Analysis and numerical tests of lifting relations to reconstruct LBM distribution functions for coupling simulations

The reconstruction operator (RO), which calculates the distribution functions of the lattice Boltzmann method (LBM) by the macroscopic variables, is important for the information exchange in the coupling simulation between LBM and macroscopic numerical methods. In the present work, different ROs are derived based on the first- and second-order expansions. The derivations show that the second-order ROs can conserve the momentum but the first-order ROs fail. The first-order models can be modified to guarantee the momentum conservation. The numerical tests are employed to evaluate the precisions of different models. The results demonstrate that the multiple-relaxation-time LBM is more stable than the single-relaxation-time LBM, and the first-order models are more stable than the second-order models. The second-order models give better reconstructions of the non-equilibrium parts of the distribution functions than the first-order models. However, the modified first-order models have similar accuracy to the second-order models when they are employed as boundary conditions or used in coupling simulations. Therefore, the second-order models are suggested for the initial conditions. The modified first-order models are suggested for the boundary conditions and the coupling simulations. Moreover, CPU time for coupling simulation decreases with the increase of the ratio between the time steps of finite volume method and LBM, and an optimized time ratio exists to minimize the errors. Finally, the flow past a porous medium is simulated by both the coupling methods and the multi-block LBM. The results show that the coupling methods are more efficient when using a large time step for the finite volume method.