The profile drag of yawed wings of infinite span

TECHNISCHE HOGESCHOOL VLIEGTUIGBOUWKUNDE

REPORT No. 38

" 9 Sep,1950

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE PROFILE DRAG OF YAWED WINGS

OF INFINITE SPAN

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

T. B. BOOTH, B.A., D.C.Ae.

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

% -

lp"

TECHNISCHE HOGESCHOOL VLIEGTUIGBOUVrKUNDE R e p o r t No. 3 8 . May, 1950. T H E C O L L E G _E OF A E R O N A U. T I C S , C R A N F I E L D

THE PROFILE DRAG OF YAWED WINGS OF INFINITE SPAN b y -A. D. YOUNG, M. -A. , A . F . R . A e . S , and T . B . BOOTH, B.A. 5 D . C . A e . ( D e p a r t m e n t of A . e r o d y n a m i c s ) oOo S U M M A R Y

A method is developod. for calcnlatlng the pi-ofile drag of a yawed v/ing of infinite span, har.ed on the assumption that the form of the spanv/ise di.striTbution of velocity in the boundary layer, whether lamitjar or turhulent, is insensitive to the chordwise pressure distrihution. The form is assumed to he the same as that accepted for the boundary layer on an unyawed plate ViTith zero external pressure gradient. Experi-mental evidence indicates that these assumptions are reasonable in this context. The method is applied to a flat plate and the NACA 6i.|.-012 section at zero inci-. dence for a range of Reynolds nurabors betv/een 10° and 10°, angles of yaw up to k3', and a range of tr'ansition point positions. It is shown that the drag coeff J^-i ^^nt

of a flat plate varies with yaw as Cos^/\(where A i s the angle of yaw) if the boundary layer is completely

laminar, and it varies as Cos^'-Z^/x ^f the boundary layer is completely turbulont. The drag coefficient of the NACA 6[|T012 section, hovifever, varies very

closely as Cos-Afor transition point positions be-tween 0 and 0' 5 c. Further calculations on v/ing sections of other shapes and. thi okrinyses and more dehRiled (^xyeriracutql checks of the banio n R sumptions nt higher Reyuni ds nnmi'ers are dosirablc,

Part of this paper appeared in the Thesis presented by T.B. Booth for the College Diploma, 191+9.

•

N O T A T I O N

x,y curvilinear orthogonal coordinates on thel See surface of a body (y is in spanwise ) para, direction for yawed infinite span wing). ) 2. 1,and

) Pig.i4. 2 normal distance from the body ) u,v,w velocity components in the boundary layer

in "I he x,y,z directions, respectively. U.,V velocity components just outside the

boundary layer in x,y directions, respectively. V velocity vector = (u,v,w).

0) curl V.

Q undisturbed stream velocity.

Ü ,V components of Q in direction of chord and ° span resoectiveïy, of yawed infinite wing. ^^ angle of yaw of wing.

u',v',w' turbulent fluctuation components of velocity in x,y,z directions respectively.

c chord of wing» measured normal to span, p pressure.

p density.

|j, coefficient of viscosity. V kinematic viscosity = i^i/p. D d.rag of unit span of the wing.

X component of D in chordwise direction, V component of D in spanwise direction.

Cp^£a,3] drag coefficient at a Reynolds n-umber a and an angle of yaw 3.

R .Reynolds number = Q c/v. Ö boundary layer thickness.

öy

I h -

ir-)

dz

^-'0 u ^ V u J'

®xx j ^ ^ 1 1 - T ^ m z n(j

yy Jo ^

/ ' Z_fi _ .4-ldz

7(^-v->

m

/ « x y

N O T A T I O N '^contd. ) "xy «yx « X « y "^xz \ 2

(^-i

('^i

K-A

y K

A

z, f v_ /• _ u_ \ ^0 v^ r ••

v^/"^

component of shear stress in chordv/ise (x) direction.

component of shear stress in spanwise (y) direction,

value of T,,^ at surface,

va'^ue of 1' at surface.

curvature of lines y = const, x - const. respectively.

n (H f 5)/2 at the trailing edge,

"T

'^xx

•

:^1

8-.

lllj. ' c X

The letters Ï.E. refer to the trailing edge L.E. to the leading edge.

1. Introduction a-nd Roview._

The increasing use of sweepback to delay compressibility troubles has aroused interest in the behaviour of boundary layers on yav/ed and swept back wings, for we require an understan..ling of this

behaviour if we are to predict and control satisfac-torily the characteristics of swept back wings. This paper is concerned only with a limited and relatively simple aspect of the problem, viz. the profile drag of a yawed ?/ing of infinite span. It

is hoped, hov/ever, that apart from the intrinsic interest of the results, the discussion will add to the basic store of data and ideas needed to tackle the more general problem.

The treatment begins with the development of the boundary layer equations and the momentum integral equations for laminar and turbulent flow. The development of the laminar boundary layer equa-tions is .not^new, having in essentials been given elsev/here,''^'^^3s4.* it is however included here for completeness. It is shown that the chordwise flow is independent of the spanwise flow and hence the corresponding boundary layer equation can be solved by any of the accepted methods. A method for

solving the spanwise flow equation is suggested which is approximate but sufficiently accurate for the purpose in mind. For a flat plate at zero incidence it is found that the spanwise velocity profile is strictly/ similar at all points to that in

the chordv/ise direction, and the resultant direction of flow is everyYi/here that of the free stream. The skin friction coefficient at any point, as well as the,overall di^ag coefficient, is reduced by the factor Cos^Aduo to yaw, and thus the skin friction at a

'00int is the same as that on an unyav/ed plate at the same distance measured parallel to the undisturbed stream direction downstream from the leading edge. This result is peculiar to lam.inar flow in the boundary layer,

The turbulent boundary layer equations also lead to the conclusion that the chordwise com-ponent of flow in the boundary layer can be treated as independent of the spanwise component. The momentum integral equation of the former component can be solved on accepted lines to yield the compo-nent of the drag per unit span in the chordwise direction. The drag of unit span of the v/ing is the resultant of this component and the spanwise component. To obtain the latter it is assumed that the form of the spanwise velocity distribution in the boundary layer is independent of the chordwise pressure distribution and everwhere satisfies a power law, of the type normally assumed to hold in the turbulent boundary layer on an unyawed flat

plate in a zero external pressure gradient. This assumption, v/hich might be based somewhat loosely if intuitively on the argument that the spanwise condi-tions of flow correspond in some res-oects to the simplifying conditions usually assumed in the theory from which the power law is derived, is shown by experiments to be reasonable. For the calculations discussed in this paper the law assumed is the l/yth. power law. The spanwise momentxim, thickness at the

trailing edge is then derived in terms of the /chordwise

-2-chordvirise momentum thickness and readily yields the S'oanwise component of the drag. An expression for the resultanï. drag coefficient is then presented in a simple and concise form. For a fully turbulent boundary layer on a flat plate at zero incidence, v;ith the velocity distributions i.n both spanwise and

chordwise directions satisfying the 1/7th. power law, the drag coefficient is reduced by yaw by the factor Cos^'^^-V\' Here we may note that although the flow is everyv/here parallel to the undisturbed stream ' direction the skin friction at a point is not the

same as that on an unyam^ed plate at the same distance measured parallel to that direction from the leading

edge. Were it so, the drag coefficient would change with yaw as the factor Cos''/5/^^,

The method has been applied to calculate the drag coe.fficient of a flat plate and. of the NACA 6i|.-0l2 section at zero incidence for various transition positions, Reynolds numbers and angles of yav/. In the case of the flat plate (Fig. lb) the factor describing the variation of the drag co-efficierit with yav/ changes steadily from Cos^/5/\

1 / 2

to Cos /\ as the transition point moves back from the leading to the trailing edge, and there is no noticeable effect of Reynolds number on this factor within the range 10 to 10 . In the case of the 12 percent thick vi/ing sect,;i^on (Fig. 1a) the factor remains very closely Cós'/"'Afor transition positions from the leading edge to 0.5 c, and the effect of Reynolds number on this factor is very slight,

Preliminary experiments made on a fl^t plate at a fairly lov/ Reynolds number (about 10^) tend to confirm the assumptions and conclusions of the theory for angles of yav/ up to l;5°s but further experimental work at higher Reynolds numbers is desirable.

„3-2. Boundary Layer Equations. 2.1. Laminar boundary layer

The equations of m.otion of a steady viscous, incompressible fluid with negligible body forces are in the vector notation of Ref. 5,,

- V X cü = - grad (— + •§• v ) - v curl co. (1 ) The equation of continuity is

div V = 0. (2) Consider the flow past a surface and take

x,y,z as orthogonal curvilinear coordinates, such that z is the normal distance from the surface and

the lines x = const.. , y = const. , define an orthogonal network on the surface. ï'.'e v/ill now make the usual assumptions for the flow inside the boundary layer, viz. that the rates of change of the velocity compo-nents normal to the surface are large compared vi/ith the corresponding rates of change parallel to the

surface and that the boundary layer thickness is small compared v\[ith a representative linear dimensiori of the surface. We will further assume that, if ^ and

H ave the curvatures of the lines y = const, ana

X = const. , then the qaantities ^{ 6, H, ,o > o d^ X y X , ^2 dk .2 dkl . 2 a,<, ^3 a^K.. ''^ 6 '^s o ^x , 5 jy , Ö ^x , 3y 9 y " 9x gy2 — 0 pnr'iv ^^^ ^11 a m a l l cc'Tipared 9x^ 9x9y -^-"^^

with unity, where ö is the boundary layer thickness. With those assumptions the equations of motion yield the boundary layer

equations:-9 u equations:-9 u , equations:-9 u i o p , . o u /-, 1 9 P p 9x 1 9p p ay + + 3z^ 9z^ 9 v 9 v , 9 " ' I o u o V /I \

^ 9i- + ^' a? ^

^

'

^

' 9z '- - - - ^ " - ^ ^^^

The equation of motion in the z direction leads to the result that the pressure may be taken as con-stant across the boundary layer. The equation of continuity is

a^i , az . M _ n

(^\

95E •" 9? ^ 9^ ~ °

'^'''

For the case of the yawed wing of infinite span, ijve will adopt the notation illustrated in Fig. i+., viz. X is measured along a section profile from the leading edge in a plane normal to the span and y is measured along the span. In this case the rate of

-k-change of any quantity with respect to y is zero; and hence equations (3)? (4) and (5) become

-

t - » If

U -r-— + \lf -r— 9x 9z = = -V P 3- 3z2 ' a 2 . ,„2 ' (6) (7) 9z and H '• 9ÏÏ = ^' ^^^ At the surface u = v = w = 0

Just outside the boundary layer u = U.,v = V.,

^—• == •^— = 0, and the flow is irrotational, consequently o z o z 9U. "l 9x = -and au^ 9V^ 9y ~ 9x Since, however, 1 9p p ax ' = 0.

au

(9) 9V^ it follows that -r— = 0, oX i . e. V. = c o n s t . = V , vv'here V i s t h e component 1 0 0 ^

of the undisturbed stream velocity along the y axis. Equations (6), (8) and (9) and the cor-responding boundary conditions are identical with those for the boundary layer over the wing in an unyawed flow and with a main stream velocity = U = Q Cos/\. Hence, the chordwise flow in the

boundary layer is independent of the spanwise flow and may be derived by any of the v/ell-established methods. An approximate method fo^ solving for the

spanwise flow will be discussed in para.

3-2. 3-2. Turbulent Boundary Layer.

V'/ith the usual boundary layer assumptions the equations of motion for the turbulent boundary layer in three dimensions become

4 1 . ..|. _ | | _ 1 II _ i ! u _ a. ^ 2 l |_ ^ ^

(10)

(u'w)

- f ^ v u ' ^ y . (11)

"^ax ""ay ^ ''9z p 9y ^ "^^^2 9 x V / 9 y v /

^ (j^^)'.

9z (12) /where . . , .

-5-where undashed letters now refer to time means, the dashes denote turbulent fluctuations and a bar denotes a time mean. The third equation of motion may again be interpreted as indicating that the variation of pressure across the boundary layer can be neglected. The equation of continuity is

|u + 11 ^ |w ^ ^_ ; (^3) ax Ty -)s

For the yavi/ed wing of i n f i n i t e span, these equations

reduce to

I

4 f - I f - ? i ^ "^^ - feC-'^^-fefe^^'-• (^«

(ïïV^)-'

^i^^li^" J-fet^'--.v

au , 9w

v'v,^j, (15)

and ^ + 4^ = 0. (16) 9x dz • ^ ^

Again we note from equations (Ik), (15) and (16) that the chordwise flovif is independent of the spanwise flow and can be calculated as for the unyawed wing in the stream component normal to its span,

2. 3. Momentum Integral Equations.

It is usual to assume that the rates of change of the mean turbulent stresses in directions uarallel to the surface can be neglected compared

with the rates of change normal to the surface. Both the laminar and turbulent boundary layer equations

for general three-dimensional flow can then be Vifritten

9 T

9u ^,,au au _ 1 a^ 1 _ _ f ,

^ .

^'ax • • ay • "3z p ax ' p az

and

where T.,_ = ^i'{rr::)} a'„„ = ul^—), in laminar flow, and

9v ^v _!. av 1 ap 1 ' 'Vz f^a\ ^ax ^ay as p ay p az ' ^ ''

"xz \ 9 z A yz

-^xz ^ = ' 4 i ) " P ^ ^ ' ^ ' ' V z = ^ ( a i ) - p ^ ' ^ ' ^ i ^

turbulent flow.'

Outside the boundary layer U. ^"l , .. 9"l 1 ap and

' 9x ^ M ay p ax * ^'^''

u 1 + V

-

= .. 1 iE

(90)

^1 ax ^ M ay p ay '

^"^^^

If we now integrate equations (17) and (18) with respect to z from z = 0 to z = o, and make use

of equations (13)> (19) and (20) v/e get eventually /'^x^Q

6

-{^..X - h (P "l'^xx) * k (P "l\y)^ W-(f "l^x)

au.

+ ay and av

CyzX = I? (^ ^ i \ y ) ^ fe (F ^ I \ J ^ 9ré ^i^y)

9 V , ...- .,,N •^ a ^ r l P U l ^ x > (22) Where ( T ^ J and (T ) r e s p e c t i v e l y , a t t h e s u r f a c e , and a r e t h e v a l u e s ^f L and x , -. rv - ^ X z y z 'O ' " ' O '^

*x=/C - ï ï j ' " ' 'y=/ i^ -^>='

\ V ^ / O ' ''' " ^0 Ai 'XX

{U-^>' v^?,0-^,>

„ „ , „ V . V V ^ / • dz o I ^ I ' ü?'o u. V

u.;

yx I v A v J ^0 ^ '• Jo '

Por t h e case of t h e yavi/ed wing of i . n f i n i t e s p a n , t h e s e e q u a t i o n s r e d u c e t o

(^xzX = fe (P ^^'^xx) ^ ^ (p '^1< ) ' (23)

and N

(V),= fi(p^i V ) '

(2L^) ïïriting H = ó ^ / e , , e q u a t i o n (23) can be c a s t i n t h e X A X X familiar form

Ci!!

•^^ + ^ . - — ^ I H + 2 TT 2 ax u / ax X p u.

u^-

ax

[y

'xx (23^)

3. The Calculation of the Prof ile. Drag_ of,^a_ Yawed Wing. 3. 1. The drag components

'•'e can resolve the drag (D) of unit span of the wing into components along the x and y axis, .X and Y. Prom the above it follows that X is the drag of unit span of the wing when unyawed but with the same transition position in a stream of velocity U = 0 cos/\, where Q is the resultant undisturbed

o ^o ^ 0

stream velocity (see Pig. Lj.). Hence X can be calculated by any of the established methods (see for example,

Refs. 6,7 and 8). It remains then to determine the spanwise component of the drag, Y. To do this, vve note that, since the undisturbed stream is uniform and the wing section is constant along its infinite

span, Y has no form drag component but is solely the resultant of the frictional stressf-r ] . Thus, per unit span of wing, and for each surface

Y = ! (1. i , dx ( ' y - . ••>o r 0 ax ^"yx-= / P '^^, I T (9..T)* 'ix, from (21+), '0

P

^0 fjx -T.E. I -L.E. L -^T.E.

where L.E. and T.TC, refer to the leading edge and trailing edge, respectively. Here we have assumed that the momentum thickness 0 is zero at the

yx

leading edge and is continuous at the transition point, it being accepted that the stress fx^^ "] could not become infinite. Our problem then re-duces to the determination ofjO 1

yx -•T.E. Novif , f^ U /. V "\

vx=j v,c -vj'

= v7 rf,(' - v)

^-"l = Y~' ^ ^xx^

"^y-Where K / ^^ (l - ^ d z / |^ ^ - H Vzj

(26) 0 • ' / yn •' V

Let us suppose for the moment that we know the value of K at the trailing edge of the wing. From equa-tions (25) and (26)

'^ " P ^V.E. ^'O S.E, ^ _XX

But (see Ref 2 XX c where n = -.6) ^ ( ~ 1 2 5 T

^P

u / c

V

V ^ / T . E , 1-. Hence Y Sin A / ^ - Cos.A ' T.E. \ 'T.E./ ' / 2 .

'°o Y"'

D„ „ ./ X.

The resultant drag D is given by D = X Cos A + Y Sin/\, and therefore C.

;^fR,Al = 2-ö~ = ~ö~

C O S 3 A +

^ - . Cos2^ . S i n A

^L J ip Q^ c ip Uj^c ^p U ^ c

X f . 2 U^ ""''I

_ _—-(Cos A + C o s A . S i n ' ^ A Km -n ü )

ip U^ c ) ^•^-

" T . E . r 1 ^ ? P / ^n

= C^ R CosA.,0 . CosAJCoa A + SinV\ K^, ^ [jr^^

Du J I ^'•^•\^T.S

(27) Here C-Q fa;'^] denotes the drag coefficient of the

wing at a Reynolds number x and an angle of yaw 3-Equation (27) enables us to calculate the drag co-efficient of the yawed v/ing in terms of the existing results for the drag coefficient of the unyawed wing, the angle of yaw, the chordwise component of velocity just outside the boundary layer at the trailing edge and the quantities n and K at the ti-ailxng edge, The quantity n is known from the chordwise

calcula-tions, our problem is therefore solved when we have determined K

T.E.

3. 2. The Spanv/ise Flow in the Laminar Boundary Layer. Por our purpose it is sufficiently accurate to treat the chordwise flovi/ in the boundary layer by the Karman-Pohlhausen method or by one of the variants based on it (see, for example, Ref.9). Thus we

assume

9

-and if we s a t i s f y the boundary conditions

^ ^ = - 1 If' ^* ^ = 0, and u = U., 1 ^ = 2fu ^ Q^

9z^ p 9x 1 az ^^2

at z = Ö, we find

I

-

KD

-

Kf)

where v3 _ J 4 ..V . ., .s3

I f ) = <f)- Kf) ^(f) 'Kfl'ef0 - f:

and

A =

,aU Ö1 9p • 9x •

1

I t then follows t h a t

9xx = '^

37 A X _{315 ~ 91+5 " 9072}

If we likewise assume that

,2 _{- 3 U}

V^

= ^2(f)+l'2(f) ^ =2(1)

* Ml) •

a^v

and we satisfy the boundary conditions v -^—^ = 0,

at z = 0, and 9z 2

V = V. , :-^ = -—^ = 0, at z = 0, we find 9Z

^, = nt )•

i.e. the form of the spanwise velocity distribution is independent of the chordwise pressure distribution, Hence, from (26) X K = Ö

. [F(n) +?\G(-i)i |l - F (:-,)] dr/ö^^

vo"

3i_ + H A

315 3021+

37 /\ A _

315 " 9U5 " 9072] K(A )s say. (28) /The X 2

Sears has calculated by a the spanwise velocity profiles an external chordwise pressure chordwise separation. The sp small changes of form up to a tion, after which the change o

marked, hovi^ever t h e s e changes

than the corresponding changes the chordwise profiles.

more accurate method for a simple case of gradient leading to anwise profiles shovif point near the separa-f separa-form becomes more are much less severe

-10-The value of H is given by

Hx

- U-

A.I//37_ J L _A!_ (29)

~ '^^10 120// \315 9i+5 " 9072 / ^ ^ ^ The usual methods of solution for the chordwise

flow yield /S as a function of x, and hence we can determine the value of K and Hx(and therefore n) at the trailing edge from equations (28) and (29)j assuming that the flow remained laminar to the trailing edge and was not too near separation any-where for the above method to be seriously in error.

In applying equation (27) we are implicitly assuming that the solution of the momentum integral equation for the wake developing from completely laminar boundary layers follows exactly the lines of the

solution for the, case when the boundary layers are turbulent at the trailing edge. On the face of it this assumption seems plausible enough, but in any case the problem of the wing with completely laminar boundary layers is at present of little more than academic interest.

We note that for a flat plate at zero incidence with completely laminar boundary layers

the above leads to v u . or K = 1. Since UmT:p=U^,

V = Ü

™-

°

o 0 . it follows from (27) that

CJ^[R,A]

= Cos A . Cj^jR CosA,qi

= Cos"^ A C^ []^,ol (30 )

= C^JR/Cos AsOj.

These results also follow directly from exact solutions of the boundary layer equations as can be seen from the following argument. The

equa-tions of motion are now 2 au ^u a u U -r— + W_{ax az} -r— _„2 = V az 2 av av a V ^ al + ^^ ai = ^ ~ 2 '

9z

and the equatioxi of conLl.nnity i s

V

(31)

—— + ~— = u.

9x a z J

Proceeding along classic lines, we infer from the equation of continuity the existence of a stream functionV such that u = ^ , w = - -f^ , and

_ _ — a z ax /"""""'T—"* assuming that'V-- ^JvxU^. f (•-,) , where -i = z^U^/vx, we derive the Blasius equation for f(n) from the first equation of (31)s viz.

f f' • + 2f' ' • = 0. (32)

We now write v = ~- , and assume / is of the form /= ,vxUo> g (q"). The second equation of (31) then reduces to

(33)

Prom (32) and (33) we see that f •" / f " = g'" /g'A

and hence f' = const, g' + const.

But f' = U/UQ, and g' - V / U Q , and since u = 0, V = 0, when z = 0, and u = U^, v = V Q , when z =00, we obtain finally

u _ V U ~ V •

o 0

As already noted, it follows that in this case the resultant X'low in the boundary layer is everywhere -oarallel to the undistui'bed flow. Equa-tion (30) indicates that the drag coefficient is the same as that of an unyawed wing with chord equal to the distance from leading edge to trailing edge

measured parallel to tie direction of the undisturbed flov'/ (i.e. C / C O S A ) in a stream of velocity QQ.

This result is -oeculiar to laminar flow in the boundary layer andAollows from the fact that for an unyawed C^-JXI/R". Any other Cj^jR relation

would not lead to this result even v/ith the flow in the boundary layer everyvirhere oarallel to the undisturbed scream flow.

3. 3• The S'oanwise Plov^f in the Turbulent Boundary Layer.

To obtain the value of (."-"^rp g when the boundary layer is partly or wholly turbulent the

following assumptions will be

made;-(1 ) The distribution of v/V is the same for all X where the boundary layer is turbulent and is independent of the chordwise pressure gradient and therefore of the distribution of u/U,. ,

(2) The distribution of v/V^ is then given by V/VQ - (z/ö)^'\ where m will be taken to be I/7.

It is highly unli-cely that these assump-tions describe exactly wha'- happens in practice, but experiiiiental evidence tending to indicate that

they are sufficiently close to the truth for our purposes will be discussed in a subsequent para-graph (para. 5). We m.ay note, however, that in the usual argument underlying the derivation of the povirer or 'log' laws for the velocity distribution

in the turbulent boundary layer on a flat plate the develonment of the boundary layer with distance from the leading eoige plays no direct part.

fg" + 2g'" = 0. with V - U g'. I

1 2 -P r o m t h i s p o i n t o f v i e w , t h e a r g u m e n t m i g h t b e e x p e c f e d t o a p p l y r a t h e r m o r e r e a d i l y t o the- span-w i s e f l o span-w i n t h e b o u n d a r y l a y e r of a y a span-w e d , span-w i n g t h a n i n t h e a o c e p t e d c a s e o f t h e chordv/ise f l o w i n t h e b o u n d a r y l a y e r o f a n u n y a w e d v/ing. W i t h t h e s e a s s u m p t i o n s w e c a n w r i t e jx 'xx V. '%ö/a XX u yi 0

/..V7 /9

x1/7

1 -ff-^ l-fi

P r o m t h e e v i d e n c e m a r s h a l l e d i n R e f . 7 w e m a y a s s u m e t h a t H is a f u n c t i o n of z/ö^x ^^'^ ^x^ o n l y , U-j

whilst 6 / 6 is a function of H^, only. we can ï/rite Hence d U / U .

yx

o ( _±

6 " V I U XX o \ o K:

v/here K is now a function of H.^-:- only, given by K = K(H^) = ^0/6

1 u

U. I

0

XX

1 -

N' ' XX .1/7 7^ XX > ^

Vxx)

(31*) The function K(H^) can be readily determined using

the curves of Ref. 7. The value of H^ can either be assumed to be a constant over the wing as in the

method of profile drag calculation described in Ref. 6, or it can be calculated as part of the chordwise profile drag calculations as discussed in Ref. 7. Hence the value of Hx at the trailing edge can be determined, and therefore we can calculate K

Sample values of K are as follows:- T.E.

H^ = 1.29, K=1 I H^=1.i|, K=0.910J H^ = 1.5, K=0.777. Having determined K and H-^ (and therefore n= (H-j^+2)/5) at the trailing edge, the drag coefficient can be determined from equation (27).

For a flat plate at zero incidence it is reasonable to assume u v and hence K = 1. In

U V o

this case (27) reduces to

Cp TR, \] = 0^ (R COS A5OJ . Cos A.

₍₃₅₎

For a f u l l y t u r b u l e n t boundary l a y e r on the p l a t e

the l / 7 t h pov'er lav/ l e a d s t o

C^

| [ R , O ]

= const. /R

1 /

Hence

^D &"^] = ^D [?'°]- Gos^/^A,

(36)

To illustrate the remarks made at the end of para.3^ 2 we note that in this case the drag coefficient of an

unyawed flat plate with chord = c/Cos ,A would be C^ [R/COS/\,OJ = C-^ [R,O'\. Cos^'/^A

in spite of the fact that the flow is everywhere parallel to the undisturbed stream direction.

• ^• Specimen Calcu 1 ati^ons and Results

The method described above has been used to calculate the profile drag coefficie.nt of a yawed wing of section NACA 6I|.-012 at zero incidences for angles of yaw up to i].5°, for Reynolds numbers of 10°, 10' and 10 and for various transition positions from the leading edge to 0.5 c. Similar calculations, but covering a range of transition positions from the leading edge to the trailing edge, have also been made for a flat plate at zero incidence. For the aerofoil the calculations v/ere made on the assumption that Hx was constant over the section and equal to 1,U* The results are shown in Pig. 1 (a) and (b).

It i.s of interest to note that although the variation with yaw of the drag coeffic^ient of the flat plate is described by the factor Cos^-^-^.-x Y/hen the

boundary layer is fully turbulent, and changes to the factor Cos /2,A as the transition moves back to the trailing edge, the corresponding f)actor for the wing section remains very closely COS''/2.,A for transition positions between 0 and 0.5 c. The latter factor

shows only slight variation with Reynolds numbers. It appears that for a wing, where there is usually a small posi.tive pressure at the trailing edge helping to reduce the chordwise drag, the factor (u /g ^ p ~ in equation (2/) weights the spanwise drag component as compared with the chordwis^e drag cojnponent, so

that the resultant factor C-Q [R , A|/Cj^ |R,c] is somewhat higher than that for a flat platc^. iviore extensive

calculations are required, however, to throw adrli t j oriFil 1 i qht on the effect of section shapf^ and thickness.

5« Some Ex-oerimental Results

A few preliminary experiments have been made to check some of the assumptions of the above

theory and the results. The experimonts were made on a flat plate of 19" chord extending across an open jet tunnel (,1et dimiensions 3'6" x 2'6'' ) of top speed about 125 f.p.s. giving a Reynolds number of about 1.2 X 10^. It was intended to check (a) whether the flow in the turbulent boundary layer of the yav/ed plate at zero incidence v/as everywhere parallel to the undisturbed stream direction,

-1k-(b) whether vfith the plate at incidence the spanwise velocity distribution v/as independent of the chord-wise pressure gradient, and (c^ whether the drag coefficient at zero incidence with a fully turbulent boundary layer obeyed the law given by equation (35).

Unfortunately, at the low Reynolds numbers of the tests it was impossible to secure a fully turbulent boundary layer or a sharp transition with-out the aid of a turbulence v/ire of such a size that its drag coefficient was an appreciable and. not

accurately known fraction of that of the plate. The experiments v/ere made v/ith the plate at zero

incidence and at 3-g° incidence with various angles of yaw up to /4.5°> and consisted of velocity and yav/meter traverses through the boundary layer at various stations aft of the leading edge. Prom these traverses the spanwise and chordv/ise velocity traverses v/ere deduced as well as the drag coeffi-cient of the part of the plate ahead of the traverse station

The yawmeter traverses made at zero inci-dence vi/ith various angles of yav/ showed no measurable deviation anywhere in flow direction from that of the undisturbed stream, thixs checking the assuinption that at zero incidence the chordv/ise and spanwise velocity distributions in the boundary layer were similar. Some of the spanwise and chordwise velocity distribu-tions measured at x/c = 0.855 at 14.5^ yaw and 0^ and 32^ incidence are shown in Pig. 2. It will be seen that whilst the chordwise distribution shows a marked change of shape with change of incidence the spanwise distribution shows comparatively little change. The latter falls between a 1/5th. and 1/7th. power law

for regions in the boundary layer near the surface but fits a somewhat smaller power than l/7th. in the outer part of the layer. Because of the lack of measurements very near the surface it is difficult

to estimate the corresponding values of Hy = (öy/6 ) v/ith satisfactory accuracy but the value appears '^

to be about 1.14.. In this connection we may note that similar velocity traverses were made in the turbulent boundary layer of a yav/ed wing of. finite span and are reported in Ref. 10. The wing was of elliptical plan form and of aspect ratio about 5 « U J

it v/as tested for an angle of yav/ of 25° and at angles of incidence of 12"-" and 11+°, at which latter

incidence part of the wing was stalled. Neverthe-less, even under these extreme conditions the

velocity profiles of the spanv/ise flow shov/ed themselves to be very nearly independent of the chordwise pressure gradients. An indication of this is the fact that the value of Hy for the

suanv/ise profiles rarely deviated from a value

v/ithin the range 1.3 to 1. 5 5 v.-hilst the corresponding values of Hx ranged from about 1.7 to 6.7. The evidence therefore tends to supDort the assumption that the form of the spanwise velocity distribution is relatively insensitive, to the chordwise pressure gradients.

-15-The measured ratios of the drag coefficients Cp [Hs A*]/Cj3[R,o] are shown in Pig. 1(b) for comparison with the theory. It will be seen that the experimental results show an even greater reduction of drag

coeffi,-cient with yaw than is given by the theoretical C O S V 5 A

law. This may well have been due to the drag contri-bution of the transition wire, this drag is almost

en-tirely form drag and would vary as Cos3./x.

It is of interest to note that the theory

as developed in this paper leads to the follov/ing result for the thickness at a given chordwise position of the fully turbulent boundary

layer:-^ - Ê layer:-^ = Cos-'/layer:-^A, ö[R,Oj

where ö )ƒ!,'^ 1 is the boundary layer thickness at the specifie'd position for a Reynolds nuraber equal to a and a yav; of angle 6. This relation is shown in

Pig. 3 for comparison with the measuTed values. The very close agreement between theory and experiment in this instance is probably in part fortuitous, but it is clear that the assumptions of the theory cannot be seriously v/rong. For comparison the factor _Cos~^/5A is also shov/n, this defines what the ratio 5 i R,'M/5 [R , Oj would be if the development of the boundary layer on

the plate depended solely on the distance from the leading edge mieasured in the direction parallel to the undisturbed stream and v/as otherwise unaffected by yaw.

These experiments cannot be regarded as completely conclusive, and further work is required particularly at higher Reynolds numbers, but the results indicate that the assumptions made in the theory developed here are unlikely to lead to serious errors.

R E F E R E N C E S

Prandtl, L. On boundary layers in three dimensional flow.

(Volkenrohde R and T No. 61+, Pile No. B. I.G. S.8I|, 191+6.

Sears, W.R, Boundary layer of yawed cylinders. Jnl. Aero. Scs. Vol. 15s J-ii. 191+8. pp.i+9-52,

"'ild, J.M.

Jones, R.T.

Goldstein (Ed)

Squire and Young

Doenhoff and Tetervin Nitzberg, G.E. Young, A.D. Kuethe, McKee and Curry Boundary l a y e r of yav/ed i n f i n i t e v/ings. J n l . Aero. S c s . , Vol. 16, J a n . 191+9. p-r;). 1+1-^5' E f f e c t of sweepback on b o u n d a r y l a y e r and s e p a r a -t i o n . NACA T.N. No. 11+02. (15ij-7). Modern Developments in Fluid Dynamics. Vol.1, Ch.Ill (Clarendon Press, 1938).

The calculation of the nrofile drag of aerofoils. A.R.C. R and M 1838 (1937).

Determination of general relations for the behaviour of turbulent boundary layers. NACA T.R. No. 772

(191+3)-A concise theoretical method for profile drag

cal-culation,. ' N A C A .4RC No,l+80f^ (191+i+). Skin f r i c t i o n i n t h e l a m i n a r b o u n d a r y l a y e r i n c o m p r e s s i b l e flow. A e r o -n a u t i c a l Q u a r t e r l y , V o l . 1 Aug. 191+9, pp. 1 3 7 - l 6 i | . Measurements i n t h e b o u n d a r y l a y e r of a yav/ed wing. NACA T.N. No, 191+6

(1949)-COLL&C-e OP A&RONAÜTICS REPORT No 3 8 l O •9 COTRJA} COCR,0) 'S IC* 2 0 * A 3 0 * 4 0 * 5 0 * COS'8 A FULL LINES R E F E R CALCULATED VALUEÏ N A C A 6 4 ^ 0 1 2 SEC FOR T.R POSITIONS

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O O'S 0-4 z / s O G O S 10 MEASURED VELOCITV DISTRIBUTIONS IN CHORDWISE AND

SPANWISE PLANES OF BOUNDARY L A V E R O F VAWEP FLAT PLATS AT ZSRO tNClDENCg A N D AT I N C I D E N C E

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FIG. 3.

SKETCH I L L U S T R A T I N G N Q T A T I ON