Analytical and numerical studies of the boundary slip in the immersed boundary-thermal lattice Boltzmann method

We analytically and numerically investigate the boundary slip, including the velocity slip and the temperature jump, in immersed boundary-thermal lattice Boltzmann methods (IB-TLBMs) with the two-relaxation-time collision operator. We derive the theoretical equation for the relaxation parameters considering the effect of the advection velocity on the temperature jump of the IB-TLBMs. The analytical and numerical solutions demonstrate that the proposed iterative correction methods without the computational cost of the sparse matrix solver reduce the boundary slip and boundary-value deviation as effectively as the implicit correction method for any relaxation time. Because the commonly used multi-direct forcing method does not consider the contributions of the body force to the momentum flux, it cannot completely eliminate the boundary slip because of the numerical instability for a long relaxation time. Both types of proposed iterative correction methods are more numerically stable than the implicit correction method. In simulations of flow past a circular cylinder and of natural convection, the present iterative correction methods yield adequate results without the errors of the velocity slip, the temperature jump, and the boundary-value deviation for any relaxation time parameters and for any number of Lagrangian points per length. The combination of the present methods and the two-relaxation-time collision operator is suitable for simulating fluid flow with thermal convection in the multiblock method in which the relaxation time increases in inverse proportion to the grid size.