Preferential concentration in the particle-induced convective instability

Heavy particles in turbulent flows have been shown to accumulate in regions of high strain rate or low vorticity, a process otherwise known as preferential concentration. This can be observed in geophysical flows and is inferred to occur in astrophysical environments, often resulting in rapid particle growth, which is critical to physical processes such as rain or planet formation. Here we study the effects of preferential concentration in a two-way coupled system in the context of the particle-driven convective instability. To do so, we use direct numerical simulations and adopt the two-fluid approximation. We focus on a particle size range for which the latter is valid, namely, when the Stokes number is $\lesssim O\left(0.1\right)$. For Stokes number above $\sim 0.01$, we find that the maximum particle concentration enhancement over the mean scales with the rms fluid velocity $u_{\mathrm{rms}}$, the particle stopping time ${\tau }_p$, and the assumed particle diffusivity ${\kappa }_p$ from the two-fluid equations, as $u_{\mathrm{rms}}^2{\tau }_p/{\kappa }_p$. We show that this scaling can be understood from simple arguments of dominant balance. We also show that the typical particle concentration enhancement over the mean scales as ${(u_{\mathrm{rms}}^2{\tau }_p/{\kappa }_p)}^{1/2}$. We finally find that the probability distribution function of the particle concentration enhancement over the mean has an exponential tail whose slope scales as ${(u_{\mathrm{rms}}^2{\tau }_p/{\kappa }_p)}^{-1/2}$. We apply our model to geophysical and astrophysical examples, and discuss its limitations.

Figure 1

Snapshots of the particle concentration $r$ (top row) and the horizontal component of the fluid velocity $u$ (bottom row) at various times in a two-fluid simulation with $T_{\mathrm{p}}=0.005$, $W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000$, ${\mathrm{Re}}_{\mathrm{p}}=1000$, ${\mathrm{Pe}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$.

Figure 2

Low $T_{\mathrm{p}}=0.005$ two-fluid simulation versus an equilibrium Eulerian simulation with $W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$, comparing various diagnostics of the particle concentration (a) and of the horizontal component of fluid velocity (b).

Figure 3

Evolution of the mean particle concentration $\overline{r}$ and of the rms fluid velocity $u_{\mathrm{rms}}$ (defined in the main text) profiles for the two-fluid $T_{\mathrm{p}}=0.005$ (red) and equilibrium Eulerian simulations (green) with $W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$. The black dotted Gaussian curve (first row) represents the purely diffusive solution (43) for comparison.

Figure 4

Top: Snapshots of the particle concentration $r$ at various times in a simulation with $T_{\mathrm{p}}=0.1,W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$. Bottom: Diagnostic properties of the particle concentration and fluid velocity as a function of time for the same simulation.

Figure 5

Measures of particle concentration at $t=54$ of two-fluid simulations for $T_{\mathrm{p}}=0.005$ and $T_{\mathrm{p}}=0.1$ with otherwise identical parameters $W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$.

Figure 6

Comparison of $r_{\sup }$ for 2D simulations with varying $T_{\mathrm{p}}$. Remaining parameters: $W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000,\mathrm{Pr}=1$. An equilibrium Eulerian simulation marked “Eulerian” is shown for comparison.

Figure 7

Snapshots of the particle concentration field for varying $\mathrm{Re}$ and ${\mathrm{Pe}}_{\mathrm{p}}$ with fixed $T_{\mathrm{p}}=0.1$ taken at $t\approx 54$. Only the vicinity of the particle layer is shown. Remaining parameters: $W_s=0.1,R_{\rho }=0.5,{\mathrm{Re}}_{\mathrm{p}}=1000,\mathrm{Pr}=1$.

Figure 8

Comparison of $r_{\sup }$ for 2D simulations with varying ${\mathrm{Pe}}_{\mathrm{p}}$ and $\mathrm{Re}$ for $T_{\mathrm{p}}=0.1$. Remaining parameters: $W_s=0.1,R_{\rho }=0.5,{\mathrm{Re}}_{\mathrm{p}}=1000,\mathrm{Pr}=1$.

Figure 9

Power spectra of the particle concentration field (a) and the fluid velocity field (b) as a function of the horizontal wave number $k_x$ for varying $\mathrm{Re}$ and ${\mathrm{Pe}}_{\mathrm{p}}$. Remaining parameters are $T_{\mathrm{p}}=0.1$, $W_s=0.1,R_{\rho }=0.5,{\mathrm{Re}}_{\mathrm{p}}=1000,$ and $\mathrm{Pr}=1$. The error bars indicate the rms temporal variability of the spectrum.

Figure 10

Power spectra of the particle concentration field (left) and the fluid velocity field (right) as a function of the horizontal wave number $k_x$ for varying $T_{\mathrm{p}}$. Remaining parameters are $W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000,$ and $\mathrm{Pr}=1$. The error bars indicate the rms temporal variability of the spectrum (as in Fig. 9).

Figure 11

Comparison of the fluid velocity $u_{\sup }$ and particle concentration $r_{\sup }$ (defined in the text) between 2D and 3D simulations with settling velocity $W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000,$ and $\mathrm{Pr}=1$. Left figure: $T_{\mathrm{p}}=0.005$. Right figure: $T_{\mathrm{p}}=0.1$.

Figure 12

Maximum particle concentration enhancement over the mean as function of $U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}$ for a simulation with parameters $T_{\mathrm{p}}=0.3,W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$. Each dot represents an instant in time, with points moving from the bottom-left corner to the top-right corner over time. The black solid line shows ${(r^{\prime }/\overline{r})}_{max}=(1/4)U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}$.

Figure 13

Maximum particle concentration enhancement over the mean as function of $U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}$, for varying $W_s$ and $T_{\mathrm{p}}$ (with fixed $R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000,\mathrm{Pr}=1$). The black solid line represents ${(r^{\prime }/\overline{r})}_{max}=(1/4)U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}$. Details of simulations can be found in Table 1.

Figure 14

Maximum particle concentration enhancement over the mean as function of $U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}$, for varying $T_{\mathrm{p}}$, $\mathrm{Re}$, and ${\mathrm{Pe}}_{\mathrm{p}}$ (with $W_s=0.1,R_{\rho }=0.5,\mathrm{Pr}=1,{\mathrm{Re}}_{\mathrm{p}}=1000$). The black solid line represents ${(r^{\prime }/\overline{r})}_{max}=(1/4)U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}$. Details of simulations can be found in Table 1.

Figure 15

Typical particle concentration enhancement for varying $W_s$ and $T_{\mathrm{p}}$ with $R_{\rho }=0.5,\mathrm{Pr}=1,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000$, unless otherwise denoted. The black solid line represents ${(r^{\prime }/\overline{r})}_{\mathrm{rms}}=(1/4)U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}$, and the blue line represents ${(r^{\prime }/\overline{r})}_{\mathrm{rms}}=(1/5){\left(U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}\right)}^{1/2}$. Details of simulations can be found in Table 1.

Figure 16

Probability distribution functions for the function $r_{\mathrm{rel}}$ (60) at various times during two simulations with $W_s=0.1,R_{\rho }=0.5,\mathrm{Pr}=1,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,$ and ${\mathrm{Pe}}_{\mathrm{p}}=1000$ for $T_{\mathrm{p}}=0.005$ (a) and $T_{\mathrm{p}}=0.1$ (b).

Figure 17

Time-averaged PDFs of $r_{\mathrm{rel}}$ (see Eq. (60)) during the peak of the mixing event. (a) At fixed $W_s=0.1,R_{\rho }=0.5,\mathrm{Re}=1000,{\mathrm{Re}}_{\mathrm{p}}=1000,{\mathrm{Pe}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$ and varying $T_{\mathrm{p}}$. (b) At fixed $T_{\mathrm{p}}=0.1,W_s=0.1,R_{\rho }=0.5,{\mathrm{Re}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$ and varying $\mathrm{Re}$ and ${\mathrm{Pe}}_{\mathrm{p}}$.

Figure 18

The slope $b$ of the exponential tail of the PDF of $r_{\mathrm{rel}}$ as a function of $U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}$ for simulations at various $T_{\mathrm{p}}$, $\mathrm{Re}$, and ${\mathrm{Pe}}_{\mathrm{p}}$ with $W_s=0.1,R_{\rho }=0.5,{\mathrm{Re}}_{\mathrm{p}}=1000$, and $\mathrm{Pr}=1$. In all cases, the the PDF is computed during the peak of the mixing event. The red solid line shows $b=7{\left(U_{\mathrm{rms}}^2{T}_{\mathrm{p}}{\mathrm{Pe}}_{\mathrm{p}}\right)}^{-1/2}$.