A simplified discrete unified gas kinetic scheme for incompressible flow

Abstract

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.

An implicit kinetic inviscid flux for predicting continuum flows in all speed regimes

“…The LU-SGS method is used as a preconditioner of the linear system (23), to be solved by the GMRES method. The GMRES method is an advanced iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace 44 .…”

“…19 , given the mean collision time and the characteristic time of flows, the direct modeling philosophy couples the macroscopic and microscopic mechanisms together, which dominates the continuum flows and rarefied flows, respectively, and their weights are determined by these two time scales. The multiscale schemes based on this direct modeling have good validity and accuracy in simulating both rarefied and continuum flows [20][21][22][23][24][25][26] , combined with different kinds of technologies [27][28][29][30] , which have succeeded in extending to more physics, like turbulence 31 , flux-structure interaction 32 , plasma 33 , multiphase flows 34 , non-equilibrium multi-temperature flows 35 and so on. Before the proposition of direct modeling, it was already implied in the numerical flux of gas-kinetic method 20 .…”

A conservative implicit scheme for three-dimensional steady flows of diatomic gases in all flow regimes using unstructured meshes in the physical and velocity spaces

“…As a result, the cell size and time step are also not passively limited by the molecular mean free path and collision time. Recently, some simplified algorithms of UGKS [18] and DUGKS [19], conserved DUGKS [20,21] have been developed. Although UGKS and DUGKS provide a unified framework for capturing the flow behaviors in all flow regimes [22,23], the model equations need to be discretized in time, physical space and particle velocity space same as the DVM method.…”

Investigation of nonlinear squeeze-film damping involving rarefied gas effect in micro-electro-mechanical-systems

“…Furthermore, as the DUGKS is an unsteady flow simulation method in nature, the ensemble averages are not required compared with the DSMC method. Currently, some improved and enhanced schemes based on the DUGKS also have been developed [25,26,27]. And, Wang et al [28] proposed an arbitrary Lagrangian-Eulerian-type DUGKS for solving the moving boundary problems in continuum and rarefied gas flows.…”