. Author manuscript; available in PMC: 2020 Nov 15.

Published in final edited form as: J Comput Phys. 2019 Jul 18;397:108830. doi: 10.1016/j.jcp.2019.07.029

Table I.

Boundary conditions used in the numerical simulations.

Boundary	Macro-variables Conditions	Meso-variables Conditions
x direction boundaries	Periodic	Periodic
Upper wall	u = 0, φ = 0 and ∆ρ_c = 0	Bounce-back for f_i [32, 38, 40, 41,48] Bounce-back for g_i [32, 38, 40, 41, 48]
Lower wall	u=0, φ=φ₀ and ρ_c = ρ₀	Bounce-back for f_i $\frac{\partial g_{i}}{\partial z} = 0$