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Two causality correlation techniques applied to jet noise 1973

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. ' 1 5 1 ' K f TWO CAUSALITY CORRELATION TECHNIQUES APPLIED TO JET NOISE by ROBERT RACKL D i p l . Ing., Technische Hochschule Graz, A u s t r i a , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mechanical E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by hi s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of H e ^ a m ( i'd^Tl'naj The University of B r i t i s h Columbia Vancouver 8, Canada Date MrSU^ (O. /<?73 i S u p e r v i s o r : Dr. Thomas E. Siddon ABSTRACT The t h e s i s d e s c r i b e s two techniques f o r i n v e s t i g a t i n g e x p e r i m e n t a l l y the g e n e r a t i o n of noise by t u r b u l e n t j e t s u s i n g a r e c e n t l y developed method of c r o s s c o r r e l a t i o n (Siddon, 7th ICA, Budapest 1971), The work i s motivated by the need t o reduce f u r t h e r the exhaust no i s e of modern j e t t r a n s p o r t a i r c r a f t . The aim i s to p r o v i d e new i n f o r m a t i o n on n o i s e g e n e r a t i o n mechanisms, to s t i m u l a t e f u t u r e t h e o r e t i c a l r e s e a r c h by f i n d i n g the d i s t r i b u t i o n and c h a r a c t e r of n o i s e s o u r c e s , and to p r o v i d e d i a g n o s t i c t o o l s f o r making n o i s e surveys of t u r b u l e n t f l o w s i n g e n e r a l . Experiments were conducted i n an anechoic chamber u s i n g a c o l d model a i r j e t at a Mach number of about 0.32. The f i r s t method used, the image technique, c r o s s c o r r e l a t e s the p r e s s u r e on a s u r f a c e c l o s e to the j e t with the r a d i a t e d sound i n the f a r f i e l d . T h i s enables the deduction of the a c o u s t i c source s t r e n g t h per u n i t s u r f a c e area which i n t u r n g i v e s an i n d i c a t i o n o f the approximate l o c a t i o n of the sound s o u r c e s i n the j e t . The method i s shown to be s e l f c o n s i s t e n t . The technique can be used i n t h e i n v e s t i g a t i o n of s u p e r s o n i c j e t s as w e l l ; t h i s i s where i t s r e a l power may l i e . The second method d i r e c t l y c r o s s c o r r e l a t e s the hydrodynamic. pressure f l u c t u a t i o n i n the t u r b u l e n t j e t flow with the f a r f i e l d sound a l l o w i n g the ded u c t i o n of the l o c a l a c o u s t i c source s t r e n g t h per u n i t volume and the a s s o c i a t e d power s p e c t r a l d e n s i t y . A s p e c i a l l y designed f o i l type sensor i s used to measure the pressure f l u c t u a t i o n s . A d i s t r i b u t i o n of source s t r e n g t h over the j e t f o r r a d i a t i o n at 45° to the j e t a x i s i s o b t a i n e d , and i s . i i not unexpectedly, s i m i l a r to the d i s t r i b u t i o n of mean v e l o c i t y shear. The a x i a l d i s t r i b u t i o n from a " s l i c e " of the j e t r e s u l t s by r a d i a l l y i n t e g r a t i n g the source s t r e n g t h s ; i t i s i n q u a l i t a t i v e agreement with the r e s u l t s of other r e s e a r c h e r s . However, d i f f i c u l t i e s are encountered i n making the method s e l f c o n s i s t e n t , i . e . , the r a d i a t e d sound as p r e d i c t e d by i n t e g r a t i n g over a l l the sources i n the j e t d i f f e r s widely from the d i r e c t l y measured sound. Reasons f o r t h i s d i s c r e p a n c y are given and s o l u t i o n s to overcome i t are proposed. A F o u r i e r transform technique a l l o w s the deduction of the c o n t r i b u t i o n to the spectrum of the r a d i a t e d pressure from a u n i t volume of t u r b u l e n c e . Thus, the s p e c t r a l c h a r a c t e r of the sound sources i n the j e t i s obtained, although the above d i f f i c u l t i e s are r e f l e c t e d here as w e l l . The- number of independent a c o u s t i c sources i n the j e t was estimated by c a l c u l a t i n g the average c o r r e l a t i o n v o l u m e . The number t u r n s out to be of the order of 100 to 200; t h i s l i e s between two widely d i f f e r e n t e s t i m a t e s of about 3 and about 2500 giv e n by other r e s e a r c h e r s . i i i ACKNOWLEDGEMENTS T h i s work was undertaken at the s u g g e s t i o n and under the s u p e r v i s i o n of Dr. T.E. Siddon who guided the r e s e a r c h with p a t i e n c e and on whose experi e n c e the author o f t e n h e a v i l y r e l i e d . Thanks go to summer stu d e n t s N. M i c h a i l o f f f o r h e l p i n g c o n s t r u c t the anechoic f a c i l i t y , and to R. Sanson and K.F. Phoon f o r h e l p i n g with the data a c q u i s i t i o n . Thanks a l s o t o Mr. John Smith of Richmond, BC, f o r f i b r e g l a s s i n g the j e t n o z z l e . T h i s r e s e a r c h was sponsored by the N a t i o n a l Research C o u n c i l of Canada, under Grant No. 67-7106. The development of the p r e s s u r e sensor r e c e i v e d support by the Defense Research Board of Canada under Grant No. 66-9603. i v N O T A T I O N ( S p e c i a l n o t a t i o n s of the appendices are not included) A a x i a l flow e r r o r c o e f f i c i e n t B c r o s s flow e r r o r c o e f f i c i e n t c D ambient speed of sound Cp s t a t i c p r e s s u r e e r r o r c o e f f i c i e n t C P u C p f o r u-component only C probe contamination r a t i o D n o z z l e diameter db d e c i b e l s h d i s t a n c e j e t a x i s to s u r f a c e S L i n t e g r a l l e n g t h s c a l e /j u n i t v e c t o r p e r p e n d i c u l a r to s u r f a c e S m meters mm m i l l i m e t e r s H Mach number n number of i n c o h e r e n t sources i n the j e t P t o t a l s t a t i c pressure ( s u b s c r i p t s have same meaning as below f o r lower case p's) p f l u c t u a t i n g part of s t a t i c pressure p o ambient p r e s s u r e pco) II pseudosound" pressure p ( i ) a c o u s t i c a l l y propagating part of the p r e s s u r e p P m measured pressure Pj t r u e p r e s s u r e p^ s u r f a c e pressure p s t a t i c p r e s s u r e at j e t e x i t V r r = | x - y j ; or r a d i a l d i s t a n c e from j e t c e n t e r l i n e rms r o o t mean square = J4 2 \ 5 s u r f a c e Sj imaging s u r f a c e S U A source s t r e n g t h per u n i t area S y V source s t r e n g t h per u n i t volume t time \. r e t a r d e d time Tjj L i g h t h i l l ' s s t r e s s tensor u component of v e l o c i t y i n a x i a l d i r e c t i o n Uj v e l o c i t y v e c t o r U 0 n o z z l e e x i t v e l o c i t y 6 mean v e l o c i t y V volume of source r e g i o n V. c o r r e l a t i o n volume Vj ef volume of sound producing j e t t u r b u l e n c e v,w components of v e l o c i t y p e r p e n d i c u l a r t o a x i s of probe x x=|x| Xj,x space c o o r d i n a t e (often i n d i c a t i n g the p o i n t of sound d e t e c t i o n i n the f a r f i e l d ) x Q d i s t a n c e from nozzle e x i t to f a r f i e l d microphone y. space c o o r d i n a t e ( u s u a l l y used i n the source r e g i o n 1 and/or as i n t e g r a t i o n v a r i a b l e ) 8.. Kronecker D e l t a •J 6 angle between j e t a x i s and f a r f i e l d d i r e c t i o n v frequency p d e n s i t y po ambient d e n s i t y T time delay t r e t a r d e d time delay v i T.. v i s c o u s s t r e s s t e n s o r •J £ s p a c e s e p a r a t i o n v e c t o r (£j,£ 2,£ 3) V 2 a 2 / a y 2 o r d % x 2 • a d o t o n a q u a n t i t y i n d i c a t e s d e r i v a t i o n w i t h r e s p e c t t o r e a l t i m e a n o v e r b a r d e n o t e s t i m e a v e r a g i n g [ . . . ] r e p l a c e i n t h e e x p r e s s i o n w i t h i n t h e b r a c k e t s t b y t = t - r / c Q , o r T b y f = T + r / c o f w h i c h e v e r i s a p p r o p r i a t e . The j e t S t r o u h a l number i s d e f i n e d by i/D/U o v i i TABLE OF CONTENTS I. Foreword and H i s t o r i c a l Background ..... 1 I I . Present Trends and I n t r o d u c t i o n to S e c t i o n s I I I and IV ..10 I I I . Image Technique .........16 3.1 Imaging System ..16 3.2 Cross C o r r e l a t i o n Formalism ......19 3.3 Experiments 21 3.4 N o n d i m e n s i o n a l i z a t i o n 23 3.5 R e s u l t s .-24 3.6 Check 29 IV. C a u s a l i t y C o r r e l a t i o n Technique ........31 4.1 I n t r o d u c t i o n .............31 4.2 Theory ...32 4.2.1 F l u i d D i l a t a t i o n s 32 4.2.2 C a u s a l i t y C o r r e l a t i o n s ........... 35 4.2.3 L o c a l Source S t r e n g t h and Spectrum ................. 37 4.2.4 C o r r e l a t i o n Volume 38 4.3 Development of the Pressure Sensor ......42 4.3.1 L i m i t a t i o n s on C o n v e n t i o n a l C y l i n d r i c a l Probes ........... 42 4.3.2 A i r f o i l Probe 44 4.3.3 S t a t i c P r e s s u r e C a l i b r a t i o n ......47 4.3.4 The Complete Pressure Sensor .....50 4.4 Experiments ..............51 4.4.1 A i r Supply, S e t t l i n g Chamber, and Nozzle ..51 4.4.2 The Far F i e l d Microphone 51 4.4.3 S i g n a l P r o c e s s i n g .....52 4.5 Data Reduction ..........53 4.5.1 Source Strength D i s t r i b u t i o n .....53 4.5.2 Check, C l o s u r e D i f f i c u l t y .55 4.5.3 S p e c t r a 58 4.5.4 Number of Incoherent Sources 60 V. Summary C o n c l u s i o n s , Recommendations ...63 References .66 App. A: Mathematical D e t a i l s Concerning Eq.(4.7) Through Eq. (4.11) ...71 v i i i App. B: N o n d i m e n s i o n a l i z a t i o n of C o r r e l a t i o n F u n c t i o n s . . . 7 4 App. C: P r e d i c t i o n of the Shape of the Cross C o r r e l a t i o n F u n c t i o n 77 App. D: Probe Contamination R a t i o 83 F i g u r e s . . 90 ix FIGURES F i g . 3 - 1 Image geometry F i g . 3 - 2 J e t n o i s e spectrum with and without a d j a c e n t s u r f a c e F i g . 3 - 3 Mounting of 1/4-inch microphone i n sheet of 1/2- in c h p l e x i g l a s s F i g . 3 - 4 T y p i c a l c r o s s c o r r e l a t i o n between s u r f a c e pressure and r a d i a t e d p r e s s u r e ; a l s o showing p r e c u r s o r F i g . 3 - 5 R e l a t i o n s h i p between s u r f a c e pressure and e x i t v e l o c i t y F i g . 3 - 6 D i s t r i b u t i o n of source s t r e n g t h over s u r f a c e ; r e f l e c t i o n path. 0 F i g . 3 - 7 Equal source s t r e n g t h contours f o r 0=90° F i g . 3 - 8 Equal source s t r e n g t h contours f o r 0=60° F i g . 3 - 9 Zone of i n f l u e n c e F i g . 3 - 10 3 t y p i c a l p s p (r) F i g . 3 - 11 D i s t r i b u t i o n of root mean squared s u r f a c e pressure F i g . 3 - 12 " S l i c e w i s e " i n t e g r a t e d source s t r e n g t h F i g . 4 - 1 Experimental setup of C a u s a l i t y C o r r e l a t i o n Technique F i g . 4 - 2 Geometry F i g . 4 - 3 D e f i n i t i o n of c o r r e l a t i o n l e n g t h F i g . 4 - 4 C l a s s i c a l c y l i n d r i c a l probe c o n f i g u r a t i o n F i g . 4 - 5 C o r r e l a t i o n f u n c t i o n contaminated by d i p o l e n c i s e F i g . i t - 6 P r i n c i p l e of a r r a n g i n g f o i l probe i n the flow F i g . it- 7 L o c a t i o n of pressure sensing holes on a i r f o i l F i g . 4 - 8 Press u r e d i s t r i b u t i o n on subsonic a i r f o i l F i g . 4 - 9 Contaminated and uncontaminated c o r r e l a t i o n f u n c t i o n s and t h e i r F o u r i e r transforms F i g . 4 - 10 V e l o c i t y v e c t o r s on a i r f o i l F i g . 4 - 11 Probe shapes t r i e d out but d i s c a r d e d F i g . 4 - 12 The f o i l type pressure sensor F i g . 4 - 13 S t a t i c p r e s s u r e c a l i b r a t i o n s F i g . 4 - 14 P r e s s u r e d i s t r i b u t i o n over f o i l with dip near t r a i l i n g edge F i g . 4 - 15 F o i l type sensor mounted on p r e a m p l i f i e r F i g . 4 - 16 Frequency response of f o i l type sensor F i g . 4 - 17 Schematic of experimental setup F i g . 4 - 18 V a r i a t i o n of pressure f l u c t u a t i o n with d i s t a n c e from j e t F i g . 4 - 19 S i g n a l flow F i g . 4 - 20 D i s t r i b u t i o n of source s t r e n g t h i n a subsonic j e t f o r 45° to the j e t a x i s F i g . 4 - 21 " S l i c e w i s e " d i s t r i b u t i o n of source stren.gth; comparison with other r e s u l t s F i g . 4 - 22a Some t y p i c a l c r o s s c o r r e l a t i o n f u n c t i o n s p<o)p F i g . 4 - 22b Some t y p i c a l c r o s s s p e c t r a ( F o u r i e r transforms of f u n c t i o n s shown i n Fig.4-22a) F i g . 4 - 23 Peak f r e q u e n c i e s versus downstream p o s i t i o n F i g . 4 - 24 Cross c o r r e l a t i o n p<o>p and f i t t e d to i t the p r e d i c t e d f u n c t i o n F i g . 4 - 25 Sears* f u n c t i o n F i g . 4 - 26 E f f e c t i v e c h o r d l e n g t h on c y l i n d r i c a l pressure probe F i g . 4 - 27 S p e c t r a l d e n s i t y of the "pseudosound" p<o) at 4D X Fig.4-28 The d i r e c t l y measured and the i n t e g r a t e d spectrum at 45° i n the f a r f i e l d Fig.B-1 Image geometry to s c a l e 1 I. FOREWORD AND HISTORICAL BACKGROUND In t h i s s e c t i o n , a b r i e f review of aerodynamic noise r e s e a r c h with emphasis on subsonic j e t n o i s e w i l l l e a d up to the present work. S e c t i o n I I w i l l say more on why the present work was undertaken and e s t a b l i s h i t s proper p l a c e i n the o v e r a l l r e s e a r c h e f f o r t . As with most d i s c i p l i n e s of s c i e n c e , theory and experiments go hand i n hand. Here, f o r reasons of c l a r i t y i n the h i s t o r i c a l development, they are sep a r a t e d . I t was p r i m a r i l y the advent of new, very n o i s y p r o p u l s i o n methods at the end of World War I I - the j e t engine and the r o c k e t - t h a t s t i m u l a t e d the development of a new d i s c i p l i n e , the study of sound generated a e r o d y n a m i c a l l y . In h i s book " A c o u s t i c s of a Nonhomogeneous Moving Medium," B l o k h i n t s e v (1946) gave a thorough account of the i n t e r r e l a t i o n s h i p between a c o u s t i c and aerodynamic motions of a f l u i d . References to t h i s work are r a r e i n the E n g l i s h l i t e r a t u r e , probably because i t was t r a n s l a t e d i n t o E n g l i s h only 10 years a f t e r i t was f i r s t p u b l i s h e d i n Russian. L i g h t h i l l (1952) combined the b a s i c equations of f l u i d dynamics ( c o n s e r v a t i o n of mass and momentum) i n such a way as t o show how to formulate an inhomogeneous form of the wave equation f o r the case o f f l u i d s d i s t u r b e d by l a r g e r than a c o u s t i c p e r t u r b a t i o n s : where: p = d e n s i t y , t=time, c c=speed of sound, X j = s p a c e v a r i a b l e , T-,j = a tensor r e p r e s e n t i n g e x t e r n a l l y a p p l i e d s t r e s s e s ( e x t e r n a l from the p o i n t of view of the a c o u s t i c i a n ) . The s o l u t i o n to Eg. (1.1) i s i n the form of r e t a r d e d p o t e n t i a l s [ s e e Eq. (1 . 2 ) ] . L i g h t h i l l h i m s e l f and then many other workers a p p l i e d , enlarged and s i m p l i f i e d the theory: L i g h t h i l l found t h a t the sound i n t e n s i t y from a f r e e aerodynamic flow should vary with the e i g h t h power of some c h a r a c t e r i s t i c v e l o c i t y and the square of some c h a r a c t e r i s t i c dimension (U 8D 2-Law). He co n s i d e r e d t u r b u l e n c e as a source of noise and suggested a model f o r j e t noise i n terms of convected a c o u s t i c guadrupoles, r e p l a c i n g the s t r e s s e s i n the f l u i d , i n an ambient f l u i d a t r e s t ( L i g h t h i l l 1954). Proudman (1952) estimated the p r o p e r t i e s of no i s e generated by i s o t r o p i c t u r b u l e n c e , mainly assuming Gaussian models f o r the s t o c h a s t i c processes. T h i s i s , however, an o v e r s i m p l i f i e d model f o r sub s o n i c j e t n o i s e because the tu r b u l e n c e i n the mixing r e g i o n , f a r from being i s o t r o p i c (see f o r i n s t a n c e Ribner 1965 or Fuchs 1972), i s c h a r a c t e r i z e d by a l a r g e mean v e l o c i t y g r a d i e n t . Proudman was a l s o the f i r s t t o show t h a t the f a r f i e l d noise from any flow can be viewed as being generated by the components of the v e l o c i t y f l u c t u a t i o n s i n the d i r e c t i o n of o b s e r v a t i o n . C u r i e (1955) i n c l u d e d the e f f e c t of s u r f a c e s i n the flow; such s u r f a c e s could be r e a l or j u s t 3 h y p o t h e t i c a l (e.g., so as to exclude an u n i n t e r e s t i n g or e x c e s s i v e l y c o m p l i c a t e d r e g i o n from the f i e l d ) . He obtained the most g e n e r a l s o l u t i o n of Eg. ( 1 . 1 ) : d P u n dS(y) I a r r l d S ( y ) l d2 r r i d V ( y + ——<r 1 — r - J T.. — - (1.2) where: x p o i n t s to the p o i n t of o b s e r v a t i o n , po=ambient d e n s i t y , S=surface, =vector p e r p e n d i c u l a r to S a t y, y=space v a r i a b l e i n the source r e g i o n , r=|x-2l# U j = v e l o c i t y , p=pressure, T j j =viscous s t r e s s t e n s o r , V=volume of sound ge n e r a t i n g flow, [...] means r e p l a c e time t by the e a r l i e r time t - r / c c . Many aerodynamic sound problems can be s o l v e d approximately by s t r a i g h t f o r w a r d a p p l i c a t i o n of Eg. (1.2) i f s i m p l i f y i n g assumptions are a d m i s s i b l e . Such may be: the r e c e i v e r i s many t y p i c a l wavelengths away from the source; the r e c e i v e r i s many t y p i c a l source dimensions away; the motions a t the source are "coherent", i . e . , the p a r t i c l e s move i n unison over a c e r t a i n r e g i o n ; heat conduction and v i s c o s i t y are n e g l i g i b l e ; the flow i s qu a s i i n c o m p r e s s i b l e . In the g e n e r a l case, however, Eq. (1.2) i s an i n t e g r a l e q u a t i o n which has not been s o l v e d I n c l o s e d form. A d d i t i o n a l d i f f i c u l t i e s a r i s e from the i n d e t e r m i n i s t i c , random c h a r a c t e r of t u r b u l e n c e . Powell (1964) suggested a d i f f e r e n t model f o r aerodynamic n o i s e g e n e r a t i o n : A vortex flow and an a c o u s t i c d i p o l e have a s i m i l a r s t r e a m l i n e p a t t e r n . He r e p l a c e s the flow f i e l d by an e q u i v a l e n t d i p o l e f i e l d . Although t h i s theory seems a t t r a c t i v e i n view of T a y l o r ' s i n t e r a c t i n g v o r t i c e s model of t u r b u l e n t t r a n s f e r mechanisms (Tennekes 5 Lumley 1972, s e c t i o n 3.3), i t has not had a s i g n i f i c a n t impact on j e t n o i s e r e s e a r c h . Perhaps the s i m p l e s t concept of flow noise was developed by Ribner (1962), a t f i r s t i n 1958, e l a b o r a t i n g on i d e a s a l s o b y a r r i v e d at i n d e p e n d e n t l y Heecham & Ford (1958): The flow f i e l d i s regarded as an a r r a y of d i l a t i n g and compressing eddies bumping i n t o each other. The pressure i s separated i n t o three p a r t s , the f i r s t r e p r e s e n t i n g the ambient p r e s s u r e independent of time, the second the hydrodynamic pressure f l u c t u a t i o n (or the "pseudosound"), the t h i r d the pressure d i s t u r b a n c e due to a c o u s t i c waves t r a v e l i n g through the f l u i d . Ribner shows t h a t : where p i s the f a r f i e l d r a d i a t e d sound and p<°> i s the second time d e r i v a t i v e of the pseudosound pre s s u r e . Although i t may not be e v i d e n t a t f i r s t s i g h t , t h i s theory of "Sound from F l u i d D i l a t a t i o n s " i s c o n s i s t e n t with the guadrupole concept and has been acknowledged as such (Appendix B of L i g h t h i l l 1963). A monopole type of source n e c e s s a r i l y changes i t s volume when r a d i a t i n g sound, a guadrupole does not. In a f r e e t u r b u l e n t flow the net change i n volume of a l l monopoles combined i s zero, the 5 net radia'tion i s not. Thus, f o u r n e i g h b o u r i n g monopoles can be thought o f as making up a quadrupole. At moderate subsonic Kach numbers the pseudosound i s very w e l l approximated by the s t a t i c p r e s s u r e f l u c t u a t i o n s w i t h i n the t u r b u l e n c e . I f the l a t t e r c o u l d be measured. Eg. (1.3) c o u l d be e v a l u a t e d . S e c t i o n IV of t h i s paper d e s c r i b e s such an attempt. L i g h t h i l l ' s U 8D 2-Law has been e x p e r i m e n t a l l y confirmed f o r t u r b u l e n t j e t s over a wide range of subsonic Hach numbers. Other s e m i - e m p i r i c a l laws d e a l i n g w i t h j e t n o i s e (Ribner 1958; L i l l e y 1958; Powell 1958) were d e r i v e d using the s e l f p r e s e r v a t i o n p r o p e r t i e s of the t u r b u l e n c e i n the mixing r e g i o n (the f i r s t 4 to 5 diameters downstream of the nozzle) and of the f u l l y developed flow f u r t h e r downstream, (The extent of the t r a n s i t i o n between these two r e g i o n s i s not yet c l e a r l y d e f i n e d ) . Thus i t was p r e d i c t e d t h a t the power s p e c t r a l d e n s i t y of the f a r f i e l d p r e s s u r e should f i r s t i n c r e a s e with the square and then decrease with the i n v e r s e square of frequency; measured s p e c t r a show a s i m i l a r shape and l o c a t e the peak S t r o u h a l number (based on e x i t v e l o c i t y , e x i t diameter and frequency) between 0.2 and 0.3 ( M o l l o - C h r i s t e n s e n , K o l p i n & K a r t u c e l l i 1963, s e c t i o n 5 ) , depending on the d i r e c t i o n of r a d i a t i o n . With r e s p e c t to the s p a t i a l d i s t r i b u t i o n of sound sources i t was p r e d i c t e d t h a t the a c o u s t i c source s t r e n g t h of " s l i c e s " of the j e t remains approximately c o n s t a n t i n the mixing r e g i o n and f a l l s o f f very r a p i d l y with the i n v e r s e seventh power of a x i a l d i s t a n c e i n the f u l l y developed r e g i o n . Although there i s some su p p o r t i n g evidence, (see d i s c u s s i o n i n s e c t i o n IV) c o n c l u s i v e experimental c o n f i r m a t i o n i s l a c k i n g . The p r e s e n t r e s e a r c h not only attempts 6 to make t h i s c o n f i r m a t i o n but w i l l go one step f u r t h e r and show a way to measure the l o c a l a c o u s t i c source s t r e n g t h d i s t r i b u t i o n over the j e t volume. In the e x p e r i m e n t a l f i e l d 1 e f f o r t s c o n c e n t r a t e d f i r s t on s u r v e y i n g the r a d i a t e d sound f i e l d from t u r b u l e n t j e t s . I t was found t h a t lower frequency sound i s r a d i a t e d more or l e s s o m n i d i r e c t i o n a l l y and t h a t higher f r e q u e n c i e s f o l l o w a heart shaped p a t t e r n with the d i p on the downstream a x i s - the higher the frequency the more pronounced the d i p of the h e a r t . P a r a l l e l to t h a t , the p r o p e r t i e s of the j e t t u r b u l e n c e were s t u d i e d i n d e t a i l , such as mean v e l o c i t i e s , t u r b u l e n c e i n t e n s i t i e s , and c o r r e l a t i o n l e n g t h s , with the aim of p r e d i c t i n g the n o i s e r a d i a t i o n from t h i s data. The p r o p e r t i e s of s e l f p r e s e r v a t i o n mentioned above were d i s c o v e r e d , but a q u a n t i t a t i v e n o i s e p r e d i c t i o n was not p o s s i b l e to an a c c e p t a b l e degree of c o n f i d e n c e . Although t h e o r e t i c a l formalisms r e l a t i n g the t u r b u l e n c e p r o p e r t i e s to the r a d i a t e d sound abound, very few experimenters have t r i e d to v e r i f y those t h e o r i e s . Chu's (1966) work was p i o n e e r i n g . Using the t h i r d i n t e g r a l of Eq. (1.2) he was able to convert i t i n t o a form such t h a t the mean squared r a d i a t e d p r e s s u r e i n the f a r f i e l d depended on the f o u r t h d e r i v a t i v e of a v e l o c i t y space-time c o r r e l a t i o n . He measured t h i s c o r r e l a t i o n a t * A l i s t of r e f e r e n c e s of e a r l y experimental and t h e o r e t i c a l work can be found i n L i g h t h i l l (1962) or Ribner (1964). The l i s t of r e f e r e n c e s of the present paper attempts to i n c l u d e many of the experimental papers p u b l i s h e d s i n c e 1963 i n the open E n g l i s h l i t e r a t u r e ; the c i t a t i o n s are preceeded by a M * " . A more rec e n t l i s t of r e f e r e n c e s of t h e o r e t i c a l • work can be found i n Doak (1972). 7 a p o i n t 4 diameters downstream f o r d i f f e r e n t a n g l e s of sound r a d i a t i o n and estimated the o v e r a l l mean squared a c o u s t i c p r e s s u r e ; the agreement was reasonable but not t o t a l l y s a t i s f y i n g f o r two reasons: 1) I t would have been p r o h i b i t i v e l y time consuming to survey the complete j e t (modern m u l t i channel c o r r e l a t o r s were not a v a i l a b l e a t the t i m e ) ; 2) t a k i n g the f o u r t h d e r i v a t i v e of an e x p e r i m e n t a l l y o b t a i n e d f u n c t i o n was s u b j e c t to a c o n s i d e r a b l e degree of u n c e r t a i n t y , as the r e s u l t s v a r i e d with the method of curve f i t t i n g used. More r e c e n t l y , Siddon has i n t r o d u c e d a means f o r t y i n g t o g e t h e r the f a r f i e l d and t u r b u l e n c e p r o p e r t i e s by implementing the " c a u s a l i t y c o r r e l a t i o n t e c h n i q u e " (Siddon 1970, 1971b). Any s o u r c e f l u c t u a t i o n appearing i n Eq. (1.2) or (1.3) can be viewed as the b a s i c sound generator. I f i t can be measured and the r e s u l t i n g sound i n the f a r f i e l d i s monitored s i m u l t a n e o u s l y , the two s i g n a l s may be c r o s s c o r r e l a t e d ; i n t h i s manner, the a c o u s t i c s t r e n g t h of the sources can be deduced q u a n t i t a t i v e l y . The word " c a u s a l i t y " i s used because the process of the cause (the source f l u c t u a t i o n ) g i v i n g r i s e to the e f f e c t (the r a d i a t e d sound) p r o v i d e s the b a s i s f o r the measurement. The technique has been a p p l i e d to the a c o u s t i c r a d i a t i o n from s m a l l a i r f o i l s embedded i n a flow (Clarke & Ribner 1969; Siddon 1970, 1973a) and to j e t 9 n o i s e by Lee (1971) and the present work (see f u r t h e r d i s c u s s i o n i n s e c t i o n I I ) . Although the mathematical t h e o r i e s have not been a b l e to g i v e c l e a r c u t guidance f o r d e s i g n i n g q u i e t j e t engines, s u b s t a n t i a l n o i s e r e d u c t i o n s have been achi e v e d , mainly by a p p l y i n g the U 8D 2-Law. Reductions of the j e t e x i t v e l o c i t y with 8 c o r r e s p o n d i n g i n c r e a s e of the n o z z l e diameter and mass flow l e d to the e v o l u t i o n of high bypass r a t i o fan engines. S i g n i f i c a n t n o i s e r e d u c t i o n s with i n c r e a s e d e f f i c i e n c y of performance r e s u l t e d . A very l a r g e number of p a s s i v e n o i s e s u p p r e s s i o n d e v i c e s were proposed and t r i e d out. In many cases the i n s e r t i o n of a suppressor decreased the nois e from the j e t t u r b u l e n c e but not the o v e r a l l n o i s e . The i n t e r a c t i o n of the flow with the d e v i c e generated a d d i t i o n a l d i p o l e n o i s e such that minimal net n o i s e r e d u c t i o n r e s u l t e d . Only two suppressor c o n f i g u r a t i o n s were found to be a c c e p t a b l e , i n terms of t h e i r a c o u s t i c b e n e f i t / aerodynamic p e n a l t y r a t i o . These are the c o r r u g a t e d n o z z l e and the multitube n o z z l e . Even these, however, produce l e s s than the expected noise r e d u c t i o n , i f one d e f i n e s n o i s e i n terms of s u b j e c t i v e l y weighted i n d i c e s such as PNdb or dbA. They are designed with the n o t i o n of r e d u c i n g the i n t e n s i t y of t u r b u l e n c e or the amount of v e l o c i t y shear, i . e . , to make the mixing process smoother, mainly by e n t r a i n i n g a i r not onl y on the o u t s i d e but a l s o on the i n s i d e thereby breaking up the mixing r e g i o n . T h e i r exact f u n c t i o n i n g i s s t i l l s u b j e c t to s p e c u l a t i o n . I t i s f e l t t h a t r e f r a c t i v e s h i e l d i n g may play a s i g n i f i c a n t r o l e i n t h e i r a c o u s t i c a l behaviour as w e l l . The two methods presented i n t h i s paper may be used as t o o l s both t o f u r t h e r our understanding of the n o i s e g e n e r a t i o n process and as a i d s i n the design and improvement of such n o i s e s u p p r e s s i o n d e v i c e s . 9 Although the n o i s e from commercial j e t t r a n s p o r t a i r c r a f t has been reduced by s l i g h t l y more than 2 0 PNdb i n the past two decades people c o n t i n u e t o be annoyed i n the v i c i n i t y of a i r p o r t s as the number of f l y o v e r s i n c r e a s e s . We can expect t h i s i n c r e a s e to c o n t i n u e and t h e r e f o r e cannot s t o p t r y i n g to reduce j e t n o i s e . 10 I I . PRESENT TRENDS AND INTRODUCTION TO SECTIONS I I I AND IV. Present t h e o r e t i c a l e f f o r t s seem to be guided by the f i n d i n g s of the Aerodynamic Noise Symposium at Loughborough i n September 1970 ( F i s h e r & Lowson 1971; Lowson 1971). E f f e c t s of c o n v e c t i o n of the s o u r c e s on the r a d i a t e d sound f i e l d are adequately understood and can be c a s t i n t o mathematical form. E f f e c t s of r e f r a c t i o n are more d i f f i c u l t to d e a l with: We have to t r y to understand the mechanisms of sound g e n e r a t i o n and p r o p a g a t i o n i n and through a shear l a y e r . The reader i s r e f e r r e d to Doak*s e x c e l l e n t review a r t i c l e (1972) i n which new approaches are suggested: He i n f e r s t h a t L i g h t h i l l ' s theory was a b i g step forward, but "... was s p e c i f i c a l l y designed to d e l i b e r a t e l y a v o i d a l l q u e s t i o n s of the i n t e r d e p e n d e n c i e s among a c o u s t i c , t u r b u l e n t and thermal types of motion ...".Recent t h e o r i e s by Crow, Doak and L i l l e y (among others) are d i s c u s s e d t h a t go back to the o r i g i n s of a c o u s t i c s and aerodynamics, i . e . , the works of Stokes, K i r c h h o f f and R a y l e i g h . In Doak's o p i n i o n , a combination of these t h e o r i e s w i l l take us another b i g s t e p forward. The Loughborough Symposium a l s o concluded "... i t appears t h a t the e x i s t e n c e of a s i g n i f i c a n t degree of order i n j e t t u r b u l e n c e i s confirmed ..." ( F i s h e r & Lowson 1971, p. 594). T h i s statement was based on e x p e r i m e n t a l evidence presented at the Meeting by Lau, F i s h e r & Fuchs, and by Crow & Champagne. However, o p i n i o n s s t i l l d i v e r g e widely on t h i s s u b j e c t . The s u p p o r t e r s of 11 the o r d e r l y s t r u c t u r e h y p othesis are making a s t r o n g case f o r t h e i r view by p u b l i s h i n g s u b s t a n t i a l amounts of e x i s t e n t i a l evidence (Crow C Champagne 1971; Davis 1971; Lau, F i s h e r S Fuchs 1972; Fuchs 1972; Fuchs S Michalke 1971). V i s u a l i z a t i o n techniques and space c o r r e l a t i o n s indeed seem t o i n d i c a t e that coherent l a r g e s c a l e s t r u c t u r e s convect f o r many diameters downstream i n the j e t before l o s i n g t h e i r i d e n t i t i e s . T h i s coherence i s e a s i l y r e c o g n i z a b l e a t s m a l l Reynolds numbers ( a f t e r t r a n s i t i o n from the laminar shear l a y e r ) ; i t i s not so well r e c o g n i z a b l e as the Reynolds number i n c r e a s e s but shows a l s o f o r Reynolds numbers over 2*10 5 which i s of the o r d e r of the Reynolds number i n the experiments d e s c r i b e d i n t h i s paper. By analogy, one may c o n s i d e r the t u r b u l e n c e s t r u c t u r e i n a wake behind a b l u f f c y l i n d e r : At moderate Reynolds numbers the Karmann Vortex S t r e e t develops, r e s u l t i n g i n a s t r o n g p e r i o d i c c o n t r i b u t i o n to the spectrum. When the Reynolds number i s i n c r e a s e d t h i s p e r i o d i c i t y gets b u r i e d i n a broad band spectrum, c h a r a c t e r i s t i c of a wide range of t u r b u l e n c e s c a l e s . The t u r b u l e n c e i n a j e t may behave s i m i l a r l y ; the shape of the p r e s s u r e spectrum (Fig.4-27) does not show a d i s c r e t e p e r i o d i c component, but a b r o a d l y peaked energy d i s t r i b u t i o n i s e v i d e n t . Even though . t h e r e may e x i s t an o r d e r l y s t r u c t u r e i n j e t t u r b u l e n c e i t i s not n e c e s s a r i l y of d i r e c t consequence to the r a d i a t e d n o i s e . I t i s w e l l e s t a b l i s h e d t h a t i t i s the r a t e of decay of the c o n v e c t i n g v e l o c i t y s t r u c t u r e (whether t u r b u l e n t or p e r i o d i c ) t h a t c o n t r o l s the n o i s e e m i s s i o n . In f a c t , the r a d i a t i o n depends on the second time d e r i v a t i v e of the source f l u c t u a t i o n q u a n t i t y and i s thereby biased to h i g h e r frequency 12 content than the t u r b u l e n c e proper. In Lee's (1971) and the present work, f o r low speed j e t s , not only was no evidence d i s c o v e r e d of an o r d e r l y s t r u c t u r e p e r t i n e n t to the j e t n o i s e r a d i a t i o n , but the r e s u l t s p o i n t to a c o n t r a r y h y p o t h e s i s of a l a r g e number of i n d e p e n d e n t l y r a d i a t i n g s o u r c e s . A "Simple P r e s s u r e Source Model of J e t Noise" has been put forward by Scharton 5 White (1972). While the i d e a of more or l e s s independent sources i s r e t a i n e d the paper a l s o draws some support from the o r d e r l y s t r u c t u r e h y p o t h e s i s by p o s t u l a t i n g a s m a l l number of l a r g e s c a l e p r e s s u r e sources (of the order of 3). These are presumed t o develop c l o s e to the n o z z l e e x i t , then f l u c t u a t e and convect downstream. The present author f e e l s that the r e s u l t i n g e m p i r i c a l model may be s u i t e d f o r p r e d i c t i o n of the o v e r a l l j e t n o i s e r a d i a t i o n , but t h a t i t does not p r o v i d e d e t a i l e d i n f o r m a t i o n on the n o i s e generating mechanisms. A more thorough c r i t i q u e can be found i n s e c t i o n IV. The c a u s a l i t y c o r r e l a t i o n technique as implemented by Lee (1971, a l s o Lee S Ribner 1972) uses the v e l o c i t y f l u c t u a t i o n s as the b a s i c e m i t t e r of sound. The v e l o c i t y was measured with a hot f i l m anemometer whose frequency dependent phase response was suspected to be q u i t e d i f f e r e n t from t h a t of the microphone measuring the sound i n the f a r f i e l d . I t was t h e r e f o r e necessary to perform the c r o s s c o r r e l a t i o n s i n narrow frequency bands i n s t e a d of i n a broad band c o v e r i n g a l l f r e q u e n c i e s of i n t e r e s t , thus i n c r e a s i n g s u b s t a n t i a l l y the amount of experimental work. C i t i n g from the a b s t r a c t of Lee S Ribner (1972): (Their work) "... y i e l d e d the r e l a t i v e i n t e n s i t y and spectrum of the n o i s e o r i g i n a t i n g from u n i t volume of a j e t (35 l o c a t i o n s ) and r e c e i v e d 13 at a f a r f i e l d p o i n t (r=96D, Q =40°); t h i s i n t u r n l e d to the r e l a t i v e e m ission of s u c c e s s i v e " s l i c e s " of a j e t versus a x i a l d i s t a n c e x over the measurement range (1D<x<7D). Q u a l i t a t i v e agreement was found with Ribner's x°-Law, and the s p e c t r a l peaks f o r each s l i c e were l o c a t e d i n frequency e s s e n t i a l l y as p r e d i c t e d by Powell (1959)." I t was a l s o r e p o r t e d that the t o t a l e f f e c t i v e number of u n c o r r e l a t e d n o i s e - p r o d u c i n g eddies i s on the order of 2500, i n sharp c o n t r a s t to the number 3 r e p o r t e d by Scharton 5 White (1972). In s e c t i o n IV of the present paper the number of i n c o h e r e n t s o u r c e s i s estimated to be on the o r d e r of 100 to 200. There, i t i s a l s o d i s c u s s e d how extraneous a c o u s t i c or e l e c t r o n i c n o i s e and bandwidth can s u b s t a n t i a l l y i n c r e a s e or decrease the apparent v a l u e of t h i s number, i n t r o d u c i n g a source o f ambiguity. L i t t l e r e c e n t p r o g r e s s has been made i n the area of n o i s e s u p p r e s s i o n d e v i c e s . Arndt (Barefoot & Arndt 1972) i s experimenting with s c r e e n s p e r t u r b i n g the j e t and Scharton 6 White (1972) have t e s t e d a suppressor c o n s i s t i n g of f i v e s taggered c o n c e n t r i c tubes f i l l i n g the p o t e n t i a l core of the j e t . In both cases some n o i s e r e d u c t i o n i s r e p o r t e d , however, the a s s o c i a t e d performance penalty (e.g. t h r u s t r e d u c t i o n ) i s not w e l l documented, and indeed i s expected to be r a t h e r h i g h . The t e c h n i q u e s presented here do not have as t h e i r o b j e c t i v e n o i s e r e d u c t i o n as such. Rather, the aim i s on the one hand to i n c r e a s e the knowledge of n o i s e g e n e r a t i n g mechanisms i n t u r b u l e n t flows and maybe provide some s t i m u l a t i o n f o r f u t u r e t h e o r e t i c a l work; on the other hand the techniques developed can be used as d i a g n o s t i c t o o l s i n s u r v e y i n g more complicated flow n o i s e phenomena and i n d e s i g n i n g and o p t i m i z i n g noise 14 s u p p r e s s o r s . The image technique d e s c r i b e d i n s e c t i o n I I I c r o s s c o r r e l a t e s the pressure on a s u r f a c e c l o s e t o the j e t with the f a r f i e l d n oise a l l o w i n g the deduction of the s u r f a c e source s t r e n g t h d i s t r i b u t i o n . T h i s i n t u r n g i v e s an i n d i c a t i o n of the approximate l o c a t i o n of the a c o u s t i c a l sources i n the j e t r a t l e a s t i n an a x i a l sense. The advantage of having the flow u n d i s t u r b e d by a probe i s traded o f f a g a i n s t a l o s s of s p a t i a l r e s o l u t i o n . The technique i s shown to be s e l f c o n s i s t e n t and may have a p p l i c a t i o n s as a quick and easy to use t o o l i n the design of n o i s e s u p p r e s s o r s . A l s o , i t i s not l i m i t e d to c o l d , subsonic f l o w s and can be used i n the i n v e s t i g a t i o n of hot and/or s u p e r s o n i c flows where i t i s q u i t e d i f f i c u l t i f not i m p o s s i b l e to i n s e r t a probe. The c a u s a l i t y c o r r e l a t i o n technique d e s c r i b e d i n s e c t i o n IV forms the main body of t h i s t h e s i s . I t complements and extends Lee's (1971) work. Instead of the t u r b u l e n t v e l o c i t y f l u c t u a t i o n s the f l u c t u a t i n g t u r b u l e n t p r e s s u r e s are taken as the b a s i c noise e m i t t e r s . P r e s s u r e , being a n o n d i r e c t i o n a l s c a l a r q u a n t i t y , i s i n p r i n c i p l e more a t t r a c t i v e to the experimenter than v e l o c i t y . D i f f i c u l t i e s a r i s e when one t r i e s to measure t u r b u l e n t pressure f l u c t u a t i o n s as the l o c a l v e l o c i t y f i e l d produces a d d i t i o n a l p r e s s u r e s at the probe, thus c r e a t i n q a p r e s s u r e e r r o r . The development of a unique a i r f o i l type pressure sensor as a means of a v o i d i n g those e r r o r s i s r e p o r t e d . The sensor i s then used to perform the pressure c a u s a l i t y c o r r e l a t i o n s by scanning the j e t r a d i a l l y at d i f f e r e n t downstream l o c a t i o n s . Thus the l o c a l source s t r e n g t h per u n i t volume and the a s s o c i a t e d power s p e c t r a l 15 d e n s i t y a r e d e d u c e d . T h e c o r r e l a t i o n s a r e d o n e i n b r o a d b a n d a s b o t h t r a n s d u c e r s a r e c o n d e n s e r m i c r o p h o n e s w i t h m a t c h e d p h a s e c h a r a c t e r i s t i c s . D i f f i c u l t i e s a r e e n c o u n t e r e d i n m a k i n g t h e t e c h n i q u e s e l f c o n s i s t e n t , a l t h o u g h r e s u l t s a g r e e q u a l i t a t i v e l y w i t h t h e r e s u l t s o f L e e ( 1 9 7 1 ) a n d o t h e r w o r k e r s . T h e t h e s i s h o p e s t o e s t a b l i s h t h a t t h e t w o m e t h o d s p r e s e n t e d h a v e e x p e r i m e n t a l a d v a n t a g e s a l l u d e d t o a b o v e a n d a r e w o r t h w h i l e t o b e f u r t h e r d e v e l o p e d . 16 I I I . IMAGE TECHNIQUE T h i s technique was born from a d e s i r e to a v o i d i n s e r t i n g probes of any kind i n t o the flow. I n s t e a d , a r i g i d plane s u r f a c e i s p l a c e d i n the near f i e l d of the flow. I t w i l l be shown that the sound r e f l e c t e d from the s u r f a c e ' i s e x a c t l y e q u i v a l e n t to the n o i s e r a d i a t i o n from an i d e n t i c a l "image" flow behind the s u r f a c e i f t h i s s u r f a c e were removed. The d e r i v a t i o n f o l l o w s the i d e a s of A. Powell (1960) who developed the imaging p r i n c i p l e with boundary l a y e r noise i n mind. Here, i t i s a p p l i e d to j e t n o i s e . In a d d i t i o n , the present work employs the c a u s a l i t y p r i n c i p l e by c r o s s c o r r e l a t i n g the r a d i a t e d sound i n the f a r f i e l d with the p r e s s u r e on the imaging s u r f a c e . T h i s pressure i s thought of as a "cause" or source of the sound i n the sense t h a t i t i s r e p r e s e n t a t i v e of a c e r t a i n l i m i t e d r e g i o n i n the j e t flow, the e x t e n t of which w i l l be d i s c u s s e d . The a c o u s t i c source s t r e n g t h per u n i t s u r f a c e area w i l l be deduced. Looking at the " a c o u s t i c f o o t p r i n t " the flow l e a v e s on the s u r f a c e a l l o w s some c o n c l u s i o n on the d i s t r i b u t i o n of n o i s e sources i n the image flow behind the s u r f a c e . 3.1 Imaging. System^ He s t a r t with L i g h t h i l l ' s (1952) equation f o r aerodynamic noise [Eq. (1.1) r e w r i t t e n ] : 17 V p - — o — = p" 3 H— I 3 * c c 2 a t2 c 0 2 ay; aVj where -T,j = />Uj U j - T ;J . • ( p - c ^ ) 8 ;J Using the i s e n t r o p i c assumption P(x) = lpi*)-p0)cZ <3'2> (which i s good i n the a c o u s t i c f a r f i e l d and i n the source r e g i o n as w e l l , i f e n t r o p i c f l u c t u a t i o n s are not present) and n e g l e c t i n g at the s u r f a c e a l l but the pre s s u r e s t r e s s e s , the s o l u t i o n f o r the r a d i a t e d sound becomes: i J v s o I where the b r a c k e t s i n d i c a t e e v a l u a t i o n at the e a r l i e r time t - |x- H/c e The f i r s t i n t e g r a l i s the d i r e c t c o n t r i b u t i o n from the j e t ; the second i n t e g r a l r e p r e s e n t s the sound r e f l e c t e d from the s u r f a c e . P s i s the s u r f a c e p r e s s u r e . i s the outward normal to the s u r f a c e S( ( F i g . 3 - 1 ) . I f S( were v i b r a t i n g or t r a n s l a t i n g a t h i r d i n t e g r a l would appear [ a s i n d i c a t e d i n the more g e n e r a l Eg. ( 1 . 2 ) ], but t h i s i s not the case f o r the r i g i d w a l l assumed here. In order to e s t a b l i s h the image system we add a j e t which i s the exact i n s t a n t a n e o u s m i r r o r image of the f i r s t j e t ( F i g . 3 - 1 ) . The s u r f a c e S( i s the plane of symmetry. Whether S( i s a r e a l r i g i d s u r f a c e or a h y p o t h e t i c a l plane does not matter because the adding of the in s t a n t a n e o u s m i r r o r image with S, removed w i l l 18 produce e x a c t l y the same boundary c o n d i t i o n s by symmetry, as with S, i n plac e as a r e a l r i g i d boundary. T h e r e f o r e the sound f i e l d r a d i a t e d from the image j e t i s e x a c t l y equal to the sound f i e l d r e f l e c t e d from S, . With the q u a n t i t i e s a s s o c i a t e d with the image system denoted by primes we can w r i t e : I f x i s l a r g e ( f a r f i e l d approximation) terms l i k e aire r e p l a c e d by - - ^ 2 f [•"] (3*5) (see L i g h t h i l l 1952 or Ribner 1964) Eq. (3.3) becomes: vo ^1 where we can say that the second i n t e g r a l r e p r e s e n t s e i t h e r the sound r e f l e c t e d from the s u r f a c e S, , or [by Eq. (3.4) ] the sound r a d i a t e d from the image j e t . In what f o l l o w s i t i s shown how i n f o r m a t i o n on the noi s e sources i n the image j e t (approximate s p a t i a l d i s t r i b u t i o n and s p e c t r a l c h a r a c t e r ) can be o b t a i n e d . S i n c e the image j e t i s an i n s t a n t a n e o u s m i r r o r image the same i n f o r m a t i o n i s good f o r the r e a l j e t . However the r e s u l t s may not be i d e n t i c a l l y v a l i d f o r a s i n g l e i s o l a t e d j e t r a d i a t i n g i n t o unbounded space as the i n s e r t i o n of the s u r f a c e S, may s l i g h t l y change the s t r u c t u r e of the j e t (see second paragraph of s e c t i o n - ^ • / " M - l 19 3.2 Cross C o r r e l a t i o n Formalism. Equation (3.6) i s m u l t i p l i e d by the f a r f i e l d sound pressure p at a d i f f e r e n t time t 1 , and then a time average i s performed. Under the assumption of s t a t i s t i c a l s t a t i o n a r i t y the r e s u l t w i l l be independent of the a b s o l u t e time and depend only on the time delay T = t - t * ( C r a n d a l l & Hark 1963, s e c t i o n 1.5). The overbar denotes a time average: 4 . p p ( T ) = ^ / [ f ^ ] p d V ^ M ^ ] p d S ( 3 - 7 ) where, agai n , the second i n t e g r a l c o u l d be r e p l a c e d by the c o n t r i b u t i o n from the image. T h i s i s the part we are i n t e r e s t e d i n : pp <*>,„«,. ,„„ = -i—ril-A^}^ (3-9) where Xj / j = - x c o s 0 ( F i g . 3 - 1 ) . Use i s made of the f a c t that d i f f e r e n t i a t i o n with r e s p e c t to time under the time average can be converted i n t o d i f f e r e n t i a t i o n with r e s p e c t to time delay i f the p r ocesses are s t a t i s t i c a l l y s t a t i o n a r y ( C r a n d a l l & Mark, 1963, s e c t i o n 1.6)» The b r a c k e t s now mean e v a l u a t i o n a t the r e t a r d e d time delay (Siddon 1973a) X = T = l x - j j | / c 0 (3. 10) where y i n d i c a t e s the p o i n t where p g i s d e t e c t e d . Notice t h a t the L i g h t h i l l s t r e s s t e n s o r Tjj has disappeared. I t s c omplicated s t r u c t u r e does not concern us here. 20 W r i t i n g Eg. (3.9) i n d i f f e r e n t i a l form and s e t t i n g x=0: d p 2 -COSfl f d ——] #_ _ • Eq. (3.11) g i v e s the p o r t i o n of the r a d i a t i o n from the image j e t a s s o c i a t e d with a u n i t area of the s u r f a c e S, a t the p o i n t where p s i s d e t e c t e d . We can regard t h i s as a measure of the " l o c a l " a c o u s t i c source s t r e n g t h i n the image j e t . S e c t i o n IV w i l l d e s c r i b e a method of determining the a c t u a l d i s t r i b u t i o n of source s t r e n g t h i n the flow by measuring an approximation of T|j . Here we are content with a l o s s of s p a t i a l r e s o l u t i o n but g a i n t h r e e advantages: 1) I t i s not necessary to i n s e r t a probing d e v i c e i n t o the flow, a simple measurement of s u r f a c e pressure p s i s s u f f i c i e n t . 2) Only one time d e r i v a t i v e i s i n v o l v e d i n Eq.(3.11); Eg. (3.7) i n d i c a t e s two time d e r i v a t i v e s and the work of Chu (1966) r e q u i r e d t a k i n g a f o u r t h d e r i v a t i v e of a c o r r e l a t i o n . Numerical e r r o r s i n c r e a s e with the order of the d e r i v a t i v e whether the d i f f e r e n t i a t i o n i s done by analog c i r c u i t r y before the c o r r e l a t i n g process or a f t e r w a r d by d i g i t a l methods. 3) Ho f u r t h e r assumptions have been made about the nature of the flow, the i s e n t r o p i c assumption (3.2) p e r t a i n s only to the r a d i a t e d sound i n the f a r f i e l d . The method could t h e r e f o r e be used f o r i n s t a n c e i n the i n v e s t i g a t i o n o f n o i s e from hot and/or s u p e r s o n i c j e t s . At higher than low subsonic Hach numbers or with l a r g e d i f f e r e n c e s between the d e n s i t i e s of the source f l u i d and the ambient a i r , the f a r f i e l d microphone would have to be placed 21 such t h a t the sound r e f l e c t e d from the s u r f a c e towards the microphone would not pass aga i n through the flow f i e l d where i t would be r e f r a c t e d or even s h i e l d e d , as c o u l d be the case with s u p e r s o n i c j e t s . 3.3 Experiments. The experiments were conducted i n an anechoic room. The setup was a s t shown i n Fig . 3 - 1 . The j e t diameter was 38.1 mm (1.5 i n c h e s ) , the Mach number at the nozzle e x i t about 0.32. F i g . 3-2 shows s p e c t r a of the a c o u s t i c pressure i n the f a r f i e l d at #=45° to the j e t a x i s ; one f o r the j e t alone (a) , the other with the w a l l i n s t a l l e d behind the j e t (b). They peak at a S t r o u h a l number of about 0.2 (corresponding to about 700 Hertz) . The "bumpy" nature of (a) i s thought to be due to the room not being completely anechoic a t a l l f r e q u e n c i e s , (b) i s of course l a r g e r as i t r e p r e s e n t s both the d i r e c t and the r e f l e c t e d j e t n o i s e . That (b) c o n t a i n s more than one pronounced peak i s expected as the d i r e c t and the r e f l e c t e d sound waves w i l l depending on t h e i r frequency - tend t o enhance or c a n c e l each o t h e r . T h i s i s a w e l l known d i f f i c u l t y a r i s i n g when j e t engines are t e s t e d near a hard ground plane. The d i s t a n c e from the j e t c e n t e r l i n e to the s u r f a c e was a compromise between c o n f l i c t i n g requirements: On the one hand the s u r f a c e should be c l o s e to the flow i n order to i n c r e a s e the s p a t i a l r e s o l u t i o n . On the other hand the s u r f a c e has to be f a r enough away from the flow i n order to a v o i d the Coanda e f f e c t ; i . e . , the tendency of the j e t to a t t a c h to nearby s u r f a c e s because the f l u i d between the j e t and the s u r f a c e i s being dragged along by entrainment, d e c r e a s i n g the pressure and thereby 22 drawing the j e t towards the s u r f a c e . & very c l o s e s u r f a c e may a l s o i n t e r f e r e with the entrainment of a i r i n t o the j e t so that i t s developed s t r u c t u r e may d i f f e r s u b s t a n t i a l l y from t h a t of a s i n g l e i s o l a t e d j e t . Furthermore, d i r e c t i n t e r a c t i o n of the flow with the s u r f a c e must be avoided; t h i s would i n t r o d u c e unwanted n o i s e by v i o l a t i n g the assumption t h a t s u r f a c e shear s t r e s s e s are unimportant [ t h i s assumption i s i m p l i c i t i n Eg. (3.3) ]. In r e c o g n i t i o n of these l i m i t i n g f a c t o r s the d i s t a n c e between the s u r f a c e and the j e t c e n t e r l i n e was chosen to be 197 mm ( l i t t l e more than 5 n o z z l e d i a m e t e r s ) . Host of the s u r f a c e S( c o n s i s t e d o f a 0.5 i n c h t h i c k sheet of p l e x i g l a s s . The s u r f a c e pressure p s was measured u s i n g a B r u e l & Kjaer 1/4-inch microphone (type 4 136) connected t o the s u r f a c e by a s h o r t 0.020 i n c h diameter hole ( F i g . 3 - 3 ) . The resonant frequency of the Helmholtz reso n a t o r formed by the h o l e and the s m a l l c a v i t y i n f r o n t of the microphone diaphragm was s u f f i c i e n t l y high above the f r e q u e n c i e s of i n t e r e s t i n the j e t n o i s e . The f a r f i e l d microphone was placed f o r most of the experiments at 45° to the j e t a x i s at a d i s t a n c e of about 1.9 meters from the j e t o r i f i c e . Some data was a l s o obtained at 60° and 90°. The s u r f a c e pressure p s was measured a t many p o i n t s on the s u r f a c e and c r o s s c o r r e l a t e d with the s i g n a l from the f a r f i e l d . 23 The s i g n a l p r o c e s s i n g equipment and the anechoic room were the same as w i l l be d e s c r i b e d i n s e c t i o n 4.4. Fig.3-4 shows an example of a c o r r e l a t i o n f u n c t i o n p s p (T). Eq. (3.11) r e q u i r e s t a k i n g the d e r i v a t i v e of t h i s f u n c t i o n a t the delay time T = |x-yJ/c 0 which was done on a d i g i t a l computer. The curve a l s o e x h i b i t s a " p r e c u r s o r " the e x i s t e n c e of which i s e x p l a i n e d as f o l l o w s : , . The flow n o i s e r a d i a t e s i n a l l d i r e c t i o n s , i n c l u d i n g to the p o i n t where the f i e l d microphone i s l o c a t e d and to the p o i n t on the w a l l where p s i s measured; i n the v i c i n i t y of the time delay c o r r e s p o n d i n g to the d i f f e r e n c e i n sound t r a v e l times from the source flow to these two p o i n t s the s i g n a l s are somewhat coherent and a secondary c o r r e l a t i o n " b l i p " r e s u l t s , i n order to separate t h i s p r e c u r s o r from the pa r t of the c o r r e l a t i o n f u n c t i o n we are i n t e r e s t e d i n (the v i c i n i t y around T=|x-y|/c o) one must not put the s u r f a c e S, too c l o s e to the flow; the f u r t h e r the s u r f a c e i s from the flow the s m a l l e r the d i f f e r e n c e i n sound t r a v e l times, and the more the p r e c u r s o r i s moved to the l e f t on the time delay a x i s . However, the presence of the p r e c u r s o r does not v i o l a t e the theory and, i n p r i n c i p l e , can be ig n o r e d . 3.4 N o n d i m e n s i o n a l i z a t i o n . (See a l s o Appendix B) In order to make the r e s u l t s independent of the p a r t i c u l a r experimental setup chosen and to make them comparable with the r e s u l t s of other workers the measured q u a n t i t i e s are nondimensionalized using c h a r a c t e r i s t i c l e n g t h s , d e n s i t y , and v e l o c i t i e s . I t i s w e l l e s t a b l i s h e d ( L i g h t h i l l 1952) t h a t the f a r f i e l d a c o u s t i c pressure of a subsonic a i r • j e t i s s u i n g i n t o a medium of i d e n t i c a l gas 24 p r o p e r t i e s obeys the f o l l o w i n g dimensional s i m i l a r i t y : p CC yoU^Dc-Zx-1 (3. 12) T h e r e f o r e the a c o u s t i c pressure p was nondimensionalized by 2 (0. 5pU£) M2D/x (3.13) M i s the j e t exit Mach number, x i s the distance to the f i e l d point. Ps i s a near f i e l d pressure at the plane s u r f a c e ; i t s v a r i a t i o n with U 0 was determined by experiment. A p o i n t s i x j e t diameters downstream from the i n t e r s e c t i o n of the e x i t plane with the r i g i d s u r f a c e was chosen as r e p r e s e n t a t i v e . In Fig.3-5 H i s the dynamic head at the n o z z l e e x i t i n meters of water. H i s p r o p o r t i o n a l to U 2. At the chosen l o c a t i o n i t turned out t h a t p s v a r i e d as H 2 or U*, i . e . i t obeyed the same s i m i l a r i t y r e l a t i o n g i v e n i n Eq. (3. 12) as does p. T h e r e f o r e p s was nondimensionalized by the same q u a n t i t y (3.13) except that x was r e p l a c e d by h, the d i s t a n c e of the s u r f a c e from the j e t c e n t e r l i n e . Time was n o n d i m e n s i o n a l i z e d by D/0 o. More d e t a i l s are to be found i n Appendix B. 3.5 R e s u l t s ^ F i g . 3-6 shows the nondimensional source s t r e n g t h d i s t r i b u t i o n on the r i g i d s u r f a c e . The a c t u a l q u a n t i t y p l o t t e d i s : (for rationale see Appendix B) - c o s 4 5 ° [ a 7 P S P ] T - T S U A = _ S T T (3.14) { 2 ( 0 . 5 , u W } ^ f f The s t r o n g e s t sources appear to be c o n c e n t r a t e d i n an area 4 25 to 14 diameters downstream of the nozzle e x i t . I t would be convenient to be ab l e t o use the r e f l e c t i o n p r i n c i p l e from a c o u s t i c ray theory, as i n d i c a t e d i n Fig.3-9, i n order to t r a c e back where the sound has emanated from the j e t . S t r i c t l y speaking, the r e f l e c t i o n p r i n c i p l e can only be a p p l i e d i f the r e f l e c t i n g , s u r f a c e i s much l a r g e r than a t y p i c a l wavelength, i f i t i s t r u l y r i g i d , and i f the i n c i d e n t a c o u s t i c d i s t u r b a n c e has f a r f i e l d c h a r a c t e r ( i . e . , i f the wave f r o n t s are approximately p l a n e ) . The f i r s t two requirements are w e l l f u l f i l l e d by the present e x p e r i m e n t a l setup. The t h i r d i s not so w e l l f u l f i l l e d as the s u r f a c e p r e s s u r e s p s are measured i n the t r a n s i t i o n r e g i o n between near f i e l d and f a r f i e l d . (Fig.4-18 shows how the pr e s s u r e f l u c t u a t i o n s vary with d i s t a n c e from the j e t ce n t e r l i n e ; h~5D i s i n an area where the near f i e l d changes i n t o the f a r f i e l d . ) At 6 diameters, p s behaves q u i t e l i k e a f a r f i e l d p r e s s u r e , as mentioned before (Fig.3-5) , but there i s some s c a t t e r i n the data. Fig.3-7 and Fig.3-8 show equal source s t r e n g t h contours f o r angles 6 ( f a r f i e l d d i r e c t i o n to j e t axis) of 90° and 60°. T h i s data was taken e a r l y i n the r e s e a r c h before c e r t a i n refinements i n the e l e c t r o n i c s were i n t r o d u c e d . I t i s however u s e f u l i n a q u a l i t a t i v e way s u p p o r t i n g the ray a c o u s t i c argument: The l o c a t i o n of the s t r o n g e s t sources a p p a r e n t l y moves from 2 diameters to 12 diameters as 8 goes from 90° to 60°. Such a s h i f t i s what would be expected i f ray theory were a p p l i c a b l e . I t appears t h e r e f o r e t h a t one can "borrow" the ray a c o u s t i c s argument when d i s c u s s i n g the problems of s p a t i a l r e s o l u t i o n a f f o r d e d by the image method. Fig.3-9 i l l u s t r a t e s the 26 h y p o t h e t i c a l "zone of i n f l u e n c e " : the r e g i o n of the j e t that w i l l predominantly c o n t r i b u t e to the source s t r e n g t h on the s u r f a c e whare p s i s measured. The zone of i n f l u e n c e i s cone shaped; the angle of a p e r t u r e of the cone i s however l e f t to s p e c u l a t i o n . The angle w i l l be s m a l l e r f o r higher f r e q u e n c i e s s i n c e the r e f l e c t i o n argument becomes more and more v a l i d as the wavelength decreases. T h i s p e r t a i n s both to the r e f l e c t i o n mechanism i t s e l f and to the f a r f i e l d c o n d i t i o n . . The j e t n o i s e spectrum peaks a t about 700 H e r t z . T h i s corresponds to a wavelength of about 0.5m which i s l a r g e r than h=0.197m. For t h i s frequency one c o u l d t h e r e f o r e not expect a very narrow angle of a p e r t u r e . S u p e r s o n i c flows g e n e r a l l y produce much hi g h e r f r e q u e n c i e s . There, the r e f l e c t i o n argument should be much more a p p l i c a b l e . Using the r e f l e c t i o n argument r e g a r d l e s s of whether i t i s s t r i c t l y a p p l i c a b l e or not, the a c t u a l sound producing r e g i o n w i l l be found to be 2 t o 8 diameters downstream of the e x i t plane. T h i s i s c o n s i s t e n t with the f i n d i n g s of other r e s e a r c h e r s {Ribner 1958, Lee 1971, Dyer 1959, Powell 1959, L i l l e y 1958, et a l . ) . T h i s statement does not, however, add much new i n s i g h t i n t o the c h a r a c t e r of noise p r o d u c t i o n from t u r b u l e n t j e t s . The method d e s c r i b e d here i s presented i n order to show i t s f e a s i b i l i t y i n the study of flow n o i s e . In p a r t i c u l a r , as a l r e a d y mentioned above, the d e r i v a t i o n of the image system does not c o n t a i n any assumption about Mach number except an i m p l i c i t d i s r e g a r d of r e f r a c t i v e e f f e c t s due to temperature or v e l o c i t y g r a d i e n t s along the path of the sound. The method could t h e r e f o r e be used to study the noise g e n e r a t i o n from s u p e r s o n i c flows where i t i s s t i l l more d i f f i c u l t i f not i m p o s s i b l e to l o c a t e the n o i s e 27 sources by probing the j e t i t s e l f . M a e s t r e l l o & HcDaid (1970) have advanced a d i f f e r e n t image technigue f o r determining j e t n o i s e c h a r a c t e r i s t i c s . There, a means i s shown of l o c a t i n g sound sources by determining the d i r e c t i o n s where sound waves are coming from by a complex F o u r i e r a n a l y s i s of two p o i n t s p a c e - t i m e - c o r r e l a t i o n s of the s u r f a c e p r e s s u r e on a r i g i d s u r f a c e i n the v i c i n i t y of the j e t . The theory i s q u i t e c o m p l i c a t e d , and so are the computations to process the data. Some r e s u l t s are presented but not enough to enable a c o n c l u s i o n on the d i s t r i b u t i o n of sound sources. The paper reads as i f the authors f i r s t expected to o b t a i n d e t a i l e d i n f o r m a t i o n on sound source d i s t r i b u t i o n i n the j e t but l a t e r encountered problems of s p a t i a l r e s o l u t i o n s i m i l a r to the ones i n the present work. The f o l l o w i n g guote from the M a e s t r e l l o & McDaid paper (1970) e q u a l l y a p p l i e s to the technigue developed here: "... sound undergoes c o n v e c t i o n and s c a t t e r i n g between the time i t i s generated and the time i t l e a v e s the t u r b u l e n c e of the j e t . Thus the p o i n t of g e n e r a t i o n c o u l d be f a r t h e r upstream ..." Only a technigue probing the flow [such as Chu (1966), Lee (1971), s e c t i o n IV of the present paper] can shed more l i g h t on t h i s problem. A l l an image technique can determine at the most i s the d i r e c t i o n from where the sound waves are coming when they impinge on the r e f l e c t i n g s u r f a c e . An advantage of M a e s t r e l l o & McDaid's (1970) approach i s t h a t s p e c t r a l i n f o r m a t i o n i s not only obtained very e a s i l y , but i s i m p l i c i t i n the method. However, i n order t o gain some s i g n i f i c a n t i n s i g h t , the amount of experimental work r e q u i r e d i s q u i t e l a r g e . In the present approach one would t h i n k i t i s easy 28 to o b t a i n s p e c t r a l i n f o r m a t i o n by computing the F o u r i e r transform of E g . ( 3 . 9 ) r e s u l t i n g i n the c o n t r i b u t i o n to the o v e r a l l power s p e c t r a l d e n s i t y from a u n i t s u r f a c e area a t the p o i n t where the s u r f a c e pressure i s measured. U n f o r t u n a t e l y , the p r e c u r s o r ( d i s c u s s e d i n s e c t i o n 3.3 and shown i n F i g . 3 - 4 ) superposes i t s e l f on the c r o s s c o r r e l a t i o n f u n c t i o n and makes the F o u r i e r transform d i f f i c u l t to i n t e r p r e t : Eg. ( 3 . 1 1 ) g i v e s the c o n t r i b u t i o n from the image only. The p r e c u r s o r does not a f f e c t s i g n i f i c a n t l y the d e r i v a t i v e of p s p(T) at r = r / c 0 because i t l i e s f a r enough to the l e f t on the time delay a x i s . The o p e r a t i o n of F o u r i e r t r a n s f o r m i n g i n v o l v e s the complete r e c o r d of p g p (T ) , i n c l u d i n g the p r e c u r s o r . Attempts to perform t h i s F o u r i e r o p e r a t i o n showed t h a t the t r a n s f o r m became negative f o r some freguency ranges. T h i s occurs because of the c o n s t r u c t i v e and d e s t r u c t i v e i n t e r f e r e n c e e f f e c t s at c e r t a i n f r e q u e n c i e s as a l r e a d y d i s c u s s e d i n s e c t i o n 3 . 3 . In view of t h i s ambiguity ways of o b t a i n i n g q u a n t i t a t i v e s p e c t r a l i n f o r m a t i o n were not f u r t h e r e x p l o r e d i n the image technique. N e v e r t h e l e s s , some q u a l i t a t i v e s p e c t r a l i n f o r m a t i o n can be o b t a i n e d by i n s p e c t i n g the c r o s s c o r r e l a t i o n s : F i g . 3 - 1 0 shows how they change i n shape as one moves downstream i n the j e t . I t can be seen t h a t the c h a r a c t e r i s t i c f r e q u e n c i e s are higher c l o s e r to the n o z z l e e x i t (the c o r r e l a t i o n has high c u r v a t u r e s a t the peaks and the d i s t a n c e between zero c r o s s i n g s i s small) and becomes lower f u r t h e r downstream (lower c u r v a t u r e and l a r g e r d i s t a n c e between zero c r o s s i n g s ) . T h i s behaviour i s of course expected as the t u r b u l e n c e s c a l e s grow i n the downstream d i r e c t i o n . Fig.3-11 shows the d i s t r i b u t i o n of the r o o t mean squared 29 near f i e l d p ressure over the s u r f a c e . I t shows a peak i n the same area as Fig.3-6, but the t a i l s decrease much l e s s r a p i d l y , both i n the d i r e c t i o n of the j e t flow and p e r p e n d i c u l a r to i t . An attempt to i d e n t i f y sound sources t h a t way would be much l e s s s u c c e s s f u l . The zone of i n f l u e n c e on p s alone i s the whole hemisphere and not the narrow cone shaped r e g i o n of the image c r o s s c o r r e l a t i o n technique (which s o r t of " p o i n t s " to the source of sound). When the source s t r e n g t h per u n i t area i s i n t e g r a t e d over t r a n s v e r s e " s t r i p s " of the s u r f a c e , i . e . , i n the d i r e c t i o n p e r p e n d i c u l a r to the j e t a x i s at each a x i a l l o c a t i o n . Fig.3-12 r e s u l t s . I t has a pronounced peak. Using the r e f l e c t i o n argument, by t r a c i n g back from the peak l o c a t i o n of 10 diameters, the s t r o n g e s t a c o u s t i c s o u r c e s would be l o c a t e d between 4 and 5 diameters from the no z z l e e x i t . T h i s i s i n c l o s e agreement with other r e s u l t s (e.g. Ribner 1958, 1962; Dyer 1959). 3.6 Check. Eq.{3.9) can be used as a check of the method: When the " s t r i p w i s e " source s t r e n g t h of Fig.3-12 i s i n t e g r a t e d along the a x i s , the s u r f a c e i n t e g r a l of Eg. (3.9) can be e v a l u a t e d , f o r r=0. The l e f t hand s i d e i s c a l c u l a t e d from a d i r e c t measurement of the j e t noise without the r e f l e c t i n g s u r f a c e . In order Jco r e p r e s e n t the r a d i a t i o n from the image j e t a c o r r e c t i o n was made f o r the d i f f e r e n c e i n d i s t a n c e to the f a r f i e l d microphone between the r e a l and the image j e t . The i n t e g r a t i o n on the r i g h t hand s i d e was c a r r i e d out g r a p h i c a l l y . With the n o n d i m e n s i o n a l i z a t i o n done as i n d i c a t e d i n s e c t i o n 3.4 and i n Appendix B, the d i r e c t l y measured value ( l e f t hand side) becomes 30 - - y = 0.177 x I 0 ~ 5 (3.15) {2 (0.5^U o 2)M 2D/x} and the i n t e g r a t e d v a l u e ( r i g h t hand side) i s 0.147*10 _ s. T h i s corresponds to a d i f f e r e n c e of o n l y 0.8 db. Apart from numerical e r r o r s i n the computation of the c o r r e l a t i o n f u n c t i o n s and t h e i r d e r i v a t i v e s , s o u r ces of u n c e r t a i n t y are the f i n i t e e x t e n t of the s u r f a c e and i t s i m p e r f e c t r i g i d i t y ( r e c a l l i n g t h a t most of the s u r f a c e c o n s i s t s of a 0.5 i n c h t h i c k sheet of p l e x i g l a s s ) . T h i s check on i n t e r n a l c o n s i s t e n c y lends f u r t h e r c o n f i d e n c e i n the use of c a u s a l i t y c o r r e l a t i o n techniques to s p a t i a l l y r e s o l v e complex source d i s t r i b u t i o n s (see a l s o Siddon 1973a). 31 IV. CAUSALITY CORRELATION TECHNIQUE 4.1 I n t r o d u c t i o n - T h i s technique was developed i n order to probe_ the j e t flow i n such a way as to o b t a i n the s p a t i a l d i s t r i b u t i o n and c h a r a c t e r i s t i c s of the elementary a c o u s t i c sources. Use i s made of a r e c e n t l y developed c a u s a l i t y c o r r e l a t i o n technique (Siddon 1966, 1970, 1971b, 1973a; C l a r k e & Ribner 1969; Lee 1971; Lee S Ribner 1972; Siddon & Rackl 1971; Rackl 1972b). As mentioned i n s e c t i o n I the f l u c t u a t i n g p r e s s u r e s are taken as the b a s i c n o i s e e m i t t e r s . By computing the c r o s s c o r r e l a t i o n between these p r e s s u r e s and the r a d i a t e d a c o u s t i c pressure i n the f a r f i e l d q u a n t i t a t i v e i n f o r m a t i o n on the a c o u s t i c source d i s t r i b u t i o n can be o b t a i n e d . S e c t i o n 4.2 summarizes the u n d e r l y i n g theory and t r e a t s the concept of c a u s a l i t y c o r r e l a t i o n s , spectrum from a u n i t volume, and c o r r e l a t i o n volume. The experimental setup was as shown i n F i g . 4 - 1 : As i n s e c t i o n I I I , a c o l d , c i r c u l a r (1.5 i n c h d i a m e t e r ) , subsonic j e t (Hachnumber about 0.32) d i s c h a r g e d i n t o an anechoic room. The t u r b u l e n t p r e s s u r e s were measured by a s p e c i a l l y developed a i r f o i l type p r e s s u r e sensor ( d e s c r i b e d i n d e t a i l i n s e c t i o n 4.3). The f a r f i e l d p ressure was monitored by a 1/2-inch microphone. The c r o s s c o r r e l a t i o n s of the two s i g n a l s were computed on a h y b r i d s i g n a l c o r r e l a t o r . T y p i c a l c o r r e l a t i o n 32 f u n c t i o n s are shown i n Fig,4-22a. More d e t a i l s w i l l be found i n s e c t i o n 4.4. The remaining s e c t i o n s a f t e r 4.4 show how the data was reduced and d i s c u s s the r e s u l t s . Comparisons are made with r e s u l t s of o t h e r r e s e a r c h e r s . At l e a s t q u a l i t a t i v e agreement i s achieved. D i f f i c u l t i e s are encountered i n t r y i n g to make the method q u a n t i t a t i v e l y s e l f c o n s i s t e n t . P o s s i b l e e x p l a n a t i o n s are g i v e n . once the v i a b i l i t y of the method has been demonstrated i t becomes a d i a g n o s t i c t o o l f o r the i n v e s t i g a t i o n of o t h e r flows. I t c o u l d , f o r i n s t a n c e , be used i n the study of the e f f e c t i v e n e s s of v a r i o u s j e t q u i e t e n i n g d e v i c e s . 4.2 Theory. 4.2.1 F l u i d D i l a t a t i o n s ^ The theory i s based on Ribner's (1962) i d e a s on viewing the t u r b u l e n t flow as an a r r a y of a c o u s t i c monopoles generated by a q u a s i - i n c o m p r e s s i b l e and i s e n t r o p i c flow. For subsonic f l o w s , L i g h t h i l l ' s wave equation (1.1) can be r e - w r i t t e n i n the approximate form: C i a2p o - V 2 p = d 2 pu^ dy-, dy. ( 4 - 1 ) The r i g h t hand s i d e of Eq. (4.1) may be regarded as a f o r c i n g term of the wave equation f o r p. A c o u s t i c a l l y , i t r e p r e s e n t s a d i s t r i b u t i o n of a c o u s t i c quadrupoles embedded in, a medium at 33 r e s t . I t can be converted i n t o a co r r e s p o n d i n g monopole f i e l d ( " d i l a t a t i o n s " ) by s p l i t t i n g the pressure d i s t u r b a n c e i n t o two p a r t s : p - P o = p C 0 3 + p C 1 D (4.2) where: p = in s t a n t a n e o u s s t a t i c pressure P o = l o c a l time average s t a t i c pressure pco) i s d e f i n e d by _ V 2 CCO = d f U i U i ( U . 3 ) It has been shown t h a t p<o) x s the a c t u a l p e r t u r b a t i o n p r e s s u r e w i t h i n an e x a c t l y i n c o m p r e s s i b l e flow and i s a good approximation f o r M 2<<1 (see Kraichnan 1956 and Ribner 1962) . p<i> i s the p a r t of the pre s s u r e a s s o c i a t e d with the prop a g a t i o n (not the generation) of a c o u s t i c waves. The r e l a t i o n s h i p of p<o> and p<i> i s expressed i n the " d i l a t a t i o n e q u a t i o n " which . r e s u l t s a f t e r combining Eg. ( 4 . 1 ) , Eq. (4.2), Eq. (4.3) : 1 6 9 — v 2 P C D = - -V ^-V <̂> c 2 a t 2 w c 2 at o o The r i g h t hand s i d e of Eq. (4.4) can be regarded as the f o r c i n g 34 term of the wave equation f o r the a c o u s t i c p r essure p<l>. The s o l u t i o n f o r the r a d i a t e d sound i n terms of K i r c h h o f f ' s r e t a r d e d p o t e n t i a l s becomes: p C O ( X i t ) = > d 3 y 47rc 0 2 Jy l x - y l L at2 Jt " where, again, the b r a c k e t s i n d i c a t e e v a l u a t i o n at the e a r l i e r time t = t - | x - y j / c 0 (See Fig.4-2 f o r geometry). In the geometric f a r f i e l d , and i f the o r i g i n i s chosen i n or c l o s e to the source r e g i o n . the term 1/|x-.Yj m a Y be approximated as 1/x = 1/1x|« Then: p C D(x,t) = - - — r J ——r—J. d y 4TTC 0 x v d\c Jt (4.6) In the a c o u s t i c f a r f i e l d (many t y p i c a l wavelengths away), p ( 1 > approximates the t o t a l p ressure f l u c t u a t i o n P ( x , t ) . Fig.4-18 shows the t y p i c a l v a r i a t i o n of the pressure f l u c t u a t i o n with d i s t a n c e from the j e t . S t a r t i n g at a p o s i t i o n on the c e n t e r l i n e 4 diameters downstream of the n o z z l e and moving a t r i g h t a n g l e s to the c e n t e r l i n e , the f a r f i e l d c o n d i t i o n of a 6 db drop i n the rms pressure per d o u b l i n g of d i s t a n c e i s reached at about 10 diameters (the c r o s s c o r r e l a t i o n measurements were made about 50 35 diameters away). The s t r a i g h t l i n e i n d i c a t i n g the 6 db drop per dou b l i n g of d i s t a n c e r e p r e s e n t s the v a r i a t i o n of p<*> alone. I f the l i n e i s extended backward i n t o the source r e g i o n one can estimate the magnitude of an e q u i v a l e n t p o i n t source r e p r e s e n t i n g the r a d i a t i o n of the whole j e t . As shown on Fig.4-18 i t s pressure f i e l d would be about 20 t o 40 db weaker than the a c t u a l measured j e t p r e s s u r e , i . e . , the approximation of p<°>. In f a c t , the j e t i s composed of many i n c o h e r e n t sources each of which w i l l of course be weaker than the s i n g l e e q u i v a l e n t p o i n t source. In the source r e g i o n t h e r e f o r e p<*> i s very much s m a l l e r than p<°>. T h i s means t h a t we do not expect the sound to have any s i g n i f i c a n t back r e a c t i o n on the flow or to d i s t u r b the measurement of p<o> which a r i s e s p u r e l y from i n c o m p r e s s i b l e t u r b u l e n t momentum exchanges. In order to o b t a i n the mean squared r a d i a t e d p r essure Ribner proceeds to square and time average Eg. (4.6), i n v o l v i n g a space- time c o r r e l a t i o n of p<o> under a double i n t e g r a l with f o u r time d e r i v a t i v e s . T h i s o p e r a t i o n i s s i m i l a r to the one employed by Chu (1966). 4.2.2 C a u s a l i t y C o r r e l a t i o n ^ In the present formalism, we o b t a i n the mean squared r a d i a t e d pressure by m u l t i p l y i n g both s i d e s of Eq. (4.6) by i t s l e f t hand s i d e , e v a l u a t e d a t a new time t+T. A f t e r time a v e r a g i n g : 36 p(x , t ) p(x,t + T) = - „ 1 2 / f p c w ( y , t ) l p(x,t+r) d 3 y ( 4 . 7 ) 4 T T C / X •(/ L - J ? p(x,t+r) can be taken under the i n t e g r a l s i g n because i t i s independent of j. A l s o , i n t h i s case time averaging and i n t e g r a t i o n over V are i n t e r c h a n g e a b l e o p e r a t i o n s . Under the assumption of s t a t i s t i c a l s t a t i o n a r i t y the l e f t hand s i d e i s a f u n c t i o n of r o n l y ; s i m i l a r l y on the r i g h t hand s i d e : P P ( * , T > ' = ~ A ' 2 / [ P C ° 3 ( y ) P ( x ) ] . d 3 y ( 4 . 8 ) ^ 7 1 C c X V T where f=t+r-t= t+r-t+ | x-y | /C D=T+ | x-y | / c G pz i s obtained by s e t t i n g T = O. The i n t e g r a n d i n Eg. ( 4 . 8 ) e s t a b l i s h e s the c a u s a t i v e r e l a t i o n s h i p between the source f l u c t u a t i o n and the r a d i a t e d sound. According to a p roperty of s t a t i o n a r y random v a r i a b l e s ( C r a n d a l l S Hark 1963) t h i s i n t e g r a n d can be w r i t t e n i n d i f f e r e n t ways: , 37 The second form turned out to be.the most convenient i n most of the experiments d e s c r i b e d h e r e i n . Fig.4-22a shows t y p i c a l c r o s s c o r r e l a t i o n s p<o)p. T h e i r shape i s very c l o s e to antisymmetric. Appendix C t r e a t s an e x t e n s i o n of the theory t h a t a l l o w s the p r e d i c t i o n of the shape of the c o r r e l a t i o n f u n c t i o n s ; good agreement i s achieved with a c t u a l measured f u n c t i o n s . Note t h a t o n l y one probe has to be i n s e r t e d i n t o the flow (Fig.4-1) i n s t e a d of two as have been normally employed i n the more c l a s s i c a l approach; only a s i n g l e i n t e g r a l i s to be e v a l u a t e d ; o n l y two time d e r i v a t i v e s are i n v o l v e d . 4.2.3 L o c a l Source Strength and Spectrum^ W r i t i n g Eg. (4.8) i n d i f f e r e n t i a l form g i v e s the r a d i a t i o n from u n i t volume: d V 4 T T C 0 2 X L ° T J T = T Again, the c o n t r i b u t i o n t o the t o t a l mean squared pressure from the p o i n t where p<o) i s d e t e c t e d i s obtained by s e t t i n g r=0. The c o n t r i b u t i o n t o the spectrum <&{v) of p(x,t) from the p o i n t of p<o^-detection i s obtained by t a k i n g the r e a l p a r t of the F o u r i e r Transform of Eq. (4.10): 38 03 — . . 0 0 C ° PP^T^ « i 2 7 r i / T . 1 f f 5 ~ C C 0 7 1 Q i 2 7 T i / T . 0 0 n 7 r ° o * - 0 0 ^{[Aw].} where means F o u r i e r Sine Transform. (See Appendix A f o r a more d e t a i l e d d e r i v a t i o n . ) 4.2.4 C o r r e l a t i o n Volume^ The c o r r e l a t i o n volume i s d e f i n e d by analogy t o the i n t e g r a l l e n g t h s c a l e i n t u r b u l e n c e . The l a t t e r i s d e f i n e d by CO L ( x ) = J ° u (x ) u(x+£) d£ (4.12) u 0 where u i s a v e l o c i t y component and £ i s a space s e p a r a t i o n v e c t o r . Fig.4-3 shows t h a t the i n t e g r a n d w i l l c o n t r i b u t e to the i n t e g r a l only over a c e r t a i n d i s t a n c e , the c o r r e l a t i o n l e n g t h L. L g i v e s an i n d i c a t i o n of the t y p i c a l l e n g t h s i n v o l v e d i n the t u r b u l e n t energy exchange. S i m i l a r l y , we would l i k e to d e f i n e a c o r r e l a t i o n volume g i v i n g an i n d i c a t i o n of the t y p i c a l s i z e s of volumes or eddies a s s o c i a t e d with the r a d i a t i o n of sound. To t h i s end we f o l l o w , f o r a moment, the more common method of s q u a r i n g and time averaging Eg. (4.6): 3 <£(»/) av 39 where t« = t - | x-y' I / c c and t«' = t - | x-jn | /c +f. The in t e g r a n d w i l l c o n t r i b u t e s i g n i f i c a n t l y to the i n n e r i n t e g r a l o n l y over the c o r r e l a t i o n volume V c. Thus Vc i s de f i n e d by v c = - = ! = - / [ ' P C O > ] . [ P £ 0 > ] » D 3 « «»•«> - c o 2 \> Jy L Jy where y_"=y'+£ and the time d e l a y T i s s e t t o zero. T h e r e f o r e , d i f f e r e n t i a t i n g Eg. (U. 13) with r e s p e c t to volume r e s u l t s i n the mean squared p r e s s u r e from u n i t volume: 5 V ,6 ,2 4 X 2 P '"•15' By a c o n s i d e r a t i o n s i m i l a r to the one i n Eg. (4.9) we can write down the a l t e r n a t i v e forms ;CCO ' a P C°V<"} = { f t P C ° V ° 5 } „ C.16) and use whichever form i s the most convenient. Combining Eg. (4.15) and Eq. (4.10) and using Eg. (4.16) we get 40 V r « 4TTC 0 2X — — 1 " r /°° (4-17) •CO)* The u s e f u l n e s s of t h i s c o r r e l a t i o n volume V. s u f f e r s s l i g h t l y i n t h a t i t i s dependent upon the c h o i c e of the d i r e c t i o n of x, the l o c a t i o n of d e t e c t i o n of the r a d i a t e d sound p. I t i n d i c a t e s the average s i z e of an eddy r a d i a t i n g i n the x- d i r e c t i o n . The j e t i s o f t e n thought of being composed by a number of i n c o h e r e n t s o u r c e s , i n essence t u r b u l e n t eddies, compressing and d i l a t i n g as they bump i n t o each o t h e r . I f V c i s evaluated a t a p o i n t such t h a t i t r e p r e s e n t s the average s i z e of a l l sound producing eddies i n the j e t , then a rough estimate of the number of eddies i n the whole j e t can be o b t a i n e d from the q u o t i e n t n=Vj e t/V c, where Vj e t i s the volume occupied by the j e t t u r b u l e n c e c o n t r i b u t i n g s i g n i f i c a n t l y to the n o i s e . S u b s t i t u t i n g Eq. (4.14) i n t o Eq.(4.13), with T=0 and V c=const, g i v e s : on the other hand, from Eq. (4.8), the f o l l o w i n g approximation r e s u l t s : 41 p 2(x) * A 1 2 [ / - p c 0 5 p l V j e t (4.19) 4 T T C 2 X L<3T Jf=r/c J o / o T h e r e f o r e , combining Eg. (4.18) and Eg. (4.19) g i v e s , as a rough e s t i m a t e : Vie. _ p W n = —^ - — - — (U.20) [ & P C 0 3 P. Here, n i s a c o a r s e measure of the number of i n c o h e r e n t s o u r c e s i n the j e t . I f the s p e c t r a are peaky [as they are for jet pressure (see Fig.4-27)]dif f e r e n t i a t i o n with r e s p e c t to time t or time d e l a y r can be approximated by m u l t i p l y i n g by a c h a r a c t e r i s t i c frequency 2irv: .ceo. n - p 2 ( 2 ™ ) 4 { p C C V ° 5 } r LP"'Pi 1--2 [ ( 2 7 T , ) 2 p C ( D p ] L PrmsPrms Eq. (4.21) shows that n can be roughly estimated by c a l c u l a t i n g t h e i n v e r s e square of the c o r r e l a t i o n c o e f f i c i e n t . 42 4.3 Development Of The Pressure Sensor.. 4.3.1 L i m i t a t i o n s on Co n v e n t i o n a l C y l i n d r i c a l Probes^ In order to e v a l u a t e e x p e r i m e n t a l l y Eg. (4.10) i t i s necessary to measure p<°>. As mentioned i n s e c t i o n 4.2.1 p<o) approximates the a c t u a l t o t a l p e r t u r b a t i o n pressure i n the flow f o r M2<<1. In the present experiments the Mach number at the e x i t was about 0.32. The Mach numbers of the c o n v e c t i n g e d d i e s w i l l be s m a l l e r ; of the order of 0.2. Th e r e f o r e the c o n d i t i o n H2<<1 appears to be well s a t i s f i e d . The d i f f i c u l t i e s i n h e r e n t i n the ac c u r a t e measurement of p r e s s u r e f l u c t u a t i o n s i n tur b u l e n c e are w e l l r e c o g n i z e d . For a probe of the c l a s s i c a l c y l i n d r i c a l c o n f i g u r a t i o n (Fig.4-4) the measured in s t a n t a n e o u s pressure p m i s g e n e r a l l y lower than the tr u e i n s t a n t a n e o u s pressure p^ ( i . e . f o r no probe i n the flow) by an amount which depends on the f l u c t u a t i n g components v and w of c r o s s flow normal to the a x i s of the probe (Siddon 1969). The e m p i r i c a l e q u a t i o n , p -p a:-Bp(v2 + w 2-v2-w2) (4.22) m t where B — 0.5 gi v e s a good approximation to the instantaneous e r r o r . Siddon (1971a) estimated t h a t the rms value of t h i s i n s t a n t a n e o u s e r r o r can be q u i t e l a r g e . In a t u r b u l e n t j e t (where t u r b u l e n c e i n t e n s i t i e s are t y p i c a l l y of the order of 0.2) t h i s rms ins t a n t a n e o u s e r r o r i s approximately 0.025 or higher when nondimensionalized by the dynamic pressure of the flow 0.5yo(J 2; the i n s t a n t a n e o u s e r r o r w i l l be l a r g e r , maybe of the order of 0.04. Although the rms value of the inst a n t a n e o u s e r r o r can be l a r g e , the d i f f e r e n c e between the rms values o f the tr u e and the 43 measured p r e s s u r e s o f t e n t u r n s out to be much s m a l l e r . In the present a p p l i c a t i o n where c r o s s c o r r e l a t i o n t e c h n i q u e s are employed i t i s the in s t a n t a n e o u s e r r o r we are concerned with. I t w i l l be necessary to s u b s t a n t i a l l y reduce i t as the measured pr e s s u r e f l u c t u a t i o n i n a t u r b u l e n t j e t a t t a i n s maximum v a l u e s of about 0.05 when nondimensionalized by 0.5^Uo2. T h e r e f o r e , using a c o n v e n t i o n a l c y l i n d r i c a l probe the in s t a n t a n e o u s e r r o r c o u l d be of the same magnitude as the pre s s u r e f l u c t u a t i o n i t s e l f . Siddon (1969) a l s o showed a way of b u i l d i n g an e r r o r compensating probe by measuring s i m u l t a n e o u s l y the pressure p m , and the v e l o c i t y components v and w, and i n s t a n t a n e o u s l y e v a l u a t i n g Eq.(4.22) thereby c o r r e c t i n g the measured pressure and a r r i v i n g at the tr u e p r e s s u r e p t . Even i f the accuracy c o u l d be improved i n t h i s way another problem appears. During the e a r l y p a r t of the r e s e a r c h i t was found t h a t c y l i n d r i c a l probes i n t e r a c t i n g with the t u r b u l e n c e generate s i g n i f i c a n t extraneous n o i s e as i n d i c a t e d i n Fig.4-5. Imagine a patch of t u r b u l e n c e c o n v e c t i n g past the probe. I n t e r a c t i n g with the nose of the probe a l o c a l i z e d f l u c t u a t i n g s i d e f o r c e i s induced, sending a d i p o l e pulse to the f i e l d microphone. A s h o r t time l a t e r the same patch of t u r b u l e n c e i n t e r a c t s with the stem of the probe sending o f f another d i p o l e p u l s e . These d i p o l e p u l s e s are coherent with the b a s i c source f l u c t u a t i o n and t h e r e f o r e make an extraneous c o n t r i b u t i o n to the c o r r e l a t i o n p<°>p. Because these p u l s e s leave the probe e a r l i e r and l a t e r than the t r u e s i g n a l , the r e s u l t i s a tendency f o r the c o r r e l a t i o n to be broadened i n a r a t h e r p e c u l i a r way near the c o r r e c t value of a c o u s t i c t r a v e l time | x - y j / c 0 . Shortening the 44 nose makes the t h r e e c o n t r i b u t i n g e f f e c t s even l e s s d i s t i n g u i s h a b l e as shown by the dashed l i n e on F i g . 4 - 5 . I t must be s t r e s s e d at t h i s p o i n t t h a t , although the o v e r a l l n o i s e i n the f a r f i e l d i n c r e a s e s by only one d e c i b e l or so when i n s e r t i n g the c y l i n d r i c a l probe, the e f f e c t on the c r o s s c o r r e l a t i o n i s very prominent because of the d i s c r i m i n a t o r y power of the method: I t s i n g l e s out the, c o n t r i b u t i o n to the o v e r a l l n o i s e from a very l o c a l i z e d r e g i o n i n the flow which i s more a p p r o p r i a t e l y c a l l e d ' c o r r e l a t i o n volume'. I f the o v e r a l l j e t noise a r i s e s from many independent ( s e p a r a t e l y c o r r e l a t e d ) source r e g i o n s , the extraneous d i p o l e e f f e c t can c o n t r i b u t e s u b s t a n t i a l l y to the apparent r a d i a t i o n from such a l o c a l i z e d source r e g i o n , without n o t i c e a b l y a l t e r i n g the o v e r a l l d e c i b e l v a l u e . 4.3.2 A i r f o i l Probe The best prospect f o r s u p p r e s s i n g the d i p o l e r a d i a t i o n r e s t s i n the p o s s i b i l i t y of red u c i n g the s u r f a c e area of the pressure sensor i n the d i r e c t i o n of the f i e l d microphone to something much s m a l l e r than the c o r r e l a t i o n area of the t u r b u l e n c e . A f l a t a i r f o i l - l i k e probe s a t i s f i e s t h i s requirement. I t i s i n s e r t e d i n t o the flow such t h a t the plane of the f o i l c o n t a i n s the d i r e c t i o n of the mean flow and of the r a d i a t i o n to the f a r f i e l d microphone as shown i n Fig.4-6. The t h i n s t r u c t u r e e n s u r e s ' t h a t d i p o l e r a d i a t i o n i n the plane of the f o i l i s v i r t u a l l y e l i m i n a t e d . In order to put the above statements on a more q u a n t i t a t i v e b a s i s i t i s h e l p f u l t o i n t r o d u c e the "Probe Contamination R a t i o " 45 C. C g i v e s the r a t i o between the mean squared a c o u s t i c pressure due t o d i p o l e r a d i a t i o n from the probe s u r f a c e and the mean squared a c o u s t i c p r e s s u r e due to quadrupole r a d i a t i o n from the eddy or c o r r e l a t i o n volume surrounding the probe. Of course we would l i k e to keep C as s m a l l as p o s s i b l e . Appendix D shows how C i s d e f i n e d mathematically, how i t can be estima t e d , and g i v e s some estimates of i n t e r e s t i n the present problem. I t would appear from these rough e s t i m a t e s t h a t the f o i l type pressure probe b u i l t d u r i n g the present r e s e a r c h has a s m a l l enough probe contamination r a t i o t h a t d i p o l e n o i s e should not be a problem. By g i v i n g the f o i l a c e r t a i n shape and s t r a t e g i c a l l y l o c a t i n g the pressure s e n s i n g h o l e s the probe can be made i n s e n s i t i v e to pressure e r r o r s of aerodynamic o r i g i n ( F i g . 4 - 7 ) . For example i n the subsonic case ( F i g . 4 - 8 ) , the p r e s s u r e d i s t r i b u t i o n s f o r a t h i n a i r f o i l suggest t h a t by l o c a t i n g the pr e s s u r e sensing holes c l o s e to the t r a i l i n g edge, the probe can be made almost i n s e n s i t i v e to angle of a t t a c k f l u c t u a t i o n s both normal to and along the span of the a i r f o i l . T h i s of course presupposes t h a t the s c a l e of the tu r b u l e n c e s t r u c t u r e i s l a r g e compared to the c h a r a c t e r i s t i c a i r f o i l dimensions. The c o r r e l a t i o n p l o t s on Fig.4-9 show th a t the e f f e c t s of extraneous d i p o l e r a d i a t i o n can be e l i m i n a t e d . Curve "a" i s a c r o s s c o r r e l a t i o n between p<o> (as measured by a f o i l shaped probe) and the r a d i a t e d sound p, while curve "b" was obtained by using a c y l i n d r i c a l BSK 1/8-inch microphone with a s h o r t nosecone, a l l other c o n d i t i o n s being kept the same as f o r curve "a". The microphone was mounted on a BSK type 2618 p r e a m p l i f i e r v i a an adaptor type UA0160 modified i n t o a r i g h t angle connector. 46 The i r r e g u l a r i t i e s i n curve "b" are a t t r i b u t e d to d i p o l e r a d i a t i o n from the c y l i n d r i c a l probe, p a r t i c u l a r l y the nosecone, and from the probe support. Curve " c " i s the F o u r i e r Sine transform of curve "a" or the cross-spectrum. Curve "d" r e p r e s e n t s the 'cross-spectrum as c a l c u l a t e d from curve "b". I t does not have as " n i c e " a shape as the one c a l c u l a t e d from curve "a", and even becomes ne g a t i v e f o r some f r e q u e n c i e s . A n e g a t i v e going c r o s s - s p e c t r a l d e n s i t y f u n c t i o n does not seem p o s s i b l e f o r a s i t u a t i o n wherein the source d i s t r i b u t i o n c o n s i s t s of a random a r r a y of u n c o r r e l a t e d phenomena: The power s p e c t r a l d e n s i t y from a u n i t volume of source flow should always be p o s i t i v e . I t i s , however, p o s s i b l e t o hypothesize s i t u a t i o n s where l o c a l i z e d s p e c t r a may e x h i b i t peaks and v a l l e y s a r i s i n g from c o n s t r u c t i v e and d e s t r u c t i v e i n t e r f e r e n c e at c r i t i c a l f r e g u e n c i e s i f s p a t i a l l y s e p a r a t e d sources are t o some degree coherent. Such a s i t u a t i o n was seen i n the image experiment ( s e c t i o n 3 . 3 ) . For the case of probe generated d i p o l e n o i s e , the d i p o l e source and the c o r r e s p o n d i n g t u r b u l e n t (eddy) source are somewhat coherent. They r a d i a t e a t d i f f e r e n t times, the d i f f e r e n c e being given by the c o n v e c t i o n speed and the d i s t a n c e from the nose of the probe to the s e n s i n g h o l e s . A f u r t h e r i n v e s t i g a t i o n of t h i s phenomenon i s l e f t f o r f u t u r e work. Great care was devoted to d e v e l o p i n g the f o i l type p r e s s u r e sensor. An e a r l y type with a sensing hole a t the t r a i l i n g edge of the wing t i p (Siddon S Rackl 1971) d i d not e x h i b i t s a t i s f a c t o r i l y low s e n s i t i v i t y to angle of a t t a c k f l u c t u a t i o n s i n the plane of the f o i l and to f l u c t u a t i o n s i n the chordwise d i r e c t i o n . To have low s e n s i t i v i t y to angles of a t t a c k i n the d i r e c t i o n normal to 47 the plane of the a i r f o i l i s d e s i r a b l e but not necessary i n the present a p p l i c a t i o n as the f o l l o w i n g argument w i l l show: i Remember t h a t the f i e l d microphone i s placed i n the same plane t h a t c o n t a i n s the f o i l probe ( F i g . 4 - 6 ) . The f a r f i e l d p r e s s u r e i n Proudman's ( 1 9 5 2 ) form i s p ( x , t = 2 ~ / f o H d 3 y ( 4 . 2 3 ) 4TTC 0 2 x if L at2 Jf (conf. F i g . 4 - 1 0 ) T h i s c l e a r l y shows t h a t only the v e l o c i t y component u x i n the d i r e c t i o n x of the f i e l d microphone c o n t r i b u t e s to p. P r e s s u r e measurement e r r o r s are due t o the t u r b u l e n t v e l o c i t i e s i n t e r a c t i n g with the probe. u x l i e s i n the plane of the f o i l probe. E r r o r s due to t u r b u l e n t v e l o c i t i e s u,v i n t h i s plane are minimized by the probe design. E r r o r s dependent on the v e l o c i t y component w p e r p e n d i c u l a r to t h i s plane w i l l not, i n p r i n c i p l e , c o n t r i b u t e to the c r o s s c o r r e l a t i o n pCo)p because w does not c o n t r i b u t e t o the r a d i a t i o n of p. 4.3.3 S t a t i c Pressure C a l i b r a t i o n ^ V a r i o u s c o n f i g u r a t i o n s of f o i l type pressure probes were t r i e d out, some t y p i c a l examples being shown i n F i g . 4 - 1 1 . They c o n s i s t of a s t a i n l e s s s t e e l tube embedded i n b a l s a wood with an a i r f o i l type c r o s s s e c t i o n . The end of the tube connects to the lower and upper s u r f a c e of the f o i l . P r e s s u r e e r r o r s due to changing i n c l i n a t i o n of the onset flow were determined s t a t i c a l l y i n the p o t e n t i a l core of a 1.5 i n c h diameter j e t about 0.5 diameter from 48 the e x i t plane, u s i n g a P i t o t s t a t i c tube as a r e f e r e n c e . The probe c o n f i g u r a t i o n t h a t gave the most s a t i s f a c t o r y performance i s the wedge shaped sensor of Fig.4-12. Fig.4-13 g i v e s the d e v i a t i o n s from t r u e s t a t i c p r e s s u r e , i n the form C p = (P- P*,) / (°-5p u£) • F o r angles of a t t a c k l e s s than ±12° ( e q u i v a l e n t to tu r b u l e n c e i n t e n s i t i e s l e s s than 0.2) the C p e r r o r i n the plane of the f o i l (v-component) v a r i e s by l e s s than .008. By way of comparison, the i n c i d e n c e e r r o r f o r a c y l i n d r i c a l probe i n 20% t u r b u l e n c e c o u l d a t t a i n an ins t a n t a n e o u s C p value of about 0.04 ( s e c t i o n 4.3.1). The out of plane e r r o r curve (w-direction) has a marked anti-symmetry. T h i s i s most probably due to one or both of 2 reasons: 1) the a i r f o i l s e c t i o n and the pr e s s u r e taps l a c k symmetry due to an i m p e r f e c t manufacturing p r o c e s s ; 2) the j e t flow development i t s e l f was i n f l u e n c e d by the probe not being on the c e n t e r l i n e a t d i f f e r e n t angles of a t t a c k . In order to minimize the s t a t i c p r e s s u r e measurement e r r o r due to f l u c t u a t i o n s i n the u-component ( d i r e c t i o n of mean flow) the se n s i n g holes are l o c a t e d i n a d i p , immediately forward of the t h i c k e n i n g t r a i l i n g edge. The r e s u l t i n g p r essure recovery o f f s e t s a tendency f o r ne g a t i v e C p due to the f i n i t e t h i c k n e s s ( F i g . 4 - 1 4 ) . The C p e r r o r due to u-component f l u c t u a t i o n s can be estimated from Fig.4-13. We d e f i n e : A = (P m-P e e)/(0.5 yoU|) (4.24) where P m i s the measured s t a t i c p r e s s u r e f o r 0° angle of a t t a c k , P̂ , i s the s t a t i c pressure t h a t would occur without the probe i n 49 the flow. At 0° angle of a t t a c k , the d e v i a t i o n from the true p r e s s u r e i s shown to be C p =-0.006. However, P^ was not e x a c t l y known. By v a r y i n g the c o n f i g u r a t i o n of the f o i l probe and the r e f e r e n c e P i t o t s t a t i c tube i n the j e t i t was found that the u n c e r t a i n t y of where the C p = 0 - l i n e l i e s i s of the order of 0.003. In the worst case: , C p = -0.009 = (P m- poo)/(°- 5/> Uo) = A (4.25) I f , as d i s c u s s e d i n the next paragraph, the s t a t i c c a l i b r a t i o n remains v a l i d i n the t u r b u l e n t flow we can w r i t e (Siddon 1969): o In 20% t u r b u l e n c e u/U = 0.2. The C p u - e r r o r w i l l t h e r e f o r e be at the most -0.0036. The s u g g e s t i o n t h a t s t a t i c i n c i d e n c e c a l i b r a t i o n s remain v a l i d i n the r e a l , t i m e - v a r y i n g flow i m p l i e s a g u a s i - s t e a d y assumption. Such an assumption holds i f the sensor dimensions are s m a l l compared with the s c a l e s of the approaching f l o w . In other words, the r a t i o i/L/0 must be s m a l l , where L i s the c h a r a c t e r i s t i c chord l e n g t h of the f o i l . For the present probe L-10 mm. Thus the upper l i m i t i n g frequency w i l l be about -U/4L-2500 Hz i n a 100 m/sec flow. T h i s i s q u i t e a severe r e s t r i c t i o n . In the zone of maximum shear the mean v e l o c i t y i s even s m a l l e r (about 70 m/sec), and we are i n t e r e s t e d i n f r e q u e n c i e s up to about 5 kHz although the j e t noise peaks at 700 Hz ( F i g . 3 - 2 ) . We may t h e r e f o r e expect to run i n t o d i f f i c u l t i e s of 50 r e s o l u t i o n . T h i s may be one of the reasons why the f i n a l check ( s e c t i o n 4.5.2) does not work out. As t h i s i s the f i r s t p ressure probe of an a i r f o i l type i t should be looked upon as a prototype. I t i s hoped t h a t f u r t h e r development w i l l f o l l o w , i n p a r t i c u l a r with regard to m i n i a t u r i z a t i o n . 4.3.4 The Complete Pressure Sensor as shown i n Fig.4-12 and Fig.4-15 c o n s i s t s of the f o i l probe mounted to a BSK 1/8-inch microphone which i s connected v i a an adaptor type UA0160 to a type 2618 p r e a m p l i f i e r . The adaptor was f i l l e d with p a r a f f i n and the p r e a m p l i f i e r - a d a p t o r combination was coated with p a r a f f i n to e l i m i n a t e mechanical resonances. The c a v i t y c o nnecting the f o i l to the microphone combines with the s m a l l p r e s s u r e s e n s i n g holes to form a Helmholtz r e s o n a t o r . The r e s u l t i n g resonant response a t about 2500 Hz was damped out with a c o t t o n i n s e r t . . Fig.4-16 shows the r e s u l t i n g frequency response of the pr e s s u r e sensor. I t i s e s s e n t i a l l y f l a t at the f r e q u e n c i e s c o n t r i b u t i n g predominantly to the a c o u s t i c spectrum. 51 4.4 Experiments The experiments were conducted i n an anechoic chamber with dimensions 2. 7m*2.7m*1.8m (measured from the t i p of the wedges) with a lower c u t o f f frequency of about 300 Hz. The j e t n o z z l e was pl a c e d near one corner of the room; the n o z z l e e x i t diameter was 38.1mm (1.5 i n c h e s ) . i 4.4.1 A i r Supply^ S e t t l i n g Chamber^ and Nozzle^ ( Fig.4-17) A r e c i p r o c a t i n g compressor was used as the a i r supply. I t f e d i n t o a l a r g e r e c i e v i n g tank. The a i r then flowed through a f l e x i b l e hose to the s e t t l i n g chamber. Host of the s e t t l i n g chamber was packed with l o o s e f i b r e g l a s s of the type used f o r heat and sound i n s u l a t i o n . T h i s e l i m i n a t e d n o i s e propagation from upstream, e s p e c i a l l y from the va l v e and the e n t r y i n t o the s e t t l i n g chamber. Immediately upstream of the n o z z l e were p l a c e d a c o t t o n wool f i l t e r to r e t a i n dust and a s e t of screens with d i m i n i s h i n g mesh s i z e t o ensure a uniform v e l o c i t y d i s t r i b u t i o n at the no z z l e e n t r y . The no z z l e i t s e l f was designed such that a uniform v e l o c i t y d i s t r i b u t i o n would r e s u l t at the n o z z l e e x i t by analogy to the magnetic f l u x through a r i n g c u r r e n t (Smith S Wang 1944) . The shape of the nozzle was generated on a computer. The mould was produced on a numerical m i l l i n g machine, the no z z l e i t s e l f being made of f i b r e g l a s s using g e l c o a t as the f i r s t l a y e r to o b t a i n a smooth i n s i d e s u r f a c e . 4.4.2 The Far F i e l d Microphone.. A B&K 1/2-inch microphone was used as the f i e l d microphone l o c a t e d at 45° from the j e t a x i s a t a d i s t a n c e of about 50 diameters (1.9 meters) from the n o z z l e e x i t . That the f a r f i e l d 52 c o n d i t i o n was met i s shown i n Fig.4-18: I t pr e s e n t s the v a r i a t i o n of the rms p r e s s u r e f l u c t u a t i o n from i n s i d e the flow r e g i o n a c r o s s the near f i e l d to the f a r f i e l d (the l a t t e r being c h a r a c t e r i z e d by the 6 db drop f o r every d o u b l i n g of d i s t a n c e ) . The t r a v e r s e was done p e r p e n d i c u l a r to the j e t a x i s and s t a r t i n g at an a x i a l l o c a t i o n of 4 diameters from the no z z l e e x i t . 4.4.3 S i g n a l P r o c e s s i n g ^ (Fig.4-19) The s i g n a l p ( 0 > ( t ) from the p r e s s u r e sensor was a m p l i f i e d , f i l t e r e d (passband 270 - 22500 Hz), d i f f e r e n t i a t e d , and f e d i n t o channel A of a PAR Model 101 S i g n a l C o r r e l a t o r . The s i g n a l p(t) from the f a r f i e l d , measured by a 1/2-inch B6K microphone, was a m p l i f i e d , f i l t e r e d the same way, and fed i n t o channel B of the c o r r e l a t o r . An x-y p l o t t e r was used to r e c o r d the r e s u l t i n g c o r r e l a t i o n f u n c t i o n s p<o>p. The raw c o r r e l a t i o n f u n c t i o n s c o n t a i n e d some n o i s e from the s i g n a l c o r r e l a t o r and a c o u s t i c contamination from r e f l e c t i o n s o f f the probe support and t r a v e r s i n g mechanism. The l a t t e r m a n i f e s t s i t s e l f by weak secondary peaks o c c u r r i n g some time a f t e r the s o u r c e - r e c i e v e r d e l a y time r = r / c 0 . Some s u b j e c t i v e judgement was used i n e l i m i n a t i n g the a c o u s t i c contamination i n the t a i l s of the c o r r e l a t i o n f u n c t i o n s . T h i s measure a f f e c t s o n l y the c r o s s - s p e c t r a r e s u l t i n g from F o u r i e r T r a n s f o r m a t i o n of the complete c o r r e l a t i o n f u n c t i o n . The u n f i l t e r e d l o c a l source s t r e n g t h , equal to the d e r i v a t i v e at r = r / c 0 ( v e r t i c a l l i n e i n Fig.4-9, curve " a " ) , i s the p r i n c i p a l q u a n t i t y e x t r a c t e d from the data as i n d i c a t e d by Eq.(4.10). In order to process the data on a computer the p l o t t e d c o r r e l a t i o n f u n c t i o n s were d i g i t i z e d using a g r a p h i c a l d i g i t i z e r . The d e r i v a t i v e was c a l c u l a t e d by l e a s t 53 square f i t t i n g a polynomial to the c o r r e l a t i o n f u n c t i o n i n the v i c i n i t y of r = r / c 0 . The F o u r i e r Sine Transform a c c o r d i n g to Eg. (4.11) was estimated u s i n g a F o u r i e r transform u t i l i t y package (Rackl 1972a). (Fig.4-22a and Fig.4-22b). Keeping the f i e l d microphone always a t the same l o c a t i o n of 45° to the j e t a x i s the p r e s s u r e sensor was moved around i n the j e t on the s i d e o p p o s i t e to the microphone, i n order to minimize i n t e r f e r e n c e of the probe support with the sound f i e l d ( F i g . 4 - 1 ) . R a d i a l t r a v e r s e s were done at v a r i o u s downstream l o c a t i o n s . 4. 5 Data Reduction 4.5.1 Source S t r e n g t h D i s t r i b u t i o n . , The procedure d e s c r i b e d i n the p r e v i o u s s e c t i o n r e s u l t s i n the d i s t r i b u t i o n of source s t r e n g t h per u n i t volume shown i n Fig.4-20. N o n d i m e n s i o n a l i z a t i o n was e f f e c t e d as shown i n Appendix B. The nondimensionalized source s t r e n g t h p l o t t e d i s : s uv ~~ 2{0.5PU2)2MZ D/x 0 (U/D) 2 (4.27) where: nozzle e x i t v e l o c i t y . 0 o / c Q = ttach number. D n o z z l e e x i t diameter, x o d i s t a n c e n o z z l e e x i t - f i e l d microphone. I t i s not s u r p r i s i n g t h a t the d i s t r i b u t i o n of bears some s i m i l a r i t y t o the d i s t r i b u t i o n of mean shear i n a j e t . I t has t r a d i t i o n a l l y been argued t h a t sound g e n e r a t i o n from t u r b u l e n t flows should be s t r o n g e s t i n the r e g i o n s o f most i n t e n s e t u r b u l e n c e o c c u r r i n g where the g r a d i e n t s of mean v e l o c i t y are l a r g e s t . Thus, over the f i r s t few diameters, i s s t r o n g e s t h a l f a diameter from the j e t a x i s ; f u r t h e r downstream the peak s h i f t s toward the a x i s and decreases r a p i d l y i n amplitude. At t h i s p o i n t i t should be emphasized t h a t t h i s d i s t r i b u t i o n does not i n d i c a t e the sound power r a d i a t e d from u n i t volume, but the' c o n t r i b u t i o n t o the mean squared p r e s s u r e i n the f a r f i e l d a t 45° to the j e t a x i s . For each downstream l o c a t i o n the r a d i a l d i s t r i b u t i o n was i n t e g r a t e d r e s u l t i n g i n the d i s t r i b u t i o n of source s t r e n g t h from a " s l i c e " of the j e t : 00 Ssiice = / S u v 2 7 r r dr (4.28) r=0 F i g . 4-21 shows the p r e s e n t r e s u l t and compares i t q u a l i t a t i v e l y o n l y w i t h r e s u l t s of o t h e r workers (Lee & Ribner 197 2; Ribner 1958, 1962; Dyer 1959; Jones 1968; see a l s o Chu, L a u f e r & Kao 1972). The p r e s e n t and H.K. Lee's works use s i m i l a r t e chniques ( c a u s a l i t y c o r r e l a t i o n s ) ; Lee's d a t a were taken f o r a f a r f i e l d l o c a t i o n of 4 0° to the j e t a x i s and a t a Mach number of about 0.3. The other curves are not experimental data; they are based on s e m i - e m p i r i c a l t e c h n i q u e s and i n d i c a t e the sound power r a d i a t e d from a s l i c e , without c o n s i d e r a t i o n of d i r e c t i o n - a l i t y . I t i s known t h a t h i g h e r f r e q u e n c i e s are r a d i a t e d more 55 preferentially i n directions i n the v i c i n i t y of 45°, this being due to source c o n v e c t i o n and r e f r a c t i o n by the shear l a y e r a f f e c t i n g the h i g h e r f r e q u e n c i e s more than the lower ones. Higher f r e q u e n c i e s are generated c l o s e r to the e x i t than lower f r e q u e n c i e s : One might t h e r e f o r e expect the a x i a l source d i s t r i b u t i o n f o r 45° r a d i a t i o n to peak e a r l i e r ( i . e . between 2 and 3 diameters downstream) than would be expected f o r the o v e r a l l power d i s t r i b u t i o n . The r e s t r i c t i o n s d i s c u s s e d i n s e c t i o n 4.3.3 must however be born i n mind. The g e n e r a l trend i s to support Ribner's p r e d i c t i o n from s i m i l a r i t y c o n s i d e r a t i o n s except very c l o s e to the n o z z l e e x i t where the flow has to r e a d j u s t very r a p i d l y from a boundary l a y e r i n s i d e the n o z z l e to a f r e e l y growing shear l a y e r . One must a l s o p o i n t out t h a t i t i s not known a c c u r a t e l y where the assumptions of s i m i l a r i t y become v a l i d downstream of the p o t e n t i a l core, i . e . the a x i a l l o c a t i o n where the x~ 7-Law takes over i s u n c e r t a i n . 4.5.2 Cheeky C l o s u r e D i f f i c u l t y F u r t h e r i n t e g r a t i o n of S s ! j c e along the j e t a x i s should recover the p 2 as measured d i r e c t l y a t x, a c c o r d i n g to Eg. (4.8). T h i s p r o v i d e s a check of the procedure. However, the p 2 as obtained by i n t e g r a t i o n i s n e a r l y 10 times as l a r g e as the measured one. The p o s s i b i l i t y of a numerical mistake i s r u l e d out as the c a l c u l a t i o n s were checked many times, a l s o by d i f f e r e n t persons. Numerous p o s s i b i l i t i e s e x p l a i n i n g t h i s d i s c r e p a n c y were c o n s i d e r e d and many were d i s c a r d e d . Among the l a t t e r were: Vortex shedding from the probe t r a i l i n g edge (the shedding frequency i s of the order of 14 K i l o h e r t z ) , mechanical shaking of the microphone. P o s s i b l y v a l i d e x p l a n a t i o n s a r e : 56 X)_ The d e r i v a t i v e of p<o>p i S to be e v a l u a t e d a c c o r d i n g to Eg. (4.10) a t r = r / c o f the time delay t h a t corresponds to the t r a v e l time of the sound from the source to the f a r f i e l d microphone. T h i s " g e o m e t r i c a l " time delay i s i n f a c t not equal to the r e a l sound t r a v e l time as the.probe i s i n s e r t e d from the f a r s i d e of the j e t (Fig.4-1) and the sound must t r a v e l through the j e t flow where i t i s a c c e l e r a t e d and r e f r a c t e d . The " e f f e c t i v e " time delay i s t h e r e f o r e s l i g h t l y s m a l l e r . The t y p i c a l p<o>p shown i n Fig.4-24 i s q u i t e a n t i s y m m e t r i c a l . According to the g e o m e t r i c a l time delay the d e r i v a t i v e would have to be c a l c u l a t e d at a p o i n t s l i g h t l y t o the r i g h t of the p o i n t of symmetry (here c a l l e d the " c e n t e r " ) , but s t i l l on the p o r t i o n with p o s i t i v e s l o p e . The e f f e c t i v e time delay would move t h i s p o i n t toward the cen t e r where the s l o p e i s s t e e p e s t . Experimental evidence was c o l l e c t e d showing t h a t the e f f e c t i v e time delay should be s h o r t e r : Some experiments were performed with the pressure sensor i n s e r t e d from the near s i d e . In t h i s case the geometric time delay was very c l o s e to the " c e n t e r " as one would expect s i n c e the sound does not t r a v e l through the j e t exhaust stream. In another s e t of experiments, a p o i n t source with a diameter of about 0.1 inches was pl a c e d immediately under the pressure probe ( i n s e r t e d from the f a r side) and d r i v e n by random noise and by pure tones. The c r o s s c o r r e l a t i o n was computed i n the usual manner with and without the j e t flow. In the l a t t e r case a peak i n the c o r r e l a t i o n f u n c t i o n was observed very c l o s e to the c a l c u l a t e d g e o m e t r i c a l time d e l a y . When the j e t flow was turned on t h i s peak s h i f t e d to the l e f t toward a s h o r t e r time d e l a y . 57 T h i s s h i f t v a r i e d somewhat f o r random nois e and f o r pure tones of d i f f e r e n t frequency; i t was i m p o s s i b l e to determine the exact s h i f t with the simple setup used. A t h i r d experiment was performed on the computer o n l y : The F o u r i e r Sine transform was c a l c u l a t e d f o r a s e l e c t e d p<o>p "cen t e r e d " a t a v a r y i n g T, i . e . the amount of o r i g i n s h i f t on the T - a x i s ( i n d i c a t e d i n Eq. (4.11) by [...].£} before F o u r i e r t r a n s f o r m i n g was v a r i e d i n the v i c i n i t y of r = f = r / c 0 . As may be expected, when p<o>p w a s cen t e r e d on the point of symmetry a s i n e transform r e s u l t e d t h at remained p o s i t i v e f o r a l l f r e q u e n c i e s . I f centered before or a f t e r the p o i n t of symmetry the transform showed negative going p a r t s , which does not seem p o s s i b l e as a l r e a d y mentioned i n s e c t i o n 4.3.2. I t was t h e r e f o r e decided t o assume that the e f f e c t i v e time d e l a y i s given by the p o i n t of s t e e p e s t s l o p e . T h i s d e c i s i o n may not have been the best. The s l o p e a t the g e o m e t r i c a l time delay i s s m a l l e r and would r e s u l t i n a s m a l l e r i n t e g r a t e d p 2 , but the change would f a l l f a r s h o r t of accounting f o r the t o t a l observed d i s c r e p a n c y . 2]_ The c a l c u l a t i o n s i n Appendix D i n d i c a t e t h at d i p o l e n o i s e should not be a problem with the present pressure sensor. However, Appendix D d e a l s only with gross t u r b u l e n c e p r o p e r t i e s f o r x/D=4. Haybe one should g i v e more c o n s i d e r a t i o n to the s m a l l e r t u r b u l e n c e s c a l e s e s p e c i a l l y when they are of the order of the probe s i z e . T h i s w i l l be the case c l o s e r to the e x i t plane; t h e r e , d i p o l e n o i s e c o u l d indeed become dominant. 3)_ The i n s e r t i o n o f the probe i n t o the flow w i l l , of course, change the flow. The s m a l l e r the probe, the l e s s t h i s d i s t u r b a n c e . The d i s c u s s i o n of s e c t i o n 4.3.3 shows t h a t the probe may indeed 58 be too l a r g e to adequately r e s o l v e the t u r b u l e n t pressure a t a p o i n t i n the present j e t , e s p e c i a l l y f u r t h e r upstream ( c l o s e r than about 2 diameters from the e x i t ) where s c a l e s are s m a l l and f r e q u e n c i e s are h i g h . The q u a s i s t e a d y assumption f o r the pressure e r r o r would no longer be v a l i d . 4}_ I t was p o i n t e d out to the author t h a t there i s a s t r o n g mean v e l o c i t y g r a d i e n t a c r o s s the s u r f a c e of the probe when i t i s i n s e r t e d from the f a r s i d e as shown i n Fig.4-1. T h i s may l e a d to an a d d i t i o n a l flow d i s t o r t i o n and pressure e r r o r . However, when the probe was i n s e r t e d at a p o i n t 1/2 a diameter d i r e c t l y above the a x i s (not behind as i n Fig.4-1) where th e r e would h a r d l y be any v e l o c i t y g r a d i e n t a c r o s s the probe, the c o r r e l a t i o n p<o>p was not s i g n i f i c a n t l y d i f f e r e n t . 5\_ The dynamic c a l i b r a t i o n of the pressure sensor was done a c o u s t i c a l l y o n l y . I t may be t h a t flow over the pressure s e n s i n g h o l e s s u b s t a n t i a l l y a l t e r s the frequency response of the probe. 4.5.3 S p e c t r a . The upper p a r t of Fig.4-9 shows a t y p i c a l p a i r of c r o s s - f u n c t i o n s : 9a the e x p e r i m e n t a l l y obtained c r o s s c o r r e l a t i o n f u n c t i o n i n the time domain, 9c the c r o s s spectrum f u n c t i o n i n the frequency domain c a l c u l a t e d from 9a a c c o r d i n g to Eg. (4.11), i . e . , the s p e c t r a l d e n s i t y of the source s t r e n g t h per u n i t volume. Fig.4-22b shows some t y p i c a l c r o s s spectrum f u n c t i o n s f o r v a r i o u s downstream l o c a t i o n s . Again i n t e g r a t i n g r a d i a l l y one o b t a i n s the s p e c t r a from s l i c e s of the j e t . T h e i r peak f r e q u e n c i e s are p l o t t e d a g a i n s t downstream d i s t a n c e from the n o z z l e e x i t i n Fig.4-23 ( f u l l c i r c l e s ) t o gether with o t h e r r e s u l t s a v a i l a b l e . (Lee & Ribner 1972;' Chu, L a u f e r and Kao 1972). 59 The empty squares of Fig.4-23 were a r r i v e d a t by a very simple argument: We know that the r a d i a t e d sound depends on the second d e r i v a t i v e of the source f l u c t u a t i o n . Bearing t h i s i n mind, the s p e c t r a l d e n s i t y of p<o) a s measured with the s t a t i o n a r y f o i l probe (Fig.4-27)' was m u l t i p l i e d . by the square of frequency and the peak f r e q u e n c i e s entered i n t o Fig.4-23. Although t h i s does not account f o r d i r e c t i o n a l i t y and i s not amenable to * q u a n t i t a t i v e a n a l y s i s of source s t r e n g t h the curves f a l l c l o s e t o g e t h e r . I t i s i n t e r e s t i n g t o note t h a t none of the curves i n F i g . 4 - 23 a t t a i n s the value of about 0.2 which corresponds to the peak S t r o u h a l number of the o v e r a l l n o i s e a t 45° from the j e t a x i s . Lee (1971) has commented t h a t t h i s i s "... due to the omitted c o n t r i b u t i o n of t h a t p a r t of the j e t extending beyond X=7D, which must dominate the low frequency sound e m i s s i o n . " In the present case the measurements were done as f a r downstream as 10 diameters; the curve s t i l l does not show much tendency to bend down to lower S t r o u h a l numbers. T h i s can be i n t e r p r e t e d i n d i f f e r e n t ways: 21 The low frequency sound i s emitted s t i l l f u r t h e r downstream. Looking at Fig.4-21 however, a l l a v a i l a b l e data suggests t h a t t h e r e i s h a r d l y any s i z a b l e c o n t r i b u t i o n to the j e t noi s e from r e g i o n s beyond 10 diameters. 2)_ The low frequency sound i s generated upstream of the e x i t p l a n e , i . e . i n the n o z z l e and the s e t t l i n g chamber a c t i n g as a c o u s t i c r e s o n a t o r s . I f t h i s were t r u e one may expect to see i n the spectrum a low frequency hump above the j e t n o i s e which would 60 vary markedly from one j e t to another; but t h i s i s not the case. 3}_ Some of the p o i n t s d i s c u s s e d i n s e c t i o n 4.5.2 w i l l , of course, be r e f l e c t e d i n the s p e c t r a . In p a r t i c u l a r p o i n t 2) : The s m a l l e r t u r b u l e n c e s c a l e s are a s s o c i a t e d with h i g h e r f r e g u e n c i e s . T h e r e f o r e , i f d i p o l e c o n t a m i n a t i o n were a problem i t would enhance the high frequency contents and thus make the s p e c t r a peak at higher S t r o u h a l numbers. In a d d i t i o n , f a r downstream, where s c a l e s are l a r g e , i n t e r a c t i o n of the flow with the stem of the probe may produce a s i g n a l which i s coherent with the measured pressure s i g n a l and again contaminate the c r o s s c o r r e l a t i o n . The d i r e c t l y measured sound spectrum i n the f a r f i e l d at 45° from the j e t a x i s and the one t h a t r e s u l t s by i n t e g r a t i n g a l l elementary s p e c t r a over the j e t volume should, t h e o r e t i c a l l y , be the same [Eq. (4.11) i n t e g r a t e d over volume]. As f o r the mean squared pressure t h i s check does not work out. Fig.4-28 compares the d i r e c t l y measured and the i n t e g r a t e d nondimensionalized s p e c t r a . The r e c o n s t r u c t e d spectrum a p p a r e n t l y i n c l u d e s an excess of high frequency energy. 4.5.4 Number of Incoherent Sources. The c o r r e l a t i o n c o e f f i c i e n t of Eg. (4.21) at a l o c a t i o n 3 • diameters downstream and 1/2 a diameter from the j e t a x i s turned out to be approximately 0.08. T h e r e f o r e , a rough estimate of the number of i n c o h e r e n t sources i n the j e t n i s n ~ (0.08) -2 ^ 150 It would appear reasonable to i n f e r from t h i s number that n i s of the order of 100 to 200. One must however i n v e s t i g a t e what 61 e x t e r n a l i n f l u e n c e s may i n c r e a s e or decrease the c o r r e l a t i o n c o e f f i c i e n t : 21 D i p o l e n o i s e o f f the probe can s i g n i f i c a n t l y i n c r e a s e the magnitude of the c o r r e l a t i o n f u n c t i o n and of the c o r r e l a t i o n c o e f f i c i e n t : In Eg. (4.21) both p<o)p and p r m s w i l l be a f f e c t e d by the d i p o l e r a d i a t i o n . However, p<o>p w i l l i n c r e a s e much more than Prms because prms ^ s t n e n o i s e coming from the whole j e t , while p<o)p r e p r e s e n t s the c o n t r i b u t i o n from the s m a l l volume where the probe i s i n s e r t e d . (This was a l r e a d y d i s c u s s e d i n s e c t i o n 4.3.1). Thus n may be c o n s i d e r a b l y underestimated. The low value of n«3 given by Scharton and White (1972) was a r r i v e d a t using a 1/8- i n c h microphone i n a 5/8-inch diameter j e t with Mach number egual to one. The probe contamination r a t i o (Appendix D) f o r t h i s c o n f i g u r a t i o n i s of the order of 1.8. T h i s high value would i n d i c a t e t h a t d i p o l e n o i s e from the probe not only p l a y s a s i g n i f i c a n t r o l e , but t h a t i t i s dominant. 2]_ F i l t e r i n g may have an i n f l u e n c e on the magnitude of the c o r r e l a t i o n c o e f f i c i e n t i f a s i g n i f i c a n t amount of the energy of one or both of the two s i g n a l s being c o r r e l a t e d i s f i l t e r e d out. In the present case, f o r i n s t a n c e , one may wish to i n c r e a s e the s i g n a l to n o i s e r a t i o by using an octave band f i l t e r c e n t r e d on the peak frequency of the j e t n o i s e spectrum. I f p < 0 ) ( t ) i s measured at a p o i n t i n the j e t where the spectrum of p<o> peaks i n the same octave band, the c o r r e l a t i o n p<o>p m a y be l e f t e s s e n t i a l l y u n i n f l u e n c e d by f i l t e r i n g . On the other hand, p(t) c o n t a i n s a s i g n i f i c a n t amount of energy o u t s i d e t h i s octave band which i s c o n t r i b u t e d from other p a r t s of the j e t . T h e r e f o r e the f i l t e r e d p r m s w i l l be s m a l l e r than the u n f i l t e r e d one, again 62 i n c r e a s i n g the c o r r e l a t i o n c o e f f i c i e n t and l e a d i n g to an u n d e r e s t i m a t i o n of n [ o c t a v e band f i l t e r i n g was employed by Scharton 5 White (1972) which may have c o n t r i b u t e d to t h e i r low value of n-3 ]. 3|_ E x t e r n a l a c o u s t i c n o i s e can s i g n i f i c a n t l y decrease the c o r r e l a t i o n c o e f f i c i e n t (but l e a v i n g the c o r r e l a t i o n f u n c t i o n u n a f f e c t e d ) : In Eg. (4.21), the a c o u s t i c pressure p r m s c a n c o n t a i n other s i g n a l s than the one r a d i a t e d from the j e t . T h i s would decrease the c o r r e l a t i o n c o e f f i c i e n t and thereby overestimate n. Care must t h e r e f o r e be taken to e l i m i n a t e e x t e r n a l noise sources. In the present work f o r i n s t a n c e , low frequency noise from the compressor s u p p l y i n g the a i r had to be f i l t e r e d out by a 4th order high pass f i l t e r with a c u t o f f frequency of about 270 Hz. 4)_ E l e c t r o n i c n o i s e may a l s o s u b s t a n t i a l l y decrease the c o r r e l a t i o n c o e f f i c i e n t by superposing i t s e l f on the p < 0 > ( t ) and p(t) s i g n a l s thereby i n c r e a s i n g p r m s and p ( 0 ) r m s i n Eq.(4.21). Again, n would be overestimated. E l e c t r o n i c noise has high freguency c h a r a c t e r and can u s u a l l y be e l i m i n a t e d by a s u i t a b l e low pass f i l t e r . Lee S Ribner (1972, p. 1289) roughly estimated n to be of the order of 2500. Although t h i s may come c l o s e t o the t r u t h i t i s not known i f i t may have been overestimated as d e s c r i b e d i n p o i n t s 3) and 4) above. 63 V. SUMMARY CONCLUSIONS^. RECOMMEND ATI ONS.. i i Two experimental techniques were developed with the aim of l e a r n i n g more about the d i s t r i b u t i o n of sound sources i n a t u r b u l e n t j e t . They both use the c a u s a l i t y p r i n c i p l e i n t h a t they measure a source f l u c t u a t i o n and r e l a t e i t to the r a d i a t e d sound by a c r o s s c o r r e l a t i o n method. In the image technique a r i g i d s u r f a c e i s put c l o s e to the j e t . The p r e s s u r e on the s u r f a c e i s taken as a f l u c t u a t i o n r e p r e s e n t a t i v e of a c e r t a i n l i m i t e d r e g i o n of the j e t . The r e s u l t i n g d i s t r i b u t i o n of source s t r e n g t h over the s u r f a c e a l l o w s some c o n c l u s i o n on the no i s e sources i n the j e t using the ray a c o u s t i c s r e f l e c t i o n p r i n c i p l e . For the low speed subsonic j e t used the s t r o n g e s t sources appear to be l o c a t e d between 4 and 6 diameters downstream. I f t h i s technique i s employed i n the f u t u r e i t would probably be of advantage to r e p l a c e the r i g i d s u r f a c e by a h y p o t h e t i c a l plane as the presence of the r e a l s u r f a c e i n f l u e n c e s the development of the flow (Siddon 1973b). I t may, however, be necessary to measure both v e l o c i t y and p r e s s u r e , as the boundary c o n d i t i o n of no normal v e l o c i t y on the s u r f a c e would then disappear [terms c o n t a i n i n g Uj i n the s u r f a c e i n t e g r a l s of Eg.(1.2) would not v a n i s h , but may be n e g l i g i b l y s m a l l i n some cases ]. In the d i r e c t t u r b u l e n c e probing technique the f l u c t u a t i n g 64 p r e s s u r e i s measured by an a i r f o i l type sensor and i s c r o s s c o r r e l a t e d with the f a r f i e l d r a d i a t e d sound. Using Ribner's (1962) d i l a t a t i o n theory l e a d s to a volume d i s t r i b u t i o n of a c o u s t i c source s t r e n g t h i n the j e t which l o o k s somewhat s i m i l a r to the d i s t r i b u t i o n of mean v e l o c i t y shear. R a d i a l i n t e g r a t i o n g i v e s the a c o u s t i c s t r e n g t h from " s l i c e s " of the j e t ; t h i s i s i n q u a l i t a t i v e agreement with r e s u l t s of other r e s e a r c h e r s . However, an attempt to r e g a i n the t o t a l r a d i a t e d pressure by i n t e g r a t i n g over a l l sources i n the j e t f a i l e d . Reasons f o r t h i s f a i l u r e are g i v e n . From these the f o l l o w i n g recommendations r e s u l t : XI To f u r t h e r develop the p r e s s u r e sensor: the shape of the present sensor was a r r i v e d at by "educated guessing". A t h e o r e t i c a l i n v e s t i g a t i o n i n the pressure d i s t r i b u t i o n over the f o i l s u r f a c e using a t h r e e dimensional theory [ f o r i n s t a n c e Smith's method (Smith 5 Hess 1967) ] should l e a d to an optimum f o i l shape which a l s o takes i n t o account the presence of the probe support. The a n a l y s i s c o u l d even be c a r r i e d f u r t h e r to i n c l u d e time v a r y i n g i n c i d e n t flow. However, the computing time r e q u i r e d using a modified Smith's method would be very l a r g e . - M i n i a t u r i z a t i o n of the pressure sensor i s very d e s i r a b l e both from the p o i n t of view of d i p o l e n o i s e contamination and of flow d i s t u r b a n c e . The f e a s i b i l i t y of using t r a n s d u c e r s o t h e r than condenser microphones should be e x p l o r e d . Such c o u l d be of the p i e z o e l e c t r i c type or K u l i t e semiconductors which could be embedded d i r e c t l y i n t o the a i r f o i l s u r f a c e . - The dynamic c a l i b r a t i o n should be done i n a known unsteady flow such as an i n c l i n e d r o t a t i n g n o z z l e which generates a flow with s i n u s o i d a l l y v a r y i n g angle of a t t a c k . 65 2)_ The l a r g e r the j e t and the s m a l l e r the probe, the l e s s problems w i l l a r i s e from extraneous d i s t u r b a n c e s . A l s o , t e s t s run at higher Mach numbers would not o n l y be more r e a l i s t i c models of j e t s used i n modern a i r c r a f t but reduce the probe contamination r a t i o i n p r o p o r t i o n to M - 2 as shown i n Appendix D, Eg. (D.12). 3]_ The u n c e r t a i n t y r e g a r d i n g the c o r r e c t time delay of the " c e n t e r " of the c a u s a l i t y c o r r e l a t i o n f u n c t i o n should be reduced [ s e c t i o n 4.5.2, p o i n t 1 ) ] : The exact sound t r a v e l time from the t r a n s d u c e r measuring the source f l u c t u a t i o n to the f a r f i e l d microphone has to be measured or c a l c u l a t e d i n some way. For i n s t a n c e , a p o i n t source of sound i n the j e t c o u l d be used at the l o c a t i o n of the t r a n s d u c e r , without the l a t t e r i n p l a c e . S i n c e i t i s next to i m p o s s i b l e to b u i l d a s t r o n g p o i n t source with a uniform phase response, t h i s phase response must 'at l e a s t be known and c o r r e c t e d f o r l a t e r . The t h e s i s e s t a b l i s h e d the f e a s i b i l i t y of e x p l o i t i n g Ribner's pressure source model of flow n o i s e using the c a u s a l i t y c o r r e l a t i o n technique. The d i f f i c u l t i e s i n h e r e n t i n the method were uncovered; some were s o l v e d , f o r o t h e r s v i a b l e s o l u t i o n s were proposed. S i n c e the c o r r e l a t i o n s can be performed i n the broad band i t i s an e f f i c i e n t method. In i t s present s t a t e i t g i v e s a good p i c t u r e of the r e l a t i v e source d i s t r i b u t i o n . I f the above recommendations are implemented h i g h l y a c c u r a t e q u a n t i t a t i v e measurements of the source d i s t r i b u t i o n should be p o s s i b l e . 66 REFERENCES ( A l p h a b e t i c a l l y a c c o r d i n g to the f i r s t author. A c i t a t i o n preceded by a "*" i n d i c a t e s an e x p e r i m e n t a l work p u b l i s h e d s i n c e 1963.) * J . 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Fuchs, "Space C o r r e l a t i o n s of the F l u c t u a t i n g Pressure i n Subsonic Turbulent J e t s , " J . Sound Vib . 23(1), pp. 77-99, 1972. Ian S.F. Jones, " S c a l e s P e r t i n e n t t o Noise P r e d i c t i o n from the F i r s t Few Diameters of a J e t , " AFOSR-UTIAS Symposium on Aerodynamic Noise 1968, Book of A b s t r a c t s , p. 8. R.H. Kraichnan, "Pressure F i e l d w i t h i n Homogeneous A n i s o t r o p i c Turbulence," J . Acoust. Soc. Am., 28(1), pp. 64-72, January, 1956. * J.C. Lau, M.J. F i s h e r & H.V. Fuchs, "The I n t r i n s i c S t r u c t u r e of Tu r b u l e n t J e t s , " J . Sound Vib . 22(4), pp. 379-406, 1972. J.C. Laurence, " I n t e n s i t y , S c a l e and Spectra of Turbulence i n Mixing Region of Free Subsonic J e t , " Lewis F l i g h t Prop. Lab. Report # 1292, 1957. * H.K. Lee, " C o r r e l a t i o n of Noise and Flow of a J e t , " UTIAS Rep. 168 (1971); a l s o AFOSR-TR-71-2572 (1971). * H.K. Lee and H.S. 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Sound Vib. 17 (1) , pp. 1-4, 1971. * P.A. Lush, "Measurement of Subsonic J e t Noise and Comparison with Theory," J . F l u i d Mech. 46(3), pp. 477-500, 1971. * L. M a e s t r e l l o S E. McDaid, " A c o u s t i c C h a r a c t e r i s t i c s of a High Subsonic J e t , " AlAA Paper No. 70-234, Jan. 1970. W.C. Meecham 6 G.W. Ford, " A c o u s t i c R a d i a t i o n from I s o t r o p i c Turbulence," J . Acoust. Soc. Am. 30(4), pp. 318-322, A p r i l 1958. * E. M o l l o - C h r i s t e n s e n , M.A. K o l p i n , and J.R. M a r t u c e l l i , " E x p e r i ments on J e t Flows and J e t Noise Far F i e l d S pectra and D i r e c t i v i t y P a t t e r n s , " J . F l u i d Mech. 18, pp. 285-301, 1964. * E. M o l l o - C r i s t e n s e n , " J e t Noise and Shear Flow I n s t a b i l i t y Seen from an Experimenter's Viewpoint," ASME J . of Appl. Mech. 34, pp. 1-7, 1967. * R.C. P o t t e r and J.H. Jones, "An Experiment to Locate the Acoust i c Sources i n a High Speed J e t Exhaust Stream," A b s t r a c t , 74th Meeting of the Acoust.Soc.Am., JJ4, 1967. A. Powell, " S i m i l a r i t y C o n s i d e r a t i o n s of Noise P r o d u c t i o n from T u r b u l e n t J e t s , Both S t a t i c and Moving," Douglas A i r c r a f t Company Rep. SM-23246 (July 1958) ; a l s o abridged i n J.Acoust.Soc.Am. 31, pp. 812-813 (1959). A. P o w e l l , "Aerodynamic Noise and the Plane Boundary," J.Acoust.Soc.Am. 32, p. 982, 1960. A. Powell, "Theory of Vortex Sound," J.Acoust.Soc.Am. 36, 177-195 , 1964. I. Proudman, "The Generation of Noise by I s o t r o p i c Turbulence," Proc.Roy.Soc. A, V o l . 214, p. 119, 1952. * R. Rackl and T.E. Siddon, " J e t Noise Study Using Image Technique," CANCAM 1971, C a l g a r y , Proceedings p. 473. R. Ra c k l , " C o r r e l a t i o n F u n c t i o n P r o c e s s i n g with F o u r i e r Transform," Dept. Mech. Eng., UBC, May 1972a. * R. R a c k l , " F u r t h e r S t u d i e s on Cross C o r r e l a t i o n s between F l u i d D i l a t a t i o n s and Flow Noise," 84th Meeting of Acoust. Soc. Am., paper XX8, 1972b. H.S. Ribner, "On the Str e n g t h D i s t r i b u t i o n of Noise Sources Along a J e t , " UTIA Rep. 51 ( A p r i l 1958); a l s o abridged i n J . Acoust. Soc. Am. 30, p. 879 (1958). 69 H.S. Ribner, "Aerodynamic Sound From F l u i d D i l a t a t i o n s , " UTIA Rep. No. 86,July 1962. Also AFOSR IN 3430. H.S. Ribner, "The Generation of Sound by Turbulent J e t s , " Advances i n Appl i e d Mechanics, V o l . 8, Academic Press Inc., New York, 1964. * T.D. Scharton and P.H. White, "Simple Pressure Source Model of J e t Noise," JASA V o l . 52, No 1 (Part 2) J u l y 1972, p. 399. * L.K. Schubert, "The Role of J e t Temperature and Sound Source P o s i t i o n i n R e f r a c t i o n of Sound from a P o i n t Source Placed i n an A i r J e t , " M.A.Sc. T h e s i s (unpublished), OTIAS, 1965. W.R. Sears, "Some Aspects of No n s t a t i o n a r y A i r f o i l Theory and i t s P r a c t i c a l A p p l i c a t i o n , " J . Aeron. Sc. 8(2), p. 104, February 1941. * T.E. Siddon, "On the Response of Pressure Measuring I n s t r u m e n t a t i o n i n Unsteady Flow," UTIAS Rep. No. 136, January 1969. Also AFOSR 68-2466. * T.E. Siddon, " C o r r e l a t i o n Study of Surface I n t e r a c t i o n N c i s e , " Trane Co. Research Dept. Report, La Cro s s e , Wise. , Dec. 1970. T.E. Siddon, " F l u c t u a t i n g Pressure Probe/Design and C a l i b r a t i o n , " B o l t , Beranek and Newman Report, J u l y 1971a. * T.E. Siddon 8 R. R a c k l , " C r o s s c o r r e l a t i o n A n a l y s i s of Flow Noise with F l u i d D i l a t a t i o n s as Source F l u c t u a t i o n " , H12, 82nd Meeting of ASA, Denver 1971. T.E. Siddon, "New C o r r e l a t i o n Method f o r Study of Flow Noise", 25 N 12, 7th ICA, Budapest 1971b. * T.E. Siddon, "Surface D i p o l e S t r e n g t h by Cross C o r r e l a t i o n Method", JASA V o l . 53 No.2, pp.619-633, Feb. 1973. Also a v a i l a b l e as Siddon 1970. 1973a. T.E. Siddon, "Noise Source D i a g n o s t i c s Using C a u s a l i t y C o r r e l a t i o n s , " a b s t r a c t submitted f o r S p e c i a l i s t s ' Meeting •Noise Mechanisms', AGARD F l u i d Dynamics P a n e l , Belgium, Sept., 1973b. A.M.O. Smith & J.L. Hess, " C a l c u l a t i o n of P o t e n t i a l Flow about A r b i t r a r y Bodies," Prog. Aeron. Sc., V o l 8, pp.1-138, 1967. R.H. Smith & C.T. Wang, " C o n t r a c t i n g Cones G i v i n g Uniform Throat Speeds," J.Aeron.Sc., Oct. 1944, p. 356. H. Tennekes 8 J . L. Lumley, "A F i r s t course i n Turbulence," The MIT Press, Cambridge, Massachusetts, and London, England, 1972. * R.B. Webster, " J e t Noise S i m u l a t i o n on Shallow Water," J . F l u i d Mech. 40(2), pp. 423-432, 1970. 7 0 APPENDICES 71 Appendix A Mathematical D e t a i l s Concerning Eg_._J4 .,7]_ Through Eg . J 4 . J . l L i Eg. (4 .7) i s repeated: where [ , . . ] ^ means: Replace the argument (t) by argument-|x-y|/c 0 ( t - | x - y | / c D ) . p (t) can be taken under the i n t e g r a l s i g n because independent, i n t e r c h a n g e a b l e o p e r a t i o n s . Dnder the assumption of s t a t i s t i c a l s t a t i o n a r i t y the time averages are f u n c t i o n s of the time delay r o n l y c The l e f t hand s i d e of Eg c (4 e 7)_ becomes the a u t o c o r r e l a t i o n of the f a r f i e l d sound pp ( r ) . The r i g h t hand s i d e can be put i n t o a more convenient form: p(t)p(t+r) = - 4TTC,fx (A.1) i t i s independent of y. Time ave r a g i n g and i n t e g r a t i o n over V are Consider: d d dr at ~ a r dt Put: t'=t+r -> [ p C 6 5 ( t , - r ) ] r p ( f ) (A. 2) where [...]£ again means: r e p l a c e the argument by 72 argument* | x-yj / c 0 ( f = T + | x-j£| / c Q ) . Eg. (4.7) t h e r e f o r e becomes i n d i f f e r e n t i a l form: app(r) I r a - av - V - [f P ( 0 3 P ] ^ CA.3) T T C „ X L dr J T /.-.. e,Z7rvTdr = ?{ ... } (A.4) Now take the F o u r i e r Transform 00 / - C O of Eq. (A.3). The r i g h t hand s i d e becomes: CO r.h.s. = — V - f\4- P C ° ' P L e i 2 ^ T d r (A.5) 4 T T C 2 X L 3 T J ? By p a r t i a l i n t e g r a t i o n : CO u . — l 2 m / f [ COD 1 \2ttvt . ,„ £. r.h.s = g — / | p " " p ] A e d r ( A . 6 ) 4 * * C o * -co T By d e f i n i t i o n , the r e a l p a r t of the F o u r i e r transform of the a u t o c o r r e l a t i o n app(r)/av i s the power s p e c t r a l d e n s i t y . T h e r e f o r e , t a k i n g the r e a l p a r t on both s i d e s r e s u l t s i n the s p e c t r a l d e n s i t y of p per u n i t volume: where T { . . . } = TC{ • • • } • i-% {• • • }» 3^ = c o s i n e transform, 3^= s i n e transform. When the F o u r i e r t r a n s f o r m i s . estimated by a numerical method i t i s u s u a l l y necessary t o c o r r e c t the r e s u l t 73 ("smoothing") because a numerical i n t e g r a t i o n cannot be extended t o i n f i n i t y . In the present case, however, the c o r r e l a t i o n f u n c t i o n s p<o>p decrease s u f f i c i e n t l y r a p i d l y to zero w i t h i n the le n g t h of the r e c o r d a v a i l a b l e so th a t such a c o r r e c t i o n was not made. -74 Appendix B N o n d i m e n s i o n a l i z a t i o n of C o r r e l a t i o n F u n c t i o n s There are two reasons f o r n o n d i m e n s i o n a l i z i n g data: 1) F l u c t u a t i o n s i n ambient c o n d i t i o n s . w i l l have l e s s degradating e f f e c t on the f i n a l data. 2) R e s u l t s are more g e n e r a l and e a s i e r to compare with r e s u l t s of other workers. Lengths are measured i n g e n e r a l i n meters [m], sometimes i n i n c h e s [ i n ] , p r e s s u r e s i n N/m2 (N=Newton), time i n m i l l i s e c o n d s [msec], and v o l t a g e s i n V o l t s [ V ] . B.1 C a u s a l i t y C o r r e l a t i o n Technique The s i g n a l flow i s shown i n Fig.4-19. The c o r r e l a t o r computes e^eg (T), where e A and eg are the v o l t a g e s a p p l i e d to the two i n p u t channels; the dimension of e A e g i s [ V 2 ] . The output C c ( T ) of the c o r r e l a t o r c o n s i s t s of v o l t a g e s the v a l u e s o f which are n u m e r i c a l l y i d e n t i c a l to the v a l u e s of e^eg. The s e n s i t i v i t y S c of the c o r r e l a t o r i s t h e r e f o r e S c = 1[1/V], What we are i n t e r e s t e d i n i s p<°>p. T r a c i n g through the e l e c t r o n i c s we get: e A ( t ) = RC G C A G M A S A p<o>( t) eg(t) = G Cg G Mg Sg p(t) where: RC = time constant of the d i f f e r e n t i a t o r [msec], 75 G Q = i n p u t g a i n of c o r r e l a t o r , G|Y| = g a i n of measuring a m p l i f i e r , S = microphone s e n s i t i v i t y [ V /N/m 2]. C c ( T ) i s recorded on an x - y - p l o t t e r ( s e n s i t i v i t y S x y [ V / i n ] ) , the p l o t t e d f u n c t i o n C (T) i s measured i n i n c h e s . Then: S x y C ( T ) = e A e Q S c , p ™ p ( r ) = S . „ C ( T ) < S A 6 M A R C G C A " S B G M B G C B ) S C N o n d i m e n s i o n a l i z a t i o n i s e f f e c t e d by d i v i d i n g the f a r f i e l d a c o u s t i c pressure by 2 (0.5^0-2) M 2D/x Q, the hydrodynamic pressure f l u c t u a t i o n by 0.5/3U2, time and time d e l a y by D/U Q. Using the second form of Eg. (4.9), Eg. (4.8) i s nond i m e n s i o n a l i z e d : p p ( r ) { 2 ( 0 . 5 ^ U 2 ) M 2 D / x o } 2 - PC0> (Uo/D) 2 r r a a ospu? P -I 3 4 T T C O 2 X o 2 M 2 D / X O Jy L a ( r ^ ) a(tuyD) { 2 ( 0 .5/>U 2)M 2D/x o } - y The nondimensional i n t e g r a n d [...]» i s e x t r a c t e d from the measured c r o s s c o r r e l a t i o n data. 76 B.2 Image Technigue The s i g n a l flow i s n e a r l y the same as i n Fig.4-19; we s u b s t i t u t e a q u a r t e r i n c h microphone f o r the f o i l p ressure sensor and omit the d i f f e r e n t i a t o r . Then: , , S x y C M p p( T ) = 1  S S A GMA GCA S B G M B G C B S C Sinc e i t was determined t h a t ps s c a l e s much the same way as p, p s i s n o n d imensionalized by the same q u a n t i t y (3.13) except that x G i s r e p l a c e d by h, the d i s t a n c e of the s u r f a c e from the j e t a x i s . Equation (3.10) becomes: 7 {2(0.5 PU o 2 ) M 2 D / x c } 4 cos 8 ( l l / D ) f \-A P§ P _ _ l d S J L.dfru/D) ?(0.5oU 2)M 2D/h 2(0.5 oM 2)M 2D/x k 4TTC OX h/xQ s (T  2 / 3U o 2) 2D/h 2(0.5 / 0U 2 )M 2D/ H = 0.325 ( t y p i c a l ) , 8 - 55°, h/D = 5.17. (See Fig.B-1.) The nondimensional i n t e g r a n d [ ... ] ^ was e x t r a c t e d from the measured c r o s s c o r r e l a t i o n data by a computer program. The i n t e g r a t i o n was done by hand. 77 Appendix C Of The Shajse Of The Cross C o r r e l a t i o n Function^ The procedure i s s i m i l a r to the one o u t l i n e d by Siddon (1970, 1973a). Eg. (4.6) i s r e w r i t t e n : - Pc"(x,t') = - ~ i r - ' / ^ ( y ' . t ' - r ^ ) d 3 y ' ( c . i , Eg. (4.7) was o b t a i n e d by m u l t i p l y i n g both s i d e s of Eg. (4.6) by p ( x , t ) . Here, we are i n t e r e s t e d i n p r e d i c t i n g the shape of the f u n c t i o n P ( 0 ) p C n , so we m u l t i p l y Eq. (C. 1) by pt o > (j, t - r / c Q ) and time average. On the r i g h t hand s i d e the f a c t o r can be taken under the i n t e g r a l s i g n because the i n t e g r a t i o n i s with r e s p e c t to j ' : p c w ( y , t - r / c o ) p ( x , t ' ) = ( C 2 ) / p C W ( y , t - r / c J p C 0 V , t ' - r ' / c o ) d 3/ 4TTC O 2X V. Retarded time d i f f e r e n c e s due to r#r' can be neglected i n low speed f l o w s , s i n c e the a c o u s t i c wavelengths are g e n e r a l l y l a r g e compared with c o r r e l a t i o n s c a l e s ( L i g h t h i l l 1962), i . e . , we s e t r=r«. Assuming t h a t a l l v a r i a b l e s are s t a t i o n a r y random f u n c t i o n s and with T=t'-t we get: 78 p c 0 )(y,t-r/c o) p(x,f) = p""p (x,yir+r/c 0) CO). (C.3) p^(y,t-r/c 0) pC05(y' tt'-r>co) = pc0> p C 0 ?'(C ,y,r) , u , _ dr where £=y' ~ Y. Hence p < 0 , p U + r/c.) = - T - 1 ! - f-jj- P C 0 5 p C O 5 , ( C , y , T ) d 3 £ ( c . 5 ) 477"C0 x v 0T We s e l e c t a f u n c t i o n a l form f o r the c o r r e l a t i o n p * 0 ^ * 0 * 1 which e x h i b i t s the property of c o n v e c t i n g decay as space s e p a r a t i o n and time delay are i n c r e a s e d : p c o y o ) . = pco>2 f l ^ - u ^ J / L ^ / L ^ ^ / L ^ r / T } (c.6) (the 1 - d i r e c t i o n i s the d i r e c t i o n of motion f o r the t u r b u l e n c e s t r u c t u r e with c o n v e c t i o n v e l o c i t y U c ) . The f u n c t i o n f has the p r o p e r t i e s : (1) f (|=0, T=0) = 1 (2) f # T ^ > 0 f o r arg —> 00 , s u f f i c i e n t l y r a p i d l y f o r o a r g a l l i n t e g r a l s to converge. (3) | f (arg=0) | > | f (| arg | >0) | (4) f (arg) = f (-arg) 79 (5) f and are continuous, d org where arg stands f o r any of the 4 arguments of f . Now put Xj = C / L j > ^ T / T , a = U C T / L | . T i s some s u i t a b l e time s c a l e of no f u r t h e r concern here. *t i s a nondimensional time delay, not the r e a l time. Eg. (C.'6) becomes: p « V w = f ( x , - a t , x 2, x 3, t) (C.7) The c o n v e c t i o n of the t u r b u l e n c e i s allowed f o r i n the X| - d i r e c t i o n which i s p a r a l l e l to the j e t a x i s . There i s a l s o some c o n v e c t i o n i n the x 2- and X j - d i r e c t i o n s due to spreading of the j e t ; t h i s e f f e c t i s very s m a l l and ne g l e c t e d here. Now c o n s i d e r : d t dr d t d r T at (C.8) d co) co}' I co} 2 d rf •>> Y\ - g - p t u V = p W J -jzz f ( x , - a t , x 2, x 3, t) (C.9) dr Put x,1 =x, - a t . In what f o l l o w s — — means p a r t i a l I I d org d i f f e r e n t i a t i o n with r e s p e c t to one of the 4 arguments of f , the o t h e r s being held c o n s t a n t . Then: 80 a f _ df_ dx]_ dj_ _dx^ dj_ d x ^ dj_ a r " ^ d T + tfx2 d T + </x3 d T + dT = - a ——•= + —773- (c. 10) tfx,1 d\ ddf d ( di ,df , , d , df ^di , a 2 . dtf dx] d\ d\ dx] d\ = a — T I . - 2 a + z ( C 11) I t i s mathematically convenient to assume that f i s separable: f = f,(x}) f 2 ( x 2 ) f 3(X 3) f 4 ( T ) (C.12) and d e f i n e : CO / f:dX; = F; -CO (C.13) Then : 81 ,2 L, U U F0 F, p c w p ( r + r / 0 = - p ( 0 ) H ; 2 ? ' 2 ' 5 X 4TT C„ x T ' V f , . . d f . °R d f, . d 2 f , 0 9 x { « 2 ^ / ^ dx, - 2 a ^ f / ^ j - dx,+ ^ / f t f j d x , } ( C U ) 1 - c o d x . ' d t - i ) d x 1 . ' d T -co' ' ' J The f i r s t i n t e g r a l i s zero by property (2) and (5). Because of p r o p e r t y (4) the second i n t e g r a l v a n i s h e s . With dx (=dx^ f i n a l l y 7 V r + r / 0 = - P L '  LlL 3 F 2 T F | ( C I S ) 4-TTC^XT* d t * p<°>p i s independent of c o n v e c t i o n f o r any f| s a t i s f y i n g p r o p e r t i e s ( 1 ) - ( 5 ) . I t depends very s t r o n g l y on T and the f u n c t i o n £4 which together d e s c r i b e the decay p r o p e r t i e s of the t u r b u l e n c e . For a nondecaying f r o z e n convected p a t t e r n £4 i s a c o n s t a n t ; no sound would be r a d i a t e d . For a flow to produce a c o u s t i c output i t i s necessary that p a t t e r n s are generated and then decay. I f we make the r a t h e r s p e c i a l assumption of a convected Gaussian f o r f : f = exp{-(x,-aT) 2 -xz - x f -T 2} (C.16) CO CO CO / f , d x , = / f 2 d x 2 = / f 3 d x 3 = J¥ (C.17) -CO -CO -CO 82 The t o t a l area under e~^ 2(1-2T 2) i s zero. T h i s means there i s no dc-component i n p<o>p as one might expect s i n c e p i s an a c o u s t i c p r e s s u r e . T h i s i s true f o r any f ^ s a t i s f y i n g p r o p e r t i e s ( D - ( 5 ) . The shape of the c r o s s c o r r e l a t i o n p<o>p can now e a s i l y be p r e d i c t e d : p - p = _ A p - p = _ £ pco> p £ T j T p c o ) 2 L, L 2 L 3 2 e - T 2 ( 3 T _ 2 T 3 } { C - i g ) 2 c c 2 x T Fig.4-24 shows a t y p i c a l measured c r o s s c o r r e l a t i o n p f C ) p and f i t t e d to i t by l e a s t squares a f u n c t i o n c f the type A e ( T> { 3 ( f ) - 2 ( f ) 3 } (C.20) I t appears t h a t the convected Gaussian f o r f i s a good assumption as the agreement i s good. Knowing the a p p r o p r i a t e c o e f f i c i e n t s from the f i t t e d f u n c t i o n one cou l d e m p i r i c a l l y estimate the r a d i a t i o n from u n i t volume. The F o u r i e r transform (equal to the c r o s s s p e c t r a l d e n s i t y ) could be obtained i n c l o s e d form without r e c o u r s e to numerical transform techniques. 83 Appendix C Probe Contamination Ratio When a probe i s i n s e r t e d i n t o a t u r b u l e n t flow the i n t e r a c t i o n of the flow with the probe s u r f a c e c r e a t e s n o i s e of a d i p o l e nature. I t superposes i t s e l f on the guadrupole sound r a d i a t e d from the flow i t s e l f . For the c r o s s c o r r e l a t i o n measurements u s i n g the c a u s a l i t y p r i n c i p l e i t i s necessary to minimize t h i s d i p o l e r a d i a t i o n . The reader may r e c a l l that d i p o l e s are more e f f i c i e n t e m i t t e r s than guadrupoles of s i m i l a r s t r e n g t h , i . e . , even a s m a l l probe s u r f a c e may be a b l e to c o n s i d e r a b l y d i s t o r t a c o u s t i c measurements. The probe contamination r a t i o C to be d e f i n e d here i s a number by which the a c o u s t i c a l performance of a probe i n a t u r b u l e n t flow with given p r o p e r t i e s can be e v a l u a t e d . D.1 R a d i a t i o n from One Coherent Eddy with c o r r e l a t i o n volume V c (no probe i n the f l o w ) . I t f o l l o w s from Eg. (4.15) and Eg. (4.23) t h a t : D. 2 Probe I n t e r a c t i o n Noise.. Assuming that the probe does 84 not move and that shear s t r e s s e s r a d i a t e n e g l i g i b l e sound (see s e c t i o n D.5), and using Eg.(3.2) and Eg.(3.5), only the second i n t e g r a l of Eq. (1.2) i s l e f t . I t becomes i n the f a r f i e l d : p. = - i f c V H p s ] ^ where the s u b s c r i p t "d" i n d i c a t e s d i p o l e type r a d i a t i o n , /? i s the angle between the normal t o the s u r f a c e S and the d i r e c t i o n of r a d i a t i o n , and p g i s the s u r f a c e normal pressure. A f t e r squaring and time averaging, and s e t t i n g cos/3 = 1 f o r the worst case, Eq. (D.2) becomes: I f the process i s s t a t i s t i c a l l y s t a t i o n a r y and the s u r f a c e i s s m a l l compared with a t y p i c a l wavelength, the brackets [..-] i n d i c a t i n g e v a l u a t i o n a t r e t a r d e d time can be dropped i n t h i s c a s e. Assuming t h a t p s i s w e l l c o r r e l a t e d ever the s u r f a c e of the probe ( i . e . i t i s at l e a s t c o n t a i n e d i n a c o r r e l a t i o n volume), the i n t e g r a l s i n Eg.(D.3) r e p r e s e n t nothing e l s e but the mean squared d e r i v a t i v e of the f l u c t u a t i n g l i f t f o r c e exerted on the probe. We can use unsteady a i r f o i l theory to estimate the l i f t L i f the d i s t u r b a n c e v e l o c i t i e s are s m a l l compared to the mean flow U, and i f the probe shape comes c l o s e to a s l e n d e r body shape. The requirements are not too s t r i n g e n t as the probe contamination r a t i o i s onl y a rough estimate designed to gain some f e e l i n g f o r the amount of d i p o l e " p o i s o n i n g " . C L i s the unsteady l i f t 85 c o e f f i c i e n t , a i s the in s t a n t a n e o u s angle of a t t a c k : L ( t ) 0 . 5 y O U 2 S p C L 0 . 5 / ) U 2 S p - ^ - a ( t ) (D.4) d C I t i s w e l l known t h a t the l i f t curve slope L- depends on the da wavenumber of the i n c i d e n t d i s t u r b a n c e [ S e a r s (1941) f u n c t i o n , see a l s o Liepmann' 1952, p. 796 ; and- see Fig.4-25]. Most probes can be g e n e r a l l y regarded as low aspect r a t i o f i n i t e wings f o r d Ci which the v a r i a t i o n of L with the wave number i s only moderate, as i n d i c a t e d t y p i c a l l y by the dashed l i n e i n Fig.4-25. k value of 2 i s t h e r e f o r e used f o r the l i f t curve s l o p e i n the subseguent c a l c u l a t i o n s . With a(t) ^ v ( t ) / U (C.5) (v i s the component of the v e l o c i t y d i s t u r b a n c e p e r p e n d i c u l a r to U i n the plane t h a t c o n t a i n s the d i r e c t i o n of r a d i a t i o n ) Eg.(D.4) becomes a f t e r d i f f e r e n t i a t i n g with r e s p e c t to time, s g u a r i n g , and time a v e r a g i n g : which i s an estimate of the product of the two i n t e g r a l s i n Eg. (D.3). T h e r e f o r e : pZ ~ , ^ 2 2 2 ( D * 7 ) d 16 7T 2 c 2 x 2 D.3 D e f i n i t i o n of Probe Contamination Ratio.. We now d e f i n e the probe contamination r a t i o as the r a t i o between the mean squared a c o u s t i c pressure due to d i p o l e type r a d i a t i o n from the probe s u r f a c e to the mean squared a c o u s t i c pressure due to guadrupole type r a d i a t i o n from the adjacent eddy of f l u i d : ' C = p j / p ^ ' (D.8) 86 Combining Eq.(D.1), Eq. (D.6) , and Eq. (B.7): , _ U _ f / p c 2 S P x- c (you/) 2c .2 -2 C 0 For low speed flows the d e n s i t y p can be regarded constant and then c a n c e l s out i n Eg. (D.9). D.4 Estimates f o r a Low Speed Turbulent J e t T h e p o i n t 4 diameters downstream and h a l f a diameter from the a x i s i s chosen as r e p r e s e n t a t i v e . There, the mean v e l o c i t y U i s about 0.6U o, where U „ i s the e x i t v e l o c i t y . I t i s assumed that v r m ( ? — u r__. — ° 1 rms rms 0.15U q, and th a t d i f f e r e n t i a t i o n with r e s p e c t to time can be r e p l a c e d by m u l t i p l y i n g by 2irv m 27r0.2U/L c, where L c i s the c o r r e l a t i o n l e n g t h , and i s qiven bv L_ — 0.13x (x i s measured i n diameters from the e x i t ) (from Laurence 1957). I f the c o r r e l a t i o n volume i s imagined as an e l l i p s o i d with minor axes 1/3 of the major then V c— L£. T h e r e f o r e Eq.(D.9) becomes: /0.6U o\2 4 S D 2 (277-i/)2 V 2 C - (-7-0) <V 5_ — = - (D.10) A l s o (from Siddon 1969): v* ̂  2v"2 (D.11) (note t h a t f o r a Gaussian: v~* = 3v* ) S 2 c 4 " C * ( 0 . 6 M ) 2 \ ^ T = L ° (277-0.2 0.6U o /L c ) 2 2(0. l5U o ) 2 14 / Sp (-tV) M 2 * L c I f we d e s i r e t h a t C be l e s s than 0.1 f o r good accuracy, we must 87 r e q u i r e t h a t For x=4D, L. —D/2. Then we need S p < D2M/48 e In the experiments d e s c r i b e d i n t h i s t h e s i s : D=38.1 mm, M=0.32: S p < 9.7 mm2 (D.13) For c y l i n d r i c a l probes S p can be r e p l a c e d by an " e f f e c t i v e " area because the pressure f l u c t u a t i o n s due to the t u r b u l e n t c r o s s v e l o c i t i e s are c o n c e n t r a t e d near the nose of the probe as shown i n Fig.4-26. Assuming t h a t S p i s egual to 2 d 2 (where d i s the diameter of the probe) d would have to be 2.2 mm or l e s s , a very s m a l l s i z e f o r a dynamic pressure sensor. In a d d i t i o n , the 90° bend downstream of the s e n s i n g elements would c r e a t e a d d i t i o n a l n o i s e by flow s e p a r a t i o n and vortex shedding. The pressure probe d e s c r i b e d i n s e c t i o n IV of t h i s t h e s i s has a c r o s s s e c t i o n a l area of about 10 mm2, thus b a r e l y s a t i s f y i n g Eq. (D. 13). However, the c a l c u l a t i o n s here are done f o r x=4D. C l o s e r to the j e t e x i t the requirements become more s t r i n g e n t and the present probe may be r a d i a t i n g a s i g n i f i c a n t amount of d i p o l e n o i s e . D.5 Is the Skin F r i c t i o n Indeed Unimportant f o r the F o i l Probe? The r a d i a t e d sound due to s k i n f r i c t i o n or shear s t r e s s i s o b t a i n e d by c o n s i d e r a t i o n s s i m i l a r to the ones a r r i v i n g at Eq. (D.3) : 88 P.r = ,, 2 ' 2 2 I [+] d S / , [ T ] ' d S ' ' " " " I 16 7r c Q x s s [from the t h i r d term i n the second i n t e g r a l of Eg, (1.2) ] Tinder the same assumptions as f o r the l i f t f o r c e case the 2 i n t e g r a l s r e s u l t i n W2, the mean square d e r i v a t i v e of the drag f o r c e W. The Reynolds number Re i s based on the chord l e n g t h of the probe b=9.85 mm, the mean v e l o c i t y U=0.6Uo= 66 m/sec, and the kinema t i c v i s c o s i t y of the a i r at 20°C z>=14.9*10~6 m 2/sec: Re = bU/i/ = 4.36*10* < 5*10^ = Re f r (D. 15) where Re t r i s the Reynolds number f o r t r a n s i t i o n , i . e . the boundary l a y e r i s laminar. For a rough estimate of the quasi steady drag f l u c t u a t i o n we use B l a s i u s ' Law f o r the laminar s k i n f r i c t i o n c o e f f i c i e n t c w , although i t i s s t r i c t l y a p p l i c a b l e only t o a f l a t p l a t e : c w = 1.328//Re~ (D. 16) then, the i n s t a n t a n e o u s drag f o r c e becomes: W=CWS 0, 5/o (D + u) 2= 1 3 2 8 S 0.5/o(U2 + 20u+~0) (D. 17) f v Re pf where Sp^ i s now the upper s u r f a c e area of the f o i l ; u i s the v e l o c i t y f l u c t u a t i o n i n the mean flow d i r e c t i o n . Eg. (D.17) becomes a f t e r d i f f e r e n t i a t i n g , s q u a r i n g , and time a v e r a g i n g : -89 (1.328 S poU)^ — W 2 - f U 2 (D.18) Re We now d e f i n e the probe contamination r a t i o f o r s k i n f r i c t i o n f l u c t u a t i o n s by analogy to Eg. (D.8) : C f r = H /Pvc" ( D ' 1 9 ) Combining Eg. (D.19) and Eg. (D.1): _ (P^f u2 JL76_ 2 Q c f r - IcJ I v c i ; — R e ( D ' 2 0 ) Assuming u 2 — v"2, using Eg. (D.9) , and p u t t i n g S p f — IOSp f o r the present probe: C„ - C ( D . 2 1 , Eg. (D.21) shows that the probe contamination r a t i o f o r s k i n f r i c t i o n i s very much s m a l l e r than the one f o r sound r a d i a t e d due to f o r c e s normal to the s u r f a c e . Shear s t r e s s e s are t h e r e f o r e unimportant i n the d i p o l e n o i s e c a l c u l a t i o n s a f f e c t i n g the present f o i l type pressure sensor. S i n c e C r̂ i s about 2 o r d e r s of magnitude s m a l l e r than C, d i p o l e noise due to shear s t r e s s e s i s probably h a r d l y ever of any importance i n low speed t u r b u l e n t a i r flo w s . 90 F i g . 3 - 1 Image geometry 91 F i g . 3 - 2 Nondimensional spectrum o f j e t no i se at 45 to j e t a x i s , (a) i s o l a t e d j e t , (b) w i th su r face S 1 behind the j e t . (computer p l o t ) 92 F i g . 3 - 3 M i c r o p h o n e m o u n t i n g . 1: % - i n c h B r u e l & K j a e r m i c r o - phone, 2: b r a s s s l e e v e , 3: % - i n c h t h i c k p l e x i g l a s s p a n e l . 93 Fig. 3-4 T y p i c a l c ross c o r r e l a t i o n f u n c t i o n between su r f ace pressure p and f a r f i e l d p ressure p. D e r i v a t i v e to be e v a l u a t i d a t T= r / c 0 . No t i ce p recu rso r to the l e f t . F i g . 3 - 5 R e l a t i o n s h i p between s u r f a c e p r e s s u r e p $ and e x i t v e l o c i t y U Q. H = s e t t l i n g chamber p r e s s u r e i n meters o f w a t e r . U1  20 IS 12 8 4 F i g . 3 - 8 Equal s o u r c e s t r e n g t h c o n t o u r s f o r 6=60° VO, 98 far field F i g . 3 - 9 Zone o f I n f l u e n c e . A n g l e o f a p e r t u r e o f t h e dashed cone n o t known. 99 F i g . 3 - 1 0 3 t y p i c a l p ^ p ( x ) a t d i f f e r e n t downstream p o s i t i o n s x. D i s t a n c e j e t a x i s t o s u r f a c e i s 5.17D. p s measured d i r e c t l y o p p o s i t e o f j e t a x i s , p measured i n f a r f i e l d a t 45° t o j e t a x i s , (computer p l o t )  F i g . 3 - 1 2 " S l i c e w i s e " i n t e g r a t e d s o u r c e s t r e n g t h f o r 9=45° 102 F i g . 4 - 1 E x n e r i m e n t a l s e t u o o f c a u s a l i t y c o r r e l a t i o n t e c h n i q u e . P r e s s u r e s e n s o r measures p(°)(t), microphone i n t h e f a r f i e l d measures D U ) ( t ) . . I) = e x i t v e l o c i t y . origin F i g . 4 - 2 Geometry uu(i) 4 space separation £ F i g . 4 - 3 D e f i n i t i o n o f c o r r e l a t i o n l e n g t h L 104 F i g . 4 - 4 C l a s s i c a l c y l i n d r i c a l probe c o n f i g u r a t i o n . Upper p a r t : p r e s s u r e d i s t r i b u t i o n due t o c r o s s f l o w , f u l l l i n e : 40 < Reynolds number < 40000, dashed l i n e : p o t e n t i a l f l e w . Lower p a r t : V i n s t a n t a n e o u s v e l o c i t y v e c t o r , V a x i a l component, V c r o s s f l o w component, a n 105 F i g . 4 - 5 T y p i c a l c r o s s c o r r e l a t i o n between f l u c t u a t i n g p r e s s u r e i n t h e j e t p(°) and a c o u s t i c p r e s s u r e p, c o n t a m i n a t e d by d i p o l e n o i s e : T u r b u l e n c e i n d u c e s a f l u c t u a t i n g l i f t on t h e nose o f t h e c y l i n d r i c a l probe s e n d i n g o f f a d i - p o l e p u l s e . Dashed l i n e : e f f e c t o f s h o r t e n i n g nose p i e c e . P r i n c i p l e o f a r r a n g i n g f o i l probe i n t h e f l o w ( d i s t a n c e s and s i z e s n o t p r o p o r t i o n a t e ) : P l a n e o f f o i l c o n t a i n s mean f l o w d i r e c t i o n and d i r e c t i o n o f r a d i a t i o n toward f a r f i e l d m icrophone ( r i g h t ) . L o c a t i o n s o f p r e s s u r e s e n s i n g h o l e s on a i r f o i l : ( a) s u p e r s o n i c , (b) s u b s o n i c . The measured p r e s s u r e i s t h e average between t h e p r e s s u r e s on t h e upper and t h e l o w e r s u r f a c e . 108 Fig.4-8 S u b s o n i c a i r f o i l . Upper p a r t : f o i l c r o s s s e c t i o n and d i s t r i b u t i o n o f p r e s s u r e c o e f f i c i e n t due t o t h i c k n e s s . Lower p a r t : c p due t o t h i c k n e s s and a n g l e o f a t t a c k , c : upper s u r f a c e , c_,: l o w e r s u r f a c e . V o C o n t a m i n a t e d and u n c o n t a m i n a t e d c o r r e l a t i o n s and t h e i r F o u r i e r t r a n s f o r m s . p(°) d e t e c t e d a t x=3D, r=0.4D; p measured a t 45 t o j e t a x i s . C urve a: fi(°)p (p(°) measured by f o i l t y p e s e n s o r ) , c u r v e b D ( ° ) D (p(°) measured by 1 / 8 - i n c h m i c r o p h o n e w i t h s h o r t nose p i e c e . Curve c ? o u r ? e r t r a n s f o r m o f a , c u r v e d: F.T. o f b. T t i m e d e l a y , v f r e q u e n c y . F i g . 4 - 1 0 V e l o c i t y v e c t o r s on a i r f o i l probe  112 F i g . 4 - 1 2 D e t a i l e d d r a w i n g o f f o i l t y p e p r e s s u r e s e n s o r . 1 b a l s a l e a d i n g edge s e c t i o n , 2 s e n s i n g h o l e , 3 s t a i n l e s s s t e e l t u b i n g , 4 c o t t o n w o o l , 5 t e f l o n c o n n e c t o r , 6 1/8-inch B r u e l & K j a e r m i c r o p h o n e . 0.020" 0.010 -- -.2 -./ w- component ./ ^ .2 tana -0.010 0.005 v- component -0.010  J- F i g . 4 - 1 3 S t a t i c p r e s s u r e c a l i b r a t i o n s o f f o i l t y p e p r e s s u r e s e n s o r shown i n F i g . 4 - 1 2 . cq = {?m - P t)/(0.5pU£). v-component: a n g l e o f a t t a c k a i n the p l a n e o f t h e f o i l , w-component: a p e r p e n d i c u l a r t o p l a n e o f f o i l . 114 location of pressure tap F i g . 4 - 1 4 E s t i m a t e d p r e s s u r e d i s t r i b u t i o n o v e r f o i l w i t h d i p n e a r t h i c k e n i n g t r a i l i n g edge.  F i g . 4 - 1 6 Frequency r e s p o n s e o f f o i l t y p e p r e s s u r e s e n s o r . 117 compressor anechoic room S c h e m a t i c o f a u x i l i a r y equipment F i g . 4 - 1 8 V a r i a t i o n o f p r e s s u r e f l u c t u a t i o n w i t h d i s t a n c e from j e t , s t a r t i n g a t 4 d i a m e t e r s downstream i n t h e j e t and p r o c e e d i n g a t r i g h t a n g l e s t o the j e t a x i s . Channel A Channel B 119 preamplifieF p (o:>(t)[N/m2] p(t)[N/m2] ^ 2 1/8 "microphone i SA [v/N/m2] measuring amplif. gain 6MA differentiator, time constant RC CmsecJ SB [^N/m2J l measuring amplif. gain G^B i HP filter Cr(z)[v] input gains 'CA 'CB *A(t)[v] yB(»[v] CORRELATOR Sr[l/V] e7e^(v)LV2J A^B graphical deck of punched digitizer cards F i g . 4 - 1 9 S i g n a l f l o w   122 F i g . 4 - 2 2 a Some t y p i c a l n o n d i m e n s i o n a l i z e d c a u s a l i t y c o r r e l a t i o n f u n c t i o n s p ^ p a t h d i a m e t e r from t h e j e t a x i s , down- s t r e a m p o s i t i o n x v a r y i n g . P l o t t e d h i n c h above i s f i t t e d 7 t h o r d e r p o l y n o m i a l , ( c o m p u t e r p l o t ) 123 x = ID I O F i g . 4 - 2 2 b Some t y p i c a l c r o s s s p e c t r a ( S i n e t r a n s f o r m o f f u n c t i o n s shown i n F i g . 4 - 2 2 a , m u l t i p l i e d by f r e q u e n c y ) (computer p l o t ) 124 F i g . 4 - 2 3 Peak f r e n u e n c i e s v e r s u s downstream p o s i t i o n . F i g . 4 - 2 4 Measured c a u s a l i t y c o r r e l a t i o n p^ 'p ( c rosses ) and f i t t e d to i t the p r e d i c t e d f u n c t i o n . 126 k&(4 chordlengthsF' waven. k steady case F i g . 4 - 2 5 L i f t c u r v e s l o p e as a f u n c t i o n o f wave number (Se a r s f u n c t i o n ) '^^pressure distribution estimated effective chordlength ~2d F i g . 4 - 2 6 E f f e c t i v e c h o r d l e n g t h on c y l i n d r i c a l p r o b e . 127 -1 1 1 —t 1 g * — 0 0.5 1.0 1.5 vD/U0 F i g . 4 - 2 7 N o n d i m e n s i o n a l s p e c t r a l d e n s i t y o f t h e "pseudosound" p(°^ a t 4D downstream, %D from j e t a x i s , as measured w i t h f o i l t y p e s e n s o r . 128 F i g . 4 - 2 8 D i r e c t l y measured and i n t e g r a t e d s p e c t r u m a t 45 i n t h e f a r f i e l d . Comparison by shape o n l y . 129 Fig.B-1 Image geometry t o s c a l e

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