Civil Engineering Reference
In-Depth Information
Equation [3.74] defines the Lyapunov exponent for finite
integration times
()
. We can show that
tends toward
T
t
x
T
0
the classical Lyapunov exponent as
[STR 00].
T
→∞
The detection of Lagrangian coherent structures is based
on the local maxima of the field
()
[HAL 01]. Integration
for negative times leads to attractive structures,
whereas the local maxima of
T
t
x
0
T
<
0
()
for denote
repulsive structures. Green
et al.
[GRE 07] applied this
strategy to the detection of vortex structures in wall
turbulence. Figure 3.31 compares the Lagrangian technique
linked to the field
T
t
x
T
>
0
0
()
with the Eulerian detection based
on the criterion analyzed in section 3.6.2. We can see that
the two methods are qualitatively similar. Identification by
the local maximum of the Lyapunov coefficient for finite
periods of time gives us more fine details in comparison to
the criterion, particularly in terms of definition of the
boundaries of the structures.
All of the type-
Q
or criteria are Galilean invariants, in
relation to a framework without rotation. Haller [HAL 05]
considers the problem of identification of vortex structures
with regard to the principle of objectivity in a changing
framework
T
t
x
0
Q
Q
Δ
()
y
=
T()x
t
+
b()
t
, where
T
t
is an orthogonal
proper tensor that depends on time and
b
is an unsteady
translational vector. The definition of a vortex given by
Haller [HAL 05] arises from Lagrangian considerations but
uses Eulerian quantities. This author [HAL 05] shows that
the trajectories are hyperbolic if the acceleration tensor of
()
(
)
S
≡
S
ij
= ∂
U
i
∂
x
j
+ ∂
U
j
∂
x
i
2
d
dt
=+∇+∇+∇
(
)
(
)
(
)
t
[3.75]
S
AS
S J
S
J
J
S
is positive definite on a surface
with zero strain
S
0
A
S
Z
=
Z
moving with the flow. This particular set is denoted by
.
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