It is easy to verify that
df
dx
¼
2γ γ À 1
ð
Þ x À 1
ð
Þ
2 2 þ γ À 1
ð
Þx
½
γÀ1
γ þ 1
ð
Þx
½
γþ1
! 0,
ð3:35Þ
so that f(x) is increasing in the range, x ! 1. The second law of thermodynamics
demands that S 2 À S 1 ! 0 (see Fig. 3.5), so it follows that M 1 ! 1 and we conclude
that the upstream flow is supersonic, that is, M 1 > 1 and the downstream flow is
subsonic (M 2 < 1). In these circumstances it is evident from Eqs. (3.25), (3.28) and
(3.29) that p 2 /p 1 > 1, ρ 2 /ρ 1 > 1 and T 2 /T 1 > 1, that is, the gas is compressed and
heated on passing through the shock.
3.10 Fluid Motion Behind the Shock in Terms of Shock
Wave Parameters
Let us establish another Rankine-Hugoniot relation by using Eq. (3.11) for the flow
velocity behind the shock, namely,
u p ¼U 1 À U 2
¼ 1 À
ρ 1
ρ 2
U s
ð3:36Þ
after using the continuity equation and noting that U 1 ¼ U s , where U s is the velocity
of the shock wave. Using Eq. (3.17a) in this latter equation gives
u p ¼ 1 À
γ þ 1
ð
Þþ γ À 1
ð
Þ p 2 =p 1
ð
Þ
γ À 1
ð
Þþ γ þ 1
ð
Þ p 2 =p 1
ð
Þ
!
U s ,
ð3:37Þ
hence,
u p ¼
2
p 2
p 1
À 1
U s
γ À 1
ð
Þþ γ þ 1
ð
Þ
p 2
p 1
,
ð3:38aÞ
which is another Rankine-Hugoniot relationship. In the limit of a very strong shock
( p 2 /p 1 ) 1) it follows that
u p !
2
γ þ 1
U s :
ð3:38bÞ
108
3 Conditions Across the Shock: The Rankine-Hugoniot Equations
df
dx
¼
2γ γ À 1
ð
Þ x À 1
ð
Þ
2 2 þ γ À 1
ð
Þx
½
γÀ1
γ þ 1
ð
Þx
½
γþ1
! 0,
ð3:35Þ
so that f(x) is increasing in the range, x ! 1. The second law of thermodynamics
demands that S 2 À S 1 ! 0 (see Fig. 3.5), so it follows that M 1 ! 1 and we conclude
that the upstream flow is supersonic, that is, M 1 > 1 and the downstream flow is
subsonic (M 2 < 1). In these circumstances it is evident from Eqs. (3.25), (3.28) and
(3.29) that p 2 /p 1 > 1, ρ 2 /ρ 1 > 1 and T 2 /T 1 > 1, that is, the gas is compressed and
heated on passing through the shock.
3.10 Fluid Motion Behind the Shock in Terms of Shock
Wave Parameters
Let us establish another Rankine-Hugoniot relation by using Eq. (3.11) for the flow
velocity behind the shock, namely,
u p ¼U 1 À U 2
¼ 1 À
ρ 1
ρ 2
U s
ð3:36Þ
after using the continuity equation and noting that U 1 ¼ U s , where U s is the velocity
of the shock wave. Using Eq. (3.17a) in this latter equation gives
u p ¼ 1 À
γ þ 1
ð
Þþ γ À 1
ð
Þ p 2 =p 1
ð
Þ
γ À 1
ð
Þþ γ þ 1
ð
Þ p 2 =p 1
ð
Þ
!
U s ,
ð3:37Þ
hence,
u p ¼
2
p 2
p 1
À 1
U s
γ À 1
ð
Þþ γ þ 1
ð
Þ
p 2
p 1
,
ð3:38aÞ
which is another Rankine-Hugoniot relationship. In the limit of a very strong shock
( p 2 /p 1 ) 1) it follows that
u p !
2
γ þ 1
U s :
ð3:38bÞ
108
3 Conditions Across the Shock: The Rankine-Hugoniot Equations