Gas-kinetic Solver Research Articles

Compact high-order gas-kinetic scheme for direct numerical simulation of compressible turbulent flows

In this paper, the three-dimensional fully compact high-order gas-kinetic scheme (HGKS) is proposed for the direct numerical simulation of compressible turbulent flows. Because of the high-order gas evolution model, the numerical fluxes as well as the point-wise conservative variables can be evaluated from the time-accurate gas distribution function at the cell interface. As a result, both the cell-averaged variables and their cell-averaged gradients can be updated inside each cell. Based on the cell averaged values and their gradients, the compact Hermite weighted essentially non-oscillatory (HWENO) scheme is developed, in which the dimension-by-dimension reconstruction is used for three-dimensional turbulences. In both normal and tangential directions, the fifth-order HWENO reconstruction is adopted. Compared with the classical WENO scheme, the stencil for the HWENO scheme only contains 33 cells for each cell. To achieve the temporal accuracy, the two-stage fourth-order temporal discretization is used. For the evaluation of point-wise variables, the simplified third-order gas-kinetic solver is used. Several classical benchmark problems are simulated, which validate the accuracy, resolution, and robustness of compact HGKS. As a comparison, the numerical results of HGKS using non-compact WENO reconstruction are also provided. Due to the compact stencil, the compact HGKS has a favorable performance for turbulence simulation in resolving the multi-scale structures.

Grad's distribution function for 13 moments-based moment gas kinetic solver for steady and unsteady rarefied flows: Discrete and explicit forms

Grad's distribution function for 13 moments-based moment gas kinetic solver for steady and unsteady rarefied flows: Discrete and explicit forms

Two-stage fourth-order subcell finite volume method on hexahedral meshes for compressible flows

As an extension of the two-stage fourth-order subcell finite volume (SCFV) method that we developed for two-dimensional compressible flows [C. Zhang et al., “Two-stage fourth-order gas kinetic solver based compact subcell finite volume method for compressible flows on triangular meshes,” Phys. Fluids 33, 126108 (2021)], this study continues our efforts toward three-dimensional (3D) simulations on hexahedral meshes. The two components of subcell divisions and two-stage fourth-order time stepping are utilized to improve efficiency and enhance compactness, which are crucial for 3D simulations. In particular, the current method subdivides each cell into a set of subcells or control volumes (CVs) to increase the degrees of freedom for high-order reconstruction, which involves only face-neighboring cells. For traditional finite volume (FV) methods, high-order reconstruction is performed on each CV individually. In contrast, the reconstruction of SCFV is shared by a set of CVs belonging to the same cell, which can be much more efficient and compact. Moreover, the SCFV framework is combined with the high-order flux evolution by adopting a robust and time-dependent gas-kinetic flux solver and an efficient two-stage fourth-order temporal discretization. The multi-stage Runge–Kutta (RK) method is thus avoided. The coupling of inviscid and viscous terms in the gas-kinetic flux enables us to directly simulate viscous flows. To capture shocks, a limiting procedure by hierarchical reconstruction is developed for effectively preserving the accuracy in smooth flow regions and suppressing numerical oscillations near flow discontinuities. Several benchmark cases are tested. The high-order accuracy and efficiency of this scheme are validated and compared to the k-exact FV method and the traditional Riemann solver combined with a multi-stage RK method. In particular, the simulation of the supersonic Taylor–Green vortex problem demonstrates the good performance of this scheme in compressible turbulence with the presence of shock waves.

Three-dimensional discontinuous Galerkin based high-order gas-kinetic scheme and GPU implementation

In this paper, the discontinuous Galerkin based high-order gas-kinetic schemes (DG-HGKS) are developed for the three-dimensional Euler and Navier–Stokes equations. Different from the traditional discontinuous Galerkin (DG) methods with Riemann solvers, the current method adopts a kinetic evolution process, which is provided by the integral solution of Bhatnagar–Gross–Krook (BGK) model. In the weak formulation of DG method, a time-dependent evolution function is provided, and both inviscid and viscous fluxes can be calculated uniformly. The temporal accuracy is achieved by the two-stage fourth-order discretization, and the second-order gas-kinetic solver is adopted for the fluxes over the cell interface and the fluxes inside a cell. Numerical examples, including accuracy tests and Taylor–Green vortex problem, are presented to validate the efficiency and accuracy of DG-HGKS. Both optimal convergence and super-convergence are achieved by the current scheme. The comparison between DG-HGKS and high-order gas-kinetic scheme with weighted essential non-oscillatory reconstruction (WENO-HGKS) is also given, and the numerical performances are comparable with the approximate number of degree of freedom. To accelerate the computation, the DG-HGKS is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). The obtained results are also compared with those calculated by the central processing units (CPU) code in terms of the computational efficiency. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation.

Two-stage fourth-order gas kinetic solver-based compact subcell finite volume method for compressible flows on triangular meshes

To meet the demand for complex geometries and high resolutions of small-scale flow structures, a two-stage fourth-order subcell finite volume (SCFV) method combining the gas-kinetic solver (GKS) with subcell techniques for compressible flows on (unstructured) triangular meshes was developed to improve the compactness and efficiency. Compared to the fourth-order GKS-based traditional finite volume (FV) method, the proposed method realizes compactness effectively by subdividing each cell into a set of subcells or control volumes (CVs) and selecting only face-neighboring cells for high-order compact reconstruction. Because a set of CVs in a main cell share the same reconstruction, it is more efficient than traditional FV-GKS, where the solution polynomial on each CV needs to be separately reconstructed. Unlike in the single-stage third-order SCFV-GKS, both accuracy and efficiency are improved significantly by two-stage fourth-order temporal discretization, for which only a second-order gas distribution function is needed to simplify the construction of the flux function and reduce computational costs. For viscous flows, it is not necessary to compute the viscous term with GKS. Compared to the fourth-stage Runge–Kutta method, one half of the stage is saved for achieving fourth-order time accuracy, which also helps to improve the efficiency. Therefore, a new high-order method with compactness, efficiency, and robustness is proposed by combining the SCFV method with the two-stage gas-kinetic flux. Several benchmark cases were tested to demonstrate the performance of the method in compressible flow simulations.

A third-order subcell finite volume gas-kinetic scheme for the Euler and Navier-Stokes equations on triangular meshes

A third-order gas-kinetic scheme (GKS) based on the subcell finite volume (SCFV) method is developed for the Euler and Navier-Stokes equations on triangular meshes, in which a computational cell is subdivided into four subcells. The scheme combines the compact high-order reconstruction of the SCFV method with the high-order flux evolution of the gas-kinetic solver. Different from the original SCFV method using weighted least-square reconstruction along with a conservation correction, a constrained least-square reconstruction is adopted in SCFV-GKS so that the correction is not necessary and continuous polynomials inside each main cell can be reconstructed. For flows with large gradients, a compact hierarchical WENO limiting is adopted. In the gas-kinetic flux solver, the inviscid and viscous flux are coupled and computed simultaneously. Moreover, the spatial and temporal evolution are coupled nonlinearly which enables the developed scheme to achieve third-order accuracy in both space and time within a single stage. As a result, no quadrature point is required for the flux evaluation and the multi-stage Runge-Kutta method is unnecessary as well. A third-order k-exact finite volume GKS is also constructed with the help of k-exact reconstruction. Compared to the k-exact GKS, SCFV-GKS is compact, and with less computational cost for the reconstruction which is based on the main cell. Besides, a continuous reconstruction can be adopted in SCFV-GKS among subcells in smooth flow regions, which results in less numerical dissipation and less computational cost for flux evolution. Thus the efficiency of SCFV-GKS is much higher. Numerical tests demonstrate the strong robustness, high accuracy and efficiency of SCFV-GKS in a wide range of flow problems from nearly incompressible to supersonic flows with strong shock waves.

High-order ALE gas-kinetic scheme with WENO reconstruction

In this paper, a high-order multi-dimensional gas-kinetic scheme is presented for both inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation. Compared with the traditional ALE method, the flow variables are updated in the finite volume framework, and the rezoning and remapping steps are not required. The two-stage fourth-order method is used for the temporal discretization, and the second-order gas-kinetic solver is applied for the flux evaluation. In the two-stage method, the spatial reconstruction is performed at the initial and intermediate stages, and the computational meshes are determined by the mesh velocity. In the moving mesh procedure, the mesh may distort severely and the mesh quality is reduced. To achieve the accuracy and improve the robustness, the newly developed WENO method [40] on quadrilateral meshes is adopted at each stage. The Gaussian quadrature is used for flux calculation. For each Gaussian point, the WENO reconstruction is performed in local moving coordinate, where the variation of mesh velocity along cell interface is taken into account. Numerical examples are presented to validate the performance of current scheme, where the mesh adaptation method and the cell centered Lagrangian method are used to provide mesh velocities.

A high-order hybridizable discontinuous Galerkin method with fast convergence to steady-state solutions of the gas kinetic equation

The mass flow rate of Poiseuille flow of rarefied gas through long ducts of two-dimensional cross-sections with arbitrary shape is critical in the pore-network modeling of gas transport in porous media. Here, for the first time, the high-order hybridizable discontinuous Galerkin (HDG) method is used to find the steady-state solution of the linearized Bhatnagar–Gross–Krook equation on two-dimensional triangular meshes. The velocity distribution function and its traces are approximated in piecewise polynomial spaces (of degree up to 4) on the triangular meshes and mesh skeletons, respectively. By employing a numerical flux that is derived from the first-order upwind scheme and imposing its continuity weakly on the mesh skeletons, global systems for unknown traces are obtained with fewer coupled degrees of freedom when compared to the original discontinuous Galerkin formulation. To achieve fast convergence to the steady-state solution, a diffusion-like equation for flow velocity, which is asymptotic-preserving into the fluid dynamic limit, is solved by the HDG simultaneously on the same meshes. The proposed HDG-synthetic iterative scheme is proved to be accurate and efficient. Specifically, for flows in the near-continuum regime, numerical simulations have shown that, to achieve the same level of accuracy, our scheme could be faster than the conventional iterative scheme by two orders of magnitude, also it is faster than the synthetic iterative scheme based on the finite difference discretization in the spatial space by one order of magnitude. In addition, the implicit HDG method is more efficient than an explicit discontinuous Galerkin gas kinetic solver, as well as the implicit discontinuous Galerkin scheme when the degree of approximating polynomial is larger than 2. The HDG-synthetic iterative scheme is ready to be extended to simulate rarefied gas mixtures and the Boltzmann collision operator.

A multi-level parallel solver for rarefied gas flows in porous media

A high-performance gas kinetic solver using multi-level parallelization is developed to enable pore-scale simulations of rarefied flows in porous media. The Bhatnagar–Gross–Krook model equation is solved by the discrete velocity method with an iterative scheme. The multi-level MPI/OpenMP parallelization is implemented with the aim to efficiently utilize the computational resources to allow direct simulation of rarefied gas flows in porous media based on digital rock images for the first time. The multi-level parallel approach is analyzed in detail confirming its better performance than the commonly-used MPI processing alone for an iterative scheme. With high communication efficiency and appropriate load balancing among CPU processes, parallel efficiency of 94% is achieved for 1536 cores in the 2D simulations, and 81% for 12288 cores in the 3D simulations. While decomposition in the spatial space does not affect the simulation results, one additional benefit of this approach is that the number of subdomains can be kept minimal to avoid deterioration of the convergence rate of the iteration process. This multi-level parallel approach can be readily extended to solve other Boltzmann model equations.

Efficient simulation of multidimensional continuum and non-continuum flows by a parallelised unified gas kinetic scheme solver

ABSTRACTWe develop an efficient, parallel, gas-kinetic solver for computing both continuum and non-continuum flows over non-Cartesian geometries by utilising the unified gas kinetic scheme (UGKS). UGKS, however, requires the computationally expensive update of a six-dimensional phase space at each time step restricting its application to canonical, laminar problems and simple geometries. In this paper, we demonstrate that the applications of UGKS can be increased by parallelising it and combining it with a recently developed, Cartesian grid method (UGKS-CGM). We demonstrate that our Cartesian grid methodology as well as UGKS parallelization perform and scale well on a range of numerical test cases even for a very large number of cores. Finally, we demonstrate that the solver accurately computes canonical turbulence at low Knudsen numbers. These results demonstrate that the parallelised UGKS code can be utilised to effectively study the non-equilibrium effects of rarefaction on laminar and turbulent non-continuum flows.

An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations

For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the second-order gas-kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around a cell interface. With the adoption of time derivative of the flux function, a two-stage Lax–Wendroff-type (L-W for short) time stepping method has been recently proposed in the design of a fourth-order time accurate method for inviscid flow [21]. In this paper, based on the same time-stepping method and the second-order GKS flux function [42], a fourth-order gas-kinetic scheme is constructed for the Euler and Navier–Stokes (NS) equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [24], the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme. In terms of the computational cost, a two-dimensional third-order GKS flux function takes about six times of the computational time of a second-order GKS flux function. However, a fifth-order WENO reconstruction may take more than ten times of the computational cost of a second-order GKS flux function. Therefore, it is fully legitimate to develop a two-stage fourth order time accurate method (two reconstruction) instead of standard four stage fourth-order Runge–Kutta method (four reconstruction). Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. In the current computational fluid dynamics (CFD) research, it is still a difficult problem to extend the higher-order Euler solver to the NS one due to the change of governing equations from hyperbolic to parabolic type and the initial interface discontinuity. This problem remains distinctively for the hypersonic viscous and heat conducting flow. The GKS is based on the kinetic equation with the hyperbolic transport and the relaxation source term. The time-dependent GKS flux function provides a dynamic process of evolution from the kinetic scale particle free transport to the hydrodynamic scale wave propagation, which provides the physics for the non-equilibrium numerical shock structure construction to the near equilibrium NS solution. As a result, with the implementation of the fifth-order WENO initial reconstruction, in the smooth region the current two-stage GKS provides an accuracy of O((Δx)5,(Δt)4) for the Euler equations, and O((Δx)5,τ2Δt) for the NS equations, where τ is the time between particle collisions. Many numerical tests, including difficult ones for the Navier–Stokes solvers, have been used to validate the current method. Perfect numerical solutions can be obtained from the high Reynolds number boundary layer to the hypersonic viscous heat conducting flow. Following the two-stage time-stepping framework, the third-order GKS flux function can be used as well to construct a fifth-order method with the usage of both first-order and second-order time derivatives of the flux function. The use of time-accurate flux function may have great advantages on the development of higher-order CFD methods.

Accuracy of Bhatnagaar-Gross-Krook Scheme in Solving Laminar Viscous Flow Problems

This paper describes the development of a gas-kinetic solver to compute laminar viscous flows in two-space dimensions via a finite difference approach. The convection flux terms of the Navier-Stokes equations are discretized by a semidiscrete finite difference method. The resulting inviscid flux function is then determined by a numerical scheme that is based on the Bhatnagaar-Gross-Krook model of the approximate collisional Boltzmann equation. The scheme is based on the direct splitting of the inviscid flux function with inclusion of particle collisions in the transport process. As for the diffusion flux terms, they are discretized by a second-order central difference scheme. The cell interface values required by the gas-kinetic scheme are reconstructed to higher-order spatial accuracy via the monotone upstream-centered schemes for conservation laws variable interpolation method. An explicit-type time integration method known as the modified fourth-order Runge-Kutta is employed for computing steady-state solutions. In the numerical case studies, the results obtained from the flux vector splitting Bhatnagaar-Gross-Krook scheme are compared with available experimental data, analytical solutions, the results from upwind schemes, and the results from central difference scheme to verify the accuracy and robustness of the gas-kinetic solver. The tests have shown that the Bhatnagaar-Gross-Krook scheme is able to resolve the shear layer, the shock structure, and the flow accurately as the results compare favorably with the available experimental and analytical data.

A BGK-Type Flux-Vector Splitting Scheme for the Ultrarelativistic Euler Equations

A gas-kinetic solver is developed for the ultrarelativistic Euler equations. The scheme is based on the direct splitting of the flux function of the Euler equations with inclusion of ``particle'' collisions in the transport process. Consequently, the artificial dissipation in the new scheme is much reduced in comparison with the usual kinetic flux-vector splitting (KFVS) schemes which are based on the free particle transport at the cell interfaces in the gas evolution stage. Although in a usual KFVS scheme the free particle transport gives robust solutions, it gives a smeared solution at the contact discontinuities. The new BGK-type KFVS scheme solves this problem and gives robust and reliable solutions as well as good resolution at the contact discontinuity. In contrast to the classical kinetic theory, we have only a finite domain of dependence due to the structure of the light cone. The scheme is naturally multidimensional and is extended to the two-dimensional case in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction. In the numerical case studies the results obtained from the BGK-type KFVS schemes are compared with the exact Riemann solution, KFVS schemes, upwind schemes, and central schemes.