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In this paper, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier–Stokes equations under the framework of two-stage fourth-order temporal discretization and Hermite WENO (HWENO) reconstruction. Due to the high-order gas evolution model, the GKS provides a time dependent gas distribution function at a cell interface. This time evolution solution can be used not only for the flux evaluation across a cell interface and its time derivative, but also time accurate flow variables at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the cell averaged slopes inside each control volume through the differences of flow variables at the cell interfaces can be updated as well in GKS. So, with the updated flow variables and their slopes inside each cell, the HWENO techniques can be naturally implemented for the compact high-order reconstruction at the beginning of each time step. Therefore, a compact higher-order GKS, such as the two-stage fourth-order compact scheme can be constructed. This fourth-order compact GKS has the same stencil and as robust as the second-order scheme. In comparison with the fourth-order DG method, they have the same stencil. In order to get a fourth-order temporal accuracy, the GKS uses two stages and DG needs four stages in the time stepping methods. The CFL number used in GKS is on the order of 0.5 instead of 0.11 in the DG. This research concludes that beyond the first-order Riemann solver the use of high-order time evolution model at a cell interface is extremely helpful in the design of robust, accurate, and efficient higher-order compact schemes for the compressible flow simulations. [ABSTRACT FROM AUTHOR]

High-order (higher-than 2nd-order) gas-kinetic schemes for solving the Navier–Stokes equations have been studied in recent years. In addition to the use of high-order reconstruction techniques, many terms are used in the Taylor expansion of the equilibrium and non-equilibrium gas distribution functions in the high-order gas kinetic flux function. Therefore, a large number of coefficients need to be determined in the calculation of the time evolution of the gas distribution function at cell interfaces. As a consequence, the high-order flux function takes much more computational time than that of a 2nd-order gas-kinetic scheme. This paper aims to simplify the evolution model by two steps. Firstly, the coefficients related to the 2nd-order spatial and temporal derivatives of a distribution function are redefined to reduce the computational cost. Secondly, based on the physical analysis, some terms can be removed without loss of accuracy. As a result, through the simplifications, the computational efficiency of the high-order scheme is increased significantly. In addition, a self-adaptive numerical viscosity is designed to minimize the necessary numerical dissipation. Several numerical examples are tested to demonstrate the accuracy and robustness of the current scheme. [ABSTRACT FROM AUTHOR]

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 ( ( Δ x ) 5 , ( Δ t ) 4 ) for the Euler equations, and O ( ( Δ x ) 5 , τ 2 Δ t ) 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. [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]

This paper presents a multi-dimensional high-order discontinuous Galerkin (DG) method in an arbitrary Lagrangian–Eulerian (ALE) formulation to simulate flows over variable domains with moving and deforming meshes. It is an extension of the gas-kinetic DG method proposed by the authors for static domains (X. Ren et al., 2015 [22] ). A moving mesh gas kinetic DG method is proposed for both inviscid and viscous flow computations. A flux integration method across a translating and deforming cell interface has been constructed. Differently from the previous ALE-type gas kinetic method with piecewise constant mesh velocity at each cell interface within each time step, the mesh velocity variation inside a cell and the mesh moving and rotating at a cell interface have been accounted for in the finite element framework. As a result, the current scheme is applicable for any kind of mesh movement, such as translation, rotation, and deformation. The accuracy and robustness of the scheme have been improved significantly in the oscillating airfoil calculations. All computations are conducted in a physical domain rather than in a reference domain, and the basis functions move with the grid movement. Therefore, the numerical scheme can preserve the uniform flow automatically, and satisfy the geometric conservation law (GCL). The numerical accuracy can be maintained even for a largely moving and deforming mesh. Several test cases are presented to demonstrate the performance of the gas-kinetic DG-ALE method. [ABSTRACT FROM AUTHOR]

Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier–Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the order of particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it still has space to develop more accurate flow solver in the region, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region where it is needed. The central ingredient of the UGKS is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface, where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled. The newly developed UGKS has the asymptotic preserving (AP) property of recovering the NS solutions in the continuum flow regime, and the full Boltzmann solution in the rarefied regime. In the mostly unexplored transition regime, the UGKS itself provides a valuable tool for the non-equilibrium flow study. The mathematical properties of the scheme, such as stability, accuracy, and the asymptotic preserving, will be analyzed in this paper as well. [ABSTRACT FROM AUTHOR]

Asymptotic preserving (AP) schemes target to simulate both continuum and rarefied flows. Many existing AP schemes are capable of recovering the Euler limit in the continuum regime. However, to get accurate Navier–Stokes solutions is still challenging. In order to distinguish physical mechanism underlining different AP schemes, an implicit–explicit (IMEX) AP method and the unified gas kinetic scheme (UGKS) are employed to solve the Bhatnagar–Gross–Krook (BGK) kinetic equation in both transition and continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is used for evaluating numerical performance of these two AP schemes. The numerical results show that the UGKS captures the viscous solution accurately. The velocity profiles converge to the classical benchmark solutions in continuum regime with the mesh size being much larger than the local particle mean free path. However, the IMEX AP scheme seems to have difficulty to get these solutions in the corresponding limit. The analysis demonstrates that the dissipation of AP schemes has to be properly controlled in the continuum flow regime through a delicate numerical treatment of collision and convection of the kinetic equation. Physically, it becomes necessary to couple both the convection and collision terms in the flux evaluation in order to recover correct Navier–Stokes limit. [ABSTRACT FROM AUTHOR]

Under a generalized coordinate transformation with inclusion of both mesh velocity and acceleration, the gas-kinetic equation is transformed from a physical space to a computational space. With the WENO reconstruction, a high-order moving-mesh multidimensional gas-kinetic method, with inclusion of both normal and tangential derivatives of macroscopic flow variables, is constructed. High-order accurate Navier–Stokes solutions are obtained in the current scheme by integrating the numerical fluxes along a cell interface and over a whole time step, without using the conventional Gaussian point integration for spatial accuracy and Runge–Kutta method for temporal accuracy. For the first time, the mesh acceleration has been explicitly accounted for in the flux evaluation. Under the generalized coordinate transformation, a three-dimensional moving-mesh gas-kinetic scheme has been constructed as well. The importance of multi-dimensionality in the flux evaluation and mesh movement is illustrated through numerical tests. [ABSTRACT FROM AUTHOR]

Abstract: Up to an arbitrary order of the Chapman–Enskog expansion of the kinetic Bhatnagar–Gross–Krook (BGK) equation, a corresponding analytic solution can be obtained. Based on such a compact exact solution, a new gas-kinetic scheme is constructed for the compressible Navier–Stokes equations. Instead of using a discontinuous initial condition in the gas-kinetic BGK–NS method Xu [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.], the new scheme starts with a continuous initial flow distribution at a cell interface which is obtained through an upwind-biased WENO reconstruction, and uses the time accurate solution for the flux evaluation. The new kinetic scheme not only preserves favorable properties of the existing BGK–NS method, such as stability, high resolution, and good performance in capturing discontinuity, but also achieves a very high efficiency, which is even more efficient than the same order well-defined classical finite difference scheme based on the macroscopic governing equations directly. The stability, accuracy, and efficiency of the new scheme are evaluated quantitatively through numerical tests. The new scheme captures sharp discontinuity without post-shock oscillation, and has high accuracy in resolving viscous solution. Due to the use of time-accurate analytical solution, the overall performance of the new scheme is superior in comparison with the finite difference or finite volume schemes which start from the same initial reconstruction and use the numerical Runge–Kutta methods for time accuracy. [Copyright &y& Elsevier]

Abstract: There is great difficulty for direct Boltzmann solvers to simulate high Knudsen number flow due to the severe steep slope and high concentration of the gas distribution function in a local particle velocity space. Local mesh adaptation becomes necessary in order to make the Boltzmann solver to be a practical tool in aerospace applications. The present research improves the unified gas-kinetic scheme (UGKS) in the following two aspects. First, the UGKS is extended in a physical space with moving mesh. This technique is important to study a freely flying object in a rarefied environment. Second, the adaptive quadtree method in the particle velocity space is implemented in the UGKS. Due to the new improvements in the discretization of a gas distribution function in the six dimensional phase space, the adaptive unified gas kinetic scheme (AUGKS) is able to deal with a wide range of flow problems under extreme flying conditions, such as the whole unsteady flying process of an object from a highly rarefied to a continuum flow regime. After validating the scheme, the capability of AUGKS is demonstrated in the following two challenge test cases. The first case is about the free movement of an ellipse flying at initial Mach number 5 in a rarefied flow at different Knudsen numbers. The force on the ellipse and the unsteady trajectory of the ellipse movement are fully captured. The gas distribution function around the ellipse is analyzed. The second case is about the study of unsteady flight of a nozzle under a bursting process of the compressed gas expanding into a rarefied environment. Due to the strong expansion wave and the huge density difference between interior and exterior regions around the nozzle, the particle distribution function changes dramatically in the particle velocity space. The use of an adaptive velocity space in the AUGKS becomes necessary to simulate such a flow and to control the computational cost to a tolerable level. The second test is a challenge problem for any existing rarefied flow solver. [Copyright &y& Elsevier]

Abstract: With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method. [Copyright &y& Elsevier]

Abstract: The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge–Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge–Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier–Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier–Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier–Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such a high-order method. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations. [Copyright &y& Elsevier]

Abstract: It is well known that for increasingly rarefied flowfields, the predictions from continuum formulation, such as the Navier–Stokes equations lose accuracy. For the high speed diatomic molecular flow in the transitional regime, the inaccuracies are partially attributed to the single temperature approximations in the Navier–Stokes equations. Here, we propose a continuum multiple temperature model based on the Bhatnagar–Gross–Krook (BGK) equation for the non-equilibrium flow computation. In the current model, the Landau–Teller–Jeans relaxation model for the rotational energy is used to evaluate the energy exchange between the translational and rotational modes. Due to the multiple temperature approximation, the second viscosity coefficient in the Navier–Stokes equations is replaced by the temperature relaxation term. In order to solve the multiple temperature kinetic model, a multiscale gas-kinetic finite volume scheme is proposed, where the gas-kinetic equation is numerically solved for the fluxes to update the macroscopic flow variables inside each control volume. Since the gas-kinetic scheme uses a continuous gas distribution function at a cell interface for the fluxes evaluation, the moments of a gas distribution function can be explicitly obtained for the multiple temperature model. Therefore, the kinetic scheme is much more efficient than the DSMC method, especially in the near continuum flow regime. For the non-equilibrium flow computations, i.e., the nozzle flow and hypersonic rarefied flow over flat plate, the computational results are validated in comparison with experimental measurements and DSMC solutions. [Copyright &y& Elsevier]

Abstract: This paper presents a Runge–Kutta discontinuous Galerkin (RKDG) method for viscous flow computation. The construction of the RKDG method is based on a gas-kinetic formulation, which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gas-kinetic BGK model and the Navier–Stokes equations, the Navier–Stokes flux is automatically obtained by the present method. Numerical examples for both one dimensional (1D) and two dimensional (2D) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method. [Copyright &y& Elsevier]

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]

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]

The present paper concerns two aspects for the Burnett equations. First, we are going to theoretically show the consistency between the traditional Chapman–Enskog expansion and the successive approximation for the BGK equation up to the super-Burnett order. Second, we will design a numerical scheme to efficiently solve the Burnett equations. The current approach is an improvement of the BGK-Burnett scheme [K. Xu, Phys. Fluids 15 (2003) 2077], where the locally constant collision frequency is considered. Based on the Burnett distribution function, a high order time accurate numerical flux is derived by different approaches. The resulting scheme is tested in the problem of normal shock wave and that of force-driven Poiseuille flow. [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]

This paper is about the extension of the gas-kinetic BGK scheme to the shallow-water equations with source terms. In the current study, the particle velocity change due to the gravitational force and variable river bottom is implemented explicitly in the flux evaluation. The current scheme is a well-balanced method, which presents accurate and robust results in both steady and unsteady flow simulations. [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]

Most plasmas are only partially ionized. To better understand the dynamics of these plasmas, the behaviors of a mixture of neutral species and plasma in ideal magnetohydrodynamic states are investigated. The current approach is about the construction of coupled kinetic models for the neutral gas, electron, and proton, and the development of the corresponding gas-kinetic scheme (GKS) for the solution in the continuum flow regime. The scheme is validated in the 1D Riemann problem for an enlarged system with the interaction from the Euler waves of the neutral gas and magnetohydrodynamic ones of the plasma. Additionally, the Orszag–Tang vortex problem across different ionized states is tested to examine the influence of neutrals on the MHD wave evolution. These tests demonstrate that the proposed scheme can capture the fundamental features of ideal partially ionized plasma, and a transition in the wave structure from the ideal MHD solution of the fully ionized plasma to the Euler solution of the neutral gas is obtained. • A novel gas-kinetic approach is formulated to describe the behavior of partially ionized plasma. • With varying fractions of neutral and ionized particles, both magneto-hydrodynamic and the Euler equations are recovered. • Analysis is conducted to examine the wave characteristics of the MHD equations and the Euler equations. • The transition from MHD waves to Euler waves in the context of the shock tube problem is obtained. [ABSTRACT FROM AUTHOR]

For the two-layer shallow water equations, a high-order compact gas-kinetic scheme (GKS) on triangular mesh is proposed. The two-layer shallow water equations have complex source terms in comparison with the single layer equations. The main focus of this study is to construct a time-accurate evolution solution at a cell interface and to design a well-balanced scheme. The evolution model at a cell interface provides not only the numerical fluxes, but also the flow variables. In the development of the well-balanced scheme, the time-dependent flow variables at the cell interfaces surrounding the cell can be utilized to update the cell-averaged gradients. These time-evolving gradients are subsequently applied in the discretization of source terms inside each control volume. Based on the cell-averaged flow variables and their gradients, high-order initial data reconstruction can be achieved with compact stencils. The compact high-order GKS has advantages to simulate the flow evolution in complex domain covered by unstructured mesh. Many test cases are used to validate the accuracy and robustness of the scheme for the two-layer shallow water equations. [ABSTRACT FROM AUTHOR]

Hypersonic flow around a vehicle in near space flight is associated with multiscale non-equilibrium physics at a large variation of local Knudsen number from the leading edge highly compressible flow to the trailing edge particle free transport. To accurately capture the solution in all flow regimes from the continuum Navier–Stokes solution to the rarefied gas dynamics in a single computation requires genuinely multiscale method. The unified gas-kinetic wave-particle (UGKWP) method targets on the simulation of such a multiscale transport. Due to the wave-particle decomposition, the dynamics in the Navier–Stokes wave and kinetic particle transport has been unified systematically and efficiently under the unified gas-kinetic scheme (UGKS) framework. In this study, the UGKWP method with the non-equilibrium among translation, rotation and vibration modes, is developed based on a multiple temperature relaxation model. The real gas effect for high speed flow in different flow regimes has been properly captured. Numerical tests, including Sod tube, normal shock structure, hypersonic flow around two-dimensional cylinder and three-dimensional flow around a sphere and space vehicle, have been conducted to validate the UGKWP method. In comparison with the discrete velocity method (DVM)-based Boltzmann solver and particle-based direct simulation Monte Carlo (DSMC) method, the UGKWP method shows remarkable advantages in terms of computational efficiency, memory reduction, and automatic recovering of multiscale solution in the high speed flow simulations. • The UGKWP method for diatomic gas is developed with the inclusion of rotational and vibrational modes. • The UGKWP becomes particle method in the highly rarefied regime and Navier-Stokes solver in the continuum regime. • The wave-particle decomposition makes UGKWP be an efficient method for hypersonic flow computation in all flow regimes. [ABSTRACT FROM AUTHOR]

Highlights • An implicit unified gas-kinetic scheme (IUGKS) has been developed for unsteady flow computation in all flow regimes. • The IUGKS bceomes much more efficient than the explicit UGKS for unsteady flow simulations. • The multiscale nature of UGKS is preserved in IUGKS with a large variation of the ratio between the local time step and local collision time. • IUGKS has second-order accuracy and presents accurate solutions in both continuum and rarefied flow regimes. Abstract The unified gas-kinetic scheme (UGKS) is a direct modeling method for multiple scale transport. Based on the ratio of time step to the particle collision time, the local evolution solution on the mesh size and time step scales is used in the construction of the multiscale method. For a flow problem covering multiple flow regimes, such as the hypersonic flow around a flying vehicle in near space, the UGKS is able to capture the highly compressed Navier-Stokes solution in one region and fully expanded free molecular flow in another region, with significant variations of the ratio between the time step and the local particle collision time around the vehicle. For an explicit UGKS, the time step in the whole computational domain is determined by the CFL condition. With implicit and multigrid techniques, the efficiency of the UGKS [1,2] has been improved by two orders of magnitude for steady state computation. However, for unsteady flow computation, due to the CFL condition the global time step used in the explicit UGKS may be limited by the smallest cell size in the computational domain. As a result, for a largely stretched non-uniform mesh the global time step becomes very small and the ratio of the time step to the local particle collision time may get a very small value. Under such a circumstance, even though the UGKS is a multiscale method, the real physics represented in the explicit UGKS may be constrained to the kinetic scale transport only, and the advantage of the multiscale nature in UGKS has not been fully utilized. In order to solve the multiscale unsteady flow problem efficiently, the time step restriction from a global CFL condition has to be released. In this paper, we will develop an implicit UGKS (IUGKS) for unsteady flows by alternatively solving the macroscopic and microscopic governing equations within a time step iteratively. With a pre-defined uniform large evolution time step, the local CFL number varies greatly in different region, such as on the order 1 in the large numerical cell size region, and 100 in the small cell size region. In order to preserve coherent flow evolution and keep the multiscale nature, the time averaged numerical flux across a cell interface is still evaluated by the explicit UGKS under the local CFL condition. Therefore, the multiscale property of the UGKS modeling has been kept over non-uniform meshes. With improved temporal discretization, the current IUGKS can automatically go back to the explicit UGKS and obtain identical solutions when the time step of the implicit scheme gets to that of an explicit one. Many numerical examples are included to validate the scheme for both continuum and rarefied flows with a large variation of artificially generated mesh size. The IUGKS has a second order accuracy and presents reasonably good results for unsteady flow computation, and its efficiency has been improved by dozens of times in comparison with the explicit UGKS. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a unified gas-kinetic scheme (UGKS) for multiphase dilute gas-particle system is proposed. The UGKS multiphase (UGKS-M) is a finite volume method, which captures flow physics in the regimes from collisionless multispecies transport to the two-fluid hydrodynamic Navier-Stokes (NS) solution with the variation of Knudsen number, and from granular flow regime to dusty gas dynamics with the variation of Stokes number. The reason for preserving the multiscale nature in UGKS-M is mainly coming from the direct modeling of the flow physics in the scales of discrete cell size and time step, where the ratio of the time step to the particle collision time determines flow behavior in different regimes. For the particle phase, the time evolution solution of the kinetic model equation is used in the construction of numerical flux, which takes into account the particle transport, collision, and acceleration. The gas phase is assumed to be in the continuum flow regime and evolves numerically by the gas-kinetic scheme (GKS), which is a subset of the UGKS for the Navier-Stokes solutions. The interaction between the gas and particle phase is calculated based on a velocity space mapping method, which solves accurately the kinetic acceleration process. The stability of UGKS-M is determined by the CFL condition only. With the inclusion of the material temperature evolution equation of solid particles, once the total energy loss in inelastic collision transfers into particle material thermal energy, the UGKS-M conserves the total mass, momentum, and energy for the whole multiphase system. In the numerical tests, the UGKS-M shows multiscale property in capturing the particle trajectory crossing (PTC), particle wall reflecting phenomena, and vortex-induced segregation of inertial particles under different Stokes numbers. The scheme is also applied to simulate shock induced fluidization problem, where the simulation results agree well with experimental measurements. Highlights • Construction of a multiscale method UGKS-M for gas-particle system. • The particle phase and gas phase are fully coupled through the momentum and energy exchange. • Different flow regimes are well captured with the variation of Knudsen and Stokes numbers. • The scheme is validated through many non-equilibrium test cases. [ABSTRACT FROM AUTHOR]

In order to extend the unified gas kinetic scheme (UGKS) to solve radiative transfer equations in a complex geometry, a multidimensional asymptotic preserving implicit method on unstructured mesh is constructed in this paper. With an implicit formulation, the CFL condition for the determination of the time step in UGKS can be much relaxed, and a large time step is used in simulations. Differently from previous direction-by-direction UGKS on orthogonal structured mesh, on unstructured mesh the interface flux transport takes into account multi-dimensional effect, where gradients of radiation intensity and material temperature in both normal and tangential directions of a cell interface are included in the flux evaluation. The multiple scale nature makes the UGKS be able to capture the solutions in both optically thin and thick regions seamlessly. In the optically thick region the condition of cell size being less than photon's mean free path is fully removed, and the UGKS recovers a solver for diffusion equation in such a limit on unstructured mesh. For a distorted quadrilateral mesh, the UGKS goes to a nine-point scheme for the diffusion equation, and it naturally reduces to the standard five-point scheme for a orthogonal quadrilateral mesh. Numerical computations covering a wide range of transport regimes on unstructured and distorted quadrilateral meshes will be presented to validate the current approach. [ABSTRACT FROM AUTHOR]

The gas dynamics under gravitational field is usually associated with multiple scale nature due to large density variation and a wide variation of local Knudsen number. It is challenging to construct a reliable numerical algorithm to accurately capture the non-equilibrium physical effect in different regimes. In this paper, a well-balanced unified gas-kinetic scheme (UGKS) for all flow regimes under gravitational field will be developed, which can be used for the study of non-equilibrium gravitational gas system. The well-balanced scheme here is defined as a method to evolve an isolated gravitational system under any initial condition to a hydrostatic equilibrium state and to keep such a solution. To preserve such a property is important for a numerical scheme, which can be used for the study of slowly evolving gravitational system, such as the formation of star and galaxy. Based on the Boltzmann model with external forcing term, the UGKS uses an analytic time-dependent (or scale-dependent) solution in the construction of the discretized fluid dynamic equations in the cell size and time step scales, i.e., the so-called direct modeling method. As a result, with the variation of the ratio between the numerical time step and local particle collision time, the UGKS is able to recover flow physics in different regimes and provides a continuous spectrum of gas dynamics. For the first time, the flow physics of a gravitational system in the transition regime can be studied using the UGKS, and the non-equilibrium phenomena in such a gravitational system can be clearly identified. Many numerical examples will be used to validate the scheme. New physical observation, such as the correlation between the gravitational field and the heat flux in the transition regime, will be presented. The current method provides an indispensable tool for the study of non-equilibrium gravitational system. [ABSTRACT FROM AUTHOR]

This paper presents an implicit unified gas-kinetic scheme (UGKS) for non-equilibrium steady state flow computation. The UGKS is a direct modeling method for flow simulation in all regimes with the updates of both macroscopic flow variables and microscopic gas distribution function. By solving the macroscopic equations implicitly, a predicted equilibrium state can be obtained first through iterations. With the newly predicted equilibrium state, the evolution equation of the gas distribution function and the corresponding collision term can be discretized in a fully implicit way for fast convergence through iterations as well. The lower–upper symmetric Gauss–Seidel (LU-SGS) factorization method is implemented to solve both macroscopic and microscopic equations, which improves the efficiency of the scheme. Since the UGKS is a direct modeling method and its physical solution depends on the mesh resolution and the local time step, a physical time step needs to be fixed before using an implicit iterative technique with a pseudo-time marching step. Therefore, the physical time step in the current implicit scheme is determined by the same way as that in the explicit UGKS for capturing the physical solution in all flow regimes, but the convergence to a steady state speeds up through the adoption of a numerical time step with large CFL number. Many numerical test cases in different flow regimes from low speed to hypersonic ones, such as the Couette flow, cavity flow, and the flow passing over a cylinder, are computed to validate the current implicit method. The overall efficiency of the implicit UGKS can be improved by one or two orders of magnitude in comparison with the explicit one. [ABSTRACT FROM AUTHOR]

This paper presents an extension of previous work (Sun et al., 2015 [22] ) of the unified gas kinetic scheme (UGKS) for the gray radiative transfer equations to the frequency-dependent (multi-group) radiative transfer system. Different from the gray radiative transfer equations, where the optical opacity is only a function of local material temperature, the simulation of frequency-dependent radiative transfer is associated with additional difficulties from the frequency-dependent opacity. For the multiple frequency radiation, the opacity depends on both the spatial location and the frequency. For example, the opacity is typically a decreasing function of frequency. At the same spatial region the transport physics can be optically thick for the low frequency photons, and optically thin for high frequency ones. Therefore, the optical thickness is not a simple function of space location. In this paper, the UGKS for frequency-dependent radiative system is developed. The UGKS is a finite volume method and the transport physics is modeled according to the ratio of the cell size to the photon's frequency-dependent mean free path. When the cell size is much larger than the photon's mean free path, a diffusion solution for such a frequency radiation will be obtained. On the other hand, when the cell size is much smaller than the photon's mean free path, a free transport mechanism will be recovered. In the regime between the above two limits, with the variation of the ratio between the local cell size and photon's mean free path, the UGKS provides a smooth transition in the physical and frequency space to capture the corresponding transport physics accurately. The seemingly straightforward extension of the UGKS from the gray to multiple frequency radiation system is due to its intrinsic consistent multiple scale transport modeling, but it still involves lots of work to properly discretize the multiple groups in order to design an asymptotic preserving (AP) scheme in all regimes. The current scheme is tested in a few frequency-dependent radiation problems, and the results are compared with the solutions from the well-defined implicit Monte Carlo (IMC) method. The UGKS is much more efficient than IMC, and the computational times of both schemes for all test cases are listed. The UGKS seems to be the first discrete ordinate method (DOM) for the accurate capturing of multiple frequency radiative transport physics from ballistic particle motion to the diffusive wave propagation. [ABSTRACT FROM AUTHOR]

The solutions of radiative transport equations can cover both optical thin and optical thick regimes due to the large variation of photon's mean-free path and its interaction with the material. In the small mean free path limit, the nonlinear time-dependent radiative transfer equations can converge to an equilibrium diffusion equation due to the intensive interaction between radiation and material. In the optical thin limit, the photon free transport mechanism will emerge. In this paper, we are going to develop an accurate and robust asymptotic preserving unified gas kinetic scheme (AP-UGKS) for the gray radiative transfer equations, where the radiation transport equation is coupled with the material thermal energy equation. The current work is based on the UGKS framework for the rarefied gas dynamics [14] , and is an extension of a recent work [12] from a one-dimensional linear radiation transport equation to a nonlinear two-dimensional gray radiative system. The newly developed scheme has the asymptotic preserving (AP) property in the optically thick regime in the capturing of diffusive solution without using a cell size being smaller than the photon's mean free path and time step being less than the photon collision time. Besides the diffusion limit, the scheme can capture the exact solution in the optical thin regime as well. The current scheme is a finite volume method. Due to the direct modeling for the time evolution solution of the interface radiative intensity, a smooth transition of the transport physics from optical thin to optical thick can be accurately recovered. Many numerical examples are included to validate the current approach. [ABSTRACT FROM AUTHOR]

Abstract: A unified gas-kinetic scheme (UGKS) is constructed for both continuum and rarefied flow computations. The underlying principle for the development of UGKS is the direct modeling for the gas evolution process from the kinetic to the hydrodynamic scale, which is used in the flux construction across a cell interface. More specifically, the physical process from the kinetic particle free transport to the hydrodynamic pressure wave propagation is recovered in the flux function. In the previous study, the UGKS has been developed mainly for monatomic gas with particle translational motion only. The construction of time evolution solution is based on the BGK, Shakhov, and ES–BGK models. The UGKS has been validated through extensive numerical tests. In this paper, a UGKS for diatomic gas will be constructed, where the gas-kinetic Rykov model with a Landau–Teller–Jeans-type rotational energy relaxation is used in the numerical scheme. The new scheme will be tested in many cases, such as homogeneous flow relaxation, shock structure calculations, hypersonic flow passing a flat plate, and the flow around a blunt circular cylinder. The analytic, DSMC, and experimental measurements will be used for validating the solutions of UGKS. [Copyright &y& Elsevier]

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]

The performance of high-order gas-kinetic scheme (HGKS) has been investigated for the direct numerical simulation (DNS) of isotropic compressible turbulence up to the supersonic regime [9]. Due to the multi-scale nature and coupled temporal-spatial evolution process, HGKS provides a valid tool for the numerical simulation of compressible turbulent flow. Based on the domain decomposition and message passing interface (MPI), a parallel HGKS code is developed for large-scale computation in this paper. The standard tests from the nearly incompressible flow to the supersonic one, including Taylor-Green vortex problem, turbulent channel flow and isotropic compressible turbulence, are presented to validate the parallel scalability, efficiency, accuracy and robustness of parallel implementation. The performance of HGKS for the nearly incompressible turbulence is comparable with the high-order finite difference scheme, including the resolution of flow structure and efficiency of computation. Based on the accuracy of the numerical solution, the numerical dissipation of the scheme in the turbulence simulation is quantitatively evaluated. Meanwhile, based on the kinetic formulation HGKS shows advantage for supersonic turbulent flow simulation with its accuracy and robustness. The current work demonstrates the capability of HGKS as a powerful DNS tool from the low speed to supersonic turbulence study, which is less reported under the framework of finite volume scheme. • A parallel HGKS code is developed as a powerful DNS tool from the low speed to supersonic turbulence study. • The parallel scalability, efficiency, accuracy and robustness of parallel implementation of HGKS are validated. • The performance of HGKS for the nearly incompressible turbulence is comparable with the high-order finite difference scheme. • As a mesoscopic method, HGKS also performs better than both lattice Boltzmann method (LBM) and discrete unified gas-kinetic scheme (DUGKS). [ABSTRACT FROM AUTHOR]

Abstract: The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier–Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier–Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero. [Copyright &y& Elsevier]

Abstract: Based on the integral form of the fluid dynamic equations, a finite volume kinetic scheme with arbitrary control volume and mesh velocity is developed. Different from the earlier unified moving mesh gas-kinetic method [C.Q. Jin, K. Xu, An unified moving grid gas-kinetic method in Eulerian space for viscous flow computation, J. Comput. Phys. 222 (2007) 155–175], the coupling of the fluid equations and geometrical conservation laws has been removed in order to make the scheme applicable for any quadrilateral or unstructured mesh rather than parallelogram in 2D case. Since a purely Lagrangian method is always associated with mesh entangling, in order to avoid computational collapsing in multidimensional flow simulation, the mesh velocity is constructed by considering both fluid velocity (Lagrangian methodology) and diffusive velocity (Regenerating Eulerian mesh function). Therefore, we obtain a generalized Arbitrary-Lagrangian–Eulerian (ALE) method by properly designing a mesh velocity instead of re-generating a new mesh after distortion. As a result, the remapping step to interpolate flow variables from old mesh to new mesh is avoided. The current method provides a general framework, which can be considered as a remapping-free ALE-type method. Since there is great freedom in choosing mesh velocity, in order to improve the accuracy and robustness of the method, the adaptive moving mesh method [H.Z. Tang, T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487–515] can be also used to construct a mesh velocity to concentrate mesh to regions with high flow gradients. [Copyright &y& Elsevier]

• The DNS solution on a turbulent Mach number up to 2 has been obtained by high-order gas-kinetic scheme. • SGS production, dissipation, dilation, and diffusion have been evaluated. • The coarse-graining analysis gives the order of magnitude of all SGS terms in compressible K s g s budget. The direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime with M a t = 1.2 has been investigated by the high-order gas-kinetic scheme (HGKS) (Cao et al. (2019) [8]). In this study, the DNS on a much higher initial turbulent Mach number up to M a t = 2.0 is performed by HGKS, which confirms the super robustness of HGKS. The coarse-graining analysis of subgrid-scale (SGS) turbulent kinetic energy K s g s budget is fully analyzed for constructing one-equation SGS model in the compressible large eddy simulation (LES). The exact compressible SGS turbulent kinetic energy K s g s transport equation is derived with density weighted filtering process. Based on the compressible K s g s transport equation, the coarse-graining processes are implemented on three sets of unresolved grids with the Box filter. The coarse-graining analysis of compressible K s g s budgets shows that all unresolved source terms are dominant in the current system. Especially, the magnitude of SGS pressure-dilation term is on the order of SGS solenoidal dissipation term within the initial acoustic time scale. Therefore, the SGS pressure-dilation term cannot be neglected as that in the previous work. The delicate coarse-graining analysis of SGS diffusion terms in compressible K s g s equation also confirms that both the fluctuation velocity triple correlation term and the pressure-velocity correlation term are dominant terms. The current analysis provides an indication on the order of magnitude of all SGS terms in compressible K s g s budget, which provides a solid basis for compressible LES modeling of high Mach number turbulent flow. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a discontinuous Galerkin BGK (DGBGK) method for both viscous and inviscid flow simulations under a DG framework with a gas-kinetic flux and WENO limiters. In the DGBGK method, the construction of the flux in the DG method is based on the particle transport and collisional mechanism which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation. Due to the connection between the gas-kinetic BGK model and the Euler as well as the Navier–Stokes equations, both viscous and inviscid flow equations can be simulated by a unified formulation. WENO limiters are used to obtain uniform high-order accuracy and sharp non-oscillatory shock transition. In the current method, the time accuracy is achieved by the direct integration of both time-dependent flux function at a cell interface and the flow variables inside each element. Numerical examples in one and two space dimensions are presented to illustrate the robustness and accuracy of the present scheme. [Copyright &y& Elsevier]

Abstract: This paper presents efficient second-order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows. For compressible unsteady flows, a time-dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables. As a result, a compact scheme on the unstructured meshes is developed. For incompressible steady flows, a continuous second-order gas-kinetic BGK type scheme is presented, for which the time-dependent gas distribution function with a continuous particle velocity is used on unstructured meshes. The efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells. Therefore, even with the stencil of a first-order scheme, a high resolution method is constructed. Numerical examples are presented which are compared with the benchmark solutions and the experimental measurements. [Copyright &y& Elsevier]

Abstract: This paper concerns the extension of the gas-kinetic scheme for the compressible Navier–Stokes equations to the flow calculation with interfaces and mixing. The objective of the current research targets mainly on the accurate capturing of Navier–Stokes diffusive interfaces, where the thickness can be resolved by the cell size. Firstly, a new BGK-NS scheme coupling with the level set type scalar function transport is constructed. Even though the scalar function is directly incorporated into the gas distribution function and it evolves according to the gas-kinetic equation, it likes more or less a color function, which has no direct contribution for the time evolution of conservative flow variables, such as mass, momentum and energy. Due to the coupling of the scalar function into the gas-kinetic formulation, the governing equations for the scalar function turns out to be an advection diffusion equations and the diffusive coefficient can be controlled by the particle collision time, which makes the current scheme suitable for the gas mixing problems with a controllable diffusion coefficients. However, for the non-mixing or sharp interface problems, such as the interface between gas and liquid, the current method can be used as a scheme similar to the level set method, where the interface location can be identified with a fixed level set value, such as Θ =0. The current method is applied to a few examples from the simple square wave propagation and diffusion to the 3D Rayleigh–Taylor instability. The supersonic mixing layer and the shock helium bubble interaction case show clearly the convergence of the current Navier–Stokes solver for the flow problems with mixing of components and interface once the interface thickness can be well resolved by the cell size. In the case of shock hitting SF6 cylinder, the computation predicts the experimental measure very well. In the current scheme, the Schmidt number can be freely chosen according to the physical reality. [Copyright &y& Elsevier]

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 cause for ray effect in rarefied flow simulation has been analyzed theoretically and validated numerically. • Several mitigation techniques have been investigated in reducing the ray effect. • Three principles are proposed to adjust the velocity space discretization in order to mitigate the ray effect. Ray effect usually appears in the computation of radiative transfer when discrete ordinates method (DOM) is used. The cause and remedy for the ray effect have been intensively investigated in the radiation community. For rarefied gas flow, the ray effect is also associated with the discrete velocity method (DVM). However, few studies have been carried out in the rarefied community. In this paper, we make a detailed investigation of the ray effect in the rarefied flow simulation. Starting from a few commonly used benchmark tests, the root of the ray effect will be analyzed theoretically and validated numerically. The mitigation strategy by adjusting the discretization in the particle velocity space is studied as well. The guidelines are proposed to optimize the velocity space discretization for effectively reducing the ray effect. However, the design of optimal velocity space discretization is problem dependent and it can be hardly obtained in a highly rarefied flow simulation with complicated geometry. Due to the intrinsic self-adaptation of particle velocity, the stochastic particle method could automatically provide an optimized velocity discretization and avoid the ray effect. In the term of ray effect mitigation, the use of stochastic particle method is recommended in the numerical simulation of highly rarefied flow. [ABSTRACT FROM AUTHOR]

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]

Compressible flow has intrinsically multiple scale nature due to the large variations of gas density and characteristic scale of flow structure, especially in hypersonic and reentry problems. It is challenging to construct an accurate and efficient numerical algorithm to capture non-equilibrium flow physics across different regimes. In this paper, a unified gas kinetic scheme with adaptive velocity space (AUGKS) for multiscale flow transport will be developed. In near-equilibrium flow region, particle distribution function is close to the Chapman-Enskog expansion and can be formulated analytically, where only macroscopic conservative flow variables are updated. With the emergence of non-equilibrium effect, the AUGKS automatically switches to the original unified gas kinetic scheme (UGKS) with a discrete velocity space to follow the evolution of particle distribution function. A criterion is proposed to quantify the non-equilibrium and is used for the switch between continuous and discrete particle velocity space. Following the scale-dependent local evolving solution, the AUGKS presents the discretized gas dynamic equations directly on the cell size and time step scales, i.e., the so-called direct modeling method. As a result, the scheme is able to capture the cross-scale flow physics from particle transport to hydrodynamic wave propagation, and provides a continuous variation of solutions from the Boltzmann to the Euler equations. Different from conventional DSMC-Fluid hybrid method, the AUGKS does not need a buffer zone to match up kinetic and hydrodynamic solutions. Instead, continuous and discrete particle velocity spaces are automatically and robustly switched, such as translating the continuous Chapman-Enskog distribution function to the discrete grid points in the velocity space. Therefore, the AUGKS is feasible for the numerical simulations with unsteadiness and complex geometries. Compared with the asymptotic preserving (AP) method which solves kinetic equation uniformly over entire computational domain with discretized velocity space, the current velocity-space adaptive unified scheme speeds up the computation, reduces the memory requirement significantly, and maintains the equivalent accuracy for multiscale flow simulations. Many test cases validate the current approach. The AUGKS provides an effective tool for non-equilibrium flow studies. • A velocity-space adaptation is implemented to solve kinetic equation and its upscaling macroscopic system. • A criterion is proposed to quantify the non-equilibrium for the continuous-discrete velocity space transformation. • Both accuracy and efficiency are validated in the current scheme. [ABSTRACT FROM AUTHOR]

• Construction of unified gas-kinetic wave-particle (UGKWP) method for rarefied flow. • The multiscale solution is captured through the coupled wave and particle evolution. • In the hypersonic rarefied flow, UGKWP becomes a particle method. • In the continuum flow regime, UGKWP recovers the gas-kinetic scheme. • UGKWP achieves high efficiency in the corresponding flow regime. The unified gas-kinetic scheme (UGKS) provides a framework for simulating multiscale transport with the updates of both gas distribution function and macroscopic flow variables on the cell size and time step scales. The multiscale dynamics in UGKS is achieved through the coupled particle transport and collision in the particle evolution process within a time step. In this paper, under the UGKS framework, we propose an efficient multiscale unified gas-kinetic wave-particle (UGKWP) method. The gas dynamics in UGKWP method is described by the individual particle movement coupled with the evolution of the probability density function (PDF). During a time step, the trajectories of simulation particles are tracked until collision happens, and the post-collision particles are evolved collectively through the evolution of the corresponding distribution function. The evolution of simulation particles and distribution function is guided by the evolution of macroscopic variables, and guarantees the conservation of the scheme in the final wave-particle formulation. A new concept of multiscale multi-efficiency preserving (MMP) method is introduced. Multiscale preserving means UGKWP method preserves the flow regime from collisionless regime to hydrodynamic regime without requiring the cell size and time step to be less than the mean free path and collision time. Multi-efficiency preserving means the computational cost of the scheme including the computational time and memory cost is on the same level as the most efficient method in the corresponding regime, such as the particle methods in the rarefied regime and hydrodynamic solvers in continuum regime. The UGKWP method is shown to satisfy the MMP requirement. The UGKWP method is specially efficient for hypersonic flow simulation in all regimes in comparison with the discrete ordinate methods, and presents a much lower stochastic noise in the continuum flow regime in comparison with the particle-based Monte Carlo methods. Numerical tests for flows over a wide range of Mach and Knudsen numbers are presented. The examples include the hypersonic flow passing a circular cylinder at Mach numbers 20 and 30 and Knudsen numbers 1 and 10 − 4 , low speed lid-driven cavity flow, laminar boundary layer, shock structure, and shock tube problems. These results validate the accuracy, efficiency, and multiscale and multi-efficiency property of UGKWP method. [ABSTRACT FROM AUTHOR]

Based on kinetic model equation, a unified gas-kinetic scheme for axisymmetric flow (UGKS-AS) is proposed in the whole flow regimes. The flux transport of UGKS-AS is based on a reduced gas distribution function of a time integral solution of the full 3D BGK model. In the cylindrical coordinate system, the local time evolution of convection and source terms for the macroscopic and microscopic flow variables is fully coupled with particle free transport and collisions. As a result, a multiple scale UGKS-AS is developed and used for the flow simulation from the free molecular transport to the Navier–Stokes solutions. In comparison with the full 3D UGKS method for axisymmetric flow, the present scheme is much more efficient and gives accurate solution in all flow regimes. Different from operator splitting methods with decoupled treatment of particle transport and collision, for the Navier–Stokes solution in the continuum regime the UGKS-AS doesn't have the constraint on the time step being less than the particle collision time. To further improve the efficiency of the scheme for steady state solution, an implicit UGKS-AS method is also developed. A large number of numerical tests for axisymmetric flow are conducted by the proposed explicit and implicit UGKS-AS, which include inner and outer, steady and unsteady, low and high speed flows. Some tests are very challenging due to the delicate capturing of very small flow variations. The good agreement among the solutions from the present scheme and other analytical, numerical, and experimental studies, validates the high accuracy and efficiency of the UGKS-AS for non-equilibrium flow simulation in all regimes. [ABSTRACT FROM AUTHOR]

Most high order computational fluid dynamics (CFD) methods for compressible flows are based on Riemann solver for the flux evaluation and Runge–Kutta (RK) time stepping technique for temporal accuracy. The advantage of this kind of space–time separation approach is the easy implementation and stability enhancement by introducing more middle stages. However, the n th-order time accuracy needs no less than n stages for the RK method, which can be very time and memory consuming due to the reconstruction at each stage for a high order method. On the other hand, the multi-stage multi-derivative (MSMD) method can be used to achieve the same order of time accuracy using less middle stages with the use of the time derivatives of the flux function. For traditional Riemann solver based CFD methods, the lack of time derivatives in the flux function prevents its direct implementation of the MSMD method. However, the gas kinetic scheme (GKS) provides such a time accurate evolution model. By combining the second-order or third-order GKS flux functions with the MSMD technique, a family of high order gas kinetic methods can be constructed. As an extension of the previous 2-stage 4th-order GKS, the 5th-order schemes with 2 and 3 stages will be developed in this paper. Based on the same 5th-order WENO reconstruction, the performance of gas kinetic schemes from the 2nd- to the 5th-order time accurate methods will be evaluated. The results show that the 5th-order scheme can achieve the theoretical order of accuracy for the Euler equations, and present accurate Navier–Stokes solutions as well due to the coupling of inviscid and viscous terms in the GKS formulation. In comparison with Riemann solver based 5th-order RK method, the high order GKS has advantages in terms of efficiency, accuracy, and robustness, for all test cases. The 4th- and 5th-order GKS have the same robustness as the 2nd-order scheme for the capturing of discontinuous solutions. The current high order MSMD GKS is a multi-dimensional scheme with incorporation of both normal and tangential spatial derivatives of flow variables at a cell interface in the flux evaluation. The scheme can be extended straightforwardly to viscous flow computation in unstructured mesh. It provides a promising direction for the development of high-order CFD methods for the computation of complex flows, such as turbulence and acoustics with shock interactions. [ABSTRACT FROM AUTHOR]

The unified gas-kinetic scheme (UGKS) is a direct modeling method for both continuum and rarefied flow computations. In the previous study, the UGKS was developed for diatomic molecular simulations with translation and rotational motions. In this paper, a UGKS with non-equilibrium translational, rotational, and vibrational degrees of freedom, will be developed. The new scheme is based on the phenomenological gas dynamics model, where the translational, rotational, and vibrational modes get to the equilibrium with different time scales with the introduction of rotational and vibrational collision numbers. This new scheme is tested in a few cases, such as the homogeneous flow relaxation, shock structure, shock tube problem, and flow passing through a circular and semi-circular cylinders. The analytical and DSMC solutions are used for the validation of the UGKS, and reasonable agreements have been achieved. [ABSTRACT FROM AUTHOR]