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In this paper, a high-order gas-kinetic scheme is developed on three-dimensional unstructured meshes for compressible Euler and Navier–Stokes equations. To achieve the high-order spatial accuracy, the three-dimensional weighted essentially non-oscillatory (WENO) reconstruction is extended to the unstructured tetrahedral and hexahedral meshes. A simple strategy is adopted for the selection of candidate stencils, and the topologically independent linear weights are used for the spatial reconstruction. The efficiency and robustness of the classical WENO reconstruction are improved. In addition to the two-stage fourth-order temporal discretization and lower–upper symmetric Gauss–Seidel method, the explicit and implicit high-order gas-kinetic schemes are developed for unsteady and steady problems. Accuracy tests on hexahedral and tetrahedral grids validate the third-order of accuracy, and various three-dimensional incompressible and compressible numerical experiments are also presented. The results validate the accuracy and robustness of the proposed scheme for both inviscid and viscous flows. In the future, the current scheme will be extended to the hybrid unstructured meshes and Reynolds-averaged Navier–Stokes simulation with high Reynolds numbers. [ABSTRACT FROM AUTHOR]

For the one-stage third-order gas-kinetic scheme (GKS), successful applications have been achieved for the three-dimensional compressible flows [Pan, L., K. Xu, Q. Li, and J. Li. 2016. "An Efficient and Accurate Two-stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations." Journal of Computational Physics 326: 197-221]. The high-order accuracy of the scheme is obtained by integrating a multidimensional time-accurate gas distribution function over the cell interface within a time step without using Gaussian quadrature points and Runge-Kutta time-stepping technique. However, to the further increase of the order of the scheme, such as the fourth-order one, the one step formulation becomes very complicated for the multidimensional flow. Recently, a two-stage fourth-order GKS with high efficiency has been constructed for two-dimensional inviscid and viscous flow computations ([Li, J., and Z. Du. 2016. "A Two-stage Fourth Order Time-accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws." SIAM Journal on Scientific Computing 38: 3046-3069]; Pan et al. 2016), and the scheme uses the time accurate flux function and its time derivatives. In this paper, a fourth-order GKS is developed for the three-dimensional flows under the two-stage framework. Based on the three-dimensional WENO reconstruction and flux evaluation at Gaussian quadrature points on a cell interface, the high-order accuracy in space is achieved first. Then, the two-stage time stepping method provides the high accuracy in time. In comparison with the formal third-order GKS [Pan, L., and K. Xu. 2015. "A Third-order Gas-kinetic Scheme for Three-dimensional Inviscid and Viscous Flow Computations." Computers & Fluids 119: 250-260], the current fourth-order method not only improves the accuracy of the scheme, but also reduces the complexity of the gas-kinetic flux solver greatly. More importantly, the fourth-order GKS has the same robustness as the second-order shock capturing scheme [Xu, K. 2001. "A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method." Journal of Computational Physics 171: 289-335]. Numerical results validate the outstanding reliability and applicability of the scheme for three-dimensional flows, such as the cases related to turbulent simulations. [ABSTRACT FROM AUTHOR]

A Cartesian grid method combined with a simplified gas kinetic scheme is presented for subsonic and supersonic viscous flow simulation on complex geometries. Under the Cartesian mesh, the boundaries are represented by a set of direction-oriented boundary points, and the computational grid points are classified into four different categories, the fluid point, the solid point, the drop point, and the interpolation point. A constrained weighted least square method is employed to evaluate the physical quantities at the interpolation points. Different boundary conditions, including isothermal boundary, adiabatic boundary, and Euler slip boundary, are presented by different interpolation strategies. We adopt a simplified gas kinetic scheme as the flux solver for both subsonic and supersonic flow computations. The methodology of constructing a simplified kinetic flux function can be extended to other flow systems. A few numerical examples are used to validate the Cartesian grid method and the simplified flux solver. The reconstruction scheme for recovering the boundary conditions of compressible viscous and heat conducting flow with a Cartesian mesh can provide a smooth distribution of physical quantities at solid boundary, and present an accurate solution for the flow study with complex geometry. [ABSTRACT FROM AUTHOR]

In this paper, for the first time a third-order compact gas-kinetic scheme is proposed on unstructured meshes for the compressible viscous flow computations. The possibility to design such a third-order compact scheme is due to the high-order gas evolution model, where a time-dependent gas distribution function at cell interface not only provides the fluxes across a cell interface, but also presents a time accurate solution for flow variables at cell interface. As a result, both cell averaged and cell interface flow variables can be used for the initial data reconstruction at the beginning of next time step. A weighted least-square procedure has been used for the initial reconstruction. Therefore, a compact third-order gas-kinetic scheme with the involvement of neighboring cells only can be developed on unstructured meshes. In comparison with other conventional high-order schemes, the current method avoids the Gaussian point integration for numerical fluxes along a cell interface and the multi-stage Runge–Kutta method for temporal accuracy. The third-order compact scheme is numerically stable under CFL condition CFL ≈ 0.5 . Due to its multidimensional gas-kinetic formulation and the coupling of inviscid and viscous terms, even with unstructured meshes, the boundary layer solution and vortex structure can be accurately captured by the current scheme. At the same time, the compact scheme can capture strong shocks as well. [ABSTRACT FROM AUTHOR]

Abstract: Under a generalized coordinate transformation with arbitrary grid velocity, the gas-kinetic BGK equation is reformulated in a moving frame of reference. Then, a unified conservative gas-kinetic scheme is developed for the viscous flow computation in the moving grid system in the Eulerian space. Due to the coupling between the grid velocity and the overall solution algorithm, the Eulerian and Lagrangian methods become two limiting cases in the current gas-kinetic method. A fully conservative formulation can be obtained even in the Lagrangian limit. The moving grid method extends the applicable regime of the gas-kinetic scheme to the flows with free surface or moving boundaries, such as dam break problem and airfoil oscillations. In order to further increase the robustness of the moving grid method, similar to the arbitrary Lagrangian–Eulerian (ALE) method, a conservative adaptive grid technique is also implemented in the current method to redistribute the mesh concentration to the rapid variational flow region and remedy the distorted moving mesh due to the coupling between grid velocity and fluid speed. Many numerical examples from incompressible flow to the supersonic shock interaction are presented. The test cases verify the accuracy and robustness of the unified moving grid gas-kinetic method. [Copyright &y& Elsevier]

A gas-kinetic model with multiple translational temperature for the continuum and near continuum flow simulations is proposed. The main purpose for this work is to derive the generalized Navier-Stokes equations with multiple temperature. It is well recognized that for increasingly rarefied flowfields, the predictions from continuum formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies may be partially due to the single temperature assumption in the standard Navier-Stokes equations. Here, based on an extended Bhatnagar-Gross-Krook (BGK) model with multiple translational temperature, the numerical scheme for its corresponding Navier-Stokes equations is also constructed. In the current approach, the energy exchange between x, y, and z directions is modeled through the particle collision, and individual energy equation in different direction is obtained. The kinetic model, newly constructed is an enlarged system in comparison with Holway’s ellipsoid statistical BGK model (ES-BGK). The detailed difference is presented in this paper. In the newly derived “Navier-Stokes” equations from the current model, all viscous terms are replaced by the temperature relaxation terms. The relation between the stress and strain in the standard Navier-Stokes equations is recovered only in the limiting case when the flow is close to the equilibrium, such as small temperature differences in different directions. In order to validate the generalized Navier-Stokes equations, we apply them to the study of Couette and Poiseuille flows with a wide range of Knudsen numbers. In the continuum flow regime, the standard Navier-Stokes solutions are precisely recovered. In the near continuum flow regime, the simulation results are compared with the direct simulation Monte Carlo solutions. The anomalous phenomena in the pressure and temperature distributions from the standard Navier-Stokes equations in the Poiseuille flow case at Kn=0.1 are well resolved by the generalized Navier-Stokes equations. This paper clearly shows that many thermal nonequilibrium phenomena in the near continuum flow regime can be well captured by modifying some assumptions in the standard Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

Abstract: This paper extends the gas-kinetic BGK-NS scheme to an adaptive grid for the viscous flow simulations. The grid movement and adaptation is controlled by a monitor function which may depend on velocity gradient or other flow variables, such as density or pressure. For the viscous flow computation, the use of adaptive mesh much improves the efficiency and accuracy of the method in comparison with the methods with static mesh points. The current method is an accurate and efficient method for the viscous flow computation, where the grid points can be easily moved and concentrated on the regions with large velocity and density gradients, such as the boundary layer and multi-material interface. Many numerical examples validate the current approach for the viscous flow simulations. [Copyright &y& Elsevier]

Abstract: This paper concerns the development and application of a gas-kinetic scheme based on the Bhatnagar–Gross–Krook (BGK) model for the Navier–Stokes equations in the study of hypersonic viscous flow. Firstly, we extend the gas-kinetic Navier–Stokes solver [K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (2001) 289–335] by implementing a multidimensional particle propagation mechanism in the flux evaluation, where the gradients of flow variables in both normal and tangential directions of a cell interface are explicitly included. With the construction of a time averaged flux function, an implicit BGK scheme with LU-SGS method is constructed. The main purpose of the current research is to pave the way to extend the current approach directly to the flow computation with unstructured mesh, where the flow gradients in both parallel and perpendicular directions around a cell interface can be explicitly taken into account in a viscous flow computation. In the numerical parts, we concentrate on the computation of heat flux in laminar hypersonic viscous flows, where complicated flow phenomena, i.e., shock boundary layer interaction, flow separation, and viscous/inviscid interaction, will be encountered. The cases studied include the type IV shock–shock interaction around a circular cylinder and hypersonic flow passing through a double-cone geometry. In the hypersonic viscous flow, in comparison with the capturing of velocity and pressure fields, the accurate computation of stress and heat flux bears large difficulties. In all cases studied here, the heat fluxes obtained across body surfaces have good agreement with the experimental measurements. [Copyright &y& Elsevier]

Both lattice Boltzmann method (LBM) and the gas-kinetic BGK scheme are based on the numerical discretization of the Boltzmann equation with collisional models, such as, the Bhatnagar–Gross–Krook (BGK) model. LBM tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process. On the other hand, the gas-kinetic BGK scheme is a finite volume scheme, where the time-dependent gas distribution function with continuous particle velocity space is constructed and used in the evaluation of the numerical fluxes across cell interfaces. Currently, LBM is mainly used for low Mach number, nearly incompressible flow simulation. For the gas-kinetic scheme, the application is focusing on the high speed compressible flows. In this paper, we are going to compare both schemes in the isothermal low-Mach number flow simulations. The methodology for developing both schemes will be clarified through the introduction of operator splitting Boltzmann model and operator averaging Boltzmann model. From the operator splitting Boltzmann model, the error rooted in many kinetic schemes, which are based on the decoupling of particle transport and collision, can be easily understood. As to the test case, we choose to use the 2D cavity flow since it is one of the most extensively studied cases. Detailed simulation results with different Reynolds numbers, as well as the benchmark solutions, are presented. [Copyright &y& Elsevier]

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows. • To accelerate the convergence for steady flows, the implicit non-compact and compact HGKSs are developed for compressible flows. • For non-compact HGKS, the third-order WENO method is adopted, and the GMRES method with Jacobian matrix is used for temporal evolution. • For the compact HGKS, the third-order HWENO method is used, a simple strategy is used to update the cell averaged gradient with GMRES method. • To accelerate the computation, the current schemes are implemented with GPU. [ABSTRACT FROM AUTHOR]

The flow in a shock tube is extremely complex with dynamic multi-scale structures of sharp fronts, flow separation, and vortices due to the interaction of the shock wave, the contact surface, and the boundary layer over the side wall of the tube. Prediction and understanding of the complex fluid dynamics are of theoretical and practical importance. It is also an extremely challenging problem for numerical simulation, especially at relatively high Reynolds numbers. Daru and Tenaud [“Evaluation of TVD high resolution schemes for unsteady viscous shocked flows,” Comput. Fluids 30, 89–113 (2001)] proposed a two-dimensional model problem as a numerical test case for high-resolution schemes to simulate the flow field in a square closed shock tube. Though many researchers attempted this problem using a variety of computational methods, there is not yet an agreed-upon grid-converged solution of the problem at the Reynolds number of 1000. This paper presents a rigorous grid-convergence study and the resulting grid-converged solutions for this problem by using a newly developed, efficient, and high-order gas-kinetic scheme. Critical data extracted from the converged solutions are documented as benchmark data. The complex fluid dynamics of the flow at Re = 1000 are discussed and analyzed in detail. Major phenomena revealed by the numerical computations include the downward concentration of the fluid through the curved shock, the formation of the vortices, the mechanism of the shock wave bifurcation, the structure of the jet along the bottom wall, and the Kelvin-Helmholtz instability near the contact surface. Presentation and analysis of those flow processes provide important physical insight into the complex flow physics occurring in a shock tube. [ABSTRACT FROM AUTHOR]

Abstract: Shock-capturing schemes often exhibit anomalous behaviors, such as the carbuncle phenomenon and the post-shock oscillations, especially in the hypersonic flow regime. This paper proposes a simple and effective remedy against these shock instabilities in the case of the kinetic Lax–Wendroff scheme, where the well-known classic scheme is reinforced by means of an equilibrium distribution function of gas molecules. The pathologies are significantly improved to an acceptable level for practical purposes without any considerable side-effect by locally bringing out the robustness of the equilibrium flux method from the kinetic scheme. The remedy is applied only to the preprocessing of the data at cell-edges. The performance of the fortified kinetic scheme is demonstrated in the problem of a hypersonic inviscid or viscous flow past a blunt body. Comparisons are also made with various advanced shock-capturing schemes at present. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

This paper presents the development of a third-order compact gas-kinetic scheme (GKS) for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme utilizes a time-dependent gas distribution function at a cell interface to not only calculate the fluxes needed for updating the cell-averaged flow variables but also to evaluate the flow variables at the cell interface. This leads to the evolution of cell-averaged gradients of flow variables. The success of this scheme heavily relies on the initial data reconstruction techniques, with an emphasis on their application to the tetrahedral mesh. Employing a conventional second-order unlimited least-square reconstruction directly on the cell-averaged flow variables of von Neumann neighbouring cells can introduce linear instability into the scheme. However, by using the updated cell-averaged gradients, the GKS with a third-order compact smooth reconstruction remains linearly stable under a large CFL number when applied to a tetrahedral mesh. To enhance the robustness of the high-order compact GKS for capturing a discontinuous solution, we propose a novel two-step multi-resolution weighted essentially non-oscillatory (WENO) reconstruction. This innovative approach overcomes the stability issues associated with a second-order compact reconstruction by incorporating a pre-reconstruction step. Additionally, it simplifies the third-order non-linear reconstruction process by adding a single large stencil to those used in the second-order one. A high-order wall boundary condition is achieved by fusing the constrained least-square technique with the WENO procedure, where a quadratic element is used in the reconstruction for cells with a curved boundary. Numerical tests involving both the second-order and third-order compact GKS are presented, encompassing both inviscid and viscous flows at both low and high speeds. The results demonstrate that the proposed third-order compact scheme possesses robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations. • A third-order compact gas-kinetic scheme on 3D unstructured tetrahedral mesh is constructed. • A two-step multi-resolution weighted essentially non-oscillatory reconstruction is proposed. • High-order curved wall boundary treatment is obtained from the constrained least-square technique and the WENO procedure. • The scheme can use a much larger time step than that used in other conventional compact schemes. [ABSTRACT FROM AUTHOR]

• The numerical performances of high-order GKS and HLLC methods are compared. • The comparison is based on both inviscid and viscous solutions. • The robustness, efficiency, and accuracy of both schemes are analyzed. In this paper, a comparison of the performance of two high-order finite volume methods based on the gas-kinetic scheme (GKS) and HLLC fluxes is carried out on structured rectangular mesh. For both schemes, the fifth-order WENO-AO reconstruction is adopted to achieve a high-order spatial accuracy. In terms of temporal discretization, a two-stage fourth-order (S2O4) time marching strategy is adopted for WENO5-AO-GKS scheme, and the fourth-order Runge-Kutta (RK4) method is employed for WENO5-AO-HLLC scheme. For the viscous flow computation, the GKS includes both inviscid and viscous fluxes in the evolution of a single cell interface gas distribution function. While for the WENO5-AO-HLLC scheme, the inviscid flux is provided by HLLC Riemann solver, and the viscous flux is discretized by a sixth-order central difference method. Based on the tests of forward Mach step, double Mach reflection, and viscous shock tube, both schemes show outstanding shock capturing property. From the Titarev-Toro and double shear layer tests, WENO5-AO-GKS scheme seems to have a better resolution than WENO5-AO-HLLC scheme. Both schemes show excellent robustness in extreme cases, such as the Le Blanc problem. From the cases of the Noh problem and the compressible isotropic turbulence, WENO5-AO-GKS scheme shows favorite robustness. In the compressible isotropic turbulence and three-dimensional Taylor-Green vortex problems, WENO-AO-GKS can use a CFL number up to 0.5, instead of 0.3 for WENO5-AO-HLLC. In terms of computational efficiency, WENO5-AO-HLLC scheme is about 27% more expensive than WENO5-AO-GKS scheme in the two-dimensional viscous flow problems, but is about 15% faster in the three-dimensional case, because WENO5-AO-GKS scheme needs multidimensional spatial reconstruction for flow variables in both one normal and two tangential directions in the 3D case. Due to the multi-dimensionality, WENO5-AO-GKS scheme shows better resolution than WENO5-AO-HLLC scheme in the laminar boundary layer and the double shear layer test. [ABSTRACT FROM AUTHOR]

• A third-order compact scheme is constructed on unstructured mesh. • The solutions of compressible Euler and Navier-Stokes equations are obtained. • MSMD improves temporal accuracy and efficiency of the scheme. • The third-order compact scheme has the same robustness as the second-order one. As an extension of a fourth-order compact gas kinetic scheme (GKS) on structured mesh [24] , this work is about the development of a third-order compact GKS on unstructured mesh for the compressible Euler and Navier-Stokes solutions. Based on the time accurate high-order gas-kinetic evolution solution, the time-dependent gas distribution function at a cell interface in GKS provides not only the flux function and its time derivative, but also the time accurate flow variables there at the next time level. As a result, besides updating the conservative flow variables inside each control volume through the interface fluxes, the cell averaged first-order spatial derivatives of flow variables can be obtained as well using the updated flow variables at the closed cell interfaces around that cell through the divergence theorem. Therefore, with the cell-averaged flow variables and their first-order spatial derivatives inside each cell, the Hermite WENO (HWENO) techniques can be naturally implemented for the compact high-order reconstruction at the beginning of the next time step. Following the reconstruction technique in [64] , a new HWENO reconstruction on triangular mesh is designed in the current scheme. Combined with the two-stage temporal discretization and second-order time accurate flux function, a third-order compact scheme on unstructured mesh has been constructed. Accurate solutions can be obtained for both inviscid and viscous flows without sensitive dependence on the quality of triangular mesh. The robustness and accuracy of the scheme have been validated through many cases, including strong shocks in the hypersonic viscous flow and smooth Navier-Stokes solution. [ABSTRACT FROM AUTHOR]