Relative diffusion of a pair of inertial particles in the inertial sub-range of turbulence

Turbulent diffusion of a pair of inertial particles in 3-dimensional homogeneous and isotropic turbulence was studied using direct numerical simulation (DNS) with $2048^3$ grid points; the Taylor micro-scale Reynolds number in the DNS is approximately 425. For each set of the inertial particles with different values of the Stokes number ($St=0, 0.1, 0.2, 0.5, 1, 2, 5, 10$), $256^3$ particles are tracked using cubic spline interpolation for the velocity data in the DNS. Here $St=0$ corresponds to fluid particles. The DNS showed that for each value of $St$, the mean square of the distance $\delta x$ between the two inertial particles grows with time $t$ as $\langle \delta x^2 \rangle \sim C \epsilon t^3$ in the inertial subrange, which is in agreement with Richardson (1926) and Obukhov (1941). Here $\epsilon$ is the mean energy dissipation rate per unit mass, and $C$ is a constant of $O$(1) depending on the value of $St$ and the initial distance between the inertial particles. The DNS shows also that large clusters of strong vortices enhance relative diffusion of inertial particles of $St>1$.