Relative velocities in bidisperse turbulent suspensions

We investigate the distribution of relative velocities between small heavy particles of different sizes in turbulence by analyzing a statistical model for bidisperse turbulent suspensions, containing particles with two different Stokes numbers. This number, $\text{St}$, is a measure of particle inertia which in turn depends on particle size. When the Stokes numbers are similar, the distribution exhibits power-law tails, just as in the case of equal $\text{St}$. The power-law exponent is a nonanalytic function of the mean Stokes number $\overline{\mathrm{St}}$, so that the exponent cannot be calculated in perturbation theory around the advective limit. When the Stokes-number difference is larger, the power law disappears, but the tails of the distribution still dominate the relative-velocity moments, if $\overline{\mathrm{St}}$ is large enough.

Figure 1

Distribution $\varrho (v_r,r)$ of relative velocities between particles with different Stokes numbers. Statistical-model simulations in two dimensions ($d=2$) for $\text{Ku}=1$. (a) Contour plot of $\varrho (v_r,r)/r^{d-1}$ for $\overline{\text{St}}=1$ and $\theta ={10}^{-2}$ (see text). Dotted lines show cut-off scales $r_{\mathrm{c}}$ and $v_{\mathrm{c}}$. (b) Dependence of $r_{\mathrm{c}}$ on $\theta$ (green $\square$), and of $v_{\mathrm{c}}$ on $\theta$ (red $\circ$), for $\overline{\text{St}}=1$. (c) $\varrho (v_r,r)/r^{d-1}$ evaluated at $r<r_{\mathrm{c}}$, for $\overline{\text{St}}=1$ and $\theta ={10}^{-3}$ (green line), $\theta ={10}^{-2}$ (solid black line), and $\theta =0.1$ (magenta line). Crossover scales $v_{\mathrm{c}}$ (arrows). (d) Power-law exponent ${\mu }_{\mathrm{c}}$ versus $\overline{\text{St}}$ for small $\theta$. Exponent for power-law tails in $v_r$ for fixed $r$ with $\theta ={10}^{-3}$ (green $\square$) and $\theta ={10}^{-2}$ (red $\circ$). Exponent for power-law tails in $r$ for fixed $v_r$ with $\theta ={10}^{-2}$ (blue $\diamond$). Numerical data for $\min (D_2,d+1)$ where $D_2$ is the phase-space correlation dimension (solid blue line).

Figure 2

(a) Distribution of $\mathrm{\Delta }v$ at small $\mathrm{\Delta }x$ from white-noise simulations of Eq. (2), for ${\overline{\varepsilon }}^2=2$ and $\theta ={10}^{-3}, \theta ={10}^{-2}$, and $\theta ={10}^{-1}$; green, black, and pink curves, respectively. (b) Exponent ${\mu }_{\mathrm{c}}$ as a function of $\overline{\varepsilon }$ from simulations of Eq. (2) for $\theta =0.01$, obtained from fit to $\mathrm{\Delta }x$ tails (blue $\diamond$), and from fit to $\mathrm{\Delta }v$ tails (red $\circ$). Result for equal $\text{St}$ obtained by numerical solution of Eq. (7) (solid blue line) and the analytical prediction, Eq. (12) (dashed black line). (c) Same data as for the solid and dashed lines in panel (b), but plotted here as $-log(1+{\mu }_{\mathrm{c}})$ versus ${\overline{\varepsilon }}^{-2}$.