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A simple distribution function-based gas-kinetic scheme for simulation of viscous incompressible and compressible flows. (English) Zbl 1351.76257
Summary: In this work, a simple distribution function-based gas-kinetic scheme for simulation of viscous flows is presented. The work applies the finite volume method to discretize the governing differential equations, and inviscid and viscous fluxes at the cell interface are evaluated simultaneously by local reconstruction of solution for the continuous Boltzmann equation. Differently from the conventional gas-kinetic scheme [K. H. Prendergast and K. Xu, J. Comput. Phys. 109, No. 1, 53–66 (1993; Zbl 0791.76059); D. Chae et al., J. Comput. Phys. 158, No. 1, 1–27 (2000; Zbl 0974.76056); K. Xu, J. Comput. Phys. 171, No. 1, 289–335 (2001; Zbl 1058.76056)], in the present work, the Maxwellian distribution function is simplified by a simple distribution function, and integrals in the infinity domain of phase space are reduced to integrals around a circle. As a consequence, the computational efficiency is greatly improved. Since the simple distribution function is defined on the circle, for simplicity, it is termed as circular function hereafter. The present work is the extension of our previous work [the authors et al., J. Comput. Phys. 255, 540–557 (2013; Zbl 1349.76752)], where the circular function-based gas-kinetic scheme is presented to simulate inviscid flows. Only the equilibrium distribution function is considered in [the authors et al., loc. cit.]. To solve viscous flows, the non-equilibrium part of density distribution function has to be considered. One of major contributions in this work is to present a simple way to compute the non-equilibrium part of the distribution function. It can be calculated by the difference of equilibrium distribution functions at the cell interface and its surrounding point. As a result, the formulations for computing the conservative flow variables and fluxes at the cell interface can be given explicitly. The present solver can simulate both incompressible and compressible viscous flows. To validate the proposed new gas-kinetic scheme, several incompressible and compressible viscous flows are simulated. Numerical results showed that the circular function-based gas-kinetic scheme can provide accurate numerical results with the same computational cost as that needed by conventional Navier-Stokes solver.

MSC:
76M28 Particle methods and lattice-gas methods
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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