Explicit formulations of G13-based gas kinetic flux solver (G13-GKFS) for simulation of continuum and rarefied flows

Abstract

In this work, the explicit formulations of the Grad's distribution function for 13 moments (G13)-based gas kinetic flux solver (GKFS) for simulation of flows from the continuum regime to the rarefied regime are presented. The present solver retains the framework of GKFS, and it combines some good features of the discrete velocity method (DVM) and moment method. In the G13-GKFS, the macroscopic governing equations are first discretized by the finite volume method, and the numerical fluxes are evaluated by the local solution of the Boltzmann equation. To reconstruct the local solution of the Boltzmann equation, the initial distribution function is reconstructed by the Grad's distribution function for 13 moments, which enables the G13-GKFS to simulate flows in the rarefied regime. Thanks to this reconstruction, the evolution of distribution function is avoided, and the numerical fluxes can be expressed by explicit formulations. Therefore, the computational efficiency of G13-GKFS is much higher than that of DVM. The accuracy and computational efficiency of the present solver in explicit form are examined by several numerical examples. Numerical results show that the present solver can predict accurate results for flows in the continuum regime and reasonable results for flows in the rarefied regime. More importantly, the central processing unit time of the present solver is about 1% of that of DVM for two-dimensional (2D) microflow problems, and it is about twice of the conventional Navier–Stokes solver for 2D continuum flows.

Construction of Non-Equilibrium Gas Distribution Functions Through Expansions in Peculiar Velocity Space

“…Numerical tests have been carried out on a recently developed platform called "novel solver" [21][22][23]. The key issue for this platform is how to accurately construct distribution function at the cell interface.…”

Grad's Distribution Function for 13 Moments based Moment Gas Kinetic Solver for Steady and Unsteady Rarefied flows: Discrete and Explicit Forms

“…However, the discrete form of VDFs greatly increases the number of variables, and the memory and computations consumed by numerical integration and algebraic operations become the major [32]. Since the Grad's distribution function expresses the VDFs as an explicit function of macroscopic variables, the moment integral could be computed in an explicit way [34]. Therefore, the numerical discretization and moment integration of VDFs in molecular velocity space can be avoided.…”

“…The local solution of the Boltzmann Bhatnagar-Gross-Krook (BGK) Equation [33] is utilized to reconstruct the numerical flux with the help of the Grad's distribution function. Benefiting from the Grad's distribution, the moment integrals in the calculation could be conducted explicitly and the G13-based gas kinetic flux solver (G13-GKFS) exhibits an efficiency close to a hundred times higher than the DVM method in previous works [34]. However, the update of shear stress and heat flux are computed by moments of distribution function at cell interfaces first, and then interpolated to get the values at the cell center in the G13-GKFS.…”

A Novel Construction of Distribution Function Through Second-Order Polynomial Approximation in Terms of Particle Mass, Momentum and Energy

“…The success of regularized moment methods implies that the gas distribution function obtained through Hermite polynomial expansion has a potential for resolving flows beyond the NS level. Then, a novel solver [23,24] and a Grad's distribution function-based gas kinetic scheme [25] were proposed, which combines the advantages of the DVM and the moment method. On the one hand, these methods adopt the Grad's distribution function as the initial one and thus circumvent the tedious evolution process of gas distribution function in DVM.…”