Moments-based method for boundary conditions in the lattice Boltzmann framework: A comparative analysis for the lid driven cavity flow

Abstract

Dealing with boundary conditions (BC) was ever considered a puzzling question in the lattice Boltzmann (LB) method. The most popular BC models are based on Ad-Hoc rules and, although these BC models were shown to be suitable for low-order LB equations, their extension to high-order LB was shown to be a very difficult problem and, at authors knowledge, never solved with satisfaction. The main question to be solved is how to deal with a problem when the number of unknowns (the particle populations coming from the outside part of the numerical domain) is greater than the number of equations at our disposal at each boundary site. Recently, BC models based on the regularization of the LB equation, or moments-based models, were proposed. These moments replace the discrete populations as unknowns, independently of the number of discrete velocities that are needed for solving a given problem. The full set of moments-based BC leads, nevertheless, to an overdetermined system of equations, and what distinguishes one model from another is the way this system is solved. In contrast with previous work, we base our approach on second-order moments. Four versions of this model are compared with previous moments-based models considering, in addition to the accuracy, some main model attributes such as global and local mass conservation, rates of convergence, and stability. For this purpose, the complex flow patterns displayed in a two-dimensional lid-driven cavity are investigated.