Issues associated with Galilean invariance on a moving solid boundary in the lattice Boltzmann method

In lattice Boltzmann simulations involving moving solid boundaries, the momentum exchange between the solid and fluid phases was recently found to be not fully consistent with the principle of local Galilean invariance (GI) when the bounce-back schemes (BBS) and the momentum exchange method (MEM) are used. In the past, this inconsistency was resolved by introducing modified MEM schemes so that the overall moving-boundary algorithm could be more consistent with GI. However, in this paper we argue that the true origin of this violation of Galilean invariance (VGI) in the presence of a moving solid-fluid interface is due to the BBS itself, as the VGI error not only exists in the hydrodynamic force acting on the solid phase, but also in the boundary force exerted on the fluid phase, according to Newton's Third Law. The latter, however, has so far gone unnoticed in previously proposed modified MEM schemes. Based on this argument, we conclude that the previous modifications to the momentum exchange method are incomplete solutions to the VGI error in the lattice Boltzmann method (LBM). An implicit remedy to the VGI error in the LBM and its limitation is then revealed. To address the VGI error for a case when this implicit remedy does not exist, a bounce-back scheme based on coordinate transformation is proposed. Numerical tests in both laminar and turbulent flows show that the proposed scheme can effectively eliminate the errors associated with the usual bounce-back implementations on a no-slip solid boundary, and it can maintain an accurate momentum exchange calculation with minimal computational overhead.

Figure 1

Configuration of a boundary node that interacts with a moving wall.

Figure 2

Sketch of a Poiseuille flow driven by a constant body force between two inclined walls.

Figure 3

The steady-state velocity profiles when different reference (moving-wall) velocities are applied: (a) with Bouzidi et al.'s [8] bounce-back scheme, (b) with Yu et al.'s [5] bounce-back scheme, and (c) with a simple bounce-back scheme [1].

Figure 4

The time-dependent total hydrodynamic force acting on the two walls calculated with different momentum exchange schemes: (a) with Bouzidi et al.'s [8] bounce-back scheme; (b) zoom-in plot of (a); (c) with Yu et al.'s [5] bounce-back scheme; and (d) zoom-in plot of (c). $L_0=\sqrt{h^2+L^2}$.

Figure 5

Sketch of a boundary node that interacts with a moving wall, used to illustrate our proposed bounce-back implementation.

Figure 6

The velocity profiles at different times using the proposed bounce-back scheme with $u_w=15u_c$: (a) based on Bouzidi et al.'s [8] bounce-back scheme, (b) based on Yu et al.'s [5] bounce-back scheme, and (c) based on a simple bounce-back scheme [1]. The dimensionless time is defined as $t^{*}=H^2/\left(4\nu \right)$.

Figure 7

The steady-state velocity profiles using the proposed bounce-back scheme with different reference velocities: (a) based on Bouzidi et al.'s [8] bounce-back scheme, (b) based on Yu et al.'s [5] bounce-back scheme, and (c) based on a simple bounce-back scheme [1].

Figure 8

The time-dependent total hydrodynamic force acting on the two walls with different wall velocities using the proposed bounce-back scheme: (a) based on Bouzidi et al.'s [8] bounce-back scheme; (b) zoom-in plot of (a); (c) based on Yu et al.'s [5] bounce-back scheme; and (d) zoom-in plot of (c). $L_0=\sqrt{h^2+L^2}$.

Figure 9

A sketch of boundary configuration in a turbulent channel flow.

Figure 10

Grid arrangement in the turbulent channel flow simulation.

Figure 11

Pressure contours in a turbulent channel flow simulation indicating the existence of checkerboard instability. Top row: pressure contours at a streamwise location $x=nx/2$ at (a) $t=7000$, (b) $t=7500$, and (c) $t=8000$. Bottom row: pressure contours at a spanwise location $z=nz/2$ at (d) $t=7000$ and (e) $t=8000$.

Figure 12

The averaged mean velocity profiles at the stationary stage in the turbulent channel flow simulation.

Figure 13

The averaged Reynolds stress profiles at the stationary stage in the turbulent channel flow simulation.

Figure 14

The averaged rms velocity profiles at the stationary stage in the turbulent channel flow simulation.

Figure 15

The normalized total hydrodynamic force acting on the two walls at stationary state in the turbulent channel flow.

Figure 16

Sketch of the turbulent pipe flow simulation.

Figure 17

The averaged mean velocity profiles at the stationary stage, relative to the pipe wall, in the turbulent pipe flow simulation.

Figure 18

The averaged rms velocity profiles at the stationary stage in the turbulent pipe flow simulation.

Figure 19

The normalized total hydrodynamic force acting on the two walls at stationary state in the turbulent pipe flow.