Civil Engineering Reference
In-Depth Information
for the wavenumber
. This relation clearly shows that
the wall-normal vorticity increases indefinitely over time in
these conditions. In reality, this increase is limited by the
viscosity. The nonlinear interactions contribute directly to
the increase in energy by way of redistribution between the
different components of the velocity field.
α =
0
The source term for the spatiotemporal evolution of the
wall-normal vorticity is
ˆ (,
+++
, and its behavior is
described by the linear equations governing the disturbances
to the streamwise velocity
vyt
)
u
and the wall-normal velocity
v
.
The complete linear equations, which govern the
disturbances of the streamwise velocity
u
, the wall-normal
velocity
v
and spanwise velocity
w
′
are expressed in inner
variables by
+
+
+
+
∂
u
'
∂
u
'
dU
∂
p
'
1
+
+
2
+
+
U
+
v
'
+
=
∇
u
'
∂
t
+
∂
x
+
dy
+
∂
x
+
Re
τ
∂
v
'
+
∂
v
'
+
∂
p
'
+
1
+
U
+
+
=
∇
2
v
'
+
[5.74]
+
+
+
∂
t
∂
x
∂
y
Re
τ
∂
w
'
+
∂
w
'
+
∂
p
'
+
1
+
2
+
+
U
+
=
∇
w
'
+
+
+
∂
t
∂
x
∂
z
Re
τ
where
. The divergence of these equations
combined with the continuity equation gives rise to
2
∂∂
+
2
x
2
∇≡
dU
+
∂
∂
v
'
+
[5.75]
∇ −
2
p
'
+
+
+
dy
x
If we take the divergence of the equation governing
v
+
in
equation [5.74], and combine it with equation [5.75] and the
adopted form of equation [5.70], we obtain
Search WWH ::
Custom Search