Preferential concentration by mechanically driven turbulence in the two-fluid formalism

Preferential concentration is thought to play a key role in promoting particle growth, which is crucial to processes such as warm rain formation in clouds, planet formation, and industrial sprays. In this paper, we investigate preferential concentration using three-dimensional direct numerical simulations adopting the Eulerian-Eulerian two-fluid approach, where the particles are treated as a continuum field with its own momentum and mass conservation laws. We consider particles with Stokes number $\text{St}\lesssim O\left(0.01\right)$ in moderately turbulent flows with fluid Reynolds number $\text{Re}\le 600$. In our previous paper [Phys. Rev. Fluids 5, 114308 (2020)], we established scaling laws to predict maximum and typical particle concentration enhancements in the context of the particle-driven convective instability. Here, we verify that the same results apply when turbulence is externally driven, extending the relevance of our model to a wider class of particle-laden flows. We find in particular that (i) the maximum particle concentration enhancement above the mean scales as $u_{\text{rms}}^2{\tau }_p/{\kappa }_p$, where $u_{\text{rms}}$ is the rms fluid velocity, ${\tau }_p$ is the particle stopping time, and ${\kappa }_p$ is the assumed particle diffusivity from the two-fluid equations; (ii) the typical particle concentration enhancement over the mean scales as ${(u_{\text{rms}}^2{\tau }_p/{\kappa }_p)}^{1/2}$; and (iii) the probability distribution function of the particle concentration enhancement over the mean has an exponential tail whose slope scales as ${(u_{\text{rms}}^2{\tau }_p/{\kappa }_p)}^{-1/2}$. We conclude by discussing the caveats of our model and its implications in a relevant cloud application.

Figure 1

Mean flow $\overline{u}$ and Reynolds stress $\overline{uw}$ as a function of height $z$ for a simulation in the absence of particles with $\text{Re}=600$. The blue curve represents the temporal average of the gray curves, which have been extracted at various times after the system has reached a statistically steady state.

Figure 2

Instantaneous power spectra of the total fluid velocity field as a function of $\left|\mathbf{k}\right|$ for simulations in the absence of particles for $\text{Re}=100$, 300, and 600. The black solid line scales as ${\left|\mathbf{k}\right|}^{-5/3}$.

Figure 3

Comparison of particle concentration snapshots for low $T_p=0.005$ (left column) and high $T_p=0.03$ (right column) simulations. Each snapshot was extracted once the system has reached a statistically steady state. (a), (b) Volume rendering of $\widehat{r}$; (c), (d) snapshots of the particle concentration enhancement $\widehat{r}-1$ at $y=0$; (e), (f) snapshots of $\mathbf{\nabla }·{\widehat{\mathbf{u}}}_p$ at $y=0$. The remaining parameters are $r_0=0.1, \text{Re}=100, {\text{Re}}_p=600, {\text{Pe}}_p=600$.

Figure 4

Instantaneous power spectra of (a) the total fluid velocity field and (b) the particle concentration field as a function of the total wave number $\left|\mathbf{k}\right|$ for varying $T_p$. The remaining parameters are $r_0=0.1, \text{Re}=100, {\text{Re}}_p=600$, and ${\text{Pe}}_p=600$.

Figure 5

Instantaneous power spectra of (a) the total fluid velocity field and (b) the particle concentration field as a function of the total wave number $\left|\mathbf{k}\right|$ for varying $r_0$. The remaining parameters are $T_p=0.01, \text{Re}=100, {\text{Re}}_p=600$, and ${\text{Pe}}_p=600$. The solid, dashed, and dotted lines represents the predicted scaling for the power at the injection scale for $r_0=0.1$, 1, and 10, respectively (see main text for details).

Figure 6

Instantaneous power spectra of (a) the total fluid velocity field and (b) the particle concentration field as a function of the total wave number $\left|\mathbf{k}\right|$ for varying $\text{Re}$. The remaining parameters are $T_p=0.01, r_0=0.1, {\text{Re}}_p=600$, and ${\text{Pe}}_p=600$. On the right plot, we mark $k_{\lambda }/3$ given by the vertical dashed line with the same color as the corresponding simulation.

Figure 7

Snapshots of the particle concentration enhancement above the mean $\widehat{r}-1$ for two different simulations with $\text{Re}=100$ and $\text{Re}=600$. The remaining parameters are $T_p=0.01, r_0=0.1, {\text{Re}}_p=600$, and ${\text{Pe}}_p=600$.

Figure 8

Instantaneous power spectra of (a) the total fluid velocity field and (b) the particle concentration field as a function of the total wave number $\left|\mathbf{k}\right|$ for varying ${\text{Pe}}_p$. The remaining parameters are $T_p=0.01, r_0=0.1, \text{Re}=100$, and ${\text{Re}}_p={\text{Pe}}_p$.

Figure 9

Temporally averaged maximum and typical particle concentration enhancement (${\widehat{r}}_{\text{sup}}-1$ and ${\widehat{r}}_{\text{rms}}$, respectively) and the temporally averaged rms fluid velocity ${\widehat{U}}_{\text{rms}}$ for (a) varying $T_p$, (b) varying $r_0$, (c) varying $\text{Re}$, and (d) varying ${\text{Pe}}_p$ and ${\text{Re}}_p$ from simulations that have reached a statistically steady state. Error bars represent one standard deviation around the mean. Unless otherwise stated, $T_p=0.01, r_0=0.1, \text{Re}=100, {\text{Pe}}_p=600$, and ${\text{Re}}_p=600$ (see main text). More details of the simulations can be found in Table 1.

Figure 10

Maximum particle concentration enhancement over the mean as function of ${\widehat{U}}_{\text{rms}}^2{T}_p{\text{Pe}}_p$ with varying parameters (i.e., $T_p, r_0, \text{Re}, {\text{Re}}_p$, and ${\text{Pe}}_p$). The black solid line represents ${\widehat{r}}_{\text{sup}}-1=(1/10){\widehat{U}}_{\text{rms}}^2{T}_p{\text{Pe}}_p$. The legend and details of simulations can be found in Table 1.

Figure 11

Typical particle concentration enhancement over the mean as a function of ${\widehat{U}}_{\text{rms}}^2{T}_p{\text{Pe}}_p$ with varying parameters (i.e. $T_p, r_0, \text{Re}, {\text{Re}}_p$, and ${\text{Pe}}_p$). The black dotted line represents ${\widehat{r}}_{\text{rms}}=\left(0.07\right){\widehat{U}}_{\text{rms}}{\left(T_p{\text{Pe}}_p\right)}^{1/2}$ and the solid line represents ${\widehat{r}}_{\text{rms}}=(1/10){\widehat{U}}_{\text{rms}}^2{T}_p{\text{Pe}}_p$. The legend and details of simulations can be found in Table 1.

Figure 12

Probability distribution functions for $\widehat{r}$, computed from simulations that have reached a statistically steady state (a) for varying $T_p$ with $r_0=0.1, \text{Re}=100, {\text{Pe}}_p=600, {\text{Re}}_p=600$ and (b) for varying $r_0$ with $T_p=0.01, \text{Re}=100, {\text{Pe}}_p=600, {\text{Re}}_p=600$. The gray lines fit the tail of each pdf and are of the form $p\left(\widehat{r}\right)\propto e^{-b\widehat{r}}$. Values of $b$ and simulation details can be found in Table 1.

Figure 13

The slope $b$ of the exponential tail of the pdf of $\widehat{r}$ as a function of ${\widehat{U}}_{\text{rms}}^2{T}_p{\text{Pe}}_p$ for simulations at various $T_p, r_0, \text{Re}, {\text{Re}}_p$, and ${\text{Pe}}_p$ (where ${\text{Re}}_p={\text{Pe}}_p$). The blue solid line shows $b\sim {\left({\widehat{U}}_{\text{rms}}^2{T}_p{\text{Pe}}_p\right)}^{-1/2}$. See Table 1 for more details.