C D A Report No. iüa
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
AERODYNAMIC CHARACTERISTICS OF A
HYPERSONIC PARACHUTE
by
-E. A. Boyd
REPORT NO. 152 November, 1961.
T H E C O L L E G E OF A E R O N A U T I C S
CRANFIELD
Aerodynamic C h a r a c t e r i s t i c s of a Hypersonic Parachute b y -E . A. Boyd, M . A . SUMMARYNewtonian theory, both in the form of the Modified-Newtonian and the Newton-Busemann p r e s s u r e l a w s , i s used to find the shape, cloth area and drag of the a x i s y m m e t r i c canopy of a hypersonic parachute, whose only load-carrying fibres are longitudinal o n e s . A s an exanaple, an estimate i s made of the s i z e of canopy needed to give a drag of 20.000 l b . in flight at a Mach number of 10 at 100,000 feet altitude.
• 5 . 6 . 7. Canopy drag Results Conclusions Figures Page CONTENTS Summary List of Symbols 1. Introduction 1 2. Equilibrium of parachute element 1
3. Shape of canopy generator 2 3 . 1 . Drogue parachute 2 3 . 1 . 1 . Uncorrected parachute 2 3 . 1 . 2 . Corrected parachute 3 3.2. Umbrella parachute 4 3 . 2 . 1 . Uncorrected parachute 4 3 . 2 . 2 . Corrected parachute 5
4. Canopy cloth area 5 4 . 1 . Uncorrected parachute 5
4.2. Corrected parachute 6
6 6
Liet of Symbols
A total area of cloth in canopy C_ drag coefficient of canopy D drag of canopy
M free-stream Mach number 00
p pressure
p stagnation pressure behind a normal shock q free-stream dynamic pressure
OP
(r, z) cylindrical polar co-ordinates r radial co-ordinate
B arc length along canopy generator, measured from the leading edge T longitudinal tension in cloth per unit radian
% longitudinal co-ordinate
6 local angle of canopy generator
Subscripts
L leading edge T trailing edge
1
-1. Introduction
The canopy of the parachute we consider here is a surface of revolution and is impermeable. Its shape, cloth a r e a and drag a r e calculated using Newtonian theory, on the assumption that the load in the canopy is carried by longitudinal fibres only. Results derived from the Modified-Newtonian p r e s s u r e law (the "uncorrected" parachute) and from the Newton-Busemann p r e s s u r e law (the "corrected" parachute) a r e compared. Two possible canopy shapes, one for a drogue parachute, the other for an umbrella-shaped parachute, are examined.
One example is given. The size of the canopy, which offers 20,000 lb, drag in flight at M„ = 10 at 100,000 ft. is calculated.
2. Equilibrium of parachute element
We assume that the parachute Is a surface of revolution and examine two possible canopy shapes, one for a drogue parachute, the other for an umbrella-shaped parachute. The drogue canopy is formed by revolving the generator shown in Fig. 1 about the z axis. The generator of the ximbrella canopy is shown in Fig. 2.
The hypersonic parachute i s in a Newtonian flow, so that a thin shock layer coincides with its forward-facing surface. To allow the flow in the shock layer to stream away, the umbrella-shaped canopy must be vented at the crown. The vent area need only be infinitesimal for the shock layer approaching it is infinitesimally thin because of the infinitely large density ratio a c r o s s the shock. The shock layer is free to leave the trailing edge of the drogue canopy. The leading edge may be considered vented or blanked off. Here, for algebraic simplicity, we take the drogue generator to have a zero radius at the leading edge.
Now consider the canopy element formed by revolving an ar c of length ds about the z axis; F i g s . 1 and 2. The p r e s s u r e p on this element of area 27rrds is normal to the forward-facing surface and depends on the local slope. For very high Mach numbers (M„ •• •») the p r e s s u r e on the rearward-facing surface is z e r o . The load on the canopy is in equilibrium with the tension in the canopy cloth. Further we assume that the load is carried by the longitudinal fibres only, A circumferential fibre will, of course, not be in tension if its length equals the optimum length given by the radial co-ordinate we calculate below. There is no friction between the shock layer and the parachute so that the longitudinal tension is constant. Thus for equilibrium of the canopy element we require that
2;rp r cos 0ds = d(2jr T sin 0) =• 2 TT T cos 6 d0 where 6 is the local slope,
p the p r e s s u r e difference a c r o s s the element,
and T the longitudinal tension in the cloth per unit radian. Simplifying we have
p r ds • T do (1) Substituting for p in terms of the local slope we can find from this equation
2
-pressure law, p ' p sin' 6, the resulting shape Is labelled In the following the "uncorrected parachute". The "corrected parachute" follows from the Newton-Busemann pressure law,
/ , «- . sin e de / , fl» J I \ p - p^ i^ain e + - ^ ^ j r'cos fl' dr' j
In this result we correct the ordinary Newtonian sine-squared law for the centrifugal p r e s s u r e difference a c r o s s the shock layer due to the curvature of the streamlines on the concave forward-facing surface of the canopy.
3. Shape of canopy generator 3 . 1 . Drogue parachute
The a r c length s is measured from the leading edge. F r o m Fig. 1, ds " d r / s l n 9. Substituting this is equation (1) we have
^ dr = sin 6 do (2)
3 . 1 . 1 . Uncorrected parachute
Substituting for p in (2) from, the Modified Newtonian p r e s s u r e law gives
— dr^ ^ 0 <3)
from which we find
V7 Now so that T using (3) and (4). Integrating (6) yields dz = dr cot 0 (5)
/2-J^ dz = ^êll4L =. (6)
sin 0M^)
3 -where erf X - j e " " du , (8) o and X
r
-u«
• / e du , o Xƒ e"
du.
erfl X =» / e du . (9) oEquation (7) defines the shape of the generator of the uncorrected drogue parachute, and this is plotted in Fig. 3.
3 . 1 . 2 . Corrected parachute
Putting the Newton-Busenaann p r e s s u r e law in equation 2, we obtain
p ^ [ s i n * 0 + — ^ ^ j r' cos O ' d r ' V dr = T s i n 0 d 0 « L **' .0 r sin 0 ^ + / r ' cos 0' ~ d0' = — . d0 ; „ d0 p 0. - Po Differentiating (10) yields d_ d0 Integrating (11) gives dr k sin 0
Now from (10) we have, when 0 = 0 T 0 = 0, ' P o ^ ' ^ ^ L
GS)
so that _ k » — sin 0^ '^o Thus dr T «^" «L r ^o sin" 0 and do Po sin*1 r* » — sin 0, ( cot 0^ - cot 0 )
2 p L \ L / (10)
('Ü) - ' - ' « ( - i ) •» <»'
, where k i s a constant. (12) (13) (14) (15)4 -Hence (16) Again dz ^ dr cot 0 T sin 0 L cot 0 d0 , , , .. J— , using (14). Pn r sin 0 o
sin 0^ cosec 0 cot 0 d0 (cos 0^ - sin 0- cot 0)'
L L
^^l^
dz .
This integrates to
P"o
, 1
fP^
(cot 0 + 2 cot 0^ ) (17)Finally eliminating cot 6 from equation (17) by using (15),
L
The shape of the generator of the corrected drogue parachute is a cubic, and is shown in Fig. 3.
3 . 2 . Umbrella parachute
As the a r c length s is measured from the leading edge it follows from Fig. 2. ds = - d r / s i n 0. Substituting this in equation (1) we have
^ dr =• - sin 0 d0 . (19) 3 . 2 . 1 . Uncorrected parachute
Integrating (19), with the p r e s s u r e given by the Modified Newtonian law, yields
(20)
1 f P ; L /tanè0,j.N
The z co-ordinate follows as in section 3.1 from dz = dr cot 0 and is given by
/ S j ^ z . c o t i 0 ^ e r f ( ^ J 5 r ) - t a n i 0 ^ e r f ( ^ J 5 O (21) where erf x and erfl x are defined in equations (8) and (9).
The generator of the uncorrected umbrella parachute, described by equation (21), is shown in Fig. 4.
- 5
3 . 2 . 2 . Corrected parachute
Substituting the Newton-Busemann pressure law in (19) yields re
r sin 0 ^ + / r' cos ^ —, d0' - - ~ (22)
^^ \ d0' PQ
The method for reducing this has already been given in section 3 . 1 . 2 . Thus we find
. _, sin 0.
r Ë £ . .2L - ^ (23)
dfl PQ s i n ' 0 so that
J r» - ~ sin e^ (cot 0 - cot 0,j,) (24) Po
J % r - (2 sin 0, )* ' - • - '* sin 0 ) (cot 0 - cot 0,j,) (25)
i
Substituting from (23) for dr in dz » dr cot0 and using (25) gives finally for the z co-ordinate
J ^ z . i j ^ r ( c o t 0 + ^y^ . , . 2cot0,j.) (26)
Using (24) allows us to elinainate cot0 and find
The shape of the generator of the corrected umbrella parachute is the cubic shown in Fig. 4.
4. Canopy cloth area
The cloth area A of the canopy is
«T «T
I 2jrr ds " 2» T ƒ ~ ,
^L ^L
using equation (1). (28)
only on the initial and final angles 0, and 0 „ . and not on whether the parachute is
1-1 X
Clearly, since pressure p depends on the local slope, the cloth area will depend only on the initial and final a
drogue or umbrella-shaped. 4 . 1 . Uncorrected parachute
p « p sin* 0 «^ *^o
6
-2v T
ƒ
d0
s i n * 0 cot 0 . - cot 0_, i-i T (29)
4 . 2 . Corrected parachute
With equation (14) or (23) and the appropriate e x p r e s s i o n for d s it i s e a s y to show that for both the drogue and umbrella parachutes
T sin 0, r ds d0 8 i n ' 0 (30) so that P o ^ 2ir T = sin 0
•[•
L cot 0ƒ
d0 s i n ' 0 sin 0, cot 0 T sin 0 . / c o t i o \-(31)The non-dimensional cloth a r e a s , p A / T , given by (29) and (31), are compared in F i g . 5, for 0 _ = 90 .
5. Canopy drag
The longitudinal tension per unit radian i s constant s o that the drag follows moat e a s i l y from resolving the attaching f o r c e s . The same result can be found by integrating the p r e s s u r e s on the canopy. Thus the drag D of the canopy i s
D =« 2r T (cos 0. - cos 0 ) (32)
The corresponding canopy drag coefficient, which r e f e r s only to the wave drag for the frictional drag i s not evaluated h e r e , i s defined by
'D q» A
o q«.
( c o s 0 , COS 0 ^ )
(p^A / 2 T T ) (33)
The drag coefficients of the uncorrected and corrected parachutes are compared in F i g . 5, for 0 = 90 .
6. Results
The calculated shape of the generator of the drogue canopy i s shown in F i g . 3 for 0- =10 , The drogue canopy i s produced by revolving this about the z a x i s , and can be seen to be practically conical along m o s t of its length.
Clearly the corrected parachute r e q u i r e s a s m a l l e r area of cloth than the uncorrected one if both work at the same tension. This i s confirmed in F i g . 5 which shows that this difference in cloth area rapidly falls as 0 b e c o m e s l a r g e .
7
-Fig. 4 shows the shape of the generator of the umbrella canopy, and compares the corrected and uncorrected parachutes for 0 =» 4 . Again the large difference
L
in cloth a r e a is clear. The effect of changing the leading edge angle from 0 = 4 L to 0^ = 10 can be seen by comparing the two corrected canopies,
JLi
Fig. 5 yields the expected result that, for a given cloth a r e a , the drag increases as the surface becomes m o r e normal and flat to the s t r e a m . However the figure also shows that at the same time the tension becomes very large, and, of course, in the limit as 0 - 90 , T - «• . The variation in T is got most simply from the
L
result that, when 0_, = 90 , the canopy drag equals 27r T cos 0 .
To achieve a given drag from the corrected and uncorrected canopies, working at the same tension, we must make 0 the same for both. In this case the cloth a r e a required for the corrected canopy i s much smaller than for the uncorrected canopy. See Fig. 5.
F o r a given drag from a given a r e a of cloth, C is fixed. Here the advantage of the corrected parachute is that the tension in the cloth is much smaller, though the differen
see Fig. 5.
the difference is less marked when q „ C ^ / p is l e s s than 0.3 or near 1.0. Again
To give some estimate of the size of the quantities involved we consider a canopy which offers a drag of 20,000 lb. We must keep 0 small to keep T small.
L
It is convenient to choose 0 = 9 where, see Fig. 5, q C „ / p =0.3 and p A / T = 20
L ** D O o
for the corrected parachute. Now for M» = », p / q „ = 1.839 and it is close to this limiting value for M„ >4. Using these values we find that for flight at M^ = 10 at 100,000 ft. the cloth area required to produce a 20,000 lb. drag I s approximately A = 20 sq. ft. Thus J p / T = 1 and we can read the size of the canopy directly from F i g s . 3 and 4, using the values for 0 = 10 , Both the drogue and umbrella
L
parachutes have a maximum radius of 1.4 ft. The axial length of the drogue
parachute is 5.3 ft. , and that of the umbrella parachute 2.7 ft. The tension around the maximum circumference would be of the order of 2,000 lb/ft.
7. Conclusions
We have compared the uncorrected parachute, derived from the Modified-Newtonian p r e s s u r e law, with the corrected parachute, derived from the Newton-Busemann p r e s s u r e law, which corrects for the centrifugal p r e s s u r e drop across the shock layer. The corrected parachute offers a better performance than the uncorrected one. If both are working at the same tension the corrected parachute requires l e s s cloth for a given drag, particularly at small leading edge angles 0 . 0 will be small so that the tension in the cloth is kept as small as possible for a
XJ
given drag.
Two canopy shapes, for drogue and umbrella-shaped parachutes, have been derived. A particular example has been included to give an estimate of the size of canopy required.
FIG. I. COORDINATE SYSTEM,
FIG.2 COORDINATE SYSTEM.
» ^ = I O f LOCAL SLOPE SHOWN ON CURVES
UNCORRECTED PARACHUTE COSRECTEO PARACHUTE »T " 9 0 L O C A L SLOPE SHOWN ON CURVES. UNCORRECTED PARACHUTE CORRECTED PARACHUTE
1^-FIG.4. UMBRELLA-CANOPY GENERATOR
UNCORRECTED PARACHUTE e, = 90' CORRECTED PARACHUTE / . / ~ - - 4 0 ' , - 9 0 T 3 5 2 0 IS lO
1
1
1
1
]
I /A
' \ \ \ \\ V
\ / \v \
A ^
/ \ // X
/ ^ / r ^X
/ / / / / / / \ \Kv
^ v ^^_.^ ^ ^ ^ ^ J -^ / / /V
DRAG CLOTH AREA / [iX ^ ""^^^^ ^ - ^ ^ l O oa 0 - 6 !•. 0 - 4 oa o lO" 3 0 ° so" 7 0 " 9 0 'FIG. 5. CLOTH AREA AND DRAG OF CANOPY FIG.3. DROGUE-CANOPY GENERATOR
