Development of an efficient gas kinetic scheme for simulation of two-dimensional incompressible thermal flows

General mechanisms for stabilizing weakly compressible models

Many weakly compressible models with intrinsic mechanisms for stabilizing computation have been proposed to simulate incompressible flows. The present paper analyzes several weakly compressible models to establish general mechanisms that incorporate them into a unified and simple framework. It is found that all these models contain some identical numerical dissipation terms, mass diffusion terms in the continuity equation, and bulk viscosity terms in the momentum equation. They are proven to provide general mechanisms for stabilizing computation. Referring to the general mechanisms and the computational procedures of the lattice Boltzmann flux solver, two general weakly compressible solvers for isothermal flows and thermal flows are proposed. They can be directly derived from standard governing equations and implicitly introduce those numerical dissipation terms. Detailed numerical investigations demonstrate that the two general weakly compressible solvers have good numerical stability and accuracy for both isothermal and thermal flows, which validates the general mechanisms further and the general approach of constructing general weakly compressible solvers.

Variant of gas kinetic flux solver for flows beyond Navier-Stokes level

In this paper, a variant of gas kinetic flux solver (GKFS) is presented for simulation of flows beyond the Navier-Stokes (NS) level. The method retains the framework of GKFS and reconstructs the numerical fluxes by the moments of distribution function at the cell interface, which is given from the local solution of the Boltzmann equation. In the conventional GKFS, the first-order Chapman-Enskog (CE) expansion is utilized to approximate the initial distribution function. By using the differential chain rule, it was found that the CE expansion form could be linked to the stress tensor and the heat flux. For flows in the NS level, the stress tensor and heat flux can be simply calculated from the linearized constitutive relationship and Fourier's law, respectively. However, for flows beyond the NS level, due to the strong nonequilibrium effect, the linearized constitutive relationship and Fourier's law are insufficient to predict the stress tensor and the heat flux. To overcome this difficulty, this paper introduces correction terms to the stress tensor and heat flux in the initial distribution function. These correction terms will take effect in the strong nonequilibrium region for flows beyond the NS level. To avoid finding complex expressions or solving complicated partial differential equations for the correction terms, a simple and iterative procedure is proposed to update the correction terms based on the framework of GKFS. The proposed method is validated by three benchmark cases which cover the flow from the continuum regime to the transition regime. Numerical results show that the present solver can provide accurate solution in the continuum regime. It is indeed the correction terms that take effect in the strong nonequilibrium region for flows beyond the NS level, which enables the present solver to capture the nonequilibrium phenomenon with reasonable accuracy for rarefied flows at moderate Knudsen number.

Multiscale kinetic inviscid flux extracted from a gas-kinetic scheme for simulating incompressible and compressible flows

A kinetic inviscid flux (KIF) is proposed for simulating incompressible and compressible flows. It is constructed based on the direct modeling of multiscale flow behaviors, which is used in the gas-kinetic scheme (GKS), the unified gas-kinetic scheme (UGKS), the discrete unified gas-kinetic scheme (DUGKS), etc. In KIF, the discontinuities (such as the shock wave) that cannot be well resolved by mesh cells are mainly solved by the kinetic flux vector splitting (KFVS) method representing the free transport mechanism (or microscale mechanism), while in other flow regions that are smooth, the flow behavior is solved mainly by the central-scheme-like totally thermalized transport (TTT). The weights of KFVS and TTT in KIF is automatically determined by those in the theory of direct modeling. Two ways of choosing the weights in KIF are proposed, which are actually the weights adopted in the UGKS and the DUGKS, respectively. By using the test cases of the Sod shock tube, the rarefaction wave, the boundary layer of a flat plate, the cavity flow, hypersonic flow over a circular cylinder, the shock and turbulent boundary iteration, and transonic flow over a three-dimensional M6 wing, the validity and accuracy of the present method are examined. The KIF does not suffer from the carbuncle phenomenon, and does not introduce extra numerical viscosity in smooth regions. Especially in the case of hypersonic cylinder, it gives quite sharp and clear density and temperature contours. The KIF can be viewed as an inviscid-viscous splitting version of the GKS. By doing this splitting, it is easy to be used in traditional computational fluid dynamics frameworks. It can also be classified as a type in the numerical schemes based on the kinetic theory that are represented by the works of Sun et al. [Adv. Appl. Math. Mech. 8, 703 (2016)10.4208/aamm.2015.m1071] and Ohwada et al. [J. Comput. Phys. 362, 131 (2018)JCTPAH0021-999110.1016/j.jcp.2018.02.019], except the weights are determined by the weights of direct modeling.

High-order least-square-based finite-difference-finite-volume method for simulation of incompressible thermal flows on arbitrary grids

In this work, a high-order (HO) least-square-based finite difference-finite volume (LSFD-FV) method together with thermal lattice Boltzmann flux solver (TLBFS) is presented for simulation of two-dimensional (2D) incompressible thermal flows on arbitrary grids. In the present method, a HO polynomial based on Taylor series expansion is applied within each control cell, where the unknown spatial derivatives at each cell center are approximated by least-square-based finite difference (LSFD) scheme. Then the recently developed TLBFS is applied to evaluate the convective and diffusive fluxes simultaneously at the cell interface by local reconstruction of thermal lattice Boltzmann solutions of the density and internal energy distribution functions. The present HO LSFD-FV method is verified and validated by 2D incompressible heat transfer problems. Numerical results indicate that the present method can be effectively and flexibly applied to solve thermal flow problems with curved boundaries on arbitrary grids. Compared with the conventional low-order finite volume method, higher efficiency and lower memory cost make the present HO method more promising for practical thermal flow problems.