169
7
Expansion Waves: Prandtl–Meyer Flow
Introduction
The discussion given in Chapters 5 and 6 was concerned with waves that involved an
increase in pressure, i.e., with shock waves. In this chapter, attention will be given to
thetypes of waves that are generated when there is a decrease in pressure. For example,
the type of wave that is generated when a supersonic ow passes over a convex corner
andthe type of wave that is generated when the end of a duct containing a gas at a pres-
sure that is higher than that in the surrounding air is suddenly opened will be discussed
in this chapter. These two situations are illustrated in Figure 7.1.
Steady supersonic ows around convex corners will rst be addressed in this chapter.
Attention will then be given to unsteady ows.
Prandtl–Meyer Flow
In the previous chapter, supersonic ow around a concave corner, i.e., a corner involving
a positive angular change in ow direction, was considered. It was indicated there that
the ow over such a corner was associated with an oblique shock wave, this shock wave
originating at the corner when it is sharp. Consider, now, the ow around a convex corner
as shown in Figure 7.2. To determine whether an oblique shock wave also occurs in this
case, it is assumed that it does occur, a sharp corner being considered for simplicity as
shown in Figure 7.2.
Consider the velocity components indicated in Figure 7.2. For the reasons given in the
previous chapter, L
1
= L
2
, and since V
2
must be parallel to the downstream wall, geometri-
cal considerations show that N
2
> N
1
. However, N
2
and N
1
must be related by the nor-
mal shock wave relations, and in dealing with normal shock waves, it was shown that an
expansive shock was not possible since it would violate the second law of thermodynam-
ics. It is therefore not possible with an oblique shock wave for N
2
to be greater than N
1
and
the ow over a convex corner cannot therefore take place through an oblique shock.
To understand the actual ow that occurs when a supersonic ow passes around a con-
vex corner, consider what happens, in general, when the ow is turned through a differ-
entially small angle, dθ, this producing differentially small changes dp, dρ, and dT in the
pressure, density, and temperature, respectively. The present analysis applies whether dθ
is positive or negative, i.e., whether the corner is concave or convex, the changes through
the differentially weak Mach wave produced being isentropic (see later). By the reasoning
170 Introduction to Compressible Fluid Flow
previously given, the velocity component parallel to the wave, L, is unchanged by the
wave. Hence, considering unit area of the wave shown in Figure 7.3, the equations of con-
tinuity and momentum give
ρN = (ρ + dρ)(N + dN)
i.e.,
ρ dN + N dρ = 0 (7.1)
higher-order terms having been neglected, and
p − (p + dp) = ρN[(N + dN) − N]
M > 1
Suddenly
opened
p > p
a
p
a
?
?
FIGURE 7.1
Flows involving a pressure decrease.
Wave
L
1
L
2
N
1
N
2
V
2
V
1
θ
FIGURE 7.2
Assumed ow around convex corner.
171Expansion Waves: Prandtl–Meyer Flow
i.e.,
−dp = ρN dN (7.2)
Substituting for dN from Equation 7.2 into Equation 7.1 gives
N
dp
d
2
=
ρ
(7.3)
Now, in the limiting case of a very weak wave that is being considered here, dp/dρ will
be equal to the square of the upstream speed of sound, i.e.,
N
2
= a
2
or N = a (7.4)
This is indicated in Figure 7.4. Further, since L is unchanged by the presence of the distur-
bance, it follows that
(V + dV) cos (α − dθ) = Vcosα
i.e.,
(V + dV)(cosαcosdθ + sinαsindθ) = Vcosα
Expanding this equation and ignoring higher-order terms then gives
Vcosα + Vsinα dθ + dVcosα = Vcosα
Therefore,
dV
V
d
d
M
=− =
−
−
tanαθ
θ
2
1
(7.5)
p
T
ρ
L
N
V
L
V + dV
p + dp
ρ + dρ
T + dT
dθ
N + dN
FIGURE 7.3
Changes produced by a weak wave.
172 Introduction to Compressible Fluid Flow
Further, since the energy equation gives
2
1
2
1
2
γ
γρ
γ
γρρ−
+=
−
+
+
p
V
pdp
d
++
()
VdV
2
i.e., ignoring higher-order terms,
2
1
2
1
1
2
γ
γρ
γ
γ−
+=
−
+
p
V
pdp
p
ρ
−−
++
d
VV
dV
ρ
ρ
2
2
The following applies
2
1
2
γ
γρ ρ
ρ
−
−
=−
pdp
p
dp
p
pd
dp
VdV
(7.6)
However, by the previously made assumptions
γ
ρρ
pdp
d
a
==
2
Thus, Equation 7.6 becomes
2
1
1
1
2
2
a
dp
p
VdV
γγ
−
−
=−
Wave
V
N = a
α
α
FIGURE 7.4
Upstream velocity components near a weak wave.
173Expansion Waves: Prandtl–Meyer Flow
i.e.,
dp
p
VdV
a
M
dV
V
=− =−
γ
γ
2
2
(7.7 )
or using Equation 7.5
dp
p
M
M
d=
−
γ
θ
2
2
1
(7.8)
Further, since
dd
dp
dp
p
p
a
dp
p
ad
p
p
ρ
ρ
ρ
ργ
γ
== =
11
2
2
it follows, using Equation 7.8, that
dM
M
d
ρ
ρ
θ=
−
2
2
1
(7.9)
Similarly,
ds
R
p
p
=
−
−
−
1
11
2
1
2
1
γ
γ
γ
ρ
ρ
ln ln
=
−
+
−
−
+
1
1
1
1
1
γ
γ
γ
ρ
ρ
ln ln
dp
p
d
=
−
−
−
−
−
=
1
1
1
1
1
0
2
2
2
2
γ
γθ γ
γ
θMd
M
Md
M
(7.10)
which shows that there is no change in entropy across the weak wave being considered.
Lastly, since
M
V
a
V
p
2
2
2
2
==
ρ
γ
the differential change in M, i.e., dM, is given by
22
2
MdMVdV
p
V
p
d
V
p
dp=
+
−
ρ
γγ
ρ
ρ
γ

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