The Calculation of the Profile Drag of Aerofoils and Bodies of Revolution at Supersonic Speeds

VLKGTUIGBÓUWKUNDi

2 7 JUNi 1953

REPORT No. 73

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE CALCULATION OF THE PROFILE DRAG OF AEROFOILS

AND BODIES OF REVOLUTION AT SUPERSONIC SPEEDS

by

A. D. YOUNG, M.A., A.F.R.Ae.S.

This Report must not be reproduced without the permission oi the Principal of the College oi Aeronautics.

TECHNISCHE HOGESCHOOL VUEG rUlGBÓU WKUNDE

2 7 m\ t953

REPORT NO. 75

APRIL. 1953

T H E C O L L E G E O F A E R O N A U T I C S

C R A N P I E L D

The Calc\ilation of the Profile Drag of

Aerofoils and Bodies of Revolution at Supersonic Speeds

-by-A. D. YOUINTG, M.A., P . R . A e . S . of t h e Department of Aerodynamics

SUIvffllART

The effects of viscosity on the aerodynamic character-istics of wings and bodies at supersonic speeds can be assessed if we can calculate (a) the development of the boundary layers in the laminar and turbulent states, (b) the interaction of the boundary layers and main stream away fran the neighbourhood of shock waves, (c) the effects of shock wave-boundary layer inter-action, Comprehensive methods are developed ajid discussed for dealing with problems (a) and (b), problem (c) is disctissed in the light of existing experimental data but more systematic data are required before quantitative prediction of shock wave-boundary layer interaction effects in any particular case can be confid-ently made. Fortunately, for many practical cases of interest these latter effects are small.

The detailed results of calculations made on the lines described in this paper for a wide range of aerofoil thickness, body fineness ratio, Reynolds number, l/Iach nimber and transition position will be given in a subsequent report.

NOTATION

X distance measured along the si:irface (or meridian profile)

y distance measured normal to the surface

r radial distance from axis in axi-syrametric flow

r radius of c r o s s - s e c t i o n of axi-syrametric body

p density

|i coefficient of viscosity

T temperature

0) exponent i n v i s c o s i t y - temperattire r e l a t i o n ( i . e. |iocT )

u velocity component in x-direction

V velocity component in y-direction

k coefficient of heat conduction

o coefficient of specific heat at constant presstire

O coefficient of specific heat at constant volume

Y y %

<r |i c /k (Prandtl number)

M Mach number

6 bovmdary l a y e r thickness

e momentvim thickness =

J o

05

M ) .dy, in two dimensions

axi-syrametric flow

5 displacement thickness

16

M j ^y in two dimensions

V Pi^i/

^JG-^)

r > . . „ . -^y» i n

H 6*/e

iS-

angle between tangent to meridian profile and axis

-3-NOTATION . (Oontd.)

G„ local skin friction coefficient

Cp, overall s k i n f r i c t i o n coefficient o w i n g chord

X frictional stress

Z standard length

R Reynolds niinber based on -Z

R Reynolds number based on x.

Ac boundaiy l a y e r atrea =

(16 2% (r^ + y cost?-)dy (axi-symmetrlc f l o w ) U A g * displacement a r e a = 291: r .8 (axi-syimnetric f l o w ) f, g fimctions defined b y e q u a t i o n 2 A f i m c t i o n defined b y eqiiation 3 h ( M ,R ) f i m c t i o n defined b y e q u a t i o n 7 C^jO„,i<'n constants (see equations 5,6 axid 7 )

P ( X ) , G ( X ) functions defined b y e q u a t i o n 8

d a reference l e n g t h .

Suffix i r e f e r s to quantities m e a s u r e d a t the outer edge of the boundary l a y e r , suffix w t o quantities a t the stirface,

suffix o t o quantities m e a s u r e d a t some reference s t a t i o n e. g, t h e imdisturbed stream o r just a f t of the leading edge shock, suffix i t o incompressible flow, suffixes p a n d a to t w o diraensionsuL a n d axi-syrametric f l o w respectively, a n d a d a s h denotes differentiation w i t h respect t o x.

1, Introduction

It is now well established that the concept of the boimdary layer is qualitatively as valid for high speed flow as for low speed flow. However, to reach the point at which we can begin to estimate the drag, lift and pitching moments of an aerofoil

or body at supersonic speeds allowing for the effects of viscosity, we mvist clarify as far as possible our ideas

on:-(a) The development of the boundary layer in both the laminar and turb\ilent states,

(b) The interaction of boundary layer and main stream away from the neighbourhood of shock waves.

(c) The nature and effects of bo^mdary layer - shock wave interaction.

Problem (a) for the laminar boundary layer can now be regarded as solved for practical purposes, for the turbulent boundary layer the position is less certain but it is possible to

suggest «acceptable lines of attack. Problem (b) can on plausible if not mathematically rigid gro\jnd3 be reduced to the familiar problem of determining the effective displacement of the surface which can be related to the displacement thickness of the boundary layer. O-ur knovdedge on problem (c) is far from complete, but some data are available from which we can draw usefiol if interim deductions.

Thus, it will be seen that though we have not yet available a ccmplete theory on which to base the estimation of viscotis effects at high speeds, we have arrived at an interesting stage at which a semi-empirical attack can be developed on same of the problems of major practical interest,

This note is concerned with describing methods that have been adopted for the calculation of the profile drag of aerofoil sections and bodies of revolution. The results of such calculations covering comprehensive ranges of Mach nxmber, Reynolds nvmber, transition position and thickness or fineness ratio will be given in a subsequent report.

It should be noted that the term profile drag is here used to denote the drag arising frcm the viscosity of the medium and is a combination of the skin friction drag and the form drag. The latter at supersonic speeds is here defined as the change in wave drag due to the effective modification of the wing or body

shape caused by the boundary layer. The inviscid wave drag, contrary to the practice adopted by some authors, is here excluded

-5-from the definition of profile drag. It may be remarked that

the profile drag at supersonic speeds is not determinable, as it

is at low speeds, from the momenttnn loss in the viscous wake,

since this loss includes in part the wave drag which is also

manifest in a loss of momentum downstream of the vdng or body.

The profile drag must therefore be determined directly as the

sum of the separately determined skin friction and form drags.

The following paragraph will discuss in this context the problems

(a), (b) and (c) above and our methods of dealing with them.

2. The development of the boundary layer

2.1. Two dimensions

It vail be appreciated that for the purpose of estimating

profile drag we require a method of follovidng the developnent of

the boundary layer which must be reliable as far as providing

assessments of overall characteristics are concerned, e.g.

mcTiientum thickness and skin fl*iction, but wliich need not provide

characteristics of the bo\mdary layer in detail. For the laminar

layer vario\is methods are available for this ptirpose, the one that

has been adopted here is that described in Ref. 1, because of its

relative simplicity and of the fact that it is not restricted to

any particular values of the Prandtl number (cr) or the exponent

of the viscosity temperattire relation (w). It requires a simple

graphical or numerical integration to determine the momentum

thick-ness at ary point, the foiroula being

k ^'

Y ~ f U, g

)x^

g-1

p^ u^^"'dx (1)

where x i s the distance measured along the surface, p. and u

are the l o c a l values of the density and v e l o c i t y , r e s p e c t i v e l y ,

j u s t outside the boundary l a y e r , 6 i s the momentum thickness of

the bovmdary l a y e r , f and g are functions of a reference Ilach

number M , e.g. the undisturbed stream tlach number or the Llach

o

niffiiber j u s t a f t of the leading edge shock, and are given by

_ „,1-00 I - 1 - 9

f = 9.072 1 + 0.365(Y-Oa^ 1'^

g = 9.18 + 1.436 M^ - I ^ +-^^^^ a^ U^l

3

2 - % J

and

\x

is the viscosity corresponding to the reference conditions.

w

The skin friction coefficient, c„ =

——^ ,

where T

' f

2 '

w

D u

o o

is the local Intensity of skin friction, is given by (see Ref. 1)

where

( A +

12)

^

I

3R f

u • e »

o

(3)

' ^ dx u .i p u

L o ^ o ^c

Z

is a standard reference length, e.g. the wing chord,

VPO

^ ^

^ i^e. the reference Reynolds number, and

w i s t h e v a l u e of ji a t t h e s u r f a c e and i s g i v e n by I f we e x p r e s s a l l q t m n t i t i e s n o n - d i m e n s i o n a l l y i n t e r m s of t h e i r r e f e r e n c e c o n d i t i o n v a l u e s (stiffix o) t h e n e q u a t i o n ( l ) becomes x^ Rf u S 1 *^1 p^u^*^ dx and e q t i a t i o n 3 becomes ^ (/\+ 1 2 ) . u ,

3 R f e

(1A) (3A) where

A = R I u^' e^ f2 p^^^ '

w

-. \

Ü)

r iCzÜM^h ^u? ( a i - l ) | L

and u = ; 1 + 2 o U. 1 1,1 /

" I J

Knowing the distribution of c„ along a siurface to the

transition point vre can then readily determine by a simple

integration the contribution of the laminar layer to the skin

friction drag.

As in incompressible flow, 0 is assumed to be

con-tinuous at the transition point, since otherwise a discontinuity

in 8 there wo\£Ld imply an infinite local viscovis stress.

For the turbulent boundary layer the following approach

2

has been adopted. The raomentura equation of the botmdary layer

can be written

e

, . ef^ (H . 2) 4 ] =

T W

2 '

P1^1

(4)

/where the ...

-7-where the dash denotes differentiation with respect to x, and

H is the ratio of the displacement thickness of the boundary

layer (5 ) to the momentum thickness (ö). To solve this equation

we require to Icnow how H is varying with x and we require a

further relation between

i

and 6. Now for incompressible

flow past a flat plate at zero incidence the velocity power law

leads to the relations

,-1/n

w

and

2 - °1 \

Pl^1

-!-

= ^ .

c, (R

) ^

V n-1 1 ^ x'^ ,

(5)

u.x

1

where C. and n are related constants, and R = —

1 * X V

Prom these relations it follows that

1_

T / u , e \ " ^ ' ^

- ^ = ^ 2 ( - ~ )

'

(^)

where

I 1

^2 - °1

[nj

v/e now generalise equation (6) for compressible flow by assuming

that ,

1

T /u. 6 \ " n-1

»i"i ^ "I

where h(M,) is an as yet lonspecified function of M . Further,

we assume equation (7) holds locally on the surface of the aerofoil,

so that (4) becomes

1

r^^' P^' 1

/'^^^\~

n-1

6'

+

S-l-^

(H+2)

+-^i

=

CJ - ^

h(Mj.

1^1 Pl|

^\ \)

^

I f we express a l l qviantities non-dimensionally i n terms of t h e i r

reference condition values (suffix o), then t h i s equation becomes

fu.' p.') " n-1 / u , 0 \ " n-1

. , 0 | ^ ( H . 2 ) . ^ ^ C ^ H ( - 1 - ) h(H,).

Assuming isentropic conditions outside the boundary layer we can

readily show that

-i

T2 W

Pi

'^ ~

^'^ ^1 '

I* + 0. where

and

P(x) F(x) 1 1 n-1 n-1 G^ R 0 G(x) G(x) =

^|(H.2)-Mfj

/ u \ " " ^

Eqxmtion (8) can be integrated to give

(e ^-0

r ftx,

exp. I

F ( X ) .

~ j -

.

dx

n-1 2

T

1

n-1

n

n-1

n^c. G ( X ) . exp.

\x

(8)

P(x). ~:j- . dx . dx (9)

where suffix T refers to the transition point. Our problem is then effectively solved when we have determined the function h(M.,), the relation between H and x and have decided on the valties of the constant C^ and n.

To determine h(M ), v/e note that for fully turbulent

flow along a flat plate at zero incidence F ( X ) = 0, and G ( X ) = h(M ), and hence equation (9) yields

n n-1 n n-1 2 1 n-1 h(M^). X ,

from which it follows that

n-1

2 - d^ - U ;

b-1 ''d

• \

^1 1 n-1 n-1

hCt!^)]/

= ^rRx^"L^(Mi)

n n-1 w '•^1 1.

Lhftvj - .

(10) where w \

2 ]

1 V 1

is the value of -r in incompressible flow (see 2

Pl^1

equation 5) at the same value of R . Thus

h(M^) w , n —, T \ j n-1 w

- P l ^ l / / l P l \

= Lff / °f i j

n I n-1 (11) /We r e q u i r e , . ,

-9-Vfe require therefore data on the ratio c~/c„. on a flat plate and its variation with l.fe.ch number. The available experimental evidence is scanty and is not as consistent as one would wish. Perhaps the most reliable data are those provided by Coles , who has deteniiined local skin friction coefficients

on a flat plate by a direct method of force measurement and his restilts for the ratio cVc_. as a function of Mach number M are reproduced in Fig. 1. These results were obtained at a Reynolds n\mber of 8 x 10 , Other experimental results showing a similar fall of cVc„. v/ith I/Iach number, although not fully agreeing quantitatively with Coles' results, have been reported

4 5

by Wilson and Eckert . There are various theories which have been developed from which values for c V c „ , may be derived, but all such theories are extrapolations from incompressible flow theory and the results depend critically on the assvimptions under-Ijring the mode of extrapolation. However, the theory that gives closest agreement with the results of Coles is that developed by Cope based on an extrapolation of the familiar 'log' law of incompressible flow, Monaghan , following what are essentiany the same lines of arg\;ment as Oope, obtained the f c m owing

interesting result, Y/rite 0.^^,, for the overall skin friction coefficient based on the density at the surface (o ), i.e.

w

nc

u

T dx w Cp„ = ^ , where ö i s t h e c h o r d l e n g t h , •ÖO u . - w 1

and d e f i n e a Re3molds number R = 1 "^w 1 T, I 1 1 ƒ '^w\ 1

W ^ ^W ^ ^1 '^

w li ' T \_ li / I p. / T

f*vv w ^ "^w ^ " ^ 1 ' v/

where T. i s t h e t e m p e r a t u r e j t i s t o u t s i d e t h e boundary l a y e r and T i s t h e t e m p e r a t u r e a t t h e w a l l . Then Monaghan's asstiraptions

w

led him to deduce that the relation between C_, and R is the Pw w

same as that between C„ and R in incompressible flow. Similarly, the relation between the correspondingly defined local values of the skin friction coefficient c.^, and the local Reynolds number R is also the sa-ie as that between c„ and R in incompressible

xw f X "^ flow. Corresponding to the accepted empirical incompressible flow relation betvreen Cj, and R, due to Prandtl and Schlichting , viz.

Cj, = 0.455 /(log R ) ^ - ^ ^ (12)

9

Schlichting has also deduced the relation

c^ = (2 logR^^ - 0.65)^-^ (13)

If we accept Monaghan's result, then we have that for compressible flow

'fw

= (2 log R _ - 0,65)"^-^

_xw

and hence °f • p^ = ^ 2 log

w

w

-0.651 1-2.3 ,0)

Using the gas law and the relation |i <x T we therefore have

Hence C f = rp

w

2 log R - 0,65 + 2 (2+cü)log(^ -2.3 2 log R^ - 0,65

i

\

c,p. ~ T < T ^^ ^ I 2 log R^ - 0.65 + 2(2+CA))log Y-w Similarly, fran equation (10) we deduce that

2.3

. . . ( 1 4 )

C.

F

' p i

l o g R

^ l o g R + ( 2 + a ) ) l o g ^ \

^- TO- J 2.58 (15)

IVe are confining ourselves t o the case of zero heat t r a n s f e r , and

10

f o r our purposes there can be l i t t l e e r r o r i n using S q u i r e ' s

relation for the ratio T./T , viz,

T w '

T w

1 , i c Ü Mf

.V3

-1 ₍₁₆₎

I n F i g . 1 t h e r e l a t i o n g i v e n by e q u a t i o n (14) f o r R = 8 x 1 0 , o" = 0.72 and Ü) = 8/9 i s compared v/ith t h e e x p e r i m e n t a l r e s u l t s of Coles and i t w i l l be s e e n t h a t t h e y a g r e e remarkably w e l l . T h i s l e a d s u s t h e n t o l o o k w i t h some c o n f i d e n c e t o the r e l a t i o n s g i v e n by e q u a t i o n s ( l 2 ) and (13) t o e n a b l e us t o deduce t h e f u n c t i o n h (M ) .

At f i r s t s i g h t i t would seem t h a t e q t i a t i o n s ( I I ) and (I4) p r o v i d e t h e answer, b u t i t Tiall be n o t e d t h a t t h e y l e a d t o t h e r e s u l t t h a t h(M ) i s a s l o w l y v a r y i n g f u n c t i o n of R and t h e r e f o r e of x, and t h e p o s s i b i l i t y of any v a r i a t i o n of h(M ) w i t h X on a f l a t p l a t e was i g n o r e d i n the d e r i v a t i o n of e q i i a t i o n ( I I ) , I f we a l l o w f o r t h i s p o s s i b i l i t y and w r i t e h ( M . , x ) i n s t e a d of h(M ) , t h e n r e v i s i n g t h e argument t h a t l e d t o e q i i a t i o n (IO) we s e e t h a t n n-1 n n-1 C2 R 1 [Xx. n-1 h(M , x ) , dx

TECHNISCHE TjOGESCHOOL

VLiEGTUiGBO ü vV KUNDE

1 1

-v/here h(M ,x) is a mean value of h(M.,x) from the leading edge to the position x. Hence, putting x = 1, the non-dimensional chord length, we have at the trailing edge

n t 1 n-1 • ^2 n-1 i n p Ü n-1 _h(r.i^,i) But 0 = 2 C„, and and therefore 1

-" . c.

R"

^-^

n-1 n n-1 ' 2 ~ P i

J

= 2 C „ . , _n .n-1 h(M^,l) = ( C p / C p . ) ^ - ^ . (17)

I f vre are t o ignore the slow v a r i a t i o n of h(M ) with R i n o\rr general method of p r o f i l e drag c a l c u l a t i o n , then i t seems l o g i c a l t o take a mean value averaged over the whole chord. Consequently, we propose t o tise equation (17) t o determine h(M.) and not

equation ( l l ) , i . e , we s h a l l take n

h(M^) = ( C p / C p . ) " - ^ (18)

the ratio (C_ / C„.) being asstmed to be given for each Reynolds number considered by eqtiation (15). This ratio is shown plotted in Pig, 2 f or R = 10 , 10^ and 10^, and the validity of equation

(l8) can to some extent be justified a posteriori by the relatively small differences between the three curves shown,

Per the variation of H with x we assumed that H is a function of M, only, the relation being the same as that for flow past a flat plate at zero incidence and lïiach number M . Maldng the plausible assumption that the total energy is constant across the boundary layer Cope has evaluated the relations

between H and M. for boundary layer velocity distributions following various power laws. These relations are reproduced in Fig. 3 and it vri.ll be seen that H is not particularly

sensitive to the power law assumed, and for the purposes of this investigation it is probably sufficiently accurate to take the relation appropriate to the l/9th power law, as this is consistent vTith the value of n equal to 6, which for reasons described in the next paragraph was the value chosen.

In order to choose suitable values of n and C„, values were sought that gave the best fit with the accepted empirical incompressible overall skin friction coefficient formula for a

flat plate at zero incidence v/ith fully turbulent boundary layer viz.

-j2.58

Cj, = 0.455 / [ l o g ^ o E j , (12)

over the range of Reynolds number considered which v/as 10 to 10 . It is clearly possible to choose values of n and C„ to give a close fit with this formula over small specified ranges of Reynolds number, allowing the values of n and Cp to change from one range to the next. However, it was found that by taking n = 6, and C_ = 0.00878, wiaich lead in the incompressible flow-case to the relation

Cp = 0,0450 R"^/^ (19)

agreement t o within 0.0001 was obtained with the r e l a t i o n given i n equation (l2) over the Reynolds number range considered. This i s i l l t i s t r a t e d i n ï i g . 4 vrfiere the two r e l a t i o n s (equations 12 and 19) are compared. I t i s doubtful whether these v a l u e s of n and C- could be a p p l i e d much outside the Reynolds number range considered vrf.thout exceeding the above order of e r r o r , and were i t decided t o extend the range of Reynolds number then other and more s u i t a b l e v a l u e s of n and C^ would be r e q u i r e d for the e x t e n s i o n s .

Having determined 0 as a function of x we can obtain the skin f r i c t i o n d i s t r i b u t i o n making use of equation (7), t h u s s

->. 1

2x

- JL

w o r> T5 n - 1 2 c = 2 = 2 ^2 ^ • Pl^1 Pl^1 ^1 = .01756 p^u^ 2 I ^1 ® ^ ^1 n-1 . h(M^)

_ ^ K

(20)

n-1 h(M^)

V/e note that the displacement thickness 5 is given by

6* = H.e (21)

2,2. Axi-symmetric flow 11

Mangier has demonstrated that for flow in the laminar boundary layer on a body of revolution a transformation exists which will correlate the flow with that in a laminar boundary layer in two dimensions. The proviso is made that the body is sufficiently slender for the boundary layer thickness to be small compared with r , the radius of cross section of the body. The following discussion reproduces his results but involves a

different approach which has some intrinsic interest.

The monentum equations of the botindary layer in two dimensional

-13-and in axi-symmetric flow

are.-and

0' + 0 P P 0' + 0 a a

uj p;

( H . 2 ) - ^ + : i

u^

u:

T _ ^ 2 Pl^1 p ;

wa

(H+2) ^ + 114.-2 U , a ' u. p, r „ , 2 ' (22) (23)

where suffix p r e f e r s to two dimensional flow and suffix a t o

axi-symmetric flow. I t should be noted t h a t i n axi-symme t r i e

flow the displacement and momentum thicknesses are defined by

05 6 * =

U

0^

2_

_{cos . <3y} and > - . . . (24)

0 =

a

•iS

^1^1 L

. . j . . . „ 3 ^ ; ( , . H . ) .

where 1^ is the angle between the tangent to the nerjdian of tha body and the axis; x and y are measured along and normal to

the meridian.

It will now be assunied that to every axi-symmetric boundary layer there exists a two dimensional boundary layer such that at corresponding stations the boundary layer velocity profiles are the same except for a change in the scale of y, u being the same. The conditions in the external flows will be assumed to be the same and isentropic, and the plausible assumption will also be made that the relation between temperature (T) and u at corresponding stations in the boundary layer are the same. Consequently, the relations between o and u as well as the

temperatures and viscosities at the wall (T and [x ) are the same. Accepting the proviso that the boundary layer thickness is small compared with r , it follows that the term '^ cos 1 ^

o r can be neglected compared with unity in equation (24) ° and

consequently

fa «

0

P P

But, it follows from the abo'/e assumptions that = r^ , and H = H

a

wa

wp

óu ~ 0U wa

wp

= V'

/ B e nee

Hence T 0 wa p T " 0 wp a (25) E q u a t i o n (23) can t h e r e f o r e be v / r i t t e n (r 6 ) ' + (r 6 ) ^ o a ' ^ o a ' u ! o! ^. , x 0 r

(H .,2) - i + - 1 = - ^ i ^ ^

(26)

Now, consider the transformation

p - d^

r^ dx ,

o a ' (27)

where d is some reference length. Equation (26) can then be written d o a p L_

^m

(H +2) du dp, u. dx p. dx 1 P • 1 P 0 .d T " r 0 • 2 • o a p,u, ^1"1 (28) Comparing t h i s e q u a t i o n w i t h e q u a t i o n (22) vre s e e t h a t t h e y a r e i d e n t i c a l i f r 0 o a d. (29)

and i f x i s the c o o r d i n a t e p a r a l l e l t o t h e s u r f a c e i n the two P

d i m e n s i o n a l flow, i . e . ccorresponding s t a t i o n s a r e r e l a t e d by t h e t r a n s f o r m a t i o n ( 2 7 ) . Summarising t h e argument we see t h a t v d t h t h e assumption of s i m i l a r v e l o c i t y p r o f i l e s f o r t h e flovra p a s t a two d i m e n s i o n a l a n d a x i - s y m m e t r i c shape, such t h a t c o r r e s p o n d i n g s t a t i o n s a r e r e l a t e d by e q u a t i o n ( 2 7 ) , and v/ith t h e same v a l u e s of t h e main s t r e a m v e l o c i t y a t c o r r e s p o n d i n g p o i n t s , t h e n t h e v a l u e s of the boundary l a j ' e r momentum t h i c k n e s s e s a r e r e l a t e d by e q u a t i o n (29). F u r t h e r , t h e f r i c t i o n a l s t r e s s e s a t the vra.ll a t c o r r e s p o n d i n g p o i n t s a r e r e l a t e d b y (from e q u a t i o n s 25 and 29) T 0 r wa p o X ~ 0 " d wp a (30)

If we define local Reynolds ntcibers R and R by xa xp then

^iVi

xa

R

^-iX p, 1 p'^1 xp

±

_1 T ^ l R / " r V x ; ~ _{wa ^ xa- o ^ a} [vx, _{a 2,} r dx o a 2 r . X o a (31) /This relation ,,,

-15-This relation was derived by Mangier, who then deduced that in

the particular case of the flow past a circular cone at zero

incidence in an otherv/ise unifona supersonic stream

° f p / ° f a = ^ / ^ - (^2)

In this case, of co'orse, the corresponding tvro dimensional flow is that past a flat plate at zero incidence vri.th the same constant pressure, velocity, etc. jtist outside the boundary layer as that behind the shock wave attached to the cone tip. Equation (32) refers to the local skin friction coefficients at points the same distance downstream from the cone nose and the plate leading edge, in terns of the density and velocity just outside the boundary layer in the two cases.

There is nothing in the above argument that refers explicitly to the laminar boundary layer and it might at first be thought that it could be applied with eqtial justification to the turbulent boundary layer. ïïe note, however, that the initial assumption of the existence of similar velocity profiles leads to equation (30) viz. T wa T v/p _ _ E -0 0 • a r o • d

Now consider the body of revolution to be a circular cylinder of large radius r . Then in the limit as r tends to infinity

o o the flow past the body must tend to that past a flat plate, and

hence the relation between x and 0 must tend to that between wp p

T and 0 . But we have accepted for incompressible flow on both theoretical and experimental grounds a relation between T and

wp

considered

T and 0 of the form given by equation (6), i.e. for the case

T = const. wp

1 n-1

where for laminar flow n = 2, but for turbulent flow n = 6. Hence for large r

0 1_ T = const, (0 ) ""^ , wa a' ' and therefore _W£ T wa 1 n-1 \ a /

We see that this relation is not consistent vd.th equation (30)

unless n=2, i.e. unless the boundary layer is laminar. It follows, therefore, that wliilst the assumption that similarity of velocity profiles exist between the tvro dimensional and

axi-symmetric flows does not lead to inconsistencies in the case of the laminar boundary layer it does so in the case of the turbulent boundary layer and the argument based on it is then invalid.

A different approach must therefore be used when v/e consider the turbulent boundary layer.

Reverting, however, for the moment to the axi-symraetric laminar boundary layer case and dropping the suffix a, it follows fran equations (IA), (27) and (29) that

Rfu. ^1 e-1 2 p.u,* r„ dx, '^l 1 o ' (33) 1 1 u

where we have put d = t, the reference length used in R. Further, from (3A) and (30)

( A + 12)U^

3 Rf 0

(34)

v/here /\ = R uj 0^ f^

_ 1

^J

In dealing with the turbulent boundary layer, we bear in mind the fact that for large r the axi-symmetric case must tend to the two dimensional case. This leads us to make the assumption that eqtiation (7) holds for axi-symmetric flow as well as for two dimensional flow. The momentum equation (equation 23) for axi-symmetric flow can then be written

(r^0)' + (r^0) F(x) = G^ 1 n-1

t^e)

1 n-1 G ( X ) . n n-1

_....

₍₃₃₎

v/here F ( X ) and G ( X ) a r e t h e f u n c t i o n s a l r e a d y d e f i n e d i n e q u a t i o n (8) when c o n s i d e r i n g t h e two dimensional c a s e . Like e q u a t i o n (8) e q u a t i o n (35) i s r e a d i l y i n t e g r a t e d t o g i v e ( r 0) ^ o ' x n n-1 _{exp. F ( x ) . ^ . dx}

-

K'^

n n-1 _ 1 _ fP^. - - • . C , R ^-^ n-1 2 n G ( X ) r " . exp

f\^

' - Ü F ( x ) . ~ r dx n-1 . dx

(36)

Having determined r 0 as a function of x by means of equation (36)

-17-we can obtain the skin friction distribution from equation (7), thus. -n -n T, Ï1-1 2 / 1 o c „ = 2 C R p.u. ' 1 n-1 = .01756 p^u^ 1 1 V V

1

„ /u.r 0 R 2 / 1 o . r^ n-1 . h(M^)" n-1 _1_ n-1 r "-' . h ( M j . o 1 (37)

3. The interaction of the boundary layer and the external flow 3.1. The effective displacement of the surface due to the

boundary layer (not in the neighbourhood of shock waves) It is usual to assume, as in incompressible flow, that the effect of the boundary layer on the external flow is equiva-lent to a displacement of the surface equal to the displacement thickness (or area) of the boundary layer. As far as the author is aware, however, there has been as yet no published justifica-tion of this assumpjustifica-tion for compressible flow. The following discussion follov/s in essentials the lines of the argument

1 2 developed for incompressible flow by Preston.

Consider first two dimensional flow. The equation of continuity is

a

T T_{d X}

^P^) +

TTr_{3 y}

(pv) =

0,

and hence, integrating vidth respect to y through the boundary layer, we have 05

[pvf

T T (p^)- <3y _{d X}

n8

a

a X

J

dS pu. dy ^ p^u^ . -d d X

_Pl"l ^S*-ö)J ^P^u^ f

Or

V-^1 " Pi V-^1 d d X (p^u^) dS' dx (38)

where V' is 'the angle, assumed small, that the streamline at the outer edge of the boundary layer (i.e. at y = 6) makes with the

direction y=0.

Now suppose the surface to be displaced outv/ards by a small distance 5 , and consider an inviscid flow past this

surface, keeping the coordinate system the same as in the original flow. The equation of continuity is unchanged and again, by integrating vidth respect to y from 0 to 5 we have

r- i^&

C^l

d X I

d p.u. / 1 - •^^ I . dy - p.u.5 •1 1 I p.u J ^ ^^ 1 o ^ ''^

1 1 dx

But f o r y < 5 * , u = 0, and for y > 5*, pu =

P^.^.,

» hence

and therefore

W 5

(5*-

5) ^ r ^ dS *

D^U^ - p^U^ d X ^Pl^V * dx

i.e. the direction of the streamlines at y = 5 in the inviscid flow with the displaced surface is the same as the direction in the actual viscous flow at y = 5 if we can assume o-u is the same in both cases. This fact does not by itself prove that the two flows for y .2>, 5 are identical. However, it follows from the above analysis that any other displacement of the surface other than 5 leads to flow conditions in the inviscid case at y = 5 differing from those in the actual case. It follov/s that if there exists an effective displacement of the surface to yield an equivalent boundary in inviscid flow, it must be a displacement equal to the displacement thickness of the boundary layer. A proof that such an effective displacement exists in the general case does not seem easy, but there are ample grounds for accepting the existence of such a displacement, as a working hyp)othesis, particularly when the equivalence of the boundary layer to an effective source distribution is borne in mind. It should be noted, however, that in the above discussion the usual assumptions of botmdary layer theory are implicit, hence the deductions would not necessarily be valid if the associated limitations required on boundary layer thickness and curvature of the streamlines and

surface did not apply. Thus, it is doubtful whether a displace-ment equal to the displacedisplace-ment thickness, as normally defined, v/ill produce an effect on the outside inviscid flov/ equivalent to that of the boundary layer in regions of high curvature of flow or surface. It is possible, however, that by modifying the

-19-definition of the displacement thickness on the lines suggested in Ref. 12 it could still be shown to be equal to the effective displacement of the surface in such cases. But regions of high flow curvature on aerofoils or bodies generally result from the influence of shock waves on the boundary layer, and such cases require special treatment.

In the case of axi-symmetric flow we note that the equation of continuity can be written

J ^ (pru) + j ^ (prv) = 0,

where x and y are the usual curvilinear coordinates parallel to and normal to the meridian profile of the body considered, and r is measured from the axis of sjrmmetry. Again integrating with respect to y between the limits 0 and 6, multiplying by 271: we get rs6 - |

27i: (r +-6 cost/-) p.v^ = - "T

o ' ^ 1 1 a

_{X L} 0

27t (r + y cos i » pu dy i

+ p u 27;(r + 5 c o s l ^ ) . ^ , .,,,(39)

where r is the radius of cross-section of the body, and !>• is the angle between the tangent to the meridian profile and the axis.

We now define the displacement area Ac* by

At* = 2-^ r . 6

'5 o p^u^

27c(r^ + y cos-ï»(p^u^-pu).^,

and we define the boundary layer area Ac try

n5

2x(r + y ooslJ*-) dy.

h

o

It follows from equation (39) that

Pi^i

^""K •" ^

°°^^) =

TH \

Pi^i ^ 8 * -

hH •"

Pi^i d T

f

(AJ.*

-

A ^ )

, dA * 7

We now consider the surface displaced outwards normal to itself a small distance e, such that the area A traversed in the

e

displacement normal t o the surface i s equal t o Ac* , and we

consider inviscid flow past this displaced surface but as before keeping the coordinate system undisplaced. From the equation of continuity we again obtain at a distance 6 from the surface that P 18 "1 2?; (r + 6 cosii^). (DV)S. = - -r— o ' ^ '8 a x %_g 27r. (r + y cosli'')ou dy

:

l

dAg •^ Pl^1 - d ^

But for y < e, u = 0, and for y 3 e, u = u , and hence 05 2% (r^ + 8 cos 1^2-) (pv)g = - Y^ '" " + P.u.

r^

J

dAc 0 271 (r^ + 8 cos 1^2)dy 1 1 dx

a

a X D.U. ( A C - A ) + p.u. — T -1 -1 8 e M '^l

1

dx

P i ^ i l — r r — diF (Pi^i) ^-^

1 1 (since Ac* = A ) . 6 e

Hence, comparing this equation with eqtiation (40) we see-, as in the two dimensional case, that the directions of the stream-lines at y = 8 in the inviscid flov/ considered are the same as in the viscous flow if p.u is the same. With the same argument and provisos as before we deduce that the required effective dis-placement of the surface eqtiivalent in effect on the external flow to the boundary layer is e, i.e. is such that the area normal to the surface traversed in the displacement is equal to the displace-ment area.

In regions other than in the neighbourhood of shock waves, we can allow then for the effect of the boundary layer on the

external flow by assessing the effect of the equivalent surface dis-placement. Strictly the boundary layer development should then be recalculated using the modified external flov/, but the calcula-tions that have been made to date indicate that at least in the case of two dimensional flow such further recalculations are unnecessary.

It is clear that the effective displacement of the surface will result in an increase of pressure over the whole

-21-surface of the wing or body, the effective slope of the -21-surface dS*

relative to some datum being increased by -rr in the case of dfi

two dimensional flow and by •^ in the case of axi-syrametric flow.

To estimate the effect of this increase of slope in two dimensional flow it is assumed that it is sufficiently accurate to consider the flow outside the boundary layer as a simple wave flow. It then follows that the pressure increase due to the pressure of the boundary layer is

2 2 YP^M^ ^ * P^u^ ^ *

VM^^

- 1

èr^-

1

From equation (4I) the resulting changes in drag, lift and pitching moment can be readily calculated. It v/ill be clear that with the boundary layer turbulent the increment in pressure

/\ p is generally greater than when the boundary layer is laminar, and this effect will become increasingly marked with increase of Reynolds number. Consequently, it is possible for the resulting

change in drag to be negative when transition occurs at about 0. 5c, although v/ith a fully laminar or fully 'turbulent botmdary layer the integrated effect of this presstire Increment on drag is generally positive. These remarks are illustrated by the results of some specimen calculations shown in Fig. 5.

In the axi-symmetric case the problem of estimating the effect of the equivalent displacement is perhaps less simple. Over the forebody any of the simple methods for calculating the

13 pressure distribution developed by Bolton Shaw and Zienkiewicz can be readily applied with adequate accuracy. Over the rear the taethod of characteristics or the second order method of 'Van

.11

Dyke is always available, but it is hoped to develop simpler methods of adequate accuracy for the problem in mind. Such methods are under investigation.

3.2. Shock wave - boundary layer interaction effects

A reviev/ of current knowledge on the complex nature and problems of shock wave - boundary layer interaction effects has

15

been given by Zienkiewicz , and only a brief summary of the main points relevant to the problem under consideration as well as of further information that has become available since Ref,I5 was VTritten need be given here,

It is clear from the available evidence * that in

practice the conditions at the leading edge of a sharp nosed aerofoil or body may differ markedly from those predicted by

inviscid flow theory, and these differences are due to interaction effects between the leading edge shock waves and the boundary layer and in some measure to the fact that the leading edge of a vdng is never truly sharp. Nevertheless, these differences appear to have relatively little effect on the pressure distribu-tion except very close to the leading edge, and the pressure distribution obtained from inviscid flov/ theory corrected for the displacement thickness effect of the boundary layer gives very close agreement with experiment except perhaps within about 0.02c of the leading edge. At the leading edge the rate of growth of the displacement thiclcness becomes infinite according to theory and the corresponding correction to the pressure according to equation (4I) becomes inacceptable. However, a plausible extrapolation of the pressure distribution forward from 0.02c can readily be made and is recommended, the resulting possible error in overall characteristics being small.

It is near the trailing edge of a wing, and also presumably at the rear of a body, that the interaction betv/een shock waves and boundary layer can have a most profound effect on the pressure distribution and hence on the overall aerodynamic characteristics. It is of course well known that in a region of interaction between a shock wave and a boundary layer, the

pressure rise across the shock is difftised upstream and downstream in the layer, and both boundary layer and shock are to some extent modified by the interaction. Thus, the boundary layer on an aerofoil section near the trailing edge will be subjected to a positive pressure gradient due to the shock wave springing frcm the rear of the wingj the boundary layer will thicken in that region as a result and may separate before reaching the trailing edge. The effective shape of the wing, however, will then be such as to modify the shock pattern and strength at the trailing edge. The resulting effect on the pressure distribution and hence on the lift, drag and pitching moment may be considerable if the separation is extensive. In the main region of separation the pressure is generally nearly constant, and our problem

therefore reduces to that of determining the surface pressure and extent of the separated region in any given case. The subject is still largely unexplored, and the data required for a simple empirical approach is not yet available. However, there are indications discussed below that the problem may be resolved v/ith the accumulation of sufficient experimental data for which a relatively modest experimental programme may suffice. The work

-23-of Holder and Gadd ' ^ Drougge and Lighthill are valuable contributions in this connection.

It is known that with the boundary layer laminar

approaching the shock separation of the layer is likely to occur even with relatively weak shocks, both theory and experiment suggest that shocks with a presstire rise ratio of the order of 1.05 - 1.1 v/ill cause separation. On the other hand with the boundary layer turbulent much stronger shocks with pressure rise ratios of the order of 1.7 - 1.8 are required for separation to occur. This latter fact indicates that for a wide range of practical cases with the boundary layers turbulent ahead of the trailing edge separation v/ill not occur. This is forttmate from the point of view of the problem discussed in this paper because it appears that the thickening of the turbulent boundary layer associated with the shock v/ave is confined, in the absence of separation, to a distance ahead of the trailing edge less than the boundary layer thickness, and therefore the effects of such thickening can be ignored. In the first instance it is proposed to consider, in the main, cases for computation based on the methods described in this paper where transition occurs in the boundary layer ahead of the trailing edge, and where the trailing edge shock presstire rise ratio is less than about 1,8, consequently the problems introduced by separation effects will not arise in such cases.

Nevertheless, some of the factsrelating to conditions when boundary layer separation has occurred are worth reviewing.

15

Zienkiewicz has analysed available data obtained with a ten per cent thick biconvex section and a nine per cent thick

symmetrical section with a 4 trailing edge angle tested at the N.P. L. The tests referred to v/ere made at Reynolds numbers in

the region of 0.5 to 1.0 x 10 and the boundary layers approach-ing the trailapproach-ing edge were laminar but became turbulent after separation. The Mach numbers covered varied from about 1.6 to 2.5. Zienkiev/icz showed that for these results the values of . the ratio p /p (where p is the nearly constant surface

sep. o sep.

pressure in the region of separated flow and p is the un-disttirbed stream static pressure) fell reasonably close to a single curve as a function of v/ing incidence up to quite large angles of incidence, the variations in trailing edge angle and Mach number covered appearing to have little significant effect

(Pig. 14, Ref. 15). It is almost certain that if a larger

range of Reynolds number had been covered scale effect would have 18 19

been revealed, thus the results of Gadd and Holder's ' work on

shock wave boimdary layei* interaction on a flat plate suggest Po - P i

that —^^ OC — A T approx. It is also possible that there

•^o R''/^

is a v/ing thickness effect in addition and further data are clearly needed. Nevertheless, the simple nature of this result is encouragi.ng. Further, Zienld.ev/icz found that the extent of the separated region of flov/ also agreed fairly closely v/ith a single cturve when plotted as a function of incidence for the same ranges of Mach number and wing shape (Pig. 16, Ref. 15). In this connection Gadd and Holder's results suggest a scale effect such that the separation distance is approximately inversely propor-tional to the Reynolds nuTiber. It v/i,ll be readily appreciated that if a set of such simple relations can be established for a practical range of Haoh numbers, Reynolds numbers and wi.ng shapes it shotild be possible to predict not only the effects of viscosity on drag, but its effects on lift, pitching moment and aerodynamic centre, etc., over a wide range of incidence. It is one of the consolations of the study of purely supersonic flow that the pressure distribution and the overall aerodynamic characteristics are not markedly more sensitive to conditions at the trailing edge than elsewhere, and no greater accuracy is needed in determining the boundary layer development for the purposes of estimating lift and pitching moment than is needed for determining the drag. In marked contrast we may note that for subsonic flow the lift and pitching moment but not necessarily the drag can only be adequately estimated when the extension of the Kutta-Jotikowski condition at the trailing edge for viscous flov/, first formulated

12

by Preston , has been properly applied. This is a process involving considerable computation, and it has so far only been applied in the absence of separation.

In the case of the turbulent boundary layer v/hen the

pressure ratio across the shock is large enough to cause separation, the available data indicates that scale effects are very small, but no comprehensive information on other effects is yet available.

op

However, Gadd'' has suggested a simple if approximate hypothesis for determining the pressure in the region of separation that gives values in reasonable agreement v/ith available experimental data. His argument is that except very close to the wall the effects of presstire gradient on the flow in the boundary layer are much greater than the effects of friction, and it is assumed that

the lower limit of the region in which this is so is v/here the velocity is 0.6 of the velocity just outside the boundary layer. This corresponds to the 'shoulder' of the zero pressure gradient boundary layer velocity distribution. It is then argued

-25-accordingly that the separation pressure is that required to bring the air at this 'shoulder' to rest isentropically, this leads to values of p /p. of 1.84 and 2.51 at values of M. of 2.0 and

s e ^ a i I

3.0, respectively, where p. and M. refer to the pressure and Mach number ahead of the shock. The corresponding values of the separation pressure coefficient c = (p - p ) / 2'P>i''^>i are 0.3 and 0,24.

The available data for the extent of the separated

region cannot however be so readily generalised. Gadd and Holaèr have investigated the interaction of oblique shock waves of various

strengths and the boundary layer on a flat plate over a range of Iviach numbers from 1.5 to 4.0 and their results show that with

increasing shock strength the separation distance increases. Expressing the pressure rise across the shock as a coefficient

p

c in terms of ip^-ii^, then with c = 0 . 5 the separation distance is about 355 whilst with c = 1 . 0 the separation distance is

* ^ 20 about 1408 for the Iviach numbers tested. However, Drougge

has investigated the flow in comers less than I8O where the shock is generated by the comer and the flow bears more direct similar-ity to that at the rear of an aerofoil than do the cases

investi-gated by Gadd and Holder. His separation distsmces are consider-ably less than those of Gadd and Holder for a given value of c ,

thus for c = 0.5 Drougge obtains a separation distance of the order of 108 and fcrc = 1 . 0 the separation distance is about 258 .

P

The two types of experiment exemplified in the work of Gadd and Holder on the one hand and Drougge on the other differ in important respects, and it is not altogether surprising that their results do not agree. However it is clear that without further

experimental evidence one could not confidently apply the results of either set of experiments to the problem of the separation distance on an aerofoil, although it is likely that the results of Drougge vriLll provide a closer estimate.

Acknowledgements

During the course of the development of the ideas described in this paper the author v/as fortunate in securing the interest and assistance of Mr. J.E. Daboo and Mr. H.K. Zienkiewicz whilst they were students at the College of Aeronautics. The work of the former represented a valuable first attempt at some of the more important problems for aerofoils discussed in this paper, it was not published becatise subsequent experimental data led to a revision of the methods adopted. The work of the latter was published in College of Aeronautics Report No. 49.

REPERSNCSS No. 1 .

2.

Author Young, A.D. T i t l e

Young, A.D, and Winterbottom N,E.

3.

4.

5.

Coles Yaison, R.E. Eckert, E.U. 6. Cope, W.P. 7 . Monaghan,R.J.

8.

9.

10. 11. 12.

13.

Prandtl Schlichting Squire, H.B, Mangier, Y^. Preston, J. H. Bolton-Shav/ and Zienkiewicz 14. Van Dyke,M.D. 15. Zienkiewicz, H.K.

Skin Friction in the Laminar Boundary Layer in Compressible Plow.

College of Aeronautics Rep. No. 20 (l948). also The Aeronautical Quarterly Vol. 1, Aug. 1949, p. 137-164.

Note on the Effect of Compressibility on the Profile Drag of Aerofoils in the Absence of Shock Waves.

R. and M. No. 2400, (l940).

Direct Measurement of Supersonic Skin Friction. J. Ae, Sc. Vol. 19, 1952, p.717.

Turbulent Boundary Layer Characteristics at Supersonic Speeds - Theory and Experiment, J.Ae,Sc. Vol, 17, 1950, pp. 585-594.

Characteristics of the Turbulent Boundary Layer on a Flat Plate in Canpressible Flow from

Measurements of Friction in Pipes, J,Ae.Sc, Vol. 17, 1950, p. 573-584.

Notes and Graphs for Boundary Layer Calculations in Compressible Plow. (l95l).

A.R.C. Current Paper No. 89.

Comparison between Experimental Measurements and a Suggested Formula for the Variation of Turbulent Skin Friction in Compressible Plow. A.R.C. Current Paper No. 45.

Goettinger Ergebnisse, 4(l932), p. 27, Aerodynamic Theory, 3(l935) p. 153.

Ingenieur-Archiv 7(1936) p. 29, Modem Developments in Fluid Dynamics, 2, p.365. Heat Transfer Calculation for Aerofoils. R. andM. 1986, 1942.

Boundary Layers on Bodies of Revolution in Symmetrical Plow.

Volkenrode R. and T, No. 55, April 1946. The Calculation of Lift, Taking Account of the Boundary Layer.

R, and K, 2740, 7^ ^ $"

The Rapid Accurate Prediction of Presstire on Non Lifting Ogival Heads of Arbitrary Shape at Supersonic Speeds.

English Electric Rep, No. L.At. 034, 1952. Also, A.R.C. Rep. No. 15,36l.

First and Second Order Theory of Supersonic PD.ow past Bodies of Revolution,

J.Ae.Sc. Vol. 18, 1951, pp.161-178.

An Investigation of Boundary Layer Effects on Two Dimensional Supersonic Aerofoils.

College of Aeronautics Rep. No, 49 (1951). /16. ...

-27-No. Author 16. Valens i, J. and

Pruden, P.W.

17. Bardsley, O,

18, Gadd, G,E, and and Holder, D.W, 19. 20. Drougge, G. 21 Lighthill, M.J. 22. Gadu, G.E. Title

Scane Observations on Sharp Nosed Profiles at Supersonic Speeds.

R. and M. 2482.

The Conditions at a Sharp Leading Edge in Supersonic Plow.

Phil.ï,iag.Vol.42, 1951, pp. 255-263. The Interaction of an Oblique Shock Wave with the Boundary Layer on a Flat Plate. Part I. Results for M=2.

A.R.C. Rep, No. 14,848, April 1952, Part II, Interim Note on the Results for M=1.5, 2, 3, and 4.

Experimental Investigation of the Influence of Strong Adverse Pressure Gradients on

Ttirbulent Boundary Layers at Supersonic Speeds, Presented to the 'V'lII International Congress

on Theoretical and Applied Mechanics, Istanbul, 1952.

On Boundary Layers and Upstream Influence, A.R.C. Rep. No. 15,297. Oct. 1952.

On the Interaction with a Completely Lamiriar Boundary Layer of a Shock Wave Generated in the Main Stream.

VUEGTUiGCÖ U V/KUNDE

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_{FIG. 3.}

VELOCITY POWER LAW ASSUMED 18 t6 14

H

(2 IO / ' ' ' ' '/sth .th h

'U'

M,

VARIATION OF H WITH Mj FOR VARIOUS VELOCITY POWER LAW

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o-cos

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FOR FLAT PLATE SKIN FRICTION WITH PRANDTL- OF SURFACE CAUSED BY THE BOUNDARY LA'ER AS FUNCTION