On surface pressure fluctuations in turbulent boundary layers

THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

ON SURFACE PRESSURE FLUCTUATIONS

IN TURBULENT BOUNDARY LAYERS

by

A p r i l , 19éO T H E O 0_ _L L E^_E O F A E R O N A U T I C S C R A N F I E L D On Siorface Pressvire F l u c t u a t i o n s i n * Turbiilent Boundary L a y e r s - b y « G. M. L i l l e y , M . S c , , D . I . C . , and T, H, Hodgson, B . S c , , D . L . C . , D.C.Ae. SUMvIARY

Ejdsting v/ork on ihe pressure fluctuaticns in turbulent shear flov7s is

briefly revievred vd-th special reference to the problem of \mll turbiolence, An approximate theory for the pressiore fluctuations on the wall under both a turbulent boundary layer and a v/all jet is given and indicates in the latter ca^e an dn.tensity many times that corresponding to tlie flow over a flat plate at zero pressure gradient, as typified by measurements on the

•nail of a \nxid tunnel. Experiments on a v/all jet confirtn these predictions

and details of the few preliminary data are presented,

The results from the v/all jet suggest that the intensity of the pressiire fluctuations in the regions of adverse pressure gradient, on ^Tings and bodies approaching and beyond separation will be higher than in regions of zero pressure gradient.

Appendices are included wiiich deetl \7ith the necessax^'' extensions to the analysis to fit the velocity correlation functions as measured by Grant (1958), the effects of time delay and eddy convection,

*

To be r e a d a t tlie AGARD qmiposium on Boxmdary Lajzer R e s e a r c h i n London, A p r i l 1960. T h i s work was performed vuader t h e M i n i s t r y of A v i a t i o n O o n t r a o t No. 7 / G E N / 1 6 6 V ^ R 3 .

Summary

List of Symbols

Introduction 1 Theory of the pressure fluctuations in a shear flovT 2

Pressure fluctuations due to the ' inner region'

shear flow 12 PDressure fluctuations due to the ' outer mixing

region' shear flow 15 The spectrum and space pressure correlation

for zero time delay 16 The wall jet ~ Theory of the mean flow 17

Apparatus 19 Preliminary RosixLts and Discussion 21

Future Work 22 Conclusions 22 References 23 Appendix A - The pressure equation for the wall jet 25

Appendix B - The evalijiation of an integral 2? Appendix C - The structure of the large eddies 30

near the V7all in a turbulent boundary layer

Appendix D ~ On space-time correlations of the 37 fluctuating velocity and pressure

Appendix E - The evaluation of the pressure 2{2 spectrum

A a m p l i t u d e

a - a , a amplitude of big eddy velocity covarianoes 0^,^71, /i-/? u^ skin friction coefficient

D jet diameter

f longitudinal velocity correlation coefficient (isotropic turbulence) G, G , G., G , G^ Green's functions

K Von Karman constant ^, •& , -6 ,1 scales of turbulence

' p ' 1 * 2

P pressure covariance

P pressi:ire spectrum function p pressure

'p* r.m.s, value of pressure fluctuations q. source density

R, 0, Z cylindrical polar co-ordinates

•^ii» -^.7. velocity covarianoe

5. ., IL velocity correlation coefficient r s (r^, r^, rj)

S area t time (•Up, UJ, U_) velocity components Up, UJ, iL mean velocity components u s ( u , u , u ) velocity

u maximum velocity at a given R u free stream velocity

VLj. shear v e l o c i t y Ü convection speed V volume X, y , z c o - o r d i n a t e s

Zi_, Z distance to mid and maximum v e l o c i t y i n the ^ outer region respectiveHy

a s^ vi^ /u?. , and anisotropy f a c t o r

a f a f a r e c i p r o c a l s of typical turbulence s c a l e s

6 Dirac function, bovindary l a y e r thickness 6 displacement thiclaiess

K wave number

II viscosity

V kinematic viscosityj reciprocal of eddy lifetime

C x' - X

p density r time delay

T, rp7 mean shear

T optimum time delay

r wall shear stress w

0) . ciroiüar frequency

( ), denotes big eddy contribution ( ) denotes sms.ll eddy contribution

{ ) denotes 'in moving frame of reference'

( ) denotes 'in fixed frame of reference'

in turbulent shear flows is required in a wide range of aeronautical and hydrodynamic problems today. Such problems range from aerodynamic noise generated by turbulent motion, the vibrations of the aircraft slcin at high speeds and the transmission of noise to the cabin and cockpit, to mention just a few.

Although work in this area was pioneered by Heisenborg (1948),

Obuklioff (1949), Batchelor (1951) and Kraichnan (I95éa a n d b ) , it is onily relatively recently that experimental data has been obtained to check the theoretical data, and to set the pattern for investigations into more

coniplicated situations, v/here the theory at the best would be very tentative. The work of Kraichnan is of particular interest for it deals v/ith pressure

fluctuations in the presence of a mean shear, and also in the case of Yrall

turbulence, and therefore has direct application to the problem of T K L H

p3:x3ssvare fluctuations under a tijrbulent boundary layer. (The earlier vrork of Heisenberg, Obiikhoff and Batchelor considered only tlie case of isotropic

turbulBnce) . Kraichnan showed that an a ¥;a,ll ^^/^ '^ Hn ~ '^^f» where /5

is a factor bebv/cen 2 and 12. Experimental results obtained by Willmarlii (1959) ajid Harrison (1958) confirmed lüraichnan's predictions and gave values of ;9 between 2.5 and 5.0.

Tho effect of Mach number on wall pressure fluctuations, although of obvious current inportance, will not be considered here. Indeed the flovY T/ill bo assumed incompressible throughout and the problem of boundary layer noise, that is the noise radiated away from the surface, vdll hardly be touched upon. Our attention \7ill mainly be restricted to problems' of v/all turbulence, including the wall jet, and will not consider in the same detail pressure fluctuations in free turbulence. A review paper covering the items omitted by tlie authors, would nat-urally be of considerable topical interest, but it was felt that, bearing in mind the considerable efforts present in this area today and the present state of flux of knowledge in the subject, a greater need was for a fundamental appreciation of relatively siniple flov7 models. The surface over which the fluid flows vd.ll be treated as rigid, and no account v/ill be given of the response of the structure to pressure fluctuations,

The theory of the pressuare flvictuations in Vira.ll turbulence, including tho wall jet, will be treated on similar lines to the method used by

Kraichnan 195éb), In tliis method the intensity of the pressure fluctuations can be obtained once the two-point velocity correlations, mean velocity gradient, and the turbiilence intensity and scale are known. Experimental results obtained by Townsend (1956) Lauf,er (1955) and Grant (1958) v/ill be used to find p^ on the wall. For the wall jet the nean flo\7 theoiy due to Glauert (1956; will be adopted, together vri-th results in the mean flov/ by BaMce (1957) and Bi-adshaw and Love (1959). The prediction in this case for p^ v/ill be compared ^vitll those obtained from measurements,

Finally in order to avoid confusion in the theoretical treatment and the discussion of resiilts, the authors stress that they are dealing with the problem of pressure fluctuations in a pseudo-incompressible turbulent shear flov/. They pixifer, and agree that it is one of personal preference and not one agreed by convention, to refer to boundary layer noise as the sound energy radiated av/av from tlie turbulent flow, Tliis latter problem considered by Curie (1955), Phillips (l95é) and Doak (l9éo) thougli connected has obvious differences from the present treatment.

2» ^Theoi:^ of the^ pressure ^fluctuations^,jji a .shear floyy

In most instances where the pressure fluctuations are significant the fluctuations in the fluid density are significant also. The present problem is no exception, Hov/ever since v/e v/ill be more concerned v/ith the pressure fluctuations inside the turbulent shear flow, v/hich in our problem v/ill bo an essentially 'low speed flow' , than the noise radiated from it, we may safely assume that the flow is incomproBsible, The equations of continuity and motion are therefore respectively

3u. 1

9xf

1 9pu. 0 (1) +

I f v/e take the tiine d e r i v a t i v e of ( l ) from the divergence of (2) v/e obtain tho follov/ing equation f o r tne p r e s s u r e d i s t r i b u t i o n

d^u. u .

' ^ P = . p ^ ^ ^ ^ (3)

Tliis shows t h a t , vdiereas i n i n v i s c i d steady flov/ the pressure a t a p o i n t folloivs immediately from uhe dynamic pressure a t the same p o i n t , the p3ressure a t a p o i n t i n a turbiilent f l a v , since i t obeys an equation of the Poisson t y p e , i s governed by f l u c t u a t i o n s i n v e l o c i t y tliroughout the e n t i r e f l a v and not j u s t a t the f i e l d p o i n t .

I n the problem of tho v/all j e t the mean p r e s s u r e i s approximately constant everywhere, and the mean v e l o c i t y f i e l d i s r a d i a l l y symmeti'ic so t h a t i n tenns of c y l i n d r i c a l p o l a r co-ordinates ( R , 0 , Z ) , the v e l o c i t y coniponents are given by TL ( R , Z ) , U , = 0 and \L ( R , Z ) ,

Since equation (3) above can be v / r i t t e n a l t e r n a t i v e l y i n "vector n o t a t i o n a s

V2 p - - p V, (u . v) u (3a)

wo f i n d on expanding tlie r i g h t hand side and making the usx^al boundary l a y e r assuniption t h a t ( s e a Appendix A)

Vi^iore (iiA> ii'j , u i ) denote tlie f l u c t u a t i n g components of v e l o c i t y and ( u p , U J , iL) the mean v a l u e s ,

But - ^ « r ^ so t h a t f i n a l l y v/e havra

9u4 9Ur,

V.p = - 2 » - 5 l - ^ (5)

9tL

vdiere T ( R , Z ) = ^ = - i s the l o c a l nisan shear,

This r e s u l t i s s i m i l a r t o t h a t fovind by ICraichnan (1956b) f o r the boundary Xayer f l a v over a f l a t p l a t e v/ith a mean v e l o c i t y i n the ( x ) d i r e c t i o n "varying with (x^) only, tlie d i r e c t i o n normal t o the p l a t e ,

I n order to simplify tlie n o t a t i o n , i n vAiat f ollov/s, vre w i l l drop the s u b s c r i p t s on r and the primes on u^. Thus p and u w i l l now r e f e r t o tlie f l u c t u a t i n g pressure and v e l o c i t y r e s p e c t i v e l y , and r i s tlio mean shear, Therefore we can xir±te equation (5a.) i n the eqirivalent f o m

"vrfbere q[ (x, t) is the soiorce density, We etxe here ass'UiTdng that the pressure fluctuations due to the ajiplitying effect of the mean shear on

the turbulence, are greater than tliat due to the interaction of the

turbulence on itself, Kraichnan (1956b) justifies the tose of this assumption by showing that, approximately, the former will give loso to a root mean

square value of pressure about 10 db higher over that due to the latter, The solution of (6) can be found by finding an appropriate Green function, G (x, x ' ) , which satisfies the boundary conditions. Since the Green ftmction satisfies

V^ G (x, x') = - 6 (x - x') (7)

"vrf-iere 5(y) is the Dirac delta function, the appropriate solution of (6) is

p (x, t) = f q (y, t) G (x, y) dy + f ("G |2 (y, t) - p(y, t) ^ \ dS(y)

\ -'s ^ ^

(8) •Vifcere V is the total volume occupied by the flow and S is the total area of the plane over which the flioid flov/s, provided at some initial time p and rr vanish. Here n is the normal to S, measured to tlie surface from the volume,

If we write G the Green function v>^ich vanishes on the plane S, G the Green function given by

3G

_-i - 0 6n

on the plane S, and G the Green function for xmbounded space, v/e have

= G , - G^

G = G + G. + o X

(9)

where G. is the Green function for the image point in the boundary. Hence we can find thi^e equivalent solutions of (6), which depend on the particular choice of Green function. Thus

P (x, t) = ƒ qG_^ dV + ƒ G_^ - g dS (10a) V f 9G q G ^ a V - / p ^ d S (10b) V S

^% ^ ^ [ (% t

-P^W

V (10c) But on S we ha-ve % = -^i 3G^ 9G. o 1 - V (11) 9n ~ ~ an

and so (10) can be written altemati"ve]y

p (x, t) = ƒ q (G^ + G^) 37 + 2 ƒ G^ ^ as (l2a)

= / q ( G o - G i ) 3 7 - 2 / P ^ as (12b) T -'s

= / ciG^ dV *. ƒ (G^ | E - p . ^ ) as (12c)

The particular choice of one of these equations must depend on -viiiat is known about p and -r^ on the plane S, Clearly if -r^ = 0 on S v/e v/ould

on on choose (12a),

Now the equation of motion in the Z-direction at Z a 0, where Up = u , Uy = 0, is

9

_"z

^^ 9Z^

for the fluottoating q u a n t i t i e s and

9 %

0 = /. ^ (14) 9Z^

for the mean flov/.

But from the equation of continuity 9u7 9u«

^ , ^ = 0 at Z = 0 (15) and tlierefore near the v/all

4

2

q

= ^

(li

)^ +

(16)

Prom the measured distribution of XL^^ near the wall of a flat plate, channel or pipe Townsend (1956) finds that, (in our notation),

'""" 2 T.

pv

vAiere r (R) is the wall meaji shear stress given by

( f ) « 3 . 4 ^ 1 0 - -2- Ci7)

W'

^ . ^ ^

V - i-az-;,^ (18)

Althoi;igh (•^ ) > as shown by (17) is a very small quantity we are not immediately justified in putting (•^ ) = 0 in (I2a),

Hovjever, i f •& i s a t y p i c a l length over which the v e l o c i t y i s c o r r e l a t e d and •& a s i m i l a r length with respect t o p r e s s u r e , v/e see "bhat tfa©

c o n t r i b u t i o n s t o p * , from the volume and surface i n t e g r a l s i n ( I 2 a ) , are i n "the r a t i o

100 ^^ : -e^ .

1 2

Nov/ if anything •& is less than •C^ and is certainly not large compared

with -C , and so we see that the surface integral in (I2a) can be neglected in agreement with Kraichnan's approximation (1956b). Thus v/e note that fo.r this problem the solutions (12b) and (l2c) are of less value than the first.

We therefore write approximately

P(ï.*) = -t I Q-~-d" * l - - / r V ( y ) ^ ( y ) ^

^ 09)

on i n s e r t i n g q (x^ t ) f r a n (6) i n t o ( I 2 a ) , and noting t h a t

%-=! =4l ( - - V * , ~ ^ ) ''°'

\ x - y x - y ' ^ fsj n^' ' /yj f\j

Viiiere y = y ( R , 5 i , - Z ) , and the volume of integration is over the half-space Z > 0.

On the surface of the plane x ^ ( R , 0, O) so that

P ^ ^ ^^z=o = ^ / — — ~ — aj/ (21)

z>o l ^ - y ' r<-J *%»

and the pressure covariance, for zero time delay betv/een the pressijres a t X and x ' r e s p e c t i v e l y , i s

9'

Kz

2 r t ^(y) T(^) — — — r - <3y dz P ( x ; x ' ) = - ^ / - ^ S 9 A „ ~ ~ /ppN

where the velocity covariance is

and y = y (R, 0, Z ) , z ^ z (R', ^', z').

J£ the turbxolencje i s s p a t i a l l y l o c a l l y homogeneous \7e oan put

^ 2 ( y ) {L2(z) J -^ ~ •\/ -^ ~

T ^ r e r = (z - y) and R„„ i s the v e l o c i t y c o r r e l a t i o n c o e f f i c i e n t .

r<j r v

Writing r (rp,, r , , r^) YTO find t h a t (22) TDecomes

r** K y JU P ( x ; x ' ) = ^ II r{y) riy + r ) — ^ ( r ) . ~ ~ Z=0 "^ JJ „ ^ ~ ~ ~ 91? ^^ Z>0 " "R

J ^' ^y^ J ^' ^l -^ £^

_{x - y | . | x / - y - r} , dy d r (24)

can malcLng the assumption t h a t iL, does not change s i g n i f i c a n t l y over the region i n which fL_, i s d i f f e r e n t from z e r o . In e f f e c t t h i s i s saying t h a t

{£ does n o t vary g r e a t l y over tlie e n t i r e flov/ a t a given value of R, which i s not a t o o unreasonable assumption. I f E„ and i t s d e r i v a t i v e s v/ith r e s p e c t tu r_ vanish a t i n f i n i t y , and on the p l a t e , an i n t e g r a t i o n by p a r t s of (24)

leads t o

P(x;.') = £ /E,,Wa. ƒ ^(llÉ..fSM) .

~ ~ Z=0 v^ J zz ^ .^ J2>0 ^ R \ | x - y I J

NcaT i t must be noted t h a t i n t h i s analysis s t a t i o n a r y , and not moving co-ordinates are being used. Also, as s t a t e d above P(x ; x ' ) > the pressure covariance on the plane Z » 0 , i s t h a t for aero time delay ajid in-rolves the instantaneous product of tlie f l u c t u a t i n g presstires a t tlie p o i n t s x and x ' r e s p e c t i v e l y . A more l o g i c a l treatment v/ould be t o considi3r the turbulence i n a frame of reference moving a t some mean, or convection, speed ü , I f i n t h i s f r a ^ of__referenqe the turbulence i s assumed to be i s o t r o p i c , n o t because i t i s a lilcelihood i n p r a c t i c e , b u t merely t o simplify the analysis and to allow resiiLts for P(x ; x ' ) t o be e a s i l y computed, and t o be governed therefore by the l o n g i t u d i n a l v e l o c i t y c o r r e l a t i o n c o e f f i c i e n t f ( r ' ; r ) , v/here r ' i s meastured r e l a t i v e to the mcf^rijig a x e s , we can express %j„ k'^ 9 '^) t r e l a t i v e to s t a t i o n a r y axes, i n terras of f ( r - u T", r^,, r ^ ; r ) , v/here r i s the time delay,

( r ' = r^ - 5^ r , r ^ , r , )

ï " ( r , , r ^ , r , ; t + r )

T ( o , o, o; t )

Hov/ever when 7" = 0 vre see t h a t f ( r ) lias the same functional form as f ( r ^ ) and f o r sijnplicity i t i s t h i s case t h a t v / i l l , i n g e n e r a l , be t r e a t e d below, The extension t o the case f o r general values of T p r e s e n t s no difficiolty i n a n a l / s i s but only r e q u i r e s p e n c i l , paper, quiet and p a t i e n c e , as \7itnessed by the analyses i n Appendices ( D ) and ( E ) ,

Equation (25) i s the important equation i n t h i s paper. I t shows tliat the press\xre covariance a t a p o i n t on the w a l l , due t o shear flow tiribulence dominated by the mean flow shear ( T ) , i s governed by '^yt the tv/o-xjoint " l a t e r a l " v e l o c i t y c o r r e l a t i o n c o e f f i c i e n t , the mean shear and the t u r b u l e n t i n t e n s i t y .

I n equation (25) the i n t e g r a t i o n over y can only be effected v/hen the vali:ies of '' u i are kna'/n. Nov/ we can e a s i l y see t h a t the g r e a t e s t valioes ot "^ u3 e x i s t i n two quite d i s t i n c t regions of the w a l l j e t a t any value of the radius R. These regions are the constant s t r e s s region and the middle of the outer mixing region r e s p e c t i v e l y . On tlie assumption t h a t the t u r b u l e n t inner region of the v/all j e t possesses the same s t r u c t u r a l

similarity as the inner region of a flat plate, channel or pipe flow we

have that the mean velocity distribution in this region (known as the "Law of the Wall") is

1 . f 2^r ü/n^ = ^ In | ^ - - i y + A (26) and — u 9 u _ 7-^ 9Z ~ KZ •sdiere u = ^ ^v/^ ^^ ^^^ shear v e l o c i t y . (27)

Eqintion (26) i s lcnoï\ii t o hold f o r '—^ > 30 and Z/Z < 0 , 2 , v/here Z i s the thickness of the shear l a y e r up t o the maximum v e l o c i t y u = u , K, laiown as the von Kaiman c o n s t a n t , has a value I'oughly equal t o 0 . 4 . Since the motion i n t h i s i n n e r region must be described by u n i v e r s a l functions of the v/aH shear s t r e s s , r . and tlie v i s c o s i t y , the t u r b u l e n t i n t e n s i t y Vi^ must be p r o p o r t i o n a l t o u^ . Prom the measurements of Laufer (1955) i n a c i r c u l a r pipe i t i s found t h a t u^ / u ^ = a (28) a constant over most of the inner r e g i o n . ( ctis of the order of u n i t y ) , Prom equations (27) and (28) we see t h a t

, 3

r • / u

z =

ïT. / i ^ (29)

for values of \~77'j > 30. Nearer the wall i t i s found tliat

T / ^ ~ 0.009 ( ^ ) ^ (50)

and it is interesting to note that the maximum value of

viy occurs j^Ist outside the laminar sub-layer.

In f act it is very close to the region v/here the total turbulent intensity /ui + u5 + ^ j reaches its maximum value. (Pig, 1 ) .

I t i s found from ( 2 9 ) , (.^O) t h a t

r

OO lie

f ( Z * ) . ^ . ^ . 0.59 (31) t ^ e r e f ( z " ) = r^ u,^

and Z = (Z u ^ y ) .

On the other hand i t i s sham i n section (8) t h a t tho value of the mean shear i n the outer mixing region i s

- 0.616 u

2 ni'

•vdiero Zi_ , Z are the distances to tlie p o i n t s of half and maximum v e l o c i t y 2 n^ i r

-r e s p e c t i v e l y . On tho asstimption t h a t i n t h i s l a t t e -r ix3gion Ï S - 0,15 u v/e find t h a t

^2 rr 0.0085 u^

This more or l e s s conipletes the formal treatment of tlie pressure f l u c t u a t i o n s i n a v/all j e t . Since the contributions to p^ from the inner region and the outer mixing region are very d i f f e r e n t i n magnitude

3 . jPressure^ fluctua-tioris^ due to the ^^itiner region*_ shesg; flow Erom (25) we found t h a t

- - z=o

^1

^2 -

-l^^R\\i-zV

'^Mï'-ïl

but in Appendix B it is shown that

'

''f \ s f

LS

x'- z

^

9yR V l x - y i y 3ZR

ay

u| ay^ (34)

v^ere S = x' - x , and T^ U ^ is assumed to be fvaiction of y„ only,

As stated above v/e will assume that the t^urbulence is isotropic in axes moving v/ith the convection speed u , If the time delay is neglected

viiere f is the longitudinal velocity correlation coefficient and

f' s ~ , This form for ^ „ does not agree too well with the experimental results of Grant (1958) and others. Hen/ever the errors involved in the

use of {35) caj^ be shov/n to be small and in any case avoids the necessity

for graphical or numerdcal integration of (25)•

Erom (25), (34) and {33) we find that

2 r 2

r2>0

(36)

wheire | x | = \'^\t and the integration with respect toy^, is over the

vdiole turbulent flov/ giving rise to large val vies of ''^'uT, and for isotropic turbulence

I f tlie ranges of i n t e g r a t i o n v/ith r e s p e c t t o r are O < r < " j O < (j) ^ TT and O < e < TT

P(x, 0) = l p p^ f "f dr ƒ r^'Z^ dy^ (38)

Z=0 •'o o

vdxere oc is an anisotropy factor v/hich is assumed to have a value of about -g-, If in place of (34) we use the alternative result from Appendix B (equation B.13) v/e find that

P(x, 0) = j ^ p^ < T"" v^ > / rf dr (38a)

Z=0 J 0

Tdiere < r iv. > denotes a stiitable mean v a l u e . Pr<Tn a comparison of (38) v/ith (38a) i t i s seen t h a t i f use i s made of (31)

O.b u i

2 "*T" T

^ V I

P

I f v/e now i n s e r t f = escp ( - /•%)) (39)

v/ith f dr = e (40) •'0 P

i n t o equation (38) we find t h a t , with the aid of ( 3 I ) *

' ^ ^ » ~ ^ ^ ^ 0.l2^V-r^V . °f - 5 c ^ (41)

* ^m ^ r

Tiiiere v/e have used •&p = 0 . I 4 Z , —=-.,=— = I6OO, and o„ = ''•n/^a' Pu^ ) i s the l o c a l sldji f r i c t i o n c o e f f i c i e n t . This r e s u l t i s s i m i l a r t o t h a t found by Kraichnan (1956b) f o r the boundary layer on a f l a t p l a t e , except our f a c t o r 5 ireplaces the range 2 to 12 as given by Kraiclman altliough

v/e note t h a t s t r i c t l y •vp(o)/-|pu^ i s proportional t o c^/'i and |Z "ïï . V

* This value is obtained from the work of Grant (1958) by noting the zero point value of R (r, 0, O) in tlie region close to tlie wall.

Equation (41) can be compared v/ith the r e s u l t s of various exoerimenters for ordinary boundary l a y e r flow, Tlie measurements of Harrison (1958) suggest a value of

"^^/i P^^ fs 0.0095 = 4.8 o^ (42)

Tiiiile Willmarth (l959) gives

^ / l - P ^ = 0.006 = 2,5 c^ (43)

Our measurement on the v/all of a v/ind tunnel with a microphone 0,14 in, diameter and a boundary layer displacement thickness of 0.238 in. gave

^"^ /i P 1^ = 0.008 = 3,6 c^ (2^)

All these results are qualitatively in fair agreenent with the theory and indicate that in a turbulent boundary layer on a flat plate, in zero pressiire gradient, tho wall pressure fluctuation arises mainly from

disturbances in the constant stress region. Since c_ changes only slov/ly with distance we see that "• p^ will only change slov/ly v/ith increase in x.

Finally we note that owing to the approximate nature of our analysis we are led to assume that < r^ uf > is inversely proportional to -Sp wi.th -É-p = 0.14 Z . However the more exact analysis given in Appendix G

2 "7

i n d i c a t e s t h a t r u^ should be averaged over a lengtli of more nearly 0,3 Z with •^p given by •6p ;^ 0.3 Z . This does not change tlie numerical resiiLts quoted above s i g n i f i c a n t l y altliough i f anything they b r i n g the theory more n e a r l y i n l i n e v/ith ODqjeriment. flov/ever the e f f e c t on the spectrum i s veoy marked prodTicing a tv/o t o one s h i f t i n the frequency parameter tov/ards the lov/oi frequencies. Purtlier discussion on t h i s problem i s continued i n Appendix E , but v/e conclude t h i s s e c t i o n by

noting t h a t i f •6p ~ 0.3 Z v/e are not c o r r e c t i n assuming t h a t the wall press^lre f l u c t u a t i o n i s dominated by disturbances i n the constant s t r e s s r e g i o n , b u t r a t l i e r t h a t the e n t i r e boundary l a y e r , and e s p e c i a l l y the b i g e d d i e s , a l l c o n t r i b u t e ,

P(, , .0 . . I J ^ ,|. f . (r) i^Lt^;, ar

^» Pressure fluetua.tions due to_t}y^_'_out_er_jnJ.jdji_gjre^^

In the outer region of the v/all j e t the s t r u c t u r e of the turbulence i a vmiiifluenced by the v/all. Nevertheless the solution of equation (6) i s s t i l l given by ( 1 9 ) , for the contribution from the 'imago' Green function must s t i l l be included. IiOT/ever the problem i s simpler because r^ vu i s approximately constant over the region centred about the middle of the mixing region. Equation (25) therefore reduces t o

9 r ^ , s (^R - ^ ^

(45)

If we again put 'the time lag equal to sero and put (x^ - x) = 0 then

P(x ; x) = ^ p^ r^ ^ ["rf dr (46)

" " Z=0 ^^ ^ Jo vdiere a is the anisotropy factor.

If further f = esq? (- V ^ o ) (47)

P(x J x) = 2 p^ r^ u j o (48)

Z=0 ^^ ^

On i n s e r t i n g the values for r^ vi^ from (33) QJ^cL p u t t i n g

^ - ^% - \ ) (^9)

then .—

" P / i p u ^ ~ 0-^ (50) ' '^ ÏÜ.

Tlie value obtained from our measurements i s

^ / 1 - p U j ^ = 0.11 (51)

and indicates that the contribution to v p ^ from the outer mixing region is some ten times that from the constant stress layer.

* This value of -6 corresponds to tlie half-v/idtli of the outer mixing region, At first sight this value seem.s unduly high but it is in keeping v/ith

the res^Jlts of the large eddy analysis in Appendix 0 and tlie measured spectra.

This result is not surprising v/hen one remembers that the small shear in tlie outer mixing region is acting over a distance many times that of the entire inner shear layer thickness.

Tlie theoretical value of * p V"2 ^"^ is independent of the radius

(R) and this is confirmed by our experimental results. This altogether interesting result is we feel most important for it shows that even in a region of flov/, v/here the velocity changes in the mainstream direction are lar*ge (u ^ /fe)» "the pressiore fluctuations are still largely deteimined by local conditions only. This agrees with the results of the mean shear

stress on the v/all by Bradshav/ and Love (1959) (ï'ig. 2 ) , They shcrned that the skin friction coefficient, although slightly higher than for a flat plate, changed very slowly with increase in radius beyond an inner radius vdiere the wall jet was being established.

5• Tlie spectrum^ and _s;)paoe pres^sure oqrrelation_for zero ^tinB j3elay In sections (3) and (4) above the formal treatment of the pressure covariance with zero time delay is given, but only the results for the moan square of the pressure are evaluated in full. Hov/ever Lilley (l958) has already evaluated trio pressure covariance in free shear flow turbulence, and as we have seen above the results for v/all turbulence v/ill be similar apart from a numerical factor. Prom Lilley's 2?esults for P(x ; x') with f = exp(- o^ r ^ ) , we find (Pig. 3 ) , that P(x ; ^ , 0, O) has a large

1

negative loop wdth a zero crossing point of o" ^.. ft* 1. On the other

hand P(x ; 0, 0, ^ ) is positive for all values of g and falls to zero

as C ^ CO much slower than f(r), as r ^ <» . The isotropic turbulence model used in evaluating these results is a little too crude to make

comparison v/itli ejqporiment justified, apart from an order of magnitude basis, yet it is interesting to note tliat Harrison (1958) found similarly

positive values for P(x ; 0, 0, B, ) , although the fall off at large

values of Z was slower than in our results. However this is v/hat one

- 3

miglit ei^ject from the anisotropy in the large scale turbulence modiiying the form of Rp_(r) at large values of |r | ,

The wall pressure spectrum function has been evaluated in Appendix G taking into account the large eddy structure, and in Appendix E allov/ing for convection. In both these cases and also in tlie free sliear flow

turbulence example the spectrum at 1cm frequencies obeys tlie law

w^ oxp(- u^), (Pig, 4 ) , Indeed ;P(u) must behave like ui' near w = 0

in an incompressible fluid. In our wall jet experiments v/e find such a rise at low frequencies, in contrast v/ith all measuiements made in pipes

and on tunnel wall boundary layers, xAieice the flatness of tlie low frequency

H a r r i s o n (1958) h a s a l s o remarked on t h i s f l a t n e s s of tho s p e c t r a a t low f r e q u e n c i e s and h a s s u g g e s t e d t l i a t , e x c e p t a t the v e r y lov/est f r e q u e n c i e s o u t s i d e t h e r a n g e of t h e measuring equijDment t h i s might be e x p l a i n e d i n teniis, of t h e i n t e r m i t t e n c y of t h e bovindary I r y e r . Hov/ever t l i i s does n o t

i n i t s e l f esqplain t h e r e l a t i v e l y liigli energy c o n t e n t i n t h e lov/er f r e q u e n c i e s . A coinplete e x p l a n a t i o n of t l i i s phonoirenon has n o t y e t b e e n found,

assuming of c o u r s e t h a t t h e f l a t n e s s i n t h e IOT/ frequeno;" end of t h e s p e c t r a i s n o t a s s o c i a t e d w i t h spuri.ous vdnd t u n n e l e f f e c t s , such a s f a n n o i s e , f l o w n o i s e e x t r a and above t h e boundary l a y e r n o i s e , o r t u n n e l c i r c u i t r e s o n a n c e , Hov/ever a c l u e t o tlie e x p l a n a t i o n ma;^'- conie from arj. a n a l y s i s of r e s u l t s s i m i l a r t o t h e s p a c e - t i m e c o r r e l a t i o n s of vra.ll p r e s s u r e o b t a i n e d b y W i l l m a r t h (1959) . These d a t a sliav, a s e x p l a i n e d i n Appendix D, t h a t

•tiic s p a t i a l p r e s s u r e c o r r e l a t i o n f o r optimum time d e l a y ( i . e . r o u g h l y tlie a u t o c o r r e l a t i o n i n a x e s moving v/i.tli t h e mean c o n v e c t i o n speed of t h e e d d i e s ) does n o t f a l l t o n e g a t i v e v a l u e s a t l a r g e s e p a r a t i o n d i s t a n c e s i n c o n t r a s t t o t h e a u t o c o r r e l a t i o n measuied i n a x e s f i x e d i n t h e w a l l . T h i s n e a n s t l i a t tlie l i f e t i n e s of t h e b i g e d d i e s a r e b e i n g e x t e n d e d on a c c o u n t

of t h e growing s c a l e a s s o c i a t e d w i t h t h e slow i n c r e a s e i n t h e boiaidary l a y e r t l i i c k t i e s s . T h i s consequent m o d i f i c a t i o n of the a u t o c o r r e l a t i o n i n moving CQSSS a t l a r g e t i m e s c o u l d jproduce t h e n e c e s s a r y l i f t t o t h e low f r e q u e n c y end of t h e s p e c t r a althougji a f a l l off l i k e w^ must e x i s t a t

t h e v e r y l a v e s t f r e q u e n c i e s ,

On t h e o-öier hand i t would a p p e a r t h a t i n the c a s e of tlie v / a l l j e t , s i n c e tlio i n c r e a s e i n s h e a r l a y e r t h i c k n e s s i s r e l a t i v e l y li^orge c o u p l e d v d t h t h e r a p i d f a l l off i n v e l o c i t y v/ith i n c r e a s e i n r a d i u s , t h e l i f e time

of t h e eddy c o u l d n o t be e x t e n d e d i n t h i s v/ay. These d i f f e r e n c e s betv/een tlie t l i e o r e t i c a l and e x p e r i m e n t p l s p e c t r a f o r botli t h e v / a l l j e t and boundary l a y e r a r e b e i n g f u r t l i e r i n - ' / e s t i g a t o d .

6» •^kS-J{.9:Ji^-jiQ."^.J-:--^^eary j ) f Jjic^ jggjyLjQigv'f

The v / a l l j e t h a s b e e n s t u d i e d t h e o r e t i c a l l y by G l a u e r t (1956) and e x p e r i m e n t a l l y by Baltke (1957) and Bradsliaw ( 1 9 5 9 ) . Altlioiogh G l a u e r t fovind s o l u t i o n s f o r b o t h t h e l a m i n a r and t - u r b u l e n t problems o n l y t h e t u r b u l e n t c a s e w i l l be r e q u i r e d h e r e . He found t h a t t h e v e l o c i t y

d i s t r i b u t i o n i n t h e v i c i n i t y of t h e v / a l l was s i m i . l a r t o -Kiat i n a boundary l a y e r T/i-öl z e r o p r e s s u r e g r a c l i e n t . I n t h e i n n e r r e g i o n one v/C(uld e x p e c t tlie ' i W of t h e

w a l l ' t o a p p l y b u t f o r s i n i p l i c i t y G l a u e r t assviraed t h e B l a s i u s d i s t r i b u t i o n - . 2 \ /"R2^" 2 V 1 / 0.0225 (52)

viiere u is the shear velocity given by u = \(T /\

* RosTolts analoguous to these have been obtained by Pavre and his oo-v/orkers (1958) for the two-point velocity correlations \/ith separation in tho streamf/ise direction,

I n tlie outer region Glauert evaluated the v e l o c i t y d i s t r i b u t i o n numerically and showed t h a t i t i s s l i g l i t l y f u l l e r than tliat given by a ( l / 7 ) t h pov/er lav/. The maximum v e l o c i t y occurs a t Z/Zi_ = 0,125, f o r a value of m (Z-t - Z ) = Z ( . x 1 0 , wiiere Z-i i s the ordinate to the

—^ 2 m _ a"

p o i n t i n tlie outer mixing region f o r v/hicli ü^/u = 0 . 5 . The value Z/Zi = 0,125, where u = u , d i f f e r s from t h a t obtained experimentally by Ealdcc, but t h i s can be explained from G l a u e r t ' s analj-sis, for Baldce's value of

of 4 X 10^

u _{m ( Z j ^ - Z ) = 3 o 5 x 1 0 , coiriparcd with our value above}

m

U9

0o5

j a

-The distribution of maximum velocity v/ith radial distance follows the law

^

R 1 ,0D

(53)

a t the Reynnolds numbers of the t e s t s reported h e r e .

The v e l o c i t y d i s t r i b u t i o n i n the outer mixing region is. found by Glauert to be given appi^oxiiiiately by

V u

= sech 0,875 m Z - Z (54) T/here Zi 2 R 0 , 9

The experiments of Bakke confirm these p r e d i c t i o n s , Bradshaw gives scsiie r e s u l t s f o r the vp-riation of wall shear s t r e s s under the v/all j e t , These show t h a t i n the region of f u l l y developed flow the w a l l shear s t r e s s i s some 25^ higher than i n the corresponding case ( i . e . equal n and Z ) f o r the flow over a f l a t p l a t e i n zero pressure g r a d i e n t ,

7 . Apparatus Wali^Tet

The t e s t r i g shov/n i n P i g . 5 ""s^ geometrically s i m i l a r t o t h a t used by Balcke (1957) for confirmation of the t h e o r e t i c a l r e s u l t s obtained by

Glauert (1956). The j e t was of 1.5 inch diaireter and air-was svtpplied from p r e s s u r i s e d r e s e r v o i r tanks v i a a 6 inch dieuieter t h r o t t l i n g valve and 200 f e e t of 6 inch diameter p i p e , p a r t of the laboratory ring-main supply which passes h o r i z o n t a l l y 18 f e e t above the t e s t s i t e . The v e r t i c a l down-pipe was of 3 inch diameter and 10 f e e t long connected t o the 6 inch main through a 3 inch i s o l a t i n g valve, This was followed by a smooth c o n t r a c t i o n containing a wire gauze and then 5 f e e t of 1 ,5 inch diameter smooth-bore pipe f i t t e d with a 6 inch diameter fliurige forming tlie j e t . The j e t flange "waa 0,75 inch above the p l a t e ,

E i t h e r of tv/o platesweieused both measuring 4 f e e t square. One was constructed of l i g h t acoustic boarding 2 inch thick l i n e d with t h i n

bake l i t e s h e e t , the other was of 2 inch thick tuf nol about 100 l b i n weight, The t e s t procedure was to open the 3 inch valve f u l l y and to c o n t r o l tlie aijoflow by the main 6 inch valve a t the r e s e r v o i r tanlcs. This v/as necessary to keep the valve and pipe noise to a minimum.

The measuring regionv/as 3-'10 j e t diameters from the j e t axis over •which the j e t half width (distance t o half-maximum v e l o c i t y ) v a r i e d from

0,4 inch t o 1,2 inch with maximum v e l o c i t i e s of I40 f / s t o 40 f / s . P_TOSsure Transducers

Ammonitmi di-hydrogen phosphate c r y s t a l transducers, type M213 and MI4I were used. These are manufactured by the American I\iassa Company.

The M213 has an outside diameter 0,22 inch and a diajAjragm diameter 0,12i. i n c h . I t s capacity i s 12 pf and has a l e v e l frequency response to 120 k c / s . I t i s used v/ith a lav/ noise and low microphone cathode-follower connected through 6 inch of low noise c a b l e . The input impedance of the cathode-follov/er i s 200 megohm shunted by 6 pf, allovdng a 3:^3sponse down t o 20 c / s . The cathode-follower noise l e v e l i s 5 /^V, although belov/ 300 c / s the noise from the transducer i s 25 l^'V, A high-^ass f i l t e r with a cut-off a t 250 c / s can be i n s e r t e d , but i s n o t normally necessary. The s e n s i t i v i t y a t the cathode-follovror i n the oulput terminals i s

The s i g n a l i s amplified by a b a t t e r y pov/ered low noise amplifier of 28 decibels voltage gain, follov/ed by an amplifier of 94 decibels voltage gain. Root mean square readings are measured on a meter of the l i n e a r averaging t y p e , The bcmdv/idth of the amplifiers i s 5 c / s to above 500 k c / s ,

The M12(.1 has a diameter 0,6 inch, a capacity 110 pf and a s e n s i t i v i t y a t the cathode-follower output t e n n i n a l s of -94 decibels r e 1 volt/microbar (approximately 20 mici-ovolts/dyne/cm^), The low frequency'' noise of t h i s transducer i s siuch lov/er than the M213 d.ue to i t s higlier capacitance, and the usable frequency response i s 20 c / s - 30 k c / s ,

The transducers are moimted i n the l i g h t acoustic-board p l a t e i n soft rubber sleeves and i n the tufnol p l a t e with a h.ea.Yy b r a s s body and "0" r i n g suspension, s i m i l a r t o tliat of Y/'illmarth (1958). The use of the two metliods of fixing i n the two very d i f f e r e n t p l a t e s was t o ensure t h a t the measurements v/ere not affected by the v i b r a t i o n of the transducers due t o the impingem.ent of the j e t on the p l a t e ,

SiDcctrum measurements v/ere made using a s e t of t h i r d octave f i l t e r s , covering the range 40 c / s - 20 k c / s .

TuAiiIenc_e .Equi-^ieiit

Tho turbulence equipment used for measurements of tho f l u c t u a t i n g v e l o c i t y i n the j e t v/as of the constant temperature kind and v/as a

modernised version of t h a t described by Laurence and Landes (1953), with a s l i g h t l y hi.gher bandwidth and lov/er noise l e v e l . Tungsten v/ires of 0,0002 inch diameter and 0,080 inch long were used. The hot v/ires v/ere mounted on the s'lpporta 'using t h e u s u e l copper p l a t i n g technique,

Correlator

The c o r r e l a t o r used i n the space and tiiiie c o r r e l a t i o n s of pressure and ^?elocity f l u c t u a t i o n s v/orked on the analogue p r i n c i p l e . The m u l t i p l i e r was of the quarter-squaring type and used two s p e c i a l squaring v a l v e s . The design was based on t h a t of M i l l e r , Soltes and S c o t t (1955). However much improved olx'am.tr.y v/as deployed so t h a t , i f necessary, m u l t i p l i c a t i o n could be acconiplished over the bandi/idth DC - 200 k c / s . The accuracy v/as not of the standard a s s o c i a t e d vrLtli coniputing m u l t i p l i e r s , but i s of the order of 1 - 2^ v/hioh i s quite acceptable for the p r e s e n t purpose. The output- was read on a DC galvanometer v/ith a time constant v a r i a b l e between 2 - 1 0 seconds.

A time delay i s accomplished on a tv.dn channel tape recorder of extremely low v/ow and f l u t t e r , using one fixed and one moving head. The recorder used 0,5 inch v/ide tape running a t 75 inch/second,

The tv/o s i g n a l s to be c o r r e l a t e d v/ere f i r s t recorded on a loop of tape giving a sample length of 2 secaids ( t h i s sample length could be increased i f e x t r a i d l i n g p u l l e y s v/er« f i t t e d to the machine) and then

played back i n s e r t i n g the required time delay by nanual contrKsl of the moving head. The s i g n a l s could e i t l i e r be recorded on a frequency modulation system bandwidth DC - 20 k c / s or on an amplitude modulation

system with a bandwidth of 250 c / s - 100 k c / s . The s i g n a l to noise r a t i o was 42 d e c i b e l s ,

The c o r r e l a t o r was capable of c o r r e l a t i n g a 100 kc/s sine-v/ave v/ith 15 p o i n t s t o the p e r i o d . The maxir.ium time delay possible v/as 40 m i l l i s e c s aiid tlie minimum increment v/as 0,67 microseconds,

8 , Prvsldminar

The measxirements of mean v e l o c i t y across the v/all j e t from 3 to 10 diameters from the j e t axis shaved good agreement v/ith tlie s i m i l a r i t y p r o f i l e s obtained t h e o r e t i c a l l y by Glauert (1956) and exxaeririTentally by Baldce (1957) ( P i g s . 6, 7 ) . The rxixitnum v e l o c i t y v a r i e d as R""**"^, v/here R i s the distance from the j o t a x i s , and the j e t half thickness Z± STaried

0 9 ^

as R * , The maximum velocity occurred at Z/Zj^ = 0,15 corresponding u ( Z i — z ) ^

to a Reynolds number m^ p- m'^ „ o ^n'*

The measurements of the wall pressure fluctuations from 3 to 10 diameters from the jet axis slia/ed that

>rif2

p ' / i P ^n = °«''''

m

and v/as approximately independent of r a d i u s ( P i g . 8) . Tho corresponding spectra ( P i g , 9) obeyed a s i m i l a r i t y lav/ on the b a s i s of the frequency parameter w(Z-t - Z ) / u ,

•^ ^ 2" m' ' m

Oomparative r e s u l t s were also obtained i n the College of Aeronautics 20 i n . X 11 i n . Low turbulence v/ind turmel a t a p o s i t i o n i n the v/orking section v/here the displacement tliickness v/as 0,286 i n , a t a freestream

speed of 126 f / s . The corresponding value of the skin f r i c t i o n c o e f f i c i e n t , c^ , obtained from the measured v e l o c i t y p r o f i l e , was 0,0022. The value of ^ p V i P ' ' ^ ^ was 0,008 which i s i n f a i r agreement vd.th the neasurements of T/illmarth and Harrison. Hov/ever t h i s r e s u l t cannot be r e l i e d upon

q u a n t i t e t i v e l y because i n t h i s tunnel the low frequency exrI;raneous noise l e v e l i s not low, although i n e v i t a b l y i t s c o n t r i b u t i o n to the t o t a l pressure energy i s small. Perhaps v/hat i s more iniportant i s t h a t i t provides a

check both on the accuracy of the instrumentation used, and tlie values of "^ P V ' 2 P ^^ as found i n the v/all j e t using the same pressure t r a n s d u c e r s ,

The large values of * p /-aPii a.s measured on the wall jet are ilölf'' special to that case aiid have been found by Owen (1958) to exist on wings in regions of separated flow. However the experimental data on the effect of pressure gradient on v/ing pressure fluctuations is so few that we can only speculate that, on the basis of our iiieory, the surface pressure fluctuations v/ill be high v/hen either the value of c^ is high, say at transition, or the mean shear layer is very thick, say approaching and beyond separation,

9, Future Work

A more extensive study of the v/all pressure f l u c t u a t i o n s on the v/all j e t i s i n progress i n which a u t o - c o r r e l a t i o n s and space-time c o r r e l a t i o n s are being made. These measurements together with two-point v e l o c i t y

c o r r e l a t i o n measurements w i l l enable the equivalent eddy convection speeds t o be e s t a b l i s h e d together v/ith the d e t a i l e d t u r b u l e n t s t r u c t u r e . I t i s hoped t h a t t h i s d a t a , together with nBasurements on the e f f e c t of pressure

graxlient on the surface pressure f l u c t u a t i o n s on v/ings, w i l l e s t a b l i s h the parameters on v/hich ' p ^ must depend, and thus provide a soimd b a s i s f o r a b e t t e r understeinding of the more urgent p r a c t i c a l problem of the natiJre of w a l l pressure f l u c t u a t i o n s on bodies t r a v e l l i n g a t higli speeds e s p e c i a l l y a t supersonic and hypersonic speeds.

10, Cqnclusiqns^ ^

A review of t h e o r e t i c a l and experimental work on w a l l pressure f l u c t u a t i o n s i n t u r b u l e n t boundary l a y e r s i s presented. The theory due t o ICraichnan i s modj-fied and extended to include the separate e f f e c t s of the large eddy s t r u c t u r e and the convection of the e d d i e s , and the t r e a t -ment covers both ti.ie turbulent boundary l a y e r on a f l a t p l a t e and the v/all

j e t . The l a t t e r case i s presented for i t provides date i n the important p r a c t i c a l case v/hen the turbulence i s being subjected t o a r a p i d v a r i a t i o n i n mean shear, not unlike t h a t a s s o c i a t e d with the flow approaching and beyond s e p a r a t i o n ,

Preliminary t h e o r e t i c a l and experimental data for the v/all j e t give values of p ^ / i " P u^ many times t h a t of corresponding measurements on a f l a t p l a t e witli zero pressure gr-adient. These r e s u l t s are explained i n terms of the g r e a t e r thickness of the shear l a y e r , tlie r e l a t i v e l y high i n t e n s i t y of the turbulence, and tlio presence of r e l a t i v e l y large eddies i n the flow.

The p r e s e n t work i s intended as a b a s i s f o r f u r t h e r v/ork a t both low speeds and i n tlie more important p r a c t i c a l areas of supersonic and hypersonic f l i g h t .

11, References 1 . 2 .

3.

4.

5 . 6 ,

7.

8 , 9 . 10. AuthOT Bakke, P . (1957) Batehelor, G.K. ( I 9 5 l ) Bradshaw, P., (1959) Love, E.M, Curie, N. Doak, P.E. Pavre, A. Grant, H.L, Harrison, M, (1955) (I960) (1958) Peynman, R.P, (1949) Glauert, M.B. (1956) (1958) (1958) 11. Heisenberg, W, (19^) Title . An experimentel investigation of a wall-jet. J n . P l u i d Mech. 2 , P a r t 5 . Pressure f l u c t u a t i o n s i n i s o t r o p i c turbulence. P r o c . Cam. P h i l , Soo, 4 7 , P a r t 2 , p.359 The normal impingement of a c i r c u l a r a i r j e t on a f l a t surface.

A.R.C. 21,268 (Sept. 1959),

The influence of s o l i d boundaries upon aerodynamic souid.

P r o c , Roy. See. A. 231 p.505

Acoustic r a d i a t i o n from a tiorbulent f l u i d containing foreign b o d i e s , P r o c . Roy. Sec. A, 254, p.129 Quelques r o s u l t a t s d'experiences sur l a turbulence c o r r e l a t i o n s spatio-temporelles s p e c t r e s . P u b l i c a t i o n s scientd^iques e t tecliniques du Ministere de 1' a i r No. N.T.73. Phys, Rev. 76, p.769 The w a l l j e t . J n . Pliiid Mech. 1 , P a r t 6, p.625 The large eddies of turbulent motion.

J n . P l u i d Mech. Vol,4 P a r t 2 , p.149, J u n e . Pressure f l u c t u a t i o n s on the wall

adjacent to a t u r b u l e n t boundary l a y e r .

Hydromechanics Laboratory Report 1260 David Taylor Model Basin.

Zur s t a t i s t i s c l i e n Tlieorie der Turbxilenz.

fnoritirojcd}_ Kraichnan, R.H. (1956a) Kraichnan, R.H. (1956b) L a u f e r , J . (1955) Lai^rence, J . C . , (1953) L a n d e s , L.G-. L i l l e y , G.M. (1958) l a i l e r , J . A . , (1955) S o l t e s , A , S . , S c o t t , R . E . Obuklioff, A.M, (1949) 0\TOn, T.B, (1958) P h i l l i p s , O.M. (1956) Tov/nsend, A.A. ( l 9 5 6 ) Tovvncend, A.A, • (1957) W i l l n a r t h , W.¥. (1958) T f i l l m a r t h , W.¥. (1959) P r e s s u r e f i e l d v/i t h i n homogeneous a n i s o t r o p i c t u r b u l e n c e . J . A c o u s t . Sec.Amor» 2 8 , p , 6 4 P r e s s u r e f l u c t ^ u a t i o n s i n t u r b u l e n t flow o v e r a f?uat p l a t e , J , A c o u s t . S o c . Amer, 2 8 , p . 3 7 8 Tlie s t r u c t u r e of t u r b u l e n c e i n f u l l y developed p i p e f l o w , NAGA R e p o r t 1174

A u x i l i a r y Equipment and techniques f o r adaptSng t h e c o n s t a j i t tempei-ature h o t - w i r e anemometer to s p e c i f i c a i r f l o w phenomena, NACA Tt^T.2843 On t h e n o i s e from a i r j e t s . ARC 2 0 , 3 7 6 U n p u b l i s h e d , Wide-band a n a l o g f u n c t i o n r n u l t i p l i e r . E l e c t r o n i c s P e b . p , l 6 0 P r e s s u r e p u l s a t i o n s i n a t u r b u l e n t f l o w . M i n i s t r y of S u p p l y , E n g l a n d . T r a n s l a t i o n P 21i|-52 T. Tecliniques of p r e s s u r e f l u c t u a t i o n measurements employed i n t h e R.A.E. low speed v/ind t i m i i e l s .

AGiSD R e p o r t 1 7 2 . S u r f a c e n o i s e from a p l a n e t u r b u l e n t boundary l a y e r . P r o c . Roy, S o c . A. 2 3 4 , p . 3 2 7 The s t r u c t u r e of t u r b u l e n t s h e a r flov/. Cambridge U n i v e r s i t y P r e s s , The t u r b u l e n t bovmdary l a y e r . Boundary L a ye r R e s e a r c h Symposium P r e i b u r g , August 1 9 5 7 . Wall p r e s s u r e f l u c t u a t i o n s i n a t u r b u l e n t boundary' l a y e r , N/iCA TN. 4 1 3 9 . Space-time c o r r e l a t i o n s and s p e c t r a of v / a l l p r e s s u r e i n a t u r b u l e n t boundary l a y e r . NASA Memo 3-17--59V/.

-^^-.Jgg^gS-i]^?lg,-6q^Q-Aion f o r tlie v/all ^^ot

The e q u a t i o n s of c o n t i n u i t y and motion f o r an i n c o n p r e s s i b l e flow a r e , i n v e c t o r n o t a t i o n : pD u Dt V , ( U ) = 0 ( A , 1 ) = - V p + / J V ^ u (A.2) m = "9t + ^ ^ • ^ • ' ^'^'^^

I f i n t h e c y l i n d r i c a l c o - o r d i n a t e system ( R , 0 , Z) v/e vnrite t h e c o r r e s p o n d i n g v e l o c i t y coroponents ('Up» '"^gi,» ^ ) * tlien t h e pressiore

d i s t r i b u t i o n e q u a t i o n , found from s u b t r a c t i n g t h e time d e r i v a t i v e of ( A , 1 ) from t h e d i v e r g e n c e of ( A , 2 ) , V 2 p = - P y . ( u . v ) u (A.4) oan be VTrltten i n f u l l a s

.y...P 1 3. pr, 1 ^ 3 f^ f s ^

" p ~ R 3R ^ T l aR R d(l> •*" ^ 3Z '' 1 1 / 3 u , u 3 u , 3 u , R d<p + _{3Z I ^}

( ^ i R ^ R^ ^ + "Z T z J (A,5)

3R "^ R 90 "^ "Z '"SZ

Now t h e mean v e l o c i t y u = (uL , 0 , u,-,) s a t i s f i e s t h e c o n t i n u i t y e q u a t i o n

(v/ith Z measured normal t o t h e p l a n e ) , so t h a t i f vie vn?ite R, "Up a s a dimension and v e l o c i t y of o r d e r u n i t y , 0 ( l ) , t h e n Z, \L a r e each 0( S) v/here

6 i s tlie t h i c k n e s s of t h e s h e a r laj^er. S i n c e terms i n v o l v i n g s q u a r e s and p r o d u c t s of f l u c t u a t i n g q u a n t i t i e s v . l l l be s m a l l compared v/ith t h o s e

i n v o l v i n g p r o d u c t s of f l u c t u a t i n g q u a n t i t i e s and mean flow d e r i v a t i v e s , t h e o n l y terros of iniportance i n (A,5) a r e f o r R » 0 ,

- p = R OR ^ \ ^ - 6 1 + ^ Tz

•^ az V"^

-FR

^ \ -az

3R 3R '^ aR 3Z az az

au^ 3iL

Tlie^ _ejyaluation ^of _an^.inte^jal

I t v/as shovm i n s e c t i o n 2 t h a t tlie pressure c o r r e l a t i o n depends on the value of

[ f(y, r) dy

I^ = 1 — ^ ^ : i ^ ^ - ^ - ^ (B.1)

Y+ 1 X - y I |x' - y - r 1

v/here the integration is to be taken over the half-space V for v/hich y^ >0.

Kraichnan (1956a) has evaluated a similar integral follos.lng the method of Peynman (1949). He found that

ƒ 4 iï-ïT'

é\y*r-:^r

^

= 2.r _ . Ü _ J l - § I (B.2)

v/here E =• x' - x , and V the volume of i n t e g r a t i o n tends to i n f i n i t y . I f i n ( B , 1 ) we put

E = x Z - x : J? = y - x ' : S = H + S ; t = S - r

tlien 1 = f f ( ^ , r )d S

— — - ^ - - ' - (B.3)

Now f ( y , r ) i s a function v/hich vanishes on the plane y„ = 0 and a t i n f i n i t y . Let us suppose, for s i n i p l i o i t y , t h a t

f ( y , £ ) « f ( S ^ , r^) , (B.4) *^"" r rr d S d ^

If we introduce a new variable t

^' = S -

with K>' > 0

(see figure belov/) and follov/ing Kraichnan (1956a) we use the i d e n t i t y

2 _,_,dr (a2 + b2 r2) (B.6) tlien

= ^ /"7T';r ƒ ^(^^^^^^^^

_{1 + 7-}

r d s' d

1 3

z'

(1 + 0

) (B,7)

If the surface integral, with respect to ^' and Z' , is taken over a circle of radius R in the plane parallel to OEP' (i.e. the v/all) T/e find that

r dS' d^' '1 2

(1 + 0 ^

«^In (1 +r^ (B,8)

Clearly we cannot continue furtlier v/ith the evaluation of I , since i t tends t o i n f i n i t y a^ R •* <x> , Hov/ever i n our a n a l y s i s i t i s not I tliat v/e rx^q-uire, but 3^ ^ I f therefore we d i f f e r e n t i a t e ( B , 7 )

a^^

/ 3 I \ '' r "^

( ^ ) = 4 t^ / f ( ^ ' , r ) d ^' / . J J A L „ „ . _ .

(B,9) I n order t o simplify the a n a l y s i s f o r the pressure oovairiance two a l t e r n a t i v e approximations v/ill be made i n the evaluation of ( B , 9 ) .

In the f i r s t of these we w i l l assimie Z' /, << 1 giving

/ f ( ^ ' , r ) d S' ( B . 1 0 ) j 2 2 2 9 1 2 7 r t 3 r ,2 1 t aiid therefore 52 9 I^ 3 / S - r 3r^

" -é ( | f r ^ 7 ) 2 - / f(y,, r^dy^ (B.11)

I n the second a l t e r n a t i v e approximation we w i l l replace f ( ^ ' , r ) i n ( B , 9 ) by a s u i t e b l e mean value f ( r ) . I t then follov/s tiiat

9 l t^ ^ ïT f -qr (B.12)

ar " t

1 and 9 1 9 / { B, - V y _ ^ - „ . j r t — . - - . J - — i ) ( B . 1 3 ) 3r^ 3^ \ I g - r _{1 1}

v/hich i s j u s t half the value found i n ( B . 2 ) above, for ? = 1 , and

v/hen I^ i s talcen over the v^hole space and not the half-space considered i n our problem.

The ^strjactmx^ Jï^^'k^^^ large eddies near the v/all i n a turbulent boundary l a y e r

The T/ork of Tov/nsend (1956), (1957) and t h a t of Grant (1958) have shov/n tliat a r e l a t i v e l y siiiiple structui'e e x i s t s for the large eddies i n a tiu.'bulent flov/, or a t l e a s t the exr];)eriniental observations are not i n c o n s i s t e n t v/ith the hypothesis tliat the large scale motions can be adequately described^in terms of r e l a t i v e l y single e d d i e s . G r a n t ' s mea-Eurements of the R, . ( r ) v e l o c i t y c o r r e l a t i o n s a t d i f f e r e n t heights from tlie surface i n a t u r b u l e n t boundary l a y e r give r i s e t o the suggestion

tliat tlie large eddies near the siirface ( i . e . i n the constant s r e s s layer) have tiie form of ' two-dimensional' j e t s of f l u i d o r i g i n a t i n g near the viscous l a y e r , and may a r i s e from the i n s t a b i l i t y of t h a t l a y e r , and are roughly alï.gned i n the d i r e c t i o n of raeEui motion. Similarly i n the ' o u t e r region' the large scale t u r b u l e n t motion again appeairs to be doiimiated by the piresence of mixing j e t s of turbulent f l u i d v,diich o r i g i n a t e in the i n t e r i o r of the flow and p e n e t r a t e to the region of the non-turbiilent flow outside the boundary l a y e r .

A siniple extension of the Tamsend-Grant de s c r i p IdLcBi of the large eddy s t r u c t u r e for such motions l e d us t o ass\jme t h a t tlie eddies randomly

d i s t r i b u t e d over a l l space may be described by

u, = è - (1 - «' x ' ) e x p ( - 2 ! | ! ) (C.I) u , = - 1 ^ a^x, (1 - a ^ x,^) exp ( - ^ ) (C.2)

S = è " ^ S ^ ^3 (1 "% ^2) 6 ^ ( - ~ f ^ ) (C.3) 3 '^ where (u^ , Ug , Uj) QXQ the components of v e l o c i t y i n the d i r e c t i o n s

(2^ , x^, X j ) , v/ith x^ i n the d i r e c t i o n of the freestream and Xg perpendicular to the w a l l , and

a^ X? s a'^ x^ + a*^ x^ + a^ x^ . (C.4)

1 1 2 2 3 3

Tlie v e l o c i t y components, given by equ, 0,1 t o G.3 s a t i s f y the equation of c o n t i n u i t y 3u vdtli ( a , a , a ) 1 2 3 3u 9u 3x 3x 2 3 = 0 (G.5) having quite a r b i t r a r y v a l u e s ,

Following Tovvnsend we next find the t\^-o-T50int velocity correlations

are given by

,,(r) = - - t V ƒ ^ ^ ' - ^ 3 ' ^ 0 "^^3 + r / ) e x p ( - y ^ / 2 ^ e x p ( - i i ^ ' )

• ' 2 3 * ' / ^N

(G.6)

^22CE)

= — 7 " f ^ y/y, - ?,)(i - yp 0 - (h - r,)^)xp(- P/^)

1 2 3 ' ' . . _ „ , 2 . exp 2

(^.li-|.?i') (C.7)

^^^ ^33^E^ = —-^ / ^ y i y3 ( y i + ^ , ) ( y , + r3) (1 - y^) (1 - ya - r^) a a a " 1 2 3 * ' . exp ( - y^^g) ^ ^ ( - (y + ^ ) ^ / 2 ) ^^'^^ where y^ = a^ y^ • y^ = a^ y^ J ^3 = S ^3 ^^

dy = dy., dyg ^^3 • After i n t e g r a t i o n these r e l a t i o n s can

be v/ritten i n the form, i f R^^etc. are the v e l o c i t y c o r r e l a t i o n c o e f f i c i e n t s , and the subscript (b) denotes the b i g eddy c o n t r i b u t i o n ,

R , , ( r ) l , = a^ (1 - ? / + ^y^^) exp ( - v^^) ( G . 9 )

^33(E)b = S (^ - V 2 ) (^ - V2^ (^ - V6^ ^ ^ ^- ^ V ^'-^'^

G r a n t ' s experimental r e s u l t s close t o the v/all suggest t h a t

1 .35 v/here as i n ttie outer region

2.5 . a a 2 <^. ,4, 2 2 . 2 0 . 6 • > • » a ^ ". 1 a a 1 - 3 . 0 ~ 1.5 • > • a. a. 1 a 1

Since and R (o) = ^A^ «^'/^ T/e see t h a t a a a 1 2 3 R^ fo) = -i A^ 7r^2 8 R33(o) = | A ' n-'/^ «1 «2 «3 (0,12) u ' U" 2 a ^ ₂ and U2 3 ,

^t

a (G.13)

•viiere ^ e t e . r e p r e s e n t the b i g eddy c o n t r i b u t i o n to u? e t c .

^b ^ I n the constant s t r e s s l a y e r Lauf,er (1955) finds t h a t

^ / u = 0.5 and u^ , _

2 / u,

1 9 v/liilst i n tlie outer region

= 0.5

u ^ / - - 2 _{and u , /•-? = 0.2}

This suggests t h a t i n the constant s t r e s s region a^, = 2.5 a^ ;

a = 0.9 a w h i l s t i n the outer region a^ = 0.4 a^ j a^ =• 0.9 a^ These r e s u l t s show c l e a r l y t h a t the r o l e of the u^ v e l o c i t y component i s changed i n passing from the constant s t r e s s region t o the outer r e g i o n .

The simple edd;^' s t r u c t u r e , i n the constant s t r e s s r e g i o n , portrayed by these r e s u l t s i s t h a t of an elongated vortex r i n g ha'/ing i t s longest

dii'ection i n l i n e v/ith t h a t of the mainstream. I t i s r o t a t i n g i n a plane p a r a l l e l to the v/all, v/hilst tlie longiti:idinal motion i s v/avelilce, being away from and next towards the v/all. The l a t t e r motion i s n o t uiililce the jot-lilce m.otion described by Tov/nsend and thus may s i m i l a r l y be i n t e r p r e t e d . I n the o u t e r region tlie osculations are dimdnished i n scale \ / h i l s t the

edd;;' spreads out s l i g l i t l y i n a d i r e c t i o n p a r a l l e l to tlie v/all and perpendicular t o tlie freestream,

The conrplctc value for Ë^^ must be the sum of the contributions from

the small and big eddies. Thus to H^^iv), we must add on (l - ag) times

the value for ^^^(r) obtained from the smaller eddies. If v/e assume that those eddies are isotropic, in contrast v/ith the strong anisotropy of the big eddies, v/e have that

R22(r)„ = (1 - a j •-'S 2 2 (G.14) f' s ^ . Thus finally 2

v/hcre f(r) is the longitudinal velocity correlation coefficient and

of

dr

Ra2 = ^ 2 2 ( 3 ) "• ^ 2 2 ( b ) ^^-^^^

6n tlie assumption t h a t

f ( r ) = exp ( - V ^ ) (0.16)

the values of R have been evaluated and are coinpared vn.th Grant's results

in Pigs. 10, 11, 12 for the measurements made close to the v/all (i.e. V Ö ~ 0,08) Althoufh the agreement is not good qi.iantitatively we have qualitatively

obtained results which predict the basic fo?Tïi of the measured results. Similar comparisons can be made for R^ ^ and R^^ both in the constant stress layer and in the outer region provided the appropriate values of a.,, ag and a^ are chosen as explained above,

With these values for 522(1^) '^^ '^^^ ^^"w fiiid. the tv/o-point pressure covariance on the v/all, with zero tiiiie delay. Prom the previous analysis we find that the pressure covariance on tlie v/all, v/ith the separation vector g, is given by

P^ 2 -3 f - / N 9 ^ ^ ^ r ^ ' i ^

P( X : ^ ; 0) = — . < r^ u^ > R,,(r) ~ — ' — « i - . dr

M 0 r2> 0 1 I e - £ |

(G.I7)

and the corresponding spectrum function is P* . .^2 "T r(x ; K 0, K ; 0) = -»- < r^ ^ u^ > R (r)dr ^ r2>0

-iC^.^+'^^y 9^ ^°''^'*

. e ^ i^.r I dg^ d^^

Thus i f ;f(x ; K J O) i s t h e s c a l a r spectrum f i a i c t i o n on t h e s u r f a c e found by a v e r a g i n g ^ over a l l a n g l e s i n the p l a n e , n o t i n g t l i a t

K = V «2 ^ K^ and P 00 P ( x ; 0 J 0) = / P ( x ; K ; 0) dK (G.19) tlien p2 __ r 2ïr r f ( x : K ; O) = — < r^ u^ > / cos^ Ö d© / d r r >0 2 r - /<rr, - i ( « r + « r ) , e ' ( 1 + ^ r ) R ( r ) e ' ' ' =' (0.20) s i n c e i n e q u a t i o n (C,18) - i ( / C £ + K E) g2 e ' ' ' =' ^ 3 _ I ^ - r i d^ d ii 9^2 I - - I 1 3 1 Zir --^ (1 + K r J e " ' ^ ^ 2 ( C . 2 l )

K-We can now find the separate contributions to P from the small and big

eddies. Thus v/ith '^^^(•u\ given by equation (C.10) we find after some

,a , V5 ,a, C^A;^ exp ( - ' ^ V p a ^ ) <^^ ( -'^Vpct^) . I. - K^ 1 + ^'w-' >^/a (1 - Va^J E r f c C^/ïJ (1 - 2 K y < ) 2 ( V ( ^ " (C.22)

S i i r i l a r l y w i t h Raafa") g i v e n by e q u . ( G , 1 4 ) v/ith f ( r ) = exp ( - /•&$ ( i n o r d e r t o s i m p l i f y t h e a l g e b r a b u t n o t n e c e s s a r i l y changing t h e o r d e r of magnitude of t h e r e s u l t s ) v/e f i n d t h a t 2 «3 ^ .^2 U2 p ( 2 ; ' ^ ; 0 ) 3 = (1 - a2) P^ ^ ^ S ^ ^ > ^^f^-W A ^ i + Vw" e^ ^ ' ^ , E r f c (—) (1 - — 2 " ) ( t ó )

A comparison between equations (G.22) and (0,23) sliov/s that whereas the

snail üddiT contribution to ? is dominant near wave nur.ibers of « = •&

(the 'tj'pical energ;;,^ bearing scale of the smaller class of eddies) the big eddy contribution is vcr^ dependent on tlie relative scales of the big eddies in the directions (x , x^, x^) respectively, i.e. l/ct , i / « 2 , l / s ^

Nov/ i n o b t a i n i n g a l l tliese r e s u l t s i t h a s been assumed t h a t 'B.^J^v) does n o t change s i g n i f i c a n t l y w i t h d i s t a n c e from t h e w a l l , v/hich i s c e r t a i n l y n o t t r u e i f , a s v/e have shov/n a b o v e , v/e p a s s from t h e c o n s t a n t s t r e s s l a y e r t o t h e o u t e r r e g i o n . A rough comparison of e q u a t i o n C.22 v/ith W i l l m a r t h ' s (1959) r e s u l t s c l e a r l y shov/s t h a t s i n c e higli f r e q u e n c y p e a k s bej/ond Kb = 1 (v/here 5 i s t h e boundary l a y e r d i s p l a c e m e n t t h i c k n e s s ) a r e n o t p r e s e n t , the p r e s s u i ' e spectrum must be d e t e rm i n e d from t h e form of tlie v e l o c i t y c o r r e l a t i o n f u n c t i o n v/hich i s n o t t o o n e a r tlio v / a l l . T h i s means i n f a c t t h a t S / « . , c a n n o t be l a r g e o r i n o t h e r words t h e p r e s s u r e s p e c t r u m i s l a r g e l y d e t e r m i n e d , up t o t h e higli f r e q u e n c y c u t off p o i n t , from t h e b i d eddy c o n t r i b u t i o n s v/ith e d d i e s v/hose t r a n s v e r s e e x t e n t ( i n p l a n e s p a r a l l e l witii t h a t of t h e w a l l ) i s n o t g r e a t l y d i f f e r e n t

from tliat i n the streaniv/ise d i r e c t i o n . This v/ould imply t h a t tlie large eddies outside the constant s t r e s s region play a more s i g n i f i c a n t r o l e i n tlie determination of the w a l l pressure f l u c t u a t i o n s , than do the l a r g e eddies i n s i d e t h a t region. This i s not a r e s u l t v/e miglit have expected from considerations of the region j u s t outside the viscous l a y e r i n v/hich the maximum values of the t u r b u l e n t i n t e n s i t y and mean shear a c t . However i t must be borne i n mind t h a t our analysis i s n e c e s s a r i l y very approxini£ite and ihe reason for t h i s r e s u l t miglit appear mci-e obvious i f we had worked tlirough the analysis .retaining tlie v a r i a b l e mean shear and

t u r b u l e n t i n t e n s i t y i n s t e a d of r e p l a c i n g them by an averaged value talcen over tlie e n t i r e shear l a y e r ,

P i n a l l y i n t h i s appendix i t i s v/orth pointing out the differences wliich e x i s t betv/een our r e s u l t s and tliose of Kraichnaai (1956b), Kraiclinan makes tlie assumption t h a t the turbulence near the v/all i s homogeneous i n plajies p a r a l l e l to tlio v/all but not i n planes normal t o the v/all. He next assumes a model f o r the turbulence i n v/hich ' m i r r o r - l i k e ' v e l o c i t y boimdary conditions are s a t i s f i e d on both s i d e s of the v/all. I n tliis model tlie u, and Uj v e l o c i t y components are f i n i t e on the w a l l b u t U2 i s zero. The e f f e c t i v e c o r r e l a t i o n c o e f f i c i e n t i s tlien taken a s

^ 2 2 ( ^ 2 » ^2 ; ^1 » ^ ) =^22^=^2 - ^^2 ' ^1 > ^ 3 ) " ' ^ 2 2 ( ^ 2 + =^2 ' ^1» ^ 3 ^

and v/hen i n s e r t e d i n t o equation (C.I7) i t leads t o the r e s u l t , i f f = exp( - o - ^ r ^ ) ,

(C.24)

f (X ; «,. 0) ~ ( f ) ' exp ^ - ( ^ ^ ; ) Vir 1 + Erfo ("2"|) e

V,

40^

- 1 (C.25) Nimerically t h i s r e s u l t i s ver^r l i t t l e d i f f e r e n t from t h a t given hy

equation (C.23) but i s considered l e s s s a t i s f a c t o r y i n viev/ of the v e l o c i t y boundar;).- conditions being e s s e n t i a l l y different' from those e:cisting i n the region outside the viscous l a y e r . lüraichnan a l s o includes the case of v a r i a b l e mean shear, follov/ing tlie la.v/ r ~ e x p ( - /?x^), and althougbj.

t i l l s treatment i s preferable t o the averaged mean shear approach v/hich v/e have adopted, the r e s u l t s are n o t , q u a l i t a t i v e l y d i f f e r e n t and do not lead

to cm ox-dor o.f magnitude difference i n the numerical' r e s u l t s . I t v/ould be of i n t e r e s t , however, t o r e p e a t our a n a l y s i s v/ith r ~ e x p ( - /9x ), and tills i s being done.