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Turbulent Particle Pair Diffusion, Locality Versus Non-locality: Numerical Simulations

Particle pair (relative) diffusion in a field of homogeneous turbulence with generalised power-law energy spectra, $E(k)\sim k^{-p}$ for $1< p\le 3$ and $k_1\le k\le k_η$ with $k_η/k_1=10^6$, is investigated numerically using Kinematic Simulation (Kraichnan 1970, Fung et al 1992). If $Δ=|{\bf x_2}(t)-{\bf x}_1(t)|$ and $σ_Δ=\sqrt{\langleΔ^2\rangle}$, (the angled brackets is the ensemble average over particle pairs), then we find that: (1) The pair diffusivity scales like $D_p\sim σ_Δ^{γ_p}$ with $γ_p$ obtained from the simulations such that $γ^l_p<γ_p<2$, where $γ^l_p =(1+p)/2$ is the Richardson locality scaling. The range of $Δ$ over which these scalings are observed diminishes from above and from below as $p$ increases towards $3$.\\ (2) $M(p)=γ_p/γ^l_p>1$ in the range $1<p<3$, and $M$ has a peak at $p_m\approx 1.8$. This suggests that for spectra close to this in the range $1.5<p<2$, which includes Kolmogorov turbulence, local and non-local correlations play comparable roles in the pair diffusion process.\\ (3) The mean square separation scales like $\langleΔ^2\rangle_p \sim τ_p^{χ_p}$ where $χ_p=1/(1-γ_p/2)$ and $τ_p$ is an adjusted travel time.\\ (4) For Kolmogorov turbulence $p=5/3$, we observe $D_{5/3}\sim σ_Δ^{1.53}$, and $\langleΔ^2\rangle_{5/3}\simτ^{4.2}$. \\ (5) At $p_*\approx 1.4$ ($E(k)\sim k^{-1.4}$) we observe $D_{p_*}\simσ_Δ^{4/3}$, and $\langleΔ^2\rangle_{p_*}\simτ^3$; these are different to the Richardson $4/3$-law and Richardson-Obukov $t^3$-regime which occur for $p=5/3$. These results are consistent with Malik's (2014) theory, and supports the principle upon which the theory is based that both local and non-local correlations are effective in the pair diffusion process inside the inertial subrange.