Page 301 - Aerodynamics for Engineering Students

Compressible flow 283

which rearranged gives
dA 5AMZ-1
-=-- (fi)
dM MM2+5
Similarly from Eqn (6.18a), withy = 1.4

E,-
Po [@: 5] 7/2

(iii)

Thus
dp dM -7MZ
-
p MM2+5
From (ii) above:
dM dA M2+5
-=-
M A 5(W- 1)
and this substituted in (iv) gives the non-dimensional pressure change in terms of the Mach
number and area change, i.e.
dp - dA -7MZ
-_-
p A 5(@ - 1)
In the question above, M = 1.4, MZ = 1.96, dA/A = 0.01, so
dP
- = -0.0286
P

6.2.3 Velocity along an isentropic stream tube
The velocity at any point may best be expressed as a ratio of either the critical
speed of sound a* or the ultimate velocity c, both of which may be taken as flow
parameters.
The critical speed of sound a* is the local acoustic speed at the throat, i.e. where the
local Mach number is unity. Thus the local velocity is equal to the local speed of
sound. This can be expressed in terms of the reservoir conditions by applying the
energy equation (6.17) between reservoir and throat. Thus:

which with u* = a* yields

(6.26)

The ultimate velocity c is the maximum speed to which the flow can accelerate from the
given reservoir conditions. It indicates a flow state in which all the energy of the gas is