The Characteristics of a Two-dimensional Supersonic Sail

lECHN' 31 DELFT

Michiel de Suyterwei 10 - DELFT C o A R e p o r t N o . 143

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE CHARACTERISTICS OF A TWO-DIMENSIONAL

SUPERSONIC SAIL

by

REPORT NO, 143 December, 1960.

T H E C O L L E G E OF A E R O N A U T I C S

C R A N F I E L D

The Characteristics of a Two-dimensional Supersonic Sail b y -E. A. Boyd, M.A. of the Department of Aerodynamics SUMMARY

The two-dimensional supersonic sail is analysed using Busemann's second-order theory. It is found to have a universal shape, which is part of a Sici spiral. Aerodynanaic characteristics are calculated for a few sails. The tension in sails flying at Mach numbers of 2 and 3 at altitudes of 20,000 and 70,000 feet is large and suggests that wire sails will be needed for flight under these conditions.

1. 2. 3 . 4 . 5. 6. CONTENTS Summary List of Symbols Introduction Sail geometry Aerodynamic characteristics Results Conclusions References

Table 1 - Comparison of Sail Loadings Figures Page 1 1 3 4 4

LIST OF SYMBOLS total length of sail

chord of sail

s

C , C coefficients in Busemann's second order approximation

D L M M 00 P q«> s T x , y a

e

moment about leading edge f r e e - s t r e a m Mach number p r e s s u r e

f r e e - s t r e a m dynamic pressur a r c length of sail

tension in sail per unit span cartesian co-ordinates incidence of sail

local angle of sail

leading edge or lower surface trailing edge

upper surface

1 -1 . I n t r o d u c t i o n T h e t w o - d i m e n s i o n a l s u p e r s o n i c s a i l i s i n v e s t i g a t e d u s i n g s i m p l e w a v e t h e o r y i n t h e f o r m of B u s e m a n n ' s s e c o n d - o r d e r a p p r o x i m a t i o n . T h i s l i m i t s t h e s a i l s c o n s i d e r e d t o t h o s e w h i c h t u r n t h e flow t h r o u g h a n a n g l e a t t a i n a b l e b y an a t t a c h e d o b l i q u e s h o c k w a v e . F o r a p a r t i c u l a r d e f l e c t i o n t h e M a c h n x i m b e r m u s t b e h i g h e n o u g h t o m a i n t a i n w h o l l y s u p e r s o n i c flow o v e r t h e s a i l . F o r a p a r t i c u l a r M a c h n u m b e r t h e d e f l e c t i o n m u s t not b e t o o l a r g e . T h e l a r g e s t t r a i l i n g - e d g e a n g l e u s e d i n t h e c a l c u l a t i o n s b e l o w i s 40 . It m i g h t b e a r g u e d t h a t t h e i n v i s c i d p r o b l e m c o n s i d e r e d h e r e i s not l i k e l y t o b e a r e a l i s t i c o n e b e c a u s e t h e c o n t i n u o u s c o m p r e s s i o n on t h e l o w e r s u r f a c e of t h e s a i l w o u l d c a u s e s e p a r a t i o n of t h e b o u n d a r y l a y e r . On t h i s p o i n t w e h a v e t h e e x p e r i m e n t a l w o r k of J o h a n n e s e n (Ref. 2) on t w o - d i m e n s i o n a l s u p e r s o n i c flow o v e r c o n c a v e s u r f a c e s . H e c o n c l u d e d t h a t t h e flow o u t s i d e t h e b o u n d a r y l a y e r w a s i n good a g r e e m e n t w i t h t h e i n v i s c i d flow if t h e b o u n d a r y l a y e r w a s t u r b u l e n t . If t h e b o u n d a r y l a y e r w a s l a m i n a r , it s e p a r a t e d , a s h o c k w a v e o r i g i n a t e d a t t h e p o i n t of s e p a r a t i o n , a n d t h e r e w a s a c o n s i d e r a b l e d e v i a t i o n f r o m t h e i n v i s c i d f l o w . On t h e a s s u m p t i o n t h a t u n s e p a r a t e d flow i s p o s s i b l e o v e r t h e s a i l , t h e r e s u l t s of t h e p r e s e n t a n a l y s i s s h o u l d b e of s o m e r e l e v a n c e , N u m e r i c a l r e s u l t s a r e i n c l u d e d t o g i v e e s t i m a t e s of t h e t e n s i o n l i k e l y t o o c c u r i n t h e s a i l . 2 . S a i l g e o m e t r y If t h e t w o - d i m e n s i o n a l s u p e r s o n i c s a i l i s a t i g h t o n e , s u c h t h a t t h e m a x i m u m d e f l e c t i o n c o n c a v e t o t h e s t r e a m c a n b e a t t a i n e d b y a n a t t a c h e d o b l i q u e s h o c k , t h e p r e s s u r e d i s t r i b u t i o n on t h e s u r f a c e of t h e s a i l c a n b e c a l c u l a t e d b y s i m p l e w a v e t h e o r y . U s i n g B u s e m a n n ' s s e c o n d o r d e r a p p r o x i m a t i o n w e h a v e C = C 0 + C e"" (1) p n a w h e r e 'J i s t h e l o c a l a n g l e of t h e s a i l ( s e e F i g . 1) a n d C . = 1 (2) ( M * - 1)^ 00 ( M * - 2 ) ^ + v M * 00 ' 0 0 C = (3) ^ 2(M* - 1 ) ^ T h e p r e s s u r e d i f f e r e n c e a c r o s s t h e s a i l i s g i v e n b y C - C = 2 C , 6 = k e (4) PT P * L "^u w h e r e

2

-k = . (5) M^ - 1

The p r e s s u r e difference a c r o s s each element ds of the sail is in equilibrium with the tension in the sail so that

(p^ - P ) cos d ds = d(T sin 6 ) L u

k q . 9

= T f l .

(7)

Equation (7) can be integrated to give the arc length of the sail, s, measured from the leading edge,

e k q „

The total length, B, of the sail is given by

(9)

j

de* . . .

e

parametric equations of the sail a r e

T ^ -T y '• <

cose' da'

c i ( d ) - c i ( e ^ )

3 -T h u s the t w o - d i m e n s i o n a l s u p e r s o n i c s a i l f o r m s p a r t of a Sici s p i r a l , (Ref. 1). T h e c h o r d , c, of t h e s a i l i s given b y k q • \ | \ T T / \ T ^ T / * c = .,1 \. — = - x^ y + \-^=r- Yrr. I (12) T and i t s g e o m e t r i c a l angle of a t t a c k , a , b y 3- A e r o d y n a m i c c h a r a c t e r i s t i c s

The t o t a l lift, d r a g , and l e a d i n g - e d g e m o m e n t of t h e t w o - d i m e n s i o n a l s a i l a r e d e t e r m i n e d d i r e c t l y by t h e a t t a c h i n g f o r c e s and t h e i r a n g l e s at t h e leading and t r a i l i n g e d g e s b e c a u s e t h e t e n s i o n in t h e s a i l i s u n i f o r m . T h u s L = T (sin 6 ^ - sin ©^ ) (14) D = T ( c o s 0 - cos e ) (15) M = T ( y ^ cos e ^ - x^ sin 6 ^ ) (16) T h e c o r r e s p o n d i n g coefficients can b e e x p r e s s e d a s ^ ( s i n Q^ - sin 9^} ^^ ^ ~%:^ " ^ (k q ^ c / ^ ) ^^^^ (cos 0 - cos 6 ) C „ = - ^ = k —T. ^ — J — (18) D q ^ c (k q ^ d^) M ^CL^^ A ^^<^^T . ^ \ q c D qoo<^ M %> c q a o C / r j . ) ^ and

- ^ = cotKe^ + V <2°^

. 4

-4, Results

Fig. 2 shows the shape of the supersonic sail for a leading-edge angle of 4 . The local sail angle is indicated on the curve, the largest angle plotted being 40 . It is known that for large trailing-edge angles simple wave theory will not give an accurate description. The leading-edge angle chosen is a r b i t r a r y because the sail shape shown is a universal one. If a l a r g e r leading-edge angle is required the sail co-ordinates can be calculated immediately by transferring the origin to the appropriate point on the curve. The sail may be terminated of coiirse at any angle up to 40 depending on the aerodynamic characteristics required.

The aerodynanaic characteristics have been calculated for sails with three patrticular values of leading-edge angle, namely © = 4 , 10 , and 20 , and for trailing-edge angles up to 40 . Where necessary a particular sail is indicated in the graphs by a number pair which refers to ( 6 , 6 ).

L 1 Fig, 3 shows the overall lift and drag coefficients obtainable and Fig. 5 the montient coefficient about the leading-edge. Fig, 4 clearly indicates the high lift-drag ratios (neglecting friction) obtainable with tight sails with small leading-edge angles, a fact which is easily deduced from equation (20), and at the same time defines the limited C^ range obtainable frona any one sail.

Table 1 compares the tensions in three particular sails flying at Mach numbers of 2 and 3 at altitudes of 20,000 and 70,000 ft. The tension in the (4, 10) sail is relatively high because it is too tight. In the limit of a flat plate the tension is infinite. Moving from a (4, 20) to a (10, 30), or a (4, 30) to a (10, 30), sail makes the sail loading about half as much again and

approx;imately doubles the tension. F r o m the figures quoted even at 70,000 ft. it would seena, taking into account the loss of strength of the naaterials

because of the high temperatures experienced, that the sails will need to be wire ones.

5. Conclusions

The universal shape of the two-dimensional supersonic sail is drawn in Fig, 2 for a leading-edge angle of 4 and a trailing-edge angle of 40 , The geometry of any other sail can be found by interpolation.

Figures a r e included to show the values of C^ » C_., C. , and C^ .^ for

L \j rjl J_i/ C _ . a range of s a i l s ,

Table 1 summarises the results of a few particular examples. Only wire sails a r e likely to support the tensions calculated, remembering that the sail material will lose strength at the high temperatures which will be met.

5 -6. R e f e r e n c e s 1, J a h n k e , E . , T a b l e s of functions. E m d e , F . D o v e r P u b l i c a t i o n s , 1945.

f

Height Mach Sail L o a d / u n i t a r e a T e n s i o n ft. No. {e^, drj,) l b . / s q . f t . l b . / f t , 20,000 2 ( 4 , 1 0 ) 723 69,600 (4,20) 1070 39,400 (10,20) 1530 91,000 3 (4,10) 996 96,000 (4,30) 1670 42,400 (10,30) 2620 80,000 70,000 2 (4.10) 44 4,200 (4,20) 65 2,390 (10,20) 93 5,520 3 (4,10) 94 9,080 (4,30) 176 3,990 (10.30) 246 7.520 Note: E a c h s a i l c o n s i d e r e d h a s a chord of 10 ft,

FIG. I. SAIL CO-OPDINATE& (fei-Zr) 0-2 0-4 0'6 0-8 l O 1-2 M 1-6 18 2 0 2-2 0-2 {kU^y o.. 0-6 0-8

0-5|

X

\<f 20» 30» 4Cf

RG 3 UFT and DRAG COEFFICENTS.

0 2 0.4

X

FIG.4. Lri—DRAG RATIOS.

' • % O' 0 2 0-3 0-4 0-5 '•^-o.. - 0 - 2 - 0 - 3 ( 2 0 ^

NUMBER PAIRS REFER TO ( \ » V

FIG. 5. MOMENT COEFFICIENT ABOUT LEADING EDGE.