Turbulent Particle Pair Diffusion Using Kinematic Simulations

Sweeping errors in Kinematic Simulations (KS) [1] have been shown to be negligible in turbulent flows with extended inertial subranges up to at least \textit{1\textless k\textless 10}$^{6} (k$ is the wavenumber) [2]. The departure from locality scaling observed in the pair diffusivity $K=$\textit{\textless }$\Delta \cdot $\textit{v\textgreater } in KS may therefore be a genuine effect, challenging previous assumptions [3] that in turbulence with generalized power-law energy spectra, $E(k) \sim k^{-p}$ for \textit{1\textless p}$\le $\textit{3}, locality would lead to, $K \sim \sigma_{\Delta }^{\gamma }$, where $\sigma_{\Delta }=$[ \quad \textless $\Delta ^{\mathrm{2}}$\textgreater ]$^{\mathrm{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }}$, $\Delta $ is the pair separation, $v$ is the pair relative velocity, \textless \textgreater is the ensemble average, and $\gamma_{\thinspace }=$\textit{(1}$+$\textit{p)/2}. For Kolmogorov turbulence this gives, $K \sim \sigma_{\Delta }^{4/3}$. A new analysis, supported by KS [4] confirms that both local and non-local effects govern the pair diffusion process, leading to, $K \sim \sigma _{\Delta }^{\gamma p}$, where now $\gamma_{p\thinspace }$\textit{\textgreater }$\gamma_{\thinspace }$; for Kolmogorov turbulence, $K \sim \sigma _{\Delta }^{1.53}$. Thus non-local diffusional processes cannot be neglected, and this may have important consequences for the general theory of turbulence. REFERENCES: [1] Fung, J. C. H., Hunt, J. C. R., Malik, N. A., {\&} Perkins, R. J. \textit{J. Fluid Mech. 236, 281 (1992).} [2] Malik, N. A. \textit{Under Review, Physics of Fluids (2015).} [3] Richardson, L. F. \textit{Proc. Roy. Soc. Lond. A 100, 709 (1926).} [4] Malik, N. A. On Turbulent Particle Pair Diffusion. \textit{Under review, Physics of Fluids (2015).}