Behavior of settling particles in homogeneous shear turbulence

In this study, particle-laden shear turbulence is investigated by performing direct numerical simulation under a point-particle approximation for spherical heavy particles with a diameter smaller than the Kolmogorov length scale of the flow. The ranges of the Stokes number and gravity factor considered in our study are $\text{St}=0.1,0.5,1,5$, and 10 and $W(=v_t/v_{\eta })=0,10,20,30$, and 40, respectively, where $v_t$ and $v_{\eta }$ are the terminal velocity and the Kolmogorov velocity scale, respectively. The behavior of settling particles is investigated by introducing a two-dimensional distribution function to quantify the anisotropic clustering. A principal component analysis of the two-dimensional distribution evidently indicates that the preferential orientation of clustering at different scales is identified well. Small-scale clustering displays a multifractal nature with an explicit angular distribution.

Figure 1

Statistics of flows in HST: (a) Reynolds number ${\text{Re}}_{\lambda }$ and shear parameter $S{q}^2/\epsilon$; (b) dissipation rate $\epsilon$ and production $P=-S〈u_1{u}_2〉$ over 200 shear timescales, $t_s=S\times t=200$. Inset in (b) shows ratio of production and dissipation.

Figure 2

Schematics showing method to interpolate data for particles located near (a) the top boundary and (b) the bottom boundary to satisfy the shear-periodic condition.

Figure 3

Particle distribution for various Stokes numbers and gravity factors. Gravity acts in vertical direction. Colored contour denotes spanwise vorticity. From top to bottom, $\text{St}=0.1,0.5,1,5$, and 10, respectively. From left to right, $W=0,10$, and 30, respectively.

Figure 4

(a) Plots of difference between mean settling velocity and terminal velocity and (b) difference between mean horizontal particle velocity and horizontal slip velocity for various Stokes numbers and gravity factors.

Figure 5

(a) Plots of particle trajectory for various Stokes numbers and gravity factors and (b) dispersion relative to terminal trajectory. Average particle trajectory is normalized by settling velocity and shear rate. Terminal trajectories for turbulence-free shear flow are expressed as $x_1^t=S{\overline{{\tau }_p}}^{2}gt-\frac{1}{2}S\overline{{\tau }_p}g{t}^2$ and $x_2^t=-\overline{{\tau }_p}gt$. Dispersions are displayed for $\text{St}=0.1,1,$ and 10.

Figure 6

Plots of average closest distances. (a) Drawn with gravity factor for $\text{St}=0.1,0.5,1,5$, and 10. (b) Drawn with average closest distance in each direction. The average closest distance is normalized by the corresponding value for a random distribution of the same number of particles in the same domain so that values smaller than 1 indicate clustering.

Figure 7

Two-dimensional distribution function (2DF) for various Stokes numbers and gravity factors. From top to bottom, $\text{St}=0.1,0.5,1,5$, and 10, respectively. From left to right, $W=0,10$, and 30, respectively. The gray-scale contour level represents the 2DF values for the same Stokes number. The blue contour denotes $g(x,y)=1.06$, and the area inside the blue line is taken as the structure of large-scale clustering. The white region indicates a relatively void region.

Figure 8

Plots of $\sqrt{{\lambda }_1\left(h\right)}$ and $\sqrt{{\lambda }_2\left(h\right)}$ [(a), (c), (e), and (g)] and principal angles of particle distributions [(b), (d), (f), and (h)] as a function of $\sqrt{{\lambda }_1}$, normalized by Kolmogorov scale, $\eta$ for various gravity factors, $W=0,10,20,40$ and $\text{St}=0.5,1,5,10$.

Figure 9

Principal angles of (a) small-scale distribution and (b) large-scale distribution vs gravity factor for $\text{St}=0.1,0.5,1,5$, and 10.

Figure 10

Plots of $g(r,\theta )$ [(a) and (b)] and the compensated RDF $g(r,\theta ){sin}^{\alpha }(\theta -{\theta }_{{\lambda }_1})$ [(c) and (d)] vs $\eta /r$ at representative angles $\theta$ in case of (a), (c) $\text{St}=1,W=0$, and (b), (d) $\text{St}=1,W=20$. Red dashed line represents the most stretching principal angle ${\theta }_S$ for ${\theta }_{{\lambda }_1}$, and blue dotted line represents the most compressive principal angle for ${\theta }_{{\lambda }_2}$.

Figure 11

Plots of $g(r,{\theta }_{{\lambda }_2})$ for $\text{St}=0.1,0.5,1$, and 5 and $\text{St}=0.1$. The RDF for $W=0$ is approximated by ${({ł}_c/r)}^{\alpha }$ well with the value of $\alpha$ in the isotropic turbulence, given by Bec et al. [69]. For $\text{St}=0.5$ and 1, in most cases, the diverging behavior of $g(r,\theta )$ as $r\to 0$ is remarkably fitted well by ${(l_c/r)}^{\alpha }$ with the corresponding $\alpha$ listed in Table 4. For $\text{St}=5$, the diverging behavior is not observed.