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2D Turbulence (PhD)

PhD thesis

Dispersion of passive tracers in an experimental  two-dimensional turbulent flow

We present an experimental study of the dispersion of passive tracers in a forced two-dimensional turbulence. The experiments are performed in electromagnetically driven flows, using thin, stably-stratified layers.  Instantaneous velocity fields are measured using particle imaging velocimetry techniques.
The dispersion of passive tracers is studied in the direct enstrophy cascade and in the inverse energy cascade, in accordance with the conjecture formulated by Kraichnan in 1967 about the double cascade mechanism in 2D forced turbulence.

In a first approach, the evolution of a blob of fluoresceine is studied in both cascade. The enstrophy cascade has the particularity  to concentrate the strain at large scales, the velocity field is then Taylor expandable at first order. Due to the simplicity of the velocity field, it is possible to developp analytic exact solutions. We show that the spectrum of the concentration fluctuations follows Batchelor's spectrum k-1. In addition, analytic solutions found by Chertkhov et al. are in good agreement with our experiment : exponential tails for both fluctuation distributions and increment distributions, logarithmic behaviour for the structure function of order 2.In the inverse energy cascade, we show that Corrsin-Obukhov analysis is in good agreemnt with our experiment concerning statistics of order 2. A further analysis shows that higher orders deviate strongly from Kolmogorov arguments, and the exponents of structure functions saturate to a constant value.

An other approach is a lagrangian one. Pairs of particles are followed along their trajectories. This study is done in the inverse energy cascade. The statistical properties of the process are obtained by numerically integrating the Lagrangian trajectories of the particles, using the experimentally obtained velocity fields.  The hyperdiffusive Richardson t3 law is observed, and strongly non-gaussian behavior is obtained for the distributions of pair separation.  The process is shown to be self-similar in time and that strong temporal correlations are present .  The observations, which fit well in the Kolmogorov framework, jeopardize the relevance of the Lévy walk approach. 

Jullien et al., Phys. Rev. Lett., 1999
Paret et al., Phys. Rev. Lett., 1999
Jullien et al., Phys. Rev. Lett., 2000
Jullien et al., Phys. Rev. E, 2001
Jullien, Phys. of Fluids, 2003

Batchelor dispersion in the 2D enstrophy cascade :

YouTube Video

Dispersion of passive tracer in the inverse energy cascade :

YouTube Video