next
up
previous
contents
Next: Numerical results
Up: Lagrangian description of the
Previous: Structure functions and scaling
Contents
It has to be noticed that the active nature of $\omega$ has been
completely ignored in the above arguments:
the prediction for the structure function given by Eq. (2.31)
is identical to Eq. (2.29) obtained for the passive
scalar with finite life-time.
The crucial hypothesis in the derivation of Eq. (2.31) is
that we have assumed that the statistics of trajectories is
independent of the forcing $f_{\omega}$:
\begin{displaymath} \langle \Omega^p e^{-p \alpha T_L(r)} \rangle_{f,T_L(r)} = ... ...mega^p \rangle_f \langle e^{-p \alpha T_L(r)} \rangle_{T_L(r)} \end{displaymath}
(2.35)
While for the passive scalar this is trivially true, because the
forcing of the passive field has no relation with the velocity field,
and cannot influence fluid trajectories, in the case of vorticity
this is quite a nontrivial assumption, since it is clear that forcing
does affect large-scale vorticity and thus influence velocity
statistics, but it can be justified by the following argument.
The random variable $\Omega$ arises from forcing contributions
along the trajectories at times [画像:$s<t-T_L(r)$], when the distance
between the two fluid particles is larger than the
forcing correlation length $L,ドル whereas the
exit-time $T_L$ is clearly determined by the evolution of the strain
at times [画像:$t-T_L(r)<s<t$].
Since the correlation time of the strain is $\alpha^{-1},ドル for
\begin{displaymath} T_{L}(r) \gg 1/\alpha \end{displaymath}
(2.36)
we might expect that $\Omega$ and [
画像:$T_L(r)$] be statistically independent.
This condition can be translated in terms of the finite-time Lyapunov
exponent as:
\begin{displaymath} r \ll L \exp(-\gamma/\alpha) \end{displaymath}
(2.37)
and thus at sufficiently small scales it is reasonable to consider
$\omega$ as a passive field.
We remark that, were the velocity field non-smooth, the exit-times would be
independent of $r$ in the limit $r \to 0$ and the above argument
would not be relevant. Therefore, the smoothness of
the velocity field plays a central role in the equivalence of
vorticity and passive scalar statistics.
next
up
previous
contents
Next: Numerical results
Up: Lagrangian description of the
Previous: Structure functions and scaling
Contents
Stefano Musacchio
2004年01月09日
