Effect of Area Variation on Flow Properties in Isentropic Flow
In considering the effect of area variation on flow properties in isentropic flow, we shall determine the effect on the velocity V and the pressure p .
From Eq . (39.11), we can write
Dividing by , we obtain
(40.8)
A convenient differential form of the continuity equation can be obtained from Eq. (39.6) as
Substituting from Eq. (40.8),
(40.9)
Invoking the relation (39.3b) for isentropic process in Eq. (40.9), we get
(40.10)
From Eq.(40.10), we see that for
Ma<1 an area change causes a pressure change of the same sign, i.e. positive dA means positive dp for
Ma<1 . For
Ma>1 , an area change causes a pressure change of opposite sign.
Again, substituting from Eq. (40.8) into Eq. (40.10), we obtain
(40.11)
From Eq. (40.11) we see that
Ma<1 an area change causes a velocity change of opposite sign, i.e. positive dA means negative dV for
Ma<1 . For
Ma>1 an area change causes a velocity change of same sign.
These results can be summarized in fig 40.2. Equations (40.10) and (40.11) lead to the following important conclusions about compressible flows:
At subsonic speeds(Ma<1) a decrease in area increases the speed of flow. A subsonic nozzle should have a convergent profile and a subsonic diffuser should possess a divergent profile. The flow behaviour in the regime of
Ma<1 is therefore qualitatively the same as in incompressible flows.
In supersonic flows
(Ma>1) the effect of area changes are different. According to Eq. (40.11), a supersonic nozzle must be built with an increasing area in the flow direction. A supersonic diffuser must be a converging channel. Divergent nozzles are used to produce supersonic flow in missiles and launch vehicles.
Fig 40.2 Shapes of nozzles and diffusers in subsonic and supersonic regimes
, we obtain 
