The axect flow behind a yawed conical shock

CoA Report No.116

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE EXACT FLOW BEHIND A YAWED

CONICAL SHOCK

by

LIST OF GQMTEISffS Page No. 1 . I n t r o d u c t i o n . 5 2. P r o p e r t i e s of t h e Plow. ' 7 3 . Method of S o l u t i o n . 8 2^, Disc\assion. 12 5 . C o n s l u s i o n s . 15 6 . Acknowledgements. 15 7 . R e f e r e n c e s . 1 6 •APPENDICES A. D e t a i l s of s o l u t i o n . 17 B. C a l c i i l a t i o n of L i f t and Drag C o e f f i c i e n t s . 1 8 FIGURES 1 - 1 5

3

-LIST OF SYMBOLS

a • Speed of sound.

0 Specific heat at constant pressure,

0 Specific heat at constant volume. 2 _,

Op Pressure coefficient = •— ( /p - 1 ) . h Enthalpy = C T.

h Interval hetT/een successive values of the independent variable x , , h = x - x (in Appendix A) p Pressure.

_g_ Flow velocity.

q ^ Absolute velocity (of gas discharging into vacuum)

M Mach Number.

r radial co-ordinate in Spherical polar system. R gas constant = 0 - C . _{e> p v}

S Entropy.

T Absolute temperature,

v. Freestream velocity.

X independent variable.

y dependent variable.

( r ^ f , di) Spherical polar co-ordinates, (u, V, w) Components of velocity.

u Conponent positive in direction r increasing. V Component positive in direction f increasing, w Ccmponent positive in directxon u increasing.

L i s t of Symbols contd,

a Angle of yaw or incidence.

Q

y Ratio of specific heats = _ £ ,

C

e Equivalent shock wave angle.

P Density.

f

Shock cone semi-apex angle.

W • i tj

SUBSCRIPTS.

w Pertaining to shock wave,

s Pertaining to solid body surface.

0 Pertaining total or stagnation conditions.

1 Pertaining to freestream (and upstream side of shock wave).

2 Pertaining to dovmstream side of shock wave.

3,4,5, etc. Pertaining to successive if*-wise steps chosen for

5

-1 . INTRODUCTION.

The problem of s u p e r s o n i c flaw around a yawed c i r c u l a r cone h a s

r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n r e c e n t t i m e s b u t i s s t i l l o n l y p a r t i a l l y s o l v e d , A comprehensive s u r v e y of t h e e x i s t i n g s t a t e of knowledge i n t h i s f i e l d was g i v e n i n 195^ by Vfoods (Refence 1 . ) . S e v e r a l methods of s o l u t i o n have been p r o p o s e d , b u t t h e f i r s t and second o r d e r t h e o r i e s developed b y Stone seem t o have had t h e vd.dest a p p l i c a t i o n .

I n a p a p e r p u b l i s h e d i n 1948 Stone (Ref. 2) d e a l t v/ith t h e problem b y t r e a t i n g t h e flow a s t h e sura of t h e non-yaw flow, e x a c t l y s o l v e d b y T a y l o r a n d Maccoll i n 1933 (Ref, 3) ancL a small p e r t i i r b a t i o n due t o yav^

( i n c l u d e d i n t h e s o l u t i o n s a s f i r s t o r d e r t e r m s i n a , t h e a n g l e of yaw). I n a l a t e r p a p e r , p u b l i s h e d i n 1951 (Ref. 4) he developed a second o r d e r t h e o r y . E x t e n s i v e t a b l e s b a s e d on S t o n e ' s t h e o r y have b e e n p r e p a r e d a t M . I . T , b y Kopal and p u b l i s h e d a s companion volumes t o t h e t a b l e s of a x i

-syirmetric f l o w around c o n e s . ( R e f s . 5 . 6 , 7 ) . These t a b l e s c o v e r a wide r a n g e of Mach number and cone apex a n g l e .

S t o n e ' s work has met w i t h a c e r t a i n amount of c r i t i c i s m mainly on two c o u n t s . One o b j e c t i o n was on t h e grounds t h a t t h e system of c o

-o r d i n a t e s ( u s i n g wind a x e s ) u s e d -o r i g i n a l l y b y St-one and s u b s e q u e n t l y

by Kopal i n t h e p r e p a r a t i o n of t h e M . I . T . t a b l e s was i n c o n v e n i e n t t o employ i n p r a c t i c e . A d e t a i l e d d i s c u s s i o n of t h i s a s p e c t of t h e problem was g i v e n by R o b e r t s and R i l e y (Ref. 8) who a l s o l a i d down a p r o c e d u r e t o modify t h e Stone s o l u t i o n s ( a s t a b u l a t e d b y Kopal) t o more p r a c t i c a l c o - o r d i n a t e s ( u s i n g body a x e s ) .

The o t h e r was an i n i p o r t a n t t h e o r e t i c a l o b j e c t i o n . S t o n e ' s f i r s t o r d e r t h e o r y i m p l i e s a p e r i o d i c v a r i a t i o n of e n t r o p y around t h e c i r c u m f e r e n c e of t h e yawed c o n e . Thus t h e e n t r o p y v a r i e s from a maximum v a l u e a t t h e "wind-ward" g e n e r a t o r t o a minimum v a l u e on t h e "lee"wind-ward" g e n e r a t o r . P e r r i p o i n t e d out i n 1950 (Ref,9) t h a t t h i s c o n t r a d i c t s t h e r e q u i r e m e n t t h a t t h e s o l i d cone s u r f a c e must be a s t r e a m s u r f a c e and t h e r e f o r e i t s e l f a s u r f a c e of c o n s t a n t e n t r o p y . He d i s c u s s e d t h e flo>? around t h e cone i n t h e g e n e r a l c a s e and showed t h e e x i s t e n c e of s i n g u l a r p o i n t s a l o n g t h e "leeward"

g e n e r a t o r on which t h e e n t r o p y i s many v a l u e d and t o which a l l s t r e a m s i i r f a c e s c o n v e r g e . P e r r i a l s o i n t r o d u c e d , f o r t h e c a s e of t h e s l i g h t l y yawed cone, t h e c o n c e p t of t h e " v o r t i c a l l a y e r " t h r o u g h -ïrfiich t h e e n t r o p y changes from i t s c o n s t a n t v a l u e on t h e s u r f a c e of t h e cone t o t h e v a l u e p r e d i c t e d b y S t o n e ' s t h e o r y and gave a method f o r c o r r e c t i n g S t o n e ' s f i r s t o r d e r s o l u t i o n n e a r t h e cone s u r f a c e . S t o n e ' s s o l u t i o n however, i s v a l i d t h r o u g h most of t h e f l o w f i e l d between t h e shock and cone s\irface. A l s o , i t h a s been p o i n t e d out b y R o b e r t s and R i l e y (Ref. 8 ) a n d T/oods (Ref, l ) t h a t a l t h o u g h t h e e n t r o p y c o r r e c t i o n s s e t out b y P e r r i a r e l o g i c a l l y n e c e s s a r y t h e y may be n e g l e c t e d i n p r a c t i c e . I t h a s been found t h a t though S t o n e ' s t h e o r y was o r i g i n a l l y i n t e n d e d t o b e a p p l i c a b l e t o o a s e s of s m a l l jraw, i t s u s e f o r c o m p a r a t i v e l y l a r g e v a l u e s of yaw g i v e r e a s o n a b l y s a t i s f a c t o r y r e s u l t s .

I n the course of h i s i n v e s t i g a t i o n of t h e supersonic flow around a

yawed cone Woods observed (Ref. 1) t h a t Stone's f i r s t order theory Itroke

down in p r e d i c t i n g the entropy behind t h e shock a t remarkably low values of

yaw. He found t h a t t h i s theory used i n conjunction with the M.I.T. cone

t a b l e s p r e d i c t s for c e r t a i n cases a decrease in entropy through t h e

shock wave, v/hich i s a phenomenon p h y s i c a l l y irripossible. This aspect of

S t o n e ' s theory does not seem t o have been noticed p r i o r t o Wood's work.

Prom t h e above considerations i t becomes obvious t h a t although S t o n e ' s

theory i s quite s a t i s f a c t o r y i n respect of many p r a c t i c a l a p p l i c a t i o n s ,

a c o r r e c t and complete s o l u t i o n of t h e supersonic flow around a yawed cone

has not yet been achieved.

The present i n v e s t i g a t i o n was c a r r i e d out in order t o make an e f f o r t

t o xmderstand the problem b e t t e r . One of t h e assumptions of Stone's

theory i s t h a t v/hen the cone i s yawed, t h e shock wave continues t o bé

c o n i c a l with the same semi apex angle as i n t h e non-yaw case; t h e only

difference being t h a t now the shock cone a x i s w i l l be yawed with respect t o

the a x i s of t h e conica.1 body a l s o . The present i n v e s t i g a t i o n was intended

t o demonstrate how f a r t h i s assumption was j i i s t i f i e d by considering t h e

exact flow behind a yawed conical shock wave. The problem e s s e n t i a l l y

c o n s i s t s of p o s i t i o n i n g a conical shock wave with i t s a x i s inclined t o t h e

free stream and then i n v e s t i g a t i n g the flow behind t h e shock cone i n order

t o determine the s o l i d body wiiich would produce t h i s p a r t i c u l a r configuration,

I n t h e present case t h i s i s achieved by using a numerical procedure

f o r solving t h e d i f f e r e n t i a l equations of motion, which a r e set out i n

Section2, The procedure adopted for t h e numerical s o l u t i o n and r e s u l t s of

the a p p l i c a t i o n of t h i s procedure t o a p a r t i c u l a r case i s set out i n

Section 3. A discussion of t h i s s o l u t i o n follows i n Section 4.

A comparison between the body shape obtained by the splution 6f

t h i s p a r t i c u l a r case and the corresponding f i r s t order j^av/ed cone s o l u t i o n

i s made i n Figure 15,

K

The terms 'fconical body" and "Conical Shockwave" vd.ll be used t o

7

-2 . PROPERTIES OF THE FLOW

2 . 1 , System of C o - o r d i n a t e s and Nomenclature

We employ s p h e r i c a l p o l a r c o - o r d i n a t e s T^,f, w b a s e d on t h e apex

of t h e c o n i c a l shock wave a s o r i g i n and t h e a x i s of t h e shock wave c o i n c i d i n g witVi t h e a x i s of t h e c o - o r d i n a t e system 1^=0» The p l a n e of yaw ( o r

syn-iiiiotry) i s d.efined b y w = 0 w = ^ , A dia.gram of t h e c o - o r d i n a t e system i s g i v e n i n PinT.ir;e_J . u , v, w a r e t h e ccmponents of t h e v e l o c i t y and t h e y a r e defijied t o b e p o c i i t i v e i n t h e d i r e c t i o n r i n c r e a s i n g , f i n c r e a s i n g and CO i n c r e a s i n g r e s p e c t i v e l y . The f r e e s t r e a m of v e l o c i t y T. i s

c o n s i d e r e d t o b e i n c l i n e d t o t h e shock a x i s a t an a n g l e a such xhat t h e p a r t of t h e p l a n e of symmetry d e f i n e d b y o) = 0 i s on t h e "leeward" s i d e and Oi = ir on t h e "windv/ard" s i d e .

2 . 2 , C o n i c a l Flow

I n a s t e a d y s u p e r s o n i c c o n i c a l f i e l d of flow no fundamental l e n g t h i s i n v o l v e d and t h e phj^sical p r o p e r t i e s of t h e flow a r e f u n c t i o n s o n l y of a n g u l a r v a r i a b l e s . The e q u a t i o n s of motion a r e independent of r , 2.3 • D i f f e r e n t i a l e q u a t i o n s of motion.

The stead.y " c o n i c a l " flow of an i n v i s c i d c o m p r e s s i b l e gas w i t h

c o n s t a n t s p e c i f i c h e a t and w i t h no h e a t c o n d u c t i o n s a t i s f i e s t h e f o l l o w i n g e q u a t i o n s , E i i l e r ' s E q u a t i o n s c a n b e e x p r e s s e d a s 9u W 9ll 2 Z n /^\ V - ^ + -: , -?-- - V - W"^ = 0 ( 1 ) of ^ sxn f of;} ^ ' 9u 9w w 9 v 2 . „ , 9 3 / o ^ - u -r- - w -T- + —:—, -r— + uv - vr c o t f = T -^—- {2) of of sxn f d(ji * df ^ ' , 9v/ 9u 9v . , , m 9 S /-,\ V sxn 1^—5— - u T— - V •^- + u w sxn f + v MJ cosi^ s T -5-— (3)

Of o(j} 9a) 9a) ^ '

where S = E n t r o p y T = TernperatiJre. E q u a t i o n of C o n t i n u i t y

u(v« + w* - 2a^ ) - a^ v c o t ^ + -21 (v^ - a^ )

- ^ ) = 0 (4) where a = speed of sound.

v« + w* 1 s i n f - 2a^ ) - a^ V c o t f + ^^ (v^ - a^ 9 w / 2 z \ / 9 w 1 g^ (w - a ) + V w ( g ^ H- ^ ^ ^

It is convenient to combine equations (l), (2) and (3) to give 9S w 9S ^ /(-s

^ -g;? + S l ^ "9^ = ° ^5)

Equation of Energy 2 C T + -3^ = constant (6) P 2 , 2 2 2 2 where q = u + v + w

C = specific heat at constant pressure.

Equation (5) is general for any conical flew and in fact defines the lines of constant entropy which correspond to the streamlines. If L is the sti-eamline projection on the sphere r = constant,

= 0, Using dS dL = e q u a t i o n _9S

91^-(5)

( ~ ) -^d^ ^L " C o n i c a l flow

i^

we Imve w V s i n ^ w i t h o u t a x i a l 9S 9cj do) dL symmetry (7) 2,4. I n r e f e r e n c e 9 P e r r i h a s d i s c u s s e d i n d e t a i l t h e p r o p e r t i e s of s i i p e r -s o n i c c o n i c a l flow w i t h o u t a x i a l -syiimetry and h a -s -shown t h a t -s i n g u l a r i t i e -s must e x i s t i n any such flow. He c o n s i d e r s a c o n i c a l body p l a c e d i n a

f r e e s t r e a m i m c l i n e d t o i t s a x i s and by p h y s i c a l r e a s o n i n g shows t h a t t h e e n t r o p y must be c o n s t a n t on t h e s u r f a c e of t h e cone o r must change i n a d i s c o n t i n u o u s manner. I t h a s b e e n shown b y P e r r i t h a t such a d i s c o n t i n u i t y o c c u r s a l o n g t h e g e n e r a t o r of t h e cone on t h e "leev^ard" m e r i d i a n p l a n e ( w = O) a t wh.ch t h e e n t r o p y i s many v a l u e d . The c h a r a c t e r of t h e flow i s such t h a t s t r e a m l i n e s downstream of t h e shock cone c u r v e round and c o n v e r g e t o t h i s s i n g u l a r g e n e r a t o r . The e n t r o p y on t h e cone sxirface i s e q u a l t o t h a t on t h e "windvv-ard" m e r i d i a n p l a n e ( O) ='Tr ) ,

3 . METHOD OF SOLUTION

3 . 1 , P r o c e d u r e f o r n u m e r i c a l s o l u t i o n

The d i f f e r e n t i a l e q u a t i o n s of motion s e t out i n p a r a g r a p h 2 , 3 can b e

i n t e g r a t e d s t e p - b y - s t e p vd.th r e s p e c t t o f making u s e of n u m e r i c a l d i f f e r e n t i a t i o n t o o b t a i n d e r i v a t i v e s w i t h r e s p e c t t o w , The method i n b r i e f , i s a s f o l l o w s : Consider t h e c i r c l e of i n t e r s e c t i o n { f = ^ ) of t h e shock cone and s p h e r e r = c o n s t a n t . Choose a l a r g e number of a z i m u t h a l s t a t i o n s around t h i s c i r c l e . The q u a n t i t i e s u , v, w, S a n d T and t h e i r d e r i v a t i v e s w i t h r e s p e c t t o o) a r e Icnown on t h i s c i r c l e from t h e shock wave e q u a t i o n s . S u b s t i t u t i n g t h e s e vsuLues

9.

-in the equations we obta-in the values of the derivatives with respect to f at each azimuthal station. Now consider

a small inv/ard step Ai^ in i^ (along the sphere r = constant). Making use of numerical integration we obtain values of u, v, w, S and T on the circle f = f -t^f , Derivatives of u, v, w, S and T with respect to 0) can be founc[ by numerical differentiation around this circle making use of the values at the various azimuthal stations. Now the derivatives with respect to f can be found by stibstitution in the equajrions of

motion. The process followed above is repeated to carry on the integration as far as is required Details of the procedure adopted are given in

Appendix A.

3, 2. Accuracy of Method

The accuracy of t h e method depends mainly on two f a c t o r s . The f i r s t

i s t h e choice of xhe i n t e r v a l betvreen azimuthal s t a t i o n s . The accuracy

of the prcxjess of numerical d i f f e r e n t i a t i o n which has t o be used t o

determine t h e d e r i v a t i v e s v/ith respect t o w,at each step in f depends

mostly on the i n t e r v a l between t h e s t a t i o n s used i n the d i f f e r e n t i a t i o n ;

the smaller t h e i n t e r v a l , the more accurate the methcsd w i l l be.

The second f a c t o r i s the magnitude of l^f which i s chosen f o r the

step-by-step i n t e g r a t i o n The accuracy of the method w i l l be enhanced

by using as small a step in i/^ as i s p o s s i b l e . By a s u i t a b l e choice of ^f

and use of the process of successive approximations described i n Appendix K i t

i s possible t o obtain a s a t i s f a c t o r y acc\iracy. I n general, the choice

of Ai^ should be consistent with t h e choice of the i n t e r v a l between

azimuthal s t a t i o n s ,

3 , 3 . D e t a i l s of t h e solution for a p a r t i c u l a r c a s e .

The numerical procedure was applied t o a p a r t i c u l a r case with t h e

following i n i t i a l conditionsi Free stream Mach No. M. = 10 Shock wave

semi-apex angle f - 30 Angle of yaw a = 20 . ^

Eleven azimuthal s t a t i o n s at i n t e r v a l s of 15 were chosen between

w = 0 and w _ TT around the shock cone. I t v/as thodght t h a t t h i s choice

of the i n t e r v a l i n w would give s a t i s f a c t o r y r e s u l t s .

Numerical d i f f e r e n t i a t i o n formulae given by Bickley (Ref,1O) vrere

used. Since t h e d e r i v a t i v e s of u, v, w and S with respect t o o) could

be obtained a n a l y t i c a l l y on the shock \7ave i t s e l f , i t provided a check

on the accuracy of the numerical d i f f e r e n t i a t i o n a t the s t a r t of the

solution. Five-point and seven-point formulae vrere t r i e d along v/ith a

c e n t r a l difference formula using up t o 7th differences. I t was fourjd

t h a t the 5-point formula v/as quite s a t i s f a c t o r y in a l l cases though the

7-point formula was found t o be more accurate i n the case of 9 S .

Hence, the 5~Point formula was used for finding —— , —— , ——

and t h e 7-point foxmiila for -^— . I n a l l cases, f o r finding the d e r i v a t i v e

a 0)

at any point an equal number of points on either side of the point were chosen and differentiation formula for the derivative at the middle ordinate were used, since this involved the minimum of error This procedure could be applied even to points near u = 0 and o) =ir by virtue of the symmetry of the flow about the plane o) = 0, o) =ir ,

Commencing at the shock wave the step-by-step procedure of f -wise nianerical integration as detailed in Appendix A was carried out using increments ^f = «0 30', It \Tas found that differences between the first approximations and second approximations obtained by invoking the

trapezoidal rule were not of great significance (the differences were, in the case of velocities much less than 0.1^ and entropy, less thanl^ and hence no attempt was ra-'.de to obtain further approximations,

The same procedure was repeated using increments of ^f = 1 and this- gives values which were found to agree very closely with those obtained \asing half this increment. Thus it wr.s observed that 1 •

f -wise increments would be quite satisfactory,

The solution of the equations proceeded in a very satisfactory manner till a value of f = 26 30* was reached. At this stage it was

observed that 'v' on the "leeward" meridian plane w = 0 had reached a value very nearly zero and tl-jat any further step would take the solutions on this plane beyond the singular point discussed in the previous

section. It was also clear that with further steps 'v' would reach zero at other azimuthal stations on the "leeward" side. Hence the solution in the neighbourhood of w = 0 and beyond f = 26 30' was difficult to obtain.

It was observed that vihen v tended to zero the derivatives with respect to f changed in magnitude rapidly This was particularly true about 9 w and 9 S . This rapid increase in the value of 9 w seemed to

d f d f 'd f

indicate reversals in the azimuthal component of velocity w for the siiiall increment of Ai^ = 0 30' from beyond f = 26°30' in the neighbourhood of w = 0.

Hence as a first step, azimuthal stations were omitted at which large magnitudes of 9 w indicated reversals in the sign of w, and the solution

9 f Q

T/as carried on for the rest of the stations in steps of i^f = -1 , By proceeding in this manner it -//-as possible to continue the process

until stages of f were reached at which the values of entropy at each azimuthal station (from w _ 4.5 to " = 180°) had reached the magnitude of the entropy on the meridian plane '^ = ^ , thereby indicating the surface of the hypothetical body. The point at which v became zero in the plane o) = TT Icxjated the position of the intersection of the solid

- ^^ _

surface with t h a t plane. This follows from the boundary condition that

the v e l o c i t y component normal t o t h e surface should be zero. On t h e

plane o) = 0, o) = TT ' V ' isl^ by symmetry, the normal component of v e l o c i t y ,

To obtain some laiowledge about the region between o) = 0 and O) = 45

beyond f z: 26 3 0 ' , the solution was s t a r t e d again from f - 2é 30'

using only the five s t a t i o n s a t o) = 0, 25 , 30 , 45 , 60 , The i n t e g r a t i o n

procedure was repeated using increments of /Sf = 0 1 5 ' . During t h i s

i n v e s t i g a t i o n i t v/as observed t h a t the i n d i c a t i o n of r e v e r s a l i n sign of

w f cnind e a r l i e r were due t o the choice of increments of f end t h a t i t

vnxB possible t o continue the s o l u t i o n without meeting t h i s d i f f i c u l t y

by proceeding i n very small s t e p s in f . This, as mentioned above, was

undertaken and t h i s set of c a l c u l a t i o n s gave reasonable r e s u l t s .

I t was found t h a t the values a t o) = 45 and a) = 60 obtained i n t h i s

l a t t e r c a l c u l a t i o n vrere in agreement with those obtained e a r l i e r ,

The v a r i a t i o n of entropy, temperature and the t h r e e v e l o c i t y

components behind the yawed c o n i c a l shock i s presented in the following

Figures, /Non-dimensional values of u, v, w, S and T as set out in Appendix

A are used.

F i g . 7

F i g . 8

F i g , 9

P i g . 10

m g . 1 1

F i g . 12

F i g . 13

-S S T T u V w / ^ J r\t « V DW * V < V f\J f 0) f 01 0) 0) 0)

for various

for various

for various

for various

for various

for various

for various

w. f. W, If. f. f. f.

By cross p l o t t i n g from the above figures t h e projections of constant

value l i n e s on a sphere with c e n t r e at t h e o r i g i n of the co-ordinate

system (apex of shock cone) and radius r = constant v/ere obtained and

are shown in the following f i g u r e s .

Lines of constant entropy ( i . e . streamlines)

Lines c3f constant temperature.

Lines of constant u,

Lines of constant v,

Lines of constant w,

From Fig, 2 we get the shape of the body surface which i s defined

by S = 1,307.

F i g .

F i g .

F i g .

F i g .

F i g .

2

?

4

5

6

Having obtained the body shape, the distribution of pressure on the surface could be found. The values of C the pressure coefficient, at the various azimuthal stations around the body surface ojre compared v/ith the values just behind the shock in Fig, 14.

The head lift and drag coefficient of the body as defined in Appendix B have been calculated.

. The lift coefficient C, =0,410

Jj

The drag c o e f f i c i e n t G_ = 0.545

4. DISCUSSION

4 . 1 . Me-f-hod of Solution,

As mentioned e a r l i e r , t h e numerical procedure was found t o work i n

a very s a t i s f a c t o r y manner up t o if = 26 3 0 ' , when the s o l u t i o n was i n

the neighbourbood of the singiilar point in the "leeward" meridian plane

w = 0. The main d i f f i c u l t y from t h i s stage onwards was t h a t the value

of V tended tov/ards zero and a subsequent change i n sign ( t h e change i n

V i t s e l f v/as quite regular throughout). This f a c t o r was h i g h l y c r i t i c a l

since t h e evaluation of 9 u , 9 w and 9 S involved d i v i s i o n by v.

d f d f d f

This meant t h a t v/hilst t h e value of v passed through zero and changed

sign, i t was possible t o get l a r g e magnitudes of the above d e r i v a t i v e s

changing i n sign quite rapidly. However, t h i s v/as found t o be highly

c r i t i c a l only in t h e case of the evaluation of 9 Y/ , I t v/as t h i s feature

9 f

which vra.s responsible for t h e extreme care necessary t o continue the

solution beyond if = 2é 30' in t h e v i c i n i t y of a) = 0,

As mentioned e a r l i e r , t h i s highly c r i t i c a l region between o) = 0

and (as i t turned out) o) = 45 was i n v e s t i g a t e d s e p a r a t e l y using simller

values of ^f than t h a t used for the remainder of the azimuthal s t a t i o n s ,

Here i t may be mentioned t h a t t h e above s;;ated d i f f i c u l t i e s encountered

when V > 0 and changes s i g n , v/ere avoided i n the case of t h e "windv/ard"

s i d e . This was Because the surface of the s o l i d body (as represented

by the l i n e of constant entropy of magnitude equal t o t h a t of t h e entropy

on the "windward" plane o) = w ) was obtained before the c r i t i c a l region

(v -^> O) v/as reached. The solution vra.s not c a r r i e d any fxjrtlicr beco.use

t h e behaviour of the flcKr i n s i d e the body surface wns of no s p e c i a l i n t e r e s t

in the present case,

13

-4 . 2 , P r o p e r t i e s of the flow.

4.2.1. Velocity,

It is found that the variation of the velocity components is quite regular and exhibit no peciiliarities. However, the variation in the values of u and particularly w in the vicinity of the singular point needs some consideration. Some difficulty v/as esqjerienced in the finding of the numeri(3al values of w and u in the region 60 > O) > 0 for

values of f smaller than 26 30'. Although it appeared that the values

of u and v/ behaved regularly in this region it v/as considered that

accurate numerical values could only be obtained if smeller intervals of w were used in the numerical method,

The component v is found to vary in a very regular manner. This is quite iHiderstandable since the evaluation of 9 v depends on (v^ - a^ ) v/i.th v^ « a^. 9 ^

4.2.2. Temperature.

The variation of temperature follows from the v/ay in v/hich the velocity changes. It is found that the variation in temperature throughout the field is quite regular,

4.2.3. Entropy and streamlines.

The distribution of entropy in the flow behind the shook cone is represented in Fig, 2^7,8, The projections of constant entropy lines

(they correspond to streamlines) on the sphere r = constant are represented in Pig. 2. The location of the singular point, on the "leeward"

meridian plane a) = 0, at which the entropy is many valued is also

indicated in the figure. It is found that the streamlines, after leaving the shock cone, curve round and converge to the singular point. The STjrface of the hypothetical solid body (corresponding to the constant entropy line having the same entropy as a plane o) = TT ) which will produce the shcxjk wave dealt v/ith here is also indicated in the figure.

One feature in the pattern of the streamlines near the singular point may be pointed out. From equation (5) we have

9 s w 9 S / \ ' ^ T 7 + "^ï^f •^"ü = ° from which v/e have as equ.(7)

/d tO\ w d f' , T. ~ V s i n ^

streamline

or more conveniently

/d i^\ _ V s i n f

^d w "" w

streamline

- H -

Kanaalstraat 10 - DELFT

Except in the meridian plane vdien v = O (and v/hen w = O) the equation is indeterminate, the above equation holds good generally. Hence, •n*ien v—* 0 and changes sign (but w ^ O) the streamlines will tend to

"flatten" out and become parallel to the line f = constant at v = 0 and then "curl up" when v becomes positive. This is illustrated in the accompanying diagram,

This happened in the case of a fev/ streamlines on the "leeward" side.

Streamline^

4.2,4. Pressure distribution on surface.

The pressxire distribution on the body was worked oub and a comparison with the values on the downstream side of the shock wave is made in Pig.14.

This indicates that thei-e is an expansion in the flow betv/een t>ie shock v/ave and body except in a small region o) = 140 to 180 on the "v/indward"

side v/here a slight compression of the flc3w takes place.

J4,2.5. Comparison with first order solution,

The body shape obtained by the present method is compared here with the first order yav/ed cone solution (Ref. 2. 6) in Fig, 15. It is

observed that the body is smaller than the corresponding cone in the fist order solution. The body is not v/holly circular; however, it is noted that it is mostly circular- with a small h-ump on the "leeward" side. The smaller size of the body as noted in the case

óf the present solution might mean that in actual practice the asstimption of the first order theory at comparatively large yaw v/ith respect to shape of the shock cone may be valid but that it may be necessary to make a correction for the change in size of the shock cone.

The head lift and drag coefficients of the conical body (of

non-circular cross section) obtained by the present method have been calciilated using expressions defined in Appendix B,

Head lift coefficient

and a Head drag coefficient

Cj^ = 0,410

1 5

-These first order values v/ere obtained only as a means of checking the orders of magnitude of C.^ and C^ obtained for the body of the present solution, A direct comparison between the two sets of values cannot be considered to have any conclusive significance,

4.3. Method of numerical solution.

It is felt that, in general, the numerical investigation v/as satisfactory. Hov/ever, the difficulties involved in carrying on the solution neeir the singular point on the "leeward" side have shown that extreme care has to be exercised in the choice of the interval betv/een azimuthal stations and steps ±n f , In the present investigation

the region betv/een o) = 0 and o) = 45 was studied separately by carrying out the solution at five equispaced azimuthal stations. It is felt that thi-s is not a very satisfactory method and could be improved upon to a considerable extent. For investigation of the flow in this region it is necessary to have azimuthal stations closer to each other than 15 . It may perhaps be best to choose a larger number of azimuthal stations

on the "leeward" side than on the "windv/ard" side,, For future v/ork it is suggested that azimuthal stations should be spaced at intervals of Ao) = 5 from 0) = 0 to 0) = 75 and at intervals of 15° from O) = 75 to o) = 180 .

5 . CONCLUSIONS

I t h a s been found t h a t t h e n u m e r i c a l m.ethod a d o p t e d f o r t h e

i n v e s t i g a t i o n of t h e e x a c t fle w b e h i n d a yawed c o n i c a l shock i s simple t o u s e and produced r e a s o n a b l y s a t i s f a c t o r y r e s u l t s . The a c c u r a c y of t h e method can b e improved by c h o o s i n g a s m a l l e r i n t e r v a l betv/een a z i m u t h a l s t a t i o n s ,

As a p a r t i c u l a r c a s e , t h e flov/ behimd a c o n i c a l shock of s e m i - a p e x a n g l e 30 i n c l i n e d a t 20 t o a f r e e s t r e a m of Mach Number 10 h a s been i n v e s t i g a t e d and t h e shape of t h e c o n i c a l body (of n o n - c i r c u l a r s e c t i o n ) v/hich v/ould produce such a shock v/ave h a s been d e t e r m i n e d and c a n p a r e d w i t h t h e yawed cone s o l u t i o n . I n t h i s c a s e , i t h a s been found t h a t t h e

shape d e p a r t s from c i r c u l a r o n l y t o a s m a l l e x t e n t on t h e "leev/ard" s i d e . More s i g n i f i c a n t l y , i t i s n o t e d t h a t t h e s i z e of t h e body i s s m a l l e r t h a n t h a t of t h e c i r c t i l a r cone v/hich a c c o r d i n g t o S t o n e ' s f i r s t o r d e r t h e o r y (Ref. 2,6) v/ould produce t h e given shock wave,

The p r o p e r t i e s of t h e flow betv/een t h e shock cone and t h e s o l i d body s u r f a c e have been d e t e r m i n e d and t h e p a t t e r n of t h e s t r e a m l i n e s

has been s t u d i e d . The e x i s t e n c e of a s i n g u l a r g e n e r a t o r on t h e body s u r f a c e i n t h e "leeward" m e r i d i a n p l a n e o) = 0, a t which t h e e n t r o p y i s many

v a l u e d h a s been w e l l brought o u t .

6 . ACKNOla'LEDGEKÏENTS.

The author acknowledges v/ith gratitiide the advice and help given by

Mr T,R,F.Nonweiler who suggested the subject of t h i s t h e s i s and supervised

1 .

2.

3.

Woods, B,A,

Stone, A.H,

Taylor G.I,, and Maccoll J.W.

4,

5.

6.

7.

8,

9.

1 0 . 1 1 . 1 2 . S t o n e , A.H. K c p a l , Z and s t a f f of Coinputing Lab. n » R o b e r t s , R , C . , and R i l e y , J . D , P e r r i , A, Bickley,W,G, Ames R e s e a r c h StafJ Young, G.B.V^., and S i s k a , C.P.

The s u p e r s o n i c flow ejround a yawed cone. C o l l e g e of A e r o n a u t i c s T h e s i s , J u n e 1956,

( u n p u b l i s h e d ) ,

On t h e s u p e r s o n i c flow p a s t a s l i g h t l y yawing c o n e . J o u r n a l of Mathematics and P h y s i c s , V o l . 2 7 . 1948,

The a i r p r e s s u r e on a cone moving a t h i g h speeds. P r o c , Roy.Soc, S e r i e s A, V o l . 1 3 9 , 1933.

On s u p e r s o n i c flow p a s t a s l i g h t l y yav/ing cone I I . J c u r n a l of Mathematics and P h y s i c s , V o l , 3 0 . 1 9 5 1 .

T a b l e s of s u p e r s o n i c flow around cones. M . I . T . Report No. 1 , 1947.

T a b l e s of s u p e r s o n i c flow around yawing c o n e s , M . I . T , R e p o r t No, 3 , 1947.

T a b l e s of s u p e r s o n i c flow around cone of l a r g e yaw. M . I . T . R e p o r t No. 5 , 1949. A guide t o t h e u s e of M . I . T , cone t a b l e s , Joiurnal A e r o . S c , Vol.21 May 1954.

S u p e r s o n i c Plov/ eiround c i r c u l a r cones a t a n g l e s of a t t a c k . NACA R e p o r t 1045 ( l 9 5 l ) T.N,2236 ( 1 9 5 0 ) .

Formulae f o r n u m e r i c a l d i f f e r e n t i a t i o n , M a t h e m a t i c a l G a z e t t e , Vo. 25 ( l 9 4 l ) , p . 22, E q u a t i o n s , t a b l e s and c h a r t s f o r

c o m p r e s s i b l e flow. NA.CA Report 1135. (1953). Supersonic flow around cones a t l a r g e yaw. J o u r n a l A e r o . S c , V o l . 1 9 No. 2, F e b . ( 1 9 5 2 ) ,

1 7

-APPENDIX A. DETAII^ OF SOLUTION.

A , 1 , P r o c e d u r e f o r n u m e r i c a l s o l u t i o n ,

The d i f f e r e n t i a l e q u a t i o n s of motion ( l ) t o (6) s e t out i n

p a r a g r a p h 2 . 3 can b e e x p r e s s e d i n n o n - d i m e n s i o n a l form b y e f f e c t i n g t h e f o l l o w i n g s u b s t i t u t i o n s . (Primes d e n o t e n o n - d i m e n s i o n a l q u a n t i t i e s ) . u ' S ' a ' = = ^ u Vi S

c

p a , V ' , T' a^ = V = Vi ' ^ T " ViVCp ( a ' V, ) ' = w " V, where V, 2 a R T' ^ 1 % and ( a ' ) ^ = ( y - l ) T' u s i n g t h e above r e l a t i o n s v/e have

^ , 9 ^ _ w l . 3 j i l _ ^2 „ ,,2 ^ 0 (1)A o f sxn r^ 9 0) ^ '

. 9 u ' 9 w' w' d^T*

- ^* T T - ^^' ^ ^ ^ . Il5^ -' -' - -• - ^ ^ =T' -f' (2)A

9w' 9 u ' 9 v '

V» Sinif-g- u ' - ^ - v ' - g ^ + u ' w' Sin\f + v ' w' cosif = T ' J ^ ' / ^ ' J A

u ' ( v ' ^ + w ' ^ - 2 [ a - i ] T ' ) - ( a_>|)T' v ' c o t i f + ^ ' ( v ' ^ - j o - l l T ' ) „ , 9 S ' w' 9 S ' _ / ^ s ,

^ W *'^ï^f TIT = ° ^5)A

T ' + - ^ = c o n s t a n t 2 (6)A + '1 = T! + i V ' 1 = T ' + N = T' + ^ H e r e a f t e r t h e s e n o n - d i m e n s i o n a l q u a n t i t i e s w i l l be u s e d and t h e primes w i l l be o m i t t e d .

The above mentioned d i f f e r e n t i a l equations can be i n t e g r a t e d s t e p

-b y - s t e p , f o r small steps in f , proceeding inwards from a l a r g e num-ber

of azimuthal s t a t i o n s on t h e shock cone defined by various values of o),

The procedure i s as f ollov/s ;

( i ) Choose a s\jfficiently l a r g e number of azimuthal s t a t i o n s (preferably

equally spaced) around the shock wave f ran a ) = O t o a) = 7r. Since

the flow i s symmetrical about the plane of yav/ i t i s s u f f i c i e n t to consider

only the region on one s i d e of t h e plane. The physical p r o p e r t i e s of the

f lov/ j u s t behind the shock v/a.ve a r e known f ran the shock wave equations.

Y/e v/ill use t h e subscript ' 2 ' t o i n d i c a t e conditions j u s t behind the

shock wave, (Subscript ' 1 ' i s used for free stream c o n d i t i o n ) . From the

shock v/ave equations t h e values of u„, v„, w„, T„, S„ and 9u„ , 9v

_ •*• 2 » 2 ' 2 ' 2 » 2 g ' Z J

9 -w 9 s dO) Töi

-x— —T :— can be calculated at each azimuthal station. These values 9a) 9a)

can be substituted in equations (l), (3), (4) and (5) (it is sufficient to use either (2) or (4)) to give 9 u^ , 9 Vg , 9 w^ and 9 S^

df df df df

( i i ) Now choose a s u f f i c i e n t l y small increment ±n. f , hf say, and obtain

a f i r s t approximation t o the values of u^, v^, w^, S^ at f^=:f^ - l^f

a t each azimuthal s t a t i o n . This i s achieved by t h e use of t h e simple

p o i n t - s l o p e formula which i n generel terms can be v/ritten as

^nfi = ^n + ^^A ' "^''''^ y = ^ ( ' ' ) ' y ' = ^

y . = f ( x . ) , y = f ( x )

h = X . - X

n+1 n

The f i r s t approximation t o t h e value of ^ can be obtained by s u b s t i t u t i n g

the values of u^, v^, w^ obtained above in equation. ( 6 ) .

( i i i ) Having obtained the values of u^, v^, w^, S^ a t the various

• o - t - T o - a . - jn , , 9 ^ 9 v 9 w 9 S

azimuthal s t a t i o n s for f =:f , 3 , 3 , 3 , 3

90) d(ii 9a) 9a)

19

-( i v ) Now the d i f f e r e n t i a l equations can be iised to obtain

9 u 9 V 9 v/ 9 S

3 3 3 3

d f ' d f ' d f ' d f

and using these values of the d e r i v a t i v e s vdth respect t o if at if ,

and those a t i^ , the t r a p e z o i d a l formula

yn.1 = ^n + I (^n^-l ' + ^n' ^

can be used to give a second approximation to the values of "^^ » "^^ > "^-x

and S at If , This also provides a check on the numerical accuracy of the fxrst approximation.

(v) The process detailed above can be repeated to give successive approximations to the values at f until no changes in the values occiir to the accuracy required ,

(vi) Having satnsfactoi-Jly conipleted the first step (from if to if ) a further step can be taken. Consider another increment Ai^ and obtain a first approximation to the -values (of u, v, w, S) at if = if - ^if by the more accurate formula

^n^l = ^n-l + 2 h y^ .

The same procedure v/hich v/as used for t h e f i r s t step i s repeated and t h e

d e r i v a t i v e s vrLth respect t o if a t each of the aximuthal s t a t i o n s c a l c u l a t e d

for f = If.. The t r a p e z o i d a l r u l e can be invoked t o give a second

approximation t o the values a t if , The v/hole process can be repeated i f

necessary t o give f u r t h e r approximations,

( v i i ) The same procedure i s used t o cari-y t b c Ro.lntinu fovwaa'd for as

many s t e p s i n ^ a.s i s required.

APPENDIX B

C / J J C U L A T I O N O F LIFT AND DRAG COEFFICIEMTS.

\ f

R

(-0

j^U_(90-a))

ctTus-JdRr f{ length X of the body

(a cone of general cross section)

measured from the apex a t o r i g i n 0, We use i n a d d i t i o n t o t h e s p h e r i c a l

co-ordinates ( r , if, w ) a c y l i n d r i c a l polar co-ordinate system

(x, R, w ) such t h a t r = x cos f and r = R s i n if. Now consider an elemental

length ds along the circumference of t h e general shaped cross s e c t i o n ,

Let ds be i n c l i n e d t o the v e r t i c a l a t an angle ^ , Then

ds = R d 0)

cos

{ji-iSO- (tijj

Force on (the triangular) elemental area (r, ds)

f V r d. 3

component of the above force perpendicular to an axis (i.e. the component force lies in the plane x = constant),

f cos if = p r d s cos if = p d s

X

coinponent of t h i s force normal t o t h e a x i s and p a r a l l e l t o the plane of

synmetry w = 0, w = 180 i s

= (f COS If) s i n 0 = -2-^— X s i n i.

|~The components perpendicular to plane of symmetry cancel each other acting from the tv/o sides of the plane,J

21

-rr ir

N = 2x / ^ ^ ^ sin 0 = 2x / ^^ " ^ j ^ sin S^ ds, say

O • O

since p is constant,

Normal Force coefficient „

TT

[ C

0„ = ~ = - —r j -^ s i n <§ ds , v/here A = base area,

XN . - 1 2 il. _•' tt A 2 P^v^ o TT 2x / p R sin ^ d 0) ~ A „J 2 o cos |^?5-(90 -w )2

The component of the force f parallel to axis

„ . , p r d s . , p d s . ,

= f s i n if = -^— s i n if = *—p— X t a n if .

Hence we get t h e

Axial force coefficient

C - 2x ƒ f £ R t a n if ^ ^

^ A o-* 2 ^^^ [<^(90- a))3

From t h e above v/e have i f /? i s t h e angle of yav/

L ü ^ c o e f f i c i e n t C,. = C.^ cos /3 - G. s i n /£? (Head Lift Coeff,)

L N A

Drag c o e f f i c i e n t C-^ = C. cos /? + C., sin f (Head Drag Coeff,)

The c o e f f i c i e n t s C^^, C. , G^ and C_^ p e r t a i n t o complete cone from

apex t o the section considered and do not include base p r e s s u r e s ,

L i f t and Drag of Equivalent Cone,

The semi-apex angle of a c i r c u l a r cone t h a t w i l l i n axi-symmetric

flow a t M, = 1 0 , produce a conical shock v/ave of semi-apex and i^ = 30

i s 1^ = 2D, 6 (approx) , This was obtained from chart 5 in Ref. 1 1 .

Making uac of t h e 1st order theory of Stone (Ref, 2) v/e have t h a t

when the cone i s yav/ed v/ith respect t o the free stream ( a t an angle ^ )

the shock w i l l r e t a i n i t s size and shape but i t s a x i s w i l l be i n c l i n e d

t o t h e free stream a t an angle ( i n general not equal t o / ? ) . Prom

p a r t I I of Ref. 6 we have t h a t for f = 2 6 , 6 , M = 1 0 (by graphical

i n t e r p o l a t i o n ) a = 1,046,

T

, ' . P = 1 9 . 1 ° . We h a v e f u r t h e r t h a t

K-j = 0,628 and K_ = 0,167 v/here K-, and TL^ a r e c o e f f i c i e n t s of normal and d r a g f o r c e s d e f i n e d a c c o r d i n g t o vrind c o - o r d i n a t e s i n Ref. 6. The t r a n s f o r m a t i o n t o t h e more p r a c t i c a l body c o - o r d i n a t e system c a n be e f f e c t e d a s f ollov/s. T h i s method was p o i n t e d out b y Young and S i s k a i n Refurence 12 vAio g i v e t h e f o l l o w i n g formulae f o r t h e t r a n s f o r m a t i o n s . Normal F o r c e C o e f f i c i e n t G^ = ( M - ) ^ = ^ - ^ 4 ^ X . 6 2 8 ^ = 19.1 ° = 0,333 r a d i a n s = 0 . 5 3 3 cos;^ = 0.945 A x i a l F o r c e C o e f f i c i e n t s i n ^ = 0.327 r 8 TT 8 X , 1 6 7 p, , „^ Cj^ = C^ c o s /3 - C^ s i n /S- = , 5 0 4 - . 1 3 9 = 0,365 Cj3 = C^ cos y9 4. Cj^ s i n /? = , 4 0 2 + , 1 7 4 = 0.576

tA^sO

y=.o

CO= TT

FIG. 2 . VARIATION OF ENTROPY BEHIND YAWED C O N I C A L SHOCK

M = IO l//^s= 3 0 ° oC = 2 0 ° LINES OF CONSTANT ENTROPY

*

FIG. 3 . VARIATION OF TEMPERATURE BEHIND YAWED CONICAL SHOCK

M =i l O Y^ = 3 0 ° oC = 20° LINES OF CONSTANT TEMPERATURE

-FIG. 5 . VARIATION OF \ > BEHIND YAWED CONICAL SHOCK

M = lO y / ^ = 3 0 ° o^ = 2 0 ° LINES OF CONSTANT

V-l * > = - 0 - 3

BODY SURFACE

- O - I

FIG. 6 . VARIATION OF C^ BEHIND YAWED CONICAL SHOCK

M = I O 1^^= 3 0 ° o C = 2 0 ° LINES OF CONSTANT

ca-. — —ca-.— —ca-.J >

N

^ ^

\i

\ \ . I \ \

v\

k ^

"^ \

fllll

h ^

^ - ^ " ^ " ^ \ ^ \ ^ ^ ^ ^ • ~ . ^ - ^ " ^ ^ ^

1

a— I60 150 lao' lOS* to' 7 S ' »o* 4 5 " yo' M « i o y ° s ao'cc^ao" s , 22 2 3 lA 2 5 ^ " It 2 7 28 29 3 0

FIG. 7. VARIATION OF ENTROPY BEHIND YAWED CONICAL SHOCK

FIG. 8 . VARIATION OF ENTROPY BEHIND YAWED CONICAL SHOCK MslO V^^sSO" o£ = 2 0 S ~ t j

o-a ' 0 1 0 • »« BODY S U H F A C E ^ ^ ^ ^ N N .^ / / / ^ ^ ^ ^ ' >> /

K

^ - ^ - - ^

K^

^ ^ S ^ ^ ^ ^ ^ ^ "~~—~« ^ . ^ ^ _______^ . -^ -^ -^ ~ - ^ — ^ : • ~ ~ — • • - ^ • — _ _ "^ 13»' 120* I 0 5 * «o° " • . O * 21 23 23 34 35 yy' 3» 3 ' 2B 2« 5 0

FIG. 9. VARIATION OF TEMPERATURE BEHIND YAWED CONICAL SHOCK

M>IO V ^ = 3 0 ° 0 6 - 2 0 ° T~Y

FIG. 10. VARIATION OF TEMPERATURE BEHIND YAWED CONICAL SHOCK

I ' O 0 - 9 0 « a 0 - 7 0 - 4

FIG. I I . DISTRIBUTION OF U BEHIND YAWED CONICAL SHOCK M - I O Y'^sSO** 0^ = 20° T X ' ^ t J

•0-- 0 •0-- 2

- 0 - 3

FIG. 12. DISTRIBUTK)N OF V BEHIND YAWED CONICAL SHOCK

1 -o 0 - 9 O - B 0 - 7 0 •» 0 - 5 0 - 4 0 - Ï 0 - 2 0- 1 'c, /

— t ^

_ SHOCK WAVE _ \ /

i

h

/ / * * • / / ^ /

—A

- ^ / ^

7

/ OY

1

SURFACE W "

FIG. 14. FLOW BEHIND YAWED CONICAL SHOCK. PRESSURE DISTRIBUTION M s 10 ^^'30° cC = 2 0 ° C P - ^ W

CJ

\

BODY SURFACE PRESENT SOLUTION

PIkST ORDFR S O L N - CONE 1 / = 26-6°-AXIS A

( y = 0 - 9 ° ) ' 8 0 CONE y/^ ? 4 " - AXfS B

FIG. 15. FLOW BEHIND YAWED CONICAL SHOCK