Consistent forcing scheme in the cascaded lattice Boltzmann method

In this paper, we give an alternative derivation for the cascaded lattice Boltzmann method (CLBM) within a general multiple-relaxation-time (MRT) framework by introducing a shift matrix. When the shift matrix is a unit matrix, the CLBM degrades into an MRT LBM. Based on this, a consistent forcing scheme is developed for the CLBM. The consistency of the nonslip rule, the second-order convergence rate in space, and the property of isotropy for the consistent forcing scheme is demonstrated through numerical simulations of several canonical problems. Several existing forcing schemes previously used in the CLBM are also examined. The study clarifies the relation between MRT LBM and CLBM under a general framework.

Figure 1

Comparison of relative errors of different forcing schemes $E_2$, as defined in Eq. (27), as a function of $s_3$, in steady Poiseuille flow.

Figure 2

The second-order accuracy of the new consistent forcing scheme, as measured by the relative errors $E_2$, defined in Eq. (27), as a function of grid sizes in steady Taylor-Green flow at $\mathrm{Re}=50,100$, and 150.

Figure 3

Steady-state density contours of single static droplet for varied $s_3$, achieved by different forcing schemes (from left to right): Premnath et al. [27] (first column), Lycett-Brown and Luo [28] (second column), De Rosis [29] (third column) and the new forcing scheme (fourth column). The additional dashed circle is the theoretical location of the droplet.

Figure 4

Isotropy defined by $I=\left|R_{\pi /4}/R_0\right|$, as a function of $s_3$ for different forcing schemes for single static droplet simulation.