Note on the application of the linearised theory for compressible flow to transonic speeds

LüCHw; 1 2 j u f i 1950

KhPA.-?3m?!?a 1 - 2629 HS D E L R REPORT MO. 2 J a n u a r y , 1 9 4 7 . 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 F I E L D Note on t h e A p p l i c a t i o n of t h e L i n e a r i s e d Theory f o r Compressible Flow t o T r a n s o n i c Speeds

By

-A. Robinson, M . S c , and -A. D. Young, M.-A.,

SU:^.:ARY

IT I S SHOWN THiT POR PlffiTE ASPECT R;lTIO THE

LINEiffilSED THEORY OP COIffRESSIBLE PLOW Rm'i/ilNS THEORETICiULY CONSISTENT IN TFIE REGION OP TafiJMSONIC SPEEDS, ALTHOUGH ITS PREDICTIONS MAY DEVIATE APERECIi3LY FROM EXPERIMEITiLL RESULTS IN THAT REGION. THE ViiEIATION OP TilE THEORETIC.:^ LIFT CURVE SLOPE OF m AEROFOIL OP FINITE SPM IS CONSIDERED ILS TKE Fi/\CH ]>IUMBER IjNGREi>SES PROM BELOW UNITY TO ,'£0W UNITY, MD IT I S

SHOi^'N THviT TFIE LIFT CURVE SLOPE REtt'JiNS FINITE SJ^JD CONTINUOUS.

INTRODUCTION.

It is well kno\m that on the basis of the linearised theory for subsonic speeds (i.e. the Prandtl-Glauert theory) the lift curve slope of an aerofoil in two diijionsions bocomos infinite as'the speed of sound is approached. Similarly, the linearised theory for supersonic speeds (i.e. the Ackeret theory) shows the lift curve slope in two dimensions to become infinite as the speed of sound is approached from above. This has led to the belief that the linearised theory breaks down in the transonic region. However, a recent application of the theory to Delta wings in supersonic flow (Ref.l) showed that with the aspect ratio finite the lift curve slope tended to a determinate

finite value as the Mach number tended to unity. Further R. T. Jones (Ref,2) has shown that for aerofoils of vanishingly small aspect ratio the lift curve slope is independent of Mach number (and hence is

continuous from subsonic to supersonic speeds).

In this note it is shovm that the lift curve slope of a wing of finite aspect ratio remains finite in subsonic flovr as the Mach

number of unity is approached from below. This is shoT,7n to be true even on the very simple basis of lifting line theory. flowever, as the speed of sound is approached the pressure distribution on a wing of finite aspect ratio can be related to that on a wing of vanishingly smtill aspect ratio in incompressible flow (see pfara.3). Hence vre must reject the quantitative results given by the lifting line theory and examine the results given by lifting surface theory. These latter results are shown to agree at a Mach number of unity with the results given by the analysis of Delta wings in supersonic flow mentioned above. It is concluded that the linearised theory gives both finite and continuous values of the lift curve slope of m n g s of finite aspect ratio as the Mach nimber

passes through unity.

This donclusion does not, of course, imply that the linearised theory will necessarily give results in close s.greement with experiment through the transonic region. The development of a shock stall in that region may be associated with disturbances v/hich are far too largo for the linearised theory to remain applicable.

NOTATION /I

2y

-^L

cL M / A -aspect ratio apex angle lift coefficient

incidence (in radians) Mach muTiber Mach angle

u

Uo

P

Po

So

<p

-' -longitudinal velocity free stream velocity pressvire

free stream pressure free stream density velocity potential The suffices i and c indicate incompressible, and compressible flow respectively.

SUBSONIC PLOW

3.1. Linearised Theory - General.

The linearised theory for subsonic speeds has been developed in some detail in Ref.3, v/here it is shoY/n that if

is the potential function for the flow round a body in a uniform incompressible flo?/- of velocity U steaming in the direction of the X axis, then

(pQ = Uo2c + L,. f (x, /3y,^3z)

is the potential function for the flow round the same body in compressible flow, the Mach number M being related to /3 ^J "the equation

M =

VI -/i^

Consider a flat plate of any plan form and of aspect ratio A set at a small angle of incidence. Suppose the x axis is along the main stream direction, and the y axis is in the plane of the plate. We note that, on the surface of the plate Yfhen z is small, in

incompressible flow

Ui = M i = Uo + fx (x, y, o) - - . (1)

ex.

to the order of aoouracy of the theory, and, in compressible flow

Uc = ^ - U o + fx (x,^y, o) > - - (2)

Hence, i n incompressible flow, the pressure coefficient on the siorface

i s given by

(P - Po ) i = -2 fx (x»y>o) _ . _ (3)

•Ijo Uo^ Uo

and in compressible flow the pressure coefficient is

(P - Po)c = -2fx i^>Ay> o) - - - (4)

It follows that at any point on the plate the pressture coefficient in compressible flow is 1 times the coefficient at the point (x, /jy ) in incompressible flow with the lateral ordinate y of the plate reduced in the ratio /Sri. Therefore the lift coefficient and lift curve slope at a Mach number M of a flat plate of aspect ratio A are 1 times the lift coefficient and lift curve slope in incompressible flow of the flat plate vri-th its aspect ratio reduced to ^ ft *

3.2. Lifting Line Theory.

If we apply the above.conclusion to the formula given by lifting line theor3' for the lift curve slope of a wing of elliptic plan form in incompressible flow, we find that in compressible flow

L\ _ CL ^ _ - _ (5)

^ '-^ J c

/3 + a Oo

Tl-A

This formula has already been deduced in Ref,4.

We note that as M — * 1.0 and f^ —-> o,

Hence / ._„„.Ji\ remains finite for finite aspect ratio. However,

Vd

di )c

as M -—-> 1.0 the equivalent aspect ratio A/3 — > o, and hence we may expect the formula given by equation (6) to become increasingly invalid as M = 1.0 is approached.

We must therefore consider the results given by lifting surface theory.

3.3. Lifting surface Theory.

The only available comprehensive results based on the linesLrised lifting surface theory for incompressible flow aj?e those derived by

lOrienas (Ref.5) for elliptic flat plates, which include as a special case the result obtained by Kinner (Ref.6) for the circular flat plate. These results give us values for /d Gj\ for aspect ratios of 0.637,

[doC

)i

1.27, 2,55, and 6.37, and these values are plotted against A in Pig.l. In addition, S. Ï. Jones-(Ref,2) has shown that for flat plates of any plan form and of vanishingly small aspect ratio

ff-SA —^llA , as A

\doL}± 2

^ o.

It vd.ll be seen from Pig.l. that the smooth curve through llrienes' and Kinner's points is quite consistent T/ith a tangent at the origin having a slope equal to If as given by Jones' theory. It is, therefore,

~2

reasonable to accept the curve of Fig.l as describing the variation of ' '-'L\ with aspect ratio given by the lifting surface theory for .dioL )±

elliptical plates. We proceed to accept this curve, with perhaps less justification, as applicable to triangular plan forms (Delta wings). In favour of this we may note that we are primarily interested in small

aspect ratios when Jones' 2 argument would lead us to expect a negligible effect due to plan fo37m, and we may also note that in practice measured differences in lift curve slope as between one plan form and another are generally mthin the order of ejqperimental accuracy,

Using this curve, therefore, and the result developed in para.2,1, above we can estimate the lift curve slope of a lifting surface of any aspect ratio at any Mach number up to 1,0.

flow

Since we have accepted the result that for incompressible

^

^ L ]

..

IT

_{ILk, as A —> o,}

.d c< /i 2 then for compressible flow

fl__ïïl\ -^ I L A , as M — # 1.0 from below.

id

U

Jc

2

This is exactly half the value given by the lifting line theory.

IThether the curve of [ "^ '-'L j against Mach nuiTiber is flat

\ d aC / c /^ \

topped or cusped at M = 1.0 depends on whether the curve

of f^ ^j]

\d ü^J

i

against aspect ratio has a point of inflection at A = o or not, and we

have not as yet sufficient evidence on this point,

1+,

SUPERSOmC FLOW.

The lift of a flat Delta wing moving at supersonic speed is

calculated in Ref.l. on the assumptions of linearised theory. The lift

coef-icient is given by

CT = 4 << tanytv , \7hen M * Y

• and by C^ = 2 TTc^. tan y -^vhen y U > T > " " ^^^

E'(cot/^ tan y )

In these formulae

yu.

is the Llach angle, 2 "y is the apex angle of

the Delta v/ing, and E'(u) is the elliptic integral defined by

It

E • (u) =

[-t'Jl

- (1 - u^) sin^ ^ d J2f

Since A = 4 tan V for a Delta wing, and cot/*- = \/M - 1, equations

(7) may be re-vnritten

\

^

^L =

^-

, when ^M^ - 1 > 4„

and y

(8)

^

^L = T T A • v/hen >/M" - 1 < 4

wnen

\n'

^ '^

2E' /VM - 1 A ,

[ ^ I J

For a given aspect ratio, v M - 1

vrill

ultimately become

smaller than —i—, as M = 1 is appraoched from above, so that the second

A

formula in (8) will apply. Now E' (u) •-'-> 1 as u — > o , and so

h ]

— ^ Z L A, as M ^ 1.0 from above.

d c< / 2

Comparison with paragraph 3 shows that t M s is exactly the same

value as obtained when a = 1 is approached from belovir. It follows

that the lift curve slo-oe does in fact vary continuously vri.th M through

M = 1 (Fig.2).

It is of some interest to calculate the slope of the curve

''^

^L)\r5 M as M--T^l from above.

^du J •

We have du l-u2 (K'(u) - S'(u)) where K'(u) = f l " - ^ ^

7 1 -

(1

-

i?y sin

^

Hence _E' M^-1A

4

- K' M 2 . M^-1 A

4

E

.

IJ^

1 A^

4

16-(M2-1)A'

N o w E ' ( u ) — • ! , as u — > o, as mentioned above, while at the same time K'(u) —,f o o as log 1+ , i.e. (K'(U) - log _4_ ) — — * 0, as u — - * 0.

u u (Ref.7, page 521), It follows that _d_ / d CT \ tends to - « ^

dM

_M

_\

_d^j

as M tends to 1 from above, although the intensity of that infinity is "comparatively weak".

The position is summarised in Pig.2 which shov/s the variation of dCj/dot Tv-ith laach number for both sub and supersonic speeds

(including M = l.O) obtained for A = O o , I4..0, 2.3, and 1.07, i.e. for semi-apex angles If = 90°, 45°, 30°, and 15° respectively.

Author Title, etc.,

A, Robinson Lift and drag of a flat Delta -^ring

at supersonic speeds,

R,A.E, Technical Note No,Aero.1791 1946,

R. T. Jones

S, Goldstein and A. D. Young.

A. D. Young

Properties of low-aspect ratio pointed wings at speeds below and above the speed of sound.

N.A.C.A.A.CR. L5 FI3.

The linear perturbation theory of compressible flov;, v/ith applications to \7ind-tunnel interference,

R ci M No. 1909, 1943.

A further survey of compressibility effects in aeronautics,

R.A.E. Rep. No,Aero,1926, 1944. K, Krienes

"ff. Kinner

E. T, Whittaker and G. N. Watson.

Die elliptische Tragfldche auf

potential-theoretischer Grundlage, Z,A.M.M. vol 20, 1940,

Die Ivreisforraige Tragflache auf potentialtheoretischer Grundlage. Ing, Arch, vol.3, 1957.

No. 2. LIFT C U R V E 4.0 S L O P E A C L dLeL 5 0 4.0 3 0 2 0 1 0 JONES > ( L I M I T I N G VALUE W H E N / A - O ) . / / / ^ / > / / / / / / / . ^•-•^'^^ ^ ^ ^ K R I E N E S A - A S P E C T C,.- LIFT COE R A T I O LFFICIENT e<. - I N C I D E N C E I N R A D l M » O 1 0 2 0 3 0 4 . 0 S O 6 0 7 0 V A R I A T I O N O F L I F T C U R V E S L O P E WITH A S P E C T R A T I O ( K R I E N E S - J O N E S ) FIG I. A - ASPECT RATIO S^- APEX A N C L E _ CL- LIFT C O C F R C I E N T I - I N C I D E N C E IN RADIANS O -2 -4 -e - 6 10 12 '•* ' 6 10 z-o ^4 VARIATION O f T H E L I F T C U R V E S L O P E O F A D E L T A WING WITH M A C H N U M B E R F O R V A R I O U S A S P E C T R A T I O S