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at the origin when is sufficiently large in isotropic
homogenous turbulence. Thus, the trajectories in the phase
plane tend essentially toward the origin in this scenario. The
behavior is more complex when is small, given the
proximity of the two critical points. Thus, we observe a dual
behavior between the trajectories, which tend, in a stable
manner, toward the origin, and others which tend toward
the saddle point
β
β
(
)
2 3
3, 2
and away from it.
−
ββ
These results are based on the fact that in
isotropic homogenous turbulence, and obviously cannot be
directly transposed to wall-bounded flows. The Lagrangian
DNS applied to a turbulent boundary layer with a low
Reynolds number , conducted by Chacin and
Cantwell [CHA 00], show that there is a qualitative
similarity between the trajectories in isotropic
homogenous turbulence and the wall flow outside of the
viscous sublayer.
19
Figure 3.28, adapted from [CHA 00],
shows the trajectories in the phase plane in the case of
isotropic homogenous turbulence with a high Reynolds
number, represented by a high-value parameter , and in
the case of turbulence in the buffer sublayer. The trajectories
have fairly similar characteristics, and tend toward
the asymptotically stable improper node at the
origin . The same similarity can be seen between
the isotropic homogenous turbulence and that of the
logarithmic sublayer. In the viscous sublayer, however, the
topology that emerges from the use of DNS is closer to the
characteristics of a focal point at the origin than a stable
node. The zone where viscosity is dominant and
characteristic scales are dissipative cannot be described by
an Eulerian model, and the hypothesis
H
ij
∝−
a
ij
Re
θ
=
300
Q
−
R
Q
−
R
β
(
Q
,
R
)
=
(0, 0)
is absolutely
H
ij
∝−
a
ij
not valid in the viscous sublayer.
19 With the exception of the outer layer of wake where the local shear
decreases drastically. See [CHA 00] for further details.
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