Eliminating spurious velocities in the free-energy lattice Boltzmann method

Spurious velocities are unphysical currents that appear close to curved interfaces in diffuse interface methods. We analyze the causes of these spurious velocities in the free-energy lattice Boltzmann algorithm. By making a suitable choice of the equilibrium distribution, and by finding the best way to numerically calculate derivatives, we show that these velocities may be decreased by an order of magnitude compared to previous models. Furthermore, we propose a momentum conserving forcing method that reduces spurious velocities by another factor of $\sim 5$. In three dimensions we find that 19 velocity vectors is the minimum number necessary.

Figure 1

The steady-state velocity profile around a droplet using (a) a standard choice of equilibrium distribution and (b) an improved choice of equilibrium.

Figure 2

The maximum spurious velocity $v$ (in units of lattice velocity $c$) as a function of (a) the equilibrium distribution parameters $p=w_{5–8}^p$ (solid line), $p=w_{5–8}^t$ (dashed line), $p=w_{5–8}^{xx}$ (dotted line), and (b) the stencil parameters $p=B$ (solid line), $p=D$ (dashed line), and the force stencil parameter $p=F$ (dotted line).

Figure 3

The variation in spurious velocity $v$ (in units of lattice velocity $c$) as a function of $\tau$ for the standard lattice Boltzmann (solid line) and for the new forcing method (dashed line).