acoustic waves generated by vortex shedding

ACOUSTIC WAVES GENERATED B Y VORTEX SHEDDING

T h e s i s by Hideki Nomoto

In P a r t i a l Fulfillment of the R e q u i r e m e n t s f o r the D e g r e e of

Aeronautical Engineer

California Institute of Technology P a s a d e n a , California

Submitted May 1980

-ii-

ACKNOWLEDGEMENTS

I wish to e x p r e s s my s i n c e r e appreciation to P r o f e s s o r

F. E. C. Culick f o r suggesting the problem and h i s constant encourage- ment throughout the project. Many thanks a l s o to P r o f e s s o r Toshi Kubota and Dr. Kiran Magiawala f o r their invaluable discussions on this problem.

I a m grateful to the people i n GALCIT who a s s i s t e d m e with this r e s e a r c h . Special thanks a r e due M r s . Karen Valente f o r h e r beautiful job of typing a s well a s many suggestions i n English.

Finally, I a m indebted to the people of Mitsubishi Heavy Industries, Ltd., who gave me a chance to study abroad and sup- ported m e f o r two y e a r s .

-

ⁱⁱⁱ

-

ABSTRACT

An internal flow self- sustained oscillation s y s tem, which consists of a two-dimensional duct and a p a i r of baffles inside, i s investigated experimentally. This s y s t e m produces a high amplitude p u r e tone when c e r t a i n flow and geometrical conditions a r e satisfied.

The frequency of this generated tone s e e m s to be determined by the longitudinal acoustic modes of the duct, while the dependence of pure tone production on flow and geometrical conditions s e e m s to be

related to the interaction between vortex shedding and acoustic feed back mechanism.

Some f e a t u r e s on self- sustained oscillation s y s t e m s a r e r e - viewed briefly and R o s s i t e r ' s idea on the cavity tone system i s applied f o r interpretation of the mechanism of the pure tone produc- tion.

Flow visualization shows stable vortical s t r u c t u r e of the flow between the baffles when a pure tone i s produced.

ACKNOWLEDGEMENTS

-iv-

TABLE O F CONTENTS

ABSTRACT

TABLE O F CONTENTS NOMENCLATURE

LIST O F FIGURES 1. INTRODUCTION

2. SELF-SUSTAINED OSCILLATION SYSTEMS 2 . 1 Edge Tone System

2.2 Hole Tone System 2.3 Choked J e t

2 . 4 Cavity Tone

2.5 Transonic Wind Tunnel Noise 2.6 Jet-Edge-Resonator System 2.7 Pipe Tone

2.8 Internal Flow System 3. INTERNAL FLOW SYSTEMS

3. 1 Definition

3.2 Dimensional Analysis

3.3 P o s sible Frequency Scales i) Longitudinal Acoustic Mode ii) Vortex Shedding

iii) Vortices /Acoustic Waves Interaction 3.4 Resonance

4. EXPERIMENTAL ARRANGEMENT

4. 1 Apparatus

4.2 Instrumentation 4.3 Flow Visualization 5. EXPERIMENTAL RESULTS

5. 1 Description of Phenomenon

5.2 Longitudinal Acoustic Modes of the Duct 5.3 Resonant Frequencies

i) h = 1.90 c m ii) h = 1. 52 c m iii) h = 2.28 c m

5.4 Non-Dimensional Frequency 5.5 P u r e Tone Production Regions

i) h = 1.90 c m ii) h = 1.52 c m iii) h = 2.28 c m

iii

viii

TABLE O F CONTENTS (Contd. ) Chapter

5. 6 Velocity Field Between the Baffles i) Centerline Velocity

ii) Velocity P r o f i l e 5. 7 Flow Visualization 6. DISCUSSION

6. 1 Mechanism of Sound Production 6.2 P u r e Tone Production Regions

6.3 Other A s p e c t s of the Experimental Results 6 . 4 Application of the F o r m u l a to the P r e s e n t ^Data 7. CONCLUDING REh4ARKS

REFERENCES TABLE

FIGURES

NOMENCLATURE speed of sound

width of the cavity

average speed of sound in the cavity l a t e r a l gap between the baffles = D

-

^2h

duct width

frequency of oscillation

longitudinal acoustic mode frequency acoustic feedback frequency

vortex/acous tic wave interaction frequency vortex shedding frequency

jet-edge distance baffle height an integer

= 1, 2.3, 3 . 8 , 5.4,

...

ratio of vortex convection velocity to reference flow velocity = distance between baffles

duct length

distance f r o m one end of the duct to the f i r s t baffle c o r r e c t e d duct length

open end correction of the duct an integer = m

+

m

v a

number of acoustic waves number of vortices

Mach number a n integer

Reynolds number

Strouhal number time

velocity of flow entrance velocity

vortex convection velocity

estimated reference velocity = U, D/d

phase difference between vortical waves and acoustic waves phase difference between vortical waves and acoustic waves a t the u p s t r e a m c o r n e r

phase difference between vortical waves and acoustic waves a t the downstream c o r n e r

wavelength of the acoustic wave wavelength of the vortical wave kinematic viscosity of the fluid phase

Figure 1. 1 1.2

-viii-

LIST O F FIGURES

Self -Sustained Oscillation Systems

Dependence of Edge Tone Frequency on J e t Velocity U and Jet-Edge Distance h Ros s i t e r t s Model on Vortex/Acoustic Wave Interaction i n Cavity Flow

Internal Flow System

Vortex/Acoustic Wave Interaction Model Schematic of the Apparatus

Blower Capability Measurement System

Overall Sound P r e s s u r e Level Re s onant Frequency

Resonant Frequency Strouhal Number

P u r e Tone Production Regions

Centerline 'Flow Velocity Between the Baffles

Velocity Profile Between the Baffles Flow Visualization

On the Mechanism of P u r e Tone Production

P u r e Tone Frequency

On the P u r e Tone Production Regions

Page 50

I. INTRODUCTION

Aerodynamic generation of sound has been discussed both ex- perimentally and theoretically f o r a long time, but the mechanism of this phenomenon i s not fully understood even f o r f a m i l i a r musical instruments such a s the flute. Studies of this sound creation by fluid flow include not only musical instruments but a l s o undesirable phenomena in aeronautics such a s a i r f r a m e noise, transonic wind tunnel noise and burning instability of solid propellant rocket motors,

a l l which a r e classified a s self- sustained oscillating systems. (Fig. 1. 1. ) One of the c l a s s i c a l problems in sound generation by fluid

motion i s the edge tone phenomenon. When a wedge type s t r u c t u r e i s placed in the s t r e a m of a jet, this s y s t e m produces a resonant l a r g e sound which has a dominant frequency component under c e r t a i n flow and geometric conditions. The main features of the edge tone phenomenon a r e summarized a s follows ( (Fig. 1. 2):

(1) F o r a fixed jet-edge distance, the frequency of the tone i n c r e a s e s linearly a s the velocity of the jet increases.

T h e r e a r e minimum and maximum jet velocities beyond which no significant tone i s generated. The frequency of the tone jumps to the neighboring "stage" a t certain flow velocities.

(2) F o r a fixed jet velocity, the tone frequency d e c r e a s e s inversely proportional to the distance between jet exit and edge. At a c e r t a i n jet-edge distance the frequency of the tone jumps to the neighboring "stageu. There a r e minimum and maximum jet- edge distances beyond which no sound production i s observed.

( 3 ) The jumping p r o c e s s e s in frequency in both c a s e s exhibit h y s t e r e s i s characteristics.

(4) The sound field has directional p r o p e r t i e s that a r e c h a r a c t e r i s t i c of a dipole field.

Most of these f e a t u r e s were observed long ago, but i t was only i n the 1950's that this pure tone generation s y s t e m was recognized and t r e a t e d theoretically a s one of the self-sustained oscillation s y s t e m s which had an acoustic feedback mechanism coupled with a periodic hydrodynamic flow field. Because of the prominent d i s c r e t e tone, theories were constructed based on the concept that the agent responsible f o r this sound generation was a vortex o r a s y s t e m of vortices shed f r o m the c o r n e r of the jet exit.

Then i t was speculated that these v o r t i c e s interacting with the wedge shaped s t r u c t u r e produced p r e s s u r e fluctuations which propagated u p s t r e a m to stabilize the vortex shedding activities to f o r m a closed loop feedback system. This idea of the vortex/acoustics interaction model on the edge tone, s y s t e m was partly supported by the flow visualization which showed vortical s t r u c t u r e in the s h e a r layer.

Another widely discussed example of aerodynamic sound genera- tion i s the cavity noise which also turned out to be one of the self- sustained oscillation systems. The study of flow p a s t a cavity i s complex and has s e v e r a l aspects of fluid mechanical problems ranging f r o m fundamental to practical; i. e. d r a g and heat t r a n s f e r problems, unsteady flow effects, formation of vortices in the cavity, and generation of sound. Escape hatches, open cockpits and bomb bays a r e p r a c t i c a l examples of the cavity flow relating to a n a i r - craft. The ventilation holes in transonic wind tunnel walls can a l s o

- 3 -

be regarded a s cavities which may contribute to noise generation.

The principal f e a t u r e s of the sound generated by a cavity s y s t e m a r e essentially the s a m e a s those of the edge tone system.

The frequency of the emitted sound i n c r e a s e s linearly with the flow speed f o r fixed cavity width, and inversely proportional to the width of the cavity f o r fixed flow velocity. A minimum width i s required to

produce sound, s o i s a minimum flow velocity. T h e r e a r e frequency jumps which f o r m s e v e r a l "stages1' of sound generation. A vortex o r a s y s t e m of vortices i s observed i n the cavity by the flow visual- ization technique.

Sound waves radiated f r o m a cavity were c l e a r l y demonstrated by Karamc he ti ( 3 3 4), by schlieren and i n t e r f e r o m e t r i c pictures f o r high speed flow. He also studied the effect of the nature of the boundary l a y e r approaching the cavity and found that the minimum width for initially laminar flow was s h o r t e r than that f o r turbulent flow.

~ o s s i t e r ' ~ ) , on the other hand, modelled the cavity flow with shedding vortices f r o m the u p s t r e a m c o r n e r and acoustic waves emanating f r o m the r e a r c o r n e r of the cavity. He then derived a formula which described the non-dimensionalized frequency of the

sound in t e r m s of the Mach number of the flow; the ratio of vortex convection velocity to external flow velocity; and the phase difference between the vortical waves and acoustic waves a t the r e a r c o r n e r of the cavity. This formula also includes an a r b i t r a r y integer which r e p r e s e n t s the "stage" of the radiated sound.

The generation of sound in d i s c r e t e frequencies by self- sustained oscillatory systems which have strong f r e e s h e a r l a y e r s leads us

to the s t r u c t u r e of the shear layer. The periodic nature of

a jet was recognized a century ago a s a sound sensitive jet o r a singing f l a m e and discussed by many r e s e a r c h e r s , including Lord

( 6 )

Rayleigh

.

A jet, once placed in an acoustic field, changes i t s rebaracteristics"according to the frequencies of the sound. Therefore,

gas f l a m e s which used to- be employed f o r illumination seemed to

dance i n harmony with the pitch of musical instruments.

In the modern study of jets, ~ a t o ' ~ ) found sinusoidal, pro- g r e s s i v e , wave-like velocity fluctuations i n the jet where the laminar flow became unstable. He a l s o excited the jet acoustically to find that the effect of the excitation was m o s t r e m a r k a b l e when the fxequency of the excitation coincided with that of the intrinsic sinu-

soidal fluctuation. Crow and Champagne!8), on the other hand, a l s o excited a jet acoustically and showed the l a r g e , o r d e r l y s t r u c t u r e i n jet turbulence by flow visualization.

The large, coherent s t r u c t u r e i n a mixing l a y e r was investi- ( 9

1

gated experimentally by Brown and Roshko

.

They showed, by flow visualization, vortex-like l a r g e s t r u c t u r e s convected a t nearly constant speed and amalgamating with neighboring ones to i n c r e a s e the s i z e s and spacings discontinuously.

These studies on the s t r u c t u r e of s h e a r l a y e r s might shed f u r t h e r light on the mechanism of the sound production since a l l

the self

-

sustained oscillation s y s tems have strong f r e e s h e a r l a y e r s and the nature of these s h e a r flows s e e m s to be a key factor in the study of sound generation.

The acoustic instability problem in solid propellant rocket motors i s another example of the self

-

sustained oscillation systems.

L a r g e solid propellant rocket m o t o r s have flange- type s t r u c t u r e s

i n s e r t e d between segments of the grains. T h e s e s t r u c t u r e s remain unburned while the propellant burns radially

.

A residual ring

-

shaped s t r u c t u r e produces a strong shear flow in the chamber then constitutes a self- sustained oscillatory s y s tern with another unburned flange s t r u c t u r e located downstream.

In the c a s e of this internal flow, the length s c a l e s of the s y s t e m include the s i z e of the chamber a s well a s the distance between the unburned flanges (baffles). The l a t t e r corresponds to the distance between the jet and the wedge in the edge tone system, o r to the width of the cavity i n the cavity system. The chamber might play the r o l e of a resonator which, i n fact, complicates the problem.

An exploratory experimental study modelling this internal flow/

(10) acoustic interaction problem was done by Culick and Magiawala

.

They found s e v e r a l acoustically excited ranges i n flow speed and distance between the baffles. A simple theoretical consideration, however, gives t h r e e d i s c r e t e frequency scales: the vortex shedding frequency, the acoustic feedback frequency and the frequency of the longitudinal mode of the acoustic oscillation. I t was, therefore, felt that m o r e detailed experimental work was n e c e s s a r y to investigate the mechanism of the sound generation in the internal self-sustained flow system.

Because of the nature of the problem which has many p a r a m e t e r s , geometric and fluid dynamic, a new apparatus has been designed f o r the work reported here. The geometrical conditions such a s the total length of the duct, baffle height, distance between the baffles and the location of the baffles i n the duct w e r e varied to investigate every influence of these p a r a m e t e r s on sound production. The duct was

isolated acoustically to obtain a c l e a r length s c a l e f o r the longitudinal acoustic mode. A rectangular c r o s s - s e c t i o n f o r the duct was chosen to get a two-dimensional flow f o r flow visualization. Frequencies of the generated tone were measured by a microphone while flow conditions were determined using a hot wire anemometer. Smoke flow visualization was a l s o performed to obtain pictures of the flow between the baffles with a strobotac a s a periodic light source.

In the next chapter, basic f e a t u r e s of the self- sustained oscillation s y s t e m s a r e reviewed i n m o r e detail while theoretical considerations particular to the internal flow s y s t e m s a r e presented i n the succeeding chapter. In Chapter 4, the experimental a r r a n g e - ment of the p r e s e n t study i s described, followed by the explanation of r e s u l t s in Chapter 5. Discussions of the r e s u l t s a r e given in Chapter 6. Our conclusions a r e summarized in the final chapter.

2. SELF-SUSTAINED OSCILLATION SYSTEMS 2. 1 Edge Tone System

Although the h i s t o r y of the study on the edge tone could be t r a c e d a century ago, G. B. Brown's experiments (11) i n the 1930's w e r e e s s e n t i a l in the e a r l y days. He demonstrated the b a s i c f e a t u r e s of this phenomenon d e s c r i b e d i n the previous chapter.

He a l s o derived a n e m p i r i c a l f o r m u l a f o r the frequency of generated sound i n the following f o r m

where j = 1, 2.3, 3 . 8 , 5.4,

---,

U i s the velocity of the jet i n c m / s e c and h i s the jet-edge distance i n c m . The number j corresponds to the "stage" of the sound. He p e r f o r m e d flow visualization a t the s a m e time which showed v o r t i c a l s t r u c t u r e s of the jet flow field. But i n spite of the flow visualization, h i s e m - p i r i c a l formula f o r the frequency does not shed any light on the mechanism of the sound generation.

N. ~ u r l e ' l ~ ) , i n 1953, proposed another formula f o r the frequency of the sound i n a s i m i l a r fashion. F r o m a n analog to the vibration of a s y s t e m with damping, he a s s u m e d the relation between jet-edge distance h and wavelength h of the v o r t i c e s a s foliows:

where i i s the number of v o r t i c e s i n the s y s t e m which a l s o r e p r e s e n t s the stage of the sound. He then derived the convection velocity of the v o r t i c e s f r o m Brown's experimental data and finally

suggested the formula f o r the frequency

where d i s the width of the jet a t the exit and a l l quantities a r e measured in CGS units. This extension of Brown's formula does not s e e m to be based on concrete support f r o m fluid mechanics and does not include the concept of the acoustic feedback mechanism.

W. L. Nyborg's theory (13), on the other hand, i s based on the idea of the displacement of the jet centerline. He derived an integral equation which determined the shape of the centerline of the jet impinging on the edge. The displacement of the centerline i s expressed by the integral of the velocity of the particle which has the f o r m

where 6 i s the t r a v e l time of the particle f r o m the jet exit to the position (x, t). On the other hand, the l a t e r a l acceleration of the particle i s proportional to the l a t e r a l p r e s s u r e gradient. He therefore assumed that this p r e s s u r e gradient could be expressed by a function of the centerline displacement a t the edge multiplied by another function of the distance f r o m the edge. Then we obtain

where

q e i s the displacement of the centerline a t the edge.

Substitution of Equation (2. 1. 5 ) into Equation (2. 1. 4) gives

Particularly, a t x = h ( a t the edge), this becomes

where 6, i s the total travelling time of the p a r t i c l e f r o m the jet exit to the edge. This i s the required integral equation which

determines the unknown function qe. Assuming some simple f o r m s f o r g and cp a s t r i a l functions, Nyborg demonstrated the oscillation c h a r a c t e r s of the centerline. The integral equation formulation s e e m s to be describing the feedback field, but the fluid dynamic, a s well a s acoustic, c h a r a c t e r i s t i c s a r e not solved i n this theory.

A. Powell (14) made a n argument introducing the concept of

"effectiveness" which includes an amplification r a t e and a phase shift of the disturbances. His idea i s a s follows. At f i r s t the effectiveness of the presence of the edge, q i s derived f r o m the

S

interaction between the jet flow and the edge. Then the consequent fluid motion, thus resulted n e a r the edge, gives a disturbance to the jet a t the exit with effectiveness q . This disturbance i s then communicated back to the edge with effectiveness qd. There, the disturbance i s amplified by a factor of q relative to the initial amplitude of the perturbation. With these effectiveness f a c t o r s , the s y s t e m i s self

-

sustained if

q S

nt

^{qd q = 1}

.

Powell then assumed that the effect of the fluid action a t the edge could be represented by a distribution of dipole s o u r c e s along the edge.

F r o m this assumption he derived the expression of q which turned t

out to have a phase shift

This means that a disturbance originating a t the edge s t a r t s a q u a r t e r cycle e a r l i e r than the disturbance a r r i v e s there.

2 . 2 Hole Tone System

A high tone produced by the s t e a m action of the tea kettle i s one of the f a m i l i a r examples of the aerodynamic sound generation observed i n our daily life. The phenomenon of the hole tone i s essentially the s a m e a s that of the edge tone. The sound i s c l a s s i - fied into s e v e r a l s e t s of frequencies which a r e called I1stagesr1. A jumping i n frequency f r o m one stage to another occurs accompanied by hysteresis c h a r a c t e r i s t i c s . T h e r e i s a minimum distance between the exit of the jet and the hole. The frequency of the tone i n c r e a s e s a s the jet velocity i n c r e a s e s o r the jet-hole distance d e c r e a s e s .

Chanaud and ~ o w e l l ' l ), using the effectiveness concept, tried to give the explanation of this phenomenon. They took into account the indirect feedback effects coming through the jet duct and s u r - rounding outer field a s 'well a s the d i r e c t effects described in the previous section. They gave a qualitative explanation of this s y s t e m but did not p r e s e n t any quantitative descriptions.

2 . 3 Choked J e t

A jet noise usually has a broad band continuous spectrum without any prominent frequency component. But when the jet p r e s s u r e r a t i o i s high enough compared to the ambient p r e s s u r e , the flow i n the nozzle chokes itself. Under this condition, a

"whistle" o r l t s c r e e c h l t sound i s observed and the overall sound p r e s s u r e level i s increased significantly. The discontinuity in

frequency when jet p r e s s u r e i s changed continuously i s a l s o observed, a s a r e the h y s t e r e s i s c h a r a c t e r i s t i c s . This phenomenon was discussed by Powell (I6' 17). He found that the wavelength of the dominant f r e - quency of the sound was related to the regular shock wave spacing in the flow. Hence he attributed the shock formation in the jet flow to be responsible f o r the feedback mechanism of the sound production s y s tem.

~ a m r n i t t ' 18), on the other hand, demonstrated the importance of the sound waves acting on the b a s e regions of the jet. He then found that the jet could be stabilized significantly by shielding i t s b a s e region f r o m the sound waves.

2 . 4 Cavity Tone

The radiated sound waves f r o m a cutout of a surface o r of a cavity a r e c l e a r l y seen in schlieren and Mach-Zehnder interferometric pictures by ~ a r a m c h e t i ' ~ ' 4). These pictures also show that the

p r i m a r y source of the acoustic waves i s n e a r the trailing edge of the cavity.

An analytical study of the mechanism of the cavity tone generation was done by ~ o s s i t e r ' ~ ) . This theory i s based on Powell's idea of the mechanism of choked jets. Their idea i s a s follows (Fig. 2 . 4 . 1). Suppose the shedding vortical waves have a wavelength

X

Also acoustic

v '

waves generated f r o m the downstream c o r n e r a r e assumed to have wavelength

'a' L e t the width of the cavity be b. The vortices a r e convected with a speed of k U, where U i s the uniform velocity outside of the cavity. At t i m e t = 0, when a n acoustic wave s t a r t s f r o m the downstream c o r n e r , a vortex i s supposed to be located a t the distance yhv downstream of the downstream

corner. A s h o r t time l a t e r , a t t = t l , a n acoustic wave reaches the u p s t r e a m c o r n e r where v o r t i c e s a r e shed. Assume t h e r e a r e m

-

¹ vortices and m acoustic waves in the cavity a t time

v a

t = 0. The frequency of these waves i s expressed a s

where c i s the mean speed of sound in the cavity. At t = t l , the geometric condition f o r the vortical waves gives

mvXY = b

+

yX _v

+

k U t l

,

and f o r the acoustic waves

F r o m these t h r e e equations we obtain a formula f o r the nondimen- sional frequency

where M = U /a i s the Mach number of the outer flow. (a i s the sound velocity in the outer flow. )

F o r a low speed c a s e where M = 0, Equation ( 2 . 4 . 4 ) reduces

where

m = m + m _{a v}

.

This i s the formula that i s compared with those of Brown (Eq.

2. 1 . 1 ) o r Curle (Eq. 2. 1 . 3 ) .

He then derived the empirical values of y and k f o r shallow cavities:

These experimental r e s u l t s mean that, according to his formulation, the vortices a r e convected in the speed that i s 57 percent of the uniform flow, and the acoustic waves begin to leave the downstream c o r n e r when a vortex i s located a q u a r t e r wavelength down-

s t r e a m of this c o r n e r . In t e r m s of the phase relation, this can be interpreted s o that the acoustic waves have 90 d e g r e e s phase lag compared to the vortical waves a t the r e a r c o r n e r of the cavity.

This r e s u l t d i f f e r s f r o m Powell's model on the edge tone s y s t e m where he distributed dipole s o u r c e s along the edge and brought 90 d e g r e e s advance in phase a s discussed i n Section 2.1. C u r l e ' s assumption on the edge tone system, which gives the relation between the jet- edge distance and wavelength, corresponds to Y =

-

^0.25.

Bilanin and Covert ( 19) s e t up a n analytical model that takes into account the stability of a vortex sheet subjected to a periodic p r e s s u r e impulse. Effects of a n acoustic monopole located on the downstream face of the cavity on the p r e s s u r e field were a l s o con- sidered. After mathematical manipulation, they derived a formula f o r nondimensional frequency which turned out to have a s i m i l a r f o r m to that of R o s s i t e r

where n i s an integer.

R o s s i t e r ' s formula corresponds to

The t e r m cp/(Z~r) comes f r o m the leading edge p r o c e s s e s , while 3 / 8 = 0.375 comes f r o m the trailing edge p r o c e s s e s which R o s s i t e r ' s experiment gave a value of 0.25. Rossiter did not give the phase shift effect a t the u p s t r e a m c o r n e r of the cavity.

2. 5 Transonic Wind Tunnel Noise

One of the c a u s e s of the noise i n transonic wind tunnels which have perforated walls a t the t e s t section i s considered to be produced by the edge tone mechanism. The boundary l a y e r on the wall s e p a r - a t e s a t the u p s t r e a m c o r n e r of the hole forming a jet-like flow.

This f r e e s h e a r layer, partly sucked into the plenum chamber, hits the r e a r c o r n e r of the hole. This f r e e s h e a r l a y e r and the hole construct a self

-

sustained oscillation system. The hole plays the r o l e of a cavity and sound i s generated near the trailing c o r n e r of the hole.

Although the detailed mechanism i s not known, some empirical suggestions have been made to prevent this undesirable, unsteady phenomenon f o r the wind tunnel technique. One of them r e l a t e s to the proposal of the diameter of the hole. ~ a b e ~ ' ~ ' ) , f r o m his ex- periment, suggested that, i n o r d e r to avoid this sound radiation, the hole diameter d and the boundary l a y e r displacement thickness h P along the wall should satisfy the relation

He also reported that edge tones could be eliminated by reducing the tunnel wall porosity.

The edge tone i n the tunnel i s a l s o reduced by covering the holes by o r inserting poles ( 2 2 ) o r splitter plate (23) ⁱⁿ the holes.

McCanless and Boone (24) derived a n e m p i r i c a l formula f o r the nondimensional frequency of the edge tone associated with the wind tunnel in the following form:

where

'

^r is the longitudinal length of the hole diameter and U L i s the convection velocity of the vortex.

2.6 Jet-Edge-Resonator System

When a resonator i s introduced in an edge tone system, the c h a r a c t e r i s t i c s of the sound generation of this new system change impressively compared to the simple jet-edge system. This system has s e v e r a l s i m i l a r f e a t u r e s which occur in the internal flow oscilla- tion systems. An experiment on the jet- edge- resonator system was performed by Nyborg e t al(25). They placed a tube a s a resonator in a jet-edge sound field and changed the natural frequency of the tube to investigate the effect on the frequency of the generated

sound. A simple edge tone s y s t e m without a resonator has s e v e r a l stages, while each stage appears a t a c e r t a i n range of the jet

velocity. Suppose we introduce a resonator in the sound field and change i t s natural frequency keeping the jet speed constant.

Frequencies of the generated sound observed in this situation change their c h a r a c t e r i s t i c s greatly owing to the n a t u r a l frequency of the resonator. In the region where the stage jump occurs f o r a simple jet-edge system, the frequency of the sound i s determined uniquely by the natural frequency of the resonator. Even a t the jet speed where only one stage of the sound i s observed, other stages of the tones which a r e not observed regularly, can be excited by adjusting the eigenfrequency of the resonator. Sometimes subharmonic s of the edge tone a r e included i n the sound field.

2 . 7 Pipe Tone

A pipe with a n orifice a t i t s downstream end produces a pure tone when the flow in the pipe satisfies c e r t a i n conditions. This phenomenon i s known a s the pipe tone o r Pfeifentb'ne. This i s

another example of the oscillatory s y s t e m with a resonator. As the mechanism for the excitation of this pipe- tone, Anderson ( 2 6 - 2 8 ) a r g u e s a s follows. A periodic fluctuation in the orifice a r e a i s produced by the periodic shedding of the v o r t i c e s f r o m the orifice edge. When the frequency of the p r e s s u r e fluctuation caused by the vortex shed- ding i s n e a r one of the natural frequencies of the orifice-pipe system, the column of a i r in the pipe resonates to give a l a r g e level of pure tone. He a l s o demonstrated the periodic vortex ring s t r u c t u r e in the jet flow by flow visualization when a p u r e tone was produced.

2. 8 Internal Flow System

One of the burning instability problems of the solid propellant rocket m o t o r s i s attributed to vortex shedding i n the combustion chamber coupled with acoustic oscillations. In the c a s e of this type

of internal flow, t h e r e a r e t h r e e time ( o r frequency) scales. Culick and Magiawala ( l o ) considered these s c a l e s a s follows. The f i r s t frequency i s related to the vortex shedding which i s supposed to be proportional to the flow velocity U and inversely proportional to the dimension of the s t r u c t u r e ( o r baffle) d f r o m which vortices a r e shed. This i s expressed a s

where S i s the Strouhal number. The second frequency scale i s derived analogous to the theory on the edge tone mechanism. A vortex shed f r o m the f i r s t baffle i s c a r r i e d downstream with the velocity of the o r d e r of U. Then an acoustic wave i s generated by interaction of the vortex with the second baffle, which propagates u p s t r e a m with the speed of sound. Therefore the round-trip time of these wave propagations i s expressed a s

where a i s the average speed of sound and

a

i s the distance between the baffles. When the flow velocity i s s m a l l compared with the speed of sound, this equation gives the second characteristic frequency

The third scale of the frequency is the natural longitudinal mode of the acoustic oscillation in the combustion chamber. This i s given by

where L i s the length scale of the chamber and

P

i s a

proportionality factor which r e p r e s e n t s the mode of the oscillation.

Assuming that the m o s t s e v e r e resonance o c c u r s when these three frequencies coincide, geometric and flow conditions a r e required to

satisfy the following relations to produce pure tone:

Based on these considerations they performed an experiment modelling the rocket chamber flow. They used a c i r c u l a r c r o s s -

section pipe and two s e t s of orifice plates placed a t s e v e r a l locations in the duct. In t h e i r experiment they found that acoustic modes w e r e excited not f o r d i s c r e t e values of flow and geometrical conditions ex- pected f r o m Eq. ( 2 . 8. 5 ) and Eq. ( 2 . 8. 6), but spread in distinct

broad ranges of these p a r a m e t e r s . They a l s o found that modes having higher frequencies were excited a t higher flow speed and the location of the orifice plates in the duct had a g r e a t effect on the

sound production activity.

Those considerations of the acoustic waves excited in an internal flow s y s t e m have motivated the work discussed here.

-

^19-

3 . INTERNAL FLOW SYSTEMS

3 . 1 Definition

An internal flow s y s t e m which sustains oscillations by itself consists of a duct and a p a i r of baffles located a t a n a r b i t r a r y position in the duct. Both ends of the duct a r e open to the surround- ing field. A flow of constant mean velocity goes through the duct f r o m one end to the other. The duct can be either axisymmetric o r two-dimensional, but we consider the two-dimensional case.

3 . 2 Dimensional Analysis

An internal flow has the following physical variables (Fig. 3 . 2):

Duct Length, Duct Width,

Distance f r o m One End of the Duct to the F i r s t Baffle,

Baffle Height,

Distance Between Baffles Velocity of Flow,

Speed of Sound,

Kinematic Viscosity of Fluid, Frequency of 0 scillation

T h e r e a r e nine variables. Since the basic dimensions for this fluid mechanical s y s t e m a r e 3 , the independent dimensionless groups a r e 6. They a r e taken a s follows:

M i s the Mach number of the flow and indicates the compressibility of the flow. R i s the Reynolds number which i s a m e a s u r e of viscous

effects i n the fluid. S, the Strouhal number, i s the nondimensionalized frequency.

5

designates the location of the baffles which might be

related to the modes of the longitudinal acoustic oscillation i n the duct. q i s the p a r a m e t e r which p r e s e n t s the contraction ratio of the flow owing to the presence of the baffles. And finally

5

i s a dimensionless m e a s u r e of the depth of the cavity which i s formed by the baffles and the duct wall.

3 . 3 P o s sible Frequency Scales

i) Longitudinal Acoustic Mode

An open duct of length L has longitudinal acoustic modes, of which frequencies a r e expressed a s

where

L t = L + A L

.

AL i s the open end correction and i s determined by the shape of the c r o s s - s e c t i o n and the conditions at the ends of the duct.

ii) Vortex Shedding

The vortex shedding frequency

i s also expressed a s

where U i s the convection velocity of the vortex,

X

i s the

C v

"wavelength" of the vortex and U i s the reference velocity of the e

flow i n the duct. The vortex shedding frequency can be assumed to be proportional to the flow velocity and inversely proportional to the dimension of the s t r u c t u r e f r o m which vortices a r e shed. In our c a s e the height h of the baffle s e e m s to be appropriate to be chosen a s this length scale. Then with the Strouhal number S, we obtain

iii) Vortices /Acoustic Waves Interaction

Following Ros siter.' s idea, the vortical and acoustical waves a r e modelled a s follows (Fig. 3 . 3 . 1). The vortices which a r e shed f r o m the edge of the u p s t r e a m baffle have a wavelength

X,J

and a r e convected downstream a t a velocity U = kU,. U i s the reference velocity

e e

of the flow in the duct. At time t = 0, when a n acoustic wave leaves the downstream baffle, there a r e m

-

1 vortices between

v the baffles. The m th vortex i s located a t

yvxv distance down- v

s t r e a m of the second baffle. T h e r e a r e m acoustic waves between a

the baffles. They have a wavelength Xa and propagate a t speed c. At time t

=

ti, an acoustic wave reaches the f i r s t baffle, but a vortex cannot be generated simultaneously. This time lag can be

expressed by introducing a distance yaXv which indicates the location of the imaginary vortex. The frequency of the vortical and acoustic waves a r e assumed to coincide a t the resonance con- dition. Therefore

The geometrical condition f o r the vortical waves a t time t = t t gives

- -

^yalV

+ a

+ yvxv + k u e t l

,

( 3 . 3 . 7 )

and f o r the acoustic waves,

R

= maXa

+

c t '

.

( 3 . 3 . 8 )

F r o m Eqs. ( 3 . 3 . 6), ( 3 . 3 . 7 ) and ( 3 . 3 . 8 ) we obtain the equation f o r the frequency in nondimensional form:

where M = U /a i s the Mach number of the flow.

e

When the flow speed i s low compared to the speed of sound, and f o r the c a s e of the long acoustic wavelength X relative to the

a baffle distance, in other words, if

Eq. ( 3 . 3 . 9 ) i s reduced to

3 . 4 Resonance

A resonance i s expected when the frequencies of the longitud- inal acoustic mode and v o r t i c e s / a c o u s t i c waves interaction coincide.

T h e r e f o r e putting

we obtain

This equation i s a l s o written a s

where

-24-

4. EXPERJMENTAL ARRANGEMENT

4. 1 Apparatus

Because of the nature of this problem, which has many geometric p a r a m e t e r s , a new apparatus was designed including the application of flow visualization. The duct c r o s s - s e c t i o n was chosen a s a rectangular type to achieve two-dimensional flow and special c a r e was taken f o r placing the baffles in the duct. The distance between the two s e t s of baffles a r e designed to be changed continuously.

The duct which has a c r o s s - s e c t i o n of 5. 1 c m x 15.2 c m i s made of 1.27 c m thick plexiglas s consisting of s e v e r a l sections to change the total length of the duct. Along the inner surface of the side walls two lines of grooves a r e cut to bury the b a r s which connect the baffles.

Downstream of the duct i s a settling chamber, 50 c m x 50 c m x 50 cm, which i s designed to reduce the turbulence level of the flow in the duct a s well a s to isolate the duct f r o m other components acoustically (Fig. 4.1.1).

The flow i s driven by a blower located downstream of the settling chamber. The flow speed i s adjusted by the speed controller (Minarik Blue Chip I1 Model BCR 290). The motor i s chosen a s a d i r e c t c u r r e n t type (Minarik Blue Chip I1 1. 5 HP) i n o r d e r to obtain fine speed adjust- ment. The motor has a capability to produce a flow up to 100 m / s e c when a p a i r of baffles i s installed (Fig. 4. 1. 2 ) . The noise level of the blower i s sufficiently low compared to the power of the resonant pure tone even a t i t s highest speed.

A 6. 4 m m thick aluminum plate i s used f o r the baffles.

The heights of the baffles a r e 1.27 cm, 1. 52 cm, 1. 90 c m and 2 . 2 8 cm, which gives the r a t i o s of the c r o s s - s e c t i o n a l a r e a s of the flow a t the entrance of the duct to those a t the baffles a s 1. 99, 2.48, 3.92 and 9.44, respectively. Data for the s h o r t e s t baffles w e r e not measured since no significant resonant pure tone was observed f o r this case. Another s e t of baffles of height 1.90 c m which had sharp edges (30 d e g r e e s ) w e r e a l s o tested.

The distance between the baffles i s adjusted via b r a s s b a r s to which baffles a r e fixed by bolts. In the b r a s s b a r a long slit f o r the bolt was made s o that the baffles ..can be attached a t any position along the b a r . These b a r s a r e buried in the grooves along the

side walls in o r d e r not to disturb the flow i n the duct.

The entrance and the exit of the duct a r e not treated a e r o - dynamically, which leaves the open end c o r r e c t i o n problem of the longitudinal acoustic mode.

4. 2 Instrumentation

A hot w i r e (TSI Model 1210-T1. 5) was employed f o r the flow measurement. The wire was placed a t the center of the inlet of the duct to m e a s u r e the entrance velocity U, and representative flow velocity Ue was calculated by a r e a ratio correction:

where D is the width of the duct and d i s the gap a t the baffles.

A constant t e m p e r a t u r e anemometer (GALGIT made Matilda) was used. Outputs of this anemometer were measured by a digital voltmeter (Hewlett-Packard Model 5326B).

A condenser microphone ( ~ r G e l & Kjaer Type 4134) was placed

u p s t r e a m of the inlet of the duct to avoid i n t e r f e r e n c e . Frequencies of tone w e r e m e a s u r e d by a counter ( H - P Model 5326B) through a n amplifier (Princeton Appl. Res. Model 189). O v e r a l l sound p r e s s u r e level was m e a s u r e d by a t r u e root mean s q u a r e e l e c t r i c voltmeter (Ballantine Model 3 20).

Signals f r o m the hot w i r e and the microphone w e r e always monitored simultaneously by a n oscilloscope (Tektronix Type 55 1) to check the wave f o r m and frequency of the outputs (Fig. 4. 2. 1).

4. 3 Flow Visualization

Flow visualization was p e r f o r m e d by injecting k e r o s e n e vapor smoke into the flow. The flow was illuminated periodically by a strobotac (General Radio Type 153 1-A) and the photographs w e r e taken by a 35mm c a m e r a (Nikon F2A) with a 135 m m telescopic lens. Kodak T r i - X pan black and white f i l m s (ASA 400) w e r e used.

Flow p i c t u r e s w e r e taken adjusting the frequency of the strobo tac illumination so that the s t r u c t u r e s of the flow looked to be r e - maining a t the s a m e po,sition.

The speed of the shutter of the c a m e r a ranged f r o m 1/60 sec to 1 / 8 sec, while the a p e r t u r e was kept constant (f = 2. 8).

5. EXPERIMENTAL RESULTS 5. 1 Description of Phenomenon

Figures 5. 1. 1

-

5. 1 . 6 show the fundamental aspects of the phenomenon. These graphs a r e f o r fixed duct length (L = 50.8 c m ) , baffle location (L1 = 20. 0 c m ) and baffle height ( h = 1. 90 cm). The distance .t between the baffles i s changed f r o m 1 c m to 6 c m . The a b s c i s s a in the estimated velocity of the flow a t the baffles by the relation

where D/d i s the ratio of the cross-sectional a r e a of the flow a t the entrance of the duct to that a t the baffles, and Uq i s the measured entrance velocity. The coordinate i s the overall sound p r e s s u r e level measured by a microphone which i s placed f a r outside the duct entrance. The unit of the coordinate i s a r b i t r a r y but i s

the s a m e f o r Figs. 5. 1. 1 to 5. 1 . 6 f o r comparison.

Figure 5. 1. 1 shows the data f o r baffle distance .t = 1. 0 cm.

When the flow speed goes up to 15 m / s e c , the sound produced shows a significant i n c r e a s e in power and then d e c r e a s e s to the level where no pure tone i s observed. The elevated sound in the neighborhood of U = 20 m / s e c i s a pure tone with c l e a r sinusoidal wave f o r m

e

when monitored by an oscilloscope. When we f u r t h e r i n c r e a s e the speed of the flow, another powerful pure tone i s detected n e a r

U

= 40 m / s e c and then the level of this sound d e c r e a s e s to the e

1

ordinary noise level with no significant frequency component.

In the c a s e of baffle distance, J?, = 2. 0 cm, the pure tone pro- duction regions grow in numbers to three a s shown in Fig. 5. 1.2.

A s i m i l a r behavior in sound generation activity i s observed f o r A = 3 . 0 c m (Fig. 5. 1 . 3 ) , but for 1 = 4 . 0 cm, the numbers of the peaks in the sound p r e s s u r e level r e t u r n to two. The maximum level of the pure tone p r e s s u r e v a r i e s depending on the baffle distance. In fact, the peak i n sound p r e s s u r e which a p p e a r s a t lower flow speeds shows i t s highest level f o r A = 3. 0 cm, while the average level does not change so much. In c o n t r a s t to this f i r s t peak, the change both in level and shape of the peaks which occur a t a higher flow speed i s d r a s t i c a s the baffle distance i s varied.

F o r the c a s e of sufficiently l a r g e baffle distance, 4 = 6. 0 c m , no peak in sound p r e s s u r e level i s observed (Fig. 5. 1. 6), and a s a consequence no pure tone generation i s achieved.

Measured pure tone frequencies f o r the c a s e of A = 3 . 0 c m a r e shown in Fig. 5. 1 . 7 . P u r e tone i s observed in three separated regions in the flow velocity, but the frequencies of sound a r e essen- tially constant throughout these regions. T h e r e i s , however, a slight i n c r e a s e in frequency in each region when the flow speed i s increased. The width of the range of the velocity where a pure tone i s generated v a r i e s according to the geometrical conditions.

Another example of the measured pure tone frequencies i s shown in Fig. 5. 1. 8. This i s the c a s e of duct length L = 92. 7 c m and baffle location L1 = 20.0 cm. Under these conditions, t h r e e modes of sound a r e observed in s e v e r a l separated regions, while in each region the frequency is n e a r l y constant but -shows a slight increase.

5. 2 Longitudinal Acoustic Modes of the Duct

Longitudinal acoustic modes of the duct, with and without

baffles installed, w e r e m e a s u r e d independently under no flow through the duct. A dynamic speaker was placed a t the s l i t of the duct, and then sound was generated by a function generator. The input signal to the speaker was of a sinusoidal type and the frequency was counted by a frequency counter. A hot w i r e was placed i n the duct to detect the fluctuations and monitored by a n oscilloscope. The exciting frequency was swept in the range of i n t e r e s t , and f r o m the observation of the hot wire output, the natural frequency of the duct was determined.

Prominent l a r g e oscillations of the hot wire were observed a t 268 and 554 Hz f o r the s h o r t duct ( L = 50.8 cm), and 168, 339, 511 a n d 6 7 4 H z f o r the long duct (L = 92.7 cm). The presence of the baffles did not r e s u l t in a n appreciable difference in these frequencies.

With geometrical considerations, i t was concluded that the longitudinal modes of the acoustic oscillation have fundamental f r e - quencies of 268 Hz (open end correction AL = 12.6 c m ) f o r the s h o r t duct and 168 Hz (AL = 8 . 5 c m ) f o r the long duct.

5.3 Resonant Frequencies

Resonant frequency measurements w e r e performed in various combinations of the total duct length L, the height of the baffle h and the location of the baffles in the duct

h .

(L1 i s measured f r o m the entrance of the duct. )

i ) h = 1.90 c m

F i g u r e s 5.3. 1

-

5 . 3 . 4 show the c a s e f o r the short duct ( L = 50.8 cm). The distance A between the baffles i s denoted by a different symbol a s indicated.

-30-

Figure 5.3. 1 i s the c a s e where the baffle i s located a t the

entrance of the duct. In general, when the flow velocity i s increased, the f i r s t longitudinal acoustic mode i s excited a t a c e r t a i n flow speed, and then i t disappears and r e a p p e a r s again. Then the second acoustic mode i s excited a t a higher flow velocity. The fifth mode i s excited only f o r = 3. 0 c m and a t a higher flow velocity.

Figure 5 . 3 . 2 i s f o r the c a s e of L, = 10. 0 cm. At this baffle location, the activity of sound production i s not strong and the second mode i s not excited a t all, but instead the third mode i s brought about f o r the s h o r t baffle spacing.

Figure 5 . 3 . 3 i s f o r L1 = 20.0 cm. Here three regions of the second mode take place but the f i r s t mode excitation i s not observed ( o r i t i s too weak to be measured with the p r e s e n t instrumentation).

The third and the fifth modes a r e a l s o excited f o r the c a s e

a

= 1 . 0 c m .

Figure 5 . 3 . 4 i s f o r L1 = 30.0 cm. The f i r s t mode reappears, but only in a narrow range of the flow velocity. Two o r three

regions of the second mode a r e observed followed by the third mode excitation a t higher speeds f o r s h o r t baffle distance.

F r o m Fig. 5.3. 5 to Fig. 5.3. 10 a r e the c a s e s of the long duct ( L = 92. 7 c m ) . Figure 5.3. 5 i s f o r the baffle located a t the duct entrance ( L = 1 . 0 cm). In general the f i r s t mode i s excited twice a t a lower speed accompanied by one o r two regions of the second mode. Higher modes a r e also observed f o r the s h o r t e r baffle

spacing.

Figure 5.3.6 i s f o r L, = 10. 0 cm. At this baffle location, sound generation activity i s not strong and regions of tone productions

a r e s m a l l compared to other c a s e s .

Figure 5.3. 7 i s f o r L1 = 20.0 cm. At this location, sound production activity i s complex and s e v e r a l interesting f e a t u r e s can be noticed. Consider

R

= 2. 0 c m (triangle symbols), f o r ex-

ample. When the flow velocity goes up to 10 m / s e c , the f i r s t mode appears and disappears quickly. Then the f i r s t mode r e a p p e a r s a t about a velocity of 16 m / s e c . After this mode vanishes, the third mode i s excited and then disappears. Then a t about 40 m / s e c the fourth mode a p p e a r s and disappears. Then a t 48 m / s e c the mode comes back to the third and disappears and then a t Ue = 61 m / s e c , the mode goes up to the fourth again and finally disappears.

Figure 5.3. 8 i s f o r L1 = 30.0 cm. F o r this c a s e the activity goes down again and s e v e r a l modes, high and low, a r e excited a t isolated s m a l l regions of the flow velocity.

Figure 5.3.9 i s f o r L, = 42.9 cm. The second mode i s excited twice and the fourth mode i s accompanied a t a higher speed.

Figure 5.3. 10 i s f o r L1 = 61.9 cm. The second and the third modes a r e excited a t lower speeds while higher modes a r e

seen a t s h o r t e r baffle distance.

ii) h = 1.52 c m

F i g u r e s 5.3. 11

-

5.3. 14 a r e f o r lower baffle height. In

general, sound production activities f o r the s h o r t e r height c a s e s a r e l e s s complex than those f o r higher height c a s e s .

Figure 5.3.11 i s f o r L = 50.8 c m and L1 = 1 . 0 cm. Only the f i r s t mode i s excited a t s e v e r a l ranges of the velocity. In con- t r a s t , f o r

k

= 20. 0 cm, only the second mode i s excited a t two velocity ranges (Fig. 5.3. 12).

Figure 5.3. 13 i s f o r the longer duct c a s e ( L = 92. 7 c m ) and L, = 1.0 cm. F o r this configuration t h r e e lower modes a r e excited a t s e v e r a l regions while f o r = 20. 0 c m only the third and f o r t h modes a r e excited a t a higher speed.

iii) h = 2.28 c m

F i g u r e s 5.3. 15 and 5.3. 16 a r e f o r higher baffle height and s h o r t e r duct length L = 50.2 em. The situations f o r this c a s e differ considerably compared to those of the lower baffle height.

Much higher resonance frequencies a r e observed. Figure 5.3. 15 i s f o r the c a s e that the baffles a r e placed a t the entrance, and Fig. 5.3.16 i s f o r L1 = 20.0 cm.

This slope, o r the r a t i o f/Ue f o r m s the Strouhal number when a c h a r a c t e r i s t i c length scale i s introduced. F o r these c a s e s of higher baffle height, the Strouhal number s e e m s to be f a i r l y constant.

5 . 4 Non- Dimensional Frequency

The non-dimensional frequency i s considered to be the slope of the line which connects the origin with the measured point i n the frequency vs. velocity graphs. Since the sound frequency i s a l m o s t constant in each tone production region, disregarding the baffle distance A , it i s appropriate to take the height of the baffle a s a r e f e r e n c e length scale. Then the non-dimensional frequency, o r Strouhal number, i s defined by

where Ue i s the representative velocity of the flow, given by Eq. (5. 1. 1).

As the observed sound i s related to the longitudinal mode of the duct, we can e x p r e s s the frequency of the sound a s follows:

where a i s the velocity of the sound, n i s an integer which r e p r e s e n t s the mode and L' i s the length s c a l e of the duct which differs slightly f r o m the actual geometric length because of the

open-end correction. (Here we assumed an open pipe configuration. ) Substitution of Eq. ( 5 . 4 . 2 ) into Eq. ( 5 . 4 . 1 ) gives

When we introduce the Mach number by

Eq. ( 5 . 4 . 3 ) i s expressed a s

Therefore, f o r fixed geometrical conditions, the Strouhal number i s inversely proportional to the Mach number, and the number which indicates the mode i s included in the proportionality constant.

Some examples f o r this Strouhal number vs. Mach number plotting a r e shown in Figs. 5 . 4 . 1

-

^5.4.XO. As can be expected f r o m the frequency data in t e r m s of velocities, the Strouhal numbers f o r the generated sound l i e in certain regions, though there seems to be some dependence on the height of the baffles. In general,

Strouhal n u m b e r s a r e i n the range:

i

0 . 1 < S

- - <

0 . 6

,

^{(h =} 1.90 c m ) 0. 1

- - <

S

<

0 . 3

,

( h = 1. 52 c m ) 0 . 5

- - <

S

<

1.0

.

^(h = 2.28 c m )

5 . 5 P u r e Tone Production Regions

P u r e tone production regions a r e shown in t e r m s of flow velocity and distance between baffles i n F i g s . 5. 5. 1

-

5. 5. 16.

Numbers denoted a t each region indicate the mode of the generated sound. Narrow regions a r e shown a s l i n e s and isolated points, instead of regions, a r e a l s o included a s c i r c l e s o r horizontal lines.

In general, higher modes a r e observed a t a higher speed and the tone production region shifts to the higher speed side when the d i s - tance between the baffles i s increased.

i) h = 1.90 c m

F i g u r e s 5. 5. 1

-

5. 5 . 4 a r e c a s e s f o r the s h o r t duct. The f i r s t mode i s about 270 Hz, .the second mode 540 Hz, and so on. In g e n e r a l the f i r s t mode a p p e a r s a t low speed and the higher modes

(up to the third) appear a t a higher speed. T h e r e s e e m s to be a minimum speed to produce p u r e tone.

F i g u r e s 5 . 5 . 5

-

5.5.10 a r e f o r the long duct. The frequency of the f i r s t mode i s about 160 Hz and the second mode 320 Hz, and so on. General c h a r a c t e r i s t i c s

--

modes a r e lower a t low speed and higher a t high s p e e d - - c a n a l s o be s e e n f o r these c a s e s . It i s c l e a r that a minimum speed i s required to produce a p u r e tone, but the r e q u i r e m e n t f o r the minimum distance i s not c l e a r .

ii) h = 1. 52 crn

The situation i s l e s s complex f o r this baffle height. The s h o r t e r baffle regions where p u r e tone i s produced a r e c l e a r l y separated f r o m each other. Only one mode i s observed in each c a s e f o r the s h o r t duct (Figs. 5. 5. 11-12), while s e v e r a l modes a r e detected f o r the long duct (Figs. 5. 5. 13-14).

iii) h = 2. 28 c m

As was expected f r o m the frequency graphs, the regions of the sound production a r e scattered, consisting of many s m a l l islands.

Higher modes a r e , i n general, excited a t a higher flow velocity (Figs. 5. 5. 15- 16).

5. 6 Velocity Field Between the Baffles i ) Centerline Velocity

Because of the complex flow field i n the duct n e a r the baffles, the c o r r e c t velocity a t the baffle cannot be estimated by a simple geometric condition expressed by Eq. (5. 1. 1).

The actual velocity m e a s u r e m e n t was done by placing a hot wire between the baffles on the duct centerline. The r e s u l t s a r e

shown f o r the two c a s e s of baffle height. F i g u r e 5. 6. 1 i s for

h = 1.90 c m o r D/d = 3 . 92, where D and d a r e cross-sectional a r e a s a t the entrance of the duct and that a t the baffle, respectively.

The centerline velocity, i n general, exceeds the value of D/d f o r a e s h o r t e r baffle spacing o r f o r lower flow velocity, while f o r the l a r g e r baffle spacgxg-it s e e m s to be a good first.approximation.

Figure 5 . 6 . 2 i s f o r h = 1. 52 c m (D/d = 2.48). F o r this s h o r t e r baffle case, the centerline velocity s e e m s to be independent

of the baffle distance and indicates slightly higher values than D/d.

But Ue given by Ep. (5. 1. 1) s e e m s to be a good f i r s t approximation.

ii) Velocity Profile

The velocity profile between the baffles was measured by traversing the hot w i r e probe a c r o s s the duct. An example f o r h = 1.9 c m i s shown in Fig. 5. 6 . 3 . As i s c l e a r l y shown f r o m this figure, a strong jet-like s h e a r profile i s achieved between the baffles.

The velocity on the line which connects the edges of the baffles s e e m s to be about 50 percent of the c e n t e r velocity.

5. 7 Flow Visualization

Flow visualization was performed f o r the long duct ( L = 92. 7 c m ) with baffles of height h = 1.90 c m located a t the entrance of the duct (L, = 1.0 cm). Both blunt and s h a r p edge baffles a r e employed for this purpose, but no essential difference was detected between them.

F o r each configuration the flow speed was adjusted so that the acoustic resonance was obtained a t the lowest flow velocity, but i t was a l s o tried f o r higher speed. Under these resonant conditions a v e r y stable vortex o r vortices were observed by adjusting the frequency of the strobotac illumination. Out of the region of pure tone produc- tion stable v o r t i c e s a r e difficult to obtain except f o r v e r y slow speeds where regular vortical shedding activity i s observed.

F i g u r e s 5. 7. 1 and 5. 7 . 2 a r e for the baffle spacing k = 1.0 cm.

At flow velocity Ue = 14 m / s e c (Fig. 5. 7. l ) , a vortex i s formed between the baffles, while a t a higher speed Ue = 23 m / s e c (Fig.

5. 7 . 2 ) , two vortices, though s m a l l e r in scale, a r e seen along each side of the flow.

Figures 5. 7.3 and 5. 7 . 4 a r e f o r the baffle spacing A = 3 . 0 cm. These pictures a l s o show the i n c r e a s e i n number of vortices between the baffles when the resonance i s obtained a t a higher flow

speed.

The vortices observed under these resonant conditions a r e symmetric along the centerline of the duct.

- 3 8 -

6. DISCUSSION

6. 1 Mechanism of Sound P r o d u c t i o n

A s c a n b e s e e n f r o m the c o m p a r i s o n of the p u r e tone frequency with the longitudinal a c o u s t i c m o d e s of the d u c t ( F i g s . 5. 3. 1

-

5.3. 16), the frequency of the g e n e r a t e d tone i s d i r e c t l y r e l a t e d to the n a t u r a l frequency of the duct. T h e a p p e a r a n c e and d i s a p p e a r a n c e of the p u r e tone production activity, however, s e e m s to be r e l a t e d to the i n t e r - a c t i o n of v o r t i c e s shed f r o m the f i r s t baffle with the a c o u s t i c waves g e n e r a t e d a t the second baffle. In o t h e r words, i n the f i g u r e of frequency f vs. velocity Ue (Fig. 6. 1. l ) , the horizontal line

i s d e t e r m i n e d by the longitudinal mode of the duct. H e r e f . ' s a r e

1

the n a t u r a l frequency of the duct. The slope f/Ue, on the other hand, i s d e t e r m i n e d b y the i n t e r a c t i o n s between the v o r t e x shedding and acoustic waves, which is given by Eq. (3.4. 5):

T h e i n t e g e r m b r i n g s i n the d i s c r e t e f e a t u r e of the tone production.

v

T h e i n t e r s e c t i o n s of the l i n e s e x p r e s s e d by Eq. (6. 1. 1 ) and the l i n e s r e p r e s e n t e d by Eq. (6. 1 . 2 ) a r e the points w h e r e p u r e tone production c a n be achieved.

An e x a m p l e of the application of this idea to the a c t u a l data i s shown i n Fig. 6. 1 . 2 f o r L = 92. 7 c m , h = 1.90 c m , L, = 20. 0 c m and R = 3 . 0 c m .

6 . 2 P u r e Tone Production Regions

Under fixed g e o m e t r i c a l conditions (4 = ll ), points of p u r e tone