fluid_mechanics

11.4 Isentropic Flow of an Ideal Gas 593
including choked flow, shock waves, acceleration from subsonic to supersonic flow, and deceleration
from supersonic to subsonic flow.
Density, crosssectional
area, and
velocity may all
vary for a compressible
flow.
11.4.1 Effect of Variations in Flow Cross-Sectional Area
When fluid flows steadily through a conduit that has a flow cross-sectional area that varies with
axial distance, the conservation of mass 1continuity2 equation
(11.40)
can be used to relate the flow rates at different sections. For incompressible flow, the fluid density
remains constant and the flow velocity from section to section varies inversely with cross-sectional
area. However, when the flow is compressible, density, cross-sectional area, and flow velocity can
all vary from section to section. We proceed to determine how fluid density and flow velocity
change with axial location in a variable area duct when the fluid is an ideal gas and the flow through
the duct is steady and isentropic.
In Chapter 3, Newton’s second law was applied to the inviscid 1frictionless2 and steady flow
of a fluid particle. For the streamwise direction, the result 1Eq. 3.52 for either compressible or incompressible
flows is
(11.41)
The frictionless flow from section to section through a finite control volume is also governed by Eq.
11.41, if the flow is one-dimensional, because every particle of fluid involved will have the same experience.
For ideal gas flow, the potential energy difference term, g dz, can be dropped because of
its small size in comparison to the other terms, namely, dp and d1V 2 2. Thus, an appropriate equation
of motion in the streamwise direction for the steady, one-dimensional, and isentropic 1adiabatic and
frictionless2 flow of an ideal gas is obtained from Eq. 11.41 as
(11.42)
If we form the logarithm of both sides of the continuity equation 1Eq. 11.402, the result is
Differentiating Eq. 11.43 we get
or
m # rAV constant
dp 1 2 r d1V 2 2 g dz 0
dp
rV 2 dV V
ln r ln A ln V constant
dr
r dA A dV V 0
(11.43)
dV V dr r dA A
(11.44)
Now we combine Eqs. 11.42 and 11.44 to obtain
dp
rV 2 a1 V 2
dpdr b dA A
(11.45)
Since the flow being considered is isentropic, the speed of sound is related to variations of
pressure with density by Eq. 11.34, repeated here for convenience as
c B
a 0p
0r b s
Equation 11.34, combined with the definition of Mach number
and Eq. 11.45 yields
dp
Ma V c
rV 2 11 Ma2 2 dA A
(11.46)
(11.47)