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that we have analyzed are virtually different for a generic
case such as the Burgers vortex. Let us first consider the
0
criterion. In light of the first equation in the system
[3.69],
Q
>
detects only those vortices whose intensity is greater than a
certain threshold. It is clear from relation [3.44] that
if
. In other words, the criterion
Q
>
0
Re
Γ
>
23
Q
>
0
Q
=
0
corresponds to the “compactness”
technique also filters the structures whose Reynolds number
is
Re
. The
λ <
0
λλ=
13
2
cR
cI
2
. Figure 3.29, adapted from [CHA 05], shows the
dimensionless radius
r
Γ
<
*
detected by different techniques, as
a function of the Reynolds number
Re
γ
of the Burgers vortex.
Each technique has its own maximum Reynolds number
of a vortex that it cannot detect, such as
Γ
for
Re
Γ
=
23
Q
>
0
or
Re
0
. The radius of the structure, which tends
toward the boundary
2
for
Γ
=
λ
2
<
is sufficiently large,
is correctly identified by both the previous techniques and
that of the criterion
*
when
Re
r
γ
≈
1.6
Γ
λ
>
0
. Controlling the compactness
cI
enables us to extend the validity zone of the
detection, but a threshold such that
λλ
cR
cI
is
necessary to identify structures of infinitely low intensity, a
result which is easy to understand from a physical point of
view. Apart from a few subtle details, all of the techniques
yield results that are qualitatively similar when
λλ
∝−
1
Re
→−∞
cR
cI
Γ
Re
Γ
>
10
.
3.12. Summary
The results arising from the detection criteria analyzed in
the above sections differ by the extent of the boundary of the
vortex core that they identify. As previously indicated, there
are no direct relations between the techniques using the
invariants , , and and . To express this fact
differently, suppose that the detection is based on
Δ
λ
λ
λ
Q
2
cI
cR
,
Q
> σ
Q
where
is a threshold remaining to be defined. Thus, it is
σ
Q
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