fluid_mechanics

11.4 Isentropic Flow of an Ideal Gas 595
Flow
Flow
(a)
F I G U R E 11.6 (a) A converging–diverging duct. (b) A diverging –
converging duct.
(b)
A converging–
diverging duct is required
to accelerate
a flow from subsonic
to supersonic
flow conditions.
A diverging–converging duct 1Fig. 11.6b2, on the other hand, would involve a maximum area.
If the flow entering this duct were subsonic, the fluid velocity would decrease in the diverging portion
of the duct and the sonic condition could not be attained at the maximum area location. For
supersonic flow in the diverging portion of the duct, the fluid velocity would increase and thus
Ma 1 at the maximum area is again impossible.
For the steady isentropic flow of an ideal gas, we conclude that the sonic condition
can be attained in a converging–diverging duct at the minimum area location. This minimum area
location is often called the throat of the converging–diverging duct. Furthermore, to achieve supersonic
flow from a subsonic state in a duct, a converging–diverging area variation is necessary. For
this reason, we often refer to such a duct as a converging–diverging nozzle. Note that a converging–
diverging duct can also decelerate a supersonic flow to subsonic conditions. Thus, a converging–
diverging duct can be a nozzle or a diffuser depending on whether the flow in the converging portion
of the duct is subsonic or supersonic. A supersonic wind tunnel test section is generally preceded by
a converging–diverging nozzle and followed by a converging–diverging diffuser 1see Ref. 12. Further
details about steady, isentropic, ideal gas flow through a converging–diverging duct are discussed
in the next section.
1Ma 12
11.4.2 Converging–Diverging Duct Flow
In the preceding section, we discussed the variation of density and velocity of the steady isentropic
flow of an ideal gas through a variable area duct. We proceed now to develop equations that help
us determine how other important flow properties vary in these flows.
It is convenient to use the stagnation state of the fluid as a reference state for compressible flow
calculations. The stagnation state is associated with zero flow velocity and an entropy value that corresponds
to the entropy of the flowing fluid. The subscript 0 is used to designate the stagnation state.
Thus, stagnation temperature and pressure are T 0 and p 0 . For example, if the fluid flowing through
the converging–diverging duct of Fig. 11.6a were drawn isentropically from the atmosphere, the atmospheric
pressure and temperature would represent the stagnation state of the flowing fluid. The
stagnation state can also be achieved by isentropically decelerating a flow to zero velocity. This can
be accomplished with a diverging duct for subsonic flows or a converging–diverging duct for supersonic
flows. Also, as discussed earlier in Chapter 3, an approximately isentropic deceleration can be
accomplished with a Pitot-static tube 1see Fig. 3.62. It is thus possible to measure, with only a small
amount of uncertainty, values of stagnation pressure, p 0 , and stagnation temperature, T 0 , of a flowing
fluid.
In Section 11.1, we demonstrated that for the isentropic flow of an ideal gas 1see Eq. 11.252
p
r constant p 0
k r k 0
The streamwise equation of motion for steady and frictionless flow 1Eq. 11.412 can be expressed
for an ideal gas as
dp
(11.51)
r d aV2 2 b 0
since the potential energy term, g dz, can be considered as being negligibly small in comparison
with the other terms involved.