Mass and momentum transfer across solid-fluid boundaries in the lattice-Boltzmann method

Mass conservation and momentum transfer across solid-fluid boundaries have been active topics through the development of the lattice-Boltzmann method. In this paper, we review typical treatments to prevent net mass transfer across solid-fluid boundaries in the lattice-Boltzmann method, and argue that such efforts are in general not necessary and could lead to incorrect results. Carefully designed simulations are conducted to examine the effects of normal boundary movement, tangential density gradient, and lattice grid resolution. Our simulation results show that the global mass conservation can be well satisfied even with local unbalanced mass transfer at boundary nodes, while a local mass conservation constraint can produce incorrect flow and pressure fields. These simulations suggest that local mass conservation, at either a fluid or solid boundary node, is not only an unnecessary consequence to maintain the global mass conservation, but also harmful for meaningful simulation results. In addition, the concern on the momentum addition and reduction associated with status-changing nodes is also not technically necessary. Although including this momentum addition or reduction has no direct influence on flow and pressure fields, the incorrect fluid-particle interaction may affect simulation results of particulate suspensions.

Figure 1

A schematic to illustrate the velocity inter- and extrapolation BB boundary treatment. Different symbols are employed to indicate the node types: circles,- fluid nodes; squares,- solid nodes; solid symbols,- boundary nodes; and open symbols,- interior nodes.

Figure 2

A schematic to illustrate the fluid volume fraction $\beta$ of each lattice node for the calculation of total fluid mass with Eq. (18). The dashed squares show the lattice cells centered at individual lattice nodes. Different symbols are employed to indicate the node types: circles,- fluid nodes; squares,- solid nodes; solid symbols,- boundary nodes; open symbols,- interior nodes.

Figure 3

Location (a) and velocity (b) of the right wall and the total fluid mass between the two plates (c). The dashed line $x=290$ in (a) intersects with the right wall location (solid line) at $t-t_0=524$, at which instant the fluid mass from Eq. (20) displays an abrupt drop in (c).

Figure 4

Fluid density (top) and velocity (bottom) profiles form simulations with (right) and without (left) LMC considered. The $t-t_0$ values (the time steps after the right wall starts to move) when the profiles are obtained are also displayed near the curves.

Figure 5

Flow velocity (arrows) and pressure (background colors) fields obtained from simulations with a moving cylinder (System A, top) and moving walls (System B, bottom) with (right) or without (left) LMC implemented. For the system symmetry, only half of the simulation domain is shown in each panel. The cylinder center locates at $x=100$.

Figure 6

Flow density (a) and horizontal velocity (b) variations along the channel centerline. The cylinder center locates at $x=100$ and there is no density for the solid region $60<x<140$.

Figure 7

Geometric structure and flow field (a) arrows for the flow field and background colors for the pressure distribution; those displayed with the color scale bar are the $\rho -{\rho }_0$ values), mass flux (b), and incoming-outgoing PDF-flux differences at fluid (${\rho }_f$) and solid (${\rho }_s$) boundary nodes across the lower boundary (c).

Figure 8

The bounce-back process at the lower flat wall in Fig. 7a. (a) The PDFs linked to the same solid boundary node at $x=i$, and (b) the PDFs linked to the same fluid boundary node at $x=i$.

Figure 9

Domain configuration (a) and local boundary region (b) for the shear flow simulation with a gap distance of $h=15$. The rectangle box in (a) indicates the region enlarged in (b). In (b) different symbol styles are employed to indicate the node types: circles, fluid nodes; squares, solid nodes; solid symbols, boundary nodes; and open symbols, interior nodes. The boundary lattice links are also displayed in (b), with the red solid lines for a particular period of the boundary links, and red dashed lines for other links.

Figure 10

Plate locations (top) and fluid-plate interaction forces [bottom, blue solid lines from Eq. (28) and red dashed lines from Eq. (17)] for the left (left panel) and right (right panel) plates. The dashed lines in the top panels are plotted to indicate the time instants when the plates cross a grid line. The insets in the bottom panels display the details of the force spikes when the plates cross grid lines indicated in the top panels.