Resolved simulations of sedimenting suspensions of spheres

Several results on sedimenting equal spheres obtained by resolved simulations with the Physalis method are presented. The volume fraction ranges from 8.7% to 34.9% and the particle Galilei number from 49.7 to 99.4. The results shown concern particle collisions, diffusivities, the mean free path, the particle pair distribution function, and other features. It is found that many qualitative trends found in earlier studies continue to hold in the parameter range investigated here as well. The analysis of collisions reveals that particles interact prevalently via their flow fields rather than by direct contacts. A tendency toward particle clustering is demonstrated. The time evolution of the shape and size of particle tetrads is examined.

Figure 1

Two examples of the time dependence of the instantaneous fluctuations of the square of the particle velocity in the vertical (orange upper lines) and horizontal directions with respect to the time-mean values indicated by the horizontal lines for (a) $\varphi =8.7\%$ and (b) $\varphi =34.9\%$, both with ${\rho }_p/{\rho }_f=3.3$. The intervals of time during which the vertical velocity fluctuation differs by $\pm 10\%$ from the mean are highlighted. The time history shown is a fraction of the total length of the simulation. The reference velocity $\tilde{u}$ is defined in (3).

Figure 2

Probability density function of Voronoi volumes normalized by the particle volume $v_p=\frac{\pi }{6}{d}^3$ for the two cases of Fig. 1, namely, (a) $\varphi =8.7\%$ and (b) $\varphi =34.9\%$, both with ${\rho }_p/{\rho }_f=3.3$. The blue lines marked with circles are the average for the periods of greater-than-average velocity in Fig. 1, while the orange lines marked with squares are the average for the periods of smaller-than-average velocity. In (b) the two lines are essentially superposed while in (a) a somewhat larger probability for smaller values of the Voronoi volumes is visible for the greater-than-average velocity periods.

Figure 3

Plot of the PDF of $n\left(r\right)/\langle n\rangle$, with $n\left(r\right)$ the average number density of particles with center in spherical shells of different radii $r$ centered on each particle during the periods of time highlighted in Fig. 1, $\langle n\rangle$ the mean number density over the computational domain, for $r/a=2.25$ (blue), 2.50 (orange), 3 (green), and 3.50 (red), and the same conditions as in Figs. 1 and 2. The solid and dashed lines are the values of $n\left(r\right)/\langle n\rangle$ during the high- and low-velocity intervals, respectively. (a) For the low-concentration case note the higher probability of larger values of $n\left(r\right)/\langle n\rangle$ during the higher-velocity periods highlighted in Fig. 1. (b) For the denser case, this effect is only apparent for $r=3.5a$ (red) and it is also present for $n\left(r\right)<\langle n\rangle$.

Figure 4

Pair distribution function $g(r,\theta )$ for (a) $\varphi =8.7\%$ and (b) $\varphi =34.9\%$ with ${\rho }_p/{\rho }_f=2$ (${\text{Re}}_t=43.27$) and for (c) $\varphi =8.7\%$ with ${\rho }_p/{\rho }_f=5$ (${\text{Re}}_t=110.8$).

Figure 5

Angular averages of the pair distribution functions shown in Fig. 4. The solid lines are the average over the entire range $0\le \theta \le \pi /2$; the dashed lines are the average over a vertical sector $0\le \theta \le \pi /12$ and the dotted lines over a horizontal sector $5\pi /12\le \theta \le \pi /2$.

Figure 6

Particle collision frequency normalized as in (7) vs volume fraction for the systems simulated in this study. The symbols denote the particle-to-fluid density ratios ${\rho }^{*}={\rho }_p/{\rho }_f=2$ (asterisks), 3.3 (squares), 4 (circles), and 5 (triangles). The dashed lines are the predictions from the theory of granular flows.

Figure 7

Examples of the time dependence of the mean-square particle displacement in the vertical $\langle r_z^2\left(t\right)\rangle /2\tilde{u}td$ (upper lines) and horizontal $\langle r_{\perp }^2\left(t\right)\rangle /2\tilde{u}td$ directions. In (b) the displacement is normalized by ${\tilde{u}}^2{t}^2$. The horizontal lines are the long-time and short-time mean values.

Figure 8

Particle diffusivities in the (a) vertical and (b) horizontal directions normalized as in (11) and (c) the ratio of the two diffusivities, illustrating the marked anisotropy of the diffusion process in the system investigated. The dashed lines are the predictions from the theory of granular flows (11). The symbols denote the different particle-to-fluid density ratios ${\rho }_p/{\rho }_f=2$ (asterisks), 3.3 (squares), 4 (circles), and 5 (triangles).

Figure 9

Normalized particle mean free path defined in (12) in the (a) vertical and (b) horizontal directions. The dashed line is the kinetic theory prediction (13). The symbols denote the different particle-to-fluid density ratios ${\rho }_p/{\rho }_f=2$ (asterisks), 3.3 (squares), 4 (circles), and 5 (triangles).

Figure 10

Two examples of the velocity autocorrelation (14) for (a) $\varphi =8.7\%$ and (b) $\varphi =34.9\%$, both for ${\rho }_p/{\rho }_f=3.3$. The orange upper and blue lower lines are for the horizontal and vertical velocities, respectively. On the lower horizontal axis time is normalized by the particle relaxation time ${\tau }_p$ defined in (5). On the upper axis time is normalized by the particle settling timescale $\tilde{u}t/d$.

Figure 11

Normalized integral timescale defined in (15) in the (a) vertical and (b) horizontal directions. The symbols denote the different particle-to-fluid density ratios ${\rho }_p/{\rho }_f=2$ (asterisks), 3.3 (squares), 4 (circles), and 5 (triangles).

Figure 12

Integral timescale defined in (15) in the (a) vertical and (b) horizontal directions normalized by the collision frequency.

Figure 13

Open symbols show the particle diffusivities in the (a) vertical and (b) horizontal directions as obtained from integration of the velocity correlation normalized by $d\tilde{u}$; the closed symbols are the same as in Fig. 8 with this different normalization. The symbols denote the different particle-to-fluid density ratios ${\rho }_p/{\rho }_f=2$ (asterisks), 3.3 (squares), 4 (circles), and 5 (triangles).

Figure 14

Particle velocity fluctuations in the (a) vertical and (b) horizontal directions; (c) the ratio of the two, which indicates a marked anisotropy. The symbols denote the different particle-to-fluid density ratios ${\rho }_p/{\rho }_f=2$ (asterisks), 3.3 (squares), 4 (circles), and 5 (triangles).

Figure 15

Time dependence of the normalized mean tetrad radius of gyration defined in (23) for ${\rho }_p/{\rho }_f=2$ (gray), 3.3 (green), 4 (red), and 5 (blue).

Figure 16

Time dependence of the normalized eigenvalues of the shape tensor (22) for ${\rho }_p/{\rho }_f=2$ (gray), 3.3 (green), 4 (red), and 5 (blue).

Figure 17

Time dependence of the shape variance $\mathrm{\Delta }$ [Eq. (24)] and shape factor $S$ [Eq. (25)] for ${\rho }_p/{\rho }_f=2$ (gray), 3.3 (green), 4 (red), and 5 (blue). Increasing hue saturation corresponds to increasing particle volume fraction (see the Supplemental Material [18]).

Figure 18

Time dependence of the alignment of the tetrad principal axes with gravity. Colors refer to different particle-to-fluid density ratios ${\rho }_p/{\rho }_f=2$ (gray), 3.3 (green), 4 (red), and 5 (blue). Increasing hue saturation corresponds to increasing particle volume fraction (see the Supplemental Material [18]).

Figure 19

Time dependence of the alignment of the tetrad principal axes with the principal rate of strain at the initial instant corresponding to the largest eigenvalue. Colors refer to different particle-to-fluid density ratios ${\rho }_p/{\rho }_f=2$ (gray), 3.3 (green), 4 (red), and 5 (blue). Increasing hue saturation corresponds to increasing particle volume fraction (see the Supplemental Material [18]).