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

TL;DR

This study uses numerical simulations to analyze particle pair diffusion in turbulence with various energy spectra, revealing the interplay of local and non-local effects and extending understanding of diffusion scaling laws.

Contribution

It provides new insights into the scaling behavior of pair diffusivity and separation in turbulence, highlighting the roles of locality and non-locality across different spectral slopes.

Findings

Diffusivity scales with separation as D_p ~ σ_Δ^γ_p, with γ_p between locality and 2.

Peak non-locality effect at p ≈ 1.8, indicating comparable local and non-local roles near Kolmogorov turbulence.

Scaling laws for mean square separation differ from classical Richardson laws at certain spectral slopes.

Abstract

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.