Forcing scheme analysis for the axisymmetric lattice Boltzmann method under incompressible limit

Because the standard lattice Boltzmann (LB) method is proposed for Cartesian Navier-Stokes (NS) equations, additional source terms are necessary in the axisymmetric LB method for representing the axisymmetric effects. Therefore, the accuracy and applicability of the axisymmetric LB models depend on the forcing schemes adopted for discretization of the source terms. In this study, three forcing schemes, namely, the trapezium rule based scheme, the direct forcing scheme, and the semi-implicit centered scheme, are analyzed theoretically by investigating their derived macroscopic equations in the diffusive scale. Particularly, the finite difference interpretation of the standard LB method is extended to the LB equations with source terms, and then the accuracy of different forcing schemes is evaluated for the axisymmetric LB method. Theoretical analysis indicates that the discrete lattice effects arising from the direct forcing scheme are part of the truncation error terms and thus would not affect the overall accuracy of the standard LB method with general force term (i.e., only the source terms in the momentum equation are considered), but lead to incorrect macroscopic equations for the axisymmetric LB models. On the other hand, the trapezium rule based scheme and the semi-implicit centered scheme both have the advantage of avoiding the discrete lattice effects and recovering the correct macroscopic equations. Numerical tests applied for validating the theoretical analysis show that both the numerical stability and the accuracy of the axisymmetric LB simulations are affected by the direct forcing scheme, which indicate that forcing schemes free of the discrete lattice effects are necessary for the axisymmetric LB method.

Figure 1

Axial velocity profile of the Hagen-Poiseuille flow obtained by using various forcing schemes within the Guo et al. model [25] framework.

Figure 2

Axial velocity profile (a), convergence of the velocity errors with respect to the mesh resolution (b) for the Hagen-Poiseuille flow obtained by using various forcing schemes within the Li et al. model [22] framework.

Figure 3

Flow structures of the cylindrical cavity flow with $R_A=1.5$ and $\mathrm{Re}=200$ obtained based on the Guo et al. model [25] with various forcing schemes. The left panel of each subplot reflects each forcing scheme: (a) the direct forcing scheme A (i.e., the sources terms in the LB equation for $\mathit{u}$ discretized by the direct forcing scheme, while the trapezium rule is applied for the evolution of the swirling velocity $u_{\theta }$); (b) the direct forcing scheme B (i.e., source terms for both distribution functions are discretized by the direct forcing scheme); (c) the semi-implicit centered scheme (which is only implemented for the LB equation for $\mathit{u}$ in the meridian plane while the source terms for the evolution of distribution for azimuthal velocity component $u_{\theta }$ are discretized by the trapezium rule). The right panel of each subplot shows the reference results from the original Guo et al. model [25].

Figure 4

Flow structures of the cylindrical cavity flow with $R_A=2.5$ and $\mathrm{Re}=200$ obtained based on the Guo et al. model [25] with various forcing schemes. The left panel of each subplot reflects each forcing scheme: (a) the direct forcing scheme A (i.e., the source terms in the LB equation for $\mathit{u}$ discretized by the direct forcing scheme, while the trapezium rule is applied for the evolution of the swirling velocity $u_{\theta }$); (b) the direct forcing scheme B (i.e., source terms for both distribution functions are discretized by the direct forcing scheme); (c) the semi-implicit centered scheme (which is only implemented for the LB equation for $\mathit{u}$ in the meridian plane while the source terms for the evolution of distribution for azimuthal velocity component $u_{\theta }$ are discretized by the trapezium rule). The right panel of each subplot shows the reference results from the original Guo et al. model [25].

Figure 5

Axial velocity profile along the symmetry axis obtained by using various forcing schemes within the Guo et al. model [25] framework: (a) for the case of $R_A=1.5$ and $\mathrm{Re}=200$; (b) for the case of $R_A=2.5$ and $\mathrm{Re}=200$.

Figure 6

Axial velocity profile along the symmetry axis obtained by using various forcing schemes within the Li et al. model [22] framework for test cases of $\left(R_A,\mathrm{Re}\right)$ equaling (a) (1.5, 990), (b) (1.5, 1290), (c) (2.5, 1010), and (d) (2.5, 2200).

Figure 7

Pressure contours obtained based on the Li et al. model [22] for the test cases of $\left(R_A,\mathrm{Re}\right)$ equaling (a) (1.5, 990), (b) (1.5, 1290), (c) (2.5, 1010), and (d) (2.5, 2200). The left panels show the profiles obtained with the direct forcing scheme B, while the right panels are the reference results obtained from the original Li et al. model [22].

Figure 8

Pressure contours obtained from the semi-implicit centered scheme within the Li et al. model [22] framework for the test cases of $\left(R_A,\mathrm{Re}\right)$ equaling (a) (1.5, 990), (b) (1.5, 1290), (c) (2.5, 1010), and (d) (2.5, 2200).