Statistics of Lagrangian velocity in fully developed turbulence

Jean-François Pinton,

Laboratoire de Physique,

Ecole Normale Supérieure de Lyon (France)

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.