Civil Engineering Reference
In-Depth Information
The vorticity at the core of the vortex is
Γ
(
)
(
)
2
2
ω
=
exp
−
γ
r
2
ν
=
Re
exp
−
γ
r
2
ν
z
Γ
2
πν
The axial velocity stretches the vortex tube and
intensifies the vorticity. That vorticity diffuses under the
influence of viscosity, but the viscous spreading is, at the
same time, restricted by radial convection. The
circumferential velo
city
reaches its maximum at
approximately a distance that can be considered
to be the boundary of the vortex. We can see that
u
z
r
l
=
1.6
νγ
is
r
l
independent of the Reynolds number
.
Re
Γ
As the velocity field is a known entity, we can calculate
the invariants of the velocity gradient tensor and
analytically determine the different detection criteria. [CHA
05] gives us the following expressions:
()
(
)
2
2
*
QRe g
=
γ
γ
−
3
Γ
()
6
2
*
γ
Re g
γ
2
()
Γ
⎡
⎤
Δ=
9
+
Re g
2
γ
*
⎣
⎦
Γ
27
⎡
Re
(
)
⎤
()
(
)
*2
2
2
*
*2
−
γ
/ 2
λγ
=
1
−
Re g
γ
−
Γ
2
+
γ
e
−
2
[3.69]
⎢
⎥
2
Γ
*2
γ
⎣
⎦
λ
λ
1
cR
=−
()
()
*
Re
g
γ
cI
Γ
λγ
=
Re
g
γ
*
cI
Γ
(
)
()
The function
is expressed by way of
g
γ
*
=
g
γ
/
ν
1
(
)(
)
()
(
)
*2
*2
*
*2
−
γ
/ 2
−
γ
/ 2
g
γ
=
1
+
γ
e
−
1 1
−
e
*4
γ
The reason why we are reiterating these expressions here
is to demonstrate the extent to which the various criteria
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