Analysis and assessment of the no-slip and slip boundary conditions for the discrete unified gas kinetic scheme

The discrete unified gas kinetic scheme (DUGKS) with a force term is a finite volume solver for the Boltzmann equation. Unlike the standard lattice Boltzmann method (LBM), DUGKS can be applied on nonuniform grids. For both the LBM and DUGKS, the boundary conditions need to be processed through the density distribution function. So researchers introduced the boundary conditions from the LBM frame into the DUGKS. However, the accuracy of these boundary conditions in the DUGKS has not been studied thoroughly. Through strict theoretical deduction, we find that the bounce-back (BB) scheme leads to a different dependence of the numerical error term in the DUGKS as compared to the LBM. The error term is influenced by the relaxation time and the body force. And it can be reduced by lowering the kinetic viscosity. Unlike the BB scheme, the nonequilibrium bounce-back scheme has the ability to implement real no-slip boundary condition. Furthermore, two slip boundary conditions incorporated with Navier's slip model are introduced from the LBM framework into the DUGKS. The tangential momentum change-based (TMAC) scheme can be used directly in the DUGKS because it generates no numerical error term in the DUGKS. For the combination of the bounce-back and specular reflection schemes (BSR), the relation between the slip length and the combination parameter should be modified in accordance with the numerical error term. Analysis shows that the TMAC scheme can simulate a wider range of slip length than the BSR scheme. Numerical simulations of the Couette flow and the Poiseuille flow confirm our theoretical analysis.

Figure 1

Schematic of the control volume near the boundary

Figure 2

Computational domain of the Couette flow.

Figure 3

Velocity distributions of the Couette flow for varied kinetic viscosities ($Ny=20, B=0$).

Figure 4

The velocity at the wall of the Couette flow for varied kinetic viscosities ($Ny=20, B=0$).

Figure 5

Velocity distributions of the Couette flow for varied meshes ($\nu =0.1, B=0$).

Figure 6

The velocity at the wall of the Couette flow for varied meshes ($\nu =0.1, B=0$).

Figure 7

The dependence of the dimensionless slip length $B$ on the combination parameter ($r$: square and $\lambda$: circle) when $\nu =0.1$.

Figure 8

Velocity profiles of the Couette flow with varied $B$ ($Ny=20, \nu =0.1$).

Figure 9

Velocity profiles of the Couette flow with varied $\nu$ ($Ny=20, B=0.2$).

Figure 10

Velocity profiles of the Couette flow with varied meshes ($\nu =0.1, B=0.2$).

Figure 11

Sketch of the Poiseuille flow driven by a constant body force.

Figure 12

Velocity profiles of the no-slip Poiseuille flow for different $\nu$; (b) is the magnified view of (a).

Figure 13

Numerical errors generated by the BB scheme for the no-slip Poiseuille flow.

Figure 14

Velocity profiles of the no-slip Poiseuille flow for different meshes; (b) is the magnified view of (a).

Figure 15

Velocity profiles of the slip Poiseuille flow for different $B$; (b) is the magnified view of (a).

Figure 16

Velocity profiles of the slip Poiseuille flow for different $\nu$ ($Ny=20, B=0.2$).

Figure 17

Velocity profiles of the slip Poiseuille flow for different meshes; (b) is the magnified view of (a).