The Busemann correction to the characteristics of the two-dimensional hypersonic sail

CoA Report No. 140

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LFvfuSGTUlGBOUWKUNDE Michiel de Suyterweg 10 - DELFT

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THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

BUSEMANN CORRECTION TO THE CHARACTERISTICS

OF THE TWO-DIMENSIONAL HYPERSONIC SAIL

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REPORT NO, 140 November, 1960. T H E C O L L E G E O F A E R O N A U T I C S

C R A N F I E L D

The Busemann correction to the characteristics of the two-dimensional hypersonic sail

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-E. A. Boyd, M.A. of the

Departm.ent of Aerodynamics

SUMMARY

The two-dimensional hypersonic sail is examined using the Newton-Busemann p r e s s u r e law. The results a r e compared with those of Daskin and Feldman (1958) who used the empirical modified Newtonian p r e s s u r e law. It is found that for a given chord length of sail a corrected sail will give a specified lift for a smaller tension in the sail.

At a flight Mach number of 10 at 100,000 ft, the tension in one particular sail considered could be supported with a working s t r e s s of about 20 tons/in .

CONTENTS

Page Summary

List of Symbols

1. Introduction 1 2. The sail equation 1 3. Sail geometry 2 4. Aerodynamic characteristics 5

5. Results 6 6. Conclusions 7 7. References 7

LIST OF SYMBOLS B total length of sail

C chord of sail C_^ drag coefficient C, lift coefficient i-i C pressure coefficient P

C value of C behind normal shock pmax p

D drag L lift

M free-stream Mach number p pressure

p stagnation pressure behind normal shock q dynamic pressure, 2P00U

s length along sail T tension per unit span U free-stream speed X longitudinal co-ordinate y normal co-ordinate a angle of attack Q local angle of sail p density

GT local sail angle, see Fig. lb Subscripts

T trailing edge L leading edge

1

-1, Introduction

Daskin and Feldman (1958) investigated the characteristics of the two-dimensional hypersonic sail using the empirical Newtonian p r e s s u r e law. As Hayes and Probstein (1959) point out, a rational theory of

hypersonic flov/ should include the centrifugal correction of Busemann. Here the two-dimensional hypersonic sail is re-examined using the complete Newtonian p r e s s u r e law, and the results compared with those of Daskin and Feldman.

The results obtained by including the centrifugal correction for a body of convex curvature a r e less accurate than those obtained by

neglecting the correction. In the theory this is a result of the singularity which occurs when the p r e s s u r e on the body falls to zero, and, as

F r e e m a n (1960) has pointed out, invalidates the assumption of the close-ness of the shock wave to the body. However the present problem deals with a body of concave curvature (lower surface of the sail) and in this case the full centrifugal correction should apply.

2. The sail equation

The tw-dimensional hypersonic sail is assumed to be in a Newtonian flow. Accordingly we assume that a thin shock layer coincides with the surface of the sail and that there is no friction between the layer and the sail.

Fluid which enters the shock layer is assumed to flow along s t r e a m -lines with its velocity unchanged downstream of the shock. Because of the curvature of the shock layer the p r e s s u r e on the under-surface of the sail, concave to the oncoming stream, is increased by a p r e s s u r e difference across the layer due to the centrifugal effect. The p r e s s u r e on the under-surface of the sail is then given by the Newton-Busemann p r e s s u r e law, (see Figs^ l a , l b for notation),

c r'<y>

p^ = s i n ' e + sine ^ ^ ^ cos<T(y')dy^ . (1)

p m a x ^ Ö L

In the limit of very high Mach number we may assume that the p r e s s u r e on the r e v e r s e side of the sail is zero. The p r e s s u r e on the front is given by equation (1), where the left hand side takes the

simpler form p/p , thus

o e

£• = sin^e + s i n e -r- J cos a dy'' . (2) P^ dy 6

2

-• Neglecting the m a s s of the s a i l , each element of the s a i l i s in e q u i l i b r i u m u n d e r the f o r c e s due to the p r e s s u r e difference a c r o s s the s a i l and the t w o - d i m e n s i o n a l t e n s i o n in the s a i l . T h u s

p cos e d s = T d (sin 6 )

P = T 2 i (3,

Substituting (2) in (3) g i v e s the b a s i c equilibrium equation of the s a i l . e • 2^ , • ^ d e , / T de ... s m e + s i n e -7— J coscr dy = — ^r- . (4) dy --Q PQ ^^ which m a y be w r i t t e n >6 • 2 A ^ de . , / T de ... s m 6 + 3— J sincr coscr ds = — -:— (5) d s Q p d s

V

o r ƒ «*s . 2- , J . d s ' T . „ . —r Sin e + a smcr coscr-r— dcr = — . (6) a e ö_ der p L o 3 . Sail g e o m e t r y P u t in (6) <P ( e ) ?5 ( e ) d s . 2 „ sin e c o t e • (pier) dcr T_ P, (7) (8) o Differentiating (8) with r e s p e c t to e y i e l d s 96'(e) + cote 54 ( e ) = 0 (9) which h a s the solution

^ ( 0 ) = A c o s e c e (10) w h e r e A i s a r b i t r a r y c o n s t a n t .

3

-T

But, at e = e ?i(e^) = / p from equation (8), so that L L o A = — sin e^ (11) Po ^ F r o m (7), (10), and (11) ^ = ^ • ~ <12) do p - 3 0 ^o s m Ö

The a r c length, s, of the s a i l m e a s u r e d from the leading edge i s given by

P Q ^ • ft / d e

sin e _{e s m Ö}

JLJ

p s sin e tan jQ

The t o t a l length of the s a i l , B , i s

p B ^ sin e tan \ e, o 1 , . „ . „ L . „ - ,

w h e r e Ö i s the t r a i l i n g - e d g e a n g l e .

The p a r a m e t r i c equations of the s a i l follow s i m p l y ,

T . - c o s e . .... — s m e_ . , from (14). P 1^ • 3„ o sin e sin e ^ ^ (15) dx dS Po'^ T dy _ d e n d s COS a r— = -de 1 / 1 2 i sine • n d s ^ ^ " Q d e = sin e sin e T ^^" ®L — , from (14). P • ^ a o s m 6

o <^ ^ S i n 61

- = - = cos 6 - COS e ±L

T L s m e

(16)

E l i m i n a t i n g e between (15) and (16) gives for the equation of the s a i l

x sin e^ 1 Po^ _{- y cos}

^' L • (17)

The shape of the s a i l i s a p a r a b o l i c c y l i n d e r with the v e r t e x of the p a r a b o l a pointed d o w n s t r e a m , a s w a s noted by Hayes and P r o b s t e i n (1959).

T h e c o - o r d i n a t e s of the t r a i l i n g - e d g e a r e P x ^

*^o T 1^ 2

s i n e ,

sine^ sin e ^ sin e ^ (18)

Po^T s i n e ,

= cos cos e

T sin e, (19)

and the c h o r d , c, of the s a i l i s

! ^ = J ( V T J . ( V T J

(20)

while the g e o m e t r i c a l angle of a t t a c k , a , i s given by

• 1

- 5

4. Aerodynamic characteristics

The tension in the sail is uniform so that the simplest method of finding the total lift and drag is to consider only the attaching forces and their angles at the leading and trailing edges. Then it follows that

L = T (sin e - sin 6 ) . (22)

1 L

D = T ( c o s e - cos e ) . (23) and the corresponding coefficients are

(24) (25) and ^ L

c

D ^ L

S

L i 0 u^'c 00 D

ip u c

CK3 L D Po ^ %. sin e^ cose -(sin e^ - s i n e ^ ) P O ^ / T (cos e - cos e ) L 1 Po^^T sin 6 LI c o s e ^ (26) = cot I (6j^ + 6^) .

The lift-drag ratio is given by the same result for both the corrected and uncorrected sail. Clearly lift-drag ratios in excess of unity are achieved if 6 + 6 < 90 ; in other words with tight sails with small leading edge angles.

F o r infinite Mach number, and a specific heat ratio of 1.4, p , = 1.839. F o r Mach numbers above 4, p . rapidly

8

-5. Results

In Fig. 2 the sail shape given by the com.plete p r e s s u r e law is compared with that found by Daskin and Feldman (1958) using the uncorrected Newtonian p r e s s u r e law. The corrected sail is much shorter and tighter than the uncorrected one when both are used at the same tension.

Fig. 3 compares the aerodynamic characteristics of the two sails with e = 10 . For a given trailing-edge angle, e the corrected

L i.

sail offers a higher lift and drag for a smaller incidence.

A better comparison of the performance of the two sails is got by requiring a given lift from a given chord length of sail. The two

sails will not work at the same incidence, for at a given incidence the corrected sail always gives a higher lift. Fig. 4. F o r a given lift the corrected sail will work at a lower incidence. Above a certain value of lift it is necessary to increase the leading-edge angle, 6 .

L

When the lifts given by a corrected and an uncorrected sail a r e equal, Fig. 6 shows that the value of p c . ^ is greater for the corrected sail. In other words, if the chord, c, is the same for both sails, there is a smaller tension in the corrected sail. When the values of p c , ^

^o / T in Fig. 6 a r e greater for an uncorrected sail than for a corrected one, it will be found in Fig. 4 that the uncorrected sail is no longer able to yield as much lift as the corrected sail.

Fig. 5 shows the increase of drag with incidence. Similar conclusions to those in the last paragraph can be drawn from the drag r e s u l t s .

To estimate the magnitude of the tension in one of these sails o o

consider a sail with Q = 10 , ©^ = 20 , at a flight Mach number

2

of 10 at 100,000 ft. The sail loading is found to be 240 lb/ft . The tension per foot span in a sail of 10 ft. chord, would be 14,500 lb. A sail 0.05 in. thick of woven high tensile steel wire could support such a tension, with a s t r e s s of about 20 tons/in .

7

-6. Conclusions

A corrected sail is much shorter and tighter than an uncorrected one, when there is the same tension in the s a i l s . For any given incidence there is a corrected sail which will give a higher lift and drag than an uncorrected one. F o r given leading-edge ana trailing-edge angles the corrected sail gives higher lift and drag at a smaller incidence. For a fixed chord length of sail yielding a specified lift there will be less tension in a corrected sail.

Flight at a Mach number of 10 at 100,000 ft. would require a sail (e =10 , a_ = 20 ) 0.05 in, thick of woven high tensile steel wire, working at a s t r e s s of about 20 tons/in .

7. References

1. Daskin, W. , Journal Aero. Sciences, Feldman, L. Vol.25, 1958, pp 53 - 55. 2* Freeman, N. Journal of Fluid Mechanics,

Vol.8, 1960, pp 109.

3. Hayes, W.D. , Probstein, R. F .

Hypersonic flow theory, Academic P r e s s , 1959.

u -»

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FIG la SAIL NOMENCLATURE _{FIG. lb. SHOCK LAYER.}

WITH \ — 10?

CORRECTED SAIL.

UNCOPRECTED SAIL. ( D A S K I N and FELDMAN.) ANGLES ON CURVES SHOW LOCAL SAIL ANGLE.

LONGITUDINAL CO-ORDINATE, « 1 T i l -a a O • L = ' « f 6 0 ' ' 9 C3° 4< ^:>, 90» 1

FIG.2. SAIL SHAPE.

;i^

CORRECTED SAL. UNCOPRECTED.

FIG. 3. AERODYNAMIC CHARACTERISTICS OF HYPERSONIC SAILS.

NUMBER PAIRS REFER TO {\^ ' T )

2 0 , 8 0

-0-6 0 7

O NOTE 0 2 0 3 O^^ o 7 FALSE ORIGIN ^ = tan

.*-FIG. 4. LIFT INCIDENCE

o-6

NUMBER PAIRS REFER TO ( ' l , ° T ) . CORRECTED SAIL. UNCORRECTED SAIL. Jl£ CORRECTED SAIL. UNCORRECTED SAIL. 3 3 / / , / 1 1 1 / f J t / / / / / / / / • / • / ^^ • •^ • L = ' ^ -»l.= ICf • L = 2 0 ° ICP 3 C 50° 7Cf 'SCf

RG.6. DIMENSIONLESS CHORD TAIL ANGLE.