An alternative second order scheme for curved boundary condition in lattice Boltzmann method

An alternative scheme to implement the velocity Dirichlet boundary condition for curved boundary in the lattice Boltzmann (LB) method is developed. For inclined arbitrarily flat wall, the local second order boundary method (LSOBM) is proposed initially by Ginzbourg and D'Humières, and we further develop it to curved boundary, therefore a generalized LSOBM is achieved. In our boundary scheme, the unknown distribution functions at the boundary nodes are locally derived from the known ones by accessing the macroscopic physical information prescribed by the Dirichlet boundary conditions. Essentially, the unknown distribution functions are represented by a linear combination of the known ones, the corresponding coefficients depend on the macroscopic constraints on the boundary wall, the geometric information of the boundary nodes and the relaxation parameters. Unlike the previous curved boundary schemes, in which the boundary nodes are characterized by the intersected lattice links, a local curvilinear coordinate system associating with the curved boundary is adopted in the present scheme, and the boundary nodes are identified directly by their coordinates. Moreover, the present boundary scheme is second order accurate, as demonstrated in the theoretical derivations and also validated by two benchmark tests, the Taylor-Couette flow in-between rotating cylinders and the flow past an impulsively started cylinder.