detachment. The p r e d i c t i o n s o f t h e s e c h a r a c t e r i s t i c s f o r s h a r p a i r f o i l s undertaken i n t h e p r e s e n t s t u d y andincludestheassessmentoftheeffectof variable gasspecific-heatratio.
The problemsimposedbyaerodynamicconvectiveheating in the hypersonic f l i g h t r a n g e havebeen met w i t h t h e u s e of b l u n t n e s s on both the noses of bodiesofrevolutionandalongtheleadingedgesofplanarsurfaces.Various t h e o r e t i c a l methods are a v a i l a b l e f o r p r e d i c t i n g t h e h y p e r s o n i c c h a r a c t e r i s tics of blunt airfoils. The s i m p l e s t i s theapproximationof Newtonianimpact i s questioned in t h e o r y which h a s a p p l i c a t i o n t o t h r e e - d i m e n s i o n a l b o d i e s b u t i t s a p p l i c a b i l i t yt op l a n a rs h a p e s. Another more e x a c t method, which is extensively used in the present report, is the accurate numerical blunt body and c h a r a c t e r i s t i c ss o l u t i o n performed on a ne l e c t r o n i c computer. Other simpler approaches are needed f o r a c c u r a t e l y e s t i m a t i n g t h e h y p e r s o n i c a e r o dynamic c h a r a c t e r i s t i c s of b o t hs h a r p and b l u n t a i r f o i l s. The p r e s e n t s t u d y w a s p u r s u e d w i t h t h e developmentofsuch a s i m p l e r method as oneof i t s p r i mary o b j e c t i v e s.F o rt h en u m e r i c a ls o l u t i o n s ,o n l ya i r f o i l sh a v i n gc i r c u l a r l y b l u n t l e a d i n g edges and f l a t a f t e r s u r f a c e s are considered. The s e l e c t i o n of a c i r c u l a r l e a d i n g edge h a d s t o s i m p l i f i c a t i o n s i n t h e t h e o r e t i c a l methodsand f l a t a f t e r s u r f a c e s a r e c o n s i s t e n t w i t h t h e optimum p r o p e r t i e so fs h a r pa i r f o i l sh a v i n g f l a t windward surfaces.Numericalsolutionsofsufficientscope were o b t a i n e d t o e v a l u a t e t h e e f f e c t s o f wedge a i r f o i l a n g l e and bluntness for specific -heat ratios of 1.400 and 1.667. Through t h e u s e ofoblique-shocktheoryandsimplehypersonicconcepts,the numericalsolutions are c o r r e l a t e d. The s o l u t i o n s and c o r r e l a t i o n sp r o v i d e means f o r making r a p i d a c c u r a t e e s t i m a t e s o f t h e h y p e r s o n i c aerodynamic c h a r a c t e r i s t i c s of b o t h b l u n t - a n d sharp-wedge a i r f o i l s.

The s o l u t i o n s were o b t a i n e d f o r as wide a v a r i a t i o n i n wedge a n g l e -. a s. t h e computingprocedure would p e r m i t. S o l u t i o n s w e r e o b t a i n e d f o r wedge angles-up t o 30' whichwere w i t h i n t h e r a n g e where t h e f l o w w a s e n t i r e l y s u p e r s o n i c over t h e wedge s u r f a c e. The a b s e n c eo fs o l u t i o n sf o rg r e a t e ra n g l e s i s of l i t t l e concern because a t t h e h i g h e r a n g l e s , i n d u c e d e f f e c t s o f t h e b l u n t l e a d i n g i s c l o s e l y p r e d i c t e d by oblique-shock edge are small a n d s u r f a c e p r e s s u r e t h e o r y. The lower limits of wedge -5 and -5' f o r s p e c i f i c - h e a t 1' anglesof r a t i o s of 1.400 and 1.667, r e s p e c t i v e l y , a r e imposed becauseofinaccuraciesin t h e b l u n t -body s o l u t i o n f o r i n p u t s n e a r the body. Pressuredistributionsoverblunt leading edges. - The e f f e c t s o f s p e c i f i c h e a t r a t i o on t h e p r e s s u r e d i s t r i b u t i o n of t h e l e a d i n g edge a r e shown i n f i g c s u r e 1. S i n c e h a r a c t e r i s t i c o l u t i o n s could not be obtained for a specific h e a t r a t i o of 1. 0 a t Mach numbers of and a, o n l yt h es u b s o n i cb l u n tl e a d i n g edge p a r t of t h e s o l u t i o n i s shown i n f i g u r e l ( a ). The p r e s s u r ed i s t r i b u t i o n f o r y of 1. 0 and i n f i n i t e Mach number ( f i g. l ( b ) ) w a s obtained by the NewtonBusemann p r e s s u r e l a w (Newtonian plus centrifugalforcecorrection)givenin r e f e r e n c e 3 and under these conditions i s a ne x a c ts o l u t i o n. For s p e c i f i c h e a t r a t i o s of 1.400 and 1.667, t h e c h a r a c t e r i s t i c s p a r t of t h e s o l u t i o n begins slightly after the sonic point a t x/R 0. 3 and ontinueso c t x/R = 1.0. The s h a p e f h e o l u t i o n o r ot s f y of 1.667 s u g g e s t ss l i g h ti n a c c u r a c i e si n the blunt-body solution which are a d j u s t e d as t h e c h a r a c t e r i s t i c s o l u t i o n proceeds. The e f f e c to fs p e c i f i c - h e a t ratio for the range from 1.200 t o 1.667 i s small. Although n o t shown, modified Newtonian t h e o r y , C = C p p s i n 2 6, st closely predicts the pressure distribution over the forward part of the nose (x/R -? 0.4),b u t s e r i o u s l y u n d e r e s t i mates t h e p r e s s u r e s o v e r t h e a f t e r p a r t (x/R 9 0. 4 ). For t h e s ep r e s s u r e sr e f erence 7 gives a b e t t e r e s t i m a t e which consists in matching pressure gradients obtained from modified Newtonian t h e o r y and a P r a n d t l "eyer expansion.

(a) M = 10 ,

_"""

" -

" "

I 000 I 200 I 400

(b) % =

F i g u r e 1.- P r e s s u r e d i s t r i b u t i o n b l u n t l e a d i n g edge.

over the

Pressure distribution over a b l u n t wedge The e f f e c t s o f wedge angle and s p e c i f i c - h e a t r a t i o on t h e p r e s s u r e d i s t r i b u t i o n s o v e r t h e s u r f a c e s of b l u n t wedges are p r e s e n t e d i n figure 2 f o r two Mach numbers. I n g e n e r a l , t h e p r e s sure d i s t r i b u t i o n s d e c a y e x p o n e n t i a l l y , t h a t is, the maximum p r e s s u r e s a t or nearthetangentpointsdecaytothe sharp-wedge v a l u e. I n c r e a s i n g t h e wedge angle reduces the extent of the decay and f o r an angle of 30 the decay i s r e l a t i v e l yu n i m p o r t a n t. The s o l u t i o n s of figure 2 f o r y = 1.667 and a 30 wedge i n d i c a t e a s t e e p c o m p r e s s i o n s t a r t i n g a t t h e t a n g e n t p o i n t similar t o an imbedded shock.Except f o r small wedge angles,changing Mach number f r o m 10 t o 03 ( f i g. 2 ) d e c r e a s e dt h ep r e s s u r ec o e f f i c i e n tc o n s i s t e n tw i t h sharp-wedge t h e o r y. However, becauseof t h e i n d u c e d e f f e c t s of b l u n t n e s s , t h i s t r e n d was r e v e r s e d a t small wedge a n g l e s. The p r e s s u r e d i s t r i b u t i o n s of f i g u r e 2 a r e p r e s e n t e d i n more d e t a i l i n c h a r t s 1 and 2 a t t h e end of t h e r e p o r t.
A comparisonof the wedge-surface maximm p r e s sc o e f f i c i e n t s ur from f i g u r e 2 with sharp -wedge values is shown ifn g u r e i 3 f o r M, = w. The

,""I

""_"""""_"""~

deg 30

5CP 4-

-1.400

" " - ? " " " " ! " ".I
" " " " " "

(a) M, = 10

Inducedeffect
---""~""""_""_

12 x/R

( b ) M,
Figure 2. - h - e s s u r e d i s t r i b u t i o n over t h e blunt-wedge surface for specific -heat r a t i o s of 1.400 and 1.667.
Figure 3. - Comparison of mx u induced ai m m pressure coefficient of blunt wedges M, = m. with sharp-wedge theory;
magnitudeand r e l a t i v e importance of t h e i n d u c e d e f f e c t s of bluntness-razgi n d i c a t e d by t h e d i f f e r e n c e between t h e maximum p r e s s u r e c o e f f i c i e n t s -oTpt,he b l u n t and sharp wedges. It i s a p p a r e n tt h a ti n d u c e de f f e c t sa r e relatively unimportant f o r l a r g e wedge a n g l e s. For wedge a n g l e s less thanabout O , i n d u c e de f f e c t sa r es l i g h t l y less for 7 of 1.667 t h a n f o r ' 1.400.
ALRFOIL-SECTION CHARAC'ITERISTICS
The p r e s s u r e d i s t r i b u t i o n s f o r t h e b l u n t wedges have been integrated graphically t o provide the contributions to l i f t , drag,andpitching-moment c o e f f i c i e n t s of theleading-edge and wedge s u r f a c e s. A method w i l l be developed f o r e v a l u a t i n g c h a r a c t e r i s t i c s of b l u n t a i r f o i l s a t a n g l e o f a t t a c k f o r the by t h e s u p e r p o s i t i o n of t h e blunt-wedge solutions separately obtained upperandlowersurfaces. Superposition of Blunt-Wedge S o l u t i o n s Figure 4 p r e s e n t s t h e components of li.ft c o e f f i c i e n t andof drag c o e f f i c i e n t c o n t r i b u t e d by t h e segments of t h e l e a d i n g edgebetween t h e s t a g n a t i o np o i n t and theupper or l o w e rt a n g e n tp o i n t s( s e es k e t c h ( a ) ). The dashed portions of t h e c u r v e s were obtained from extrapolations of p r e s s u r e f o r Mach numbers f r o m 10 t o w d i s t r i b u t i o n s. The r e s u l t s a r e a p p l i c a b l e sincetheeffect of Mach number i s n e g l i g i b l e. It i s notedfromfigure 4 t h a t t h e lift anddrag f o r x/R g r e a t e rt h a n 1. 0 i n c r e a s e s v e r y l i t t l e SO t h a t c o n t r i b u t i o n s t o l i f t anddrag of s u r f a c e s t h a t do not "see" the flow a r e small.

E f f e c t s of A i r f o i l Geometry
Two p a r a m e t e r s t h a t d e f i n e t h e p r o f i l e o f c i r c u l a r l y b l u n t a i r f o i l s a r e thebluntnessratio, R/c, and t h e wedge angles, h oftheupperandthe , or r e f e r e n c e axis. S i n c e v a r i a t i o n s l o w e rs u r f a c e sw i t hr e s p e c tt ot h eb o d y of either parameter may have important effects on aerodynamic c h a r a c t e r i s w i l l b ec o n s i d e r e ds e p a r a t e l y. Only t h e t i c s ,t h ee f f e c t so fv a r y i n ge a c h it i s a p p a r e n t t h e e f f e c t o f symmetrical a i r f o i l w i l l be considered, since of a t t a c k. A asymmetrycanbeaccounted forby a shiftinreferenceangle b a s e p r e s s u r e c o e f f i c i e n t of zero w i l l beassumed.
E f f e c t o f wedge a n g l e. - F i g u r e 5 presents the effect of wedge a n g l e on t h e l i f t , drag,andpitching-moment c o e f f i c i e n t s and l i f t - d r a g r a t i o of b l u n t a i r f o i l s h a v i n g a b l u n t n e s s , R/c, of 0.05. A t hypersonic Mach numbers, it i s e v i d e n t from f i g u r e 5 t h a t i n c r e a s i n g t h e wedge angle i s a n e f f e c t i v e means f o r i n c r e a s i n g l i f t -curve slope of blunt airfoils within the a n g l e - o f - a t t a c kr a n g e shown. It may be observedthatthe , l i f t curve of the b l u n t a i r f o i l of 0 wedge angle i s w e l l approximated by oblique -shock f l a t plate theory a t thehigherangles of a t t a c k. Although increasing wedge a n g l e increasedthedragcoefficient,the maximum value of lift-drag ratio was practically unaffected because of the accompanying i n c r e a s e i n l i f t c o e f f i c i e n t ; however, t h e a n g l e of a t t a c k f o r m a x i m u m l i f t - d r a g r a t i o was reduced. For a g i v e n a n g l e o f a t t a c k , p i t c h i n g moment c o e f f i c i e n t i n c r e a s e d w i t h wedge a corresponding a n g l e. Because of i n c r e a s e i n l i f t c o e f f i c i e n t , however, i n c r e a s i n g t h e wedge a n g l e i n c r e a s e d thestaticlongitudinalstability, - ( dcm/dc ) o n l y s l i g h t l y.

Flat plate,

L / D :cot

w------\

a , deg
Effect of varying bluntness.Leading-edge bluntness may be considered toexertthreeinfluences on t h e a e r o dynamic c h a r a c t e r i s t i c s of wedge a i r f o i l s.F i r s t ,t h ep r e s s u r e s on t h e a leading edge differ from those of s h a r p a i r f o i l and e f f e c t i v e l y c a u s e concentratedforces a t theleadingedge. Second,bluntnesshas a carry-over or

( a ) M, = 10, y = 1.400 F i g u r e 5. - E f f e c t ofvarying wedge a n g l e on t h e aerodynamic c h a r a c t e r i s t i c s o f blunt a i r f o i l s ; R/c = 0.05.
i n d u c e d e f f e c t on t h e p r e s s u r e d i s t r i b u t i o n o v e r t h e f l a t a f t e r s u r f a c e s or t h ea i r f o i l.T h i r d l y ,b l u n t n e s sc a u s e s a f o r e s h o r t e n i n go ft h ee f f e c t i v e chordoverwhichtheoblique-shockpressurespredominate.Whiletheseeffects cannot be separated readily, their separate effects can be observed at cert a i n a n g l e s of a t t a c k and w i l l bedemonstratedbyfigures 5 and 6. A s shown a largeincreaseof i n f i g u r e 6, t h e f i r s t influenceofbluntnessproduces drag a t ' a n g l e of a t t a c k.A l t h o u g hn o tr e a d i l ya p p a r e n t ,t h ed r a gi n c r e a s e 0 i s accompaniedby a small l o s s i n l i f t t h a t develops with increasing angle o fa t t a c k. The loss i n l i f t is due t oa nu n b a l a n c eo fp r e s s u r ef o r c e sa c t i n g on t h e l e a d i n g - e d g e s u r f a c e s b e t w e e n t h e s t a g n a t i o n p o i n t a n d t h e u p p e r a n d lowertangentpoints. The second or i n d u c e de f f e c to fb l u n t n e s s i s demonof t h e b l u n t a i r f o i l of ' wedge a n g l e r e l a t i v e 0 strated by the increased lift t o t h a t of the flat plate in the lower angle-of-attack range of f i g u r e 5. The t h i r d e f f e c t of b l u n t n e s s i s demonstratedbythe loss i n l i f t of t h e b l u n t e r a i r f o i l s a t h i g h a n g l e s of a t t a c k - of f i g u r e 6.

a, deg

(b) M ,

Figure 5. - Concluded.

Sbarp wedge, L/D=cot(a+8,,,)->~

L : 4i 'd 8 i 0

I Sharp wedge, L / D = c 0 l ( a t 8 ~ ) \ -~

- 8 32

r , I 28 32

(a) M ,

y = 1.400

(b) M = ,

6.- Effect of varying bluntness on the aerodynamic characteristics of blunt airfoils; = 50.
Ingeneral,inducedeffects of b l u n t n e s s a r e i m p o r t a n t o n l y f o r a i r f o i l s of small wedge a n g l e a t small angles of attack and are overpowered by oblique5 shock e f f e c t s f o r wedge angles of a t l e a s t ' o r g r e a t e r a t a l l angles of a t t a c k. A t higheranglesofattack, where induced e f f e c t s a r e small, t h e l o s s i n l i f t due t o i n c r e a s i n g b l u n t n e s s i s a p p r o x i m a t e l y p r o p o r t i o n a l t o R/c. I n c r e a s i n g b l u n t n e s s was accompaniedbyincreaseddragand a reduction in lift-drag ratio. The variationofpitching-momentcoefficientwithangle of a t t a c k w a s o n l y s l i g h t l y a f f e c t e d b y i n c r e a s i n g b l u n t n e s s. E v i d e n t l y , t h e c o n t r i b u t i o n of nose drag t o p i t c h i n g moment w a s l a r g e l y compensated by an opposing moment due -to loss i n L i f t of thenose. It shouldbenotedthat this balance between nose-lift and nose-drag moment c o n t r i b u t i o n s i s a c h a r a c t e r i s t i c of t h e c i r c u l a r b l u n t n o s e a n d may n o t e x i s t f o r o t h e r t y p e s of bluntness.

Y + 1 s i n 2 e - Mc2 o

of equations ( 4 ) and ( 6 ) The p o s i t i v e and n e g a t i v e s i g n s b e f o r e t h e r a d i c a l s c o n s t i t u t e t h e s t r o n g and weak s h o c ks o l u t i o n s ,r e s p e c t i v e l y. If Ma, = equations ( 4 ) and ( 6 ) reduce t o t h e e x a c t r e s u l t s

and, i n t h e values

Newtonian limit of

0, 3 and

1, s i m p l i f y t o t h e

Newtonian

which r e p r e s e n t t h e s t r o n g

sin2 6

and weak s h o c k s o l u t i o n s , r e s p e c t i v e l y.
Conditions for shockdetachment are c l o s e l y approximated by equating the ( 4 ) t oz e r o ,g i v i n g radicalinequation
For i n f i n i t e Mach number, exact detachment c o n d i t i o n s are
These r e s u l t s c o n s t i t u t e a n e x p l i c i t hypersonic wedge t h e o r y of o b l i q u e shocks. The oblique-shock problem has 9 beenconsideredalsoinreference with an analysis that parallels small d i s t u r b a n c eh e o r y. t The equations d e r i v e d i n r e f e r e n c e 9, w h i l e u s e f u l , are cumbersome t o a p p l y and t h e e x a c t s o l u t i o n a t i n f i n i t e Mach number i s not achieved. F i g u r e 7 p r e s e n t s a comparison of equations ( 4 ) and ( 6 ) w i t he x a c t s o l u tionsoftheobliqueshockequation (3) in order to demonstrate the applicab i l i t y o ft h ep r e s e n tt h e o r y. The accuracy and the range of deflection a n g l e s f o r which t h e t h e o r y i s a p p l i c a b l e improves w i t h i n c r e a s i n g Mach number, as i n d i c a t e d by t h e e r r o r boundary of f i g u r e 7. Figure 7 demonstrates that the theory can hardly be d i s t i n g u i s h e d from t h e e x a c t c u r v e s at a Mach number of 20 f o r d e f l e c t i o n angles greater than about '. The present hypersonic wedge t h e o r y complements t h e s m a l l - d i s t u r b a n c e t h e o r y of r e f e r e n c e s 1 and 10 f o r deflection angles greater than about (where small d i s t u r b a n c e t h e o r y i s ' no l o n g e r v a l i d ) as i s shown i n f i g u r e 8. The judicious of use both t h e o r i e s p e r m i t s r a p i d and a c c u r a t e p r e d i c t i o n s of oblique-shock properties a t a l l hypersonic Mach numbers. Figure 8 c a l l s a t t e n t i o n t o t h e i n a b i l i t y ofNewtonian t h e o r y t o a c c u r a t e l y p r e d i c t p r e s s u r e c o e f f i c i e n t s or t h e detachmentofobliqueshocks f o r Mach numbers o t h e rt h a n m , even i f t h e specific-heatratio i s 1.0.

Hypersonlc-wedge theory (eq.(411 Obhque-shocktheory (Ref 81
F i g u r e 7. - Comparison of oblique-shock solut i o n s w i t h h y p e r s o n i c wedge t h e o r y ; r = 1.400.

/<-Newtonian

IO 8 6
Figure 8. - Comparisonof wedge t h e o r i e s f o r a Mach number of 5.
The s i g n i f i c a n t e f f e c t s o f s p e c i f i c -heat r a t i o on oblique-shock c h a r a c t e r i s t i c s a r e demonstrated by t h e 9 for infie x a c ts o l u t i o n so ff i g u r e n i t e Mach number. If t h es p e c i f i c h e a t r a t i o i s 1. 0 , t h e weak-shock s o l u t i o n r e d u c e s t o Newtonian t h e o r y , w h i l e the strong-shock solution yields a p r e s s u r e c o e f f i c i e n t of 2 c o n s i s t e n t with normal-shock the solution. Simple r e l a t i o n s a r e shown ( f i g. 9 ) f o r t h e p r e s s u r e c o e f f i c i e n t a.nd shock-wave f o r any s p e c i f i c angle for detachment heat r a t i o. The a p p l i c a t i o n o f e q u a t i o n (6) w i t h t a n g e n t -wedge approximations pro v i d e s a method f o r r a p i d l y e s t i m a t i n g t h e p r e s s u r e d i s t r i b u t i o n s of s h a r p , curved airfoils a t angle of attack. The accuracy of t h i s method i s comparable t o t h a t of t h e shock-expansion theory,but the present method i s more rap id.

60 8.deg

F i g u r e 9.- E f f e c t of s p e c i f i c - h e a t r a t i o on exactpressurecoefficientsandshockwave a n g l e s p r e d i c t e d by hypersonic wedge , t h e o r y ; M = m.
Development of Correlation Parameters and y = 1. 0. - S i n c e t h e s h a r p a i r f o i l c a n b e c o n s i d e r e d t h e l i m i t arises as t o i n g c a s e as a i r f o i l b l u n t n e s s d e c r e a s e s t o z e r o , t h e q u e s t i o n what e x t e n t s h a r p - a i r f o i l t h e o r y i s u s e f u l f o r p r e d i c t i n g aerodynamic
c h a r a c t e r i s t i c so fb l u n ta i r f o i l s.F o rt h e Newtonian limit of M = 00 and , is simpleandwellunderstood. For t h e s ec o n d i t i o n s , t h e Newton-Busemann p r e s s u r e l a w p r e d i c t s t h a t t h e f l o w s e p a r a t e s from 47 the circular leading edge a t an angle of 5. ' from t h e s t a g n a t i o n p o i n t (x/R = 0.422) a n d t h e p r e s s u r e c o e f f i c i e n t is zero until the flow "sees" the f l a t s u r f a c e so ft h ea i r f o i l( s e er e f. 11). The p r e s s u r ec o e f f i c i e n to ft h e f l a t s u r f a c e i s givenby $ = 2 s i n " w. For M = 00 and y = 1, t h e c i r c u l a r , l e a d i n g edge c o n t r i b u t e s t o t h e d r a g b u t n o t t o t h e l i f t of t h e a i r f o i l f o r a widerangeofangles of a t t a c k. Theseconcepts when a p p l i e d t o b l u n t a i r f o i l s (provided CL > %) givethefollowingequations,

y = 1.0, t h e r e l a t i o n s h i p

(1 - s i n 6,)

cos 6w

E' - -R+ I c c

cos 6,

R (1 ; cos 6,

sin 6 , )

2 sin2(a

and x ' /c i s t h e moment r e f e r e n c e c e n t e r. A s a na i dt oi n t e r p r e t i n gt h e e q u a t i o n s ,t h ea i r f o i lf o r c e s and geometry a r e shown i n s k e t c h ( c ). I n summary, equations ( ) through (19) r e p r e s e n t e x a c t asymptoticsolutionsofthe aerodynamic properties of any circularly bluntor sharp-wedge a i r f o i l i n t h e Newtonian limit M, = and y = 1.0. The s h a r p wedge s o l u t i o n sa r e ,o fc o u r s e ,s p e c i a l c a s e s f o r which R/c and CdLe (15) are zero. Although equations through ( ) a r e m a i n l y ofacademic interest, by suitable modifications they can be applied to the more p r a c tical case of M, < a and y > 1. 0. These modifications w i l l now be made.

Sketch ( e )

M, < 03 and y > 1.0. - Departure from the Newtonian limit i n t r o d u c e s : t h r e e e f f e c t s whichmustbeconsidered.Inorderofincreasingimportancethese (1)a negative l i f t c o n t r i b u t i o n of t h el e a d i n ge d g e , (2) e f f e c t sa r e : induced effects of bluntness on t h e f l a t s u r f a c e s of t h e a i r f o i l , and ( 3 ) t h e f a c t t h a t 2 sin2 w is a relativelypoorestimate of t h e c o r r e c t l e v e l o f p r e s s u r e c o e f f i c i e n t on t h e f l a t windward s u r f a c e. These d i f f e r e n c e sf r o m t h e Newtonian l i m i t are i l l u s t r a t e d by s k e t c h (d).
Since it has been previously d e m o n s t r a t e d( f i g. 3 ) t h a t induced a t small effectsareimportantmainly f l o w d e f l e c t i o n s , no c o r r e c t i o n f o r i n d u c e de f f e c t s w i l l be made. C o r r e c t ing equations (15 ) and (17 for l i f t of ) (6) t h e l e a d i n g edgeandusingequation with 6 replaced by a + 6w r a t h e r thanequation (19) givesthemodified equations
- Mco=co, y = l(Newton-Busemann) "- Mao<rn, y >I

2 sln2w

Sketch
These equations can be put in a form more s u i t a b l e for c o r r e l a t i n g t h e numeri c a l s o l u t i o n s by d e f i n i n g t h e c o r r e l a t i n g p a r a m e t e r s

cp c o s ( a

&)
a f l a t p l a t ei n c l i n e d
It may b eo b s e r v e dt h a t Cp i s t h e l i f t c o e f f i c i e n t o f a t an angle, a + S, ( u p p e r s u r f a c e e x p a n s i o n n e g l e c t e d ).
L i f t - c u r v es l o p e ,d c l / d a ,a n dl o n g i t u d i n a ls t a b l i t y , - ( dCm/da), may b e e s t i m a t e d from equations (20), ( ) , a n d( )b yd i f f e r e n t i a t i o n ,c o n s i d e r i n g e l and em as dependent variables and a as the independent variable.
l i f t , drag,andpitching-moment coeffiThe n u m e r i c a l s o l u t i o n s f o r t h e c i e n t s of f i g u r e s 5 and 6 are c o r r e l a t e d b y t h e s e p a r a m e t e r s i n f i g u r e 10. With t h e e x c e p t i o n of low a n g l e s of d e f l e c t i o n , whereinduced e f f e c t s are important and where t h e o r y d o e s n o t a p p l y f o r f i n i t e Mach numbers, e x c e l l e n t c o r r e l a t i o n i s demonstrated for a v a r i e t y of a i r f o i l s and flow conditions. The e f f e c t s o f wide v a r i a t i o n s i n Mach number and s p e c i f i c - h e a t r a t i o are w e l l correlated for the angle-of-attack range wherein maximum l i f t - d r a g r a t i o i s l i k e l yt oo c c u r. Even f o r t h e b l u n t e r a i r f o i l s , f o r which R/c > 0.05 ( f i g. ( b ) ) and f o r whichinduced e f f e c t s a r e more important, good c o r r e l a tion is retained.
From t h e s e r e s u l t s , it i s e v i d e n t t h a t t h e i n d u c e d e f f e c t s of b l u n t n e s s on t h e p r e s s u r e d i s t r i b u t i o n o f t h e u p p e r andlower a i r f o i l s u r f a c e s are largelyself-compensatingandhypersonic wedge t h e o r y ( e q. ( 6 ) ) c l o s e l y a c c o u n t sf o ri n t e g r a t e df o r c e s and moments. To d e m o n s t r a t et h i s more c l e a r l y , a single computation using Newtonian t h e o r y ( e q. ( ) ) i n p l a c e ofhypersonic wedge t h e o r y i s shown i n figure l O ( a ). The d e v i a t i o n from p e r f e c t c o r r e l a t i o n of l i f t c o e f f i c i e n t i s about 20 p e r c e n t i f Newtonian t h e o r y i s used as compared t o a b o u t 2 p e r c e n t f o r h y p e r s o n i c wedge t h e o r y.

$ " b '

1.400 1.667 1.400 1.667

( a )o ff f e c t E

wedgee(c tn g l e. Eff a ) b

of b l u n t n e s s

Figure 10.- C o r r e l a t i o n o f a e r o d y n a m i c c o e f f i c i e n t s o f b l u n t a i r f o i l s e v a l u a t e d characteristicsolutions; a >

&.

Sinceinduced e f f e c t s d e c r e a s e w i t h d e c r e a s i n g Mach number, t h e c o r r e l a tions are believed applicable for all hypersonic Mach numberswhere hypersonic wedge t h e o r y i s adequate - generallyfrom M = 5 t o 03, ,
Equations (20) through ( ) c o n t a i n t h e g e o m e t r i c a l p a r a m e t e r [ 1 (R/c) (1 - s i n % ) ] / c o s h. TO f a c i l i t a t e r a p i d e s t i m a t e s ofaerodynamic c o e f f i c i e n t s of b l u n t a i r f o i l s , t h i s p a r a m e t e r is presented in chart
F o re v a l u a t i o n s of b l u n t a i r f o i l c h a r a c t e r i s t i c s ( e q s. (24), ( ) , and ( ) ) t h e l i f t and d r a g a c t i n g on t h e l e a d i n g edge may be taken from the num-
4 for gas specific-heat ratios of 1. 2 , erical solutions presented in figure 1. 4 , and 1.667. These q u a n t i t i e s may a l s o be e s t i m a t e dw i t hl e s sa c c u r a c y by modified Newtonian theory using the following expressions

a >

P r e d i c t i o n of A i r f o i l C n a r a c t e r i s t i c s
Maximum l i f t c o e f f i c i e n t The e f f e c t of s p e c i f i c - h e a t r a t i o on t h e e x a c t lift c u r v e s o f f l a t - p l a t e a i r f o i l s a t i n f i n i t e Mach number i s p r e s e n t e d i n f i g u r e 1 where 1 cz = -1 + y s i n 2 a - cos a d 1 - 72 s i n 2 a cos a 2 ( ) (30)
Flat-plate lift curves exhibit decidedly nonlinear variations with angle of a t t a c k and d e p a r t s i g n i f i c a n t l y fromimpacttheory as maximum l i f t c o e f f i c i e n t i s reached. The l i f t c o e f f i c i e n t a t shockdetachmentincreaseswithincreasing y , reaches a maximum a t 7 = and t h e r e a f t e r d e c r e a s e s w i t h f u r t h e r i n c r e a s ei n 7. It i s i n t e r e s t i n gt on o t et h a tt h e maximum l i f t c o e f f i c i e n t corresponds t o t h a t f o r shock detachment f o r Y of about 1.15, b u tf o r y < 1.15, t h e maximum l i f t c o e f f i c i e n t o c c u r s w e l l below t h e a n g l e f o r shock detachment.These same e f f e c t s and t r e n d s p r e v a i l a t lowerhypersonic Mach numbers,exceptthecurvesshift totheleftinaccordancewiththereduction i n shock-detachment.anglewithreductionin Mach number. It should be noted the results presented in figure 1 a p p l y t o anysharp-wedge a i r f o i l f o r 1 I2o ra g>hS, i f i t ne i is shifted a d i s t a n c e 6w t o h e i g h t. t r

The e x t e n t t o which t h e f l a t - p l a t e l i f t curves of f i g u r e 1 n e a r d e t a c h 1 ment a p p l y t o b l u n t a i r f o i l s c a n n o t be determined by n u m e r i c a l s o l u t i o n s. However, b e c a u s e o f t h e c l o s e c o r r e l a t i o n p r o v i d e d by hypersonic wedge t h e o r y a t h i g h a n g l e s of a t t a c k ( e q. (24) and f i g. 10) it i s reasonable to expect that the flat-plate solutionspredictthetrendscaused a. deg Figure 11.- Effect of specific -heat ratio by v a r i a t i o n s i n b o t h s p e c i f i c - h e a t on the lift characteristics of sharp flat rat i o and Mach number.
airfoils at infinite Mach number.

pace es,

L i f t - d r a g r a t i o. - While it i s generally recognized t h a t s p e c i f i c - h e a t r a t i o h a s a small e f f e c t on t h e l i f t - d r a g r a t i o of s l e n d e r c o n f i g u r a t i o n s ' having very small minimum d r a g c o e f f i c i e n t s ( r e f. 121, t h e e f f e c t on blunter shapeshasnotbeenclearlyestablished.Sinceanaccurateestimate of maximum lift -drag ratio i s e s s e n t i a l t o p r e d i c t 7; performance and reentry aracteristics ch of it i s worthwhile t o 30" examine t h e e q u a t if o r on l i f t -drag 6r a t i o of f l a t - p l a t e t h e o r y i n some at infinite d e t a i l. The l i f t - d r arg t i o a Mach number of a f l a t - p l a t e a i r f o i l whose minumum d r a gc o e f f i c i e n t i s cdo i s g i v e n e x a c t l y by

L D os (cdo/5 c

-t t a n

',. '. 1667. I 400

where Cp i s a f u n c t i o n of t h a n g l e e of a t t a c k and may be obtained from equat i o n ( 8 ) w i t h 6 r e p l a c e d by a. An examination of e q u a t i o n ( ) shows t h a t as cdo approaches departures zero, from inviscid f l a t - p l a t e t h e o r y (L/D = c o t a ) a r e n e g l i g i b l e. Thus t h e e f f e c t of s p e c i f i c - h era tttih r o u g h a o i t s dependence on p r e s s u r e C o e f f i c i e n t i s insignificant However, i f cdo i s s u f f i c i e n t l yl a r g e ,t h ec o n v e r s e is true and a dependence of L/D on s p e c i f i c h e a tr a t i o i s i n d i c a t e d.

Nurnerlcal solutions

T h e o r e t i c a l f l a t - p l a t e v a l u e s of maximum l i f t - d r a g r a t i o and t h e l i f t coefficient r fo maximum L/D have been o b t a i n e d from e q u a t i o n s (31) and ( 8 ) t o show t h e e f f e c t of s p e c i f i c - h e a tr a t i o , and t h e r e s u l t s are p r e s e n t e d i n f i g ure 1 2. Values from numerical computa5 t i o n s of v a r i o u s b l u n t a i r f o i l s ( f i g s. and 6 ) w i t h t h e l i f t c o e f f i c i e n t s c o r r e c t e d f o r loss i n l i f t of t h e l e a d i n g edge are shown as d a t ap o i n t s.E x c e l l e n t agreement of t h e n t - a i r f o i l blu solutions w i t h f l a t -plate theory i s demonstrated i f v a l u e s a r e p l o t t e d v e r s u s minimum d r a g c o e f f i c i e n t. Although t h e r e s u l t s p r e s e n t e d i n f i g u r e 12 are f o r i n i t e inf Mach number, t h e maximum l i a t i-o s a g rf dr shown closely approximate those for t h e Mach 10 t o 03. number rangefromabout While t h e f e c t s ef of s p e c i f i c - h e a t

. , T h e o r y '. Y. 1667

'I 000

FiLure 1 2. - E f f e c t o f s p e c f i c - h e a t r a t i o on t h c pe1,folmance p a ~ a m c t c r s o f b l u n t a i r f o i l s a t hypersonic hlnch numbers.
rat i o on maximum l i f t - d r a g r a t i o a r e r e l a t i v e l y u n i m p o r t a n t for v e r y small minimum d r a g c o e f f i c i e n t s , t h e e f f e c t i s more i m p o r t a n t f o r l a r g e v a l u e s o f minimum d r a g c o e f f i c i e n t c h a r a c t e r i z i n g b l u n t a i r f o i l s. The l a r g e e f f e c t s of s p e c i f i c - h e a t r a t i o on t h e l i f t c o e f f i c i e n t f o r maximum l i f t - d r a g r a t i o ( f i g. 12) are consistent with those shown e a r l i e r f o r t h e l i f t curvesofboth Newtonian b l u n t and s h a r p a i r f o i l s ( f i g. 5 ). It i s r e a d i l y a p p a r e n t t h a t t h e o r y would s i g n i f i c a n t l y u n d e r e s t i m a t e b o t h t h e l i f t c o e f f i c i e n t f o r maximum l i f t -drag rat i o and t h e maximum l i f t - d r a g r a t i o.
CONCLUDING REMARKS Numerical solutions have been obtained for blunt wedges i n a p e r f e c t g a s. The e f f e c t s of wedge angle, nose bluntness, and s p e c i f i c - h e a t r a t i o havebeen d e t e r m i n e df o rt h e Mach number rangefrom 10 t o m. The p r e s s u r e d i s t r i b u t i o n s have been integrated t o provide curves from which inviscid l i f t , drag, andpitching-moment c o e f f i c i e n t s of c i r c u l a r l y b l u n t - w e d g e a i r f o i l s c a n be evaluated. The n u m e r i c a l s o l u t i o n s i n d i c a t e t h a t i n c r e a s i n g t h e wedge angle of t h e bluntairfoilssignificantlyincreasedthelift-curveslope,increasedthe d r a g , and had l i t t l e e f f e c t on maximum l i f t - d r a g r a t i o. These t r e n d s a r e consistent with those predicted by oblique-shock theory for sharp airfoils. I n c r e a s i n g t h e b l u n t n e s s of t h e a i r f o i l s i n c r e a s e d t h e d r a g b u t a l s o c a u s e d a l o s s i n l i f t whichdevelopedwithincreasingangleofattack. The loss i n l i f t was g e n e r a l l y p r o p o r t i o n a l t o t h e loss i n l i f t of t h el e a d i n ge d g e. For wedge a n g l e s n e a r Oo, t h e i n c o r p o r a t i o n of b l u n t n e s s t e n d e d t o l i n e a r i z e t h e l i f t curves a t small a n g l e s of a t t a c k. The s i g n i f i c a n t e f f e c t s of i n c r e a s i n g gas specif ic-heat ratio were g r e a t e r l i f t - c u r v e s l o p e s and l i f t - d r a g r a t i o s.
A hypersonic wedge theory, based on e x p l i c i t o b l i q u e - s h o c k e q u a t i o n s , is shown t o provide a r a p i d and a c c u r a t e p r e d i c t i o n of t h e aerodynamic characteri s t i c s of a i r f o i l s w i t h a t t a c h e d s h o c k s f o r a n y v a l u e of t h e s p e c i f i c - h e a t r a t i o.m et h e o r ya l s op r o v i d e sa na c c u r a t ep r e d i c t i o n of c o n d i t i o n s a t shock detachment f o r s h a r p a i r f o i l s. For t h e l i m i t i n g c a s e of i n f i n i t e Mach number, thetheory i s exact for any specific-heat ratio and i s i n agreementwith Newtonian t h e o r y f o r a s p e c i f i c - h e a t r a t i o of u n i t y. By a c c o u n t i n g f o r t h e l i f t and d r a g of t h e l e a d i n g e d g e , t h e t h e o r y i s shown t o be a p p l i c a b l e t o blunt airfoils for an angle-of-attack range wherein maximum l i f t - d r a g r a t i o o c c u r s. When leading-edgebluntness i s t a k e ni n t oa c c o u n t ,t h et h e o r yp r o videsanexcellentcorrelation of t h e e f f e c t s of wedge a n g l e , b l u n t n e s s , and s p e c i f i c - h e a t r a t i o a t hypersonic Mach numbers. The t h e o r y i s u s e f u l f o r p r e d i c t i n g t h e maximum l i f t - d r a g r a t i o and t h e l i f t c o e f f i c i e n t a t maximum lift-drag ratio for both blunt and sharp airfoils.

Ames ResearchCenter National Aeronautics and Space Administrat ion M o f f e t t F i e l d , C a l i f. Jan. 14, 1964

REFERENCES

% &

'i>

1 Linnell, RichardD.: Two-Dimensional Airfoils in Hypersonic Flows. Jour. Aero. Sci., vol. 16, no. 1 Jan. 1949, pp. 22-30. ,
2 Dorrance, WilliamH : Two-Dimensional Airfoils at Moderate Hy-personic. Velocities. Jour. Aero. Sci., vol. 19, no. 9, Sept. 1952, pp. 593-600.
3. Truitt, Robert Wesley: Hypersonic Aerodynamics. Ronald Press, N. Y., 1959
4 filler, FranklynB.: Numerical Solutionsfor Supersonic Flow of an Ideal. Gas Around Blunt Two-Dimensional Bodies. NASA D-791, 1961. TN
D., D.: Supersonic Flow Past a Family 5. Van Dyke, Milton and Gordon, Helen of Blunt hisymmetric Bodies. NASA TR R-1, 1959.
6. Inouye, Mamoru, and Lor,mx, Harvard: Comparison of Experimental and
Numerical Results for Floslr of a the Bodies. NASA TN D -1426, 1962. Perfect Gas About Blunt-Nosed
8. Ames Research Staff: Equations, Tables, and Charts for Compressible Flow. NACA Rep. 1135, 1953.
9. Carafoli, Elie: On a Unitary Formula for Compression-Expansion in
Supersonic -Hypersonic Flow. Revue DGMecanique Appliquge, vol. no. 5 , 1962, pp. 867-876.
1. Hayes, Wallace D., and Probstein, Ronald 1 F.: Academic Fress, N. Y., 1959.

Hypersonic Flow Theory.

1. Love, Eugene S., Henderson, Arthur, Jr., and Bertram, Mitchel 2 H.: Some Aspects of Air-Helium Simulation and Hypersonic Approximations. NASA TN D-49, 1959.
(b) M = , Chart 1.- P r e s s u r e d i s t r i b u t i o n o v e r t h e b l u n t - w e d g e s u r f a c e f o r
a s p e c i f i c - h e a t ratio of 1.400.
Chart 2. - P r e s s u r e d i s t r i b u t i o n o v e r t h e
bl-unt-wedge s u r f a c e f o r a s p e c i f i c - h e a t r a t i o of 1.667.

W"

( b ) Moo =, Chart 3.- C o n t r i b u t i o n o f t h e b l u n t l e a d i n g e d g e a n d wedge s u r f a c e s t o l i f t c o e f f i c i e n t f o r specific-heat ratio of 1.400. a

wL= 30

( a ) M,

w. = ' 5

1 ' 25'15' 0 20'

w,, ' 5

- Inn."

(b) M,

Chart coefficient; 7
5. - Contribution of the blunt leading edge and
wedge s u r f a c e s t o p i t c h i n g - m o m e n t

1.400.

w u =5 O

wu= 5O

(b) M = , Chart
C o n t r i b u t i o n of t h e b l u n t l e a d i n g edgeand wedge s u r f a c e s t o l i f t c o e f f i c i e n t s p e c i f i c - h e a t r a t i o of 1.667.

w u =20'

wu=200 0
C o n t r i b u t i o n of t h e b l u n t l e a d i n g edgeand wedge s u r f a c e s t o d r a g c o e f f i c i e n t s p e c i f i c - h e a t r a t i o of 1.667.