Corrected momentum exchange method for lattice Boltzmann simulations of suspension flow

Standard methods for lattice Boltzmann simulations of suspended particles, based on the momentum exchange algorithm, might lack accuracy or violate Galilean invariance in some particular situations. Aiming at simulations of dense suspensions in high-shear flows, we motivate and investigate necessary correction terms. We propose an approach which, combining accurate treatments of fluid-structure interaction and moving boundaries, is able to preserve Galilean invariance in relevant orders and to improve the physical behavior of the system. We validate the approach in a comparison with standard methods in simple test problems.

Figure 1

(Color online) Simulating sheared suspensions particles are immersed in a varying velocity field. Galilean invariance plays a relevant role: dynamical properties of the system must be maintained independent from the flow velocity or gradient.

Figure 2

(Color online) Sketch of computational fluid and solid domains. Boundary conditions (12) are applied at the boundary links, intersecting the solid-fluid interface, and the momentum exchange algorithm (13) is used to evaluate the fluid-particle interaction.

Figure 3

(Color online) LBM approaches for suspended particles. Left: Ladd method: Particles are represented by a solid shell. Inner points, belonging to the physical solid domain, are treated as normal fluid nodes. Center: ALD method. The inner fluid has no physical meaning, but it is still used to initialize new fluid nodes. Solid nodes of different particles are turned into fluid nodes if they are in contact. Bridge links are created via which lubrication force elements are applied. Right: In the present work we use an improved method applying a corrected momentum exchange (21) in combination with a refill method (22). The latter removes the need for an explicit simulation of inner fluid.

Figure 4

(Color online) (a) Applying Lees-Edwards boundary conditions, periodic copies of the system move with horizontal velocity. (b) Zoomed into the region where the black arrow points to in (a), the propagation of densities $f_5$ from nodes at the top of the original reference frame in the case a particle crosses the LE boundary are shown.

Figure 5

The benchmark $D_1$: single particle crossing the boundary between different reference frames.

Figure 6

The vertical speed $v_y$ of a single particle in a sheared system $(\dot{\gamma }=5\times {10}^{-5})$ crossing a LEbc for different superposed velocities $u_{s,x}$. At about $t=12300$ the particle reaches the LEbc and crosses it within a time $t_c\approx 2000$.

Figure 7

The vertical speed $v_y$ of a single particle in a sheared system crossing a LEbc for different shear rates $\dot{\gamma }$. The particle is moving with the fluid at a superposed background velocity $u_s=(0.05,0.005)$.

Figure 8

Test $D_2$ a: Gap size between two particles that feel horizontal forces pushing them together for different absolute values of the superposed velocity ${\mathbf{u}}_s$. The direction is ${\mathbf{u}}_s\Vert {\mathbf{r}}_{1,2}$. The three plots show results obtained with different suspension models.

Figure 9

Test $D_2$ b: Gap size between two particles for different angles $\alpha =\angle ({\mathbf{u}}_s,{\mathbf{r}}_{1,2})$. Results for the three suspension models are shown.