Consistent Boundary Conditions for 2D and 3D Lattice Boltzmann Simulations

Consistent formulations of 2D and 3D pressure and velocity boundary conditions along both the stationary and non-stationary plane wall and corner for lattice Boltzmann simulations are proposed. The unknown distribution functions are made function of local known distribution functions and correctors, where the correctors at the boundary nodes are obtained directly from the definitions of density and momentum. This boundary condition can be easily implemented on the wall and corner boundary using the same formulation. Discrete macroscopic equation is also derived for steady fully developed channel flow to assess the effect of the boundary condition on the solutions, where the resulting second order accurate central difference equation predicts continuous distribution across the boundary provided the boundary unknown distribution functions satisfy the macroscopic quantity. Three different local known distribution functions are experimented to assess both this observation and the applicability of the present formulation, and are scrutinized by calculating two-dimensional Couette-Poiseuille flow, Couette flow with wall injection and suction, lid-driven square cavity flow, and three-dimensional square duct flow. Numerical simulations indicate that the present formulation is second order accurate and the difference of adopting different local known distribution functions is as expected negligible, which are consistent with the results from the derived discrete macroscopic equation.