Asymptotic equivalence of forcing terms in the lattice Boltzmann method within second-order accuracy

We show the asymptotic equivalence of two forcing schemes in the lattice Boltzmann method (LBM) within second-order accuracy through the asymptotic analysis instead of the Chapman–Enskog analysis. We consider the single relaxation time LBM with the following two forcing schemes: the simplest scheme by He et al. [J. Stat. Phys. 87, 115 (1997)] (referred to as He forcing); the most popular scheme by Guo et al. [Phys. Rev. E 65, 046308 (2002)] (referred to as Guo forcing). It has been shown by using the Chapman–Enskog analysis that the He forcing leads the unphysical terms in the macroscopic equations due to the spatial and time derivatives of the body force, whereas the Guo forcing does not lead such terms. However, we find by using the asymptotic analysis that the order of the unphysical terms is comparable to or less than ${\left(\mathrm{\Delta }x\right)}^3$ for the continuity equation and ${\left(\mathrm{\Delta }x\right)}^4$ for the Navier–Stokes equations (where $\mathrm{\Delta }x$ is the lattice spacing). Therefore, not only the Guo forcing but also the He forcing give the macroscopic flow velocity and pressure for incompressible viscous fluid with relative errors of $O\left[{\left(\mathrm{\Delta }x\right)}^2\right]$. To verify the result of the asymptotic analysis, we simulate two benchmark problems in which the body force is changed in space and time: a generalized Taylor–Green problem and a natural convection problem. As a result, we find that the calculated results of macroscopic variables by the He forcing converge to those by the Guo forcing at the second-order convergence rate. Therefore, we can conclude that the He forcing and the Guo forcing are equivalent within the second-order accuracy even for the space- and time-dependent body force.

Figure 1

Time variations of the maximum errors of the pressure $p$, the flow velocity $\mathit{u}$, and the divergence of velocity $\mathbf{\nabla }·\mathit{u}$ relative to the analytical solution $p_{\mathrm{a}}$ and ${\mathit{u}}_{\mathrm{a}}$ for (a) the He forcing and (b) the Guo forcing.

Figure 2

Maximum errors of the pressure $p$, the flow velocity $\mathit{u}$, and the divergence of velocity $\mathbf{\nabla }·\mathit{u}$ relative to the analytical solution $p_{\mathrm{a}}$ and ${\mathit{u}}_{\mathrm{a}}$ at the nondimensional time $\omega t/\pi =10$ as functions of the lattice spacing $\mathrm{\Delta }x$ for the He forcing and the Guo forcing.

Figure 3

Computational domain for a natural convection problem.

Figure 4

Time variations of the averaged Nusselt number $\mathrm{Nu}$ obtained by the He forcing and the Guo forcing for $H=100\mathrm{\Delta }x$. The benchmark solution $\mathrm{Nu}=4.5275$ obtained by Hortmann et al. [16] is also shown.

Figure 5

The difference between the results by the two forcing schemes, where $p_{\mathrm{He}}, {\mathit{u}}_{\mathrm{He}}$, and $T_{\mathrm{He}}$ are the pressure, flow velocity, and temperature obtained by the He forcing, respectively, and $p_{\mathrm{Guo}}, {\mathit{u}}_{\mathrm{Guo}}$, and $T_{\mathrm{Guo}}$ are the results by the Guo forcing at the nondimensional time $t/(H/u_0)=20$.