Model of Turbulent Relative Dispersion: A Self-Similar Telegraph Equation

Abstract

Turbulent relative dispersion is theoretically studied using the probability distribution function (PDF) of the relative separation of two passive particles in the inertial range of homogeneous, isotropic, and stationary turbulence. Taking into account a finite separation speed and finite correlation of relative velocity by multiple-scale consideration, a telegraph equation with scale-dependent coefficients, which has a similarity solution, is derived. The tail of the solution falls off to zero at a maximum separation due to the finite separation speed. For slowly separating particle pairs, the equation is approximated by Richardson's diffusion equation with a drift term. The drift term plays an important role in the description of the main part of the separation PDF and tail part of the exit-time PDF. The relationship between the drift term and coherent structures is discussed for two 2D turbulence cases.