### Statistics of Lagrangian velocity
in fully developed turbulence

*Jean-François
Pinton, *

*Laboratoire
de Physique, *

*Ecole
Normale Supérieure de Lyon (France)*

pinton@ens-lyon.fr

The
understanding of the dynamics of turbulent flows has been a major
goal for fundamental and applied fluid dynamics research for almost a
century now. On the fundamental side, turbulence is the head figure
of a non-linear dissipative system with a very large number of
degrees of freedom. On the applied side, the properties of turbulent
flows govern the dispersion of pollutants, the physics of mixing,
etc. In very recent years, analytical and numerical studies have
shown that progress can be made by analyzing the flow properties in
the reference frame of a moving fluid particle (the Lagrangian
viewpoint), instead of considering the velocity field at a fixed
point in space (the Eulerian viewpoint).

In
order to completely address turbulence in the Lagrangian frame, one
needs to describe the dynamics over the entire range of scales of
motion. We have developed such technique, based on sonar principles,
to measure directly the velocity of individual small tracer particles
over long times. We have analyzed the statistics of the Lagrangian
velocity of single particles for flows with turbulent Reynolds
numbers between 100 and 1100. We observe that the Lagrangian
spectrum has a Lorentzian form in agreement with a Kolmogorov-like
scaling in the inertial range. The probability density function (PDF)
of the velocity time increments displays a change of shape from
quasi-Gaussian a integral time scale to stretched exponential tails
at the smallest time increments. This intermittency, when measured
from relative scaling exponents of structure functions, is more
pronounced than in the Eulerian framework.

Another
important observation is that in the erratic course of the particle
motion, infinitesimal changes of velocity occur with `random'
decorrelated directions but with a correlation of magnitude which
persists over the longest times of the flow. Using an analogy with
the properties of Multifractal Random Walks, we propose that this
feature is essential in the development of intermittency in
turbulence.