446 Introduction to Compressible Fluid Flow
i.e.,
∂
∂
+
+
+
∂
∂
+
+
x
ucT
uv
y
vcT
u
pp
ρρ
() (
22 2
2
vv
2
2
0
)
=
(14.8)
This equation can be written as
cT
uv
x
u
y
vu
x
cT
pp
+
+
∂
∂
+
∂
∂
+
∂
∂
()
() ()
22
2
ρρ
++
+
+
∂
∂
+
+
=
()
()
uv
v
y
cT
uv
p
22 22
22
0
However, by virtue of the continuity equation 14.2, the rst term in this equation is zero.
Hence, the conservation of energy equation can be written as
u
x
cT
uv
v
y
cT
uv
pp
∂
∂
+
+
+
∂
∂
+
+
=
()
()
22 22
22
0
(14.9)
Now consider the change in any quantity Z over a short length ds of a streamline as
shown in Figure 14.3.
The change will be given by
dZ
Z
x
dx
Z
y
dy=
∂
∂
+
∂
∂
(14.10)
The rate of change in Z as it moves along the streamline is dZ/dt where dt is the time
taken for the gas particles to move from A to B. Hence, the rate of change in Z is given by
∂
∂
=
∂
∂
∂
∂
+
∂
∂
∂
∂
Z
t
x
t
Z
x
y
t
Z
y
(14.11)
A
B
dy
ds
Streamline
dx
FIGURE 14.3
Changes over length of streamline considered.
447An Introduction to Two-Dimensional Compressible Flow
i.e., since dx/dt = u and dy/dt = v
∂
∂
=
∂
∂
+
∂
∂
Z
t
u
Z
x
v
Z
y
(14.12)
If the quantity Z is not changing along the streamline, i.e., if dZ/dt = 0, this equation
shows that
u
Z
x
v
Z
y
∂
∂
+
∂
∂
= 0
(14.13)
Comparing this with Equation 14.9 then shows that the conservation of energy equation
indicates that the quantity
cT
uv
p
+
+
()
22
2
is constant along a streamline. Now, in most ows, the ow can be assumed to originate
from a region of uniform ow as indicated in Figure 14.4.
In such cases, the value of [c
p
T + (u
2
+ v
2
)/2] will initially be the same on all streamlines.
The conservation of energy equation then shows that this quantity will remain the same
everywhere in the ow. Hence, if the velocity components are determined from the continu-
ity and momentum equations, the energy gives the temperature at any point in the ow as
cT
uv
cT
uv
cT
pp p
+
+
=+
+
=
22
1
1
2
1
2
0
22
(14.14)
the subscript 1 referring to conditions in the initial uniform ow and T
0
being the stagna-
tion temperature in this initial uniform ow.
Uniform
velo
city and
temp
erature
Uniform
velocity and
temperature
FIGURE 14.4
Uniform upstream velocity.
448 Introduction to Compressible Fluid Flow
Vorticity Considerations
As previously discussed, attention is here being directed to ows in which the effects of
viscosity are negligible. With this in mind, consider the uid particles in the ow. Only if
there are tangential forces acting on the surface of these particles can there be a change in
the net rate at which the particles rotate. This is illustrated in Figure 14.5.
However, the only source of a tangential force is viscosity. Hence, if viscous effects are
neglected, there can be no change in the rate at which the uid particles rotate. If the ow
originates in a uniform freestream (see Figure 14.4), the uid particles will have no initial
rotation, and so they will have no rotation anywhere in the ow.
Consider a uid particle that is initially rectangular in shape with side lengths dx and dy.
As it moves through the ow, this particle will distort as indicated in Figure 14.6.
T
angential
forces
Angular
rotation
Gas
particle
Streamline
FIGURE 14.5
Changes in rotational motion of particles produced by the viscous stresses acting on them. Tangential stresses
are required if the rotational motion is changing as the particle moves along the streamline.
B
C
D
A
x
A
dx
dy
y
B
D
C
∆x
∆
y
φ
y
φ
x
FIGURE 14.6
Distortion of uid particles.
449An Introduction to Two-Dimensional Compressible Flow
The net amount by which the particle has rotated in the counter clockwise direction is
(ϕ
x
– ϕ
y
)/2. The net rate at which the uid particles are rotating (i.e., the vorticity) is then
given by
ω
φ
φ
=
∂
∂
−
∂
1
2
x
y
td
t
(14.15)
Now, consider the velocities of corner points A, B, and D as indicated in Figure 14.7.
From this, it follows that
∂
∂
=
∂∂
∂
=
+∂ ∂−
=
∂
∂
φ
x
t
yt
x
vvxv
dx
v
x
()
()
∆/
/
(14.16)
and
∂
∂
=
∂∂
∂
=
+∂ ∂−
=
∂
∂
φ
y
t
xt
x
vvyu
dy
u
y
()
()
∆/
/
(14.17)
Substituting these into Equation 14.15 then gives the rate of uid particle rotation as
ω=
∂
∂
−
∂
∂
1
2
v
x
u
y
(14.18)
D
A
B
v
u
x
v
y
u
+
+
∂u
∂v
∂y
dy
∂x
dx
∆y
∆x
φ
x
φ
y
FIGURE 14.7
Velocities of corner points of uid particle considered.
450 Introduction to Compressible Fluid Flow
If viscous forces are negligible and if the ow initially has no rotation, it follows that
everywhere in the ow
∂
∂
−
∂
∂
=
v
x
u
y
0
(14.19)
Using this, the x momentum equation, Equation 14.5 can be written as
ρρu
u
x
v
v
x
p
x
∂
∂
+
∂
∂
=−
∂
∂
i.e.,
∂
∂
+
=−
∂
∂x
uv p
x
22
2
1
ρ
(14.20)
Similarly, the y momentum equation, Equation 14.6, can be written using Equation 14.19 as
ρρu
u
y
v
v
y
p
y
∂
∂
+
∂
∂
=−
∂
∂
i.e.,
∂
∂
+
=−
∂
∂y
uv p
y
22
2
1
ρ
(14.21)
However, since isentropic ow is being considered,
TCp=
−γ
γ
1
Hence,
∂
∂
=
−∂
∂
=
−∂
∂
=
−∂
∂
−
T
x
Cp
p
p
x
T
p
p
xR
p
x
γ
γ
γ
γ
γ
γ
γ
γ
1
11 11
ρ
i.e.,
∂
∂
=
∂
∂
T
xc
p
x
p
1
ρ
(14.22)
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