Geometry and violent events in turbulent pair dispersion

Abstract

The statistics of Lagrangian pair dispersion in a homogeneous isotropic flow is investigated by means of direct numerical simulations. The focus is on deviations from the Richardson eddy-diffusivity model and in particular on the strong fluctuations experienced by tracers. Evidence is obtained that the distribution of distances attains an almost self-similar regime characterized by a very weak intermittency. The timescale of convergence to this behavior is found to be given by the kinetic energy dissipation time measured at the scale of the initial separation. Conversely the velocity differences between tracers are displaying a strongly anomalous behavior whose scaling properties are very close to that of Lagrangian structure functions. These violent fluctuations are interpreted geometrically and are shown to be responsible for a long-term memory of the initial separation. Despite this strong intermittency, it is found that the mixed moment defined by the ratio between the cube of the longitudinal velocity difference and the distance attains a statistically stationary regime on very short timescales. These results are brought together to address the question of violent events in the distribution of distances. It is found that distances much larger than the average are reached by pairs that have always separated faster since the initial time. They contribute a stretched exponential behavior in the large-value tail of the inter-tracer distance probability distribution. At large times this tail is found to be closer to a pure exponential than to the form obtained from the Richardson diffusive approach. At the same time, the distance distribution displays a time-dependent power-law behavior at very small values, which is interpreted in terms of fractal geometry. It is argued and demonstrated numerically that the exponent converges to one at large time, again in conflict with Richardson’s distribution.

Keywords:

• turbulent transport
• relative dispersion

Acknowledgements

We are grateful to L. Biferale, M. Cencini, G. Falkovich, A. Frishmann, G. Krstulovic, A. Lanotte, and S.S. Ray for useful discussions and remarks. Access to the IBM BlueGene/P computer JUGENE at the FZ Jülich was made available through the XXL-project HBO28. The research leading to these results has received funding from DFG-FOR1048 and from the European Research Council under the European Community’s Seventh Framework Program (FP7/2007–2013, Grant Agreement No. 240579).