Quantifying Turbulence-Induced Segregation of Inertial Particles

Particles with different density from the advecting turbulent fluids cluster due to the different response of light and heavy particles to turbulent fluctuations. This study focuses on the quantitative characterization of the segregation of dilute polydisperse inertial particles evolving in turbulent flow, as obtained from direct numerical simulation of homogeneous isotropic turbulence. We introduce an indicator of segregation amongst particles of different inertia and/or size, from which a length scale $r_{\mathrm{seg}}$, quantifying the segregation degree between two particle types, is deduced.

Figure 1

Slice $400\eta \times 400\eta \times 10\eta$ of heavy $\beta =0$ (red or light gray) and light $\beta =3$ (blue or dark gray) particle positions. From left to right: $\mathrm{St}=0.1$, 1, and 4.1. Data refer to the simulation at ${\mathrm{Re}}_{\lambda }=180$.

Figure 2

$S_{{\alpha }_1,{\alpha }_2}(r)$ vs $r$ for ${\alpha }_1=({\beta }_1=0,{\mathrm{St}}_1=1.1)$ (heavy type) and ${\alpha }_2=({\beta }_2=3,{\mathrm{St}}_2=1.1)$ (light type) for different particle numbers $N(=N_{{\alpha }_1}=N_{{\alpha }_2})$: (○) $N={10}^5$, (×) $N={10}^4$, and (+) $N={10}^3$. Dashed and dotted lines refer to homogeneously distributed particles samples at various $N$. The expected Poisson scaling behavior, $S_{{\alpha }_1,{\alpha }_2}\propto r^{-3/2}$, is also reported. Inset: $r_{\mathrm{seg}}$, defined by $S_{{\alpha }_1,{\alpha }_2}(r_{\mathrm{seg}})=1/2$, as a function of $N$ for both heavy vs light particles cases and Poissonian samples. While for the former $r_{\mathrm{seg}}$ saturates as $N$ increases, for the latter $r_{\mathrm{seg}}$ goes to zero like $r_{\mathrm{seg},h}\propto N^{-1/3}$, as expected for Poissonian samples (see text for details).

Figure 3

$S_{{\alpha }_1,{\alpha }_2}(r)$ for ${\alpha }_1=({\beta }_1=0,{\mathrm{St}}_1=1.1)$ and ${\alpha }_2=({\beta }_2,{\mathrm{St}}_2={\mathrm{St}}_1)$, i.e., a heavy particle with ${\beta }_1=0$ and a given St vs those having the same St but different densities ${\beta }_2$. From bottom to top: ${\beta }_2=0$, 0.5, 1, 1.5, 3. Inset: $r_{\mathrm{seg}}$ vs ${\beta }_2$, with $r_{\mathrm{seg}}$ defined as in Fig. 2, i.e., $S_{{\alpha }_1,{\alpha }_2}(r_{\mathrm{seg}})=1/2$. The straight dashed line shows $r_{\mathrm{seg},h}\approx 4\eta \approx 0.3\lambda$.

Figure 4

$r_{\mathrm{seg}}$ between particle distributions with $\beta =0$, $\mathrm{St}=1.1$ (left panel) and $\beta =3$, $\mathrm{St}=1.1$ (right panel) vs distributions with generic $\beta$, St. (●) indicates the reference particle type, and (×) the location of the maximal segregation length $r_{\mathrm{seg}}^{(\max )}$. The solid contour line, traced at $r_{\mathrm{seg}}=r_{\mathrm{seg},h}\equiv L/N^{1/3}$, sets the sensitivity level to distinguish between segregated and unsegregated particle distributions. Dashed and dotted lines are drawn at $r_{\mathrm{seg}}=n{r}_{\mathrm{seg},h}$, with $n=2,\dots ,6$. The color scale codes the value of $r_{\mathrm{seg}}$ in units of the Kolmogorov length, $\eta$.

Figure 5

From left to right, segregation length $r_{\mathrm{seg}}$ between distribution of particles with $\mathrm{St}=(0.31,0.6,1.1,4.1)$ vs the densities ${\beta }_1$, ${\beta }_2$ and (last two panels) for particle class pairs having the same densities ${\beta }_1={\beta }_2=1.5$ (and $=3$) and different St. The color scale and the contour lines are, as in Fig. 4, at $r_{\mathrm{seg}}=n{r}_{\mathrm{seg},h}$, with $n=2,\dots ,4$.