TECHNISCHE HOGESCHOOL
VUEGTUIGBOUWKUNDE
REPORT No. 22
12 Juli 1950
THE COLLEGE OF AERONAUTICS
CRANFIELD
NOTE ON THE EFFICIENCY OF ADIABATIC SHOCK
byA. W . MORLEY, Ph.D.
of the Department of Aircraft Propulsion.TECHNISCHE HOGESCHOOL VLIEGTUIGBOUWKUNDE REPORT KO. 22 SEPTMBSR. 1946. 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
Note on t h e Efficiency of Adiabatio Shock by
-A. W. Morley, Hi. D.
of the Department of Aircraft Propulsion
SUMMARY
This note records some calculations of the efficiency of adiabatic shock in air (X" = 1.40) ; where the efficiency is defined as the ratio of the work used to compress the air in
the shock wave, from the inlet static to the outlet total head
pressure, to the work required if the compression v/ere isentropic. This is the efficiency usually of interest when considering
intakes of propulsive units for supersonic flight.
2
-Introduction
It was desired to calculate the isentropic efficiency of
tho ordinary adiabatic shock wave as a compressive process from
inlet static to outlet total hoad pressure over a wide range of
possible conditions.
For this purpose, the parameters defining the wave, found
most suitable, were the wave angle ( // ) to the incident flow, and
the wedge angle ( ^ ) through which the gas is deflected.
The well-known equations for conservation of energy, '
equilibrium of forces, continuity of flow and constant tangential
velocity applied to adiabatic shock may be written:
Tl (1 +
X-
1 M^^) = T2 (1 + r - 1
li^ )
(1)
2 2
P^ (1 + y M ^ ^ sin^/A. ) = P2 (1
+ iT M.'^axn^ jZ^S )
(2)
PjMi = P2M2 sin
'AT"- $
(3)
!ir" -— sin
i/^
Ml
^"T-1 C O S /L OM2
/^2
cos yC<. - S (4)
From these equations, expressions for M-| , M2 and
P2/P1 were obtained in terms of the basic quantities y x and
8
as follows:
M^^ = 2 cos
M - S
(5)
A sin JUL.2
M2 *• 2 cos /-^ ••• (6)
B sin
JA"-"5
and ^ = S. (7)
?•! Awhere A = sin 2/-t--
% - \
sin
$
•• (8)
B = sin
2JJ'- S
+
ysln S
(9)
With these relationships the required efficiency is calculated
quite easily.
- 3 "
Efficiency of Compression
The work required per lb. to compress the gas from P-j
to P2t is proportional to {T2-t, - T^ ) whereas ideally it would be
proportional to (T2t' •" ^.j). Here T2-t' is
"^^^
hypothetical
downstream temperature corresponding to isentropic compression from
T-| over the same pressure ratio R2t/Pl • Then we define the efficiency
as:
. T2t'' - Ti
•^^t - Ti t « » « « > « * 9(10)
or
• / - 1(Pgt/Pj) ^ - 1
n = T2t/Ti
~
•>..-... (11)
Now
Y"
V'— 1^ = ] ^ • !2 =
("MS ' ?2
P1 P2 Pi
\ T 2J
Pi
,y"
= (1 + V - 1 M2^) ^ ". P2
V 2 / ^
and, since T^^^ = T^^ ,
ï*1
^
2
T2t = 1 + y - 1 M-i
T ^ -2.
Substitution for
'F2i/^^ ^^^ '^21^'^]
-^^ equation (11) gives for
y \
the expression:
Y-
1
-1 + r - 1 Mp^L ( p p ] V _ ^
I lPiP
•
)
1 = ^:
n. ^JÈJJ.
(12
2
V - 1 M i
This last equation may be rewritten, so as to expres
r
in ;t9rms of wave angle and
,
}JL2~
and P2/P1 from equa
as a working formula, the expression:
efficiency in ;t9rms of wave angle and wedge angle only, by substituting
for M^ ,
JSQ.'^and P2/P1 from equations (5), (6) and {7). V/e obtain,
r
y-
1
^'
1
VI = A sin ./^U
i l^\ ^ - \+ L
sin 2 yu. __Jl3)
F ^ T c o s ^ ^ T l^V^/ J 2 ^in2^:^--^
where A euid B are as given in (8) and (9)« Tables of the function
y - 1/ir
X were already available.
_ 4 -Results
The values of efficiency were worked out for 5 steps in a from 0 to 30° and 5° steps in /t from 25° to 85° for atmospheric air ( is^ = 1.40). These are given, with the corresponding values of M-| , M2 and P2/F1 i" Table I and curves, Fig. 1, 2, 3 and 4.
Prom these curves the points of maximum efficiency may be located. This efficiency is plotted in Fig. 5 and compared with that obtainable with normal shock* at the same entry Mach number.
It is seen that with oblique shock the efficiency is nearly 100^ until M1 reaches about 1,8 wiien an appreciable decrease occurs with increase in M-j. Though the pressure ratio P2/PI is lower tho
same pressure ratio can be obtained at a higher efficiency with oblique than with "normal" shock.
FIGURES ATTACHED
Pig. 1. Relationship between VI , / A and &
ïig. 2. Relationship between Mi, ^,M. and S . Fig. 3* Relationship between M-(, M, A''- and ^
Fig. 4. Relationship between P2/^1, M and S . Fig. 5» Relationship between 1 ^ ^^^ and M-| .
TABLE ATTACHED Table 1. Oblique Shock Data.
* With "normal" shock:
U2' = 2 + r " - 1 M-i^
2 ^ M i ^ - èr+ 1
which may be substituted directly in equation (l2) to obtain the of-P-i A-i Rnnv at. the corresnondine" Mi.
C O P A REPORT N ° 2 2 I 5 © ' 4 e >
FIG.
O R D i N A Q Y S H O C K R E L A T I O N S M I » esETNA/EEM I S E N T R O P I C E F F I C I E N C Y OF C O M P R E S S I O N ( F R O M U P -S T R E A M -S T A T I C T O O O V V N ' -S T R E A M T O T A L , H E A D ) A N D NA/AVE A N D W E D G E A N G L E S DRAVVNJ F O R ^ = • ' 4 P . S t1-A ^'JÜ—r-rrrrrrrh '<r
y - 1 CO:
:os(p S) 1 \A y J V B /
8iN
2
()4-
S)
W H E R E A = S I N ( 2 ) X - S ) - ^ 5 I N SB = S I N { 2 . ) J I - S ) - I ^ 8 ( N S
7 0
9 C
C O P A REPORT N® 2 2 l 5 - 9 ' 4 &
PIG.
1
ORDINARY' SHOCK R E L A T I O N S H I P BETWEEN ISENTROPIC EFFICIENCY OF COMPRESSION ( F R O M U P -ST REAM S T A T IC TO O o WN ' S T R E A M T O T A L H E A D ) A N O V V A V E A N D W E D G E A N G L E S DQAVVKl FOR ^ = 1-4 P , S t > -^ ^ "V"f A
S ^ L ; ; / & \ ^ , I, f A \ . ^ s . N 2 y
^ y-i cos('t^s)i\A/ j
V B / S , ^ ,2Ü'
WHERE A = S I N ( 2 ^ - S ) - ^ S I K l S B = SIN(2.)i-s)-<^6IN S 9 C W A V E ANIGLE U (DESQEES)J5 9 4 8
FIG. 2
C R D I N A R V SHOCK
R E L A T I O N S H I P B E T W E E N ENTRY MACH MUMBËR
VVAVE ANGLE A N D W E D G & A N 6 L E
a
UJ <QI
D Z I Ü <I
> • h Z UJMf -
2cosCju^-S)
SIN jJl(siN ?)J.-5- ^ S I M § )
( D R A W N F O R ^ = 1-4^ 4 Q5 0 é o 7 o SO
Y/AVt ANGLE y<A
( D E G R E E S )C O F A PEPOCtT N 0 2 2
I 5 9 - 4 S
FIG.3
R E L A T I O N S H I P BETWEEN E N T R V M A C H . NUMBER AND E X I T MACH. NUM&ER FOR QSLKPUE SMOCK WAVE IN TE.RMS OF WAVE ANGLE U A N D WEDGE ANGUE S.
( D R A W N F O R y - i - 4 )
M^ =
2 e C O S U -$ COSEC^ S I N ^y-% - 2 J S i N $ 2 COSEC j T ^ g C 0 5 ^ S I N 2 T F § + a S I N S M l E N T R V M A C H N U M B E RC O F A REPORT MO £2 15' 9- 4a
OCDINARY SHOCK
FIG. 4
RELATIONSHIP BETWEEN PRESSURE RATIO W A V E A N G L E A N D WEDGE ANGLE Ë2 ~ SIN(g>N-S) i ^ S I N S P. S i N ( 2 j J - S ) - X S I N J S ( D R A W K » F O R X « 1-4) Q h < a; D 10 «0 UJ
a
or2^
S5 3 o 4-0 5 o Go 7 0WAVE ANGLE U (DeeRE.es)
C OF A REPOQT N o , 2 2 1 5 ' 9 ' 4-8
FIG. 5
O B L I Q U E S H O C K . M A X M Ü M tSENTCOPiC E F F I C I E N C Y A H O C O M P A R I S O N W I T H N O R M A L S M O C K ( F O R^ = I-4-)
i
lOO7c
lUu
a
Ula
>-u
z
Ul Ü iT tl. Ul Üo
It
z
Ul U3 9 080
7 0 G O5 0
S-o ^, ^ s
=. c
!/^o°
y i 5 »Xo=
6 = 5" 1 S « W E \ \ , 0 ^ 2 5 " OÖECS EMI) A M G L E ( • / > / Vy^"
X
DE6C • 87.
5_ 4- 5-2 1-E N T R V M A r W KILJMP.FO 2'O3 0
The College of Aeronautics
Eeport Ifo. 22 .
TABLE 1
OBLIQUS SHOCK DATA
For a d i a b a t i c flow and '* = 1.4
y^-L = wave angle
S
t
wedge (semi) angle
i s e n t r o p i c e f f i c i e n c y of compression from 1 t o P, 2 t !—
