High-order lattice Boltzmann: moments-based boundary conditions

Resumo

Dealing with boundary conditions (BC) was ever considered a puzzling question in the Lattice Boltzmann 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 challenging problem and, to the author's 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 node. Recently, moment-based BC models were proposed. These moments replace the discrete populations as the unknowns, regardless of the number of discrete velocities required to solve a given problem. In this work, we investigate the boundary conditions for the D2V17 LB equation by using a moments-based approach. This discrete form of the Boltzmann equation was introduced by Philippi et al. (Philippi, P. C., Hegele Júnior, L. A., Dos Santos, L. O. E., & Surmas, R. (2006). From the continuous to the lattice Boltzmann equation: The discretization problem and thermal models. Physical Review E, 73(5), 056702) and, in contrast with the popular second-order D2Q9, retrieves the Navier-Stokes without errors for isothermal problems. Third-order non-equilibrium moments are expressed in terms of the second-order viscous-stress tensor using recursive properties. The algorithm is then applied to solve the flow development inside a channel.