Two-time Lagrangian velocity correlation function for particle pairs in two-dimensional inverse energy-cascade turbulence

TL;DR

This study numerically investigates a self-similar two-time Lagrangian velocity correlation function in 2D inverse-cascade turbulence, revealing initial-separation dependence and questioning the universality of the Richardson-Obukhov $t^3$ law at high Reynolds numbers.

Contribution

It proposes a new self-similar form of the correlation function with scaling exponents dependent on initial separation, verified through direct numerical simulations.

Findings

Scaling exponents depend on initial separation.

Reynolds number effects influence the correlation function.

Potential non-universality of the Richardson-Obukhov $t^3$ law at high Reynolds numbers.

Abstract

We numerically investigate a two-time Lagrangian velocity correlation function (TTLVCF) for particle pairs in two-dimensional energy inverse-cascade turbulence. We consider self similarity of the correlation function by means of incomplete similarity. In this framework, we propose a self-similar form of the correlation function, whose scaling exponents cannot be determined by only using the dimensional analysis based on the Kolmogorov's phenomenology. As a result, the scaling laws of the correlation function can depend on the initial separation. This initial-separation dependency is frequently observed in laboratory experiments and direct numerical simulations of the relative dispersion, which is directly related to the correlation function, at moderate Reynolds numbers. We numerically verify the self-similar form by direct numerical simulations of two-dimensional energy inverse-cascade turbulence. The involved scaling exponents and the dependencies on finite Reynolds number effects are determined empirically. Then, we consider implication of the scaling laws of the correlation function on the relative dispersion, i.e. the Richardson-Obukhov $t^3$ law. Our results suggest a possibility not to recover the Richardson-Obukhov $t^3$ law at infinite Reynolds number.