Diffusion of Particles in Fully-Developed Turbulence

Abstract

The relative diffusion of a pair of dye particles in homogeneous steady turbulence is studied from the viewpoint that the turbulent transport is due to the vortex stretching of eddies in the inertial range. Thus, with the aid of Mori's (1980) formula for the vortex stretching, it is shown that the eddy diffusion rate is given by dL_*²/ dt=AL_*^{(4+2 µ)/ 3} with A=\hatAε^{1/ 3}L^{-2 µ/ 3}, where L_*²(t) is the mean square of the relative distance of the pair at time t, ε the mean rate of the energy transfer from large eddies to small eddies, L the length scale of the production range and µ the intermittency exponent. It turns out that \hatA \fallingdotseq1.39 for µ=0 and 1.22 for µ=1/ 3, where Kolmogorov's constant is taken to be 1.5. This diffusion law is valid for L>L_*(t) ≫l_d, where l_d is the length scale of the dissipation range. Hence it turns out that Richardson's (1926) empirical 4/ 3 law holds if µ is very small. If L_*(t) ≫L_*(0), then L_*²(t) ∼t^{3/ (1-µ)}. Therefore, if 0.25 ≲µ≲0.5 as experiments indicate, then the intermittency correction leads to an appreciable deviation from the t³ law. The condition L_*(t)≪L is written as t≪t_L, where L_*(t_L)=L and the upper limit of t_L is given by \hatt_L ≡t_L/ (L²/ ε)^{1/ 3} \fallingdotseq2.16 for µ=0 and 3.68 for µ=1/ 3 in the limit L_*(0)/ L →0.