Page 116 - Aeronautical Engineer Data Book

92 Aeronautical Engineer’s Data Book
2
Momentum p + p u = p + u 2
2 2
2
1 1
1
2 2
u 1 u 2
Energy c T + = c T + = c T 0
p
1
p
2
p
2 2
Pressure and density relationships across the
shock are given by the Rankine-Hugoniot
equations:
+ 1
2
– 1
p 2
–1 1
=

+ 1
–
p 1 2
–1 1
( + 1)p 2
+ 1
2 ( –1)p 1
=
p
+ 1
1 +
2
–1 p 1
Static pressure ratio across the shock is given
by:
2
2 M – ( – 1)
p 1 2
=
+ 1
p 2
Temperature ratio across the shock is given by:
T 2 p 2 2
=
T 1 p / 1
1
2 �
�
��
2
2
2 + ( – 1)M
2 M – ( + 1)
T 2
1
1
=
T 1 + 1 ( + 1)M 1
Velocity ratio across the shock is given by:
From continuity: u 2 /u 1 = 1 / 2
2 + ( – 1)M 2
u 2 1
so: =
u 1 ( + 1)M 2
1
In axisymmetric flow the variables are indepen­
dent of so the continuity equation can be
expressed as:
2
1 ∂(R q ) 1 ∂(sin q )
R

+ = 0
R 2 ∂R R sin ∂
Similarly in terms of stream function :