451An Introduction to Two-Dimensional Compressible Flow
Similarly,
∂
∂
=
∂
∂
T
yc
p
y
p
1
ρ
(14.23)
Substituting Equations 14.22 and 14.23 into Equations 14.20 and 14.21, respectively, gives
∂
∂
+
+
∂
∂
=
x
uv
C
T
x
p
22
2
0
i.e., since c
p
is assumed constant,
∂
∂
+
+
=
x
cT
uv
p
22
2
0
(14.24)
and
∂
∂
+
+
=
y
cT
uv
p
22
2
0
(14.25)
These two equations together indicate that the quantity
cT
uv
p
+
+
22
2
remains constant in an irrotational isentropic ow. This is the same as the result deduced
from energy considerations. Thus, in irrotational isentropic ow, momentum, and energy
conservation considerations give the same result. This was discussed in Chapter 4.
The Velocity Potential
The equations governing the velocity eld in two-dimensional irrotational ow are the
continuity equation (14.2) and the irrotationality equation (14.19), i.e.,
Equation 14.2:
∂
∂
+
∂
∂
=
x
u
y
v() ()
ρρ
0
452 Introduction to Compressible Fluid Flow
Equation 14.19:
∂
∂
−
∂
∂
=
v
x
u
y
0
The boundary conditions on these equations are that u and v are prescribed in the initial
ow, e.g., u = u
∞
, v = 0 well upstream of the body considered, and that the velocity compo-
nent normal to any solid surface is zero.
If a quantity, Φ, termed the velocity potential, is introduced, Φ being such that
u
x
v
y
=
∂
∂
=
∂
∂
ΦΦ
,
(14.26)
then it will be seen that the left-hand side of Equation 14.19 becomes
∂
∂∂
−
∂
∂∂
22
ΦΦ
xy
yx
This will always be zero so it follows that the velocity potential function, as dened by
Equation 14.26, satises the irrotationality equation (14.19). The continuity equation (14.2)
must then be used to solve for Φ. This equation gives
∂
∂
∂
∂
+
∂
∂
∂
∂
=
xx yy
ρρ
ΦΦ
0
i.e.,
ρ
ρρ
∂
∂
+
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
2
2
2
2
0
ΦΦ ΦΦ
xy
xx yy
(14.27)
However, since isentropic ow is being considered,
ρ=Cp
1
γ
(14.28)
Hence,
∂
∂
=
∂
∂
=
∂
∂
=
∂
∂
ρρ
x
Cp
p
p
xp
p
x
a
p
x
11
1
2
γγ
γ
Similarly, it can be shown that
∂
∂
=
∂
∂
ρ
y
a
p
y
1
2
(14.29)
453An Introduction to Two-Dimensional Compressible Flow
Using these in the momentum equations (14.5) and (14.6) and using Equation 14.26, then
gives
∂
∂
=−
∂
∂
∂
∂
+
∂
∂
∂
∂∂
ρρ
x
a
x
x
yxy
2
2
2
2
ΦΦ
ΦΦ
(14.30)
and
∂
∂
=−
∂
∂
∂
∂∂
+
∂
∂
∂
∂
ρρ
y
a
xxyy
y
2
22
2
ΦΦ
ΦΦ
(14.31)
Substituting these two equations into Equation 14.27 then gives
∂
∂
+
∂
∂
−
∂
∂
∂
∂
+
∂
∂
∂
∂
∂
∂
2
2
2
22
2
2
2
2
1ΦΦ ΦΦΦΦ Φ
xya
x
x
xy
xxy
a
yxxy y∂
−
∂
∂
∂
∂
∂
∂∂
+
∂
∂
∂1
2
2
2
2
ΦΦ ΦΦΦΦ
∂
=
y
2
0
i.e.,
∂
∂
+
∂
∂
−
∂
∂
∂
∂
+
∂
∂
∂
∂
∂
2
2
2
22
2
2
2
2
1
2
ΦΦ ΦΦΦΦ Φ
xya
x
x
xy
∂∂∂
+
∂
∂
∂
∂
=
xy y
y
ΦΦ
2
2
2
0
(14.32)
This can be written as
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂∂
+
∂
∂
2
2
2
22
2
2
2
2
2
2
2
1
2
ΦΦ ΦΦΦ
xya
u
x
uv
xy
v
y
= 0
(14.33)
Beside Φ, this equation contains the speed of sound a. An expression relating a to Φ is
therefore required. This is supplied by the energy equation, which, as discussed above,
gives (see Equation 14.14)
cT
uv
cT
pp
+
+
=
22
0
2
(14.34)
where T
0
is the stagnation temperature, which is a constant throughout the ow. Hence,
since
a
2
= γRT and C
p
= γR/(γ − 1)
454 Introduction to Compressible Fluid Flow
Equation 14.34 gives
au
va
222
0
2
1
20
+
−
+=
γ
()
i.e., using Equation 14.26,
aa
xy
2
0
2
2
2
1
2
=−
−
∂
∂
+
∂
∂
γ
Φ
Φ
(14.35)
Equations 14.32 and 14.35 together describe the variation of Φ in irrotational, isentropic,
ow. In low-speed ow (i.e., M ≪ 1), the density variation is negligible and Equation 14.27
gives
∂
∂
+
∂
∂
=
2
2
2
2
0
ΦΦ
xy
(14.36)
Hence, in low-speed ow, the variation of Φ is governed by Laplace’s equations. In
incompressible ow, then, it is relatively easy to determine Φ, Equation 14.36 being a linear
equation. In compressible ow, however, the compressibility effects give rise to the nonlin-
ear terms in Equation 14.32, i.e., terms such as (∂Φ/∂x)
2
(∂
2
Φ/∂x
2
), which involve the product
of functions of Φ. This makes the determination of Φ in compressible ows signicantly
more difcult than in incompressible ows. To solve for Φ in compressible ows, the fol-
lowing methods can be used:
• Full numerical solutions
• Transformation of variables to give a linear governing equation
• Linearized solutions
The second method is only applicable in a few situations and will not be discussed here.
Linearized solutions will be discussed in the next section and a very brief discussion of
numerical methods is given in a later section.
Linearized Solutions
To keep the drag low on objects in high-speed ows, the objects are usually kept relatively
slender to minimize the disturbance they produce in the ow. With such slender objects,
the differences between the values of the ow variables near the object and the values of
these variables in the freestream ow ahead of the object are small, e.g., consider the situ-
ation shown in Figure 14.8.
455An Introduction to Two-Dimensional Compressible Flow
If the components in the x- and y-directions of the velocity V are u
∞
+ u
p
and v
p
then, for
a slender object, the perturbation velocities u
p
and v
p
will be very small compared with u
∞
.
This assumption is the basis for the analysis given in the present section.
Now, in the undisturbed ow ahead of the object, u = u
∞
and v = 0, so the velocity poten-
tial is, by virtue of Equation 14.26, given by
∂
∂
=
∂
∂
=
∞
ΦΦ
x
u
y
an
d0
i.e.,
Φ = u
∞
x
The velocity potential in the ow will therefore be written as
Φ = u
∞
x + Φ
p
where Φ
p
is the perturbation velocity potential, which must be such that
u
x
v
y
p
p
p
p
=
∂
∂
=
∂
∂
ΦΦ
, (14.37)
Substituting the above relations into the potential function equation in the form given
in Equation 14.33 leads to
∂
∂
+
∂
∂
−+
∂
∂
++
∂
∞∞
2
2
2
22
2
2
2
2
1
2
ΦΦ Φ
pp
p
p
pp
xya
uu
x
uuv() ()
Φ
ΦΦ
p
p
p
xy
v
y
∂∂
+
∂
∂
=
2
2
2
0
i.e.,
∂
∂
+
∂
∂
−
+
∂
∂
+
∞
∞
2
2
2
2
2
2
2
2
1
ΦΦ Φ
pp pp
xy
u
a
u
u
x
221
2
2
2
+
∂
∂∂
+
∂
∞∞ ∞
u
u
v
uxy
v
u
pp pp
Φ ΦΦ
p
y∂
=
2
0
(14.38)
V
y
x
u
∞
v
∞
= 0
FIGURE 14.8
Flow situation considered.
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