In this paper, the simplified discrete unified gas-kinetic scheme presented in the former paper is extended from incompressible flow to compressible flow at a high Mach number. In our earlier work, a simplified discrete unified gas–kinetic scheme was developed for low-speed flow in which the Mach number is small for keeping the incompressible property. To simulate compressible flow, the governing equation of the internal energy distribution function presented as potential energy including the Prandtl number effect is introduced to the present method. The velocity field is coupled with density and internal energy by the evolution of distribution functions related to mass, momentum, and temperature. For simplification and computational efficiency, the D2Q13 circular distribution function is applied as the equilibrium model. Compared to our earlier work, higher Mach number flows can be simulated by the proposed method, which is of the ability to simulate compressible flow. A number of numerical test cases from incompressible to compressible flows have been conducted, including incompressible lid-driven cavity flow, Taylor vortex flow, transonic flow past NACA (National Advisory Committee for Aeronautics) 0012 airfoil, Sod shock tube, supersonic flow past a circular cylinder, and isentropic vortex convection. All simulation results agree well with the reference data.
The discrete unified gas kinetic scheme (DUGKS) is a new finite volume (FV) scheme for continuum and rarefied flows, which combines the benefits of both the lattice Boltzmann method and UGKS. By the reconstruction of the gas distribution function using particle velocity characteristic lines, the flux contains more detailed information of fluid flow and more concrete physical nature. In this work, a simplified DUGKS is proposed with the reconstruction stage on a whole time step instead of a half time step in the original DUGKS. Using the temporal/spatial integral Boltzmann Bhatnagar–Gross–Krook equation, the auxiliary distribution function with the inclusion of the collision effect is adopted. The macroscopic and mesoscopic fluxes of the cell on the next time step are predicted by the reconstruction of the auxiliary distribution function at interfaces along particle velocity characteristic lines. According to the conservation law, the macroscopic variables of the cell on the next time step can be updated through its flux, which is a moment of the predicted mesoscopic flux at cell interfaces. The equilibrium distribution function on the next time step can also be updated. The gas distribution function is updated by the FV scheme through its predicted mesoscopic flux in a time step. Compared with the original DUGKS, the computational process of the proposed method is more concise because of the omission of half time step flux calculation. The numerical time step is only limited by the Courant–Friedrichs–Lewy condition, and a relatively good stability has been preserved. Several test cases, including the Couette flow, lid-driven cavity flow, laminar flows over a flat plate, a circular cylinder, and an airfoil, and microcavity flow cases, are conducted to validate the present scheme. The observed numerical simulation results reasonably agree with the reported results.
The discrete unified gas kinetic scheme (DUGKS) is a new finite volume (FV) scheme for continuum and rarefied flows which combines the benefits of both Lattice Boltzmann Method (LBM) and unified gas kinetic scheme (UGKS). By reconstruction of gas distribution function using particle velocity characteristic line, flux contains more detailed information of fluid flow and more concrete physical nature. In this work, a simplified DUGKS is proposed with reconstruction stage on a whole time step instead of half time step in original DUGKS. Using temporal/spatial integral Boltzmann Bhatnagar-Gross-Krook (BGK) equation, the transformed distribution function with inclusion of collision effect is constructed. The macro and mesoscopic fluxes of the cell on next time step is predicted by reconstruction of transformed distribution function at interfaces along particle velocity characteristic lines. According to the conservation law, the macroscopic variables of the cell on next time step can be updated through its macroscopic flux. Equilibrium distribution function on next time step can also be updated. Gas distribution function is updated by FV scheme through its predicted mesoscopic flux in a time step. Compared with the original DUGKS, the computational process of the proposed method is more concise because of the omission of half time step flux calculation. Numerical time step is only limited by the Courant-Friedrichs-Lewy (CFL) condition and relatively good stability has been preserved. Several test cases, including the Couette flow, lid-driven cavity flow, laminar flows over a flat plate, a circular cylinder, and an airfoil, as well as micro cavity flow cases are conducted to validate present scheme. The numerical simulation results agree well with the references' results.
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