On the Closure of Diffusion Current in the PDF Kinetic Equation for the Separations and Relative Velocities of High-Inertia Particle Pairs

Relative motion of inertial, monodisperse particle pairs in stationary isotropic turbulence is investigated by evolving the Langevin stochastic differential equations governing pair relative velocities and separations in the limit of Stokes number $\gg 1$. The stochastic force in the Langevin equations is evaluated from the relative-velocity-space diffusivity that is equal to $1/\tau_v^2$ multiplied by the time integral of the Eulerian two-time correlation of fluid velocity differences seen by particle pairs that are nearly stationary ($\tau_v$ is the particle viscous relaxation time). The Eulerian two-time correlation of fluid velocity differences is computed using direct numerical simulation (DNS) of isotropic turbulence seeded with fixed particles. Since numerical forcing of energy-containing eddies is employed to achieve stationarity in DNS, the impact of forcing scheme on the two-time correlation is quantified as a function of separation $r$ and Taylor micro-scale Reynolds number $Re_\lambda$. As the DNS-computed correlation is needed to compute the stochastic force in Langevin simulations (LS), a key objective is to quantify the effects of forcing scheme on the LS predictions of pair relative motion.