Cos o Ksztalcie Dyszy z NASAf

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- . . .

* < .~

., . ,I I!

N A S A T E C H N I C A L N O T E 2 3 4 1

CCIAN COPY: RETURN TO

K! Rf LAND AI'B, N MI-X

J4I-VdL (VVUL-2)

A M E T H O D OF CHARACTERISTICSFOR STEADY THREE-DIMENSIONAL

SUPERSONIC FLOW W I T H A P P L I C A T I O N

TO INCLINED BODIES OF R E V O L U T I O N

by John V. Rdkich

A m e s Reserzrch Center

Moffett Field, Crzhy

N A T I O N A L A E R O N A UT I C S A N D S PA CE A D M I N I S T R A T I O N W A S H I N G T O N , D . C . OCTOBER 1 9 6 9



TECH LIBRARY KAFB, NM

19 . Secur i ty Class i f . (of th i s repor t)

Unrlassified

20. Secur i ty Class i f . (of th i s page)

Unrlassified

1. Report No.NASA TN D-5341

4. T i t le ond Subti t le

2. Governmeni Access ion No.I7. Author(s)

John V. Rikich

9. Performing Organizat ion Name and Address

NASA A m e s Rcscarch Ce nterMoffett Field, Calif. 94035

2. Sponsoring Agency Name and Address

Natinnal Arrmi.iotirs and Spare AdiiiiiiistrationWasningtoii, D. C. 20546

15. Supplementary Notes

16 . Abstract

3. Rec ip ien t ' s Cata log No.

-

5. Repor t Date

O r t ~her 1969-

6. Performing Organizat ion Code

-

8. Performing Organizat ion Repor tA-3325

I O . Work Unit No.129-01-03-05-00-21

11 . Cont rac t or Grant No.

13. Type o f Report and Per iod COVI

Terlinical Note

14 . Sponsoring Agency Code

..L-,.-"

1 7 . Key Wards Suggested by Author

Method of rhara cte ris tic sThree-di niensional flowSupersonic flowFluid merhanicsPeronyna mics

18 . Distr ibut ion Statement

Unclassified - Unliiiiited

*For sale by the Clcai inghouse for Federal Scientific andSpringfield, Virginia 22151

Tcrlinical Tniormation

22. P r i c e *

8 3.00



TABLE OF CONTENTS

Page

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

P R I N C I P A L SYMBOLS. . . . . . . . . . . . . . . . . . . . . . . . . . . 3

THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Basic Equat ions o f I n v i s c i d F lo w. . . . . . . . . . . . . . . . . 5

C h a r a c t e r i s t i c s T h e o r y . . . . . . . . . . . . . . . . . . . . . . 7Nota t i on . . . . . . . . . . . . . . . . . . . . . . . . . . 7

C o o r d i n a t e t r a n s f o r m a t i o n . . . . . . . . . . . . . . . . . . 8

C h a r a c t e r i s t i c d i r e c t i o n s and c o m p a t i b il i t y e q u a t io n s . 8

B i c h a r a c t e r i s t i c M e t h o d . . . . . . . . . . . . . . . . . . . . . 9

C h a r a c t e r i s t i c c on e. . . . . . . . . . . . . . . . . . . . . 10

C o m p a t i b i l i t y e q u a t i o n s . . . . . . . . . . . . . . . . . . . 11

Fundamenta l compl ica t ions . . . . . . . . . . . . . . . . . . 1 3

Reference Plane Method. . . . . . . . . . . . . . . . . . . . . . 14F i n i t e d i f f e r e n c e mesh . . . . . . . . . . . . . . . . . . . 1 4Refe rence p l ane equa t i ons . . . . . . . . . . . . . . . . . 15

C h a r a c t e r i s t i c d i r e c t i o n s . . . . . . . . . . . . . . . . . . 1 7

C o m p a t i b i l i t y e q u a t i o n s . . . . . . . . . . . . . . . . . . . 18

. . .

NUMERICAL METHODS. . . . . . . . . . . . . . . . . . . . . . . . . . . 19F i n i t e D i f f er e n c e E q u a ti o n s . . . . . . . . . . . . . . . . . . . 20

Computat ional Procedure . . . . . . . . . . . . . . . . . . . . . 2 2L o c a l i t e r a t i o n . . . . . . . . . . . . . . . . . . . . . . 2 2

G l o b a l i t e r a t i o n . . . . . . . . . . . . . . . . . . . . . . 2 2S t e p s i z e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

S t a b i l i t y c o nd i ti o n . . . . . . . . . . . . . . . . . . . . . 23Accuracy and computing t ime. . . . . . . . . . . . . . . . . 23

N u m e r i c a l D i f f e r e n t a t i o n . . . . . . . . . . . . . . . . . . . . . 2 4S t r e a m w i s e a n d r a d i a l d e r i v a t i v e s . . . . . . . . . . . . . . 2 4C i r c u m f e r e n t i a l d e r i v a t i v e s . . . . . . . . . . . . . . . . 25

S t a r t i n g Data . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

C o n e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Discon t inuous de r iva t i ve s . . . . . . . . . . . . . . . . . 31

RESULTS A N D DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . 32

Pointed Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Sphe r i ca l l y Blun t ed Cone . . . . . . . . . . . . . . . . . . . . . 33Comparison With Experiment. . . . . . . . . . . . . . . . . . . . 37

C O N C L U D I N G REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iii



TABLE OF CONTENTS - Concluded

Page

APPENDIX A.- DIRECTION COSINES FOR NONORTHOGONAL COORDINATES . . . 40

APPENDIX B.- SHOCK-BOUNDARY CONDITIONS FOR THREE-DIMENSIONAL FLOW. e . 45

APPENDIX C.- SURFACE BOUNDARY CONDITIONS FO R BODIES WITHOUT AXIALSYMMETRY . . . . . . . . . . . . . . . . . . . . . . . . . 49

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1

T A B L E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4



A METHOD O F CHARACTERISTICS FOR STEADY

THREE - DIMENSIONAL SUPE RSON I C FLOW

WITH A PPL I CA T I O N TO INCLINEDBODIES OF REVOLUTION

By John V . Rakich

Ames Research Center

SUMMARY

C h a r a c t e r i s t i c s t h e o r y f o r t h re e - di m e n si o n al c o m pr e ss i bl e f lo w i s

r ev ie we d a nd t h e c o m p a t i b i l i t y e q u a ti o n s a r e d e r i v e d i n terms o f p r e s s u r e a n d

s t r e a m a n g le s as d ep en de nt v a r i a b l e s . The r e l a t i v e m e r i t s o f b i c h a r a c t e r i s t i c

and re f e r en ce plan e methods a r e d i s c u s s e d .developed and demonst ra ted fo r po in t ed and b lun t ed bod i e s a t ang l e o f a t t a ck .

A re fe rence p l ane me thod i s

The fundamenta l compl i ca t i ons a r i s i ng i n a three-dimensional method of

c h a r a c t e r i s t i c s a r e t h a t : (1) t h e c o m p a t i b i l i t y eq u a t io n s c o n t a i n " cr o s s -

d e r i v a ti v e s " i n a n o n c h a r a c t e r i s t i c d i r e c t i o n ; and ( 2 ) t h e r e i s an i nc rea sed

n ee d f o r i n t e r p o l a t i o n t o p r e v e n t t h e computed d a t a s u r f a c e s fro m b eco min gskewed. These problems a r e minimized by us i ng a re f e re nce p l an e method wi th apre sc r ibe d and un ifo rm spac i ng o f mesh po i n t s . The f i n i t e d i f f e r en ce mesh

employed con si st s of an equ al number of po in ts between t h e body and shock su r-

f a c e s i n ea ch r e f e r e n c e p l a n e . N um er ica l p r oc e du r es f o r d i f f e r e n t i a t i o n an d

in t e rp o l a t i o n ensure second-orde r accuracy i n t erms of mesh spac ing . Four i e ra n a l y s i s i s employed i n t h e c i r c u m f e r e n t i a l d i r e c t i o n and i s f ou nd t o b e e f f e c -

t i ve i n r educ ing t h e number o f r e f e re nce p l an es and computing t imes . A t y p i -

ca l mesh c on s i s t s o f 7 p l a n e s w i t h 1 5 p o i n t s i n ea ch p l a n e . The u n i t

computation time i s about 0 .0013 minute pe r poi nt on an I B M 7094 computer.

R e s u lt s f o r t h e f lo w ar ou nd i n c l i n e d , c i r c u l a r c o n es , b o th b l u n t e d an d

p o i n t e d , a r e p r e s e n t e d t o de m o ns t ra te t h e method d e s c r i b e d . P r e d i c t i o n s o f

s u r f a c e a nd p i t o t p r e s s u r e s a r e i n g ood a gr ee me nt w i t h a v a i l a b l e e x p e ri m en t alr e s u l t s f o r a 15' sphere-cone a t 1 0" a n g l e o f a t t a c k . B l un t ne s s e f f e c t s on

s h o c k - l a y e r p r o p e r t i e s f a r f rom the nose a r e r ea sonab ly w e l l p r e d i c t e d .

INTRODUCTION

The method of c h a r a c t e r i s t i c s ha s b ee n w e l l known f o r many ye ar s an d

e x c e l l e n t t h e o r e t i c a l d ev el op me nt s c a n b e f ou nd i n numerous t e x t s a nd r e p o r t sof which re fe r enc es 1 through 3 gi ve t h e most complete reviews o f mult idimen-

s i o n a l t h e o r y . However, u n t i l r e c e n t y e a r s t h e r e h av e b e en r e l a t i v e l y f e w

p r a c t i c a l a p p l i c a t i o n s o f t h e m etho ds t o t h r ee - d im e n s io n a l f l o w s . T h is i s

p ro b ab ly due p a r t l y t o t h e l a r g e number o f o p e r a t i o n s i n vo l ve d i n f i n i t e d i f -

f e r e n c e c a l c u l a t i o n s i n t h r e e d i me ns io ns , and p a r t l y t o t h e e x t r a d e gr ee o ffreed om t h a t r e s u l t s fr om t h e e x i s t e n c e o f c h a r a c t e r i s t i c s u r f a c e s r a t h e r th a n



c h a r a c t e r i s t i c l i n e s .

t o per fo rm c a l cu la t io ns f o r th r ee -d imens iona l s t e ady and uns teady f lows and

se ve ra l g roups have r epo r t e d such work.

on t h e method o f c h a r a c t e r i s t i c s a r e r e p o r t e d i n r e f e r e n c e s 4 through 10, and

a s u c c e s s f u l u s e o f a n o n c h a r a c t e r i s t i c f i n i t e d i f f e r e n c e method i s d e s c r i b e d

i n r ef er e nc e 11.

d e s cr i b ed and d i s cu s s e d c r i t i c a l l y i n r e f e r e nc e 1 2 .

High-speed computers have made it t e c h n i ca l l y f e a s i b l e

For example, many re ce nt va r i a t io n s

A number o f th e p roposed ch ar ac te r i s t i c s schemes have been

A l l th e proposed di f fe re nc e schemes f o r t h r e e - d i m e n s i o n a l c h a r a c t e r i s t i c s

can be p laced i n one o f two b road ca t eg or i e s : (1) r e f e re nce p lane methods

( c a l l e d s e m i - c h a r a c t e r i s t i c methods i n r e f . 1 2 ) ; a nd (2 ) b i c h a r a c t e r i s t i c

me th ods . I n r e f e r e n c e p l a n e methods t h e c h a r a c t e r i s t i c l i n e s a r e o b t ai n e d

f rom th e pr oj ec t i on o f Mach cones and s t re am lin es onto a p r e s c r i b e d p l a n e . I n

b i c h a r a c t e r i s t i c methods t h e c h a r a c t e r i s t i c l i n e s are t h e g e n e r a t o r s of t h e

Mach cone and th e ac t ua l s t r eam l in es . S in ce , i n th re e d imensions , t h e equa-

t i o n s w r i t t e n a lo ng th e s e c h a r a c t e r i s t i c l i n e s a r e p a r t i a l d i f f e r e n t i a l equa-

t i o n s , a n um er ic al e v a l u a t i o n o f " c ro s s - d e r i va t i v e s " o f f t h e c h a r a c t e r i s t i c

l i n e s i s n e c e s sa r y . ( I n t h i s s e n s e , t h r ee - d im e n s io n a l c h a r a c t e r i s t i c m eth ods

are s imi la r t o non ch ar ac te r i s t i c methods f o r two-dimens ional f low .)

l e m e n co u nt e re d w i t h mos t b i c h a r a c t e r i s t i c m ethod s a r i s e s from t h e n ee d t oemploy a two-dimensional a r r a y o f d a t a t o e v a l u a t e t h e c r o s s - d e r i v a t iv e s and

t o p erfo rm t h e r e q ui r ed i n t e r p o l a t i o n i n t h e i n i t i a l d a t a s u r f a c e . F ur th er -

more, i f a c o n v en t i on a l c h a r a c t e r i s t i c s mesh i s u se d , t h e d i s t r i b u t i o n o f mesh

po i n t s i n th e d a t a su r f ace s t ends t o become very nonun iform, which causes

a d d i t i o n a l c o m p l i c a t i o n s . For t h es e r easons one i s l e d t o r e f e r en c e p la n e

methods, where a b e t t e r c o n t r o l c an b e m ai nt ai ne d on t h e f i n i t e d i f f e r e n c e

mesh and where cu rv e f i t s can be made wi t h r es pe ct t o a s i n g l e . v a r i a b l e .

Refe rence p lane methods have been c r i t i c i z e d on th e the or e t i c a l g rounds t h a t

t h e i n i t i a l d a t a may b e o u t s i d e t h e domain o f d e pe nd en ce o f t h e c a l c u l a t e d

po in t . However, prob lems r e l a t e d t o t h i s c r i t i c i s m h av e n o t m a t e r i a l i z e d . I n

f a c t , t h e u se o f su ch d a t a i s p r e c i s e l y w h a t i s r e qu i r ed by t h e wel l-known

Courant-Friedrichs-Lewey s t a b i l i t y c on d it io n .

The prob-

I n t h e p r e s e n t work , t h e c o m p a t i b i l i t y e qu a ti o ns o f c h a r a c t e r i s t i c t he o ry

a r e d e r i v e d f o r b o th t h e b i c h a r a c t e r i s t i c a nd r e f e r e n c e p l a n e me th ods.

t i c a l d i f f i c u l t i e s w i t h t h e b i c h a r a c t e r i s t i c method a r e d i s c u ss e d and a f i n i t e

di f f er en ce scheme based on the ref ere nce p la ne method i s proposed.

proposed method abandons t he us ual c h a ra c t e r i s t ic mesh and employs mi fo rm ly

spaced po i n t s a long l i ne s ly ing in equa l ly spaced mer id iona l p l a nes . Numeri-

c a l i n t e r p o l a t i o n and d i f f e r e n t i a t i o n are accomp lished by means of second -

d eg re e p ol yn om ia ls i n t h e r a d i a l d i r e c t i o n and b y F o u r i e r a n a l y s i s i n t h e

c i r c u m fe r e n t i al d i r e c t i o n .

Prac-

The

P r e li m i na r y r e s u l t s b y t h e p r e se n t m eth od w er e d e sc r i b e d i n r e f e r e n c e 1 3and compared wi t h c al cu la t i on s by th e method of re fe re nc e 4.p a r i so n s w i t h e x p er im e nt we re shown i n r e f e r e n c e 1 4 , e s t a b l i s h i n g t h e r e l i -

a b i l i t y o f t h e n um er ic al m et ho ds . I n t h e p r e se n t r e p o r t t h e f lo w e q u a ti o n s

and numerical t ec hn iq ue s a r e d e s c r i be d i n g r e a t e r d e t a i l t h a n was p o s s i b le i n

re fe re nces 13 and 14 .

Extensive com-

2



P RINCIP AL SYMBOLS

a

‘1*JC2*

cP

cPP

e x Y e r Y e @ J X i

h

H

k

K-

P

Rn

S *

’ J n Y t

v

X

,.Zi

speed of sound

b i c h a r a c t e r i s t i c d i r e c t i o n s

c h a r a c t e r i s t i c d i r e c t i o n s i n m e ri di on al p l a n e s

p r e s su r e c o e f f i c i e n t

p i t o t - p r e s s u r e c o e f f i c i e n t

u n i t v e c t o r s a lo ng x , r , Q,

e n t h a l p y

t o t a l e n th a lp y

smoothing co ns tan t (eq . (84) )

c o e f f i c i e n t o f n um e ri ca l d i f f u s i o n t e rm ( e q . ( 8 4) )

Mach number

shock normal and tan ge nt vec to rs

p r e s s u r e

body nose radius

p r o j e c t i o n o f s t r e a m l i n e s on m e r i di o n a l p la n e s

s t r e a m l i n e c o o r d i n a t e s

v el o ci ty components a long e . e e

t o t a l v e l o c i t y

a x i a l d i s t a n c e fr om b l u n t n ose (body a x i s )

u n i t v e c t o r s a lo ng s , n y t

c y l i n d r i c a l c o o r d i n a t e s

u n i t v e c t o r s a l o ng c h a r a c t e r i s t i c c o or d i na t e s

u n i t v e c t o r s a lo ng 5 , n , 5

a n gl e o f a t t a c k , d eg

,. A ,.

x’ r ’ @+v = vS^ = vj;,

3



t r ans fo rma t ion mat r ix ( eq . (16) )

J iKT

c1i j

B

s p e c i f i c h e a t r a t i o

shock an g le i n cross pla ne (eq . (A6))

t r ans fo rma t ion mat r ix ( eq . (A18) )i j

s,rl,s nonor thogonal shock- l a ye r coo rd ina tes ( f ig . 3)

f lo w a n g l e f ro m x a x i s i n m e r id i o na l p l a n e , t a n - l ( v / u ) ( s e e f i g . 1 )e

angle between & and {r

Mach angle , sin- (1/M)-I

1-I" p r o j e c t i o n o f p on mer id ional p lane

t r ans fo rma t ion mat r i x ( eq. (A13))i j

D d e n s i t y

shock ang le i n mer id iona l p l ane ( eq . ( A 5 ) )

c r o s s f l o w a n g l e , s i n - l W / V ( s ee f i g . 1)

a zi mu th al a n g l e , c y l i n d r i c a l c o o r d i n a t e s ( s e e f i g . 1 ) , deg

Subs c r i p t s

i n i t i a l a nd new d a t a l i n e s

body

i n d i c e s f o r r a d i a l p o s i t i o n o f mesh p o i n t s

( j = 1 , 2 , . . . , J - 1, J ) ( s e e f i g . 5)

i n d e x f o r c i r c u m f e r e n t i a l p o s i t i o n o f mesh p o i n t

(k = 1 , 2 , . . . , L - 1, L)

k

body nose

shock

f r e e - s t r e a m c o n d i t i o n

4



S u p e r s c r i p t s

i

v i n d ex d e n o ti n g v a r i a b l e s p , 8 , 9 , p f o r v = 1, 2 , 3 , 4

*

i nd e x f o r a x i a l p o s i t i o n o f mesh p o i n t

q u a n t i t y o r component r efe ren ced t o mer id ional p l ane

A u n i t v e c to r

+ g e n e r a l v e c t o r

Note: See equ at i on (14) f o r d ef n i t i o n o f i nd ex n o t a t

THEORY

4

on.

Bas ic Equa tions o f In v i sc id F low

Th e d e r i v a t i o n o f t h e c h a r a c t e r i s t i c r e l a t io n s w i l l b e c a r r i e d ou t i nt h i s s e c t i o n s t a r t i n g w it h t h e e q u at io ns f o r i n v i s c i d e q u il i br i um flow w r i t t e n

i n t h e i r i n t r i n s i c form w i t h p r e s s u r e a nd f lo w a n gl e s as dependen t va r i ab les .

Th is c h o i ce o f v a r i a b l e s e l i m i n a t e s e n t r o p y d e r i v a t i v e s from t h e f lo w eq ua -

t i o n s , t h e re b y s i m p l i f y i n g t h e a n a l y s i s . F ol lo w in g t h e de ve lo pm en t o f r e f e r -

ence 15 , th e combined con t i nu i t y and momentum equat ions ar e w r i t t en i n vec to r

form .

2 grad p + d i v i = 0

PV2

1;grad p + t - c u r l < = 0

Pv2

_ A , . A

where s , n , t a r e o r th o go n al u n i t v e c t o r s w i t h s t a n g e n t t o t h e s t re am -

l i n e s . F or r o t a t i o n a l , i n v i s c i d , no n he a t- c on d uc t in g , a nd n o n r e a c t i n g f lo w ,

en t ropy i s c on se rv ed a l o n g s t r e a m l i n e s a nd t h i s r e q u i r e s

- grad p - a2s grad p = 0 (4)



G = cos + cos e G + cos + s i n e & + s i n + G Q (7)X r

ii = - s i n e G + cos 0 g rX

A A n n

t = - s i n + cos e e - s i n + s i n 8 eX r

+ cos I$ eQ (9

Using equat ions (7) through (9) wi th t he s tand ard ve ct or formulas i n c y l i n d r i -c a l coord ina t es , one can ob ta in th e fo l lowing f rom equa t ions (1) th rough (4) :

a e s i n 2 I$ cos 0L*+os $,s=p v 2 a n r

C h a r a c t e r i s t i c s Th e o r y

Nota t ion . - C h a r a c t e r i s t i c s t h e o r y i s most conveniently developed,

e s p e c i a l l y f o r t h e m u lt i di m en s io n al c a s e , i f one uses index no t a t i on . There-

fo re , t h e no ta t io n and development o f Couran t and F r i ed r i ch s ( r e f . 1, p . 75)

w i l l b e fo l lowed i n t h i s review. Summation convent ion i s used wherein a sum-

mation symbol i s impl i ed by a r epea ted index . Thus equ ati ons (10) t o (12) may

b e w r i t t e n s im pl y as

Vi au - f

a - -V V axi ?J

7



where t h e i n d i c e s r e f e r t o :

v dependen t va r i ab l e (p , 8 , f o r v = 1, 2 , 3)

i

l~ equa t i on (eqs . (10) - (12) fo r p = 1 , 2 , 3)

i ndependen t va r i ab l e ( s , n , t f o r i = 1, 2 , 3)

Coord ina t e t r ans f orma t ion . - Equat ion (14) can b e expre s sed i n terms of

new coord ina t e d i rec t i o ns y j

na te s. Thus one can w r i t eo b t a i n e d b y a r o t a t i o n o f t h e o r i g i n a l coord i -

h A

where x i and y j a re u n i t v e c to r s al on g x i and y j , r e s p e c t i v e l y , a nd t h e

e le me nt s o f t h e t r a n s f o r m a t i o n m a t r i x a r e t h e d i r e c t i o n c o s i n e s d e f i n e d as f o l

lows i n terms o f t h e s c a l a r products of :i and p j :

21 $2

g2 * $2

With th i s t ra nsf orm at io n, equa t ion (14) becomes

where

i

pv'i j' = a

_ _ -h a r a c t e r i s t i c d i r e c t i o n s a nd c o m p a t i b i l i t y e q u at i on s .- I f i n i t i a l d a t a

a r e g iven f o r u" on t he sur fac e y1 = 0 , equat ion ( 1 7 ) can be so lv ed fo r

i n g d e r i v a t i v e s wi th r e s p e c t t o y2 and y3 on t h e r i g h t s i d e o f e q u at i o n ( 1 7 ) ,auV/ayl i n o r d e r t o g e n e r a t e d a t a on an a d j o i n i n g s u r f a c e y1 = Ayl. Plac-

one ob t a ins

8



where

= f - bk (k = 2 , 3)g?J ?J Flv a Y k

Equation (19) can be cons ider ed as a s e t o f a l g e b r a i c e qu a ti o ns f o r

auV’/ay1, with a so lu t i on by Cramer ’s r u l e g i v i n g

auv DV

where Ib;t)I i s t h e d e t e r m i n a n t o f b

ob ta ined by r ep lac ing th e v th column of b ( l ) wi th g

and D V i s t he de te rminan t1-1v

lJv 1-1’

I f Ibi:) va ni sh es , th e n y1 i s normal t o a c h a r a c t e r i s t i c s u r f a c e and

the f low equat ions g i v e no i n fo r ma t io n a b ou t d e r i v a t i v e s i n t h i s d i r e c t i o n ;

t h a t i s , auV/ayl may be d is con t in uou s. Th ere for e , th e equat ion of th e char -

a c t e r i s t i c s ur f ac e i s given by

On th e o th e r hand, i f equa t ion ( 2 1 ) i s s a t i s f i e d , t h e n t h e nu me ra to r of

e q u a ti o n (20) must a l so v a n ish i n o r d e r f o r a so l u t i o n t o e x i s t . Th e r ef o re ,

DV = 0 (v = 1 , 2 , 3) (22)

g i v e s t h e so - c a l l e d c o m p a t i b i l i t y e q u a t i o n s . I t may b e n o te d from e q u a ti o n

(19) t h a t t h e c o m p a t i b i l i t y r e l a t i o n s c o n t a i n o ne l e s s d im en si on t h an t h e

o r i g i n a l d i f f e r e n t i a l e q u a t i o n s . F or t h e t h r ee - d im e n s io n a l p ro bl em t h e com-p a t i b i l i t y e qu at io ns i n v ol v e d e r i v a t i v e s i n two d i r e c t i o n s .

B i c h a r a c t e r i s i c Method

The c h a r a c t e r i s t i c r e l a t i o n s r ev ie we d i n t h e pr e vi o us section can now be

sp ec ia l i ze d t o th e p roblem of equ i l ib r iu m th ree -d imens iona l gas flow. Con-

s id e r equa t ions (10) th rough (13) wi th t he supplementa ry cond i t ion s (5) and

( 6 ) . Note th a t equ a t ion (13) i s a l r e a d y i n c h a r a c t e r i s t i c f orm s i n c e i ti n vo l ve s d e r i v a t i v e s i n one d i r e c t i o n ; t h e s t r e a m l i n e , s , i s a l i n e a c r os s

which t h e d e n s i t y g r a d i e n t , a p / a n , may be d is con t in uou s. Therefore , one needo n l y c o n s i d e r e q u a t i o n s ( l o ) , ( l l ) , a nd ( 12 ) f o r w hic h t h e i c o e f f i c i e n tmatrices may be wr it te n

9



where t iki i s t h e K ro ne ck er d e l t a

0 k # i

1 k = i

6 k i = {The t ransformed co ef f i c i en t mat r ix given by (18) becomes

03 j 2

PV

C h a r a c t e r i s t i c c on e. - The c h a r a c t e r i s t i c de t e rm i n a n t, e q ua t io n ( 2 1 ) , ca n

now b e e v a l ua t e d t o g i v e t h e e q u a ti o n o f t h e c h a r a c t e r i s t i c s u r f a c e i n termso f t h e c o e f f i c i e n t s o f t h e go ve rn in g d i f f e r e n t i a l e q u a t i o n s , U sing eq u at io n

(24) one obta ins

where B 2 = M2 - 1 .

recognized asNow, from eq u at io n (16) t h e t er ms i n e q u a ti o n ( 2 5 ) are

10



whi ch_ are t h e components of a longBlcharocter ts t ic

direct ion t h e x coo rd in a t e s . Thus, equa t i ony 2 (25) ddsc r ibe s a cone a round th e x1

o r s a x i s , as i n d i c a t e d i n f igure2 ( a ) , m akin g t h e a n g l e 90 - ut h e s d i r e c t i o n . T h is co ne i s nor-

m a l t o t h e c h a r a c t e r i s t i c co ne. The

cone

v e c t o r 71 l i e s a l o n g a g e n e r a t o r o ft h e normal cone , and th e vanish ing

determinant (21) means t h a t t h e de r iv -

cone a t i v e s w i t h r e s p e c t t o y1 may b ed i s c o n t i n u o u s ; t h a t i s , t h e d i f f e r e n -

t i a l e q u at i on s ( l ) , ( 2 ) , and (3)

cannot give any informat io n about

w i t h

Chorocteristic

y‘\

1,

Normal

(a) Charac teristi c cone and bicharacteristics.

Figure 2.- Characteristic coordinates. d e r iv a t iv e s i n t h i s d i r e c t io n .

D 1 =

I t f o l l o w s , t h e n , i f t h e yi c o o r d i n at e s a r e o r t h o g o n al , t h a t 72 and

i 3 a r e t an g en t t o t h e c h a r a c t e r i s t i c co ne. I f 72 i s ch os en t o l i e a l o ng a

g e n e r a t r i x o f t h e co ne , t h e n y2 i s c a l l e d a b i c h a r a c t e r i s t i c d i r e c t io n .

gl a21 cos $ “ 3 1

g2 “11 cos 4 0 = o (26)

8 3 0 “ 1 1

Expanding (26) one obtains

where

+ “ 3 3 ay3

)4

- ( “ 3 2 ay2

11



g, = f 3 -- a p + a 3 3 -)p - (a12 ay24 + a 1 3z)42 (a32 ay2 a Y 3

P V

The b i c h a r a c t e r i s t i c d i r e c t i o n y2 can b e a r b i t r a r i l y ch osen t o l i e along any

r ay o f t h e c h a r a c t e r i s t i c co ne.

t i o n s c a n b e o b t a i n e d f r o m ( 2 7 ) .

d et er mi ne t h e s o l u t i o n f o r t h e t h r e e d ep en de nt v a r i a b l e s p , e , 4 .

Thi s means t h a t an i n f i n i t e number o f equa -

However , th re e eq ua t ion s are s u f f i c ie n t 1 t o

The b i c h a r a c t e r i s t i c s c an b e c h os en so as t o s i m p l i f y t h e c o m p a t i b i l i ty

e q u a t i o n s .

wi th ( s , n , t ) , t h a tF i r s t it i s n o te d from e q u a t i o n ( 1 6 ) , i d e n t i f y i n g ( x l , x 2 , x 3 )

n * Y 2

A , .

* Y1 t - Y 2 t * Y 3

Thus a number of th e e lements of a .w i l l b e z e r o if y2 i s i n t h e s - n

p l a n e , a n d 7 3 l i e s a l o n g t h e fa x i s . I n t h i s c a se ( s e e f i g . 2(b) )

t h e r e a r e two p o s s i b i l i t i e s g iv en by

l j

-~~

(31)

s i n p cos p

? s i n 1-1

:$0 0

y 2

2( b ) Bicharacteristics in the s- n plane. w here t h e u p p er s i g n r e f e r s t o t h e

l e f t -r u n n i n g c h a r a c t e r i s t i c C 1 , and

t h e l ow er s i g n t o t h e r i g h t - r u n n i n g

c h a r a c t e r i s t i c C 2 .

L e t t i n g 9 2 l i e i n t h e s - tp l a n e s o t h a t p 3 l i e s a lo n g t h e f ia x i s g i v e s

0

( 3 2 )s i nos p

-9

Fw= ( o i n p

i jc) Bicharacteristics in the S-t plane.

Figure 2 . - Concluded. -cos p

where t h e s i g n s c o rr es po nd t o t h e b i c h a r a c t e r i s t i c d i r e c t i o n C 3 which

i n c r e a s e s w i th i n c r e a s i n g s and t as shown i n f i g u r e 2 ( c ) .

'Redundant schemes which make us e of more th an th r e e equ ati on s a r ed i s cu s s e d i n r e f er e n c es 8 , 10 , 1 2 , and 16.

1 2



S u b s t i t u t i o n o f c1 from equ ation s (31) and (32) in t o equ atio n (27)i j

r e s u l t s i n t h r e e eq ua ti on s a l o ng t h e b i c h a r a c t e r i s t i c s C 1 , C 2 , an d C 3 .

Using equ atio n (31) f o r C1 an d C 2 , one ob ta ins

where y2 = C1 f o r t h e u pp er s i g n an d y2 = C 2 f o r t h e l ow er s i g n . Simi-

l a r l y , o ne o b t a i n s f ro m e q u a ti o n (3 2) t h e f o l l o w in g e q u a t i o n f o r t h e d i r e c t i o n

c3 :

g1 + c o t 1-1 g3 = 0

def ined i n equa t ions (28) t o (30)g i

t r a ig h t fo r w ar d s u b s t i t u t i o n o f t h e

wi th appro pr i a t e e l ement s o f a . . from equa t ion (31) o r equa t ion (32) y ie ld s ,

with some rearrangement , t h e f o l i o w in g c o m p a t i b i l i t y e q u a ti o n s :1

where , from the r i g h t s id e o f equa t ions (10) th rough (12 ) ,

s i n 2 4 cos e and s i n 4 s i n 8f 3 = -

os s i n er r rf 2 =l = -

For two-dimensional flow (4 = 0 ) , equat ions (33) and (34) reduce t o the usual

co mp at ib i l i t y equa tion s and eq ua ti on (35) becomes th e streamwise momentum

e q u a t i o n .

Fundamental complicat ions .- In compar ison wi th a x i a l l y symmetr ic f lows,

equa tion s (33j-- throu gh (35) have two com plicat ing fe at ur e s which were men-

t i o n e d e a r l i e r . The se a r e (1 ) t h e p r e sen c e o f " c r o s s - d e r i v a t i v e s " a $ / a t anda e/ an on t h e r i g h t s i d e , an d ( 2 ) t h e need t o pe rfo rm a two-paramete r in t e r -

p o la t io n f o r d a t a i n t h e i n i t i a l d a t a s u r f a ce . The second problem a r i s e s

1 3



b ec au se b i c h a r a c t e r i s t i c s t hr ou gh a g e n e r a l mesh p o i n t ( i n a p r e s c r i b e d sur -f a c e ) do n o t , i n g e n e r a l , p a s s t hr ou gh mesh p o i n t s i n t h e i n i t i a l d a t a

s u r f a c e .

Many schemes have been proposed us in g equat io ns of t h i s form along

b i c h a r a c t e r i s t i c s ( se e , e . g . , r e f s . 5-8 and 1 2 ). However, t h e programming of

such methods t ends t o be cumbersome, and th e accuracy o f e va lua t in g th e c ross -

d e r i v a t i v e s i s i n most cases l e s s t h an t h a t o f t h e b a s i c c a l c u l a t i o n s. (A n

excep t ion i s t h e r e c e n t work r e p o r t e d i n r e f e r e n c e 1 0 . ) The se p ro bl em s areminimized i f c h a r a c t e r i s t i c s l y i n g i n p r e s c r i b e d r e fe r e n c e pl an es a reemployed. Then t h e i n t e r p o l a t i o n f o r i n i t i a l d a t a and e v a l u a t io n o f c r o s s -

d e r i v a t i v e s c an b e r e du c ed t o a s e t o f one-parameter cu rve f i t s with second-

o r d e r a c c u r a c y .

d e r iv e d f o r c h a r a c t e r i s t i c l i n e s w hich are t h e p r o j e c t i o n s o f b i c h a r a c t e r i s -

t i c s C 1 , C 2 , C 3 o n to r e f e r e n c e p l a n e s .

I n t h e ne xt s e c t i o n , t h e c o m p a t i b i l i t y e q u at i on s w i l l b e

Reference Plane Method

I n t h e f o l l ow i n g d ev el op me nt , t h e f lo w e q u a t i o n s a r e w r i t t e n i n terms of

two components ly i ng i n p redete rmined r e f e re nce p l anes and a th i rd component

d i r e c t e d o u t of t h e s e p l a n e s . F or most pr ob le ms e n c o un t er e d i n e x t e r n a l a e r o -

dynamics i t i s c on v en ie nt t o s p e c i f y t h e r e f e r e n c e p l a n e s as t h e a x i a l p l a n e s,

@ = c o n s t a n t , o f a c y l i n d r i c a l c oo r d in a t e s ys te m (s e e f i g . 1 ) . The s o l u t i o n

of problems with a x i a l symmetry i s determined by c a l c u l a t i o n a l o n g a s i n g l e

p l a n e , b u t t h r e e- d im e n si o na l p ro bl em s r e q u i r e c a l c u l a t i o n a l o n g s e v e r a l

p l a n e s s i m u lt a n e o us l y . C h a r a c t e r i s t i c t h e o ry i s em ployed t o c a l c u l a t e t h e

f low along each pla ne . To a c hi e ve t h i s , t h e c o m p a t i b i l i t y e qu a ti o ns a l on g

t h e p r o j e c t i o n s o f t h e b i c h a r a c t e r i s t i c s on t h e r e f e r e n c e p l an e s must b e

d e r i v e d .

c o m p a t i b i l i t y r e l a t i o n s f ro m t h e i n t r i n s i c momentum e q u a t io n s a p p l i c a b l e t o

t h e r e f e r e n c e p l a n e s . F i r s t , however, i t w i l l b e n ec e ss a ry t o d i s c u s s t h e

co or di na te mesh which w i l l b e u s ed t o d e s c r i b e t h e s h oc k l a y e r .

The procedure w i l l b e t o f i n d t h e p e r t i n e n t c h a r a c t e r i s t i c s and

F i n i t e d i f f e r e n c e m esh. - The c y l i n d r i c a l c o o r d i n a t e sy s t em u sed t o

expand t h e v e c t o r r e l a t i o n s i n e q u a ti o ns (1) through ( 3 ) , a nd t o d e f i n e t h e

r e f e r e n c e p l a n e s , i s n o t i d e a l from t h e c o m p u ta t io n a l s t a n d p o i n t . Th i s s te m s

from the f a c t t h a t s p e c i a l t r e a t m e n t would b e r e q u i r e d f o r bo d ie s w i t h non-

c i r c u l a r c r o s s s e c t i o n s an d, more i m p o r t a n t l y , f o r sho ck su r f a c e s . One i st h e r e f o re l e d t o a f i n i t e d i f f e r e n c e mesh which d i v i d e s t h e s ho ck l a y e r i n t o

a number o f annu la r r in gs which inc lude bo th th e shock and body s u r f a ce s , asshown i n f ig ur e 3 . The r e su l t in g mesh po i n t s a r e connected by a nonorthogonal

5 , n, sy st em o f c o o r d i n a t e s : 5 and n l i e i n t h e r e f e r e n c e p l a ne w i th n

u su a l l y n orm al t o t h e body s u r f a c e ; 5 i s d i r e c t e d o u t o f b u t n o t g e n e ra l l ynormal t o t h e r e f e r e n c e p l a n e .

_ _

'Conver se ly, b i ch ar ac te r i s t i c s th rough known po in t s on a p la ne i n i t i a l

d a t a s u r f a c e w i l l , i n g e ne r al , i n t e r s e c t a t p o i n t s l y i n g i n a nonp lanar su r -

f a c e . The su bse q ue n t d a t a su r f a c e s become i n c r e a s i n g l y d i s t o r t e d as a compu-

t a t i o n p ro ce ed s away from t h e i n i t i a l d a t a p l a n e .

1 4



"3,Shock c ;

Bo d

rReference

x p l a n e s

' ' ( W ' n d w o r d '

S h o c kVm

S vmmet rv

a ? - - -

plane

Figure 3.- Nonorthogonal shock-layer coordin ates.

A r e l a t e d s ys te m i s one i n which

5 i s r e p l a c e d l o c a l l y by t h e p r o j e c -

t i o n , s * , o f s t r e a m l i n e s o nt o t h e

r e f e r e n c e p l a n e s . I t i s t h i s s * , r l ,

< system which i s u s ed t o e x p r e s s

t h e i n t r i n s i c fl ow eq u a ti o ns (10 )

through (13) i n a f or m n ee de d f o r t h e

p r e s e n t r e f e r e n c e p la n e a n a l y s i s .The u n i t v e c t o r s , j ; = (G, 6 , t ) , i n

s t r e a m l i n e c o o r d in a t e s a r e r e l a t e d t o

t h e new s y s t e m b y d i r e c t i o n c o s i n e s ,

E $ d e f i n e d b y

i

ij'

A A

where ^z * = (;* , r \ , z;) .j

W r i t te n i n te rm s o f i t s components , equat ion (36) gives

A

; E T 1 $ * + ET2: + E T 2 <(37)

(38)

A

(391-t = E;+* + E 3 2 r l +

Appendix A de sc r ibe s how E ? i s c a l c u l a t e d i n t e rm s of th e body and

s ho ck -w av e s h a p e s . The d e t a i l e d e x p r e ss i o n s f o r t h e d i r e c t i o n c o s i n es a r e

n o t n e ed ed f o r t h e d ev el op me nt o f t h i s s e c t i o n b u t it s h o ul d b e n o t e d t h a t ,s i n c e t h e sh ock s ha pe i t s e l f i s obta ined f rom a s o l u t i o n o f t h e p ro ble m ( i . e . ,

a d i r e c t as opposed t o an i nv e r se approach), E $

t h e e n t i r e f l ow. However, i n a l o c a l l y s u p e r s o n i c r e g i o n , w he re t h e s ho ck c an

b e c a l c u l a t e d s t e p by s t e p , E X

proceeds downstream from an i n i t i a l d a t a l i n e .

lj

i s no t known beforehand f o rij

can always be determined as t h e c a l c u l a t i o n

i j

Re fe rence p l ane equa t i ons .- R e c a ll i n g t h e r u l e s f o r a d i r e c t i o n a l

d e r i v a t i v e , e q u a t io n s ( 37 ) a nd (39) y i e l d t h e f o l l o w i n g e x p re s si o ns f o r

d e r iv a t iv e s i n t h e s an d t d i r e c t i o n s .

15



These d i f f e r e n t i a t i o n r u l e s a ll ow one t o w r i t e t h e i n t r i n s i c f low

e q u a t i o n s i n terms of t h e des i re d p l a na r component s.

( 4 0 ) and (41) i n t o equa t ion s (10) through (13) and regrou ping te rms, one

S u b s t i t u t i n g e q u a t i o n s

o b t a i n sa e

E l l 27 ancos ($ - f , *

82 aP

pv as

1 ap * a e

PV as* - f 2 *an + E l l cos ($--

(43)

(44)-a + - f 3 *

as*

- -p - f 4 * (451as*

The l e f t s i d e s o f t h e s e e q u at io n s c o n t a i n d e r i v a t i v e s i n t h e r e f e r e n c e p la n esand the remaining te rms a re a l l lumped in t o th e f i x , which a r e given by

16



The f i * t h u s e x p re s se d c o n t a i n d e r i v a t i v e s a l o n g t h e T and 5 c o o r d i n a t e

d i r e c t i o n s a n d , as a r e s u l t , c an b e e a s i l y e v a l u a t e d w i t h s t a n d a rd one -p ar am et er d i f f e r e n t i a t i o n r o u t i n e s . I t w i l l a l s o b e s e e n , when t h e f i n i t e

di f ference scheme i s d e s cr i b ed , t h a t t h e d e r i v a t i v e s a c $ / a s * a p p e a r i n g i n

fi

equat ions (42) through (45) c ont a in no ad di t i on al approximat ions beyond th os e

i n t h e o r i g i n a l e q u a t i o n s . Only t h e u s ua l a ss um pt io ns p e r t a i n i n g t o v i s c o s i t y

and he at conduct ion ar e made. The bra cke ted [ ] te rm s i n t h e * a l l van i sh

f o r a x i a l l y sym me tr ic fl ow s ( i . e . , f o r c$ = 0 ) , and the equa t ions r educe t ot h e f a m i l i a r i n t r i n s i c e qu at io ns f o r z er o a n gl e o f a t t a c k .

* and ap/as* appear ing i n f ,* and f 4 * a r e e a s i l y t r e a t e d .

I t i s i mp or ta nt t o n o t e t h a t , a l t h o u g h w r i t t e n i n a s i m p l i f i e d f o r m ,

fi

C h a r a c t e r i s t i c d i r e c t i o n s . - C h a r a c t e r i s t i c s t h e or y , as o u t l i ne d i n aprevious s ec t i on , can now be app l ie d t o equat ions (42) through (45) . I t i s

f i r s t o bser ve d t h a t e q u a ti o n s (4 4) an d ( 45 ) a r e a l r e a d y i n t h e d e s i r e d c h a r-

a c t e r i s t i c fo rm. These equa t ions g ive no in fo rma t ion %bout th e normal de r iv -

a t i v e a /a n; t h e r e f o r e , s* i s a c h a r a c t e r i s t i c d i r e c t io n f o r c$ and p . This

i s i n co nt ra s t t o equat ions (42) and (43) , which can be combined to g i ve

d e r i v a ti v e s i n d i f f e r e n t d i r e c t i o n s . W ri ti ng t h e l a t t e r i n t h e form o fequa t ion ( 1 4 ) , one has

VL f *

P V a x . * LI1

Expressed i n terms of new co ord ina tes , *, equ at i on (50) becomes (c f . eqs .

(17) and (24)) :Y i

where

and where a.. a re t h e e le m en ts of t h e t r a n s f o r m a t io n m a t r i x r e l a t i n g x1 3 i

and y . The c h a r a c t e r i s t i c d i r e c t i o n s f o r e q u a t i on ( 51) a r e o b t a i n e d from

the de te rminan tj

17



which yie lds

Wri t i ng a i j i n terms of a r o t a t i o n a ng l e as i n e q u a t i o n ( 3 1 ), o ne o b t a i n s

f o r t h i s c as e

where p* i s t h e ang l e between t h e

s t r e a m l i n e a n d t h e c h a r a c t e r i s t i c s

i n t h e r e f e r en c e p la n e as shown i n

f i g u r e 4 . Equat ion ( 5 2 ) shows that

u* i s re l a t e d t o t h e Mach ang l e

accord ing t o

c o t ll* = E T 1 c o t p ( 5 3 )Figure 4 . - Characteristic directions for the

reference planes.

Compa t ib i l i t y equa t i ons . - The c o m p a t i b i l i t y e q u at i on s a p p l i c a b l e t o t h e

d i r e c t i o n s C 1 * an d C 2 *

t h e b i c h a r a c t e r i s t i c s .c a n b e o b t a i n e d i n t h e way p r e v i o u s l y d e s cr ib e d f o r

F o r t h e p r e s e n t c a s e , one h a s i n p l a c e o f e q u a t i on ( 2 7 )

a 2 1

g1* - g2* = 0

where

(54)

and

1 8

A l g eb r a ic d e t a i l s a r e s t r a i g h t f o r w a r d a nd ne ed n o t b e r e p e a t e d .

c o m p a t i b i l i t y e q u at i on s a r e :T h e r e s u l t i n g



--p + cos 47e - ( f l * + Bf2*)sin p*2

p v acl* ac1( 5 5 )

Equations (55) and (56) , t o ge th er wi th equa t ions (44) and ( 4 5 ) , a r e th e f i n a l

s e t o f d i f f e r e n t i a l equa t ions programmed f o r numerica l ca l c u l a t io ns . They a resupplemented by t he energy equat ion (eq . ( 5 ) )

h + V 2 / 2 = H

and the equa t ions o f s t a t e ((6a) and (6b))

an d

De ta i l s of th e numerical methods used i n th e computer program ar e descr ib ed

i n t h e n ex t s e c t i o n .

NUMERICAL METHODS

The g e n e ra l t h e o ry o f c h a r a c t e r i s t i c s was d ev el op ed i n t h e p r e vi o u s

se c t i o n , w h e r e it was shown t h a t th e th ree-dimens ional problem can be reduced

t o an equ iva len t two-dimensional form. A numerica l s o l u t io n o f th e prob lemcan then be accompl ished i n th e usual way by numer ical l y e va lu at in g der iva -

t i v e s i n o ne d i r e c t i o n i n o r d e r t o c a l c u l at e a s t e p f or wa rd i n t h e s e co nd

d i r e c t io n . The p rob lem i s ana logous t o t he numer ica l s o l u t io n o f two-d imensiona l hyperb o l i c equa t ions by nonc ha ra c t e r i s t i c s methods.

, There fo re , i n fo rmula t ing a p r a c t i c a l method f o r c a l c u l a t i n g t h r e e -

d im e ns io na l f l ow , t h e n u me r ic a l d i f f e r e n t i a t i o n p r o c e s s i s of primary: impor-

t a n c e . I f t h e d i f f e r e n t i a t i o n i s t o b e pe rf or me d e f f i c i e n t l y a nd a c c u r a t e l y

( i . e . , a t l e a s t second o r d e r i n a ty pi ca l mesh dimension) , th e mesh po in tss h ou l d b e c o n s t r a i n e d t o l i e a lo n g s im pl e c o o r d i n at e l i n e s . S ec on dl y, t h e

b ou nd ary c a l c u l a t i o n s a r e s i m p l i f i e d i f t h e c o o r d i n a t e s l i e on t h e sho ck a nd

body su r f a ce s . These cons idera t ions sugges t th e shoc k- l ay er -o r i en te d , non-or thogonal coord ina te s shown i n f i gu re 3 . The r e su l t in g computat iona l p roce-

dure i s s i m p l i f i e d s i g n i f i c a n t l y c ompared w i t h p r e v i o u s l y p ro po se d t h r e e -

d imehsiona l ch ar ac te r i s t i c s methods ( s ee , e .g . , r e f . 1 2 ).

19



P r e s en t e d n e x t a r e t h e d i f f e r e n c e e q u a t i o n s , c o mp u t at i on a l l o g i c , an d

n u me ri ca l d i f f e r e n t i a t i o n p r o c e du r e s d e v el op e d on t h e b a s i s o f t h i s n on or th og -

ona l (< , n , 5) mesh. The problem may b e s t a t e d as f o l l o w s :

Given d a t a on a n i n i t i a l n - 5 s u r f a c e a t 5, where the f low i sl o c a l l y s u p e rs o n ic , it i s r e q u i r e d t o g e n e r a t e , by means o f t h e

f low equat ions and boundary con di t io ns , new da ta on th e adja cen t

n - 5 s u r f a c e a t 5, + A t .

F i n i t e D i f f e r e n c e E q u at io n s

Figure 5 shows th e mesh po in t arrangement f o r a t y p i c a l r e f e r e n c e p la ne

( i - 1 ) a nd ( i ) t h e i n i t i a l d a t a l i n e a t SAand new da ta l i n e a t < B . Le t t he

s u b s c r i p t s j and k d en o te t h e r a d i a l

and c i r c u m f e r e n t i a l p o s i t i o n s o f t h e

mesh po in t s . However, t h e su bs c r ip t

k w i l l b e o m i t te d f o r b r e v i t y i n

w r i t i n g t h e d i f f e r e n c e e q u a t io n s .

The method adopted3 t o c a l cu la te- I , T + I --.

c o n d i t i o n s a t a t y p i ca l mesh po i n t

i , j on Sp, makes use o f i n t e r po l a t ed

d a t a a t t h e p o i n t s o f i n t e r s e c t i o n on

5 , o f t h e c h a r a c t e r i s t i c s th ro ug h

p o i nt i , j . A t h ree -po in t Lagrange

i n t e r p o l a t i o n i s employed t o de te rmine

the necessary da ta f rom known condi -

t i o n s a t neighb or ing mesh p o in t s .Figure 5.- F i n i t e d i f f e r en c e m e s h .

T hree i n t e r s e c t i o n s a r e r e q u i r e d f o r e ach f i e l d p o i n t on S B . To i d e n t i f y

t h e s e p o i n t s t h e c o nv e nt io n i s adopted whe reby t he su bs c r ip t

t h e i n t e r s e c t i o n w it h <A o f t h e s t r e a m l in e p r o j e c t i o n s* t h rough po in t

i , j , and t h e s u b s c r i p ts T - 1 an d ~ + le p r e s e n t i n t e r s e c t i o n s of c h a r a c t e r i s -

t i c s C 1 * an d C 2 * , r e s p e c t i v e l y ( s e e f i g . 5 ) .

T r e p r e s e n t s

With t h i s c o n ve n ti o n t h e c o m p a t i b i l i t y e q u a t i o n s (55) and (56) can be

w r i t t e n i n t h e f o l l o w in g f i n i t e d i f f e r e n c e f orm:

ei- ’ ) = F1 A C l *T-

i

‘ i pi-1)-+l B 2 ( O j i - B T + l-l) = F 2 A C 2 *

(57)

(58)

.._ - -

3This method i s c a l l e d t h e H a r t r e e method ( r e f . 1 7) an d a l s o t h e i n v e r s e

method ( re f . 12) ; it was used by Katskova and Chushkin (ref . 9 ) .

2 0



Equat ions (44) and (45) a r e s im i l a r ly wr i t t e n as

an d

The system o f equa t ions i s completed by t h e energy and s t a t e e q u a t i o n s , (5)

and (6 ) . They apply t o a l l o f t h e f i e l d p oi n ts - t h a t i s , f o r t h e i nd ex j

running from 2 up t o J -1 . For the body po i n t , j = 1 , equat ion (57) i s r e p l a c e dby th e equa t ion o f t he body,

rll =

iwhich p e rm i ts t h e c a l c u l a t i o n o f 81

appendix C . I t i s shown the re tha t

where

f (x1 , Qk)

by means o f e q u a ti o n s d e r i v e d i n

For bod ies o f r evo lu t ion equa t ion (6 la ) r educes t o

A t t he shock , j = J , equa t ions (58) , ( 5 9 ) , and ( 6 0 ) a re r e p l a c e d b y t h e

obl ique shock equ at i ons . The jump con di t ion s f o r a genera l th r ee -d imens iona lshock su r f a ce a r e deve loped i n append ix B a nd c a n b e w r i t t e n i n t h e f o l l o w i ng

f u n c t i o n a l f orm f o r f i x e d f r e e - s t r e a m c o n d i t i o n s :

2 1



Vw he re u r e p r e s e n t s p , 8 , @, p f o r v = 1, 2 , 3 , 4 . Here 0 and 6 aret h e shock-wave an g les i n th e p lan es CP = c o n s t a n t and x = c o n s t a n t , r e s p e c -

t i v e l y , a n d a i s t h e - a n g l e o f a t t a c k . Equa t ion (62) depends d i r e c t l y on th e

unknown shock angle ol, a nd i n d i r e c t l y on t h e p a r a m e t e rs 6 k, @ k , a nd a. The

shock ang le 6k i s d e te rm i ne d b y n u m e ri c a l d i f f e r e n t i a t i o n o f sho ck c o o r d i -

n a t e s as d e s c r i b e d i n a pp en di x A . The fou r equa t i ons ob ta ine d f rom (62) f o r

v = 1 through 4, t o g e t h e r w i t h e q u at i on ( 57 ) , a r e s u f f i c i e n t t o d et erm in e t h eshock angle 01.

Computat ional Procedure

L oc al i t e r a t i o n . - The d i f f e r e n c e e q u a t i o n s a r e s o l v e d b y a s t a n d a r d E u l e r

An i n i t i a l g u e s s , s a y

_ _ _ _ - . ~ -p r e d i c t o r - c o r r e c t o r method i n which t h e c o e f f i c i e n t s a r e t r e a t e d a s c o n s t a n t s

l o c a l l y a nd e qu a l t o t h e i r a ve ra ge v a l ue o v e r t h e s t e p .

v i - 1(u'): = (u )

v a l u e s o f ( u ' ) ~

p r e d i c t i o n s .

s p e c i f i e d a c c u r a c y .

t i v e e r r o r t o l e s s t ha n l ~ l O - ~ . t i s shown i n s ta n d a r d t e x t s t h a t t h e i t e r -

a t e d r e s u l t h as a t r u n c a t i o n e r r o r o f t h e o r d e r o f t h e s t e p s i z e c ubed.

, i s u se d t o s t a r t an i t e r a t i v e p r o c e d u re b y w hic h c o r r e c t e dj

a r e c a l c u l a t e d u si n g c o e f f i c i e n t s e v a l u a te d w it h t h e p r e vi ou s

F or t y p i c a l mesh p o i n t s t h r e e c o r r e c t o r s r ed uc e t h e r e l a -

jT h e i t e r a t i o n i s c on ti nu ed u n t i l t h e p r e s s u r e r e p e a t s t o a

I n t h i s method t h e c o e f f i c i e n t s i n e q u at i on s (57) t hr ou gh

(60) a r e a ve ra ge d a lo ng t h e c h a r a c t e r i s t i c d i r e c t i o n a p p r o p r i a t e t o e ac h

e q u a t i on . F or e xa mp le , t h e c o e f f i c i e n t F 1 i n d i f f e r e n c e eq u a ti o n ( 57) r e p r e -

s e n t s t h e a ve ra ge o f t h e r i g h t s i d e of d i f f e r e n t i a l e q u a t i on ( 5 5) , t ak en a l on gt h e c h a r a c t e r i s t i c C 1 * , and i s w r i t t e n as

A,,, B y , and F1-I

F 1 = 1 / 2 [ ( f l * + B f 2 * ) ] f + 1 / 2 [ ( f l * + Bf2*)]:::1

i n t h e p r e s e nt n o t a t i o n . S i m i l a r l y , a v er ag es a re e v a l u a t e d u s i n g d a t a a tp o i n t ~ + lo r e q u a t io n (58) and a t p o i n t T f o r e q u a t i o n s (5 9) and ( 6 0 ) .

G lo ba l i t e r a t i o n . - The s e t o f e q u a ti o n s ( 57 ) t hr o u gh ( 60 ) a r e so l v e d

s u c c e s s i v e l y on L r e f e r e n c e p l a n e s , @k (k = 1, 2 , . . . , L) . Th e d i f f e r -

ence equa t ions govern ing th e flow a long 'va r ious r e f e r en ce p lanes a r e coup led

by c r o s s - d e r i v a t i v e s which a r e i n c lu d e d i n t h e f u n c t i o n s

r i g h t s id e o f each equa t ion and by th e shock ang le 6k appear ing i n equa-

t i o n ( 6 2) . I n o r d e r t o s o l v e t h e s e e qu at io ns i n an e x p l i c i t manner,i t i st h e r e f o r e n e c e ss a ry f i r s t t o approximate

a t e d o n < A . Then, us ing ca l cu la t ed va lues on a l l o f t h e L p l a n e s t o

e v a l u a t e c r o s s - d e r i v a t i v e s o n

and 6 k and t h e e n t i r e p r o ce s s c a n b e r e p e a t e d . T h i s i s r e f e r r e d t o as ag lo ba l i t e r a t i o n , i n c o n t r a s t t o t h e p o i n t- b y- p oi n t i t e r a t i o n employed i n t h e

l o c a l s o l u t i o n o f t h e d i f f e r e n c e e q u a ti o n s .

F,, appear ing on the

F,, and 6 k w i t h d e r i v a t i v e s e v a l u -

F,,; B , o ne o b t a i n s t h e n e x t a p pr ox i ma t io n t o

2 2



Since computing t imes are g e n e r a l l y l a r g e i n t h r e e - d im e n s i on a l p ro bl em s

( s e e n e xt s e c t i o n ) , t e s t c a se s we re r u n t o d et er mi ne t h e n ee d f o r g l o b a l

i t e r a t i o n . T ab le I shows t h a t e x c e l l e n t r e s u l t s a r e o b t a i n e d w i th o ut i t e r a -

t i o n , t h a t i s , u s i n g d e r i v a t i v e s e v a l u a t e d on E A . I t i s t h e r e f o r e e x p e c t e d

t h a t t h i s i t e r a t i o n would n o t be r e q u i re d i n most problems.

S t e p S i z e

Since t he mesh po in ts are n o t c o n s t ra i n e d t o f ol lo w c h a r a c t e r i s t i c l i n e s

wi th th e p re se n t method, th e s i z e o f a f o r w a r d s t e p , Ag = cB - S A , can be

s t a b l e n um e ri ca l p r o c e s s , t h e s t e p sho u l d n o t e x ce ed a c e r t a i n maximum d et er -

mined by t h e r e g i o n o f i n f l u e n c e o f t h e i n i t i a l d a t a . T h is s t a b i l i t y c on di -

r e q u i r i n g sm a l l e r f or wa rd s t e p s . The l a t e r a l s t e p A5 d oe s n o t a f f e c t t h e

s t a b i l i t y d i r e c t l y , a lt ho ug h i t d o es , o f c o u r se , a f f e c t t h e accuracy . The

e f f e c t of s t e p s i z e on a cc ur ac y i s d i s c u s s e d a f t e r t h e f o ll o wi n g p ar ag ra ph s

on s t a b i l i t y c o n di t io n .

a r b i t r a r y t o some e x t e n t . The o n ly l i m i t a t i o n i s t h a t , i n o r d e r t o have a

t i o n w i l l make A g d ep en d on t h e r a d i a l s t e p A n , s m a l l e r r a d i a l s t e p s

S t a b i l i t y c o nd i t io n .- Although thea n a l y s i s o f n u me ri ca l s t a b i l i t y ha s n ot

been per fo rmed f o r th e genera l non l in -

e a r e q u a t i o n s , h e s t a b i l i t y c r i t e r i o n

f o r l i n e a r h y p e r b o l i c e q ua t io n s ( s e e ,e . g . , r e f . 17) i s a p p a r e n t l y s u f f i c i e n tf o r t h e n o n l i n e a r e q u a t i o n s . T hi s i sthe well-known C - F - L (Courant ,

F r i e d r i c h s , Lewey) c o n di ti on which

e s s e n t i a l l y s t a t e s t ha t t he domain o f

dependence o f t h e c a l cu la t e d po in t mus t

b e i nc lu de d i n t h e i n i t i a l d a ta . T hi s

means t h a t t h e c h a r a c t e r i s t i c C 1 *t hr ou gh p o i n t ( i , 2 ) i n f i g u r e 6 must

p a ss t h ro ug h o r above p o i n t ( i - l , l ) .

S i m i l a r l y, t h e c h a r a c t e r i s i c C 2 *

t hr ou gh p o i n t ( i , - 1) must fa11 through

Figure 6 . - S t e p s i z e limitation. o r below p o i n t ( i - 1 ,J).

S t r i c t l y f ol lo we d, t h e C - F - L c o n d i t i o n w ould r e q u i r e t e s t i n g ev e ry

shock and body p oi nt t o determ ine th e maximum allo wab le s t e p s i z e , Agm, which

would i n s u r e s t a b i l i t y . However, e x p e ri e nc e i n d i c a t e s t h a t t h e c o n d i t i o n i sn ot o v er ly r e s t r i c t i v e i n t h e s e ns e t h a t s t e p s s l i g h t l y l a r g e r t h an do

n o t u su a l l y c a u se a v i o l e n t i n s t a b i l i t y . T h er ef or e, i t i s a de qu at e t o t e s t

o n l y a t t h e body a n d , f o r b o d i e s w i t h c i r c u l a r c r o s s s e c t i o n , on t h e windward

s i de where A g m i s l ik e ly t o be sm al le s t , due t o th e low Mach number. Thes t e p s i z e i s t hen t aken s l i g h t l y l e s s than t he maximum; a v a l u e o f

A < = 0 . 8 A< , works w e l l i n most problems.

A<,

Accuracy and computing t ime . - The accuracy of a numerical computat ion i su su a l l y e s t i m a t e d b y c o mp ar in g r e su l t s o b t a i n e d w i t h v a r i o u s mesh sp a c i n g s .

23



S i nc e t h e e x ac t s o l u t i o n i s us ua ll y unknown, one can only compare with th e

r e su l t s o f t h e f i n e s t mesh a nd o b se r ve how t h e e r r o r d e c r e a se s . I f t h e t r u n -

c a t i o n e r r o r s a r e s e co nd o r d e r i n t erm s o f t h e mesh s i z e , t h e n h a l v i n g t h e

mesh s i z e sh o u ld r ed u ce t h e e r r o r by o n e - q u a r t e r .

To t e s t t h e accuracy o f th e p rese n t method , ca lc u l a t io ns were per fo rmed

from X/Rn = 2 t o x/Rn = 3 , u s i n g 3 , 5 , an d 7 p l a n e s a n d 5 , 10, and 15 po in ts

a long each p lane .X/Rn = 2 . The r e s u l t s o f t h i s s t u d y a re shown i n t a b l e 11. T a b l e I I ( a ) p r e -

s e n t s t h e sh oc k a n gl e s a nd t h e su r f a c e p r e s su r e on t h e

X/Rn = 3 . These r e s u l t s show t h a t th e method i s of second-order accuracy.

Also, i t s ho ul d b e n o te d t h a t t h e r e s u l t s w i t h k = 5 and 7 agree very well

f o r J f i xe d (moving ho r i z on ta l l y in the t a b l e ) . The good accuracy wi th on ly

a f e w p lanes i s a t t r i b u t e d t o t h e u se o f t r i g o n om e t r i c a n a l y s i s f o r t h e

c r o s s - d e r i v a t i v e s .

I n each case the same s t a r t i n g c o n d i t i on s were u s ed a t

@ = 90" plane a t

The computing time n a t u r a l l y i n c r e a s e s as the mesh i s r e f i ne d ; t h i s i si l l u s t r a t e d i n t a b l e I I ( b ) , which l i s t s t h e t o t a l number o f p o i n t s computed,

t h e t o t a l e x e c ut e t i me , an d t h e t im e p e r p o i n t i n m i n ut e s . The c a l c u l a t i o n

was performed on an I B M 7094 Model 1 2 i n FORTRAN I V ( v e r s i o n 1 3 I B J O B proces-s o r ) .

f i n e s t mesh. The u n i t time inc re ases as t h e number o f po i n t s d ecrea ses , p rob-

ab ly because o f f ix ed inpu t /ou tp u t t imes . The ac tu a l and u n i t t imes a r e

almost doubled when one g lo bal i t e r a t i o n i s made a t e a c h s t a t i o n .

A u n i t t im e o f a b ou t 0 .1 3 ~ 1 0 - min p e r p o i n t was o b t a i n e d w i t h t h e

Numeri ca l Di f f e r e n t a t on

Discussed next i s t h e p ro ble m o f e v a l u a t i n g c r o s s - d e r i v a t i v e s a p pe a ri n g i n

t h e e q u a ti o n s. P a r t i a l d e r i v a t i v e s i n t h r e e d i r e c t i o n s a pp ea r i n t h e f u n c t io n s

f i * def ined by equ at i ons (46) through ( 49) . They ar e of th e form a/as*,a / a r , , an d

a / a r ,i n t h e s t re a m, r a d i a l , a nd c i r c u m f e r e n t i a l d i r e c t i o n s . S p e c i a l

t r ea tmen t i s g iv en t o t h e c i r c u m f e r e n t i a l d e r i v a t i v e , f o l l o w i n g t h e d i s c u s s i o n

o f t h e f i r s t two.

S tr ea mw ise a nd r a d i a l d e r i v a t i v e s . - Th e se d e r i v a t i v e s a r e t a ke n t o g e t h e r

s i nc e they a re bo th eva lua ted by th e s t and ard po lynomial approach .

i d e a i s t o employ a d i f f e r e n t i a t i o n fo rm ul a c o n s i s t e n t w i th t h e a cc ur ac y o f t h e

b a s i c c a l c u l a t i o n .

The main

F or t h e s t re a m wi se d e r i v a t i v e t h e approximation

i s c l e a r l y e q u i v a l e n t t o t h e form o f t h e d i f f e r e n c e e q u a t io n s em plo yed .

Eq ua ti on (64) r e p r e se n t s t h e d e r i v a t i v e of

v a l , As*/2, w i th an e r r o r o f t h e o r d e r

each s t e p i n t h e l o c a l i t e r a t i o n pr oc es s p r e v i o u s l y de s c ri b e d.

uv a t t h e mi dp oi nt o f t h e i n t e r -

T h i s d e r i v a t i v e i s e v a l u a t e d a ts * 2 .

24



Because of t he eq ua l l y spaced coo rdi nat e mesh pr es en t l y employed, t h e

r a d i a l d e r i v a t i v e i s s i m i l a r l y d et er mi ne d t o t h e same a c cu r ac y b y t h e c e n t r a l

d i f f e r e n c e

Equat ion (65) approx imates th e de r iv a t iv e a t p o i n t ( i , j ) w i th an e r r o r o f

p r e v io u s ly d e s c r ib e d .

o r d e r A n 2 .

SA and SB

Th e a v e r a g e r a d i a l d e r i v a t i v e a t th e midpo int ( i +1 /2 , j ) between

can be o bt ai ne d by means of t h e g l o ba l i t e r a t i o n p ro ce du re

Standar d end-poin t formulas o f e q u iv a l e n t a c c u r a c y a re used a t th e body

and shock where cen t ra l d i f fe rences are n o t p o s s i b l e . The se n ee d n o t b e

w r i t t e n o u t , as th ey can be found i n many books (e .g . , r e f . 1 8 ) .

C i r c u m f e r e n t i a l d e r i v a t i v e s . - I n t h e a er od yn am ic s o f b o d i es i t i s w e l lknown, fro m l i n e a r i z e d an d p e r tu r b a t i o n t h e o r i e s as w e l l as from experiment,

t h a t t h e s o lu t i o n o f most p ro bl em s c an b e r e p r e s e n t e d b y a t r i g o n o m e t r i c

s e r i e s i n t h e m e ri d io n al a n gl e @ . When information such as t h i s i s a v a i l -

a b l e it s h o u ld b e p o s s ib l e , b y c h o ic e of a n a p p r o p r i a t e f u n c t i o n a l form, t o

improve a numerica l proc ess . For example, wi t h da t a known t o have a ne ar ly

c o s in e v a r i a t i o n , it i s c l e a r t h a t f ew er sam ple p o i n t s a r e r e q u i r e d t o

approxima te th e da ta wi th a c o s in e s e r i e s t h a n w i t h a polynomial . A

Four ie r - se r ie s approx ima t ion i s t h e r e f o r e u se d t o e v a l u a t e d e r i v a t i v e s w i t h

r e s p e c t t o 5 . Symmetry cond i t ions , wh ich usua l ly a r i se a t @ = 0 and @ = T ,

a r e e a s i l y s a t i s f i e d i n t h i s way. T h is t e ch n iq u e makes it p o s s i b l e t o c a l -

c u l a t e w i th f e w er r e f e r e n c e p l a n e s , t h e r eb y i n c r e a s in g t h e c o m p uta ti o na l

e f f i c i e n cy ( s e e r e f . 1 3 ) .

F or t h e p r e s e n t a p p l i c a t i o n it i s n e c es s a r y t o d e t er m ine F o u r i e r

approximat ions f o r p re ssu re p , f low ang l e 0 , c ross f low ang le $ , and

d e n s i t y p from t h e i r v a l u e s on E p l a n e s @ k (k = 1 , 2 , . . . , E) w i t h

@ 1= 0 an d @ L = T . Symmetry c o n d i t i o n s f o r t h r e e v a r i a b l e s ( r e p r e s e n t e d by

UkV (v = 1 , 2 , 4 ) ) a r e s a t i s f i e d by a c o s i n e s e r i e s o f t h e form

1.- 1

.\n=o

For th e c ross f low ang le (v = 3) , which i s zero a ts e r i e s

@ = 0 and @ = n , a s i n e

L- 1F

k, =L b n s i n n @

n=1

i s n e c e s s a r y .

25



When e qu a l l y spaced meridiona l p la ne s 'k = i ~ ( k 1 ) / (L - l ) ,

(k = 1, 2 , . . . , L) a r e us ed , t h e c a l c u l a t i o n o f t h e F o u r i e r c o e f f i c i e n t s

i s p a r t i c u l a r l y s im p le b ec au s e t h e u s u a l o r t h o g o n a l i t y c o n d i t i o n s a r e

e x a c tl y s a t i s f i e d by a f i n i t e sum ( s e e , e . g . , r e f . 1 8 o r 19 ) . I n t h i s caset h e c o e f f i c i e n t s a r e g i ve n b y

and(n = 0, 1, . . ., L - 1)

( n = 1, 2 , . . . , L - 1)k

b = -n L - 1

k=2

(69)

D e r i v a t i v e s w i t h r e s p e c t t o t h e a n g l e

F o u r i e r s e r i e s , e q u a t io n s ( 6 6) a nd (67) , t o g iv e

CP a r e o b t a in e d by d i f f e r e n t i a t i n g t h e

v L - 1

du k V

__@ =En a n s i n n O

n=1

and

L- 1

d'k

= x n b cos F on=1

The d e s i r e d d e r i v a t i v e s i n t er ms o f d i s t a n c e a lo n g t h e 5 d i r e c t i o n a r e

ob t a ined f rom equa t i ons (70) and (71) ac cordin g t o

where i t i s u n de rs to o d t h a t t h e d e r i v a t i v e on t h e r i g h t s i d e of (72 ) i se v a l u a t e d u s i n g d a t a on t h e 5 c o o r d i n a t e . The s c a l e f a c t o r g c r e l a t e s

th e d i s t an ce a long cor re sponding t o an i nc rementa l change i n @ ,

26



and c an b e w r i t t e n i n t er ms of t h e d i r e c t i o n c o s i n es d ev el op ed i n a p pe nd ix A .

From t h e se c o o r d i n a t e r e l a t i o n s o n e ha se A -

r (d@/dy) = e@ * < : 33

a n d t h e r e f o r e ,

g< = ( r /Y33) (73)

S t a r t i n g Data

The method o f c h a r a c t e r i s t i c s , b e i n g a method f o r i n i t i a l v al u e

p ro bl em s, r e q u i r e s s t a r t i n g d a t a w hic h a re usual ly obta ined f rom boundary

c o n d i t i o n s o r from an i n i t i a l s o l u t i o n o b t a i n e d by o t h e r m et ho ds .

These are (1) t h e s p h e r i c a l l y b l u n t e d body w i t h t h e s o n i c l i n e l o c a t e d on t h e

sp he r i ca l nose , and (2) poin ted bodies which can be approximated by a cone i nsome small r e g io n n e a r t h e t i p .

For pres-

e n t a p p l i c a t i o n s two ty p es o f s t a r t i n g c o n d it i o n s a re o f p a r t i c u l a r i n t e r e s t .

S h e r e . - S i n c e a sphere does no t have a2-pr e f e r r e d o r i en ta t i on , t he f low remains symmet-

r i c a ro un d t h e w in d a x i s f o r any a n g le o f a t t a c k .The re fo r e , ax i symmet r ic b lun t-body s o l u t io ns

o b t a i n e d , f o r example , by t h e in ve rs e method of

r e f e r e n c e 20 c an b e use d t o p r o v id e i n i t i a l c on-

d i t i o n s . I t i s n e c e s sa r y , h ow ev er , t o g e n e r a t e

t h e s e ax is ym me tr ic s t a r t i n g d a t a on an i n i t i a l

v - < su r f ace which i s d e f i n e d i n a body axis

sys te m ; t h e d a t a w i l l not be symmetr ic wi th

r e s p e c t t o t h e body a x i s .

data lBlunt body

Bow shock

(a) Wind axes.

Sonic l lne

In i t i a l data

Shockurfoceurfacej

\

c

"lo

,I

\

\ /\

-4X

(b) Body a x e s .

F i g u r e 7 . - S t a r t i n g d a t a for a

s p h e r i c a l n o s e .

The c h a r a c t e r i s t i c s p ro gr am i s s e t up t o

g e n e r a t e body a x i s d a t a from w in d a x i s d a t a

ob ta ined from a b lunt -body so lu t i on . Giventh es e wind axi s d at a on one body normal ( se e

f i g . 7 ( a ) ) wh er e t h e f lo w i s s u p e r s o n i c , s a y

M > 1 .OS, t h e c h a r a c t e r i s t i c s c a l c u l a t i o n i s

performed i n a wind axis system and for a = 0

t o t h e p o s i t i o n X k' , l o c a t i n g t h e body n orm al s

vk ' . (Pr imes denote wind axes .) This i s i l l u s -

t r a t e d i n f i g u r e 7 ( a ) . S i n c e t h e fl ow i s a x i -symmetric i n terms of wind axes , th e normals

may be p la ce d a t an a r b i t r a r y c i r c u m f e r e n ti a l

p o s i t i o n . The v a l u e s o f q.' a re chosen s ot h a t t h e n o r m a l s v k ' match t he r i ng o f normals

'Ik emanat ing f rom the sphere a t ( x o , ro ) i nterms of body axes ( f i g . 7 (b ) ) . They a rer e l a t e d t o t h e m e r i di o n a l a n g l e @k and t h e

a n g l e o f a t t ack c1 by

xk ' = R, + (xo - Rn) cos c1 - ro s i n a cos @ k

where R, i s t h e r a d i u s o f t h e s p h e r i c a l n os e .

rik'

(74)

27



S c a l a r d a t a d e r i v e d i n t h i s way on t h e b ody n o rm al s tlk(k = 1, 2 , . . . , L) are r ea dy f o r u s e i n t h e g e n e r a l t hr e e- d im e ns i on a l ca l -cu l a t i o n . However, t he f l ow ang l e s , 0 an d 4 , must b e r e c a l c u l a t e d i n terms of

body axes b y means o f t h e t r a n s f o r m a t i o n r e l a t i o n s h i p s f o r v e l o c i t y c om po nen ts .

By t h e d e f i n i t i o n o f 0 and 4 ( f i g . 1 ) .

- 1 V

v cos (p

= s i n (75)

(76)1 w

(p = s i n -v

The magni tude of t h e ve l oc i t y i s u nc ha ng ed by t h e c o o r d i n a t e r o t a t i o n , s o

t h a t

v = V ' (77)

From ref er en ce 15 , t h e ve l oc i t y components

lows i n terms of wind ax is components:

v , w can be expressed as f o l -

v = ( u ' s i n a + v ' c o s a c o s @ ' ) c o s @ + v ' s i n 0' s i n @ (78)

w = v ' s i n @ ' os @ - ( u ' s i n a + v ' c o s a cos @ ' ) s i n 4, (79)

where 0 ' i s obta ined f rom

(x' -( 8 0 )o t @ s i n @ ' cos a cos a ' - ~-i n a = 0

r '

Equat ions (75) through (80) determine e an d (p, r e l a t i v e t o b od y- ax isc o o r d i n a t e s , fr om u ' a nd v ' c a l c u l a t e d i n t h e w i n d- a xi s s y st em .

Cone. - Conica l so lu t i o ns a r e d e f i n e d as th os e which a r e independept of

5 - t h a t i s , a l l d e r i v a t i v e s w i th r e s p e c t t o 5 van ish . Conica l f lows can

be o bta ine d i n two ways: (1) by so lv in g t h e bounda ry va lu e p rob lem fo r t h e

reduced equa t ions wi th 2/25 = 0 ( s e e, e . g . , r e f . 2 1), o r (2) by the asymp-

t o t i c s o l u t i o n of t h r ee - di m en s io n al i n i t i a l value problem wi th a conica l body

( s ee r e f . 11 o r 2 2 ) . The l a t t e r a pp ro ac h i s p r e s e n t l y t a ke n s i n c e i t f i t s

t he gene ra l computat ion scheme wi th l i t t l e change .

A pproximate i n i t i a l d a t a f o r a co ne a r e s p e c i f i e d an d t h e c a l c u l a t i o n

downstream i s c a r r i e d on u n t i l t h e c o n i c a l f lo w c o n d i t io n i s met t o a s p e c i -

f i e d a c ~ u r a c y . ~ h i s a pp ro ac h h as t h e d i sa d v an t ag e o f b e i n g g e n e r a l l y more

t ime-consuming tha n t h e boundary val ue method, b u t i t avoids many d i f f ic u l -

t i e s i n h e r e n t i n b ou nd ar y v a l u e p ro b le ms . The s i t u a t i o n i s q u i t e s i m i l a r t o

the ca l cu l a t i o n o f t h e supe rson i c b lun t -body prob lem by an a sympto ti c

_ - _ _ ._ ~ - _ _ _ - - - -

41t i s c l e a r p h y s i c a l l y t h a t t h e c o n i c a l c o n d i t i o n must b e o b t a i n e d

f a r downs tream f rom the approx ima te s t a r t i n g co ndi t i o n .

28



u ns te ad y c a l c u l a t i o n . A pp ro xim ate s t a r t i n g c o n d i t i o n s a r e o b t a i n e d e i t h e r

from t h e p e r t u r b a t i o n s o l u t i o n f o r c on es a t small a n g l e o f a t t a c k ( r e f . 23)

o r f rom a prev i ous t h ree -d i m ens i ona l so l u t i on f o r a d i f f e r e n t Mach number o rcone angle .

Two mod i f i c a t io ns t o th e gen era l computat ional p rocedure a r e requ i re dt o o b t a i n t h e c o r r e c t c o n ic a l f low s o l u t i o n s f o r c i r c u l a r c o ne s. F i r s t , it

was found neces sa ry t o accoun t f o r t h e F e r r i v o r t i c a l l a y e r ( r e f . 24) bys e t t i n g t h e e n t ro py a t a l l body poin t s , except a t t he l eew ard p l ane o f

symmetry, equ al t o th e ent ropy a t th e windward shock p o in t . The entro py a tt h e l eeward body s t a t io n , which i s a s i ng u l a r po i n t , m ust equa l t he en t ropya t t h e l eeward shock po in t . 5

cones :

Thus t h e fo l l ow i ng cond i t i on s app ly t o c i r c u l a r

and

s = s (k = 2 , 3 , . . . , L )l , k J , L

s = s1,1 J , 1

Secondly, as a r e s u l t o f t h e en tr op y s i n g u l a r i t y a t the leeward body point ,t h e c i r c u m f e r e n t i a l d e n s i t y d e r i v a t i v e th e r e i s a l s o s i n g u l a r . T h er e fo r e,d a t a from t h e s i n g u l a r p o i n t must b e e xc lu de d i n t h e n um er ic al d i f f e r e n t i a t i o n

f o r a p / a g .

Smoothing

Under c e r t a i n cond i t i on s where t he d ens i t y o r crossf low angle deve lop

l a r g e r a d i a l g r a d i e n t s , t h e n um er ic al c a l c u l a t i o n f o r t h e s e q u a n t i t i e s a pp ea rs

t o b e u n s t a b l e . T hi s s i t u a t i o n c an a r i s e i n t h e re g io n o f t h e v o r t i c a l l a y e r

on po i n t ed cones and i n t he so -ca l l e d en t ropy l a ye r ove r b l un t ed cones. A

s t a b i l i z i n g d i f f e r e n c e scheme s i m i l a r t o tha t known as t he Lax , o r Lax-Wendroff method ( r e f s . 26 and 27) i s t h e r e f o r e u se d i n t h e s e c a s e s .

Equat ions (59) and (60) a r e modi f ied fo r t h i s purpose accord ing t o the

fo l l ow i ng d i f f e re nce approx im a ti on:

where 4 o r p , r e s p e c t i v e l y .b r a c k e t i s p r o p o r t i o n a l t o t h e s eco nd d e r i v a t i v e o f

v = 3 o r 4 r e p r e s e n t s The second term i n th euv , so t h a t t h e

5 A dd it io n al s i n g u l a r p o i n t s a r i s e i n more g e n e r a l c o n i c a l f lo ws ( s e e ,

e . g . , r e f . 24 o r r e f . 2 5 ) .

29



d i f f e r e n c e e q u a t i o n (8 2) a p pr ox ima te s t h e d i f f e r e n t i a l e qu a t i o n6

where

and n J i s t h e v a l ue o f q a t the shock riJ = (J - 1 ) A q .

The add i t iona l term i n e q ua t io n (83) r e p r e s e n t s a d i f f u s io n p ro c e s s a nd

i s t h e s t a b i l i z i n g term i n th e type o f d i f fe r enc e scheme g iven by equa-

t i o n ( 82 ) . The a r b i t r a r y c o n s t a n t k i n e q u a t i o n ( 84) i s i n c l u d e d t o a ll ow

c o n t r o l o f t h e d i f f u s i o n term. I f k = 0 , equa t ion (83 ) reduces t o equa -

t i o n (59) o r equa t ion (60) . For k = ~ J / A T - I , equa t ion (83 ) i s e s s e n t i a l l y t h esame as the Lax scheme (ref . 2 6 ) .

I t i s known ( t h i s i s demonst ra ted below) t h a t t he Lax d i f fe re nc e scheme

prov ides too much d i f f us io n ; th e r e f o r e , i t i s d e s i r a b l e t o choose k between

0 an d q ~ / A r l i n o r d e r t o p r ov i de n um er ic al s t a b i l i t y w i t ho u t undue l o s s o f

accu racy . In most app l ic a t i ons k has been s e t e q ua l t o 1 , and th i s cond i -

t i o n i s p r e s e n t l y t e r m e d a second-o rder Lax smoothing because i n th i s case

-K = A n 2 = O ( A n 2 )

"15

Thus , t h e s e c o nd - or d er d i f f e r e n c e scheme ap pr oa ch es t h e d i f f e r e n t i a l e q u a ti o nl i k e A q 2 as the mesh i s re f i ne d , whereas t he Lax method approaches l i n ea r l y

i n A n . For la rg e Mach numbers , an order-of-magnitude an al ys is rev ea ls t h a t

- = 0 ( - )

-which gives K a v a l u e of about 1 /1000 for a t y p i c a l m e s h .

- - . _.-___

'This argument i s n ot s t r i c t l y r ig o ro u s, as p o i nt e d o u t i n r e f e r en c e 2 7 ,

i f t h e d i s s i p a t i v e t er m i s of t h e same or de r as t h e t r u n c a t i o n e r r o r o f t h e

o t he r t e rms . Never the le ss , th e analogy i s q u a l i t a t i v e l y c o r r e c t , e s p e c i a l l y

when t h e s t e p s i z e i s n o t smal l . A lso , when t h e i n t e g r a t i o n i s c a r r i e d o v er

l a r g e d i s t a n c e s t h e t r u n c a t i o n e r r o r m ig ht t e n d t o b e random, w h i l e t h e

d i s s i p a t i v e t e r m i s a lways add i t ive .

30



Curve xb/R,t h e i n i t i a l d a t a l i n e . T hi s problem

@ 0.4i5450 w a s t h e r e f o r e i n v e s t i g a t e d a nd i t was0 .505 found t h a t th e proposed method does@ ~ 4 5 p r o v i d e a n a c c u r a t e r e so l u t i o n o f known

flow d e t a i l s o b t a i n e d w i th a s t a n d a r d

0 .735 method o f c h a r a c t e r i s t i c s f o r ax isym -

m e t r i c f l o w ( r e f . 29) .

@ ,605

@ 6 7 0

D i sc o n t i n u o u s d e r i v a t i v e s a r i s ep r i m a r i l y on t h e body su r f a c e a t p o i n t s

w he re t h e s u r f a c e c u r v a t u r e i s n o t

30 a n a l y t i c . A s an example , f ig u re 9

shows a map o f t h e p r e s s u r e i n t h e

f o r a b l u n t e d c o n e a t 5" a n g l e o f

s u r e v a r i a t i o n a l on g a body normal, n ,between t h e body and th e shock wave.

The o r i g i n f o r ea ch l i n e i s d i s p l a c e d

i n p r o p o r t i o n t o t h e x c o o r di n a te o f

th e body po i n t s o t h a t t h e c i r c u l a r

sym bols r e p r e s e n t t h e a c t u a l s u r f a c e -

p r e s s u r e d i s t r i b u t i o n . The s u r f a c e

p r e s su r e d e c r e a s e s un i fo r m ly o n t h e

s p h e r i c a l n o s e u n t i l t h e s p he re -c on e

j u n c t u r e i s r eached , a t

w he re t h e p r e s su r e g r a d i e n t , a p / a s ,

c ha ng es a b r u p t l y . The d i s c o n t i n u i t y

f i e l d a lo ng a Mach wave.

mate p o s i t i o n o f t h e wave i s i n d i c a t e d

by the a r rows . )con s tan t beh ind th e wave and var i e sa c c o r di n g t o t h e b lu n t - no se f lo w a h ea d

mals, q ; 30" sphere-cone, M_ = 10, a = 5 " . o f t h e w av e. The c u rv e s t h e o r e t i c a l l y

A s s ee n i n t h e f i g u r e , t h e c u rv es

- 40Pa)

v i c i n i t y o f t h e s ph e re -c on e j u n c t u r e0 0 0 0 0 0 0 .2 .4 .6 .8 1.0

7)-7 s a t t ack . Each l i n e r e p r e s e n t s t h e p r e s -

0000 8 0Zeros for curve

(a) Leeward plane, 0 = 0 " .

xb = 0 .5 ,

8 O - r i i r i I I I 1 I

Shock point

7 0

P

Pa)-

40 -

I I ! I l I I 1 1 I fro m t h a t p o i n t moves o u t i n t o t h e f l ow

(The approxi-

The pressure i s n e a r l y

0 0 0 0 0 .2 .4 .6 .87)-7)s

00000 @ 0Zeros for curve 0

(b) Windward plane, Q = 180".

Figure 9 . - pressure distribution on surface nor-

should have a d i sc o n ti n uo u s s l o p e t h e r e ,

are o nl y s l i g h t l y r ou nd ed by t h e q u a d r a t i c i n t e r p o l a t i o n p r e s e n t l y e mployed.

RESULTS A N D DISCUSSION

Many numerical so lu t i on s have been obt a in ed wi th t h e de sc r ib ed method of

c h a r a c t e r i s t i c s , and t h e s e r e s u l t s compare f a v o r a b l y w i th o t h e r p u b li s he dworks (see, r e f . 1 3 ) . P re se n te d i n t h i s s e c t i o n a re t h e d e t a i l s o f a t y p i c a l

c a l c u l a t i o n f o r a 15" sphere-cone a t 10" ang le o f a t t ack and a Mach number

of 10.

t e s t o f t h e t h e o r y .

i l l u s t r a t e d a nd compared w i t h p r e d i c t i o n s o f p e r t u r b a t i o n m eth od s f o r small

ang les o f a t t ack .

po in ted cone so lu t ion , which i s d e s c r i b e d f i r s t .

This i s c on s id e re d t o b e s u f f i c i e n t l y n o n l i n e a r t o p ro v id e a good

The dominant three-dimensional fe a tu re s of th e flow are

B l u n t n e s s e f f e c t s are i l l u s t r a t e d b y c om pa ris on w i t h a

32



Po int ed Cone

The s o l u t i o n f o r a poin ted 15" cone a t a Mach number of 10.6 and 10"

a n g l e o f a t t a c k i s p r es e nt e d i n t a b l e 111.

0 = c o n s t a n t , a re the shock ang les , 0 and 6 . Below th e shock angle s a r e

tab u la ted th e x , r coo r d ina te s o f each mesh po i n t runn ing f rom t he shock t o

th e body and th e co r respond ing f low va r ia b l es a t t h o s e p o i n t s , as l a b e l e d .

The quan t i ty M* i s t h e component o f Mach number d ef in ed by t h e Mach a ng le

p* i n equat ion (53) . Remaining va r i ab le s are d e f i n e d i n t h e l i s t o f

L i s t e d f o r e a ch p l a n e ,

symbols.

T hi s c o n i c a l s o l u t i o n was o b t a i n e d w i t h a coor d ina te mesh co ns i s t i ng o f

9 mer id iona l p l anes and wi t h 11 po in ts on each p l an e . Second-order smooth-

i n g was u s e d o n t h e d e n s i t y p and the c ross f low ang le $ (smooth constant

k = 1 ) . I n i t i a l d a t a f o r t h e p r e s e n t case were ob ta ined f rom a prev ious

s o l u t i o n a t M = 7 w hich a g r ee d w i th t h e t a b u l a t e d r e s u l t s o f r e f e r e n c e 11.

The fr ee -s tr e am Mach number w a s changed from 7 t o 10 and th e computa t ion w a sc a r r i e d dow nstr eam u n t i l t h e f lo w r e l a x e d t o t h e new c o n i c a l s o lu t i o n . Th is

was accomplished i n a number o f s t a g e s , w i th e a ch s t a g e c o n s i s t i n g o f a com-

p u ta t i o n f r o m

i n i t i a l d a t a .

x = 0 . 8 t o 1 . 0 , u s i n g t h e o u t p u t of t h e pr e vi o us s t a g e as

Final Star l ing

I9t ,O 0" (leeward) i -/q I

I I 1

0 .2 .4 .6 .8 1.0

I /N

Ii

Figure 10.- Relaxation of conical solution a f t e r

N stages of calculation from x = 0.8 t o

x = 1 . 0 ; 15" cone, M_ = 10.6, a = 10".

The a c cu r ac y o f t h e s o l u t i o n i sin d i ca te d by f i gu re 10 which shows

th e s ho ck a n g l e i n t h r e e m e r idio n a l

p l a n e s as a f u n c t i o n o f t h e r e c i p r o c a l

o f t he number o f s t ag es . I t i s s e e n

t h a t t h e a pp ro xi ma te s t a r t i n g c on di -

t i on s cause th e shock ang les t o change

a b ru p tl y i n t h e f i r s t s t a g e w i t h aslow decay t o a l i m i t i n g v a l ue as 1/M

t e nd s t o zero. Condit ions on the Pee

s i d e a r e s l o w e st t o a p pr oa ch a l i m i t .The shock angle a t @ = 0 r e p e a t e d t o

an accuracy of

t h e l a s t s t a g e o f c a l c u l a t i o n and

r e p e a t e d t o a n a c c ur a cy o f l ~ l O - ~

d u r i n g t h e l a s t s t e p of t h e t e n t h

s t a g e .

Aa/o = 0 . 4 5 ~ 1 0 - ~n

The m ain f e a tu r e s o f t h i s c o n i c a l f lo w w i l l b e i l l u s t r a t e d i n c on ju nc ti on

w i t h t h e b l u n t e d c one r e s u l t s d e s c r i b e d n e x t .

Sp her ica l ly Blun ted Cone

The c a l c u l a t i o n f o r a s p h e r i c a l l y b l u n t e d cone w i t h a 15" semiver tex

a n g l e i n a i r (y = 1.4) a teach p lane .

o f - a t t a c k s o lu t i o n when

s o l u t i o n s p r e s en t e d .

M, = 10 used 7 m e r id io n a l p l a n e s w i th 1 5 p o i n t s i n

Second-order smoothing, k = 1 , w a s employed fo r t he 10" angle-

x/R, > 10; no smoothing w a s n e ce s sa r y f o r t h e o t h e r

33



'1I I I I I I

I I I I I I I I d

0 2 4 6 8 I O 12 14 16 18 20X

KFigure 14.- Axial distribution of surface cross-

flow angle; 15' spherc- cone, M_ = 10,

a = 100, 0 = g o o .

f a m i l i a r c r os s fl o w v e l o c i t y ac c o rd in gt o @ = s i n - l ( w / V ) . Figure 14 shows

t h e a x i a l d i s t r i b u t i o n of c r o ss f lo wa n g le a lon g t h e s i d e m e r idi a n @ = 90" .

The crossflow angle i s a minimum a t t h es ph er e- co ne j u n c t u r e a nd t h e n r i s e s t o

a maximum value about 2 . 4 t imes the

a n gl e o f a t t a c k . Thu s, t h e s u r f a c e

upwash f o r b lu n te d bod ies can be g r ea te r

th an t h e maximum va l ue of 2ci given by

s lender -body the o ry , wh ich ap p l ie s t o

po in t ed bod ies a t low supe rson i c speeds .

I t i s i n t e r e s t i n g a l s o t o compare t h e

r e s u l t of t h e l i n e a r iz e d c h a r a c t e r i s t i c s

30 -1

4 0Ltneorized(Ref. 13)

20

Th r e e dimensional

0 2 0 40 60 80 I O 0 120 140 160 I80

0- de g

Figurc 15.- Circumfcrent ial distri bution of

surface crossflow angle; 15" sphere-cone,

Mm = 10 , = loo.

method (ref . 15) which i s i n good agreement near t h e nose. Agreement extends

t o abou t X/Rn = 10, where t he l i n ea r method s t a r t s t o break down.

The reason f o r t he breakdown i s ev ide n t i n f ig u r e 15 which shows t he

c i r c u m f e re n t i a l v a r i a t i o n of @ f o r ci = 10" a t v a ri o us a x i a l s t a t i o n s . Neart h e no se t h e v a r i a t i o n i s n e a r l y s i n u s o i d a l , as assumed i n t h e l i n e a r i z e d

method. However, th e exa ct three-d imen sional ca lc u l a t io ns show a l a r g e

d e v i a t i o n f rom th e s i n u s o id a l v a r i a t i o n beyond x/Rn = 10.

Figu re 16 shows th e va r i a t io n o f c ross f low ang le no rmal t o t he body.

The curve f o r

shock, n/vs = 1 , b u t d e v i a t e s by a la rg e amount ne ar th e body. This i s duet o t h e low d e n s i t y o f t h e f lo w i n t h e e n t ro p y l a y e r which i s genera ted nea r

the body su r fac e by th e b l un t nose . The f low i n th i s r eg ion has l e s s momen-

tum tha t tha t away f rom the su r face , and i s t h e r e f o r e t u r n e d more by t h e c i r -

c u m f er e n t ia l p r e s s u r e g r a d i e n t . The s i t u a t i o n i s analogous t o boundary- layer

xb/Rn = 20 a pp ro ac he s t h e p o int e d- c on e s o lu t i o n n e a r t h e

35



IO 15 20

Figure 16.- Radial distribution of crossflowangle; 15 ' sphere-cone, bl_ = 10, a = l o " ,

0 = 9 0 " .

-Po0

Figure 18.- f ladinl distribution of d c n s i t y ;

xB/Iln = 16.7; 15 " sphere-conc, &lm = 10,

11 = i o " .

I.o

.a

7) .6-

.4

.2

7 s

0 .I .2

CP

.3 .4

Figure 17.- Radial distribution of static p r essu r e ;

xg/Rn = 16.7, 15" sphere-cone, M_ = 10, u = 1 0 " .

f low over an in c l in ed body.7 There fo re ,

b e ca us e o f t h e e n tr o py l a y e r , t h e b l u n t -cone crossf low does not approach the

p o in t e d- c on e d i s t r i b u t i o n u n i fo r m ly .

The en t ropy l ayer i s c h a r a c t e r i z e d

b y a n e a r l y c o n s t a n t p r e s s u r e , as i sshown i n f i g u re 17. On th e o t he r hand,

t h e d e n s i t y v a r i e s s t r o n g l y on t h e wind-

ward s id e o f th e body as may be se en i n

f i g u r e 1 8 f o r xb/Rn = 16.7 . The th ic k-

ness o f t h e en t ropy l a ye r may be t aken

as t h e d i s t a n c e t o t h e p o i n t where t h e

de ns it y has a l oc a l maximum. Thise n t r o p y l a y e r i s s i m i l a r t o t h a t fo und

i n ax i symmetr i c flow excep t t h a t i t

develops f a s t e r ( i . e . , c l o s e r t o t he

nose) on th e windward s i d e and more

slowly on t h e l e e s i d e o f t h e b od y.

Smoothing, which was used t o s t a b i l i z e

t h e c a l c u l a t i o n , c a us e s t h e pea k i n d e n s i t y t o b e r ou nd ed o f f . A t h e o r e t i c a l

maximum f o r = 180" c an b e c a l c u l a t e d b y u s i n g t h e e n tr o py c o r re sp on d in g t o

th e minimum shock an gle and by making use o f th e f a c t t h a t th e pre ss ur e i snea r ly con s ta n t . The th eo re t i ca l maximum i s a b ou t 1 5 p e r c e n t h i g h e r t h a n t h e

c a l c u l a t e d v a l u e . I t i s e s t i m a t e d t h a t t h e s mo ot hi ng h ad n e g l i g i b l e e f f e c t on

t h e d e n s i t y a nd c r o s s fl o w a n g l e f o r rl/qS g r e a t e r t h a n a bo ut 0 . 3 ( s e e f i g , 8 ) .

~ . -. . .. . .. I .~ . .

7Comparison i s made i n refe re nc e 1 4 between in v i sc id and exper imen ta l

( v is c o us ) s u r f a c e s t r e a m l i n e s .

36



Comparison With Experiment

De ta i led exper imen ta l su rveys o f th e f low f i e l d around b lu n te d cones

have been pre vio usl y re po r t ed by Cleary ( r ef s . 30 and 31) . Experimental

r e s u l t s f r o m r e f e r e n c e 30 a re u se d t o c ompare w i th t h e p r e s e n t t h e o r e t i c a l

r e s u l t s i n o r d e r t o c o nf ir m th e v a l i d i t y o f t h e pr op os ed n um er ic al m etho ds .More complete comparisons between theo ry and experim ent can be found i n

r e f e r e n c e s 31 and 1 4 , f o r b o th a i r a nd h e li u m f l ow s .

.61 I I I I I I I I r--I

0 1800xperiment (Ref. 30)

Vmrb /vm = 0.6 X IO6

(0 180") 0 I

L+.++2 4 6 8 I O 12 14 I6 18 2 0

" = 0 " )

x /R,

F i gu re 1 9 . - S u rf a ce p r e s s u r e d i s t r i b u t i o n ; 15"

s phere -cone , M _ = 10 , a = 1 0 " .

Fig u re 19 p r e s e n t s t h e s u r f a c e

p r e s s u r e d i s t r i b u t i o n a lo n g each o f t h e

7 merid iona l p lanes a long which t h eca lc ul at io ns were made. Theory and

exper iment a re i n exc e l le n t agreement

a l though the exper imen ta l d a t a t end t o

b e s l i g h t l y above t h e t h e o r y . Th is i sc o n s i s t e n t w i th t h e u s u a l b ou nd ar y-

l a y e r d i sp la ce me nt e f f e c t . A t

x/Rn = 20 t h e blu n t- c o ne p r e s s u r e i s

very nea r l y equa l t o th e poin ted-cone

va lue shown on the f a r r i g h t o ff i g u r e 1 9 .

Flow proper t ies o f f t h e body

s u r f a c e a r e most e a s i l y s t u d i e d

experimentally by means of the impact

o r p i t o t p re ss u re , as shown i n f i g -

u r e 20. The p i t o t - p r e s s u r e d i s t r i b u t i o n

normal t o t he body i s p r e s e n t e d f o r a l l7 mer id iona l p lanes and f o r an ax i a l

s t a t i o n of xb/Rn = 16 .7 . The theo ry

ex tends a l l th e way t o th e shock i n

each case , wh i le t he d a t a do no t , excep tf o r @ = 60" and @ = 150" where e xp er i-

m en ta l s hoc k p o s i t i o n s a r e i n d i c a t e d by

a s h a r p dr op i n p r e s s u r e .

I n t h e v i s c ou s bo un da ry l a y e r t h e p i t o t p r e s s u r e i s low, going t o zero a t

On the remain-th e body su r f ac e . Ev idence o f a v e ry t h i c k v i s co u s l a y e r i s i n d i c a t e d by t h e

e xp er im en ta l d a t a f o r t h e l e e s i d e of th e body, @ = 0 and 30".

in g mer idional p lan es t he boundary la ye r i s very th in . The en t ropy la ye r i sc l e a r l y i n d i c a t e d i n t h e e xp er im en ta l d a t a by t h e peak i n t h e p i t o t p r e s s u r e

j u s t o f f th e windward s i de of th e body . The theo ry underp r ed ic t s th e peak

va lue because o f th e p rev iou s ly desc r ibed smoo th ing employed i n t he ca l cu l a -

t i o n . A theoretical maximum of C = 9 . 0 3 f o r @ = 180" i s determined by t he

u se o f t h e t o t a l p r es s u r e a t th e ca lc u l a t ed minimum shock angle ( f i g . 13 ) .PP

37



3.2 I I

2.4

2.0 t

I I I I I I I

0

0 P 6 0 0 Experiment (Ref. 30)V c O r b / ~ m = 0 , 6 x 1 0 6

.8

.4

0 0 0 0 0 0 I 2 3 4 5 6 7 8

PPZeros for 0 = C

0" 30" 60° 90" 120° 150" 180'

Figure 20.- Shock-layer pitot-pressure distribution; 15" sphere-cone, M _ = 10,

a = lo", xB/R, = 16.7.

Aside f rom th e no ted d i f f e re nce s due t o t he boundary l a y e r a n d t h e e n t r o p y

la ye r , th e proposed numerica l method appears t o g i v e a n a d eq ua te p r e d i c t i o n

o f s ho ck - l a y e r p r o p e r t i e s .

C O N C L U D I N G REMARKS

C h a r a c t e r i s t i c s t h e o r y f o r t h r ee - d im e n s io n a l s t e a d y f lo w was reviewed and

th e c o m p a t ib i l i t y e qu a t i on s w er e d e r iv e d i n t e rm s o f p r e s s u r e and s t re a m

ang les as dependen t va r iab le s . I t was a rg ue d t h a t t h e m aj or p r a c t i c a l d i f -

f i c u l t y e n co un te re d i n b i c h a r a c t e r i s t i c methods r e s u l t s from t h e n ee d f o r

n um er ic al d i f f e r e n t i a t i o n and i n t e r p o l a t i o n o f randomly sp a ce d d a t a . Fu r th e r -

more, i t was o b se r ve d t h a t s h o c k - la y e r c o o r d in a t e s a re e s s e n t i a l t o a s imple

t r e a t m e n t o f c i r c u m f e r e n t i a l d e r i v a t i v e s on t h e s ho ck s u r f a c e , A r e f e r e n c e

plane method was t h e r e f o r e a do pt ed i n w hich an equa l number o f po in t s a re

equ al l y spaced between th e body and shock a long each re fe re nc e p la ne . This

mesh in t rod uce s th e added complica t ion of nonor thogonal coo rdi nat es i n t o the

equa t ions bu t a l lows the use o f s imp l e r numer ica l t echn i ques .

With t h e c o n s t r a in t o f a u ni fo rm ly sp a ce d mesh, p a r t i c u l a r c h a r a c t e r i s t i c

l i n e s a r e no t f o l l ow e d as i n more s t a n d a r d c h a r a c t e r i s t i c s m etho ds . The

r o l e o f c h a r a c t e r i s t i c s t h eo r y i n t h e r e f e r e n c e p la n e method i s t o de te rmine

how t h e f i n i t e d i f f e r e n c e e q ua t io ns a r e t o b e l o c a l l y s o l v ed . S i nc e t h e

c h a r a c t e r i s t i c s a r e no t t r a c e d t hr ou gh ou t t h e sh ock l a y e r , t h e p o s s i b l e

coal esce nce of waves t o form embedded shocks i s no t de te rmined du r ing the

38



APPENDIX A

D I RE CT I O N COSINES FOR NONORTHOGONAL COORDINATES

I n t h e d e r i v a t i o n o f t h e c o m p a t i b i li t y e q ua t i on s f o r t h e r e f e r e n c e p l a nemethod it was n e c e s sa r y t o t r a n s fo r m f ro m s t r e a m l i n e c o o r d i n a t e s t o a non-

or thogona l sys t em co ns i s t i ng of s* , TI, and 5. Th e d i r e c t i o n s* i s t h e

p r o j e c t i o n o f t h e s t r e a m l i n e on t h e m e r i di o na l p l a n e , n runs from body t o

shock i n t h e m e ri d i o na l p l a n e , a nd 5 i s t h e ou t -o f -p l an e d i r e c t i o n ( s e e

f i g . 3 ) . Th i s t r a n s f o r m a t i o n i s expressed as

A ,.

x = & * Z * (Alli i j j

where

and

,. A , . , .

x = ( s , n , t )i

I t i s t h e p ur po se o f t h i s a pp en di x t o w r i t e o u t t h e e x p r e s s i o n s f o r t h e

d i r e c t i o n c o n s i n e s E: . The ana lys i s i s made f o r t h e more ge ne ral E , n , <

c o o r d i n a t e s a nd i s l a t k r s p e c i a l i ze d t o t h e case where 5 = s * .

The d i r ec t i on s o f th e nonorthogona l S,q,c c o o r d i n a t e s a r e d et er mi ne d

s t e p by s t e p d u r in g t h e c a l c u l a t i o n as t he shock shape i s c o n s t r u c t e d ( s e e

se ct io n on Computational Procedure) , When th e lo ca t i on of th e shock i s known

i n te rm s o f x , r , @ , t h e l o c a t i o n s o f f i e l d p o i n t s be tw ee n t h e sh oc k a nd body

a r e a l s o d e te rm in e d b y t h e p r e sc r i b e d sp a c i n g o f t h e p o i n t s a l o n g t h e bodynormals . The d i r e c t io n cos ines can then be ob ta in ed by numer ica l d i f f e r e n t i -

a t i o n as descr ibe d below. This process i s des cr ibe d f o r th e shock wave bu t

a p p l i e s g e n e r a l l y t o e ach 5 curve .

Let th e shock su r f ac e be g iven by

The i n t e r s e c t i o n o f t h e c o o r d i n a t e s u r f a c e 5 = con s tan t wi th th e shock

s u r f a c e d e f i n e s a c ur ve d l i n e i n s p a ce (a 5 c ur ve ; s e e f i g . 3 ) . The x ,rcoord ina tes of t h i s space cu rve depend on th e mer id iona l a ng le , @ , and on

the body s t a t io n , Xb, which lo ca tes th e 5 = cons tan t su r f ac e . For each body

s t a t i o n t h i s d ep en den ce i s denoted by

40



Equati ons (A3a) and (A%) d e f i n e two ang les

1 %t a n Ax = --

r d@S

1 drs

t a n A r = --r d@S

which can be eva lua ted numerica l ly .

d e s c r ib e d i s ? s ed . ) I n f i g u r e 2 1 t h e

u n i t v e c to r 5 i s r e l a t ed t o e

th r o u g h tw o r o t a t i o n s , t h e f i r s t r o t a -

t i o n by Ax and t h e second by 6 .I t may a l s o b e v e r i f i e d from t h i s f i g -

6 er

7 (The F ou r i er method pr evi ous ly

@-

E^, u r e t h a te,

Shock surface

r = r s ( x . Q)

-t a n 6 =-r cos Ax

r d@

and, by equation (A4),

F i g u r c 2 1 . - C oorJ i na t c v ec to r s a n d a n g l c s . t a n g = t a n A r cos Ax (A6)

,.The remaining, .two unit , .vec tors a r e def in ed by angles

by X from e and 5 i s r o t a t e d by cr from ex . Thus, one may w r i t e

X and 0 ; rl i s r o t a t e d

r

,. ,. ,.

5 = cos cr e + s i n cr e (A71X r

,. h

r) = - s i n x e + cos AX r

A

The angle

s p e c i f i e d . I t i s u s u a l l y s e l e c t e d s o t h a t

The angles 0 and 6

shaee, and a t th e body by t h e gi ven body sh ape.

between t h e body and shock, 0 and gx , r ,Q coord ina te s o f th e mesh po in t s . These coord ina te s a r e known f o r any

sp ec i f ie d mesh sp ac ing once th e shock po s i t i on i s determined.

X , which de te rmines the d i r ec t i on of q , may b e a r b i t r a r i l y,.

i s normal t o th e body su r f ac e .

are de termined a t th e shock by t he ca lc u l a t ed shockA t i n t e r m e d i a t e p o i n t s

can be s i m i l a r l y c a l c u l a t e d f rom t h e

I t must be no ted , however , th a t6 i s

d i s t i n c t f r o m t h e a n g l e 6 whichi s used i n th e shock con d i t i ons deve loped i n append ix E .

i s found tha t

From figure 2 1 it

o r

d r - dx tan 0

r d@a n 6 = - - -

t a n 6 = t a n A r - t a n 0 t a n Ax 1-

When 6 i s chosen t o be normal t o t he body ax is , Ax = 0 and 6 = 6 .

4 1



n . . n

H av ing s p e c i f i e d t h e d i r e c t i o n s o f 5 , n ,C by equat ions (A7) , (A8) , and

(A9), th e t a sk of de te rmin ing " f j , a p p e ar in g i n e q u a t i o n ( Al ) , ca n now be

completed. Equat ions (A7) , (A8) , and (A9) a re more conven ien t ly wr i t t e n wi th

i n d e x n o t a t i o n as

n - A

z = v xi

i j jwhere

cos (5 s i n (5 0

- s i n X cos A 0" i j

- = ( cos i s i n A X s i n c o s 6 co s Ax

The in ve r se t r a ns fo rma t ion may be wr i t t e n

h n

i j ' j. = v1

where

-1v i j = (Vij)

A ltho ug h t h e m a t r i x i n v e r s io n c an b e p er fo rm e d n u m e r i c a l l y , f o r t h i s p ro bl em

it i s more e f f i c i e n t t o do t h e a lg e b r a b e f o re h a n d, oncehand fo r a l l . This i sdone by taking s c a l a r produc ts -f e q u a t i o n ( A l l ) w i th Xj t o o b t a in t er ms

l i k e z 1 - X1 = v11, 2 1 * R2 = v12, an d so on . On th e o the r hand , th e s ca la r

products of equation (A13) with 2 g iv e e x p r e s s io n s s u c h asj

h n n n n

. x 1 = v11 + v12(5 17) -I-v13(5 * 5).

i2 - i 1 = v& * i) + VI2 + V l 3 ( f i - aand s o on.

v i j .w r i t t e n as ( s e e, e . g . , r e f . 18)

I n t h i s way, o ne o b t a in s n in e e q u a t i o n s f o r t he n in e unknowns

The s o lu t i o n o f t h i s s e t o f e q u a t i o ns b y C ra m er 's r u l e c an b e f o r m a l ly

-v f

D

Z i Z jv =

i j



where

f = fZ j j Z

and the de terminant D i s given by

A A

D = 1 - (6 * i ) 2 - (i - ;)2 - (i ;)’ - 2 ( i * i ) ( i C ) ( n * t ) (A16)

Equation (A14) provides th e d i re c t io n cos ine s be tween S , n ,c and

x , r ,@ c o o r d in a t e s . The c o rr e sp o nd ing r e l a t i o n s h i p s w i th s , n , t c o o r d in a t e s

are ob ta ined by use o f equa t i9ns 5 7 ) , A ( 8 ) ’ and (9) which exp ress t he s t ream-l i n e d i re c t io n s i n terms of e e e and may b e w r i t t e n

x’ r J CP

where

COS + C O S e cos + s i n 0 s i n +- s i n 0 COS e

- s i n + cos 0 - s i n + s i n 0 co s 0

The su bs t i t u t io n of equat ion (A13) i n t o (A17) y i e l ds

A A

x -i - ‘ik’k

where E . i s ob ta ined f roml k

Eik = T i j V j k

Equation (A20) g ive s th e d ir ec t i on cos i nes between € , , n , ~ and s , n , t c o o r d i -

n a t e s . The e x p l i c i t e x pr e ss i on s f o r t h e s e d i r e c t i o n c o s i n e s a r e i n vo l ve d , s o

th e f i n a l combina tion o f t e rms i nd i ca t ed by equa t ion -A20) i s l e f t f o r t he

computer. The proc edur e i s as fo l l ows . The ma t r ix v i j , de fi ne d by (A121, i s

ca lc ul at ed with equa tion s (A4), (AS), and (A6). The va ri ou s s c a l a r produc ts

i n equation s (A15) and (A16) a r e ca lc ul at ed from

i s then ca lc u l a t ed f rom equa t ion (A14) . F i na l l y , E ik i s determined fromequations (A18) and (A20).

V i j , and t h e m a t r i x

v i j

The p rocedure desc r ibed app l ie s t o ge ne ra l nonor thogonal coo rd ina te s

S , r l , < .5 d i r e c t i o n i s t h e s t r e a m l i ne p r o j e c t i o n s * . I n t h i s case, equation (A9) i s

rep laced by

43

The t rans f o rma t ion used i n equa t ion (36) i s a s p e c i a l c a s e w he re t h e



i * = cos e + s i n 0 GX r

Then th e subsequent r e l a t i on s w i l l b e m o d i f ie d a c c o r d i n g l y a nd t h e r e i s

ob ta ined

E" = T *i k i j ' jk

which d e te rm i ne s t h e d e s i r e d t r a n s f o r m a t i o n r e l a t i o n s h i p i n d i c a t e d by

equa t ion ( A l ) .

44



APPENDIX B

SHOCK-BOUNDARY CONDITIONS FOR THREE-DIMENSIONAL FLOW

Con2 ider an e lemen ta l po r t i on o f th e shock su r fac e wi th an ou t e r un i t

normal N , as shown in f i gu re 2 2 , and wizh un i t vec to rs ? and completingan o r thogona l s e t . The tangen t ve c t o r T can be chosen such tha t the N-T

p lane i s p a r a l l e l t o t h e d i r e c t i o n of t h e

f r e e - s t r e a m v e l o c i t y . Th i s c h oi c e w i l l

permit eva lu a t ion o f t he jump cond i t ions

./- i n t he N-T plane with two-dimensional

s ho ck r e l a t i o n s . Let zm be a u n i t v e c -

cl,; ” ‘ A X t o r p a r a l l e l t o t h e f r e e- s tr e am v e l o c i ty

vector and with components

yPyf.- k~ -,f”

c.-

_ d l

hh haQf”~*

F i gu re 2 2 . - Shock-wave normal and tangent

sw = cos c1 ex + s i n ci cos @ e,

(B1)@

e c t o r s . - s i n ci s i n Q

i n a c y l i n d r i c a l c o o r d in a t e f ra me . The d e s i r e d t a n ge n t v e c to r ?? can be

cons t ruc ted f rom G m and fi by means of two vec to r - o r c ross -p ro duc ts . The

f i r s t product

h h

aL = x N (B2)

p ro du ce s a v e c t o r p a r a l l e l t o f., and the second product

h h

r e s u l t s i n a vector which i s normal t o both N and L , and which l i e s i n t h e

s w - N p lan e . Tbe fa c t o r a i n equa t ion (B2) i s e qu al t o t h e s i n e of t h e

angle between sw and fi and may b e e v a lu a t e d from th e s c a l a r p r o du c t

h h

b = sw * N = COS (sm,N)

t o g iv e

a = ,,/l - b 2 = / l - ( g w - $2

Combining equations ( B 2 ) and (B3) and expand ing the r e s u l t i ng ve c t o r t r i p l e

product one obta ins

o r

,. 1 ^T = - [Nx(^S, x f i ) ]

a

- 1T = - ( g - bN)

a m



In terms of components T,, Tr , and TQ, , equation (B6) becomes

Equation (B7) p e rm i ts e v a l u a t i o n of t h e t r u e i n c l i n a t i o n o f t h e s ho ck s u r f a c e ,

and th e r e f o re a l lows th e jump c ond i t ions t o be de te rmined wi th s t a ndard p lana r

sh o c k r e l a t i o n s ( see , e . g . , r e f . 3 2 ) . The angle cr ' between and ? i sgiven by

h h

C O S 0' = sa, - T

o r

+ s Tos 0 ' = s x T x + s r T rC P Q ,

In the fol lowing development 0' i s expressed i n terms o f two angles

m easured i n t h e c y l i n d r i c a l c o o r d i n a t e s .

Let 0 be t he angle between g x and t h e t r a c e o f t h e sho ck su r f a c e on

t h e p l a n e @ = c o n s t a n t , a n d l e t 6 be th e angle between eQ, and th e shock*

t r a c e o n t h e p l a n e x = c o n st a n t a s i l l u s -t r a t ed in f i gu re 23 . The shock ang le 6 i so b t ai n e d by nu me ri ca l d i f f e r e n t i a t i o n asd e sc r i b e d i n a pp en di x A ( s e e e q . (A10) and

f i g . 2 1) . I t i s e a s i l y v e r i f i e d t h a t t h e f o l -

lowing r e l a t i on s ho ld be tween th e ang les"'N .

e?.

N = N cos 0

r m

N = - N cos 0 t a n 6 (B11)Q, m

..

x = Constantwhere N m i s t h e p r o j e c t i o n o f N on the

x- r p lane and s a t i s f i e s t h e r e l a t i o n= Constant

F i g u r e2 3 . -

Shock a n g l e s .N m 2 + N = 1

Q,

S u b s t i t u t i o n o f N @ f rom equat ion (Bl l ) g ives

46



and the shock normal v ec to r may f i n a l ly be wr i t t en i n t e rms o f CJ and 6 as

*

( - s i n o e + cos a G r - cos o t a n 6 e a )A_ _ XN =

41 + cos2 CJ tan' 6

The t r u e shock ang le can now be e va lua ted i n terms of IS and 6 by equa t ions(Bl ) , (B7) , (BS), and (B13), and th e jump con di t ion s can be ca lc ul a t ed .

I t i s now neces sary t o determine th e f low angles 8 and (p measuredr e l a t i v e t o t h e m e ri di on a l p l a n e s .

The s t r e a m l i n e d i r e c t i o n , m ea su re d i n

t h e N-T p l a n e ( f i g . 2 4 ) , i s t u rned f rom the

s t r e a m l i n e t a n g e nt G2 l i e s , by d e f i n i t i o n , i n

t h e N-T pl an e, one may w r i t e

I f r e e stream by the ang le 82'. S i n c e t h e

,. * *

s 2 = - s i n ( a ' - 0,')N + c o s ( a l

-0,')T

Figure 24.- Shock and flow angles in the

n-t plane.

Using equ atio ns (B7) and (B13), th e ve ct or s

o f t h e i r c om ponents t o g i v e

and ? c an be w r i t t e n i n t er ms

: 2 = (ANx + BTX)gx + (AN, + BTr)er + (ma + BTa)ga (B15)

where

A = - s i n ( a ' - 8'1)

B = cos(^' - e,!)

Equat ion (7 ) , on th e o th er hand , g ives s 2 i n t erms o f 8 and 4 asfo l lows :

G 2 = cos $ 2 cos 0 2 ex + cos 4 2 s i n 0 2 G r + s i n $ 2 2 @16)

Thus, by equa ting components of eq uat ion s (B15) and (B16) one ob ta in s f i n a l l y

- s i n ( o ' - e 2 ' ) N r + c o s ( o ' - e2 ' )Tr

- s i n ( o ' - e2 ' )Nx + c o s ( a ' - e21)Txa n 8 = (E3171

and

I f f o r g i ve n free-stream co nd i t i on s , p,, p,, V,, t h e s t a n d a r d p l a n a r s ho ck

c o n d i t i o n s are w r i t t e n i n t h e f orm ( se e r e f . 32) :

47



t h e n t h e e q u a t i o n s d ev e lo p ed i n t h i s a pp en di x a l l o w t h e t h r ee - d im e n s io n a l

s ho ck c o n di t i on s t o b e f u n c t i o n a l l y w r i t t e n as

7= P(a ;6 ,@,4

This i s t he fo rm of th e shock condi t ions employed i n equa t ion (62) .

The overa l l shock ca lcu la t ion i s performed i n a s t r ai g h tf o rw a r d i t e r a t i v e

manner . This proc edur e i s s t a r t e d w i t h known f i e l d d a t a , i n c l u d i n g 0 and 6 ,

on an i n i t i a l d a t a s u r f a c e . A shock poi nt on the new dat a sur fa ce i sdetermined by t h e average val ue of 0 between th e i n i t i a l and new shock

p o i n t s ,

-0 = 1/2(01 + 0 2 )

To beg in , 0 2 i s s e t e q u a l t o 01 and th en subsequen t es t im ates a r e made

(e i t h e r by t h e Newton method o r by th e b i s ec t i ng method) u n t i l t h e p r es s ur e

f rom equat ion ( B 2 0 ) a g r e es w i t h t h e p r e s s u r e c a l c u l a t e d f rom t h e c o m p a t i b i l i t yequa t ion ( 5 7 ) . D uring e ac h i t e r a t i o n t h e c u r r e n t v a l u e o f CJ = 0 2 and the

o l d v a l u e o f 6 = 61 a r e use d t o d e te r mi n e t h e t r u e sh oc k a n g l e C J ' by

equa t ion ( B 8 ) . The p r e s su r e a nd o t h e r v a r i a b l e s a re t hen ca lcu la t ed f rom

equa t ions (B19) . When th i s has been done fo r a l l t he shock po i n t s on the

new dat a sur f ac e , new valu es f o r S can be dete rmined by numer ical d i f f er en -

t i a t i o n wi th r e s p e c t t o 0 as descr ibed i n append ix A . The e n t i r e p r o c e s s

can t he n b e re p ea t ed t o r e f i n e t h e r e s u l t s ; t h i s i s d i s cu s s ed i n t h e s e c t i o n

on G lo ba l I t e r a t i o n .

4 8



AP P ENDI X C

SURFACE B O U N D A R Y CONDITIONS FOR BODIES WITHOUT A X I A L SYMMETRY

For nonc i r cu la r bod ies t h e su r f ac e boundary i s compl ica ted by th e f a c t

t h a t t h e s u r f a c e n ormal i s n o t i n t h e m e r i d i o n a l p l a n e . Th i s means t h e

boundary condi t ion w i l l i nvo lve bo th f low ang les 8 an d 4 . ( I n terms o f

v e l o c i t i e s , a l l th re e components, u , v , w , e n t e r i n t o t h e c o n d i t i o k s i n c e

none of t h e components a r e p a r a l l e l t o t h e body s u r f a c e . ) I n t h i s a ppe nd ixthe boundary c ond i t ion f o r 8 , which was g i v e n p r e v i o u s l y i n e q u a t i o n ( 6 1 a ),

i s d e r i v e d from t h e t an g e nc y c o n d i t i o n on t h e v e l o c i t y v e c t o r .

Let t h e e q u a t i o n o f t h e bod y b e g i v e n by

g ( x , r , @ ) = r - f ( x , @ )

The un i t o u te r normal t o th e s u r f a ce may be expressed

where V i s t h e v e c t o r g r a d i e n t o p e r a t o r . I n terms o f c y l i n d r i c a l

c o o r d i n a t e s , o n e o b t a i n s

h

N =I

I (@ constant)

( x cons t an t )

Figure 25.- Surface inclination angles.

The der iva t ives o f f i n equa t io n (C3)a r e r e l a t e d t o t h e s u r f a c e i n c l i n a t i o n

From equation (7) t h e s t r e a ml i n e d i r e c t i o n i s e x p r es s e d i n terms o ff low angles 0 and cp

< = cos cp cos e 2. + cos cp s i n e 2X r

+ s i n cp

49



The tangency condi t ion ,

o f equa t ions (C3) and ( C6 ) . This g ives

- f i = 0 , may now be obt ai ne d from th e s c a l a r prod uct

N cos + cos 0 + N . cos + s i n 8 + N s i n + = 0 (C7)X r cp

A h *where N x , N r , N Q a re the components of along e,, e,, e @ , and are

i d e n t i f i e d by e q u a ti o n (C3 ) . Equation (C7) may be writ ten as a q u a d r a t i cr e l a t i on i n t a n 8

(Nr2 - N o 2 t a n 2 + ) t a n 2 9 + 2 N x N r t a n 0 + (Nx2 - N c p 2 t a n 2 +) = 0 (C8)

The so lu t ion i s

Rewri t ing equat ion (C9) i n t er ms o f t h e s u r f a c e i n c l i n a t i o n a n gl e s g iv en

by equations (C4) and (CS), one ob ta ins

( 1 - t a n 2 & Q t an2 +)

w her e t h e p o s i t i v e r o o t i s chosen s o t h a t 0 i s decreased when $ < 0 and

6@ ’ .

Equations (ClO), (C4), and (C5) de te rmine the f low ang le 0 i n t er ms o f

the c ross f low ang le + and t h e body geometry. I t i s e a s i l y v e r i f i e d t h a t

f o r z e r o c r o s s f l o w , (9 = 0 , equa t ion (C10 ) reduces t o

t a n 0 = t a n 6x

which i s t h e u su a l c o n d i t i on f o r c i r c u l a r b o d ie s .

50



REFERENCES

1 . Courant, R . ; a n d F r i e d r i c h s , K . 0 . : Supersonic Flow and Shock Waves.

I n t e r s c i e n c e , N e w York, 1948.

2 . Courant , R . ; a n d H i l b e r t , D . : Methods of Mathematical Ph ys ic s. Vol.

I1 P a r t i a l D i f f e r e n t i a l E q ua ti on s . I n t e r s c i e n c e , N e w York, 1962.

3. F e r r i , Anton io : Ch ar ac te r i s t i c s Methods f o r P roblems i n Three

Independent Va ri ab le s. Ch. 5, s e c . G of General Theory of High Speed

Aerodynamics, W . R . S e a r s , e d . , P r i n c e t o n U n i v e r s i t y Pres s , 1954,

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Strom, Charles R . : The Method of Ch ar ac te r i s t ic s f o r Three-DimensionalReal-Gas Flows. F in a l Report 1 , Apr 1963-Nov. 1966. A i r Force Sys-

tems Command, AFFDL-TR-67-47, J u l y 19 67 .

Sauerwein , Harry: Numerical Calc ula t io n of Mul t id imensional and

Unsteady Flows by th e Method of Ch ar a c te r i s t i c s . J . Computat ional

Phys . , vo l . 1, Feb. 1967, pp. 406-432.

B u t l e r , D . S . : The Numerical Solution of Hyperbolic Systems o f P a r t i a l

D i f f e r e n t i a l Eq u at i on s i n Th re e -I n de p e nd e n t V a r i a b l e s . P r o c . Roy.

SOC. A 255, no. 1281, Ap ri l 1960, pp. 232-252.

Katskova, 0 . N . ; and Chushkin, P . I . : Three Dimensional Supersonic

Equ il ib r ium Flow of a Gas Around Bodies a t An gl e' of At ta ck . NASA

TT F-9790, 1965 .

10. Ransom, V . H . ; Thompson, H . D . ; and Hoffman, J . D . : Analys i s o f Three-

Dimensional Scramjet Exhaust Nozzle Flow Fields by a New Se co nd -O rd er

M e t h o d o f C h a r a c t e r i s t i c s . A I A A Paper 69-5 , p res en te d a t t h e S e v e n t h

Aerospace Sc ien ce s Meeting , Jan. 20-22 , 1969.

11 . Babenko, K . I . ; Voskresenskiy , G . P . ; Lyubimov, A . N . ; and Rusanov,

V . V . : Three-Dimensional Flow of I d e a l Gas P a s t Smooth Bodies . NASATT F-380, 1966 .

12. Chushkin, P . I . : Numerical Method o f C h a r a c t e r i s t i c s f o r Th re e-

Dimensional Supers onic F lows. Prog. i n Aeron. Sc i . , vo l . 9 ,

K . Kcchemann, ed., Pergamon Pr e s s , 1968.

51



13.

14 .

15.

16 .

17.

18 .

19 .

20.

2 1 .

22.

2 3 .

24.

25.

26.

Rakich, John V . : Three-Dimensional Flow C al cu la t i on by t h e Method of

C h a r a c t e r i s t i c s . A I A A J . , v o l . 5, no . 10, Oct. 1967, pp. 1906-1908.

Rakich, John V . ; and Cleary , Joseph W . : Theore t i ca l and Expe r imenta l

Study of Super sonic Steady Flow Around In c l in ed Bodies o f Revolu t i on .

Seven th Aerospace Scie nce s Meet ing, Ja n. 20-22, 1969, Paper 69-187.

Rakich, John V . : Numer ical Ca l cu l a t i on o f Supe rs on i c Flows o f a Perfec t

Gas Over Bodies of Revolut ion a t Small Angles of Y a w . NASA TN D-2390,

1964.

Chu, Chong-Wei: Co mp ati bi l i ty Re la t i on s and a G e n e r al i ze d F i n i t e

Dif fe rence Approximation f o r Three-Dimensional Stea dy Super son ic Flow.

A I A A J . , v o l . 5, no. 3 , March 1967, pp. 493-501.

Fox, L . : N um er ic al S o l u t i o n o f O r d i n ar y a nd P a r t i a l D i f f e r e n t i a l

Equat io ns . Ch. 18 of Fin i te - Di ffe ren ce Methods f o r Hyperbol ic

Equ at io ns. Pergamon Pr ess , 1962.

Hildebrand, F . B . : In t ro du c t i on t o Numerica l An alys i s . McGraw-Hill

Book Co., I n c ., 1956.

Hamming, Richard W . : Numer ical Methods f o r S c i en t i s t s and Enginee rs .

McGraw-Hi 11 Book Co. , Inc . , 1962.

Lomax, Ilarv ard ; and Ino uye , Mamoru: Numerical An al ys is of Flow Proper-

t i e s About Blunt Bodies Moving a t Su per son ic Speeds i n an Equi l ibr ium

Gas. NASA TR R-204, 1964.

Briggs, Benjamin R . : The Numerical C al cu la t io n of Flow Past Conica l

Bodies Su ppor t i ng El l i p t i c Conica l Shock Waves a t F in i t e Angles o f

In ci d en ce . NASA TN D - 3 4 0 , 1960.

Ylore t t i , G . : I n v i s c i d F l o w F i e l d P a s t a Po in ted Cone a t an Angle of

At t ack . A I A A J . , vol . 5 , no . 4 , Apri l 1967, pp. 789-791.

Rakich, John V . : C al cu la ti o n of Hypersonic Flow Over Bodies of Revolu-

t i o n a t Small Angles of A t t a c k . A I A A J . , v o l . 3 , n o . 3, March 1965,

p p . 458-464.

F e r r i , Anto nio: Co ni cal Flow. Ch. 3 , s e c . H of General Theory o f High

Speed Aerodynamics, W . R . S e a r s , e d . , P r i n c e t o n U n i v e r s i t y P r e s s ,1954.

Gonidou, Ren6: Su pe rs on ic F l o w s Around Cones a t Inc iden ce. NASA TTF-11,473, 1967.

Lax, P . ; and Wendroff, B . : Systems of Conservat ive Laws. Commun. Pure

Appl . Math. , vol . 13, May 1960, p p . 217-237.

5 2



27. Richtmyer, Robert D . : A Survey of Di ffe re nc e Methods f o r Non-Steady

Flu id Dynamics. Nat ion al Center f o r Atmospheric Research Technical

Note 63-2, Bo uld er, Colorado, Aug. 1962.

28. Gal lo , W i l l i a m F . ; and Rakich, John V . : I n v e s t i g a t i o n o f M ethods f o r

Pr ed ic t i ng Flow i n th e Shock Layer Over Bodies a t Smal l Angles o f

A t t a c k .NASA TN D-3946, 1967 .

29. Ino uye , Mamoru; Rak ich , Joh n V . ; and Lomax, Harvard: A D e s c r i p t i o n o f

Numerical Methods and Computer Programs f o r Two-Dimensional and Axisym-

m et r i c Super son ic F l o w Over Blun t-Nosed and F l a r e d Bod ie s. NASA TN

D-2970, 1965.

30. Cleary , Josep h W . : An Exper imen ta l and Th eo re t i ca l Inv es t ig a t i on o f t h e

Pressu re D is t r ib u t io n and Flow F ie lds o f Blun ted Cones a t Hypersonic

Mach N h b e r s . NASA TN D-2969, 1965.

31. Cleary , Joseph W . : E f f e c t s of Angle of At tack and Bluntness on the

Shock-Layer Pr op er t i es of a 15' Cone a t a Mach Number o f 10 .6 . NASA

TN D-4909, 1968.

3 2 . A m e s Research S t a f f : Equa t ions , Tables , and C har t s f o r Compress ib le

Flow. NACA Rep. 1135, 1953.

53



TABLE I . - EFFECT OF GLOBAL ITERATION O N SHOCK ANGLE

[15" sphere-cone; M = 10 , a = l o o , 1 5 p o i n t s a nd 7 p l a n e s ]

( a ) C a l c u l a t i o n from x/Rn = 2.0 t o X /Rn = 3.0

.- . .

r I

I t e r a t i o n s

@, 1

'lane'eg k d e g t ' x / b = 3.0

120

150i80

29.0763

27.6271

23.7124

18.8404

15.5193

14.2633

13.9813

29,0763

27.6275

23.7137

18.8417

15.5194

14.2633

13.9813. - ..

(b) C a l c u l a t i o n from x/Rn = 1 0 . 0 t o x /R ,

P l a n e ,@ ,

de g

0

30

60

90

12 0

150180

-- - I t e r a t i o n s . _ _

1I .0~~.- . . -

c r , de g a t x/Rn = 11 .0

21.2083

19.5249

16.1617

16.0245

17.4741

17.4404

17.4416- -._

.- _ _ _ _ _

21.2083

19.5273

16.1651

16.0260

17.4756

17.4407

17.4415-. - . . .

= 1 1 .0

54



TABLE 11.- ACCURACY AND COMPUTING TIME

[15" sphere-cone; M = 10, 01 = 10". Ca lc u l a t io n f rom x/Rn = 2 t o X / R n = 3.1

19.3114

18.9847

18.9138

(a ) S hock a n g l e s a n d s u r f a c e p r e s s u r e o n @ = 90" plane, X/Rn = 31-10.3313 840.42

-10.3961 861.15

-10.4090 864.56

[ P o i n t s

J = 5

J = 10

J = 15

19.5919

18.9337

18.8482

-10.1783

-10.5549

-10.6017

Uni t

t ime , t / N. .

(b) Computing t ime

Tota l T ime , Unit

min

p o i n t s ,t y t ime , t / N

Planes

K = 3 I K = 5

iI

P o i n t s

To ta l T ime ,

p o i n t s , t ,N min

J = 5

J = 10

J = 15

1 - I

90 0.36

420 .75

1035 1.35

853.39 18.8 404 -10.6294 853.44I 1 I I

. 1 7 9 ~ 1 0 - ~ 1 00 1 1. 17 I .167x10-*

. 1 3 0 ~ 1 0 - ~ 1725 I 2.30 1 . 1 3 3 ~ 1 0 - ~

K = 7

Tota l T ime ,

t i m e , t / N

! !

55



TABLE 111.- 15' POINTED CONE SOLUTION

PERFECT GAS

GAS CONSTANT = 0.1716OE 04 GAMMA = 0.140OOE 01

FREE-STREAM CONDITIONS

M- 0.1DbD3E D2 V= 0.396b2E 04 P= 0.10000E 0 1 RHO- 0 . 10030E- 04 1- 0.58275E

EUUISPACEQ STARTING DATA 11 POINTS 9 PLANES

PLANES EQUALLY SPACED

STLRTING DATA NORHAL TO BODY SURFACE

ANGLE OF ATTACK 10.00 DEG

CHARLCTERlSTIC METHOD

I N l 2 1 I = 3 COR. N2 2 = 0 I T R . I N I 2 5 1 = . 2 BC , I N X 1 6 1 = 3 BODY OENSITV

F L 1 4 ) = D . 18WE 01 STEP SIZE. FL ( 8 1 = 0.1009E 01 SMOOTH

0 2 0.35000E Ob HT= 0.82152E 07

PLAN t I AN6L = 0.00 DEGLEEYARO PLANE

FI ELD D ATA

SHOCK ANGLtS, D€G SIGMA= 18.5124 DELTA= -0.0003

x R THbTA PH I P RHO H V M M "

0 . 9 8 3 9 5 t 30 W.32785E 00 0.24764E 00 -0.OOOOOE-38 0.27055E 0 1 0 . 19796E- 04 0 . 47836E 06 0.39337E 04 0.89927E 0 1 3.89927E

0.98555t OW 0.32186E 00 0.24945E 00 -0.00003E-38 0.27456E 0 1 0 . 1 9 9 8 9 E - 3 4 0 . 4 80 1 4 E O b 0.39331E 04 0.89690E 01 0.89690E

0 . 9 8 l l b E OD 0.31587E 00 0 . 2 5 1 0 l E 00 -0.00000E-38 0.27781E 01 0.20145E-34 0.48267E 06 0.3932bE 04 0 . 89499E 0 1 0 . 09499E

3.9807f& 00 0.30988E 00 0.25257E 00 -0.000006-38 0 - 2 8 0 4 7 E 01 0.20269E-04 O.48432E O b 0.39321E 04 0.89337E 0 1 0.89337E

D.99031E OW W.30389E 00 0.254OOE 00 -0.OOQOOE-38 0.28266E 01 0.20366E-34 0.48577E 06 0.39318E 04 0 . 89196E 0 1 0 . 89196E

3.991976 OW 0.29790E 30 0.25543E OD -0.OOOOOE-38 0.28L45E 01 0 . 20441E- 0 4 0 . 48704E 06 0.39315E 04 0.89072E 01 O a 0 9 0 7 2 E

0 . 9 9 3 5 8 t 00 0 . 2 9 1 9 l E OD 0.25b90E 00 -0.00000E-38 D.28589E 0 1 0.20505E-04 0.48799E O b 0.39312E 04 3.88983E 0 1 0.88980E

0.99518E OD W.28592E OD 0.25844E 00 -3.00000E-38 0.28701E 01 0.20570E-04 3 -48836E 06 0.39311E 04 0.88944E 0 1 0.88944E

0.99b7PE OW O.27993E DO 0.25994E 00 -0.00000E-38 0.28786E 01 0.20640E-04 0.48813E 06 0.39312E 04 0.88966E 0 1 0.88966E

0. 9983% OW 0.27394E 00 0.26116E 00 -0.0000OE-38 0.28855E 01 0.20706E-34 0 . 4 8 7 7 4 t O b 0 . 3 9 3 1 3 t 0 4 0.89004€ G l J . 8 9 0 0 4 6

O.IODO3F 01 0.26795f OD 0.2bL8OE 00 -0.DOODOE-38 0.28922E 0 1 0 . 2 0 1 6 2 E - 3 4 0 - 4 8 7 5 7 E 06 0.39313E 04 0 . 8902 I E 0 1 3 . 8 9 0 2 l E

PLANE 2 AN

FI ELD D ATA

SHDCK ANGLES ,

X

0.98339€ 000.98505E 03

0.98671E 093.98837E OD0.99033E OD0.99169E OD0.99335E 00

0.99531E OD0.99668E OW

0.9983- OD0.100ODE 0 1

PLAVE 3 AN

FIELO DATA

SHUCK ANGLES.

X

0.98331E 00

0.9849TE OD0.98664E OD0.98831E 000 . 9 8 W 8 E 00

3.99165E 000.99332E 000 . 9 9 4 9 9 t OD0.9966bE O W

0 . 9 9 8 3 3 t 90

O.lOOD36 01

GL - 22.50 Of

DEG S I G M A .

R

0.32995E 00

0.32375E DO0.31755E 00

0.31135E 000.30515E 00

0.29-095E 00O.29275E 003.28655E 00

0.28035E 000.27415E 00

0.26795E Do

it

18.5966

THETA

0.24898E

0.24930E

0.24990E

0 . 2 5 0 7 5 E

0.25182E

0.25312E

0 . 2 5 6 b l E

0.25629E

0.2581 1E0.26000E

0. 2 b l8O E

GL = 45.10 1

O t G S IGMA .

R

W.33025E 00

D.32402E OD0.31779E 00

0.3115bE 00

0.30533E DO9.2P910E 000 + 2 9 2 8 7 E 00

0.28664E 00W.28041E 00O.27418E 000 . 2 6 7 9 5 t DO

)EG

18.5775

THETA

0.255D4E

0.2 5413E0.25357E

0.25335E

0.25349E

3.25399E

0.25605E

0.256086

0.2576bE

0.25959E

0 - 2 6 1 8 3 t

D EC TI - 1 . 3917

P H I

00 - 0 . 6 9 2 3 6 E - 0 1

00 - 0 . 6 9 7 4 3 t - 0 1

00 -0. lOO2OE-31

00 -0.10021E-31

00 - 0 . 6 9 6 7 3 E - 0 1

00 -0 .68884E-01

00 -0.67720E-01

00 - 3 . 6 6 9 4 3 E - 0 1

00 -0.68538E-01

00 -5.74793E-01

00 - 0 . 8 5 4 2 2 E - 0 1

DELTA- -1.5218

P H I

OD -3.12197E 00

0 0 -0.12138E00

00 - 0 . l 2 0 5 5 E 0030 -0.11942E 00

00 -0.11793E 0000 -0.1160LE 00

00 -0.11385E 00

O D -0.11203E 00OD -0.11231E 0000 -0.11741E 0000 -0.1278bE 00

PLANE 4 ANGL - 67.50 DEG

F I E L O D l T A

SHOCK ANGLES, O€G SIGMA- 18. 29 94 DELTA= -2.5439

X

0 . 9 8 4 6 4 t O W

0 . 9 8 6 1 8 t OD0.98771E 00

0 . 9 8 9 2 5 t DO0 . 9901% 00

0.99232E OW

0.9938bE OD0.99539E OW

0.996Y3E OD

0.9984bE ODO. l d00M 0 1

R

0.32526E 00

0.31953E 000.31380E 00

0.30806E DO0.30233E 000.29bbOt OD0.29087E 0 00.28514E 000.21941E 00

0.27368E 000 - 2 6 7 9 5 E DO

THETA

0.25140E 00

0.25099E 00

O.2508bE 00

0.25103E ODD.25150E 00

0.25228E 000.25339E 000.25684E 00

0.25170E OD

0 . 2 5 9 0 l E 00O.Zbl8OE 00

P H I

-0.15815E OD-0.15560E 00

-0.1528OE 00

-0.14971E 00

-0.14630E 00

-0.14252E 00

-0.138k5E 00

-0.13660E 00

-0.13230E 00

-0.13318E 00-0.13705E 00

P

0.32071E

0.3176PE

0. 3 152 9E

0.31339E0.31182E

0.31343E

0.3091 1E

0.30774E

0.3061%0 .30438E

0.30238E

P0.50911E

0.49424E0.4802bE

D.46169E

0.45bkOE

0 .44624E

0.k3108E

0.42875E

0 - 4 2 0 9 7 E

0.41344E

3.40623E

0101010101010101010101

01

010101010101010 1

0 1

0 1

P

0.82316E 010.79933E 0 1

0.78012E 01

0.76251E 0 10.74638E 010.13160E 010.71800E 010 . 70548E 0 1

0.69389E 0 1

0.68305E 0 10 - 6 7 2 1 5 E 0 1

RHO

0.21986E-04

0.21793E-04

0.21615E-04

0.21C39E-04

0.21239E-04

0.2D962E-04

0 . 2 0 4 4 7 E - 0 4

0.19309E-04

0.17025E-D4

0.13579E-04

0.10202E-04

RHO

0.20462E-04

0.27594E-05O . 26739 t - 04

0.25879E-04

0 . 2 4 9 9 L E - 0 4

0.24013E-O4

0.22985E-34

0.21 70SE -3 4

0.19913E-04

O . 1 l O b l E - 0 s

0.11596E-04

H

0.51055E 06

0 . 51019E 06

0.51052E 06

0.51163E O b

0.51385E Ob

0.51832E 06

0 . 52910E 06

0 - 5 5 7 8 1 E O b

0.629CBE O b

0.10452E 060.10374E 07

H

0.62687E

0.62689E0.62864E

0.63253E

0.63918E

0.64960E

0.66556E

0.69138E

0.73992E

0.84018E

0 . 1 1 2 8 7 t

O b06

Ob06

0 6

U606

56

06

06

01

RHO H

0. 35355E- 04 0 . 81193E 06

0.34195E-04 0.81815E 06

0.33049E-34 0.82617E 06

0.31909E-04 0.83637E 06

0 . 3@759E- 04 0 . 84930E Ob0.29566E-04 0.86607E 06

0.28239E-04 0.88991E 06

0.26552E-04 0.92994E O b0.24175E-04 O.lOOC6E 07

0 . 21092E- 04 0 . 11335E 070 . 18061E- 04 0 . 13031E 01

3 1

3131313 13 13 1313 1

3 131

V Y5.39255E 04 D.BbBb4E 0 10.39256E 04 0.868976 0 10 . 3 9 2 5 5 t 34 0.86867E 01

D.39252E 04 0.86761E 010.39246E 0 4 3.86567E 0 10.39235€ 04 0 + 8 6 1 6 8 E 010.39201E 94 0.85225E 010.39134E 04 O.82848E 0 1

0.38951E 04 0.77623€ 01D.38550E 04 0 .68817E 010.37889E 04 0.588176 01

M*

0.86659E 3 10.86689E 3 13 . 8 6 6 5 7 € 3 1

0.86557E 313.863636 31

0.85032E 31

0.82665E 0 15.17444E 3 1

3.68629E 3 15.58609E 3 1

3.85966E 31

V M M*

0.38957E 04 0.77798E 01 0.77233E 31

0.38957E04

0.71797E 01 0 . 7 1 2 3 4 E 3 10.38953E 04 0.77680E 01 0.771256 3 1

0.38943E 0 4 0.7lk2OE 31 0.76878E 3 1

0.38899E 04 0.76311E 01 0.15836E 5 10.38858E 0 4 0.75311E 0 1 0 . 1 4 8 3 1 6 J 1

0.38191E 0 4 0.73164E 0 1 0.13313E 11

0.38666E 04 0.71073E 0 1 3.70633E 3 10.38385E 04 0.65900E 01 0.654566 3 10.37647E 04 0.56028E 0 1 3.55585E 3 1

0.38926E 04 0 . 7 6 9 e x 01 0.764576 3 1

V n M *

0.38479E 04 0.67521E 0 1 0.66697E 0 10.38463E 04 0.67236E 0 1 3.66443E 310.38442E 04 0.66872E D l 3.66109E 310.38416E 0 4 0166417E 01 0.65690E 310.38382t 0 4 0.65852E 0 1 3.65164E 0 1

0 - 3 8 3 3 8 t 04 0.65137E 0 1 3.64491E 010.3827bE 04 0.64154E 0 1 0.63554E 3 10.38171E 04 0.6258bE 01 3 . 6 2 3 3 4 E 3 10.37975E 04 O.59907E 01 0.59398E 01

0.37634E 04 0.55892E 0 1 5.55413E 310.37119E O L 0.51485E 0 1 3.51323E 3 1

56



TABLE 111.- 15' POINTED CONE SOLUTION - Concluded

PL4NE 5 ANGL = 90.03 UEG

F I E L O 0 4 1 4

IHUCK 4t ibLESr U E G SI(.HA= lL1.(1Io6 D EL14= - 2 . 3858

PH I00 - 0 . 1 7 3 2 4 t 03

90 - 0 . 16915E 30

03 - 0 . l b 4 8 9 E J O

05 -0.16042E 30

90 - 0 . 15575E 30

00 -0.15C85E 30

00 - 7 . 14568E 0 500 - 0 . 1432bE 00

00 -0.13503E 00

01 - 0 . 1 3 1 2 l E 00

03 - 3 . 12965E ,I 3

U H O1. 41 1 3 4 E - 3 4

1 . 40064E - 04

3 . 39019E- 04

0 . 37958E- 04

0 . 30868E- 34

0 . 35739E- 34

9 . 34566E- 04

0 . 33317E- 04

0 . 31777E- ) 4

9 . 2 9 3 9 1 t - J +

O . 2 5 7 l l E - 0 4

M *j . 2 7 - 5 0 t

3.567s4E

O.ZblR8E

0.55981E

3. 5551- E

3.5496GE

1,5433bE

3. 53593E

O.52515E.53532E

3. 46933E

X0 . 9 1 2 9 1 t

3 . 9 8 7 3 2 t

0 . 98873E

0. )931+E0.99155E

J3O W

3 0

1000

00

7 0

00

R0.32252E

0 .31526E

3 .31L) OLlE

0.30475E

3. 29949E

0. 29423E

0.28898E

00

00

0 3

000000JO

on

THE140. 24354C

0. 24163E

3.24287E

0.24433E

0. 24599E

O.2478bE

0.24997E

0.25235E

H0.10663E0.10778E

3.10913E

0 . l l O b l E

0.11238E

3.1 I 446E0.11689E

0.11984E

V9.3781ZE7 . 3 7 7 8 2 t

9.37747E

0.37707E

0.37660E

9.3 7605E9.37540E

3 . 3 7 4 b i t

9.37345E

14Z . 7 7 d 9 9 t

0.57542E

J.57141E

O.Sbb87E

3 . 56169E

G. 5 5 5 7 b t

0.54899E

D.54137E

0 . 5 2 9 f l l E

3.57,924€

3.47282E

0 2

0 2

02

02029 2

0 20 2

0 202

02

9 7

0 7

0 7

0 7

0 7

0 7

0 7

0 7

0 7

0 7

0 7

5 4

04

04

04

0 404

04

04

0 4

3 4

04

31

91

01010 191

310 10 1')I0 1

:I

3 1

31

112 13 1

3 13 1

3 1

3 1

3 1

~ . ~ -3 - 9 9 2 Y b k

0.99436E

0.99577r: 0.28372E

0.99718E 30 0.27f l46E 90 0.255ObE

0.9985pf 9 0 O.27321E 3 0 O.2581flE

0.10@13€ 9 1 3 . 26795E 30 O.Zbl8Ot

PLANE 6 A NGL i l l Z . 5 3 DE G

F I E L D J ~ T A

5 H O C K 4NCCES1 OtG SIGMA= 17.7465

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PLANE 9 ANGL 1180.03 OEG

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