Sedimentation of inertial monodisperse suspensions of cubes and spheres

Particle-resolved direct numerical simulations of monodisperse settling suspensions of cubes and spheres are performed for Galileo numbers $\text{Ga}=70$ and $\text{Ga}=160$, and solid volume fractions in the range $0.01⩽\varphi ⩽0.2$. The solid-to-fluid density ratio $m={\rho }_s/{\rho }_f$ is taken to be fixed at 2, representing liquid-solid suspensions. Strong columnar clustering is observed for $\text{Ga}=160$ and $\varphi =0.01$ in a sphere suspension, whereas similar vertical structures are not present as prominently in a cube suspension. We find that in all cases, cube suspensions tend to be more homogeneous compared to sphere suspensions, as indicated by both their microstructure and momentum transfer properties. The enhanced homogeneity is associated with the pronounced angular velocities of cubes and the resulting orientation- and rotation-induced lift forces, which promote transverse motions and the likelihood of escaping from clusters. Higher rotation rates of cubes thus play a major role in the transfer of momentum from the gravity to the transverse direction, demonstrated by the lower anisotropy of particle velocity fluctuations in cube suspensions. In more dilute cases, cubes induce significantly stronger pseudoturbulence in the flow, especially in the transverse direction. The drag of dynamic cube suspensions is found to be generally similar to static beds of cubes, the reason for which is speculated to be relevant to their motion freedom and the more homogeneous microstructure.

Figure 1

The five regular convex polyhedrons, also known as Platonic solids, in the order of increasing number of faces.

Figure 2

Computational domains used for PR-DNS of cube and sphere suspensions with different volume fractions.

Figure 3

Suspension settling velocities normalized by the terminal velocity of an isolated particle as a function of solid volume fraction for the two Galileo numbers of $\text{Ga}=70$ and $\text{Ga}=160$. The dashed lines in (a) and (b) show the Richardson and Zaki [6] formula presented in Fig. (2), with the prefactor $k=0.85$. (c) The vertical velocity of an isolated cube as a function of time at $\text{Ga}=160$ normalized by the gravitational velocity scale.

Figure 4

Velocity fluctuations (a) in the gravity direction and (b) normal to the gravity direction. The particle and fluid data are shown by solid and open symbols, respectively, whereas red and black lines and symbols demonstrate the results for cube and sphere suspensions, respectively. The inset of each plot shows the same data on a linear scale.

Figure 5

Time evolution of (a) vertical and (b) horizontal particle velocity fluctuations. In these plots, red and black lines demonstrate the results for cube and sphere suspensions, respectively. Additionally, solid lines represent simulations with $\text{Ga}=160$, whereas dotted lines show data for simulations with $\text{Ga}=70$. In order to enhance visual representation of the plots, darker lines show a filtered version of the data, while the original data are plotted with a lighter color.

Figure 6

Time evolution of (a) vertical and (b) horizontal fluid velocity fluctuations. Interpretation of line styles and colors is given in the caption of Fig. 5.

Figure 7

Anisotropy of (a) particle and (b) fluid velocity fluctuations indicated by the ratio of the vertical to horizontal velocity variance. The insets show extended ranges of values on the vertical axis.

Figure 8

Distribution of the horizontal component of particle angular velocities for different solid volume fractions. The insets show the same data with the $y$ axis scaled logarithmically, while the mean value of each distribution is shown using short vertical lines on each plot. Interpretation of line styles and colors is given in the caption of Fig. 5.

Figure 9

Joint probability distribution functions of (a) cube and (b) sphere suspensions obtained for $(\text{Ga},\varphi )=(160,0.05)$, (c) Variation of Pearson's correlation coefficient as a function of the solid volume fraction for the simulated cases. Interpretation of line styles and colors is given in the caption of Fig. 5.

Figure 10

Pair and radial distribution functions obtained for microstructure characterization of suspensions at $\text{Ga}=160$. The leftmost and center columns in the figure pertain to cube and sphere suspensions, respectively. On the rightmost plots, radial distribution functions of hard spheres [66] are also given as a reference in addition to those obtained from the suspension simulations.

Figure 11

Pair and radial distribution functions obtained for microstructure characterization of suspensions at $\text{Ga}=70$. The leftmost and center columns in the figure pertain to cube and sphere suspensions, respectively. On the rightmost plots, radial distribution functions of hard spheres [66] are also given as a reference in addition to those obtained from the suspension simulations.

Figure 12

Time evolution of ensemble-averaged local solid volume fraction ${\varphi }_{\text{loc}}$. Interpretation of line styles and colors is given in the caption of Fig. 5.

Figure 13

Drag force of the cube and sphere suspensions as a function of solid volume fraction for two Galileo numbers of $\text{Ga}=70$ and $\text{Ga}=160$. Open and solid symbols represent data for dynamic suspensions and static beds, respectively. Along with the data obtained from the present PR-DNS, we also show the drag correlations proposed by Wen and Yu [69], Di Felice [73], and Bogner et al. [79] for spheres, and that of Chen and Müller [81] for fixed beds of cubes.

Figure 14

Pair distribution functions of cube and sphere suspensions with extended ranges of the radial distance compared to Fig. 10 and Fig. 11. The plots pertain to cases with Galileo numbers of $\text{Ga}=70$ and $\text{Ga}=160$ and solid volume fractions of $\varphi =0.1$ and $\varphi =0.2$.