Double distribution function-based discrete gas kinetic scheme for viscous incompressible and compressible flows

Abstract

In this work, we propose a flux solver that relies on a double distribution function (DDF) based discrete gas-kinetic scheme (DGKS) for incompressible and compressible viscous flows. By decomposing the phase space of the Maxwellian distribution function, the double distribution function (DDF) model is established to remove the phase energy variables and develop more compact formulations. It utilizes the density distribution function to recover the macroscopic mass and momentum conservations, and the energy distribution function to derive the macroscopic energy equation. Associated with a modified collision operator for the energy distribution function, the Prandtl number in the recovered energy equation becomes adjustable. The DDF model can be further simplified with the idea of the improved circular function and by replacing the continuous integration with the quadrature algorithm, which yields the discrete DDF model. This model can then be adopted in the reconstruction of numerical fluxes on the cell interface within the framework of the finite volume method, and the improved discrete gas-kinetic scheme (DGKS) is established. In contrast to the previous DGKS [39], the improved flux solver recovers the correct macroscopic equations and allows free adjustment of the Prandtl number. Besides, the introduction of the energy distribution function abbreviates the expression of the energy flux and thus makes the final formulations more compact. Numerical tests, including the compressible Couette flow, lid-driven cavity flow, transonic flow around the airfoil, shock-boundary layer interaction and supersonic flow around a ramp segment, are presented to validate the proposed flux solver in various flow conditions and compare its performance with the previous DGKS methods [39,43].