A statistical conservation law in two and three dimensional turbulent flows

TL;DR

This paper provides evidence for a statistical conservation law involving the inverse power of particle separation in 2D and 3D turbulence, with implications for understanding flow properties and validating experimental conditions.

Contribution

It demonstrates the conservation law in 2D and 3D turbulence through simulations and experiments, extending previous theoretical proposals.

Findings

Conservation of $ angle R^{-d} angle$ in 2D and 3D turbulence.

Loss of conservation when particles exit linear flow regime.

Existence of a fluctuation relation in 2D turbulence.

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 2D we show that, as an extension of the conservation law, a Evans-Cohen-Morriss/Gallavotti-Cohen type fluctuation relation exists. We also analyse 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 $\left<R^{-3}\right>$, 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 $\left<R^{-d}\right>$, demonstrated here, can be used as a check of isotropy, incompressibility and flow dimensionality in numerical and laboratory experiments that focus on small scales.