin-film strip which excites the Tollmein-Schlichting waves

in the boundary layer upstream of the gap. For a sufficiently large gap, selfsustained periodic oscillations are observed,while for smaller gaps, which do not

oscillate naturally, periodic oscillations can be obtained by external forcing

through the strip heater. In the latter case resonance is observed whenever the

forcing frequency satisfies phase criterion -^— = iV, and amplitude exceeds
-Vcertain threshold levels, but the phenomenon, is non-self-supporting. The drag

of the cavity can be increased by one order of magnitude in the non-oscillating

case through external forcing. For naturally occurring oscillations, it is possible

for two waves to co-exist in.the shear layer (natural and forced). Also, it is possible to completely eliminate mode switching by applying external forcing. For

the first time a test is performed to cancel or dampen the amplitude of KelvinHelmholtz wave in the cavity shear layer. This is done through introducing an

external perturbation with the same frequency of the natural component but

having a different phase. Reduction by a factor of 2 is obtained in the amplitude

of the oscillation.
vii

TABLE OF CONTENTS

Chapter Title Page

1 INTRODUCTION 1

1.1 Background 1

1.2 Statement of the Problem 5

1.3 Approach 6

2 EXPERIMENTAL DETAILS B

2.1 Water Channel 8

2.2 Model 9

2.3 Pressure Measurements 10

2.4 Frequency, Phase and Correlation 11

Measurements

2.5 Velocity Measurements by Laser 12

Doppler Velocimetry

2.5.1 Signal Processor and Data 13

Acquisition

2.5.2 Operating Conditions 14

2.5.3 Operating Procedures 15

2.5.4 Data Processing 15

2.6 Flow Visualization 16

2.7 Operational Procedures. Accuracies 17

and Flow Parameters

2.8 Reference Quantities 17

PRECEDING PAGE BUNK NOT FILMED
viii

FLOW DOCUMENTATION 19

3.1 Cavity Flow Oscillations as a 19

Function of b/6o

3.2 Phase Measurements 21

3.3 Overall Features of Cavity Flow Regimes 23

3.3.1 Non-Oscillating Cavity Flow 23

3.3.2 Self-Sustaining Cavity Flow 23

3.3.2.1 General Characteristics of 24

the Self-Sustaining Modes

3.3.2.2 Similarity to Externally 25

Forced Free Shear Layers

3.3.3 Wake Mode of Cavity Flow 25

3.4 Mean Velocity Distributions 26

in the Shear Layer

3.5 Growth Rate of the Shear Layer 27

3.6 Shear Stress and Velocity Fluctuation 26

Distribution in the Shear Layer

3.7 Mean and Fluctuating Velocity Field 29

of the Cavity Flow

THE DRAG COEFFICIENT OF THE CAVITY 31

4.1 Momentum Balance of the Cavity 31

4.2 Measurement of the Cavity Drag 33

Coefficient

4.2.1 Cavity Drag Coefficient 34

in Non-Oscillating Mode
ix

4.2.2 Exponential Growth of the 34

Cavity Drag Coefficient

During Oscillating Modes

4.3 Cavity Pressure Distribution 34

4.3.1 Pressure Distribution for 35

Non-Oscillating Mode b/00 < 82

4.3.2 Pressure Distribution for 35

Oscillating Modes

4.3.3 Pressure Distribution for 36

the Wake Mode

4.3.4 Overall Characteristics of the Cavity 36

Pressure Distribution

4.4 Observation of Sudden Drag and Pressure 38

Change in Other Flows

4.5 Estimate of CD by Evaluation of 39

rti^T ni *!}

—=-and —z~

U* U*

Terms Along y = 0 Line

4.5.1 Confirmation of Exponential 39

Growth for CD

40

4.6 Distribution of

Ug

Along y = 0 Line

4.7 Distribution of -~j— 41

Ue

Along y = 0 Line

4.8 Evaluation of -

 42

Along y = 0 Line

4.9 Shear Stress Distribution 44

Along y = 0 Line

4.10 A Possible Flow Model 45
5 CONTROL OF CAVITY FLOW OSCILLATIONS 48

5.1 Forcing of the Shear Layers 48

5.2 Surface Heating Technique 49

5.2.1 Application of T-S Waves 50

as a Means of

Forcing of the Shear Layer

5.2.2 Strip Heater Set-up 51

5.2.3 Important Parameters of 51

Strip Heater Technique

5.3 External Forcing of the 52

Non-Oscillating Cavity Flow

5.3.1 Effect of External Forcing 54

on Cavity Drag in

the Non-Oscillating Mode

5.4 Effect of External Forcing on a 55

Naturally Oscillating Cavity Flow [b/0>b/0 I mln]

5.5 Cancellation Experiment 57

6 CONCLUSIONS 59

6.1 Summary of Results 59

6.2 Concluding Remarks 63

APPENDIX A. Determination of Overall Phase 64

Difference for the Cavity

APPENDIX B. Laser Doppler Velocimeter 66

Optical Arrangement

APPENDIX C. Velocity Bias and Bragg Cell 68

APPENDKD. LDV Signal Processing 69

APPENDDCE. LDV Data Processing 71

REFERENCES 74

FIGURES 78
XI

LIST OF FIGURES

Figure Title Page

1 Water tunnel schematic 78

2 Power spectrum of the velocity fluctuations 79

(a) In the absence of the model

(b) Typical cavity flow oscillation

3 Schematic of the experimental model 80

4 Schematic of the pressure sensing set-up and opera- 81

tional table of the valves (Table 1).

5 Photograph of the laser Doppler velocimeter set-up 82

and the test section.

6 Relative location of the LDV with respect to the model 83

and characteristics of the measuring volume (Table

2).

7 Frequency variation, of cavity shear layer with varying 84

cavity width, constant free stream velocity.

8 Velocity fluctuation spectra for increasing cavity 85

width from Mode II to Mode III.

9 Oscilloscope traces of cavity shear layer oscillations. - 86

10 Correlation test. 87

11 Phase angle of various downstream locations for 87

Mode II of the oscillations at b/e0 = 85, F = 5.9 Hz,

C/B = 23cm/6ec.

12 Overall phase difference between two corners of the 88

cavity for Modes II and III of the cavity flow

oscillations.

13 Flow visualization of the cavity flow in non-oscillating 89

mode; b/e^ = 66.
Xll

14 Plow visualization, of the cavity shear layer; flow is 90

frozen at 1/1000 sec.

15 Frequency variation of the cavity shear layer with 91

Uc —-—, for different cavity geometries and free stream.

velocities.

16 Flow visualization of the cavity flow; b/9o = 170 wake 92

mode; (a) and (b) 1/1000 sec; (c) 5 sec exposure.

17 Non-dimensional mean streamwise velocity profiles; 93

b/90 = 85, F = 5.9 Hz, Mode II.

18 Non-dimensional mean streamwise velocity profiles; 94

b/e0 = 103. F = 5.1 Hz. Mode II.

19 Non-dimensional mean streamwise velocity profiles; 95

6/<90 = 130, F = 6 Hz. Mode III.

30 Non-dimensional mean transverse velocity profiles; 96

b/00 = 85, F = 5.9 Hz, Mode II.

31 Non-dimensional mean transverse velocity profiles: 97

&/90 = 103. F = 5.1 Hz, Mode II.

32 Non-dimensional mean transverse velocity profiles; 98

b/00 = 130, F = 6 Hz, Mode III.

33 Non-dimensional momentum thickness growth with 99

streamwise distance.

34 Locus of points (y.5) where H/U, - .5 for cavity 100

widths. 6/00 = 85 and 6/6b = 103.

25 Time traces of velocity fluctuations and their 101

respective correlation.

36 Shear stress profiles; 6/9b = 85, F = 5.9 Hz, Model! 102

37 Shear stress profiles; b/e0 = 103, F = 5.1 Hz, Mode II. 103

38 Shear stress profiles; b/e^ - 130. F = 6 Hz, Mode III. 104
xiii

29 Growth of the maximum shear stress in the cavity 105

shear layer with downstream distance; Mode II,

30 Growth of the maximum shear stress in the cavity 106

shear layer with downstream distance; Mode III.

IT*

 3

-

31 Growth of the maximum streamwise and transverse 107

rms velocity fluctuations in the cavity shear layer

with downstream distance; Mode II, ——» 2.

32 Growth of the maximum streamwise and transverse 108

rms velocity fluctuations in the cavity shear layer

Fb with downstream distance; Mode III, 77—* 3.

33 Non-dimensional streamwise velocity profiles of the 109

Fb

cavity flow; b/e0 = 103, F = 5.1 Hz. Mode II ^-^ 2.

34 Non-dimensional transverse velocity profiles of the 110

Fb

cavity flow; b/60 = 103, F = 5.1 Hz, Mode II ^-^ 2.

35 Streamwise velocity fluctuation profiles of the cavity 111

flow; b/90 = 103. F = 5.1 Hz, Mode III |p» 2.

36 Non-dimensional transverse velocity fluctuation 112

profiles of the cavity flow; b/9^ = 103, F = 5.1 Hz,

Mode II |p-fti 2.

37 Shear stress profiles of the cavity flow; b/0o = 113

103, F = 5.1 Hz, Mode II, -|p- * 2.

38 Effect of the downstream corner on the velocity 114

fluctuations and shear stress profiles of impinging

cavity shear layer; b/90 = 103, F = 5.1 Hz, Mode II

~u7*

 2>
xiv

39 Variation of the axisymmetric cavity drag coefficient 115

with non-dimensional cavity width, b /<%.

40 Exponential dependence of the cavity drag coefficient 116

on b/90. © CDP\ A CD. ^-estimated CD based on

boundary layer friction in the absence of the cavity.

41 Pressure coefficient distribution in Mode II of the 117

oscillations.

• Upstream face pressure coefficient distribution.

© Downstream face pressure coefficient distribution.

42 Pressure coefficient distribution in Mode HI of the 119

oscillations.

•Upstream face pressure coefficient distribution.

©Downstream face pressure coefficient distribution.

43 Cavity pressure coefficient distribution in the wake 120

mode. • Q ,b/90 = 123; A A , b/e0 = 165; • a,

b /BQ = 203; solid and open symbols represent

upstream and downstream face, respectively.

44 Variation of the upstream corner pressure coefficient 121

with6/6b45 Variation of maximum pressure coefficient with 6/1%; 122

• maximum pressure coefficient; © pressure

coefficient at the downstream corner.

46 Variation of pressure coefficient at the middle of the 123

downstream face with

47 Variation of the cavity drag coefficient with b/R;© 124

present study; • Koenig (1978) optimum drag

coefficient of disk-cylinder combination.

Distribution of the mean transverse velocity •£—"•

along y = 0 line in Mode II.

Distribution of the mean transverse velocity -rr—"9

along y = 0 line in Mode III.
XV

Distribution of the mean streamwise velocity jralong y = 0 line in Mode II.

Distribution of the mean streamwise velocity ~—

along y = 0 line in Mode III.

Distribution of the -^-term along y = 0 line in Mode

II.

co ftifi 1 T^

Distribution of the — g-term. along y = 0 line in Mode

"9

III.

54 Distribution of the shear stress along y = 0 line in 134

Mode II.

55 Distribution of shear stress along y = 0 line in Mode 136

III.

56 A Shear stress contribution to the net cavity drag 137

A ^-term contribution to the net cavity drag

"e

57 (a) Variation of the shear stress at a fixed 138

x/60 with b/0Q.

(b) Variation of the net positive outflow from

the cavity with

58 Schematic of cavity flow illustrating the strip heater 139

technique.

59 Pressure distribution over the ellipsoidal nose cone 140

and respective location of the strip heaters.

60 Variation of maximum shear stress with forcing fre- 141

quency at four different forcing power levels; Case A

b/e0 - 66, U, = 23cm/fcec.

61 Variation of the maximum shear stress in the shear 142

layer with forcing frequency, constant power.
xvi

62 Distribution of the maximum shear stress with the 143

streamwise distance along the cavity shear layer; a)

three different forcing frequencies, constant power;

b) 2 different forcing powers, constant frequency;

6/90 = 77.

63 Flow visualization of a naturally non-oscillating cavity 144

flow, 6/flb = 66:

(a) unforced

(b) forced at F = 4 Hz, simulating Mode I of oscillations,

64 Oscilloscope traces of forced fluctuations, 145

(a) turn on; (b) turn off;

&/6?o = 66.F = 7Hz. ~« 2.

"e

65 Distribution of (a) -^-term, (b) shear stress along

"e

the y = 0 line in the non-oscillating mode;

©unforced, + forced at 6 Hz.

66 Velocity fluctuation spectra, presenting interaction of 147

the natural and forced frequencies in the cavity

shear layer for different power levels (P/), (h) scope

trace of Case (d).

67 Possible interactions with the cavity flow through 146

external forcing.

1) Mapping of Mode II into Regions 1 and 2.

2) Mapping of Mode III into Region 3.

3) Extension of

4) Delay of the wake Mode.

5) Simulation of Mode 1.

68 Schematic of the Cancellation Set-up 149

69 Damping of natural oscillations through 149

external forcing.

(a) natural oscillation

(b) strip heater on.
xvii

B-l Measuring volume formation. 150

Valid burst processing and coincidence window opera- 159

tion.
xviii

LIST OF SYHBOLS

Symbol Description

A frontal area for the model

AI cavities peripheral area

AZ downstream face area (=A)

A3 stings peripheral area

j44 upstream face area

b cavity width

6min minimum cavity width for self-sustained oscillations

&„. cavity width for transition to the wake mode

c speed of sound

Cjj cavity drag coefficient

CDS sting drag coefficient

pressure drag coefficient of the cavity

drag coefficient based on uv term

Cw drag coefficient based on edge velocity dynamic pressure

Cp pressure coefficient based on edge velocity dynamic pressure

?Pmax maximum local pressure coefficient of the cavity

d model diameter

dr length element along the model radius

dx length element along the cavity width

ds sting diameter

F frequency of the cavity flow oscillation

h two-dimensional cavity depth

H boundary layer shape factor
xix

ic LDV conversion factor

Kg LDV conversion factor, blue channel

KG LDV conversion factor, green channel

ZL direction cosine for the cavity surface

P static pressure

Pg free stream static pressure oustide the upstream boundary layer

r radius from center line of the model

R model radius

/?„ Reynolds number based on model diameter

Re5» Reynolds number based on boundary layer displacement thickness

Eeg Reynolds number based on boundary layer momentum thickness

S streanrwise distance over the model nose

location of first strip heater

location of second strip heater

SH3 location of third strip heater

u stream-wise velocity component

Uc convection or phase velocity of the perturbation

Ue free stream velocity outside the upstream boundary layer

v transverse velocity component

w strip heater width

x streamwise direction

y outward direction of the model perpendicular to x-z plane

u

ytS locus of —— = .5

Ug

z vertical direction of the tunnel

a angle between laser beams

<5* boundary layer displacement thickness

A<p phase difference between two signals
XX

A£< time between data points in LDV signal

AT time shift in cross correlation calculation

e boundary or shear layer momentum thickness

\ wave length of perturbations

Xa acoustic wave length

p fluid density

T^ viscous shear stress

tp overall phase difference between two corners

o Doppler frequency

( }0 upstream condition

( )min minimum value

( )max maximum value

(~) time average or mean value

( )' fluctuation about the mean

(-)' rms value

Q vector quantity

()e free stream condition outside the upstream boundary layer
-1 -

Chapter 1

Introduction

1.1. Background

Cavity flow phenomena occur in a variety of structures; slots between movable parts of control surfaces of aircraft, ships, tandem arrangements of bluff

bodies such as adjacent tall buildings or tractor-trailer combinations, gas

dynamic lasers, hydraulic gates, and control valves. The presence of cavities can

give rise to drastic changes in the steady and unsteady loading character of

these structures. The flow past cavity-type geometries (cut-outs) may introduce

large amplitude velocity and pressure fluctuations in and around the cavity

which are often damaging to the structures. Flow oscillation over gaps between

insulation tiles used on the space shuttle outer surface, caused by evaporation

of the filler material at reentry, is suspected of being responsible for the loss of

a portion of the tiles. Emission of acoustical noise is another unwanted aspect

of cavity flow-type oscillation. On the other hand, flow over cavities can be

effectively utilized to alter heat transfer (White 1971) and drag characteristics of

aerodynamic configurations (Koenig 1978).

Charwat, et al. (1961) categorized cavity flows into open or closed types :

"open" when the separated shear layer spans the gap,while "closed" refers to

flows where the separated shear layer reattaches on the bottom of the cavity

and separates again ahead of the downstream corner. For supersonic speeds

and for subsonic speeds with turbulent boundary layers, closed cavities are

Tound to have a width-to-depth b/h ratio of more than 11 (Charwart, et al.

1961). Sarohia (1975) shows a demarcation line between an open and closed

cavity to be around b/h » 7-8 for low subsonic speeds with laminar upstream
- 2 -

boundary layers. Furthermore, open cavity flows are divided into shallow and

deep cavities, depending on the type of acoustic resonance present in the flow.

When an acoustic resonance is present in the flow, the standing acoustic wave is

oriented longitudinally for shallow cavities (b/h. > 1). The standing wave is of

transverse type for deep cavities (b/h < 1) (Charwat et al., 1961. Plumblee et al.,

1962, East 1966. Tarn and Block 1978). Such a definition for the shallow and

deep cavities loses its meaning in the absence of the acoustic resonance in the

cavity. Tani et al., (1961) define cavity flows as deep and shallow cavities based

on their respective low and high pressure drag values. The demarcation line

based on Tani's definition is about b/h. ^ 1.4.

Sketch 1 illustrates the physically significant parameters of the cavity flow.

Boundary Layer

Cavity

Shear Layer-7

where U0 is the upstream edge velocity outside the boundary layer, <J*0 is the

displacement thickness and 60 is momentum thickness. For fixed flow parameters (U9 _ 90, 5*0). cavity flow shows a strong oscillation of the shear layer when

the width b exceeds a critical minimum width. 6,^, (Karamcheti 1956,

Sarohia 1975, Rockwell and Naudascher 1978). The existence of this minimum
-3 -

width for the onset of oscillation is linked to the nature of the instability growth

in the shear layer. Previous investigators noticed that the main driving force of

the acoustic resonance condition in the cavity was the amplification of instabilities by the respective cavity shear layer. In the shear layers, the final growth

and saturation of the instability waves result in roll-up of the shear layer and

appearance of large vortical structures (Brown and Roshko 197A). The velocity

fluctuation associated with the cavity flow oscillations can be attributed to the

presence of these large vortical structures (Sarohia 1975, Rockwell and Knisely

1979). In his study of the frequency selection process, Sarohia (1975) concluded

that the frequency at which the cavity chooses to oscillate is such that:

(1) certain phase criteria necessary for the self-sustainment of the flow are

satisfied; and

(2) maximum integrated amplification through the cavity shear layer occurs.

The cavity flow oscillation was described as a self-sustained type. Rockwell and

Naudascher (1978) suggest the following chain of events to be responsible for

the self-sustaining nature of the oscillation: impingement of organized vortical

structures upon the downstream corner; resultant upstream influence (interpreted as Biot-Savart induction); reception and conversion of this upstream

influence into velocity perturbation by the sensitive region of the shear layer

close to the upstream corner; amplification of these fluctuations by the shear

layer; growth into vortex structures. Consideration of the acoustic speed

becomes important in the nature of the upstream propagating waves for high

Mach number flows or when the impingement length, b, is of the same order of

magnitude as the acoustic wave length (6 » Xa) (Heller and Bliss 1975, Rossiter

1964, and Rockwell and Naudascher 1978).
-4 -

The self-sustained nature of the cavity flow oscillation is similar to that of the

jet-edge oscillation (edge tone phenomenon). Self-sustained oscillation of cavity

flow has the following properties:

(1) for a fixed cavity depth and fixed flow parameters ( l/e.^b.^o) there exists a

minimum width (&mjn) below which no oscillation occurs;

(2) for fixed cavity depth (h) and fixed boundary layer thickness there exists

a minimum Mach number or free stream velocity ( C/e) below which no oscillation

occurs ;

(3) for 6 > brain or Ue > U0m^ and fixed boundary layer thickness the fre9

quency of oscillation decreases as the parameter — — decreases.

At certain values of -r2-, the frequency undergoes a jump to a higher mode

o

of oscillation with a corresponding higher frequency. Hysteresis is observed in

Ue

the transition region. In the new mode, frequency also decreases as -—

decreases. Sarohia (1975) has found that for fixed values of U«, b, and h, there

exists a maximum boundary layer thickness f^j or <50*maxl above which no I ID&X J

oscillation occurs.

The ladder-type behavior of frequency versus — p (which has the dimension

of frequency) is remarkably similar for self-sustained flows and externally

forced free shear layers (Ho and Huang 1982). However, there is a significant

difference between the nature of perturbations introduced at the sensitive

region of the shear layer by the downstream corner and that of an external

forcing source. External forcing subjects the shear layer to a constant amplitude perturbation at one frequency. But flows excited by the impingement of
-5-

the shear layer at the corner are subjected to variation in strength of events at

the corner (Knisely I960). Often this variation gives rise to the amplitude modulation of the fluctuations,which shows as a low frequency component in the

spectrum of fluctuations. Knisely (1980) shows that transverse positioning of

the successive vortices as they approach the downstream corner yields varying incident structure severing degrees. It is not clear what sort of mechanism is responsible for the presence of these low frequencies in the cavity shear

layer. Also, the precise role of the downstream corner as a source of external

forcing for the shear layer and generation of different frequency components of

oscillation is not fully understood.

As mentioned before, the presence of cavities (gaps) can be used effectively to

change the drag characteristics of bluff bodies. A novel experiment by Roshko

and Koenig (1978) demonstrated that by properly choosing dimensions, a gap

generated by placing a disk in front of a circular cylinder can drastically reduce

the drag coefficient of the system. A similar observation for different

configurations of bluff bodies in tandem was made by Mair (1965) and Zdravkovich and Pridden(1977). It is unclear into which category these oscillations fall. Scattered information about the frequency of the shear layer oscillation shows behavior characteristic of self-sustained flow for the low drag cases

of flow over gaps generated by bluff bodies in tandem.

1.2. Statement of the Problem

A large portion of cavity flow investigations are devoted to the study of instability aspects of this problem. Quite independently, a second group of studies

can be found which deal only with the pressure field and drag characteristics of

this flow (Roshko 1955, Tani 1961, Fox 1964, Koenig 1978). To our knowledge no

investigator has tried to fill the gap between the first and the second group of

experiments. It was this gap that motivated the present investigation into the
role of instabilities in drag and.pressure field formation of cavities.

Another important aspect of cavity flow study is to develop a technique to

control the frequency and amplitude of oscillations. Previous work involving

control of the cavity flow oscillation has dealt with changing the geometry of the

downstream or upstream corner of the cavity by rounding the corners (Ethembabaoglu 1973) or by placement of external objects in the cavity such as flaps,

deflectors or perforated plates (Heller and Bliss 1975 and Ball 1957). Usually

these devices introduce their own complexities into the problem. There was a

need for a clean technique that can control the oscillations by purely introducing external perturbation to the shear layer.

The following were objectives set for the present investigation.

(1) To understand the relation between the state of the shear layer and the

drag of the cavity.

(2) To determine the influence of initial conditions on the onset of selfsustained oscillation and to obtain a better understanding of the feed-back

mechanism.

(3) To develop and use a technique which is capable of controlling (enhancing

or damping) established oscillations.

1.3, Approach

The connection between the state of the shear layer and the drag of the cavity system can be made by studying the quantities that are responsible for the

transfer of the momentum in and out of the cavity (tfiT and-uU). These quantities carry the necessary information about the nature of the instabilities in the

shear layer and play an inherent role in the formation of the pressure distribution inside the cavity system. Thus, accurate measurements of these quantities
-7 -

for different width-to-height ratios (b/h) in conjunction with the pressure distribution measurements would provide the necessary information for fulfilling

the first objective. External forcing by strip heating was used to study the feedback mechanism and control of cavity flow oscillation .thus, fulfilling the second

and third objectives.
-8-

Chapter 2

Experimental Details

2.1. Water Channel

The experiments were performed in a newly constructed free surface water

tunnel, which is located at the Jet Propulsion Laboratory, California Institute of

Technology. This tunnel is specially designed to attain a low turbulence level for

the purpose of carrying out sensitive instability-type experiments. The essential

features of the free surface tunnel are shown in figure 1. Water is recirculated

in a closed loop by a 50 horsepower axial flow pump, located 5m below the free

surface, in the lower leg of the tunnel. The desired flow speeds are achieved by

throttling flow through an on-line and a by-pass valve. The upper portion of the

tunnel consists of a diffuser, a straight wall section comprised of flow manipulators and a test section. The high speed flow from the pump is then diffused by

the 22° half-angle difluser. In order to prevent severe flow separation, the effect

of an unfavorable pressure gradient is reduced by matched pressure drops of

three perforated plates installed in the diffuser section. Before entering the test

section, the flow passes through different flow manipulators and a final 4:1 fifttr

order polynomial contraction. Each manipulator and contraction is designed to

break up and suppress different scales of the turbulence.

The test section is 45.7 cm (18 inches) wide and 63.5 cm (25 inches) deep.

The wetted depth can be increased up to 56 cm (22 inches). The test section is

bounded by glass windows on sides, bottom and top to permit visual and optical

study of the flow throughout its 1.83 m (6 feet) length.
-9-

The existing arrangement made it possible to achieve a maximum velocity of

1 m/sec in the test section. Turbulence levels and velocity profiles at 20 cm

from the entrance to the test section and outside of the wall's boundary layer

were obtained using hot film probes and an LDV system. The uniformity of the

flow within a square area (30 cm x 30 cm) centered at the center of the test section was below 1% of the center line mean velocity. The free stream turbulence

level at 25 cm/sec on the centerline was near .05%. The power spectrum of

free stream velocity fluctuations, a sample of which is shown in figure 2a, indicates most of the energy was concentrated at frequencies below .5 Hz, whichare

primarily below the range of frequencies of interest in this study, 2 to 10 Hz

(figure 2).

2.2. Model

An axisymmetric model, constructed of anodized aluminum and Plexiglas,

was used throughout the investigation. The model was composed of a rear portion and a front portion which were connected by a sting as shown schematically

in figure 3. The rear portion, a hollow cylinder 10.16 cm (4 inches) in diameter

and 91.4 cm (36 inches) in length, was connected by a strut to a traversing

mechanism which was installed on top of the test section. This allowed for

movement of the model up and down and also along the test section axis. The

front portion consisted of a half portion 2:1 ellipsoidal cone with a 10.16 cm (4

inch) minor diameter. The ellipsoidal cone was extended in the downstream

direction by a straight section which also was 10.16 cm (4 inches) in diameter

and 5.08 cm (2 inches) in length.

The sting was made from a 1.27 cm (1/2 inch) O.D. thick wall stainless steel

tube. Twelve brass tubes were installed inside of its walls to carry pressure signals from the front nose while the interior of the sting was devoted to dye tubes

and electrical wires for the strip heaters. The front end of the sting was fixed to
-10-

the inside well of the front nose while the rear end was connected to a vertical

rod by means of a 90° gearing mechanism. The vertical rod was then brought

out of the model through the strut. By manual rotation of the vertical rod, the

sting could be moved longitudinally relative to the rear portion of the model.

With the aid of this mechanism, plus a dial and a helipot at the end of the vertical rod, it was possible to change and read the cavity length. The solid inner

faces of the cavity were equipped with 14 pressure taps on the downstream face

and 8 pressure taps on the upstream face. The diameter of each tap opening

was .1 cm (.041 inch ).

2.3. Pressure Measurements

The low velocities in the range of interest introduced the well known difficulty

of measuring low differential pressures under high head. A free stream velocity

of 25 cm/sec generated a dynamic head of .38 cm (.15 inch) of water, while a

typical value of total head at the point where the pressure transducer was

located was 63.5 cm (35 inches) of water. Further, since differential pressures

inside the cavity are much smaller than for a free stream dynamic head, the

demand on the pressure measuring system for high resolution and repeatability

is very severe.

Another common problem in pressure measurements with water as a medium

arises from the presence of air bubbles in the system, which may cause erroneous pressure readings. The following approach was taken to solve the air bubble

problem and to achieve high resolution. Each line, Tygon tube ID = .16 cm (1/16

inch), from the model pressure taps or from the pitot-static probe was connected to a common drain through a toggle valve (figure 4). Each line then

extended and manifolded into a 24 port fast switch (Scanivalve type ¥2460).

The common end of the Scanivalve was connected to the low pressure side of a

pressure transducer (Validyne type Wet-Wet DP-103-12) which was able to
-11 -

measure differential pressures in the range of 0 to .762 cm (.3 inch) of water

with a resolution of 1% of the total range. The high pressure side of the transducer was connected to the total pressure tube in the test section, which was

chosen as the reference. The common drain was connected to a vacuum line and

was used as a sink into which air bubbles could rise. Also, it was used to fill the

lines from the model to the toggle valves. A special calibration tank was

designed to provide a differential head of 0 to 10.16 cm (4 inch) of water with a

resolution of .00254 cm (.001 inch) of water. One port of the Scanivalve was

devoted to the low head side of the calibration tank, while the higher head side

was connected to the high pressure side of the transducer through a 3-way

valve. A schematic of the pressure sensing system is given in figure 4. The average height of the calibration tank was set close to that of the free surface. This

was done to prevent overpressurizing the transducer during switching. Table 1

gives different operational positions of the valves. The pressure measurement

process was automated by using an HP data logger. A proper software program

was written for the data logger in order to actuate the Scanivalve and to digitize

the analog signal from the transducer. Prior to each run the digitization rate,

the lag time between each point, and the averaging time were specified to the

machine. After each run, the program stored the pressure distributions, and

the calculated drag coefficient as well as other parameters of each run on a

floppy disk.

2.4. Frequency. Phase and Correlation Measurements

Velocity fluctuation measurements were made with a TSI hot-film probe

(1277-1OA) using a TSI 1050C anemometer in conjunction with a TSI true rms

meter (1076). The output of the hot film anemometer was fed to a digital power

spectrum analyzer (SL20). A bandwidth of .1 Hz was used for locating and distinguishing frequency content of the velocity signals on line. A hard copy of the

spectrum was obtained by either taking pictures of the spectrum analyzer
-12-

screen or by plotting the spectrum on an x-y plotter (HP model 7001A).

For phase angle and cross correlation measurements a second hot-film probe

was mounted on a traversing mechanism which was able to traverse the entire

width of the cavity. Throughout this experiment the first probe was used as a

reference probe. This probe was placed above the upstream corner at x = b and

y.9s. the latter being the y-location of u/Ue = .95. Phase measurements were

made by moving the second probe relative to the reference probe along the edge

of the shear layer where U/U0 = .95 (y = y.g5). The signals from the reference

and the second probe were fed to a digital probability and correlation analyzer

(SAICOR 43A). Throughout the phase angle or cross correlation measurements a

sampling rate of 200 per second, limited by 400 samples and a corresponding

Nyquist frequency of about 30, was used. Again a hard copy of the output was

obtained on an x-y plotter. See Appendix.A for the details of phase angle measurement.

2.5. Velocity Measurements by Laser Doppler Velocimetry

Due to sensitivity of the cavity flow oscillations to the presence of external

objects in the flow, hot-film anemometry is not suitable for velocity measurements. Severe interference of the probes with cavity flow mode switching is

observed in the present study. Existence of large regions of reversed flow is

another reason for making hot-film anemometry unappealing for this flow. The

only instrument currently available to measure velocity field non-intrusively and

unambiguously in such a flow condition is the laser Doppler velocimeter (LDV).

Other useful properties of the LDV include the small sampling volume, high frequency response and linearity of the instrument. In this study, the velocity field

of the cavity flow was measured by a two-component, two-color, frequency

shifted laser Doppler velocimeter (TSI 9100-7 system). It is a dual beam (forward) scatter configuration and processor, which was operated in single-burst
-13-

mode. The LDV processor operated in conjunction with, a minicomputer system

(PDP-11), which was also used for processing the results. A forward scatter

configuration was used in these experiments, with the transmitting and receiving optics located at the bottom and top side of the facility as shown in figure 5.

The physical arrangement of the LDV optics relative to the facility and information regarding focal volume are given in figure 6 and table 2. The transmitting and receiving optics are mounted on an optical table with their optical axes

perpendicular to the flow direction. The focal volume has its longitudinal axis in

the z direction of the facility. The streamwise velocity component is measured

by the fringe system formed in the z-x plane of the tunnel >while velocity components perpendicular to the mean free stream are measured by the fringe system formed in the z-y plane of the channel. The velocity bias technique (Bragg

cell) is used to detect velocity reversals. For details of the optical setup and

Bragg shifting see Appendices B and C, respectively.

The focal volume can be moved within the test section by means of remotely

operated stepping motors. Motion on a fixed horizontal plane was obtained by a

corresponding motion of the optical table. Motion along the z-axis resulted

from the displacement of the focussing lens and receiving optics along their optical axis. The table position was controlled to a resolution better than 100

2.5.1. Signal Processor and Data Acquisition. The measurement of the velocity can be obtained only after suitable processing of the outputs of the photomultiplier tubes. Two counter processors were used, one for each velocity

component. They were operated in the single-burst mode, for which only one

particle is present in the focal volume when the measurement is made. In addition, the system was operated in coincident mode, in which the output of the

processors is tested for coincidence in time so that the two components of the

velocity vector as determined by the optical set-up are measured on the same
- 14-

particle.

The signal processing is accomplished in four stages: analog processing, burst

processing, coincidence test and transfer to the computer. The analog processing involves suitable filtering and amplification of the signal for optimum signalto-noise ratio. Only low pass niters were used. A built-in filter in the Bragg cell

circuitry removes pedestal noise (low frequencies). The low pass filter is used

for removal of high frequency noise. Cut-off values from 30 kHz to 100 kHz were

used, depending on flow velocity. The output of the filter is processed by an

amplifier. In the present experiments the gain was set at 2 or 4 depending on

the condition of the signal. A detailed description of the valid burst processing

is given in Appendix D.

2.5.2. Operating Conditions. A typical signal-to-noise ratio, as observed from

high sampling rate measurements in a quiescent fluid, was in excess of 100. An

equivalent background noise level of .4% and .6% was detected for the streamwise and the transverse velocity fluctuations »respectively. When the signals

from two channels were correlated for the purpose of shear stress measuret ment, the corresponding noise level for ~\/ — g— measurements was ,07%. The

better noise level value was due to the fact that much of the uncorrelated noise

from two independent processors was eliminated during the correlation process,

resulting in an extremely low noise level for shear stress measurement.

A time window of 50 /^sec was used in all velocity measurements. The amplitude threshold level and the gain of the amplifier were adjusted to maintain the

data rate between 60 to 200 particles /sec, which corresponds to a minimum

Nyquist frequency of 10. It was found under the present experimental conditions that particles less than 15 ftm had sufficient response frequency. Thus Jor

the purpose of LDV measurements, particles larger than 15 /nm were filtered
- 15-

from the water.

2.5.3. Operating Procedures, Prior to every experimental run with the LDV

system, a variety of tests were conducted to verify adequate operation of the

system, e.g., the transmitting optics alignment and overlapping of the green and

blue focal volumes. The alignment of the receiving optics was verified with the

facility in operation. Adjustment of the location of the pin-hole and photomultiplier assembly was often necessary for maximum data acquisition rate at the

operating conditions of the water channel.

After the high-pass and low-pass filters were set to their predetermined

values, the gain of the amplifier stage was adjusted for optimum signal-to-noise

ratio. After these tests were completed, the actual measurements were initiated. A 16,000 word buffer was obtained at every point. The data were stored

temporarily on disk. At the end of the test run, all the data were transferred to

a magnetic tape for permanent storage and processing at a later time. The size

of the data buffer combined with the data rate to give a total sampling time

between 15 and 60 sec.This sampling time was considered adequate to properly

characterize mean turbulent quantities.

2.5.4. Data Processing. The velocity data generated by the data acquisition

programs was processed using the minicomputer system. Processing involved

several steps, e.g., error detection, computation of flow characteristics in the

frame of reference of the optics, transformation of those quantities to the physical frame of reference, and the computation of integral quantities in the physical reference frame.

There was one condition that could have led to a significant error in the

measurements. This condition arises from transmission line noise, resulting in

a spurious data word being included in the stream of data as it is transferred to
-16-

the computer. Because catastrophic effects could result from this condition,

the data processing programs had to verify that the data were in the correct

order. If an error was detected, the data point would be discarded. Less than

1% of the data was removed by this test. At every measurement point, a frequency histogram was constructed for each channel. This histogram provided

information about the proper operation of the system, i.e., filter setting and the

electronic noise.

The Doppler frequency and the time between data points were utilized to calculate the flow velocity using the appropriate Doppler constant. The mean,

second moment and cross-correlation between the two velocity components

were evaluated. For a detailed description of the expressions used for the data

processing see Appendix E.

2.8. Flow Visualization

The model was equipped with two dye injection tubes whose axes each made

an angle of 1° with local curvature of the nose cone at the point of the injection.

The injection points were 2.54 cm (l inch) off the stagnation point. Two visualization techniques were employed in this study. In both cases, diluted coloring

dye or fluorescein dye was gravity fed to the injection tubes and then discharged

into the boundary layer over the cavity.

In the first method, using blue dye, a uniform background illumination was

provided specifically for photographic purposes. In the second technique

fluorescein dye was used and excited by a laser sheet. The laser sheet was

obtained from a 2-watt argon-ion laser with a thickness less than 1mm. The

technique was used first by Dewey (1976). Photographs of the resulting flow patterns were made using a 35 mm SLR camera with a focal length F3.6 lens at

shutter speed 2 to 1/1000 sec and either TRI-X black and white print film or

high speed Ektachrome tungsten color slide film.
-17-

2.7. Operational Procedures. Accuracies and flow Parameters

The pressure and velocity measurements procedure was as follows. Prior to

each run with the tunnel running at idle, the nullity of the pressure readings

was checked. The desired speed of the tunnel was set. The tunnel speed,

extracted from the dynamic pressure measurement, was checked against the

one measured by laser Doppler velocimeter. The cavity gap was set to the

desired width and the frequency of cavity flow oscillation was checked. Depending on the degree of the flow unsteadiness, the pressure averaging time was set

from 30 to 130 seconds. The alignment of the model was checked by pressure

reading differences of taps on the four sides of the downstream face. A cross

check of the model alignment was made through flow visualization.

At the end of a test run, with the tunnel again set at idle, the tap pressures

were checked to insure they had returned to zero. For velocity field measurements, the measuring volume was fixed at a reference point on the model. This

was done by observing the position of measuring volume with respect to the

model through the eyepiece on the receiving side of the LDV. Measurements

were made at speeds covering the range of 16 to 27 cm/sec. The corresponding

Reynolds number, Re, based on the model diameter, d = 10.16 cm (4 inches),and

at Ug = 23cm/£ec was 24 x 10~3 . Most of the pressure data points were measured at least twice and at some points up to 5 times. The repeatability of the

calculated drag coefficient was about 4% of the average value.

2.B. Reference Quantities

Throughout the experiment, the origin of the reference coordinate system

was located at the upstream corner. The x and y axes lie in the symmetry

plane. The x-axis was in the direction of flow; the y-axis was measured positive

outward with respect to the center of the cavity. The components of velocity

vector in the direction of the x and y axes were denoted u and v.
-18-

The cavity width, b, was non-dimensionalized by normalizing it with respect

to GO , the momentum thickness of the upstream boundary layer at the point of

separation. Momentum thickness for the shear layer is denned as

where £/„ is the free stream velocity on the outside edge of the boundary layer

at the point of separation. Using Ug instead of U*,, the free stream velocity in

front of the model nose eliminated the need for the blockage correction. This

also made the results independent of the front nose shape (provided there

existed no substantial pressure gradient over the length of the cavity).

The boundary layer upstream of the cavity was considered laminar at

Ug = 23cm/&ec and had a shape factor of H = —= 2.5 where <5* = .1cm (displacement thickness), 60 = 0.041 cm and Ree()

 = 95. For the shear layer, the

[y -y.s]

transverse distance y was non-dimensionalized as , where y.5 is the

IT

location of the non-dimensional velocity jj— with a value of .5, and 6 is the

local momentum thickness of the shear layer as defined by equation (2.1), The

radius of the axisymmetric cavity, which is 5.08 cm (2 inches) was used to normalize the radial distance of pressure taps (r) on the inner surfaces of the cavity. All drag forces were normalized by — pU*A, where p is water density and

to

A is the cross- sectional area of the model.
- 19-

Chapter 3

Flow Documentation

The measurements of various parameters relevant to the cavity oscillations

reported in this chapter are not all novel, some of them having been reported in

references cited above and in others. The purpose here was to provide a solid

reference condition for the measurements to be described in Chapters 4 and 5.

Nevertheless, some new definitions, clarifications and features are introduced .

3.1. Cavity Flow Oscillations as a Function of 6 /e0

It is known [Karamcheti (1956), Sarohia (1975), Rockwell and Naudascher

(1979)] that shear layers over cavities undergo transition from a steady layer to

an unsteady layer at a critical dimensionless width o/^jltn^. For

6 /BO < b /BO | min, the shear layer bridges smoothly over the cavity with no distinct

oscillation in it. The onset of unsteadiness is always associated with the presence

of a distinct fundamental frequency, which is amplified by the shear layer. In

this study, the fundamental frequency is identified as the one that has the

highest peak in the power spectrum of the velocity fluctuations. Velocity

fluctuations were measured at x = b and y =y.gs-(y.9s being the locus of

77—= .95) throughout the frequency measurements.

Figure 7 presents the variation of fundamental frequency, F, with b/0Q at a

fixed edge velocity (Ug = 23cm/fcec). Oscillation starts at b/e0 « 81, which

represents the minimum cavity width b/e^l^^, for the present experimental

condition. The fundamental frequency of oscillation decreases as b/e0

increases. This course of frequency variation is interrupted by a sudden jump
-20-

of frequency from F = 4.6 to 6.2 Hz. This new frequency initiates another

course of decreasing frequency as b /e$ is further increased. Anon-dimensional

rn.

frequency can be denned for these two modes of oscillation as -rr— (Strouhal

Ug

Number). This Strouhal Number has a value of « 1 over the entire range of the

starting mode, then, jumps to a value of « 1.5 which it maintains over the next

mode. According to Sarohia (1975) and Rockwell and Knisely (1980) these two

modes correspond to Modes II and III of the cavity flow oscillation.

A sequence of power spectra of the velocity fluctuations as a function of 6 /do

is shown in figure 8. Several features of these spectra are noteworthy. First is

the presence of peaks both at 4.6 Hz and 6.2 Hz during the transition region.

between the modes. From these average power spectra it is not possible to

determine whether the two frequencies alternate intermittently or whether both

coexist simultaneously. To clarify this matter, figure 9 shows three oscilloscope

traces of the velocity fluctuations in Mode II, in the transitional region, and in

Mode III of the oscillations, respectively.

Figure 9b confirms the coexistence of two frequencies by showing a strong

modulated trace of the signal. A region of hysteresis, as shown in figure 7, is

associated with the transition region. Presence of frequencies lower than fundamental is a prominent feature of Mode III (figure Be). Such low frequencies have

been observed by Knisely (1980) and were shown to be associated with the

interaction of successive shear layer vortices with the downstream corner. A

possible explanation for the existence of such a low frequency is given in the

flow model section (Section 4.10). End of Mode III is marked by a loss of coherence in the oscillation and the disappearance of the fundamental frequency. At

this stage (6/9 > 160) , cavity flow begins to exhibit bluff body wake behavior,

and a distinct frequency for this regime cannot be detected.
-21 -

3.2. Phase Measurements

Information about the overall phase difference between fluctuations at the

upstream corner (x = 0) and the downstream corner (x = b) is needed for

improving our understanding of the cavity flow oscillation. This is the subject of

the following discussion.

The existence of a strong phase gradient across laminar and turbulent shear

layers is well known [Freymuth (1966), Sarohia (1975), Elder (1978), Ziada and

Rockwell (1981), and Hussain (1981)]. This gradient can be well approximated by

inviscid theory (Michalke, 1965), except for regions close to the downstream

corner. Existence of such a phase gradient introduces difficulties in choosing a

proper criterion to define an overall phase difference between the two corners of

the cavity. Knisely (1980), Ziada and Rockwell (1981), Rockwell and Schachenmann (1982), in their effort to define an overall phase difference for different

types of self-sustaining flows, concluded that tracking values of phase along the

edge of the shear layer ~{u/Ue - .95) will result in an overall phase difference of

~ 2N rr . In contrast, other criteria .e.g., following one of the disturbance amplitude maxima within the shear layer, yield different overall phase differences for

different geometries of self-sustaining flows.

In the present study, phase measurements were obtained by following the

outer edge of the shear layer along the locus of u/Ue = .95. The reference probe

was located at x = b, y = y,^5, while the second probe was moved along yr95 of

the shear layer. Before each set of measurements was to be taken, the autocorrelation of the reference signal with respect to itself or to that of the second

probe (cross-correlation) was obtained. In this way, any initial phase difference

due to circumferential or longitudinal distances between two probes was

detected. A sample of such a measurement is given in figure 10. For the case of

b/e0 - 85 and Ug = 23cm/sec a typical phase angle distribution of the velocity
-22-

fluctuations as a function of X/OQ is presented in figure 11. The cavity shear

Fb

layer has a fundamental frequency of 5.9 Hz with corresponding 77-= 1> which

Ug

identifies it as the second mode of oscillations. Figure 11 depicts an overall

phase difference of 4n between two corners of the cavity. Similar measurements

to determine the overall phase difference between the two corners were carried

out for the entire range of Modes II and III of cavity flow oscillation. Figure 12

represents the distribution of the overall phase difference as a function of b/90-

It is important to note that the cavity flow retains a 4rr phase difference during

Mode II, while having a &n phase difference during Mode III. This fact suggests

that overall phase difference between the two corners be formulated as

%--N (3.1)

where N represents the mode of the cavity flow oscillation. Using the relation

Uc = FX , where Uc is the convection velocity (phase velocity) of the fundamental frequency of oscillation and \ is the wavelength, equation 3.1 can be rearranged as

9 _ p__ Fb ,3 gv

£n A. Uc

Equation 3.2 indicates that cavity during Mode II contains two wavelengths of

the fundamental frequency, while Mode III contains three wavelengths. Also,

emergence of the Strouhal Number.based on the phase velocity in equation 3.2,

Fb

directs us to define the mode of the oscillations based on 77-= JV rather than

L/C

-rr—= —. This convention defines each mode by the number of wavelengths

L/C 2

that it contains and establishes a relation between the mode's name and the
-23-

corresponding Strouhal Number.

It is important to note that close to the transition region, an important phase

deviation from •£—w 2 occurs (figure 12). Such a deviation of overall phase

GiT\

difference also was observed by Rockwell and Schachenmann (1982) for impinging turbulent jets at low Mach numbers.

3.3. Overall Features of Cavity Flow Regimes

3.3.1. Non-oscillating Cavity flow. The non-oscillating regime of the cavity

flow is marked by a fairly steady and weak vortex stationed inside of the cavity.

Unsteady mass transfer in and out of the cavity is minimal due to lack of any

substantial oscillations in the shear layer. Figure 13 presents flow visualization

of the cavity flow in this regime. Often, color dye trapped in the cavity remained

there for a long time. The effect of the cavity presence on the outer flow field is

minimal. Branching of the shear layer at the downstream corner is another

feature of this mode observed in flow visualization studies. It should be mentioned that for 6/00 < 6/e§ \^^hot-film signals indicated the presence of weak

periodic fluctuations in the shear layer. Due to lack of a wide enough cavity,

these perturbations do not amplify enough to make the flow self-sustained. This

matter will be discussed in more detail in Chapter 5.

3.3.2. Self-Sustaining Cavity Flow. The definition of self-sustaining flow and

the description of the chain of events that result in flow self-sustainment is

given in the introduction. As mentioned before, natural cavity flow oscillation is

by definition self-sustaining.
-24-

3.3.2.1 General Characteristics of the Self-sustaining Modes.. A selfsustaining oscillation appears in the cavity when o/00 reaches &/9olmin- Tb-e

wavelength of the fundamental frequency scales itself with the width of the cavity (figure 12). Flow in this regime is stable. Even severe attempts to perturb

the course of oscillations is resisted, depending on the strength of the selfsustainment. In this study, the most stable case is denned as the one that generates the highest level of organized fluctuations for a given b/90.

Overall phase difference between the two corners of the cavity satisfies the

relation -£— = —= 7:— = N, where N is the number of wavelengths contained

27T A t/c

by the cavity width during Nth mode of oscillations. Shown in figure 14 are flow

visualizations of Modes II and III of the cavity flow oscillations. In the present

study, the cavity always started the oscillation with Mode II, omitting the first

C7j

mode —— ra 1. Sarohia (1975) observed similar omission of the first mode in

Uc

his study of cavity flow for some free stream velocities.

Instantaneous flow visualization pictures (figure 14) show that the wave

amplitude grows as it approaches the downstream corner. This growth, which is

exponential according to the linear theory, results in a vortex roll up. This roll

up is very similar to that observed by Brown and Roshko (1974), Winant and

Browand (1974) in the free shear layer. Mode III shows, usually, two complete

roll ups of the shear layer. Unlike the case in a free shear layer, no vortex pairing is observed during the cavity self-sustaining modes. Thus, presence of low

frequency peaks in the power spectra of Mode III of the present study cannot be

associated with vortex coalescence.

Presence of a strong stable vortex is another characteristic of the selfsustaining mode.
-25-

3.3.2.2 Similarity to Externally Forced free Shear Layers.. The ladder-like

variation of cavity shear layer frequency shows striking resemblance to the

externally forced free shear layers (Miksad 1973, Ho and Huang 1982). One can

imagine that the downstream corner plays the role of external forcing source.

Also the term 6 /Uc can be assumed to be equivalent to the forcing frequency in

an externally forced free shear layer experiment. Figure 15 shows the variations of the fundamental cavity shear layer frequency as a function of b/Uc. In

this figure, several cases of the cavity flow geometries and velocities fall on the

corresponding mode lines of externally forced experiments. In general, one

must treat this resemblance with caution, because at high subsonic or supersonic cavity flows, the wavelength will not scale with cavity width b. As suggested by Laufer and Monkowitz (1980) such a scaling should follow the more

complicated feedback equation

•*- "

 <3'3)

where C is the sound velocity. In the subsonic case C» Uc. equation (3.3)

Fb

reduces to our familiar relation for the Strouhal Number -TT— = N.

Uc

3.3.3. Wake Mode of Cavity Flow. The last Mode (III) of oscillations, observed

in this study, shows strong intermittency as the cavity width is increased. It

seems that eventually an intermittency overcomes the coherent oscillation and

the feedback mechanism becomes ineffective. The flow visualization studies

revealed that when the self-sustaining regime ends, the shear layer no longer

reattaches onto the downstream corner. The cavity flow becomes semiperiodically unstable in a larger scale, comparable to the scale of the model

(figure 16a, b). Time-averaged flow visualization (figure 16c) shows that the

mean flow reattaches below the downstream corner. Instantaneous flow
-26-

visualization indicates that the flow is highly asymmetrical (figure 16a). The

flow field in this regime shows some resemblance to a three-dimensional bluff

body wake (Werle,1974), even though the presence of the rear body is still

strongly being felt by the wake of the front body. A detailed exploration of the

flow for cavity-diameters ratio b/R > 1.65 was not carried out.

3.4. Mean Velocity Distributions in the Shear Layer

For three values of cavity width b/e^ = 85 ,6/6^ = 103 and b/6o = 130, mean

velocity profiles were obtained at a number of streamwise stations. Edge velocity (Ug) was fixed at 23 cm/sec. The cavity width b/e^ = 85 marks the beginf*

ning region of Mode II - = 2 , while the cavity width b/6^ = 103 marks the

middle of the same mode. For 6/^3 = 130 -j— - 3 , the cavity was oscillating in

[Ue J

Mode III. Figure 17 presents nondimensionalized mean velocity profiles at

selected downstream stations for 6/flb = 85. As shown in figure 17, in the early

stages of the shear layer growth, the velocity profile changes from a boundary

layer profile to a shear layer profile. Free-shear-layer similarity in the -TT—Ug

profiles is attained for 6/%=i 30. In figures 18 and 19, which present nondimensionalized mean velocity profiles for b/eo = 103and6/9b = 130, respectively, self-similarity in the rr—profiles is also observed.

Ue

Figures 20 through 22 present the nondimensionalized mean transverse component of the velocity, jj—, in the shear layer. These figures reveal that unlike

Ug

the -jy—profiles, mean transverse velocity profiles do not scale with the downUg

stream distance. Thus>no self-similarity can be found for yr—profiles. Gorree,

Ug

Dimotakis and Koochesfaham (1980) were able to show that for free shear
-27-

layers with very high Reynolds numbers,self-similarity exists for -r— profiles.

Ug

The lack of similarity in the -T-— profiles of the cavity shear layers might be due

Ug

to existance of the strong recirculating flow inside of the cavity. Figure 23

presents the locus of yf values as a function of —. As in previous studies by

Browand (1966), Sarohia (1975), and Knisely (1980), the y.5 values were found

initially to lie above the level of the separation edge. Depending on the stage of

the cavity flow, the locus of y^5 approaches the y=0 line as the profile

develops. The shear layer shows a stronger deflection into the cavity for the

b/e0 = 103 than for b/e = 85.

3.5. Growth Rate of the Shear Layer

For four values of cavity width, 6/6b = 85 (Mode II),6/6b = 103 (Mode II),

b/do = 119 (Mode II) and b/Qo - 135 (Mode III), the momentum growth of the

shear layer is given in figure 24. As in the studies by Sarohia (1975) and Knisely

(1980), the momentum thickness is found to grow linearly with x/e^. Because of

this linear growth, one might assume a self-similarity for the cavity shear layers,

but it is interesting to mention that in both the previous studies and the

present experiment, the value of the shear stress is not constant (figure 24) as

is required for the self-similarity of the free shear layer.

de

dx

For all cases with b/d$ > 103, the growth rate is fairly constant = .031

and does not expenence a sudden change during the mode switching region. It

should be noted that the cavity shear layer growth rate is comparable to that of

U\

turbulent mixing layers. For -JT—= 0, the high Reynolds number shear layer of

Uz

Liepmann and Laufer (1947) had a growth rate momentum thickness of 0.035.
-28-

3.6. Shear Stress and Velocity Fluctuation Distribution in the Shear Layer

Measurements of shear stress and fluctuating components of the velocity

field are essential to an understanding of the overall nature of the cavity shear

layer. As discussed previously, due to the nature of the present LDV measurements, shear stress measurements show fairly low background noise levels.

Such an advantage was not available for jj— or -^—measurements, (figure 25) .

Ue UB

Figures 26 through 28 depict the streamwise evolution of the transverse

profile of the shear stress for b/6b = B5 (Mode II), b/6fc = 103 (Mode II), and

n*

b/e = 130 (III Mode). For large — ; shear stress profiles tend to show a second

3o

peak in the lower portion of the shear layer. Existence of such a second peak

has been observed [Freymuth (1966) ,Knisely (1980)] ; Ziada and ^Rockwell

(1981) showed that it can be predicted by Stuart's (1967) theoretical vortex

model. Also important for our consideration is the fact that, even though the

mean streamwise velocity profiles scale with — ; the maxima of the shear stress e

o

profiles do not have a constant value as would be required for self-similarity,

they show an exponential growth with X/BQ (Figures 29 and 30). Streamwise

growth of the maximum shear stress for several cavity widths is shown in figures

29 and 30. Short cavity cases, 6/6b = 85 (Mode II) and 6/6b = 103 (Mode II).

show a region of exponential growth of the maximum shear stress. For long cavities b/6Q > 103, the region of exponential growth is followed by a saturation

region. In Mode II, the rate of growth for — g — l,^ increases as b/6o increases

, U e

(figure 29). The transition region does not show a noticeable change in the

growth rate, but the general level of the shear stress decreases for the transition region (figure 29).
-29-

Figure 30 gives the distribution of maximum shear stress growth for Mode III.

This mode, too, shows exponential growth for the maximum shear stress m the

shear layer. Again,growth rate increases as b/fy increases. It is important to

mention that the general level of the maximum shear stress distribution is

higher for b/8o - 151.8 and b/do = 158.5.which are close to the end of Mode III.

The maximum shear stress achievable during Mode II, 0.0086, is lower than

that in Mode III. which is .013. For comparison the value of maximum shear

stress for a self-similar, two-dimensional turbulent mixing layer as determined

by integrating the velocity profiles presented in kepmann and Laufer (1947) is

.0125. Integration of the velocity profiles measured by Wygnanski and Fiedler

(1970) gives rm!a/pUz

e = .015. Kistler and Tan (1967) reported a maximum

value of 0.012 for two-dimensional cavity shear layers.

Figures 31 and 32 depict the variation of the maxmium streamwise and

transverse fluctuation as a function of b /e$. Again an exponential growth of the

disturbances is evident. It is clear from these figures that rms fluctuations

suffer from a high percentage noise level of either flow background or electronics.

3.7. Mean and Fluctuating Velocity Field of the Cavity Plow

Figures 33 through 37 present traverses of the velocity field through the cavity for one selected cavity width (6/flb = 103) in Mode II

Fb

= 2 . The particuU.

lar case selected has a fundamental frequency of 5.1 Hz. Figure 33 presents the

mean streamwise velocity profiles of the upstream boundary layer at the

ue

upstream corner, at four stations within the cavity and two stations downstream of the downstream corner. Similarly figure 34 displays the mean j-—

profiles.
-30-

The change in flow direction inside the cavity in figures 33 and 34 is due

to the recirculating vortex flow visible in the flow pictures. The upstream

boundary layer is laminar and has a shape factor of 2.5. The downstream boundary layer profiles at x/60 = 110 and x/80 = 142 give respective boundary layer

shape factors of 1.54 and 1.46. These shape factors are close to that of a turbulent boundary layer.

Figures 35 and 36 reveal that significant fluctuations occur inside the cavity,

but the flow field inside the cavity has a much lower shear stress level than that

of the shear layer (figure 37), possibly revealing, that most of the u' and v'

fluctuations are not well correlated. Perhaps the most important difference

fti ' f*J*

between -jj— and -jj— behavior is shown in the downstream boundary layer

Ue Ue

profiles (figure 38). The maximum value of the yr—profiles in the shear layer

Ug

continues to increase, even after the shear layer transforms to a boundary

fTTj"

layer. On the contrary>the maximum value of the -jj— profiles shows a decrease

Ug

after the downstream corner. The shear stress profile shows a monotomc

decrease in the maximum value of the profile as the shear layer transforms to

the boundary layer.
-31 -

Chapter 4

The Drag Coefficient of the Cavity

Throughout this study the term "drag coefficient of the cavity" refers to the

net force in the flow direction being experienced by the gap. The overall changes

in the net drag coefficient of the model.which might arise from cavity presence,

will not be discussed in this study.

4.1. Momentum Balance of the Cavity

A momentum balance for the fluid enclosed by the control surface shown in

sketch 2 gives

(4.1)

where = Zitrdr. = nddxs and The x-

<

n*-

[1* V"

,- y = 0 Line

i l

^-Control Surface A2

-ML

r

A3 H

SKETCH 2
-32-

component of momentum balance will be

+ pnd fuv d*! = -2JT/ \PZ - P^rdr + ndfrxydx1 + TT^/T^dxg (4.2)

Introducing u = u + u' ,v = v + v' to eq. (4.2), taking the time average and

neglecting the viscous shear stress on the free surface will give

+/>7r d f^av + vXr^dx-L = -2n f \PS - Pjnfr + ird f r^ dx3 (4.3)

Using the following definition will reduce equation (4.3) to equation (4.4)

COP = cavity pressure drag coefficient =

CDS = drag coefficient of the sting = -

= + CDP + CDS (4.4)

However, by definition, the drag coefficient of the cavity can be written as

CD - CDp + CDS

The combination of equations (4.4) and (4.5) results in a new relation for the

drag coefficient of the cavity:
-33-

CD = - rrd / (WO + 'ifTr)dxl/^-puz

9 7rd2

/4 (4.6)

Relation (4.6) reveals that it is possible to determine the cavity drag coefficient

by evaluating the mean and turbulent momentum transfer terms along a

straight line connecting two corners of the cavity. Throughout this section, this

line will be called y = 0 line. Roshko (1955) found that the skin friction

coefficient along the walls of two-dimensional cavities were two orders of magnitude less than the pressure and pressure drag coefficient. Therefore, skin friction in the cavity as a contribution to the drag is entirely negligible and

CD w CDP (4.7)

4.2. Measurement of the Cavity Drag Coefficient

The drag coefficient of the cavity is determined by measuring the integral of

the pressure distribution over the solid walls of the cavity. Figure 39 presents

the variation of drag coefficient with 6/90 in the three distinct regimes of cavity

flow as denned in chapter 3. This figure reveals two rather significant characteristics of the cavity drag. First, the average level of the drag coefficient stays

low as long as the flow remains in either the non-oscillating mode or the selfsustained oscillating mode. A second feature is the existence of a critical b/e$,

in which, by exceeding its value, the drag coefficient of the cavity jumps to a

value which is one order of magnitude higher than the typical average drag

coefficient during oscillating modes. During the course of the wake mode the

drag coefficient increases monotonically and reaches a value of 0.39 at

b/00 = 250. No more measurements were made beyond this point due to model

limitations.
-34-

4.2.1. Cavity Drag Coefficient in Non-Oscillating Made. The drag coefficient

in non-oscillating mode has a typical value of 1.3 x 10"4 at 6/90 = 77- When the

cavity was replaced by a solid surface, the drag coefficient of this surface based

on the upstream boundary layer shape factor was estimated to be .7 x 10~3. In

turn, this drag coefficient is equivalent to that of b/e0 = 98 of Mode II of the

oscillations. However, based on observations in Chapter 3, the presence of the

cavity reduces the shape factor of the downstream boundary layer and consequently may increase the skin friction drag on the rest of the model. At this

point, it is not clear that a net drag reduction can be obtained for the whole

model.

4.2.2. Exponential Growth of the Cavity Drag Coefficient During Oscillating

Modes. Distributions of the drag coefficient in Modes II and III, plotted on a loglinear scale, are presented in figure 40. At the beginning of the oscillating

modes (Mode II in this study), the drag coefficient has a value of 0.9 x 10~3 at

b/e0 = 85. The drag coefficient then increases exponentially and reaches a peak

value of 0.02 at b/eD = 108. Toward the end of Mode II, the drag coefficient

decreases and reaches a minimum value of 0.004 at b/90 - 120. The transition

region between Mode II and Mode III is marked by an increase in the drag

coefficient value; COP = .0072 at b/60 = 128. Mode III exhibits another course

of exponential growth for the drag coefficient value. A final value of 0.39 for the

drag coefficient is reached at b/e$ = 250 before transition to the wake

mode occurs.

4.3. Cavity Pressure Distribution

A better understanding of the drag generation mechanism can be obtained

from pressure distributions along the solid surfaces of the cavity.

Figures 41 through 43 present the pressure coefficient, \CP

{ PC. =

p,

, along
-35-

a radius of the upstream and downstream faces of the cavity for several

selected values of b/e$. With each distribution the corresponding value of CDP

is shown.

4.3.1. Pressure Distribution far Non-oscillating Mode b/8$< 82 . Figure 41a

reveals that in the non-oscillating region, the cavity is slightly under pressure,

i.e., Cp < 0. No significant differences were observed between the upstream and

the downstream face pressure distribution,which corresponds to closed and

open symbols, respectively.

4.3.2. Pressure Distribution for Osculating Modes.

1) Mode II [82 ^ b/fy < 12o|

Figures 41b through 41 g show the cavity pressure coefficient distribution for

Mode II of the oscillation. All cases have negative pressure distribution over the

entire upstream face. The most important point to note in all cases is the steep

pressure rise near the outer edge of the downstream face. In each case, Cp

reaches its maximum positive value (Cp ) at the corner. The magnitude of

C0 reaches a maximum at b/90 - 108, then decreases toward the end of Mode

^inax

II. The downstream face experiences a negative pressure below the downstream

corner with the same level and trend as that of the upstream face. Thus, the

contribution to the total drag from these two portions of the cavity walls cancel

each other. This leaves the outer edge of the downstream face with Cp > 0 , the

main contributor to the cavity drag. It is also interesting to mention that in all

cases, the average pressure on the upstream face is negative; and thus this face

experiences thrust during Mode II of the oscillations.

The pressure buildup near the upstream corner is evidently due to the

deflection of the shear layer into the cavity. This deflection of the shear layer
-36-

causes a strong inflow to the cavity and contributes to the generation of a stable

vortex. A large portion of the cavity experiences negative pressure coefficient

due to the presence of strong recirculating flow inside of the cavity. Similar

types of observations were made by Johannesen(1955) , Roshko(1955), McGregor

and White (1970), Charwat, et el. (1961), Fox (1964) and Rossiter (1964).

2) Mode III 120 < b/<90 < 155

Figures 42a through 42d present pressure distribution for Mode III. Except

for a smoother pressure rise near the outer edge, the pressure distribution on

the upstream face shows behavior similar to that of Mode II,

4.3.3. Pressure Distribution far the Wake Mode. Pressure distributions for

three cases of the wake mode are shown in figure 43 . Two main features distinguish the wake mode from the oscillating mode. First, the entire upstream

face of the cavity experiences positive pressure, i.e., Cp > 0. Second, the stagnation point, the position of CPmgx, steadily migrates toward the center of the cavity. We expect that for a large enough value of b, the stagnation point should

settle at the center of downstream face. While pressure on the rear face is

increasing, the average base pressure level decreases as b/90 increases (figure

43 ).

4.3.4. Overall Characteristics of the Cavity Pressure Distribution. Figure 44

presents a collection of typical upstream corner pressure distributions for

different regimes of the cavity flow. The start of the self-sustained oscillations is

accompanied by an overall drop of the upstream face pressure distribution. The

transition from Mode II to Mode III (b/do = 128) is marked by a substantial

increase in upstream face pressure. A flow visualization study revealed

that the absence of a strong recirculating flow inside the cavity might be responsible for the pressure rise of the upstream face. In Mode III, low upstream face
-37-

pressure recovers and shows a level similar to that of Mode II.

Upon the loss of the self-sustained oscillations, another drastic decrease in

the -average level of the upstream face pressure distribution occurs. At

6/9^~E"2io (b/R » l.~7). the average upstream face pressure coefficient has a

value of-0.13.

Figure 45 represents variation of maximum pressure, Q,^^, with b/8$. The

value of Cp increases as b/eo increases until it reaches a local maximum

at b/8o = 108. This maximum is immediately followed by a valley which marks

the last point of Mode II. The transition region and Mode III show a monotonic

increase in the value of CpmMI. Appearance of the wake mode is marked by a sudden increase in the value of CL from +0.17 to +0.43 at 6/flb *" 162. The value " tn&x

of Cp shows slight decrease during the wake mode. The postion of the stagnation point shifts toward the center of the cavity as b/eQ increases. It is important that the location of Cp is fixed in the vicinity of the upstream corner in

the self-sustained modes. Migration of this point toward the center of the cavity

starts only when the cavity loses the self-sustained oscillation to the wake mode.

The rather dramatic change in the pressure field of the cavity caused by exceeding a critical value of b/e$ has been observed by Roshko (1955), Charwat (1961),

<» Fox (1968) and Maul and East (1963). However, the role of self-sustained

oscillations in determining this critical b/0.. was in general unnoticed.

Variation of CDP at the middle of two-dimensional cavities studied by Roshko

(1955) and Fox (1968) as well as those of the present study are shown in

fig. 46. It is interesting that in all three cases the location of steep

jump in the value of C agrees fairly well.
-38-

4.4. Observation of Sudden Drag and Pressure Change in Other Flows

Several other investigators have reported a sudden change from a steady lowdrag flow to a highly non-steady high-drag flow over cavities, cutouts, or bluff

bodies in tandem. The intention in this section is to compare some of their

results with those of the present study, and to give a possible explanation for

the existence of the critical length, b/00.

Koenig (1978) in his study of flow over the combination of a disk and a

cylinder, defined two regimes based on the drag behavior of the system, namely,

sub-and super-critical. Drag in a subcritical regime has values one order of

magnitude lower than that of a supercritical regime. Separated shear layer in a

subcritical regime smoothly joins onto the rear body, while in the supercritical

regime the shear layer becomes unsteady on a much greater scale. Koenig's flow

visualization study revealed that the optimum low drag case appears to be a

more general example of flow over cavities. Figure 47 presents optimum drag

measurements (Koenig, 1978) superimposed on the present drag measurements.

Despite the basic differences in the geometry and the curvature of the shear

layer, figure 47 confirms Koenig's flow visualization conclusions.

Koenig also reports the presence of a steady and stable flow field in the

supercritical region. Production of sharp acoustic tones was another feature of

subcritical regime that was reported by Koenig. The symptoms of subcritical

regime are similar to those of simple>self-sustained»cavity flow oscillations.

Mair (1965) reported a sudden change from steady low-drag flow to nonsteady

high-drag flow near the base of a streamwise oriented cylinder as the gap

between the cylinder and a downstream disk was changed. Zdravkovich (1977)

observed an abrupt start of vortex shedding from an upstream cylinder accompanied by an increase in drag of a system comprised of two cylinders in tandem

when spacing between them was changed.
-39-

The presence of a feedback loop actively doing the task of crosscommunication between two corners in the gap generated by two bluff bodies in

tandem may possibly explain extra-stability and low-drag characteristics of such

flow systems.

The lock-on of the separated shear layer, or the position of Cp^^ to the

downstream corner due to feedback mechanism, has been overlooked by previous investigators and has often been considered a secondary effect. A further

discussion of this matter will be given in section 4.10.

4.5. Estimate of CD by Evaluation of — g-and — g- Terms Along y = 0 line

An estimate of CD can be obtained by integrating the shear stress and mean

momentum term over the periphery of the cavity (equation 4.6). LDV technique

was used to measure the x and y component of the velocity vector, to determine

*d7 'Lb'lj'

— =- and — =- terms along y = 0 line. For each case between 20 to 30, measureUf U£

ment points were selected along the y = 0 line, depending on cavity length.

Smaller spacing between measurement points was selected close to the downstream corner. Due to the flow field symmetry of the wake mode, no repeatable

ni tit

measurements could be done in that regime. Thus, evaluation of — a~

Ue »

terms were limited to the oscillating regime of the cavity flow.

4.5.1. Confirmation of Exponential Growth for Cj). Figure 40 presents the

Iff '"J I ' Ti'O'

variation of CD based on the integration of —jr and — g- along the y = 0 line

Ug Ue

with b/9o- It is important to note that the exponential growth of CD during oscillating modes has been confirmed by this independent method. Except for the

cases close to the end of Mode II and close to the transition region between

Modes II and III, the absolute values of CDp and CD agree fairly well. Study of
-40-

17 1Z

the 77—, -rr—, —g- and —5- distributions along the y = 0 line is essential in

t/e ^e 1/8 t'e

further understanding of the mechanism which is responsible for the CD

behavior. This is the subject of the following chapters.

4.6. Distribution of -jj— Along y = 0 line

"8

Provided that the distribution of 77 at a fixed x position is uniform on the

periphery of the cavity, the integral of v along the y = 0 line should be minimal

for a satisfactory mass balance of the cavity.

77

Starting from figures 48a through 48g, a sequence of -rr— distributions for

t/e

a few values of b/e0 in Mode II is given. Each figure is accompanied by its

corresponding drag coefficient.

17 A common feature of all -rj— distributions is the existence of a narrow region

MB

of inflow to the cavity -U close to the downstream corner. The negative i7

region is balanced by a broader region of the outflow from the cavity +77 close to

the upstream corner. The stronger the inshooting of flow into the cavity, the

higher is the peak of the positive 77 region. The peak of -v increases as b/e^

increases and reaches a maximum at 6/9= 112.5, which corresponds to the

maximum drag case. This peak eventually decreases toward the end of Mode II,

In Mode III the v region becomes slightly broader and shows a small change

(figure 49).

In terms of stable vortex inside the cavity, ——; distribution can be inter-

"e

preted as the induced velocity of cavity vortex on the y = 0 line. Therefore, it

can be concluded that the strength of the vortex increases during the Mode II as

b/6o increases. The strongest and most stable vortex appears at the drag
-41 -

maximum case; i.e., at b/&o = 112.5, F = 5 Hz. The cavity vortex becomes weaker

during transition and recovers its strength during Mode III.

A check of mass balance showed that between 93

 an<* 97 percent of mass

entrance, f vdx l^, REGION*exits the cavity. Either a slight angle of attack or

presence of three-dimensionalities in flow are responsible for imperfections in

the mass balance. It remains to mention that it is logical to assume that positive

entrainment to the shear layer, at the upstream region, is the primary process

for generating +U region and consequently causing the deflection of the shear

layer into the cavity at the downstream corner. Relation between the positive

entrainment and amplification of the perturbations in the initial region of the

shear layer will be discussed in the flow model section.

4.7. Distribution of -pr— Along y = 0 lane

"e

The ~- distribution along y = 0 line (figures 50a through 50g) exhibits some

Ue

interesting features. A typical distribution shows that ——increases from zero

Ue

at x = 0 to a peak value, and is usually followed by a stagnation flow-like

behavior where velocity experiences a sharp decrease down to zero at x = b.

The stagnation region often shows several local peaks and valleys. In Mode II,

the peak value of yp— shows an increase until it reaches its maximum at

t/e

6/e0 = 112.5 , corresponding to the maximum drag case. The end of Mode II and

u

the transitional region are marked by lower values for the peak of -:— similar

ue

v

to that of the — distribution.
-42-

Mode III does not show a substantial change in jj— distribution or value of

peak^velocity during its course (figure 51). It is important to note that jj—lmax

9

changes from 0.3 to 0.6 during its rise in Mode II, which is a clear indication of

deflection of shear layer into the cavity. Consequently this deflection will cause

the stagnation of the high velocity portion of the shear layer velocity profile at

the downstream corner. It seems that the natural diffusion of the shear layer

due to the cavity width increase (shear layer length), plays aless important role

in bringing higher velocity fluid into the stagnation region. Decrease in the peak

value of ——|max during the transitional region, despite the increase of cavity

t/e

width, supports the previous argument. In Mode 111, the averaged level of the

1

I ma (whil x e having a value of 0.5) stays constant (figure 51). Diffusion of the

shear layer into the cavity in Mode III probably plays a more important role than

the deflection of the shear layer in bringing high velocity fluid to the stagnation

region.

4.8. Evaluation of —g- Along y = 0 lane

The —£-term is evaluated along the y = 0 line, using the independent measn j <5T

urements of ——and —— and the results are shown in figures 52a through 52g.

t/e Ug

A typical distribution of —g- shows a wide region of positive values for —gwhich is followed by a sharp negative peak close to the downstream corner. The

region of positive values of —— contributes thrust to the net force balance while

the region of negative values of —j- contributes drag, (figure 53)
-43-

6 _

/

1L1} — ~-dx is always negative during Modes II and III of the

0

~

t

7

V

oscillations. Even though —— distribution showed a fairly good mass balance

Ug

for the cavity, the -v region was weighted by higher values of -HZ during

integration along the y = 0 line, resulting in a positive net drag for the cavity.

For convenience, the following conventions are introduced. The net contribution of ^-term to total drag coefficient

B *

=

 CDOV ~ fj J

u- n

Net contribution of —^-term to total drag coefficient

=-

 u c

Total drag coefficient= CD.

Therefore

Cn —

A comparison between C^ and Cjxnr contributions to the net drag will be

given in the next section.
-44-

4.9. Shear Stress Distribution! Along y=0 line

Figures 54a through 54g present shear stress distribution for Mode II. Shear

stress grows along the y = 0 line, reaches a peak and vanishes steeply as the flow

approaches the downstream stagnation point. Often two peaks appear before

the decrease in the stagnation region starts. The peak of — g-reaches the maxC/8

imum value of .0063 at b/Oo = 112.5, which corresponds to the maximum drag

case. During the transition region the peak value of the shear stress decreases

to 0.003.

The start of Mode HI is marked by the appearance of a wide region of high

shear stress value close to the downstream corner. In Mode III (figure 55), the

peak value of ^\- increases from .007 at 6/<?0 = 132 to 0.0098 at b/e0 = 146.

"e

C Crunr

Figure 56 presents the variation of — — and — — — versus 6 /%• It also reveals

that in Mode II, especially in the beginning stages, C^ contributes up to 80% of

the total drag coefficient. As b/e^ increases, the role of C^mr becomes more

important but its contribution never exceeds that of Cy^. Mode III shows a

dominant role for C^^,. Immediately after transition, C^^ constitutes 60% of

CD and increases to 70% at the critical gap (6/90 = 146). The message of figure

A/'IJI*

56 is that consideration of the shear stress, — g-, as the only means of

turn transfer along the cavity y = 0 line will result in a 30 to 80% underestimation of the cavity drag depending on cavity length.

Kistler and Tan (1967) made such estimations of the drag coefficient of a

two-dimensional cavity with a turbulent upstream boundary layer. Using hot

wire anemometry, he determined the shear stress distribution along the cavity

mouth. His results show up to 60% underestimation of Cp with respect to COP.
-45-

It is important to recall that, due to insensitivity of hot-wire techniques to the

ft titJ

direction of the flow, it is impossible to accurately estimate the term —j- along

"e

the y = 0 line.

4.10. A Possible Flow Model

The drag coefficient of the cavity stays low as long as self-sustained oscillations are present in the flow. Flow visualization studies have revealed that the

shear layer (time averaged) re-attaches onto the downstream corner. Only during the wake mode does the shear layer stagnate below the downstream corner.

The fact that the stagnation point (average position of maximum pressure) is

always fixed at the downstream corner supports the flow visualization conclusion. There remains one question to answer: what mechanism keeps the flow

from stagnating below the downstream corner and consequently results in a low

drag value for the cavity? What follows is a possible answer to the above question.

It is logical to assume that the volume of flow that is entrained by the shear

layer is proportional to the strength of the shear forces in the shear layer. Figure 57 reveals that the amount of fluid that is entrained by the shear layer (or

exits the cavity If vdx \^\ follows the trend of maximum shear stress of the

shear layer at a fixed streamwise position; e.g., x/90 = 65.

On the other hand, mass balance of the cavity requires that the shear layer

deflect into the cavity in order to supply fluid for positive entrainment to its initial region. Figure 23 presents the location of ~—= 0.5 in the shear layer. At

C/B

the beginning of Mode II (b/60 = 83, F = 6 ), where shear stress level and growth

is low, the shear layer displays mild curvature. At the maximum drag case

(b/e0 = 112.5), where the shear stress level throughout the shear layer is
-46-

higher. the shear layer shows steeper curvature at the initial region and a sharp

deflection into the cavity. Therefore, the degree of the deflection of the shear

layer depends on the nature of the instability growth in the shear layer. The

severity of the shear layer deflection is decisive in the nature of the shear layer

interaction with the downstream corner.

Rockwell and Knisely (1979) show that if the respective location of the downstream corner is raised by —= 6. the rms transverse fluctuations, v

r

. will

&o

drop by a factor of 2. Sarohia (1976) demonstrated that by external mass injection to the cavity, it is possible to deflect back the shear layer at the corner and

weaken the strength of the cavity flow fluctuations. Based on these experiments

it can be concluded that for given 6 /&Q the displacement of the impinging shear

layer on the corner due to deflection should occur in such a way that the feedback loop would not lose its strength. Any severe external interference with the

established self-sustained oscillations would result in an overall drop of the

fluctuation level throughout the shear layer. These two experiments provide the

missing link between the role of the corner and the fluctuation level. Once the

feedback loop due to the frequency selection process is established, any inboard

displacement of the velocity profile caused by the natural tendency of the wake

mode will be resisted by the following mechanism. A negative displacement of

velocity profile at the downstream corner will result in the lowering of the shear

stress level which will be followed by a decrease in the volume of positive

entrainment. However, a less positive entrainment would result in reduced

deflection of the shear layer and restoration of the initial position of the shear

layer velocity profile at the downstream corner. This mechanism will not allow

the full velocity profile at the impinging corner to be displaced more than its

total thickness, and as a consequence, keeps the location of CPmax fixed at the

downstream corner. The strength of this restoration mechanism depends on
-47-

thie stability of the cavity flow oscillations, which was already established by the

frequency selection process. It is possible that because of the existence of such

a restoration mechanism, some low frequency oscillations appear in the shear

layer. According to the above process these frequencies need not be the harmonics of fundamental frequency. As stated, Knisely (1980) observed

0.5, 0.4 and 0.6 of fundamental frequency in the shear layer over cavities. In

the present study severe low frequency components were observed in Mode III

(last mode) of the oscillations. Toward the end of Mode III, as a result of a

decrease in the effectiveness of the feedback loop, the restoration force became

weaker. Consequently, the natural wake mode of the front body overcame the

restoration mechanism, and the shear layer had more freedom to move below

the downstream corner and stagnate inside the cavity.
-48-

Chapter 5

Control of Cavity Flow Oscillations

One of the objectives of the present study was to develop and use a technique

which is capable of controlling the frequency and amplitude of established flow

oscillations. However, such a technique can be used to obtain information

regarding the necessary conditions for the onset of self-sustained oscillations.

This chapter will report on and discuss a series of experiments regarding the

application of a new technique for forcing the cavity shear layer. Throughout

these experiments, the response of the cavity shear layer, both in nonoscillating and oscillating modes, is studied against different forcing frequencies

and amplitudes.

5.1. Forcing of the Shear Layers

Forcing of the unstable and stable modes of laminar or turbulent shear

layers has been the topic of many previous studies. Basically, forcing techniques

fall into two catagories:

1) Perturbations to the bulk of either flow stream (Miksad, 1972, and Ho and

Huang, 1982).

2) Localized perturbations in the initial region of the shear layer (Oster and

Wygnanski, 1982).

In the first type of forcing, e.g., by oscillating one of the streams, the whole

streamwise length of the shear layer experiences forcing effects. As a result no

accurate information can be deduced about the influence of perturbations on

the initial regions of the shear layer.
-49-

Wygnanski and Oster showed that forcing effects can be localized by

sinusoidal oscillation of a pivoted thin flap at the leading edge of the shear layer

splitter plate. The other technique used to localize the effect of forcing is to

introduce controlled excited (T-S)* waves, amplified by the boundary layer, into

the shear layer. The idea of actively forcing the boundary layer was first proposed by Liepmann (1943). The most popular technique to facilitate the idea

was the vibrating-ribbon developed by Schubauer and Skramstad (1947). The

difficulty with either the vibrating-ribbon or the flapping edge technique for the

present experiments is that it cannot be adapted to axisymmetric flows.

5.2. Surf ace Heating Technique

Liepmann, Brown and Nosenchuck (1982) recently showed that timedependent surface heating can be used to introduce perturbations into laminar

boundary layers. Such a technique can be easily adapted to excite T-S waves in

an axisymmetric boundary layer such as that of the present study. The idea of

periodic heat release into the boundary layer for the purpose of exciting T-S

waves was also realized by Kendall (1967) in supersonic boundary layers. By

applying periodic current to two closely spaced parallel wires flush mounted on

the wall, Kendall was able to initiate glowing between the two wires and consequent heat release into the boundary layer. As a result of this periodic heat

release into the boundary layer, T-S waves of up to 30 kHz were excited.

Requirement of low pressure to attain glowing makes this technique of limited

use for subsonic flow applications.

For cases in water, liepmann, et al. (1982) show that the effect of "heating

and cooling of the surface is roughly equivalent to suction and blowing or negative and positive surface displacements," with

* Tollmien-Schlichting
-50-

v' Prl* far ]

*• ~ »i- R

11/3

 d LOGu. AT

T T

where // is dynamic viscosity, !T is the temperature of the fluid, Pr is the

Prandtl number, vr is the width of the strip, and Rx is the Reynolds number.

Such an equivalent normal surface oscillation would periodically redistribute

vorticity. Depending on the state of the boundary layer, the external perturbations would either be dampened or amplified exponentially. Linear theory can

ffi

predict the exponential growth of the (T-S) waves, based on the parameters -TT— Ug

and Re<5* for a given pressure distribution.

5.2.1. Application of T-S Waves as a Means of Forcing of the Shear Layer.

Forcing T-S waves in boundary layers has been practiced for a long time. However, to our knowledge, no use has been made of T-S waves to force a separated

shear layer resulting from such a boundary layer. In the present study,

inspired by the experiment of Liepmann, et al., periodic heating of a thin film

was used to excite the T-S waves in the boundary layer upstream of the cavity

shear layer, benefitting from amplification of the waves before their introduction into the shear layer (figure 58). One can imagine that the effect of the T-S

waves or v* component at the upstream corner of the cavity would be similar to

that of the oscillating flap at the leading edge of the splitter plate in Oster and

Wygnanski's experiment. It will be seen later that the frequencies needed to

excite the free shear layer are available in the range of boundary layer T-S

waves. Thus, a transformation of boundary layer T-S waves to the shear layer

Kelvin-Helmholtz waves is possible.
-51-

5.2.2. Strip Heater Set-up. Initially three strip heaters were mounted on the

ellipsoidal nose of the model (figure 59) at s = 9.2 cm, 12 cm, and 15.5 cm where

s is the distance on the model surface from the nose of the model. Double stick

scotch tape was used to electrically isolate heaters from the model. The first

and third heaters were stainless steel ribbons 3.7 mm (.125 inch) wide, .051 mm

(.002 inch) thick, wrapped around the front nose of the model. The second

heater was narrower, ,08 cm (.2 inch) wide. Each heater entered the model from

a flush mounted round adaptor, designed to connect the heaters to the current

supply lines. The resistance of the strip heaters was 2-4 ohms.

A high quality commercial dual channel power amplifier (Hafler DH-500) was

used to supply current to the strip heaters. DH-500 is a 500 watts per channel

amplifier which can be easily modified to provide frequencies even below 1 Hz

while delivering adequate power to the strip heater. An HP Function Generator

(3325A) was used to provide the selected frequency to the power amplifier. It

should be mentioned that due to the quadratic dependence of Joule heating on

the voltage, the input frequency to the power amplifier had to be half of the

desired forcing frequency.

5.2.3. Important Parameters of Strip Heater Technique. Several parameters

are important for the efficiency of a strip heater in forcing both the T-S waves

W and the shear layer, liepmann, et al. (1982) stated -7- (where W is the strip

o

heater width and 6* is the boundary layer displacement thickness) is an important parameter in effectiveness of strip heater operation on the local forcing of

T-S waves. The state of the free stream pressure gradient is another important

W factor. In the present study, the width-to-boundary layer thickness ratio, —o

was 5.6 for the first strip heater and maintained at 3 for the second and third

heaters.
-53-

The preliminary experiments indicated that for a fixed input frequency and

power level, the first heater, s = 9.2 cm, was far more effective in forcing the

free shear layer than the second and third heaters, set at s =12 cm and s = 15.5

W cm, respectively. Also, it was noticed that having the same ratio —?-, the

o

second heater operated more effectively than the third heater. Figure 59 reveals

that (T-S) waves initiated by the first and second heaters benefitted from a positive pressure gradient as well as a longer period of amplification than that of the

third heater.

In the final version of the experimental model, a single strip heater was flush

w

mounted at s = 9.2 with —r = 5.3.

6

5.3. External Forcing of the Non-Oscillating Cavity Plow

The receptivity of the cavity shear layer to externally imposed disturbances

was studied. Two different cavity widths were examined, with corresponding

b/e = 6Q< 6/elmin and 6/e=77< &/e|min- Throughout this section these

selected widths will be called case A and B respectively. In both cases edge velocity was fixed at Ug = 22cm/feec. Cavity flow response was measured in terms of

the maximum shear stress that was developed in the shear layer at a station

.254 cm (.1 inch) upstream of the downstream corner.

Spectral analysis of the response velocity fluctuations indicated a one to one

relation between forcing and the response frequency. For Case A, figure 60

presents the response of the cavity shear layer to a wide band of the forcing frequencies at different levels of forcing power. The results indicate that the level

of shear layer response increases as the forcing power increases. For forcing

levels above 9 watts, resonance peaks appear at the forcing frequency of 7.2 Hz.
-53-

An overall phase measurement revealed that Case A satisfies the phase criterion •£— = -77- w 2, at the resonance (F = 7.2 Hz), thus verifying it as the

STT C/c

second mode of oscillation. The absence of forcing caused the resonance mode

to disappear, thus revealing that the flow oscillations are not self-sustaining.

Figure 60 also reveals that the existence of resonance depends on a minimum

amplitude threshold.

Figure 61 presents the response of the cavity shear layer for Case B at a fixed

forcing level of 9 watts. This case shows a resonance peak at the forcing frequency of 6.2 Hz with r— = 77— ^ 2. Appearance of resonance peak for Case B filT\ UQ

at a lower forcing level than that of case A indicates that the threshold level

decreases as b/e0 increases. In Case B, the effect of initial lower amplitude of

the forcing was compensated by longer cavity length in such a way that reduced

the required forcing level for the resonance condition. The effect of higher forcing power on the exponential growth of the perturbations was to increase the

general level of fluctuation growth (figure 62a). Figure 62b presents the growth

behavior of the different frequencies for a fixed forcing level (9 watts) for Case

B.

It can be concluded that external frequencies can be imposed on naturally

non-oscillating cavity flows, within the receptivity range of the shear layer.

These frequencies can cause the cavity flow to resonate if they satisfy the following conditions:

1) •£—= N for the given width (b); and

2) A > AIK where A is the amplitude of external frequency at the downstream corner and Af^ is the threshold amplitude. It is logical to propose that

as the cavity width increases, the threshold level decreases to such an extent
-54-

that a flow background frequency, which satisfies the phase criterion and has

sufficient amplitude, will initiate the self-sustained oscillation.

It is now clear that due to the absence of proper frequencies with sufficient

intensities in the flow background, the first mode did not appear naturally in

this study. However, by selecting a proper forcing frequency and initial amplitude, the first mode of oscillations

JL_= was simulated for Case A.

2rr £/c

This was accomplished at a forcing frequency of 4 Hz, a power level of 35 watts.

A flow visualization of this case is shown in figure 63b. In comparison to figure

63a, which is the same case but unforced, two features are noteworthy. The first

is the appearance of a small amplitude wave in the shear layer of the forced

case. The second is the interaction of this wave with the downstream corner and

the resultant strong recirculating flow inside the cavity.

It is interesting to note that the response of the cavity flow to external forcing is not instantaneous. Figures 64a and 64b present simultaneous traces of

X X the velocity fluctuations taken at stations corresponding to —= 10 and — = 60,

0 &

respectively. The cavity was in the non-oscillating mode with 6/9= 66. Figure

62a indicates that for an external forcing frequency of 7 Hz, simulating Mode II,

it takes 3 cycles of the oscillation before the peak amplitude establishes itself.

When the forcing is turned off it takes many more cycles for the amplitude of

oscillations to diminish.

5.3.1. Effect of External Forcing on Cavity Drag in the Non-Oscillating

Mode. Case B, at the resonance, had a drag coefficient of 0.0012. This drag

coefficient was one order of magnitude higher than that of the same case but in

'ii'lj T//771

the absence of forcing. Distributions of —g- and —g- along the y = 0 line for

both the forced and unforced cavity are given in figures 65a and 65b. These
-55-

figures reveal that the difference in the drag coefficient of forced and unforced

cases can be contributed to the development of a strong shear stress level in the

forced case.

5.4. Effect of External Forcing' on a Naturally Oscillating Cavity Flow \b/d> b/e\

Once the cavity flow established the self-sustained oscillations

> 6/^jlminj, the flow resisted any external disturbances. Thus in order to

interact with the flow oscillations, it was expected that the amplitude level of

external forcing must be comparable to that of the self-sustained fluctuations

in the flow.

A non-dimensional cavity width of 6 /e§ - 95 was selected for an external

forcing experiment. The cavity flow was oscillating at 5.4 Hz in Mode II. An

external forcing frequency of 6 Hz was introduced to the shear layer. Depicted

in figures 66a through 66g are the spectral distributions of velocity fluctuations

for several levels of forcing as the forcing power was changed from 2 to 27 watts.

Spectra reveal that as the amplitude of the forcing increases, the amplitude

of natural oscillation decreases and eventually disappears. At the forcing power

of 40 watts, the forced frequency becomes the dominant frequency of the oscillations. Oscilloscope traces of the signal for the case when both natural and

forced frequency have the same amplitude, as seen in figure 66d, are given in

figure 66h. Strong modulation of the signal indicates that both frequencies are

present simultaneously. Note that in the co-existence region, figures 66b

through 66f, the peak amplitude of both frequencies is lower than that of the

single frequency regions of figures 66f and 66g.

Miksad (1973) in his study of dual excitation of a free shear layer observed a

similar interaction of two frequencies. He reported that if the excitation
-56-

amplitudes were adjusted so that both disturbances reached finite amplitude

together, the final amplitudes of both components tended to be reduced, but

not as strongly as when one component was clearly dominant. Miksad concluded

that the distortion of the mean flow by the stronger frequency made velocity

field unsuitable for the growth of the weaker frequency. Similar behavior had

also been observed by Sato (1970) in symmetric wakes.

Important to our study is the fact that the amplitude of the forced frequency

in figure 66g, where natural frequency just disappears, is equal to that of the

natural frequency in the absence of the forced frequency (figure 66a). This fact

establishes the amplitude of the natural frequency as the level which should be

exceeded by the amplitude of forced frequency to change the frequency of the

oscillations. With this rule it is possible to impose forced frequencies on the cavity shear layer within the receptivity range of the shear layer. These forced

oscillations are not, in general, self-sustaining. During the mode switching

region (hysteresis region), self-sustaining oscillation in a desired mode could be

achieved by providing a proper frequency within the hysteresis region of that

mode. Thus, the mode switching phenomena could be eliminated by external

forcing. Transition to the wake mode could be prevented by enhancing the

amplitude of natural frequency through external forcing. It is possible to

extend Mode III, last mode in this study,by a 20 momentum thickness through

external forcing. However, once the cavity entered the wake mode, it was not

possible to reestablish cavity flow oscillations by external forcing.

A map of possible interaction with the cavity flow oscillations observed in this

experiment is given in figure 67.
-57-

5.5. Cancellation Experiment

Cavity width, of b/6b = 82, with frequency of 6 Hz/was selected for the cancellation experiment. The signal from a downstream location was fed to a function

generator (Tektronix FG504) which had a built-in phase lock loop. It was possible to lock to lower harmonics of the input signal. Phase variation of the output

signal with respect to the first subharmonic of the input signal was possible

through an independent manual adjustment knob. The power amplifier was then

supplied by the output of the function generator. The experimental set up is

sketched in figure 68.

The following procedure was conducted in the cancellation experiment.

1) Output amplitude of function generator was set at 0.

2) Phase lock loop was locked to first subharmonic of the input signal.

3) Knowing the threshold amplitude, the power was fixed at a level

which was enough to simulate conditions as depicted in figure 66d.

4) Step (3) was repeated for different phase angles between input and

output of the function generator (phase lock loop).

Due to the small rapid phase changes of the cavity flow oscillations it was not

possible to have a phase lock situation for an extended period of time. Occasionally a correct phase angle could be found where the forced and natural frequencies would interact in such a way that wide band rms fluctuations were
-58-

reduced. A sample of such a cancellation is shown in figure 69. A reduction in

rms fluctuation was achieved by a factor of 2 during 30 cycles of the cavity

shear layer oscillations. Liepmann and Nosenchuck (1982) report a factor of 3-3

in the reduction of rms fluctuations of the natural (T-S) waves in the flat plate

boundary layer.
-59-

Chapter 6

CONCLUSIONS

8.1. Summary of Results

(a) Investigation of the flow field in the oscillating mode reveals that the

momentum thickness of the cavity shear layer grows linearly with x/e$. The

linear growth of the momentum thickness may indicate the existence of selfsimilarity in the flow. However, the self-similarity argument for the free shear

layer requires a constant maximum shear stress along the layer. In contrast,

cavity shear layers show an exponential growth of the maximum shear stress

with X/QQ. The cavity shear layer in the non-oscillating mode does not develop

any noticeable shear stress along its span. The velocity measurements also verify the existence of a recirculating flow inside the cavity. Streamwise and

transverse velocity fluctuations as high as those of the shear layer are detected

in the recirculating region.

(b) Upstream laminar boundary layer has a shape factor value of 2.5. The

effect of the cavity on this profile is found to decrease the shape factor to a

value of 1.54, which is close to that of the turbulent boundary layer.

(c) The drag measurement results presented in figure 39 indicate that the

drag coefficient of an axisymmetric cavity with laminar upstream boundary

layer remains low as long as the cavity flow stays in a non-oscillating or selfsustained oscillating mode. Results also reveal that the drag of the cavity in the

non-oscillating region is less than the case in which the cavity was replaced by

a solid surface.
-60-

(d) Flow visualization study (figure 18) in conjunction with the drag measurements reveals that for a long enough cavity width, b/R> 1.2, cavity flow exhibits

a wake-type behavior.Also, the average drag coefficient in this mode, CD » 0.35,

is one order of magnitude higher than the highest drag coefficient in the oscillating mode, CD - .036.

(e) The most important result is presented in figure 40. The drag coefficient

of the cavity exhibits an exponential dependence with b/e$ in Mode II of the

oscillations followed by a decrease in the same mode and another course of

exponential growth in Mode III. An independent estimate of the drag coefficient

through measurement of the shear stress and mean momentum transfer terms

wu along the y = 0 line confirmed the exponential growth of the cavity drag.

Furthermore, these measurements revealed that the uv term is responsible for

30% to 80% of the net drag coefficient of the cavity.

(f) Study of the cavity pressure distribution revealed that the pressure rise

close to the outer region of the downstream corner is responsible for the drag

coefficient of the cavity.

(g) In the oscillating modes of the cavity flow, the maximum pressure

coefficient (Cp ) always occurs at the upstream corner and stays so until the

wake mode appears. In the wake mode the position of maximum pressure

coefficient (Q,mwt) moves toward the center of the cavity as b/do increases.

(h) The existence of the feedback loop introduces an extra stability to the

cavity flow field which includes a fixation feature of Cp at the upstream

corner and a resultant low drag for the cavity.

(i) The distribution of the shear stress along a straight line connecting two

corners of the cavity is shown to have direct effect on the positive transverse
-61 -

velocity (-K;) distribution on the same line. It is also shown that the deflection

of the shear into the cavity is related to the distribution of positive transverse

velocity through the mass balance of the cavity.

(j) Stronger deflection of the shear layer into the cavity causes a higher wJ

region to develop close to the downstream corner and the drag of the cavity to

increase. Thus, shear stress development along the shear layer has a direct

effect on the drag of the cavity as well as an indirect effect through its role on

the development of the wv term.

The type of shear layer interaction with the downstream corner which

depends on the shear layer deflection is decisive in the development of the shear

stress throughout the cavity shear layer. Based on the findings of previous

experiments, it is proposed that inward displacements of the shear layer with

respect to the downstream corner will result in an overall drop of the shear

stress level. As a result of this overall shear stress drop, a less positive region of

transverse velocity develops. Consequently, the mass balance of the cavity

requires less deflection of the shear into the cavity. It also requires that the

shear layer deflect back and restore the position of the shear layer. This

mechanism keeps the position of C^>aaa fixed at the upstream corner and it

might also introduce some low frequency components to the oscillations.

(k) The control of the cavity flow oscillation is facilitated by external forcing

of the cavity shear layer. It is shown that the excited T-S waves can be used to

force the shear layer generated from separation of respective boundary layers.

Thus, a transformation of the T-S waves of the boundary layer to the KelvinHelmholtz waves of the shear layer is possible.

(1) For short cavities, i.e., non-oscillating mode b/90 :£ 82, periodic flow oscillation at a wide range of frequencies was obtained. A resonance condition was
-62-

observed for a selected forcing frequency if it

1) satifled the phase criterion -£— = N

and

2) its amplitude exceeded a certain threshold amplitude.

For a fixed amplitude the threshold level increases as b decreases. The forced

resonance cases are not self-sustaining.

(m) Previous conclusions indicate that, for a given b/e0, the first frequency

in the flow background that satisfies the phase criterion and has sufficiently

high amplitude will start the self-sustained oscillations. Thus the determination

of the 6/0o I mm strongly depends on the turbulence content of the flow background.

(n) Cavity drag in the non-oscillating mode can be increased by one order of

magnitude through external forcing. Non-oscillating cavity flow at o/<90 = 77

produces a drag coefficient value of 1.3 x 10"4. External forcing at 6 Hz and

power level of 9 watts increases the drag coefficient to a value of 1.2 x 10~3. It is

also possible to generate shear stress levels several orders of magnitude higher

than that of the laminar shear layer by external forcing.

(o) For naturally occurring oscillations it is possible for two waves to co-exist

in the shear layer (natural and forced). Spectral analysis, however, shows that

the amplitude of the forcing frequency has a direct effect on the growth of the

natural frequency. External forcing made possible the ability to completely subsidize the natural self-sustained frequency by a non-self-sustained frequency. It

is shown that the amplitude of the established self-sustained oscillation is the

amplitude level that should be exceeded by the amplitude of the forced frequency for complete switching of self-sustained to forced frequency.
-63-

(p) Mode switching or transition to the wake mode can be delayed by enhancing the self-sustained oscillations through external forcing. Once the wake

mode appears, the self-sustained oscillations cannot be reestablished. Different

mode extension and simulation is possible by providing the proper forcing frequency to the cavity shear layer.

(q) The most important result of external forcing is the possibility of cancelling or damping the established self-sustained oscillations. An external forcing

frequency equivalent to that of the cavity with an amplitude less than threshold

level which has a proper phase angle respective to that of the self-sustained one

was able to reduce rms fluctuations by a factor of 2.

6.2. Concluding Remarks

This study has disclosed that the presence of self-sustained oscillations helps

in keeping a low level of drag coefficient for the cavity. A more detailed study of

the role of the downstream corner on the overall development of the shear

stress along the shear layer and on the drag of the cavity is desirable. It is

important to clarify the effect of the cavity presence on the net drag coefficient

of the system.

With respect to the active control of the cavity flow oscillations, if a more elaborate feed-back system is used, a high percentage of fluctuation cancellation

might be attainable.
-64-

Appendix A

Determination of Overall Phase Difference for the Cavity

For the purpose of obtaining phase information between two time dependent

signals, the cross-correlation function between these two signals should be carried out. By cross-correlating two signals, the relative non-dimensional phase

change between the location of the reference probe and the second probe can be

determined. This is done, as shown in sketch (A-l), by measuring the time

shift, along the positive r-axis, between two corresponding peaks in the crosscorrelation function of two signals and dividing this time by the average period

between peaks.

T

Period

x=0

x=Ax

SKETCH A-l

Then the non-dimensional phase difference between two locations separated by

Az will be
-65-

—TSML— (1)

2jr rAVERAGE PERIOD

The wavelength of the oscillation can be determined as

* =

 A

* ' TT (2)

Knowing the fundamental frequency of oscillation, the phase velocity can be calculated from

(3)

Overall phase difference between the two corners is given by

6 .V (4)
-66-

Appendix B

Laser Doppler Velocimeter Optical Arrangement

A two watt argon-ion laser is used as the light source. Past a collimating lens,

a color separator provides two lines of the laser, the 514 nm green line and 488

nm blue line. Each beam is split into two pairs of equal intensity beams by a

polarization rotator and a beam splitter. Each pair passes through its designated frequency shifter. Non-desired diffracted beams are blocked by two slit

mechanisms. The measuring volume is formed by focussing the four parallel

beams by a single 598 mm focal length lens. The collimator is used to minimize

the lateral extent of the focal volume.

The scattered light from the particles is collected by a 600 mm focal length

lens. A dichroic mirror is used for color separation, with a focussing lens used

to focus the scattered light of each of the two colors into a pin-hole (250 fj, m)

and photomultiplier tube assembly. A first-surface flat mirror is used to rotate

the beam 90°, A second mirror is used to deflect the scattered light into the collecting optics.

The component of the velocity vector measured by the LDV and the Doppler

constant, the conversion factor from frequency to velocity, is determined by the

transmitting optics. The green beams'pair enters the test section in a plane

forming an angle of 0° with the downstream direction, while the blue beams'

plane forms an angle of 90°. Therefore, streamwise velocity component is measured by the fringe system formed by the green beams while velocity components

perpendicular to the mean free stream are measured by the fringe system

formed by the blue beams. Figure B-l presents the schematic of focal volume

formation. The Doppler constant which is related to the fringe spacing within
-67-

the focal volume is given by the equation

2sinfwhere A is the light wavelength and a the angle at which the beams cross in the

focal volume. The angle a = 4.72° was used which gives Doppler constants for

the green and blue components KG = 6.24 (m/fcec) (MHz)~l and

KB - 5.92 (m/feec) (MHz)~l respectively.
-68-

Appendix C

Velocity Bias and Bragg Cell

Fringes of the measuring volume are formed by the interfering effect of the

two beams crossing. These fringes are stationary and do not produce Doppler

frequency for a stationary particle in the measuring volume.

Shifting the frequency of one beam can produce a system of moving fringes.

Then, a stationary particle in the measuring volume will give an output frequency from the photodetector. If the velocity increases in the direction of the

fringe movement, the frequency will decrease. If it is opposite to the fringe

movement, the frequency will increase. Therefore knowing the direction of the

fringe system movement will discern the particle direction.

In this experiment, two acousto-optic cells (Bragg cell) were used to frequency shift one beam of each color by a fixed 40 MHz. Each cell is installed in a

cylindrical housing that aligns with the optical axis. A compensating wedge is

used to correct for shifted beam deflection, and a tilt adjustment allows shifting

light intensity to the deflected beam, as much as 80%. The cell is derived by a 40

MHz, 2 watt signal provided by a power amplifier. A frequency of 40 MHz is often

too large. A downmix circuit permits decrease of the effective frequency shift of

the photodetector output to values from 2kHz to 10 MHz in a 2, 5, 10, sequence.

Bragg shifts of nominally 30 and 50 kHz were used. In this study the green (U

component) channel was shifted against the mean flow, while the blue channel

(V component) was shifted in the positive y direction (figure B-l).
-69-

Appendix D

LDV Signal Processing

The burst processing encompasses the required electronics for the actual

measurement of the particle velocity. A particle burst is considered valid if it

satisfies the following conditions:

1) Its amplitude crosses zero and exceeds the threshold level (about 50 mV),

2) For zero crossing (with hysteresis), the signal should be more than 10 mV.

The pulse that occurs on a zero crossing after the threshold has been exceeded

is the signal of the interest. At this point the leading edge of a time envelope is

marked (by the processor) and gates a high speed clock (250 MHz) into a

counter. The length of the envelope is determined by the total number of cycles

exceeding the threshold. The first cycle that exceeds the hysteresis level but

not the threshold level determines the trailing edge of the envelope. The trailing

edge stops the counter and the clock. The input frequency is then:

INPUT FREQUENCY («) = ENVELOPE CYCLE NUMBERS

TOTAL ENVELOPE TIME

The resolution of time measurement is 1 nsec. In this experiment only bursts

with more than 16 cycles were used for processing. A schematic of valid burst

processing is given in figure D-l The output of the burst processor is in digital

form. It feeds into the computer interface that incorporates the coincidence

test between the two processors. The coincidence test consists of the initiation

of a time window after completion of processing by one of the processors. If
-70-

data from the other processor are not received before the end of the window, the

data are discharged; otherwise coincidence is attained and the data from both

processors are transferred to the computer.

A minicomputer was used to acquire and store the data. A computer program

was developed for data acquisition using the LDV system. This computer program accomplishes two tasks. It generates a header which contains information

relevant to the conditions of the experiment and the LDV setup. It also acquires

a buffer of 16,000 words of data. This buffer size corrssponds to a total of 3200

velocity measurements in two-component measurements, since two words are

generated for each component and one word was used for the time interval

information between data points.
-71 -

Appendix E

LDV Data Processing

The following expressions are used to calculate, the mean, second moment

and cross-correlation, between two velocity components.

* I W

vc = E U J

t=1

2 E

t=i

2 AT,

(E-2)

(E-3)

(E-4)

i=1
-72-

where Ar» is the time between data points and a is the frequency. The subscripts G and B refer to the frequencies measured by the green and blue

beams,respectively. The equations relating the mean and the correlations of the

various velocity components are

u = \uG - FSG\ KG, v = \u>B - FSB\ KB (E-6)

- UZ

G - 2o> FSG + FS2

c K?G (E-7)

V* = FSg + FSj] KZ

g

wu = ucwp - uGFSB - ~5BFSG + FSGF SB KB x KC (E-9)

where FSG and FSB are frequency shifts of green and blue channels,

respectively, and /c is the frequency to velocity conversion factor.

Rms fluctuations and shear stress values can be obtained using the following

relationships:

- u

z

 (E-10)

(E-ll)
-73-

= uv —uv (E-12)

No corrections were made to the data presented in this study.
-74-

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-77-

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7

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Fig. 41 Pressure coefficient distribution in Mode II of the

oscillations

• Upstream face pressure coefficient distribution.

© Downstream face pressure coefficient distribution.
118

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119

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Fig. 42 Pressure coefficient distribution in Mode III of the

oscillations

• Upstream face pressure coefficient distribution.

O Downstream face pressure coefficient distribution.
120

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Fig. 48 Distribution of the mean transverse velocity — along

y = 0 line in Mode II. e
126

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i .. . . I . I . I , I . I

20 40 60 80 100 120 140

Fig. 48 Continued.
127

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0

" F = 5.8

"• 1 . ! . I . I . 1 , 1 • ! . 1 . 1 . I . 1 . 1 •

(o)

i

(b)

(c)

V

i ! , !

0 20 40 60 80 100 120 140

Fig. 49 Distribution of the mean transverse velocity -^- along

y = 0 line in Mode III.

 e
128

(a) .3

.2

.1

0.0

(b) .4

.2

i i1,1,1,1,1,1, i 1 , 1

b

/00

= 83

CD =0.8x10-3

I l_l 1 I I I

0.0 i/i I i I i I i I i I » I . I i I i 1 i I i I i I

(c)

.4

.2

0.0

(d) .6

.4

.2

= 2.3x I0~

s 94

= 4.3x 10-3

I I I I I I I I L 1 I I *i I . I 1 I 1 I I » I

= 107

CD = 0.01

nnL/i I . I . I . I i I . I . 1 . I . I . I .11 i I . I i I

20 40 60 80 100 120 140

Fig. 50 Distribution of the mean streamwise velocity TJ— along

y = 0 line in Mode II.
129

(e) .6

.4

.2

= 112.5

- CDS-OI7

(f) -6

u

IT-

.4

.2

0

(9) -4

.2

b

/00

s|21

- CD = .012

s

 126

CD= 6.3 x I0~3

1,1, 1

,,1,1, 1

Fig. 50 Continued.

140
130

(Q) .6

.4

.2

0.0

(b) .6

- CD= 1.2 x IO'3

I i I i I i I i I i i i i i I , I , I • |L I , I

u

Ue

.4

.2

b/0o= 139

CD=l.55xlO~3

noi i I i I . I . I , I , I . I . I . I . I . I .JLl . I

.2

nnhfi I i i i I i i t )

0 20 40 60 80 100 120 140

u

Fig. 51 Distribution of the mean streamwise velocity — along U

y = 0 line in Mode III.
131

(a) +.02 C

0.0

-.02

(b) +.02

0.0

A A A A AAAAAAAA .

-.04 U

(d) +.01

0.0

-.02

-.04

-.06

•A—A-^A,

= 83

CD »-8 x io~3

87

CD' 2.3 x IO'3

ID"

 3

= 107

>

CDS-OI

,1,1,1,1,1 , 1,1,1,1 ,

0 20 40 60 80 100 120 140

Fig. 52 Distribution of the ^- term along y = 0 line in Mode II,

e
132

(e)

Ue

(f)

(9)

+.02

0.0

-.02

-.04

-.06

-.08

-.1

+.01

0.0

-.02

-.04

+.01

0.0

-.02

-.04

112.5

b/^121

1 CD=.OI2

h b/p = 126

- /t/0

r

 CD=6.3xlO~3

I , I , I . I . I . I , I , I . I . I .

20 40 60 80 100 120 140

Fig. 52 Continued.
133

(a) +.02

0.0

-.02

-.04

'32

CD=l.2xKT2

(b)

u'v1

V.2

0.0

-.02

-.04

(c) +.02

0.0

-.02

-.06

= 1.55 x!0

~

2

= 2.1 x I0

. I . I . I . I . I . I . I • I • I i I i I i I i 1 i

20 40 60 80 100 120 140

Fig. 53 Distribution of the term along y = 0 line in Mode III.
134

(0)

(b)

0.0

.002

0.0

-.003 L

(c)

u v

0.0

-.003

(d)

0.0

-.002 -

-.004 -

-.006 -

s 83

CD = .8 x io"3

s 89

= 94

Cp = 4.3 x I0~3

107

CD=.OI

-008 r. i . I . I . I . I . I . I . i . I . I . I . I . I i i

0.0 20 40 60 80 100 120 140

Fig. 54 Distribution of the shear stress along y = 0 line in Mode II.
135

(e)

0.0

.002

.004

.006

b/0 = 112.5

CD= .017

(f)

(g)

0.0

.002

.004

s l26

I0~

3

i i I .1.1.1.1. 1

20 40 60 80 100 120 140

X

'6L

Fig. 54 Continued.
136

(0)

(b)

(c)

0.0

.002

.004

.006

.008

0.00

.002

.004

.006

.008

0.0

.002

.004

.006

.008

.01

b/g = 132

CD

5 1.2 x JO"2

L

 b/00s139

CD= 1.55x10-2

= 146

»

'2.1x10-2

Ll_l i I < I I I T I . I , I , I . I , I . l . l , I

20 40 60 80 100 120 140

Fig. 55 Distribution of shear stress along y = 0 line in Mode III.
137

o

m

o

CJ

O

O

O

O)

o

CO

oc

TO

S-i

>.

4J

•H

TO

U

0)

C

c

o

•H

•H

I4-1

C

O

o

en

to

d)

oc

03

4-^1

•H

TO

O

(D

C

d

o

C

O

O

^

QJ

CO

2 l> CS Q)

00 to CVJ

60

•H

o

o

Q

O O

O

O

O
IO'3

xL5

1.4

1.3

1.2

7~7 1

u v

-it 2 Ue .9

.8

.7

.6

4

0.0

i. JO

,' \

/ \

s \ / \

/ \

s

j''

_

f

/

/

t

-

- /

A 1 il l

90 100 110 120 130

a. t/g

o

.9

.8

.7

Arbitrary

Unit

w >3

./•• ..

.3

.2

.1

0.0

—

A

s ' \

' \

S *

' \

/

' \

*' \

/ \

/ \A

/

*

-

V

 90 100 110 120 130

b. b/o

Fig. 57 (a) Variation of the shear stress at a fixed x/60 with b/80.

(b) Variation of the net positive outflow from the cavity with
13

9

0)o*o<Dto0)

.C3ex •HM

4-1

toa)

.fi

4-100c•H

4-1

rtM4JCO35o•Hn)o

M-l O0)oCOm60 •H
140

Q.

O

I

.8

.6

.4

.2

0.0

-.2

-4l. i j i

024 6 8 10 12 14 16 18

x cm

Fig. 59 Pressure distribution over the ellipsoidal nose cone and

respective location of the strip heaters.
141

I0"2x.3

.28

.26

.24

.22

.2

.18

u* v' -16

Ue

2 .14

.12

.1

.08

.06

.04

.02

• 18 watts

A 12.5 watts

• 9 watts

0 1.12 watts

(Case A)

456 7

Ff (Forcing Frequency) (Hz)

8

Fig. 60 Variation of maximum shear stress with forcing frequency

at four different forcing power levels; Case A b/60 = 66,

U = 23 cm/sec,

e
142

KT2x.3

.28

.26

.24

.22

.2

.18

• , .16

u v

U6

2

 .U

.12

.1

.08

.06

.04

.02

* b/g =66

(Case A)

1 l ' "o

(

'

 (

( (Case B)

/ i

/ i

' i i . ,

/ i

/ \ i \

i >

;

j"

 v>

/ \

s'' \

/ / \ X

r*. * 1 . 1 . 1 . 1 . ^* M 1

34567 8

Ff (Forcing Frequency) (Hz)

Fig. 61 Variation of the maximum shear stress in the shear layer

with forcing frequency, constant power.
143

en

£ ?

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144 ORIGINAL PAG£

JQE POOR QUALITY

c

.. o

v£> -H

VO 4J

CD O

"^ CO

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14-1 14-1

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« T3

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c e>o

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60
145

QUALIT

Horizontal scale .5 sec/div

Horizontal scale .5 sec/div

Fig. 64 Oscilloscope traces of forced fluctuations

(a) turn on; (b) turn off;

b/90 = 66, F = 7 Hz, |^ * 2.

c
146

Dv

.002

0.0

.002

.004

.006

.008

.01

:».,.,,

•

» 0

0 10 20 30 40 50 60 70 80

x/

0.0

.02

u v

2

U

.04

.06 , i i »,

0

(b.)

10 20 30 40 50 60 70 80

Vfl

U V Fig. 65 Distribution of (a) 773- term, (b) shear stress

a

along the y = 0 line in the non-oscillating mode; 0 unforced,

* forced at 6 Hz.
147

<J

2

1

vu;

-

-

^-^ j

-j.i ni.

Pf = 0 watts

— A

3

2

(e) Pf=13 watts

3

2

8

o

k_

io 3

2

u

3

2

u

(b)

(c)

(d)

= 5.4 Hz

Ff=6Hz

Pfs2watts

Pf =6.5 watts

3

2

(f)

3

2

u

(g)

Pf = 20 watts

Pf = 27 watts

j ~~-*^ _j\

0 2 4 6 8 10 12 14 16

(h) Frequency(Hz)

024 6 8 10 12 1416

Frequency(Hz)

Fig. 66 Velocity fluctuation spectra, presenting interaction of the

natural and forced frequencies in the cavity shear layer for

different power levels (Pf), (h) scope trace of Case (d).
148

10 r

8

I

2 4

Extension of

Mode III /

Mode III

Wake Mode

(not J

reversible]

by forcing)

/ Extension of

/ Mode II

Simulation

of Mode I

Naturally

non-oscillating

Fig. 67 Possible interactions with the cavity flow through

external forcing

1) Mapping of Mode II into Regions 1 and 2.

2) Mapping of Mode III into Region 3.

3) Extension of Modes II and III.

4) Delay of the wake Mode.

5) Simulation of Mode I.
149

ORIGINAL PAGS !3

Of POOR QUALITY

HOT-FILM

PROBE

1

CJ K- MODEL /^

STRIP

HEATER

v«X

SCOPE

SPECTRUM

ANALYZER

POWER ^ FUNCTION GENERATOR

AMPLItitK " PHASE LOCK LOOP

I T

•

AMPLITUDE

1/2 F

Fig. 68 Schematic of the cancellation set-up.

(a)

(b) WMW.W

Fig. 69 Damping of natural oscillations through

external forcing

(a) natural oscillation

(b) strip-heater on.
150

DIRECTION OF MEAN

FLOW

BIAS DIRECTION

FOR GREEN

BIAS DIRECTION

FOR BLUE

Fig. B-l Measuring volume formation.

DOPPLER SIGNAL

FILTERD

SIGNAL

THRESHOLD

HYSTERESIS

OUTPUT OF ZERO

CROSSING

TOTAL BURST MODE

DATA READY FOR _

GREEN CHANNEL

COINCIDENCE WINDOW

DATA READY FOR

BLUE CHANNEL

DATA TRANSFER _

TO THE COMPUTER

_n

Fig. D-l Valid burst processing and coincidence

window operation.