Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case

Abstract

This paper is a continuation of our earlier work [Z.L. Guo et al., Phys. Rev. E 88, 033305 (2013)] where a multiscale numerical scheme based on kinetic model was developed for low speed isothermal flows with arbitrary Knudsen numbers. In this work, a discrete unified gas-kinetic scheme (DUGKS) for compressible flows with the consideration of heat transfer and shock discontinuity is developed based on the Shakhov model with an adjustable Prandtl number. The method is an explicit finitevolume scheme where the transport and collision processes are coupled in the evaluation of the fluxes at cell interfaces, so that the nice asymptotic preserving (AP) property is retained, such that the time step is limited only by the CFL number, the distribution function at cell interface recovers to the Chapman-Enskog one in the continuum limit while reduces to that of free-transport for freemolecular flow, and the time and spatial accuracy is of second-order accuracy in smooth region. These features make the DUGKS an ideal method for multiscale compressible flow simulations. A number of numerical tests, including the shock structure problem, the Sod tube problem with different degree of non-equilibrium, and the two-dimensional Riemann problem in continuum and rarefied regimes, are performed to validate the scheme. The comparisons with the results of DSMC and other benchmark data demonstrate that the DUGKS is a reliable and efficient method for multiscale compressible flow computation.

Numerical simulation of flows from free molecular regime to continuum regime by a DVM with streaming and collision processes

“…The first test case is the 1D shock structure, which was also studied by Xu and Huang [22] using the UGKS, and by Guo et al [24] applying the DUGKS. The reference viscosity μ ∞ is determined by…”

“…For flows in continuum flow regime, this requirement will lead to enormous computational costs. Other popular approaches to simulate rarefied flows include the discrete velocity method (DVM) or discrete ordinate method (DOM) [2,3,[17][18][19][20][21], the unified gas-kinetic scheme (UGKS) [4,6,22,23] and the discrete unified gas-kinetic scheme (DUGKS) [5,24]. In these methods, the particle velocity space is discretized into a finite set of points and the discrete velocity Boltzmann equation (DVBE) is solved.…”

“…As indicated in [36], LBM can also be considered as a special form of DVM. However, different from the conventional DVM [2][3][4][5][6][17][18][19][20][21][22][23][24][33][34][35], the LBM is intended to seek a minimal set of velocities in the phase velocity space rather than directly discretizes the whole velocity space. This is an important feature of LBM for effective reduction of the computational cost.…”

“…By applying the above approaches, a number of flow problems from free molecular regime to continuum regime have been well resolved. Different from the above mentioned kinetic schemes [2][3][4][5][6][17][18][19][20][21][22][23][24], the semi-Lagrangian method [33][34][35] and lattice Boltzmann method (LBM) [36][37][38][39][40] have been devised and applied to simulate rarefied flows with streaming and collision processes. The semi-Lagrangian method is indeed a DVM.…”

Numerical investigation on performance of three solution reconstructions at cell interface in DVM simulation of flows in all Knudsen number regimes

“…Within the framework of DVM, the unified gas kinetic scheme (UGKS) and the discrete unified gas kinetic scheme (DUGKS) have been developed to overcome the defect of poor accuracy of the conventional DVM in the continuum flow regime. Different from the conventional DVM, both UGKS and DUGKS calculate the distribution function at cell interface by the local solution of Boltzmann equation with collision term.…”