Inertial effects in shear flow of a fluid-particle mixture: Resolved simulations

This paper presents results of the resolved simulation of particles suspended in a fluid in shear flow. Unlike most of the existing work on this subject, the ratio of the particle density to that of the suspending fluid is not 1 but varies between 2.5 and 10, with a focus on inertial effects rather than gravity, which is not considered. Two particle Reynolds numbers are considered, 20 and 40, while the average particle volume fraction varies from 5% to 33%. The study focuses on the effects of inertia on this class of flows. It is shown that the memory of the particles' earlier state of motion associated with inertia is responsible for significant effects. Among others, the collision rate is increased and, with it, the rate of momentum transfer from the moving walls into the bulk of the fluid-particle mixture; normal stress differences become larger and the particle-rich layer near the walls becomes denser. Comparable effects are found by increasing the particle density and the particle Reynolds number. The two normal stress differences are found to be both negative and comparable in spite of their very different physical origin. Additional results concern the wall slip, the stresslet, the effects of a nonzero collisional tangential force, the wall stress, and others. The great importance of particle-particle and particle-wall collisions reveals itself in influencing the rheology of the mixture, the normal stress differences, and the transport of momentum.

Figure 1

Local time-averaged particle volume fraction between the plates. (a) Effect of the average volume fraction for ${\mathrm{Re}}_p$ = 20, ${\rho }_p/\rho$ = 5 and, in ascending order, $\varphi$ = 5% (orange, case 3), 21% (blue, case 1), and 33% (green, case 5); (b) Effect of the particle density for ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21% and in ascending order near the walls, ${\rho }_p/\rho$ = 2.5 (orange, case 7), 5 (blue, case 1), and 10 (green, case 9); (c) Effect of the Reynolds number for $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 and, in ascending order near the wall, ${\mathrm{Re}}_p$ = 20 (blue, case 1) and ${\mathrm{Re}}_p$ = 40 (orange, case 2).

Figure 2

Average normalized collision rate as a function of the dimensionless time for, in ascending order, ${\mathrm{Re}}_p$ = 20, $\varphi$ = 5%, ${\rho }_p/\rho$ = 5 (red, case 3); ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 2.5 (brown, case 7); ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (blue, case 1); ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (black, case 2); ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 10 (green, case 9); ${\mathrm{Re}}_p$ = 20, $\varphi$ = 33%, ${\rho }_p/\rho$ = 5 (orange, case 5).

Figure 3

Collision rate normalized according to Eq. (6), the collision rate in granular flow, shown by the dashed line. Circle: ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 1); triangle: ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 2); square: ${\mathrm{Re}}_p$ = 20, $\varphi$ = 5%, ${\rho }_p/\rho$ = 5 (case 3); star: ${\mathrm{Re}}_p$ = 20, $\varphi$ = 33%, ${\rho }_p/\rho$ = 5 (case 5); pentagon: ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%%, ${\rho }_p/\rho$ = 2.5 (case 7); diamond: ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 10 (case 9).

Figure 4

Phase velocities normalized by the plate velocity $U$; the dashed line shows the linear Couette velocity profile. (a) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 1); (b) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 33%, ${\rho }_p/\rho$ = 5 (case 5); (c) ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 2); (d) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 10 (case 9). The particle and fluid velocity are indistinguishable except very near the walls, an area enlarged in Fig. 7.

Figure 5

Sketch of the velocity distribution in a system in which the effective viscosity near the walls is lower than in the bulk.

Figure 6

Root-mean-square normalized difference $u_{\text{devi}}$ between the mean fluid velocity and the velocity of the linear Couette profile at the same position for the cases considered in this study. Circle: ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho =5$ (case 1); triangle: ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho =5$ (case 2); square: ${\mathrm{Re}}_p$ = 20, $\varphi$ = 5%, ${\rho }_p/\rho =5$ (case 3); star: ${\mathrm{Re}}_p=20$, $\varphi$ = 33%, ${\rho }_p/\rho =5$ (case 5); pentagon: ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%%, ${\rho }_p/\rho =2.5$ (case 7); diamond: ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho =10$ (case 9).

Figure 7

Detail of phase velocities near the walls. The dotted line is the linear Couette velocity profile, the solid line is the fluid velocity, and the dashed and dash-dotted lines the particle material velocity including and excluding the particle rotation, respectively. (a) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 1); (b) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 33%, ${\rho }_p/\rho$ = 5 (case 5); (c) ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 2); (d) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 10 (case 9).

Figure 8

Distribution of the normalized particle velocity fluctuations in the gap between the plates; the rightmost group of points (blue) are the fluctuations in the streamwise direction and the others (green and yellow) those in the orthogonal plane. (a) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 1); (b) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 33%, ${\rho }_p/\rho$ = 5 (case 5); (c) ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 2); (d) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 10 (case 9).

Figure 9

Particle velocity correlation coefficient vs. the dimensionless delay time $\dot{\gamma }\tau$; the left panel is the correlation coefficient in the flow direction and the right panel that in the direction normal to the plates. Here ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, and ${\rho }_p/\rho$ = 2.5 (orange, case 7), 5 (blue, case 1), and 10 (green, case 9).

Figure 10

Particle velocity correlation coefficient in the direction normal to the plates vs. the dimensionless delay time $\dot{\gamma }\tau$. The left panel highlights the effect of the average volume fraction for ${\mathrm{Re}}_p$ = 20 and ${\rho }_p/\rho$ = 5; the lines are from left to right, for $\varphi$ = 33% (green, case 5), 21% (blue, case 1), and 5% (orange, case 3). The right panel shows the very weak effect of inertia for $\varphi$ = 21%: ${\mathrm{Re}}_p$ = 40 ${\rho }_p/\rho$ = 5 (orange, case 2), ${\mathrm{Re}}_p$ = 20 ${\rho }_p/\rho$ = 5 (blue, case 1), and 10 (green, case 9).

Figure 11

Contributions to the bulk stress: fluid stress (green), particle stress Eq. (11) (blue) and collisional stress, Eq. (12) (orange). (a) ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho =5$ (case 1); (b) ${\mathrm{Re}}_p=20$, $\varphi$ = 33%, ${\rho }_p/\rho =5$ (case 5); (c) ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho =5$ (case 2); (d) ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho =10$ (case 9).

Figure 12

Time-averaged distribution of the three contributions to the stress in the gap between the plates: fluid stress (green), particle stress (blue), and collisional stress (orange). (a) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 1); (b) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 33%,${\rho }_p/\rho$ = 5 (case 5); (c) ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 2); (d) ${\mathrm{Re}}_p$ = 20, $\varphi$ = 21%, ${\rho }_p/\rho$ = 10 (case 9). The stress components are normalized by division by $\mu \dot{\gamma }$.

Figure 13

Contributions to the wall stress from the fluid (orange) and particle-wall collisions (blue), nondimensionalized by division by $\mu \dot{\gamma }$, vs. nondimensional time $\dot{\gamma }t$. (a) ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho =5$ (case 1); (b) ${\mathrm{Re}}_p=20$, $\varphi$ = 33%, ${\rho }_p/\rho =5$ (case 5); (c) ${\mathrm{Re}}_p=40$, $\varphi$ = 21%, ${\rho }_p/\rho =5$ (case 2); (d) ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho =10$ (case 9).

Figure 14

Fluid contribution to the wall stress as a function of the nondimensional time $\dot{\gamma }t$. In ascending order the curves are for ${\mathrm{Re}}_p=20$, $\varphi$ = 5%, ${\rho }_p/\rho$ = 5 (red, case 3); ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho =5$ (blue, case 1); ${\mathrm{Re}}_p=20$, $\varphi$ = 21%%, ${\rho }_p/\rho =2.5$ (brown, case 7); ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho =10$ (green, case 9); ${\mathrm{Re}}_p=40$, $\varphi$ = 21%, ${\rho }_p/\rho =5$ (black, case 2); ${\mathrm{Re}}_p=20$, $\varphi$ = 33%, ${\rho }_p/\rho =5$ (orange, case 5).

Figure 15

Projection on the plane of shear of the pair distribution function in the neighborhood of the test particle. (a) ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 1); (b) ${\mathrm{Re}}_p=20$, $\varphi$ = 33%, ${\rho }_p/\rho$ = 5 (case 5); (c) ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (case 2); (d) ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho$ = 10 (case 9).

Figure 16

Examples of the angular dependence of the pair distribution function at contact. Both figures are for ${\mathrm{Re}}_p=20$ and ${\rho }_p/\rho$ = 5, in (a) for $\varphi$ = 21% (case 1) and in (b) for $\varphi$ = 33% (case 5).

Figure 17

Frontal three-dimensional view of the pair distribution function on contact with the test particle. The pixels are colored according to the frequency of contacts. The test particle moves out of the page, the gradient direction is up and the vorticity direction horizontally to the right. Both images are for ${\mathrm{Re}}_p=20$ and $\varphi$ = 21%; (a) ${\rho }_p/\rho$ = 5 (case 1); (b) ${\rho }_p/\rho$ = 10 (case 9). A similar picture is found on the back of the sphere with the contacts occurring in the upper hemisphere.

Figure 18

Local time-averaged particle volume fraction between the plates for the reference case with $\varphi$ = 21%, ${\rho }_p/\rho$ = 5, ${\mathrm{Re}}_p=20$. Orange: zero tangential collision force; blue: nonzero tangential collision force; the latter results are the same as in Fig. 1.

Figure 19

Time-averaged particle phase velocity normalized by the plate velocity $U$ with (dashed) and without (solid) a tangential component of the collision force; the dotted line shows the linear Couette velocity profile. Here ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5. The fluid velocity is nearly coincident with that of the particle phase except very near the walls (see Fig. 7).

Figure 20

Normalized particle angular velocity ${\mathrm{\Omega }}_y^{*}$ defined in Eq. (13) vs. normalized time $\dot{\gamma }t$. In descending order the curves are for ${\mathrm{Re}}_p=20$, $\varphi$ = 5%, ${\rho }_p/\rho$ = 5 (red, case 3); ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho$ = 10 (green, case 9); ${\mathrm{Re}}_p$ = 40, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (black, case 2); ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho$ = 5 (blue, case 1); ${\mathrm{Re}}_p=20$, $\varphi$ = 21%, ${\rho }_p/\rho$ = 2.5 (brown, case 7); ${\mathrm{Re}}_p=20$, $\varphi$ = 33%, ${\rho }_p/\rho$ = 5 (orange, case 5).