59
4
One-Dimensional Isentropic Flow
Introduction
Many ows that occur in engineering practice can be adequately modeled by assuming
them to be steady, one-dimensional, and isentropic. The equations that describe such
ows will be discussed in this chapter. The applicability of the one-dimensional ow
assumption was discussed in Chapter 2 and will not be discussed further here. An isen-
tropic ow is, of course, an adiabatic ow (a ow in which there is no heat exchange)
in which viscous losses are negligible, i.e., it is an adiabatic frictionless ow. Although
no real ow is entirely isentropic, there are many ows of great practical importance in
which the major portion of the ow can be assumed to be isentropic. For example, in
internal duct ows there are many important cases where the effects of viscosity and
heat transfer are restricted to thin layers adjacent to the walls, i.e., are only important
in the wall boundary layers, and the rest of the ow can be assumed to be isentropic as
indicated in Figure 4.1.
Similarly in external ows, the effects of viscosity and heat transfer can be assumed
to be restricted to the boundary layers, wakes, and shock waves, and the rest of the ow
can be treated with adequate accuracy by assuming it to be isentropic as indicated in
Figure 4.2.
Even when nonisentropic effects become important, it is often possible to calculate the
ow by assuming it to be isentropic and to then apply an empirical correction factor to
the solution so obtained to account for the nonisentropic effects. This approach has been
frequently adopted in the past, for example, in the design of nozzles.
Governing Equations
By denition, the entropy remains constant in an isentropic ow. Using this fact, it was
shown in Chapter 2 that in such a ow:
p
c
ρ
γ
=
()
aconstant (4.1)
60 Introduction to Compressible Fluid Flow
If any two points, such as 1 and 2 shown in Figure 4.3, in an isentropic ow are consid-
ered, it follows from Equation 4.1 that
p
p
2
1
2
1
=
ρ
ρ
γ
(4.2)
Hence, since the general equation of state gives
p
T
p
T
T
T
p
p
1
11
2
22
2
1
2
1
1
2
ρρ
ρ
ρ
==
,i.e.,
(4.3)
it follows that in isentropic ow
T
T
p
p
2
1
2
1
1
2
1
1
=
=
−
−
ρ
ρ
γ
γ
γ
(4.4)
Flow
Nonisentropic
boundary layer
Isentropic
core flow
FIGURE 4.1
Region of duct ow that can be assumed to be isentropic.
Boundary layer
(nonisentropic)
Shock wave
(nonisentropic)
Wake
(nonisentropi
c)
Sup
ersonic flow
(isentropic)
Expansion waves
(isentropic)
Isentropic
Isentropic
Isentropic
Isentropic
FIGURE 4.2
Region of external ow that can be assumed to be isentropic.
61One-Dimensional Isentropic Flow
From this, it then follows, recalling that
aR
T=γ , that
a
a
T
T
p
p
2
1
2
1
1
2
2
1
1
2
2
1
1
2
=
=
=
−
−
ρ
ρ
γ
γ
γγ
(4.5)
The steady ow adiabatic energy equation is next applied between points 1 and 2. This
gives
cT
V
cT
V
pp
1
1
2
2
2
2
22
+= +
i.e.,
T
T
Vc
T
Vc
T
p
p
2
1
1
2
1
2
2
2
12
12
=
+
+
(/ )
(/ )
However,
V
cT
V
RT
R
c
M
pp
22
2
22
1
2
=
=
−
γ
γγ
Thus, it follows that
T
T
M
M
2
1
1
2
2
2
1
1
2
1
1
2
=
+
−
+
−
γ
γ
(4.6)
1
2
Steady, isentropic flow
p
2
, T
2
, V
2
, ρ
2
p
1
, T
1
, V
1
, ρ
1
FIGURE 4.3
Flow situation considered.
62 Introduction to Compressible Fluid Flow
This equation applies in adiabatic ow. If friction effects are also negligible, i.e., if the
ow is isentropic, Equation 4.6 can be used in conjunction with the isentropic state rela-
tions given in Equation 4.5 to give
p
p
M
M
2
1
1
2
2
2
1
1
1
2
1
1
1
2
1
=
+−
+−
−
()
()
γ
γ
γ
γ
(4.7)
and
ρ
ρ
γ
γ
γ
2
1
1
2
2
2
1
1
1
1
2
1
1
1
2
1
=
+−
+−
−
()
()
M
M
(4.8)
Lastly, it is recalled that the continuity equation gives
ρ
1
V
1
A
1
= ρ
2
V
2
A
2
which can be rearranged to give
ρ
ρ
2
1
2
1
1
2
=
V
V
A
A
(4.9)
The above equations are, together, sufcient to determine all the characteristics of one-
dimensional isentropic ow. It will be noted that the momentum equation was not used in
the above analysis of isentropic ow. As discussed in the previous chapter, in isentropic
ow, the momentum equation will always give the same result as the energy equation.
This can be illustrated using the integrated Euler equation (2.9), i.e.,
VV dp
2
2
1
2
1
2
22
0−+ =
∫
ρ
(4.10)
Since the relation between ρ and p is known in isentropic ow, the integral can be evalu-
ated as follows
dp dp
pp
p
ρ
ρ
ρ
γ
γ
γ
γ
γ
1
2
11
1
2
1
1
1
1
1
∫∫
==
−
()
(
/
p
pp
p
2
1
1
1
1
1
1
1
1
−
−
−
=
−
γγ
γ
γρ
)
−
−
p
p
2
1
1
1
1
γ
(4.11)

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