Statistics of the relative velocity of particles in bidisperse turbulent suspensions

We calculate the joint probability distribution function (JPDF) of relative distances ($R$) and velocities (${\bf V}$ with longitudinal component $V_R$) of a pair of {\it bidisperse} heavy inertial particles in homogeneous and isotropic turbulent flows using direct numerical simulations (DNS). A recent paper (J. Meibohm, {\it et. al.} 2017), using statistical-model simulations and mathematical analysis of an one-dimensional white-noise model, has shown that the JPDF, $\mathcal{P}(R,V_R)$, for two particles with Stokes numbers, $St_1$ and $St_2$, can be interpreted in terms of $St_M$, the harmonic mean of $St_1$ and $St_2$ and $\theta \equiv \mid St_1-St_2 \mid/(St_1+St_2)$. For small $\theta$ there emerges a small-scale cutoff $R_c$ and a small-velocity cutoff $V_c$ such that for $V_R \ll V_c$ and $R \ll R_c$ the JPDF, $\mathcal{P}(R,V_R)$, is independent of $R$ and $V_R$. Beyond these two small-scale cutoffs the JPDF for the bidisperse case shows the same scaling behavior as the JPDF for {\it mono-disperse} particles with $St = St_M$. Our DNS demonstrate that this is true and the scales $R_c$ and $V_c$ are proportional to $\theta$ for small $\theta$.