Meshfree method based on discrete gas-kinetic scheme to simulate incompressible/compressible flows

A meshfree method based on the discrete gas-kinetic scheme (DGKS) (called the meshfree-DGKS) for simulation of incompressible/compressible flows is proposed in this work. In this approach, the governing equations are discretized using the meshfree method based on the least squares-based finite difference approach. To simulate compressible problems with discontinuities, the virtual mid-points between adjacent nodes, which are regarded as Riemann discontinuities, are established. Then, the concept of numerical flux is introduced, which enables computing both compressible and incompressible problems. The fluxes at the mid-points are calculated using the DGKS based on the discrete particle velocity model. The corresponding particle velocity components and distribution functions are integrated based on moment relations to obtain the flux. The meshfree-DGKS maintains the advantages of the meshless method as it is implemented at arbitrarily distributed nodes. This breaks through the limitations of the grid topology and is suitable to handle complex geometries. More importantly, the fluxes at the mid-point are reconstructed with the DGKS using the local solution of the Boltzmann equation, which can describe its physical properties well, thus easily and stably capturing the shock wave. In addition, the DGKS can simultaneously calculate inviscid and viscous fluxes when simulating viscous flow problems, which gives an improved algorithm consistency. Several representative examples, such as shock tube problems, implosion problem, couette flow, lid-driven cavity flow, flow in a channel with a backward-facing step, supersonic flow around a ramp segment, and flow around staggered NACA0012 biplane configuration, are simulated to validate the proposed meshfree-DGKS.