Force methods for the two-relaxation-times lattice Boltzmann

The two-relaxation-times collision benefits the steady lattice Boltzmann method by yielding viscosity-independent numerical errors. We present in an intuitive way how to incorporate popular force methods into the two-relaxation-times collision. We subsequently rewrite force methods into a generic equation to reveal commonalities and differences. We prove that force methods with a second-order velocity moment of the force break the viscosity independence. A force method with only a first-order velocity moment of the force averts this breakage. We validate our proof numerically.

Figure 1

Four force methods affecting the viscosity dependence of the pressure error and velocity error. $\mathrm{Re}=1$ and $\mathrm{\Lambda }=\frac{1}{5}$.

Figure 2

Four force methods affecting the $\mathrm{\Lambda }$ dependence of the pressure error and velocity error. $\mathrm{Re}=1$ and $\nu =1$.

Figure 3

Velocity streamlines of the benchmark flows.