Pressure fluctuations in an incompressible turbulent boundary layer

ESCHCOL D E L F T H i.330UWKU!-;DE

17 SGp.1960

'•/i.

HS DELR

THE COLLEGE OF AERONAUTICS

CRANFIELD

PRESSURE FLUCTUATIONS IN AN INCOMPRESSIBLE

TURBULENT BOUNDARY LAYER

by

PvEPORT NO. 133 J u n e , I960.

T H E COLL. EG E OF A E R O N A U T I C S

C R A N F I E L D

S u m m a r y P a g e 3 P a g e 5 P a g e 6 P a g e 7 P a g e 8 P a g e 9 P r e s s u r e F l u c t u a t i o n s in an I n c o m p r e s s i b l e Turbulent Boundary L a y e r b y -G. M. L i l l e y , M. Sc. , D. L C. , A. F . R. Ae. S. , A. M. I. Mech. E. CORRIGENDA Line 8 Equation (10) Equation (19) Equation (25) Equation (29) Line 17 Line 17 L i n e s 9 to 11 Since Replace 4.6 C^ by 3.1 Cf Replace X^ by x^ Replace e^ '^^2 by e^ '^ ^^ the v a r i a b l e of i n t e g r a t i o n is '^z „ , -0.31y , -Ö.31 y / 6 , R.eplace e ' by e Replace y5= Ö.31 by /S6^ = 0.31

I n s e r t 'wave n u m b e r ' before s p e c t r u m function Delete and r e p l a c e by,

^ d£ = 0.27367 we see that

V£

i P o ^ e = 3.1 C, (37)

The author v/ishes to thank D r . N. C u r i e for pointing out the e r r o r in the t e x t , a,nd showing that the i n t e g r a l can be evaluated from the tabulated v a l u e s of a s i m i l a r i n t e g r a l by Goodwin, E. T. and Staton, J . , Q. J . M. A. M. Vol. 1, 1948, p . 319.

BEPORT NO. 153 J u n e . 1960 — — • — -•- • ^ • • i i » mt\m •!•• T H E C 0 L L. _E.G.E 0 F A E R O N A U T I C S P r e s s u r e F l u c t u a t i o n s i n an Incorripressible T u r b u l e n t Boundary L ay er b y -G. M. L i l l e y , M . S c , D . I . G . , A . P . R . A e . S . , A.H.I.Mech.E. of t h e Department of Aerodynamics SIMIARJ

In a recent paper hy Lilley and Hodgson an approxiraato analysis is given of the pressure fluctuations en a rigid v/all under a turbulent boxmdary layer. One of the approxiuiate results given in that paper was

r *^ /*! 2

that \ p /2 P u » 5 0^:.» although strictly the analysis gave ' o e X

1 —p /1 _ 2 „ ^A u o P^ / E f„ U ^ ~ C / e \l -^ ' '^ o e f \ "tr~"

' o

The present paper presents a more exact analysis by using the meidiod of generalised Fourier transforms. The final result is that

\v^ /"k P "^^ *** 4.6 0_ and is independent of the boundary layer

OOIJTENTS Page Stmmary Notation 1 . I n t r o d u c t i o n 1 2 . /Analysis 1 3 . A simple r e l a t i o n f o r W 6 2 2

2f, The mean velocity distribution 7 5. The surface pressure spectrum function 7

6. The mean value of the fluctuating pressure 9

NOTATION

B constant in Coles' 'Law of the wake'

T

0- = /^ P vL local skin friction coefficient r ' "- o e

£ longitudinal velocity correlation coefYiciont K von Karman constant

X transverse scale of turbulence z

k wave number vector in the plane (x , x^) P pressure covariance

p fivetuating pressure r separation vector R22 velocity covariance

s Laplace trcmsfonn operator t time X co-ordinates X^ co-ordinates i n plane ( x , x ) 5 mean v e l o c i t y Ü v e l o c i t y outside boundary l a y e r e u ...turbulent v e l o c i i y component Tw/ U^ -\l /P shear v e l o c i t y aZ Fourier c o e f f i c i e n t of v e l o c i t y /9 mean shear parameter

6 boundary l a y e r thickness 6 displacement thickness

K three-dimensional wave number vector V kinematic v i s c o s i t y

c inverse turbulent scale p density

mean shear

T mean shear parameter

T wall shear stress w

dcj Fourier c o e f f i c i e n t of pressure

TT pressure spectrum function

# energy spectrum function

Other sjTnbols, not l i s t e d above, are defined where they appear i n the t e x t ,

1, Intr oduc t j- on

In a recent paper^ ^ a brief review is given of the theoretical and es^ïerimental research on the pressixre fluctuations in incompressible turbulent shear flows. On the theoretical side the methods used by Kraiclman ' are discussed, although the analysis in that paper depended xipon a sliglatly different model of the turbulent flov/ tlian that used by Kraichnan, In order to obtain numerical results a certain integral had

to be evaluated approximately and the accuracy of the resulting expression for p2 as a function of the skin friction coefficient cannot therefore be easily established. The present paper employs a different method

of approach and thereby avoids this difficult integral. It is shoam that the final results show good agreement with the earlier approximate

results.

2, ^ a ^ ^ i s

The equation f o r the f l u c t u a t i n g pressure i n an i n c a n p r e s s i b l e txjrbulent shear flo?/ is^ '

au^Cx, t)

^ ^ p ( x , t ) = . 2 p ^ r » ^ g ^ p ^ (1) whe3?e x s ( x , x , x ) , p i s the constant d e n s i t y , r i s the mean shear

8 5

•g^ v/ith u, a function of x^ only, and u^ i s the turbulent v e l o c i t y 2

conponent i n the d i r e c t i o n x ,

Lot us consider the s p e c i a l case Y/here the shear flov/ i s the boundary l a y e r flowing over tlic surface a t x^ = 0 vri.th co-01-dinates (x , X ) i n the plane of the s\;irface. Then, i f X 4,x f x ) i n the plane p a r a l l e l t o the surface and distance x from i t we can w r i t e the three-dimensional F o i o r i e r - S t i e l t j e s transforms of p(x: , t ) and u (x , t ) r e s p e c t i v e l y a s

f i(k^ . X + wt)

PCX^; X , t) =j e ~ ~ d^x^; k ,w) (2)

and f i ( k . X + ut)

u / x ^ j X j t ) = j e ~ ~ dZ^Cx^j k,a)) (3) T^iere k i s the wave number i n the plane and w i s tlie frequency,

2

-I f we s u b s t i t u t e for p and u i n equation (1) i n terms of the Fourier c o e f f i c i e n t s dw and dZ defined i n (2) and (3) vre obtain

dx^ ( d w ) - k 2 ( d ^ = - i 2 p ^ r ( x ^ k^ ( d Z j (4) 2 -2 _ 1-2 . 1,2 tiiiei^ k = k + k , 1 3

The boundary ccoiditions for p aï« p = 0 as x •• oo

and ^ = 0 as x ..0, Thus by means of the Laplace transform method, writing Lim (dw) = ( d w ) and Lim "=.^r^ = (dw) = 0

O O /> CuC _ 1 X -» 0 2 2 • 2 ___ /-oo * ^ & / / ~ \ —SX and dw = / ( d a ) ) e 2 a x , \ 7 e f i n d (s^ - k ' ) ( d ^ = s(dw^) - / 2 i PQ k,T(cE^)e dx^ (5) o —-r (dw)^ (dw)^ i P„ 1^1 /"" -sac, i D k «"SX * Vré^ f-'rCdZ J e ^dx, _{k ( s + k ) / ^ 2 ' 2} o vAiich on i n t e r p r e t i n g gives X (dS) kx^ -kx„ i ^ k , r ' k ( x ^ ; ) dw = — A " " ( e + e ) - - = . : ^ - - e r ( d Z ) a x ' 2 v^-. - - / ^ 2 2 i p k r^2 - k ( x - x ' ) 0 1 / ^ 2 2 ' k . ^ 2 ' 2 + •"•'^«i- / e ' ' r ( d Z ) dx' (7) But dw •• 0 as Xg -• co so t h a t 2 i p ^ k CO - k x ' {6.'^^ = - ^ ^ . . ° ^ . [ e r ( x ^ ) dZ^(x;) dx^ (8)

3

-Hence from (7) and (8) a f t e r some rearrangement i p k f"' ^c(x2+ xj du (x ) = — . ° — i / e 7<dZ ) dx' k o i p k f °° Jc lx - x ' I - ° ^ ' e ' ^ ^ ' r ( d Z j dx' (9) o

a r e s u l t previously derived by Kraiclman.

The ]pressure spectrum function Jr{x ; k ,u) i n tlie plane x i s therefore r e l a t e d to p by^ ' ^ ( x ; k ,0)) = - 1 ^ JTx ; X ^ ' , " t l ^ x j X + ' r ' ; t + V) ^ 87r j j ~ - i ( k . r + u t ' ) , , , , , e ^ ~ ~ ' dr d t ' <^^x^; r^ »'*') <!'" ( x j k ,0)) (10)

die die dco

1 3

end s i m i l a r l y the energy spectrum function i s

* > 2 ' <J ^ »'') = " 4 ff u/x^; X ,t)"r(x^; x T r , t + t')

Bvr' - i ( k , r + cjt') _, ,, , . o ^ ~ ~ ' dr d t ' -"ST" dZ ( x ; k ,u) d7. ( x ' ; k ,0)) . . dk dk^ dco

Hence on the surface x = 0 the thrce-diKieusional pressure spectrum function is _ „ oo 00 , 4 p^ k^ r / -k(x' + x") ^ ( 0 ; k ,0,) = - - i l ^ L . e ^ ^ r(xO r(x") k'' o o , $ (x', x"; k ,aj) dx' dx" (12) 2 2 2 ' 2 ' ~ ' ' 2 2 ^ '

2^

-and the tvro-dimensional pressure spectrum function i s ( i f t i s the time delay)

if p2 k ^ re» - 2 k x ' r " - i y ^

^ ( 0 ; k , t ) = ~ ^ ^ e r(x^O ax^ j e r (x^' + y^)

o -c:

2

We note in passing that the velocity covariance

u^(x^; X , t) u^(x;j X^+ r , t + tO -R22(^a' ''2' ^ ' ^'^

iU-so the two-dimensional energy spectrum function is

2 2

(5)

and so tlie more conventional throe-dimensional energy'" sipectrum function i s - ^ 2 ^ 2 *22(x2; k , t ) = — r § ( x ' ; y , ; k , t ) e ' ^ dy, / u ( x ' , t ' ) u (7'+ r , t ' + tj e"^^ -• ^dr (16) CXC — .CO = i y k *22 ( < J y2J ^£ » *) = ^^^ K ' !C, t ) e die, (17) where K ^ ( K , K . K ) and k. ^ K : k = K ; k = K , ~ 1 2 3 1 1 ' 2 2 ' 3 3

- 5 From ( 1 3 ) and ( 1 7 ) 4 p^^ i^ r -2k 3< ^.„ = TT ( 0 ; k , t) = —-ïi"-^ / e T ( X ' ) dx/ . /• § (x' ; /<, t)d/c ^ ' ~ ' ' ^ J 2 2 / 22 2 2 o «-co - k y , i /c^y, (18) e e r(x^+ y^) dy,

and for zero time delay the spectrum function of the surface pressure fluctuations i s 4 P^ l5^, r -2k x' TT (0; k) = =-.-2^-1 e ' r(x') dx' ~ k^ •' ^ 2 ' 2 o $ ( x ' : K) &K (19) 2 2 2 ' ~ ' 2 . -Jqyg i'<y2 e e r(x^ + y j dy^ -x' 2

which can be evaluated rAien the turbulent energy sxoectmm function i s given as v/ell as the distribution of the mean velocity ü as a function

of X . ••

2

We see, following lüraichnan, that (19) i s sdjT:iplified v/hen the mean shear, r , i s expressed as

- /5x

7-(xJ = r^ Q ^ (20)

vAiere r a n d /? a r e c o n s t a n t , o

é

-3 . A simple relation for * ₂₂

Let r s ( r , r , r ) then if f(r) is the conventional longitudinal velocity correlation coefficient in isotropic turbulcnoo a possible form for Rggis given by

^J<; r) = u:(x:) with 2^ 2' ^(x') r ^ + r ^ 2r Qir J Plenoe if f(r) = exp(-o^ r^) then /^/ 2 2 2" '^^ *f(x') k«e"/^'^ 2 2 (21) (22) (23) (24) and *2^^2J/S)e d =

g(^.l^ e-^Ao^3>oy

(25)

Thus with T ( X ) given by (20) and § by (24) v/e find from (19) that

^(0;fe) = - ^ . . o . J . 1 ^ ^ / e-'2(k-^/5)==; ? ( x O d x : .

2 2

^-(k+^y -0^2 dy,

2 ^ ' 2 (26)

7

-4, The mean velocity distribution

In the case of a boundary layer in zero pressure gradient the mean velocity distribution in both the inner and outer regions is given by

(6)

-Coles^ ' as (replacing ^x^ by ü and Xg by y )

x = i i . ( : i i : ) . B . | ( i . . d . f (^.1)

(27)

where the last term is Coles 'Law of the wake', and K is the von Karman constant. If v/e choose the values of K and w to be rospeotively, 0.4 and 0.55, then

è —^ = ^ + 4.33ooaf ( f - 1 ) (28)

^ dVö /^

vAiBve u aood Uj. arc r e s p e c t i v e l y the e x t e r n a l v e l o c i t y and the shear

v e l o c i t y , I W/ , and the mean shear, T , as defined above, i s given by

V / P Q

If therefore we define ras given by (20), we find that a reasonable fit with (28) is obtained if

3.7 u,

1

r = - ^ e-°-^'' y (29)

provided y is outside the laminar sub-layer, so that r = r

° 5

and /9 = 0.31. ^

5* The sinrface pressure spectrum function

In order to have a value of the velocity covariance R^, 'which can be compared with experiment we will modify slightly the value of R as given by (21). The modification is to replace e" "^2 by e"'''^'^

ti^ere 1 (x ) is the scale of the energy containing eddies in the direction normal to the surface. Thus we will put

^J<i 72? r) = ^ (X2) e-y«/^» e - ^ ^ ' (i - 0 ^ r^) (30) Tiiiere here r a i/r^ + r^ «

8

-If k = *k^ + k^ then the two-dimensional energy spectrum function is

-k2/4 o^

lé TT O^

which is similar to (25) except for the inclusion of e exclusion of e""^ ,

(31)

"^2/12 and the

The surface pressure s p e c t r i n function i s novr ( a f t e r i n t e g r a t i o n v/ith r e s p e c t t o y ) TT ( O j k ) = . 4 _{k +;9 +} 1 ( x ' ) 2^ 2 ' (32)

and when u and 1 are c o n s t a n t s ,

~ P^ r^ , 2 -k AcT^ 2

'^ (0; tó = ..° ° _ j ^ _

47r cj^ ( k +/9 + l / l 2 ) ( k + ^ - l / l g )

(33)

I f v/e no\/ i n s e r t the values u 6 r 1 o u_ _ / U 2 3.7 — ; - ^ = 0.8 e r 0-6 = 1 / 2

'A,

= 0,31 J /?6 = 0.31

vAiere the first is found follov/ing (29), the second from the results of Laufer ', the third and fourth from the results of Grant ' and the last from (29) also, (6 is the displacement thickness) then the surface

pressure spectruia function becomes

ir{0', (ip u"6)' ^•^ o e i ' (k 6^)^ + 0.62 (k 6,) (34) where ( u w ^ ) ' e

9

-6, The mean square^ '..^lue^ of _the f l u q t m t i n £ p^,^,^'^^^

I f v/e w r i t e

P( o) = p^ and E = k 6

then on i n t e g r a t i o n of (34) i n the plane over a l l wave numbers v/e f i n d t h a t

^ Ai ''o < )' = ^5 °f r ^ - ^ r r ^ (35)

° ® ^ i (E + 0.62) o o r , Since v/e see ^ / p ^ ^ ^ o 0,72 t h a t \ | p ^ ^ 0 ^ u2 e > ^e CC 0 JL 5.9

2 _

E . 4 . 6

. ; r i ? e-^ dE

^i•'o

72 f 0,62

°f

(E + 0,62) > 0.5 (36) (37)

in agreement v/ith the results given in Ref, 1, This suggests that the pressure fluctuations under a turbulent boundary layer are proportional to the external dynamic pressuare, and the skin friction coeffiaient and are independent of the boundary layer thickness except in so far as the skin friction coefficient is a function of boundary layer thickness, Equation (37) is based on the assumptions (a) that equilibrium conditions prevail in the turbulent boundary layer, and (b) that the external

10 -7 , References 1 . L i l l e y , G.M,, Hodgson, T.H, 2 , Kraichnan, R.H, 3 , P h i l l i p s , 0,M. 4 . L i g l i t h i l l , M.J, 5 . Batchelor, G.K, 6, Coles, D. 7 . Laufer, J , 8 , Grant, H.L. On siorface pressure f l u c t u a t i o n s i n turbulent boundary loi'^ers,

College of Aeronautics Note No. 101, I96O. Pressure f l u c t u a t i o n s i n t u r b u l e n t flow over a f l a t p l a t e .

JournsJ. Acoust. Soo. /jner,, V o l , 2 8 , 1956, pp.3780

On the generation of vraves by turbulent v/ind.

Journal P l u i d Me oh. V o l , 2 , 1 9 5 7 , pp 417-445.

Fourier Analysis and Generalised Functions, C,U,P,, 1958.

The theory of homogeneous turbulence,

c , u , p , , 1956,

The lav/ of the v/ake i n the t^urbulent boundary l a y e r ,

J o u r n a l Fluid Mech,, V o l . 1 , 1956, p.191 . The stiTicture of tTjrbulcnce i n f u l l y developed pipe flovT,

II,A„C.A. Report 1174, 1955.

The large eddies of t u r b u l e n t flow, J o u r n a l Fluid Mecdi,, V o l , 4 , 1958, p , 1 4 9 .