Statistical conservation law in two- and three-dimensional turbulent flows

Anna Frishman, Guido Boffetta, Filippo De Lillo, and Alex Liberzon
Phys. Rev. E 91, 033018 – Published 26 March 2015

Abstract

Particles in turbulence live complicated lives. It is nonetheless sometimes possible to find order in this complexity. It was proposed in Falkovich et al. [Phys. Rev. Lett. 110, 214502 (2013)] that pairs of Lagrangian tracers at small scales, in an incompressible isotropic turbulent flow, have a statistical conservation law. More specifically, in a d-dimensional flow the distance R(t) between two neutrally buoyant particles, raised to the power d and averaged over velocity realizations, remains at all times equal to the initial, fixed, separation raised to the same power. In this work we present evidence from direct numerical simulations of two- and three-dimensional turbulence for this conservation. In both cases the conservation is lost when particles exit the linear flow regime. In two dimensions we show that, as an extension of the conservation law, an Evans-Cohen-Morriss or Gallavotti-Cohen type fluctuation relation exists. We also analyze data from a 3D laboratory experiment [Liberzon et al., Physica D 241, 208 (2012)], finding that although it probes small scales they are not in the smooth regime. Thus instead of R3, we look for a similar, power-law-in-separation conservation law. We show that the existence of an initially slowly varying function of this form can be predicted but that it does not turn into a conservation law. We suggest that the conservation of Rd, demonstrated here, can be used as a check of isotropy, incompressibility, and flow dimensionality in numerical and laboratory experiments that focus on small scales.

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  • Received 12 January 2015

DOI:https://doi.org/10.1103/PhysRevE.91.033018

©2015 American Physical Society

Authors & Affiliations

Anna Frishman1, Guido Boffetta2, Filippo De Lillo2, and Alex Liberzon3

  • 1Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
  • 2Department of Physics and INFN, University of Torino, via P. Giuria 1, 10125 Torino, Italy
  • 3Turbulence Structure Laboratory, School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel

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Issue

Vol. 91, Iss. 3 — March 2015

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