Collision model for fully resolved simulations of flows laden with finite-size particles

We present a collision model for particle-particle and particle-wall interactions in interface-resolved simulations of particle-laden flows. Three types of interparticle interactions are taken into account: (1) long- and (2) short-range hydrodynamic interactions, and (3) solid-solid contact. Long-range interactions are incorporated through an efficient and second-order-accurate immersed boundary method (IBM). Short-range interactions are also partly reproduced by the IBM. However, since the IBM uses a fixed grid, a lubrication model is needed for an interparticle gap width smaller than the grid spacing. The lubrication model is based on asymptotic expansions of analytical solutions for canonical lubrication interactions between spheres in the Stokes regime. Roughness effects are incorporated by making the lubrication correction independent of the gap width for gap widths smaller than $\sim 1\%$ of the particle radius. This correction is applied until the particles reach solid-solid contact. To model solid-solid contact we use a variant of a linear soft-sphere collision model capable of stretching the collision time. This choice is computationally attractive because it allows us to reduce the number of time steps required for integrating the collision force accurately and is physically realistic, provided that the prescribed collision time is much smaller than the characteristic time scale of particle motion. We verified the numerical implementation of our collision model and validated it against several benchmark cases for immersed head-on particle-wall and particle-particle collisions, and oblique particle-wall collisions. The results show good agreement with experimental data.

Figure 1

Schematic representation of an oblique interparticle collision. For the sake of clarity we considered in this figure a reference frame moving with the light gray particle, which implies that the velocities sketched are relative velocities. $F_n$ and $F_t$ denote the normal and tangential component of the collision force.

Figure 2

Numerical solution of the model of Maw et al. [26] for collisions between glass spheres, compared with experimental data of Foerster et al. [27] and the model of Walton [28]. The curve and experimental data were extracted from a curve ${\mathrm{\Psi }}_{\mathrm{out}}$ vs ${\mathrm{\Psi }}_{\mathrm{in}}$ of Ref. [27] and rescaled to ${\psi }_{\mathrm{out}}$ vs ${\psi }_{\mathrm{in}}$ with the parameters of their homogeneous $3\mathrm{mm}$ glass spheres, $\nu =0.22$ and ${\mu }_c=0.092$. The two vertical dotted lines delimit the three different types of impact and are given by ${\psi }_{\mathrm{in}}=1$ and ${\psi }_{\mathrm{in}}=4\chi -1\approx 4.2$.

Figure 3

(a) Linear spring-dashpot model. (b) Notation and reference frame adopted for an interparticle collision.

Figure 4

Schematic representation of the lubrication model. We illustrate the case of particle-wall interactions for the sake of simplicity. The model is analogous for particle-particle interactions.

Figure 5

Lubrication corrections for the cases of normal particle-wall (a) and particle-particle (b) interactions, compared against the analytical solution of Brenner [31] and Cooley and ONeill [35]. Results shown for two different resolutions, $D_p/\mathrm{\Delta }x=16$ and 32.

Figure 6

Trajectory (a) and time evolution of the particle velocity (b) in the bouncing motion of a steel sphere colliding onto a planar surface in silicon oil RV10.

Figure 7

Trajectories obtained from simulations of particles colliding onto a planar surface in silicon oil, for different impact Stokes numbers with (a) and without (b) closure for lubrication interactions. Experimental data from Gondret et al. [9].

Figure 8

Sensitivity analysis to the time step, substepping, and stretching of the collision time (a) and outcome of cases SA3 and SA4 when problematic Lagrangian forcing points are excluded from the IBM forcing scheme (b). Computational parameters in Table 3.

Figure 9

Normal, wet coefficients of restitution for particle-wall collisions. The experimental data were normalized with the value $e_{n,d}=0.97$ measured in the reference.

Figure 10

Trajectories of the particles' contact points (results of the numerical simulations were shifted vertically for clarity). The solid line was extracted from Ref. [25] (a). Wet coefficients of restitution for particle-particle collisions (b). The experimental data were normalized with the value $e_{n,d}=0.97$ measured in the reference.

Figure 11

Results for oblique collision simulations in a dry system (a), and in a viscous liquids (b). Experimental data of Joseph and Hunt [23].