In this work, a discrete gas-kinetic scheme (DGKS) based on the arbitrary Lagrangian–Eulerian (ALE) method is proposed for the simulation of moving boundary problems. The governing equations are the ALE-based Navier–Stokes equations, which are discretized using the finite volume method. Starting from a circular function-based Boltzmann equation, a grid motion term is introduced to obtain the Boltzmann equation in ALE form. Based on the moment relations and Chapman–Enskog analysis, the moment of particle velocity and distribution function are summed to obtain the fluxes. The DGKS expression in the ALE framework can then be derived. In this method, the flux at the cell interface can be calculated from the local solution of the Boltzmann equation, which is physically realistic and makes the algorithm more stable. As DGKS is based on a multidimensional particle velocity model, it is not necessary to use approximate values for the reconstruction process. In addition, DGKS can simultaneously handle inviscid and viscous fluxes when simulating viscous flow problems, resulting in a higher degree of consistency. Finally, several moving boundary examples are simulated to validate the ALE-DGKS method. The results show the algorithm was observed to achieve second-order accuracy and can solve moving boundary problems effectively.
In this work, a meshfree Lattice Boltzmann Flux Solver (LBFS) is proposed to resolve compressible flow problems based on scattered points without mesh connections. The new method employs the Least Square‐based Finite Difference (LSFD) scheme to discretize the governing equations. In order to simulate discontinuous problems such as shock wave, the mid‐point between two adjacent nodes is regarded as a discontinuous interface over which the Riemann problem is established. The local fluxes at this interface point are reconstructed by LBFS using the local solution of the Lattice Boltzmann Equation (LBE) as well as its correlations to macroscopic variables and moment relations. The LBFS is constructed based on the non‐free parameter D1Q4 model: the normal component of the particle velocity on the interface is retained, while the tangential component is reconstructed by the macroscopic variables on both sides of the interface. The meshfree LBFS expects some intriguing merits. On one hand, it inherits the physical robustness of the LBFS: the local fluxes are reconstructed from the physical solutions instead of mathematical interpolations. On the other hand, it allows the implementation at arbitrarily distributed nodes, which credits to the flexibility of the method. Representative examples of compressible flows, including Sod shock tube, Osher‐Shu shock tube, flow around NACA0012 airfoil, flow around staggered NACA0012 biplane configuration and shock reflection problem, are simulated by the proposed method for comprehensive evaluation of the meshfree LBFS.
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