Based on the principles of bouncing back the non-equilibrium parts of particle populations in the normal direction and momentum preservation, Zou & He [Phys Fluids 9(1997)] proposed and validated a set of analytical pressure and velocity boundary conditions for the D2Q9 and D3Q15 LBGK models. In this article, the analytical pressure and velocity face boundary conditions are explicitly extended to the D3Q19 LBGK model, following the same methodology. The correctness and accuracy of the derived boundary conditions are verified in steady duct flow problem and unsteady Stokes’ second problem; in both problems the numerical solutions are compared well with the analytical solutions.


Over the last two decades, the lattice Boltzmann method (LBM) has been emerging as an important computational fluid dynamics (CFD) technique for modeling nearly incompressible fluid flows and simulating physics of fluids, especially when interfacial dynamics and complex boundaries are involved [1,2]. Unlike the direct discretization of Navier-Stokes equations in the traditional CFD approaches, such as finite difference, finite volume and finite element methods, in LBM packets of fluid particles move across the lattice following simple but special kinetic rules at the mesoscale such that the relations among macroscopic variables of fluids automatically recover Navier-Stokes equations [3].

The mesoscopic kinetic nature of LBM brings many advantages over the conventional CFD methods, such as simplicity of the implementation algorithm, capacity of handling complex physical mechanisms, ease of parallelization, etc. On the other hand, the mesoscopic nature of LBM makes it awkward to implement boundary conditions. In traditional continuum-based CFD methods, the common velocity and pressure conditions (or their derivatives) can be imposed to the boundaries directly (or after simple straightforward discretization). In LBM, however, the boundary conditions described by macroscopic variables have to be appropriately translated into the equivalent conditions in term of mesoscopic quantities and enforced at the mesoscale. In this translation procedure, the mathematical relations among the macroscopic variables (e.g., density and velocities) and the mesoscopic quantities (particle distributions) must be satisfied and the known particle distributions need to be respected. Since the number of unknown particle distributions is greater than the number of mathematical macroscopicmesoscopic relations, additional physical or numerical rules need to be introduced to determine the unknown particle distributions. Examples of such additional rules include momentum preservation, local thermohydrodynamic equilibrium, finite difference extrapolation [4-8].

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