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 $k_1\le k\le k_\eta$ with $k_\eta/k_1=10^6$, is investigated numerically using Kinematic Simulation (Kraichnan 1970, Fung et al 1992). If $\Delta=|{\bf x_2}(t)-{\bf x}_1(t)|$ and $\sigma_\Delta=\sqrt{\langle\Delta^2\rangle}$, (the angled brackets is the ensemble average over particle pairs), then we find that:   (1) The pair diffusivity scales like $D_p\sim \sigma_\Delta^{\gamma_p}$ with $\gamma_p$ obtained from the simulations such that $\gamma^l_p<\gamma_p<2$, where $\gamma^l_p =(1+p)/2$ is the Richardson locality scaling. The range of $\Delta$ over which these scalings are observed diminishes from above and from below as $p$ increases towards $3$.\\ (2) $M(p)=\gamma_p/\gamma^l_p>1$ 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 $\langle\Delta^2\rangle_p \sim \tau_p^{\chi_p}$ where $\chi_p=1/(1-\gamma_p/2)$ and $\tau_p$ is an adjusted travel time.\\ (4) For Kolmogorov turbulence $p=5/3$, we observe $D_{5/3}\sim \sigma_\Delta^{1.53}$, and $\langle\Delta^2\rangle_{5/3}\sim\tau^{4.2}$. \\ (5) At $p_*\approx 1.4$ ($E(k)\sim k^{-1.4}$) we observe $D_{p_*}\sim\sigma_\Delta^{4/3}$, and $\langle\Delta^2\rangle_{p_*}\sim\tau^3$; 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. 
Nadeem A. Malik
physics.flu-dyn math-ph math.MP