Two-particle diffusion and locality assumption

Abstract

A three-dimensional kinematic simulation (KS) model is used to study one- and two-particle diffusion in turbulent flows. The energy spectrum E(k) takes a power law form E(k)∼k−p. The value of this power p is varied from 1.2 to 3, so that its effects on the diffusion of one and two particles can be studied. The two-particle diffusion behaves differently depending on whether the two-particle separation is larger or smaller than the smallest scale of turbulence (Kolmogorov length scale η). When the two-particle mean square separation 〈Δ2(t)〉 is smaller than η2 it experiences a time exponential growth 〈Δ2(t)〉=Δ02eγ(t/tη) but for a very short time. For longer times, when η2<〈Δ2(t)〉<L2, the locality assumption is revisited in terms of two-particle mean diffusivity d/dt〈Δ2(t)〉. In this inertial range we observe that d/dt〈Δ2(t)〉={a ln(〈Δ(t)2〉1/2/η)+b}u′L(η/L)(p+1)/2(〈Δ(t)2〉1/2/η)(p+1)/2 for p⩽3. For Δ0/η≫1 a=0, but a≠0 for Δ0/η⩽1 and as a consequence the pair diffusion cannot have lost its dependence on the initial separation during the exponential growth, i.e., γ is a function of Δ0/η. Our modified Richardson law is compared with two other proposed modifications to Richardson’s power law, namely the virtual time [G. K. Batchelor, Proc. Cambridge Philos. Soc. 48, 345 (1952)] and the correction factor [F. Nicolleau and J. C. Vassilicos, Phys. Rev. Lett. 90, 245003 (2003)]. Further investigations on two-particle diffusion when p=3 give an excellent agreement with the experimental results in P. Morel and M. Larchevêque, J. Atmos. Sci. 31 (1974) for atmospheric turbulent flows. Finally, using two different combined power law energy spectra in KS, the isotropic small scales are found to have no significant role when their largest scale lT is less than 10 times the Kolmogorov length scale η.