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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]

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: 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]

• 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]

Computational fluid dynamics is a direct modeling of physical laws in a discretized space. The basic physical laws include the mass, momentum and energy conservations, numerical scale-consistent transport process, and similar domain of dependence and influence in the physical and the numerical wave propagations. Therefore, a physically soundable numerical scheme should be a compact one which involves the closest neighboring cells within the domain of dependence for the solution update under a CFL number (∼1). In order to keep the compactness of the scheme and be high-order, subcell flow distributions or the equivalent degrees of freedom beyond the cell averaged flow variables are needed, such as the gradients of the flow variables inside each control volume. Under the above consideration, the direct use of Riemann solver as the evolution model doesn't provide such a mechanism. High-order dynamic evolution process should be used to update the time accurate cell-averaged flow variables and their gradients. The direct modeling of flow evolution under generalized initial condition will be developed. In order to provide reliable cell averaged flow variables and their gradients for the compact data reconstruction, the evolution process is constructed to provide possible discontinuous time-dependent flow variables across a cell interface. At the same time, the time accurate flux function at a cell interface can become a discontinuous function of time in its evolution process. Same as the spatial limiter in the conventional computational fluid dynamics methods, such as the total variation diminishing and weighted essentially non-oscillatory methods, the temporal limiter for the time accurate flux function is designed as well. The direct modeling method in this paper will update flow variables differently on both sides of a cell interface and nonlinearly limit the high-order time derivatives of the flux function in case of shock wave passing through a cell interface within a time step. The direct modeling method unifies the concepts of nonlinear limiters in spatial data reconstruction and temporal flux transport. The equivalent treatment of space and time makes the compact gas-kinetic scheme be super robust and accurate for the compressible flow simulation. At the same time, the 4th to 8th-order compact schemes can use a large CFL number (∼ 0.8) in flow computation, even for the hypersonic flow simulation. • A general principle for the development of high-order compact scheme is presented. • Same as the nonlinear limiter in space, the nonlinear limiter in time for the flux function is proposed. • According to the principle the high-order compact gas-kinetic scheme is developed. • Accurate solutions of compressible Euler and Navier-Stokes equations are obtained. [ABSTRACT FROM AUTHOR]

• Development of unified gas-kinetic wave-particle method for diatomic gas. • The multiscale solution is captured through dynamic adaptation of wave and particle formulation. • UGKWP becomes particle method in rarefied regime and hydrodynamic solver in continuum regime. • UGKWP keeps optimal efficiency in different flow regimes for the hypersonic flow simulation. In this paper, the unified gas-kinetic wave-particle (UGKWP) method is further developed for diatomic gas with the energy exchange between translational and rotational modes for flow study in all regimes. The multiscale transport mechanism in UGKWP is coming from the direct modelling in a discretized space, where the cell's Knudsen number, defined by the ratio of particle mean free path over the numerical cell size, determines the flow physics simulated by the wave particle method. The non-equilibrium distribution function in UGKWP is tracked by the discrete particle and analytical wave. The weights of distributed particle and wave in different regimes depend on cell's Knudsen number, where distinguishable macroscopic flow variables of particle and wave are updated inside each control volume. The UGKWP becomes a particle method in the highly rarefied flow regime and converges to the gas-kinetic scheme (GKS) for the Navier-Stokes solution in the continuum flow regime without particles. In comparison with discrete velocity method (DVM)-based unified gas-kinetic scheme (UGKS), the computational cost and memory requirement in UGKWP have been reduced by several orders of magnitude for the high speed and high temperature flow simulation, where the translational and rotational non-equilibrium can be captured accurately in the transition and rarefied regime. As a result, for the hypersonic flow around a flying vehicle, the computation can be conducted using a personal computer for the studies in all regimes. The UGKWP method for diatomic gas will be validated in various cases from one dimensional shock structure to three dimensional flow over a sphere, and the numerical solutions will be compared with the reference DSMC results and experimental measurements. [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]