Civil Engineering Reference
In-Depth Information
2
∂ω
⎛
∂ ω
1
∂ω
⎞
x
=
ν
x
+
x
[5.4]
⎜
⎟
2
∂
t
∂
r
r
∂
r
⎝
⎠
We find the solution
Γ
⎛
r
2
⎞
ω
=
exp
−
⎜
⎟
x
4
νπ
t
4
ν
t
⎝
⎠
⎛
⎞
Γ
⎛
r
2
⎞
u
=
1exp
−
−
[5.5]
⎜
⎟
⎜
⎟
⎜
⎟
θ
2
π
r
4
ν
t
⎝
⎠
⎝
⎠
u
=
0
r
on the initial condition of a filament of vorticity triggered at
time
in
the
center
of
the
structure
t
=
0
(
) (
) ( )
. In these relations,
is the
Γ
ω
rt
,
==Γ
0
2
π
r
δ
r
x
()
recirculation of the vortex and
is the Dirac function.
δ
r
(
)
The
streamwise
disturbance
ur
++
,;
θ
t
+
rendered
dimensionless by the viscosity and the shear
k
is finally
governed by the linear differential equation
⎛
⎞
+
⎛ ⎞
+
2
+
2
+
+
2
+
∂
u
Re
r
∂
u
∂
u
1
∂
u
1
∂
u
+
Γ
1exp
−
−
−
−
−
⎜
⎟
⎜ ⎟
⎜
⎟
∂
t
+
r
+
2
4
t
+
∂θ
∂
r
+
2
r
+
∂
r
+
r
+
2
∂θ
2
⎝ ⎠
⎝
⎠
[5.6]
⎛
⎞
⎛ ⎞
+
2
Re
r
(
)
=−
Γ
+
1exp
−
−
cos
θα θ
−
in
⎜
⎟
⎜ ⎟
⎜
⎟
r
4
t
+
⎝ ⎠
⎝
⎠
where the ratio of the different shears
appears as
an additional parameter. The scales emanate from the
viscosity and the shear
k
. The rendering in dimensionless
form discussed above must not be confused with the
conventional inner variables. It must again be stressed that
the walls are infinitely distant, and that no length scale is
imposed in direction
y
. However, the disturbance
engendered by the vortex over short times
α =
kk
z
y
should
be representative of the physics in proximity to the wall. We
+
t
<<
O
(1)
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