Boundary conditions for lattice Boltzmann simulations in curved geometries

We present prescribed velocity boundary conditions taking into account the local curvature of the system. First, we evaluate the strain rate tensor through the computation of the second-order non-equilibrium moments. Second, the original frame of reference is rotated in such a way to match the local frame of reference, composed by the two vectors spanned by the tangent plane of the boundary site plus the normal vector to the tangent plane. Third, the resulting linear system of equations is solved analyticallyand the moments are then rotated back to the original frame to reconstruct the outgoing populations from the boundary site streamed to the fluid. The method is entirely based on a regularized version of the lattice Boltzmann equation, using and incompressible formulation for the BGK collision operator. Some of the components of the strain rate tensor at the rotated frame of reference are null, and we use this to compose the whole set of moments at the boundary. The lattice Boltzmann schemes considered here are the D2Q9 and D3Q19 in two and three dimensions, respectively. As a benchmark, the flow between concentric cylinders --the Taylor-Couette flow-- driven by the inner cylinder is simulated, both in two and three dimensions. We compare the solutions obtained in this study with analytical solutions from the literature for the two-dimensional case, and with numerical solutions from the literature for the three-dimensional case at various Taylor numbers.