Lees-Edwards boundary conditions for lattice Boltzmann suspension simulations

When sheared suspensions are simulated, Lees-Edwards boundary conditions allow more realistic computational setups as they remove the need of a domain bounded by shearing walls (as in Couette-type flow) which bias typical flow structures. Lees-Edwards boundary conditions therefore allow investigation of pure bulk properties in a quasi-infinite system. In addition, they improve the computational efficiency of the simulations as the whole domain can be used to calculate averages. We propose an implementation of Lees-Edwards boundary conditions for lattice Boltzmann simulations of particulate suspensions, combined with an accurate treatment of fluid-particle interactions. The algorithm is validated using a simple single-particle benchmark and further applied to a fully resolved suspension flow. Shear-thickening behavior, which is prolonged to higher shear rates as compared to Couette flow results, could be observed.

Figure 1

(Color online) Illustration of the cases to be distinguished during the propagation of $f_5$ densities in a model case. Shown are some of the nodes of the upper boundary layer and a part of the lower layer at a Lees-Edwards boundary. Yellow (light gray) nodes belong to a particle crossing the boundary; blue (mid-gray) are fluid nodes. $\mathcal{G}$ denotes the application of the Galilean transform (17), $\mathcal{R}$ the application of the reflection operator (10).

Figure 2

$y$ component of the particle’s velocity ${\mathbf{v}}_{p,y}$ while the particle is freely moving at a ${\mathbf{v}}_{f,x}=0$ position of a shear flow with rate $\dot{\gamma }=1.25\times {10}^{-5}$. The whole system is superposed with a velocity ${\mathbf{v}}_s=(0.025,0.005)$, therefore expecting a constant ${\mathbf{v}}_{p,y}=0.005$. A systematic violation of Galilean invariance can be observed for the ALD method over the whole simulation time. During the period $t\approx 12000–15000$, when the particle crosses the LE boundary, strong deviations can be observed for both methods not applying the correction to the MEA. The maximum Mach number did not exceed $\mathrm{Ma}=0.0487$ in the course of the experiments.

Figure 3

(a) Snapshot of a sheared suspension of 133 disks of radius $R=8.0$ in a domain of $259\times 259$ lattice sites. The particle-shear Reynolds number ${\mathrm{Re}}_p\approx 1.5$. The grayscale code indicates low- (dark) and high-pressure (bright) areas. Note the typical particle and pressure field structures that would be inhibited by walls in a Couette scheme. (b), (c) Comparison of profiles as obtained by application of the Couette scheme (dotted line) and Lees-Edwards boundary conditions (solid line). (b) The local average particle density $\varphi \left(y\right)$ shows strong wall effects in the Couette case. Using LEBCs particles are homogeneously distributed over the system height. Both profiles were measured at ${\mathrm{Re}}_p=1.0$. (c) The velocity profiles $v_x\left(y\right)$ as obtained using Couette boundary conditions show increasing slip effects near the wall for ${\mathrm{Re}}_p=0.1$, 1.0. Using LEBCs at ${\mathrm{Re}}_p=1.0$ no systematic deviations from the Newtonian profile can be seen.

Figure 4

Increase of the relative apparent viscosity ${\nu }_{\mathrm{app}}$ as a function of the solid-fluid ratio $\varphi$ for $\mathrm{Re}⪡1$ as obtained by a simulation using CMES compared to the theoretical behavior proposed by the Krieger-Dougherty and Einstein approximations for small $\varphi$. A comparison between Couette and LEBC results shows no significant difference in this regime.

Figure 5

Apparent viscosity ${\nu }_{\mathrm{app}}$ as a function of the particle shear Reynolds number ${\mathrm{Re}}_p$ obtained using the Couette flow scheme and Lees-Edwards boundary conditions. For a comparison also shear-thickening results obtained by applying the ALD method with the Couette flow scheme [8] are shown. Using LEBCs the increase of ${\nu }_{\mathrm{app}}$ continues over the range of reachable ${\mathrm{Re}}_p$.