Chyu, W. J.

Davis, R. T.

English

NASA Technical Reports Server

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ff 653 July 65 
BY 
R. T .  Davis‘ and W .  J .  Chp3 
N 65 - 33 126 
c 
I 
’This research was supported i n  part by NASA 
2Assistant Professor , Engineering Mechanics Department, Virginia 
3Graduate Student, Engineering Mechanics Department, Virginia 
Polytechnic Inst i tute  
Polytechnic Inst i tute  
https://ntrs.nasa.gov/search.jsp?R=19650023525 2020-03-17T00:53:48+00:00Z
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1. Ictroduction 
Tke high speed f lov  of a cwpressible  f l u i d  over a blunt body at 
zoeerate t o  l o w  Reynolds numbers has a t t rac ted  considerable a t tent ion due t o  
i t s  &?plication t o  re-entry problems, The f l o w  r e g h e  where t h e  Reynolds 
i s  high enough f o r  boundary-layer theo,y t o  apply c80 be handled with- 
oat too  Euch d i f f icu l ty .  T h i s  i s  done by obtaining a n m e r i c a l  solution t o  
the  inv isc i2  flow equations describiag t h e  flow outside the  boundary-layer 
mc using t h i s  solution t o  obtain the pressure d is t r ibu t ion  on t h e  body 
surfnce. 
l ayer  equations bj one of several  nethods available such as those of Flflgge-Loto . 
and Blottner (19621, Snith and Clutter (1963), and Davis and Flugge-Lotz (1964) 
(Z!e last  zeYzoL 
3 io t tzer .  
by usicg the  a5ove methods. 
blur,% body solutioas by Davis and Flugge-Lotz (1964). 
One caa then use t h i s  pressure d is t r ibu t ion  t o  solve t h e  boundary- 
is actual ly  a nodification of t he  method of Flsgge-Lotz and 
Bocndzry-layer calculations have been made f o r  several  flow cases 
I n  par t icular  one is  re fer red  t o  t h e  hypersonic 
As one encounters lower Reyaolds nunbers one must  contend with the  
f a c t  t h a t  one cannot expect t h e  first-order boundary-layer equations t o  give 
reasoca'ole results. 
solving the so-celled second-order boundary-layer equations. This has also 
jeer, done by Davis and Fl~gge-Lotz (1964) . 
soze znd requires a considerable a n o u t  of coziputing time. If one needs t o  go 
t o  third-order boundary-layer theory t o  ge t  suf f ic ien t  accuracy one would f ind  
the  s i t ua t ion  even more d i f f i cu l t .  One of the  d i f f i c u l t i e s  encountered i s  t h e  
problesl of calculating t h e  flow aue t o  displaceslent thickness. 
d i r ec t  solut ion f o r  t he  inviscid flow past  a body consisting of t he  or ig ina l  
body thickened by the  displacement thickness. Davis and Flsgge-Lotz (1964) 
epproximated t h i s  f l o w  by sh i f t i ng  and expanding t h e  o r ig ina l  body surface. 
Boffman (1964) has approached the  problem i n  a nore exact manner by using t h e  
sethod of iEtegral  re la t ions  t o  calculate the  flow f i e ld .  
e n t i r e l y  sz t i s fec tory ,  t h e  f irst  because of icaccuracy and the  second because of 
coxpGtr-Liond l i f f i c u l t i e s .  Another d i f f i c u l t y  with higher order boundary-layer 
theory is the Tact t h a t  at moderate Reynolds nunbers t h e  boundary-layer begins t o  
This can be corrected at t h e  high Reynolds number end by 
This, however, becmes qui te  cumber- 
This requires a 
Neither'nethod is  
I 
, .  
1 .  . 
c 
-3- 
spre2.d i a t o  the  en t i r e  skock layer  preventing one f ro3  c lear ly  distinguishing 
septate in-riscid and viscous regions [see Kao(1964)I and therefore l imi t ing  
'caunciq-lzyer theory t o  higher 3eynolds nwbers  than one night  expect. 
For t h e  reasons above one is lead t o  t h e  i eea  of t ry ing  t o  solve 
the  cozqlete Eavier-Stokes equations o r  a s implif icat ion t o  then wfiich is 
valid i n  the whole shock-layer. 
such a s b p l i f i c a t i o n  acd a method f o r  solving t h e i r  simplified equations. 
purpose of t h i s  paper is t o  present the r e su l t s  of an invest igat ion which uses 
the  method suggested by Davis and Flugge-Lotz (1964). 
Davis and FlUgge-Lotz (1964) have suggested 
The 
For s i q d i c i t y  we w i l l  consider t he  constant density flow pas t  a 
sphere. 
Zue t o  Lighth i l l  (1957) is  avai lable  f o r  cozlprison f o r  t he  inviscid part of t h e  
flow field in the high Reynolds nunber flows. 
will r e t a in  d l  of the  e s sen t i a l  features f o r  t h e  n w e r i c a l  procedure f o r  t h e  
solut ion of t h e  flow of t h e  Pore conplicated compressible fluid. 
constant density case has been solved t h e  extension t o  the  coxqressible case 
is d i r e c t  w i t h  no conplications ar is ing due t o  theory. 
also assume t h a t  the  shock is  a discontinuity even i n  the  low Reynolds nunber 
cases. 
There a re  several  reasons f o r  t h i s .  F i r s t ,  en exact inv isc id  solut ion 
Second, the constant density model 
Once t h e  
For s implici ty  we Will 
T h i s  i s  again a simplification which can be r a o v e d  [see Cheng (196311. 
I n  order to start the  n u e r i c a l  procedure one must have a solut ion which 
is v z l i d  near t he  s t e n a t i o n  point. The i d e a l  nethod f o r  finding t h i s  solut ion 
i s  to use the  se r i e s  truncation method developed by Van Dyke and co-workers. I n  
pa r t i cu la r  the  truncated series method used by Kao (1964) i n  t he  conpressible 
viscous flow past  a sphere i s  useful. The truncated s e r i e s  results should be 
pa r t i cu la r ly  good since the  form of the truncation i s  taken t o  be t h e  same as t h e  
form of the  inviscid constant density solut ion . These results a r e . a l s o  used f o r  
cor?parison with the  n m e r i c a l  f i n i t e  - difference resu l t s .  
For the  purpose of comparison w i t h  tke high Reynolds number cases 
the f i r s t -order  boundary-layer equations are also solved f o r  t h e  constant density 
flow, In t h i s  case Lighth i l l ' s  (1957) constant density solut ion is  used f o r  
, t  -.. I ,  . 
4 - I  c 
r , .  .
Lete-Tining the  surface pressure distribution. 
2. ForJulat ion of t he  problez 
2.1 Co-orciizate systcz 
Consider l m i n a r  hypersonic f l o v  of a viscous f l u i d  past  t h e  sphere 
of radius a* shown i n  f igure  1. 
strerrr, Mach nm3er Xm is i n f i n i t e  a d  t h a t  the  density p; and viscos i ty  
For s h p l i c i t y  we w i l l  assuine t h a t  t he  f r e e  
li* i n  tne  f l o w  f i e l d  be3ir.d the  shock are  constasts given by t h e i r  values 
S 
k z c d i z t e l y  jehicd t'le noma3 shock. The veloci ty  cmponents u* 8nd v* are  
taz,cr.t azd nome1 t o  the  boCy surfece respectively. The coordinate n* i s  
xzeasured no-mal t o  tke  body surface a d  t he  zzgle t$ is  neasured from t h e  
s tqna t ioc-poin t  t o  the  radius  vector. 
2.2 Dizensionless o w n t i t i e s  
For sis lpl ic i ty  the  following &ensionless quant i t ies  are introduced. 
These quant i t ies  ere of order one i n  t he  bountary-layer region near t h e  sur- 
face of tke  sphere. 
n* x = -  Taii , bomdary-layer nomal  coordinate 
2% 
a = -  a" = I ,  nose redius 
, the  veloci ty  component p a r a l l e l  t o  t h e  body 
surf ace 
U* u = -  u* 
OD 
i 
DP * 
? =  , the  pressure 2 CY: 
P; 
- -   au , shear stress at the  body surface (T$),< 
6 , the  Censity behind the  shock f o r  14- - % - F =  Pm 
a E - ( T J W  - p q'i 2 
- O D  
-5- 
U," 2% Pf 
-. 
5.3 = 7 x  , shock Remolds nmber 
S s 
Tie quantity 7 used i n  the above r e l z t ions  i s  defined 3y 
1 T =  - 
& 
S 
2.3 
(2.1 h) 
We w i l l  8ssu3e t h a t  the  constmt de3sit.j =ode1 i s  ap3licable. T h i s  
w i l l  be t rue  ne= the  stegna:ion-;oict for 2 nearly insulated bow. We w i l l  
2., _-_ w-... " ~ r  a s s ~ s  t h a t  t he  no-sli? condi:ions a2,ly at the  body surface. These 
cor?di-,lo=ls can be nodified t o  t & e  c u e  of sli? w i t h  l i t t l e  d i f f i c u l t y  [see S t r e e t  
(195G)I. 
thoz,-'r, a t  LGV 3eynolds n u b e r s  t h i s  is  not tme. 
rezGveL i n  a n e a e r  sbi lar  to t h a t  of Chez; (1963). 
ari? iz.2oseC t o  allow a t ten t ion  t o  3e focused on the  nunerical  procedure. 
res t r ic t ior -s  o l  coastant density, zo-sli? exc., c a  be reEoved with l i t t l e  
ckanze i n  t h e  aethod of solution. 
 re closely w i t h  the  r e a l  si tuztior! a r e  2resect ly  being zlade. 
\:'e w i l l  z l so  a s s u e  t h a t  the bow shock wave i s  a discontinuity even 
This r e s t r i c t i o n  ccn be 
A l l  of these r e s t r i c t i o n s  
The 
3 e s e  zo re  geaeral  calculat ions which conforn 
The lzst  a s s u q t i o n  is tka t  t h e  'oou shock wzve angle i s  the  s m e  as 
the  bow angle fo r  a given v d u e  of t . 
shock exce2t t h a t  t h e  distance Setweelz the 306y and tine shock is allowed t o  grow. 
Since the  shock is spherical  i n  the inviscici case t h i s  assumption is very good 
f o r  high Reynolds cunbers. On the Sasis oi the  n E e r i c a l  calculat ions w e  will 
see t h c t  t h i s  i s  not a bad a s s u q t i o n  even at -low Reynolds numbers. 
trlec t o  bui ld  up t he  shock w v e  i n  t he  viscous case as the  caputationsproceed 
domstreas, towever t h i s  has lerd t o  i n s t a b i l i t i e s  i n  t he  numerical procedure. 
This goint  i s  Sresently m d e r  study md an a t t ezp t  i s  being nade t o  inpose t h e  
shock coneizions i n  such a way tfiat i n s t ab i l i t i e s  do not occur. 
nxst 'oe overcoxe before other  boey shapes a re  coasi6ered along with a conpressible 
f l u i d  . 
T'nis  i s  sinilar t o  assuning a q h e r i c a l  
Me have 
This d i f f i c u l t y  
-6- 
f u l l  X2vi 
?eyr,ol2s 
cear  t h e  
2. k Gover?.ir,c 3 c r ; - . t i ~ ~  a.z< 30unchnr ConCitior,s 
Introdxcing the  dic3ensionless q x m t i t i e s  (2.la) - (2.1~) i n to  the  
.er-Stokes equations en& r.eg1ectir.g e l l  t e n s  of higher order i n  
xxz5er th,a seconl the% will a?>e2r iz  both the bornday-layer re&ion 
bocy er.6 &so i n  t he  iaviscid reglo3 oxtside t h i s  layer  we CM obtain 
2 scz of eqcztioxi siziler t o  Close give2 by k v i s  zcd Plfigge-Lotz (1964). 
t”--? ..c&- 
lezsizy q 2 r o x i s z t i o z  ve obtain the  following set of p a t i d  d i f f e r e n t i a l  
eq122tions a d  b o m d c f  conditiozs, 
[See 
---. =+er for E Liscussion of this a?proxi-,ation.] U ~ O Z  adking t h e  constant 
.... D 
S x r f a c e  Condit5or.s 
-La v = 0 rt B = 0 (2.2d) 
-Subscrl:.zs s &cote conditions behizL t h e  c o r z d  shock and subscr ipts  r$ and N 
6er.ote di f fc rzn t ia t ion .  
. .  
-7- 
u = s i n  0 
S 
The posi t ion of the shock w i l l  be located by t h e  requirezlent t h a t  
t h e  conditions (2.2e) - (2 .2g)  a30ve be sa t i s f i ed .  I n  zdai t ion t o t d  nass 
ciinservetion 'cetween the  bociy a d  tke shock is checked by the  conciitioa t h a t  
.- 
:vp -.e velue of X rt which t h i s  cocdition is s z t i s f i e d  can also be used t o  de- 
t e r s ine  tke  shock posit ion.  
S 
3. Xethods of Solution of the  Governing EQuation 
3 . 1  S e r i e s  t x n c c t i o n  method 
I n  orCer  zo start t h e  c u e r i c a l  f in i te -d i f fe rence  xzetklod it is  necessary 
ZG k-ve z? cccurcte solut ion for the f l o w  r , e z  the  stagnation-point. The 
se r i e s  t r m c a t l o n  zetkod developea an,& e.?plied by VLT Dyke and co-workers i s  
i c e d  f o r  coicz t h i s .  I n  the  cozs tmt  density flow Fast a sghere it i s  
obvious tkat tke  f o m  thzt one should tzke  f o r  t h e  t runcat ion i s  t k e  form of 
tke constmz Lensity inv isc id  solutior, of L:g?zhill (1957'). 
p r t i c a l z r l y  zccurete i n  t h e  high I).eyr,olds nE$er r m g e  when t3e boundary- 
l aye r  I s  this z d  t he  shock i s  nezrly spkericzl .  The fom or" t he  t runcat ion 
x s e 6  Sy Kto (1964) f o r  t he  coz2ressibl.e cese is  exact ly  t h e  sane as t h e  fonb 
wkich w i l l  be used here. Trobstein mu Ken? (1960) uld other  authors have 
cade siziiler trimcations i n  solving the  sl-e p r o b l a  of constant densi ty  
This should be 
flow pas t  a sohere. 
As s c e  
2 
? ( N  ,Q) = ?p + P p )  si: + ... 
S-Abstitutirg t k e s e  exsressioss i ~ t o  the coct inui ty  m d  cozen tm 
ec_lsz:iocs (2.2a - c )  a d  collecting t e r z s  we obtain the  rol lowing s e t  of 
o r d i s s j  d l f f e r s c t h l  equetions, 
1 y-l 
I s  T y+1 
v t - -  
- 2 - - -  
p2s Y'l 
"i,?L.lly, cocservction of miss requires t h a t  
h: 
= 2rp  U1(1+7?:)aq 
0 
S 
(3.2f) 
(3.2g) ' . 
(3.2h) 
(3.2i) 
(3.23) 
I 
-9- I 
. -  - .b zcz :ZVG~.~C f- t ~ 2  cz t ' rerefere >e soLv2L :zcs;ez&ect of p,. After the  
A 
scL.-: ,,-cc is sStc i~zc5 ,  I? c a  52 Eete ,z5xC frcz equazion (3.26). - 
The s e t  of ez_ll&tions (3.2~-c)  i s  fourth order. Only two bouzd2ry 
cozCitiozs are giver, et %:?e body surfece (!:=e). The cethod of solut ion i s  t o  
guess values f o r  P et the  -i~o<y surfece cui  then in tegra te  n w e r i c d u  
s t c t i r , g  f ro3  t h e  body surr'ece. 
;I cn t he  bzsis of bomdarj-12yer theory. L i g k t h i l l ' s  (1957) inviscid 
co;?s;r?t censi ty  so lz t ion  dorig r;it:? oce t e r?  of t h e  3lasius  s e r i e s  expns ion  
:sr -x.e ks,zizry-layer equ&tio-n,s near t h e  stegz.etioa-;oint orovide a f a i r l y  good 
2-.d 1: 2 1: 
Goo2 i n i t i a l  gcesses can be case f o r  P2 end 
1: 
; *<z i  &.__ _-- 0  g ~ e s s  even i n  t h e  l o w  Re-yxclds nuzher cases. The in tegrs t ion  is  
czrriec OUt ....- < T  
L.k-- t h e  shoc'l cozCition 03 i). i s  sa t i sc ied .  Values of v aad ? 
~ill i.?.er- cc c,izer,ine15. Ir?ter,clzzion ~ s i c ~  :;e-,.rton's c e t t o d  w i l l  allow 
the shock cc;?Pitlor,s or, v, ~ , ? 2  ? t o  be szz l s f l ed  d t e r  a few t rys .  It w a s  
2 1 1 
- -  
I 2 
'o"" -*Ai .< t h x  z-css cor,servation equc:icz (3.2J) - V E S  s a t i s r i e d  'to su f f i c i en t  
C:=C,L^LCY. 
F1.. 
-.-& m z e r i c a l  schene csec  was t k e  3 u g e  - K c t t a  - G i l l  xethod on an 
I. 3. !.: ?CL3 e lec t ronic  d i g i t a l  c c s p t e r .  
C-Q?, Y_rlA x e  r ,Lzzte 
' rcvir .z t i  -AL 1 s t q  s i z e  c x i l  EO c'?.c--.,e i n  the  r e s c l t s  w e s  noted i n  t h e  first four  
c:,,:-..ql ic.-- ru- places.  7 Tie riuy.ericzl resclts were c r r r i e d  out f o r  values of Re of 
Each integrat ion required less  
co=I;uting tizie aiv2 ' suf f ic ien t  accuracy w z s  assured by 
S 
w i l l  S c  cisc:sse& l z t e r  d o z g  witS t h e  r e s u l t s  :koa t h e  f ini te-difference method. 
The r e s - d t s  ere i n  agreezent v i tk  those ol' Probstein m d  Y e p  (1960). 
3 . 2 ? i 2 iz e - D i  r'r' e r c 3  r. c e !.?et ho d 
Thz z a h  s i q l i f  ica t ion  i s  t o  reeuce t h e  :<.=vier-Stokes equations 
p v e r n i n g  t k e  $ h i d  motion t o  2 s e t  of parabolic p a r t i a l  e i f f e r e n t i a l  equations 
s o  - c h t  backw~-d influeace is  elioinated,  and so t h s t  in tegra t ion  c m  be 
jzrl'or=.lec by sxz r t i zg  fron the  stqoat ion-point  er.d in tegrz t ing  downstrem 
Lol?; the  bow S u r f x e .  
f o r .  of t h e  Jzvier-stokes equ2tior.s (2.2a-c) which r e t a i n  only  terns up t o  
secori&GrLcr for l a g e  9eynolds r G b e r .  The s ignif icance of these  eqcat ioss  
. 
?fie first s t ep  i n  doing t h i s  is  t o  use t h e  s b p l i f i e d  
8 s  I'ar es c m e r i c e l  ic tegra t ion  i s  concerr.ed kcs been discussed by Eavis a d  
? ~ 2 ~ ~ e - I , o t , z  ( 1 5 6 k )  [ A  s k i l a  s e t  of eqxations has been used by Cheag (1963) .] 
-10- 
(3.4) 

-13- 
Ci..er.;S, i i .  K. 1963 "The Blunt-3ody Problea i n  Hypersonic Flow at Low Reynolds 
Xunber." Cornell Aero. Lab. Rep. no. AF-1285-A-10. 
Dzvis, 3. T. and I. FlClgge-Lotz 1961; "Second-Order Boundary-Layer Effects  i n  
i iuersonic  Flow Past  Axisymetric B l u n t  Bodies." J. Fluid Mech. vol. 20, 
part 4. 
FlUzEc-Lotz, I. and F. G. Blottner 1962 "Cmputation of t h e  Compressible 
S a i n a r  aoundary-Layer Flow Including Displacement-Thickness Ic te rac t ion  
Using Finite-Difference Methods." Div. Engng. Mech., Stanford Univ., 
Tech Rep. no. 131. (Abbreviated version published i n  J. Mechanique, 2, pp. 
397-423 ) 
30ffr.m., G. H. 1964 "Solution of the  Icv isc id  Flow Due t o  Displacement by 
t h z  Xethod of In tegra l  Relations." Lockheed Missiles and Space Capany, 
?ech Rep. no. E-61i-017. 
-. rosh izdc i  , E.  1959 "Shock-Generated Vorticity Effects  at Low Reynolds 
1:w.ber." Lockheed Aircraft Corp. , Missiles and Space D i v .  Rep., USD-48381 
vol. 1, pp. 9-';3. 
Kao, H. C. 1964 "Hypersonic Viscous Flow N e a r  t he  Stagnation Streamline of a 
Blunt Body: I. A Test of Local Similarity." AIAA Journal, vol. 2 ,  no. U. 
Lighth i l l ,  K. J. 1957 "Dynamics of a Dissociating Gas, Part I, E q u i l i b r i a  Flow." 
2. Fluid Mech. 2 ,  pp. 1-32. 
OKdchi, H. 1958 "Flow Near the  Forward Stagnation Point of a Blunt Body ' 
or' Revolction." Journal of the Aero/Space Sciences, vol. 25, no. 3.2, 
789-7900 
?robstein,  R. F. and N. H. Xenp 1960 "Viscous Aerodynsnic Character is t ics  i n  
Z-bnersonic Rarefied C a s  Flow." Journal of Aero/Space Sciences, vol. 27, 
no. 3, pp. 174-192. 
SGinar  Boundary-Layer Equations." AIM Journal, Vol. 1, no. 9. 
Snith, A. M. 0. and D. W. Clut ter  1963 "Solution of t h e  Incanpressible 
., 
St ree t ,  R. E. 1960 "A Study of Boundary Conditions i n  Slip-Flow Aerodynamics." 
Rcrefieci G a s  Dynanics (F. M. Devienne, ea.) , pp. 276-92, Pergmon 
Press , London . 
. 
Figure 1 Co-ordinate System 
1.18 
4 
0 z z  
E- 
LL 
11 
u) z 
900 I .14- 
LIGHTH ILL'S CONST. DENSITY SOLUTION I. 10 
1.04 
'""i . . . 
1.021 
I I I I : I : 
.2 .3 .4 .5 .6 .? ..8 0 1  . 
Q) 
Figure 2 Radial Distance to the Shock at Various Reynolds Numbers 
-17- 
0 
atL 
CC 0 1 . 0 4  
Re, 49 
-  FINITE D I F F E R E N C E  METHOD 
--- T R U N C A T E D  SERIES METHOD 
.2 .4 .6 .8 
U 
Figure 3a Velocity Distribution in the Shock-Layer for Res = 49 
i- 
z .  u 
0 1. IE 
Res= 100 - F I NIT€ - OlFF E RE N C E MET HOD 
--- TRUNCATED SERIES METHOD 
+so .2 .4 
/' 
.6  .8 
.2 .4 .6 
U 
3 
Figure 3b Velocity Distribution in the Shock-Layer for ReS = 100 
-19- 
W W  
Q S  
n 
2? 5 1.0 
5% 
< =  
n 
Q L L  
E O  
a 
- 1 -  
a i  
Res= 900 
- FI N I T €  0 D1 F FERENCE hl ETH OD 
-0- TRUNCATED SERIES METHOD 
---- LI GHTHILL'S CONSTANT DENSITY 
SOLUTION 
.4 . 6  . 8  
.8 
U 
Fibpre 3c Velocity Distribution in the Shock-layer for R e s  = 900 
-20- 
I .6 
I .4 
= 1.2 
- I.0 
' .8 
.6 
0 
J- 
0 
E 
LL 
- 
z 
y: 
- 
18 
2 .4 
.cI 
t-, 
Y 
.2 
TRUNCATED SERIES METHOD 
I 1 1 1 1 1 
. I  .2 .3 .4 .5 -6 .? 8 0 0 
Figure.48 Variati'on of Skin-Friction for Res = 49 
. 
1.6 
1.4 
1.2 
1.0 
.8 
.6 
.4 
-2 
.O 
Res= 100 
TRUNCATED SERIES 
.I .2 . 3 .4 .5 .6 .I .8 
9 
Figure 4b Variation of Skin-Friction for R e s  = 100 
-22- 
I. 6 
0 
0 . I  .2 .3 .4 .5 .6 .7 -8 + 
J 
Figure bc Variation of Skin-Friction for Res = 900 
1.6 
1.2 
.8 
.4 
~ ~~ 
FULL SHOCK LAYER, 
FI MTE-DIFFERENCE METHOD 
0 
0 .2 .4 
.I 
Figure 4d Variation of Skin-Friction at Various Reynolds Numbers 
. 
-22;- 
I 
n 
3 c . 
I- 
T? 
I 
0 
0 
rt 
0 c 
3 
-20- 
0 
0 
0 
5 
I I - 
CD 
z 
3 
i = ‘  . 
a 
I 
I I 1 1 I I I I 0 
(0 N 00‘ 
0 
0 
o\ 
w 
K 
3 tn 
cr) w 
Q 
a 
F i p r e  
-9 
1 
.e 
.7 
.6 
.5 
.4 
93 
LIGHTHILL'S 
( Re, 
INITE- DlfFERENCE 
AND TRUNCATED 
ERlES METHOD 
' \  
**0 8 1  .2 .3 .4 .6 .7 .8 
5d Variation of Surface-Pressure Along the Body Surface 
for Various Reynolds XunSers 


Laminar flow past a sphere at high mach number

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