Statistics of the relative velocity of particles in turbulent flows : monodisperse particles

We use direct numerical simulations to calculate the joint probability density function of the relative distance $R$ and relative radial velocity component $V_R$ for a pair of heavy inertial particles suspended in homogeneous and isotropic turbulent flows. At small scales the distribution is scale invariant, with a scaling exponent that is related to the particle-particle correlation dimension in phase space, $D_2$. It was argued [1, 2] that the scale invariant part of the distribution has two asymptotic regimes: (1) $|V_R| \ll R$wherethedistributiondependssolelyon$R$;and(2)$|V_R| \gg R$wherethedistributionisafunctionof$|V_R|$ alone. The probability distributions in these two regimes are matched along a straight line $|V_R| = z^\ast R$. Our simulations confirm that this is indeed correct. We further obtain $D_2$ and $z^{\ast }$ as a function of the Stokes number, $\mathrm{S}\mathrm{t}$. The former depends non-monotonically on $\mathrm{S}\mathrm{t}$ with a minimum at about ${\rm St} \approx 0.7$andthelatterhasonlyaweakdependenceon${\rm St}$.