Mechanics of Fluids

of eqns 11.2 and 11.20
T0
T
u2
= 1 +
2cpT = 1 + M2γ RT
2cpT
� �
γ − 1
= 1 + M
2
2
Since stagnation pressure is defined as that pressure which would be
reached if the fluid were brought to rest by a reversible adiabatic, that is, an
isentropic, process, we have from eqn 11.6,
p0
p =
� T0
T
� γ /(γ −1)
=
�
1 +
� γ − 1
2
�
M 2
�γ /(γ −1)
(11.32)
The reduction of stagnation pressure across a shock is indicated by the ratio
(p0)2
(p0)1
= (p0)2
p2
p2
p1
p1
(p0)1
which, on substitution from eqns 11.32, 11.24 and 11.29, becomes
�
(p0)2 (γ + 1)M2 1
=
(p0)1 2 + (γ − 1)M2 �γ /(γ −1) �
γ + 1
1 2γ M2 �1/(γ −1)
(11.33)
1 − γ + 1
The greater the departure from isentropic conditions the greater the loss
of stagnation pressure. This is illustrated in Fig. 11.8 where the expression
11.33 is plotted – together with M2 and the ratios of pressure, temperature
and density – against values of M1 for air (γ = 1.4). These functions are
also tabulated in Appendix 3, Table A3.1. It may be noted from eqn 11.29
that, as M1 →∞, M2 →{(γ − 1)/2γ } 1/2 = 0.378 for air; the density ratio
tends to (γ +1)/(γ −1) (= 6 for air) and the velocity ratio to (γ −1)/(γ +1)
(= 1/6 for air).
The equations may also be solved for u1 in terms of the pressure ratio:
� �
p1
u1 = γ − 1 + (γ + 1)
2ρ1
p2
��1/2 p1
Fig. 11.7
Shock waves 503