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With the WENO reconstruction, a third-order multidimensional gas-kinetic scheme, with inclusion of both normal and tangential derivatives of macroscopic flow variables, is constructed for the three-dimensional inviscid and viscous flow computations. For the flux evaluation, a time and space dependent gas distribution function from an initial piece-wise discontinuous polynomials around a cell interface is presented, where a multiple scale physical process from the kinetic to the hydrodynamic one is used for the flow evolution process. A high-order accuracy is achieved in the current scheme by integrating the space and time dependent flux function over the surface of a cell interface and a whole time step, without using the conventional Gaussian points integration for spatial accuracy and Runge–Kutta method for temporal accuracy, which have been used in many other high-order schemes. This scheme is applied to both inviscid and viscous, and low and high speed flow computations. The numerical tests clearly demonstrate that the current scheme is robust for the flows with strong discontinuities and accurate for viscous smooth flow solutions. The continuous effort on the gas kinetic scheme (GKS) is to develop a third method which has the same reliability and robustness as the well-developed second-order shock capturing schemes, but is simply much more accurate in all flow applications. The current scheme seems achieve such a target. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a gas-kinetic scheme to solve the multiple temperature kinetic model (MTKM), which was proposed in J. Comput. Math. 29(6) (2011) 639–660, for the study of non-equilibrium flows. The MTKM is a two-stage particle collision model possessing an intermediate quasi-equilibrium state with a symmetric second-order temperature tensor. A gas-kinetic finite volume scheme is developed for the numerical solution of the MTKM in the continuum and transition flow regimes. The gas-kinetic scheme is designed for the updating of macroscopic variables, which include the conservative flow variables and the multiple temperature field. In order to validate the kinetic model, the gas-kinetic scheme is used in the study of lid-driven cavity flows in both continuum and transition flow regimes. The numerical results predicted by the MTKM are compared with those from the direct simulation Monte Carlo (DSMC) method, the Navier–Stokes equations (NSE), and the early three-temperature kinetic model (TTKM) proposed in Phys. Fluids 19, 016101(2007). It is demonstrated that the MTKM has obvious advantages in comparison with the NSE and the TTKM in capturing the non-equilibrium flow behavior in the transition flow regime. One distinguishable phenomenon captured by the MTKM is that in the transition flow regime the heat flux direction can be from a low temperature to a high temperature region, which violates the Fourier’s law of continuum flows. The MTKM provides a more accurate physical model than the NSE for the non-equilibrium flows. [Copyright &y& Elsevier]

• A simple and efficient third-order WENO reconstruction is developed for 3D flows. • The WENO-AO reconstruction is adopted, which simplify the classical WENO schemes. • Consider the grid velocity, the scheme is extended to the moving-mesh computation. In this paper, a simple and efficient third-order weighted essentially non-oscillatory (WENO) reconstruction is developed for three-dimensional flows, in which the idea of two-dimensional WENO-AO scheme on unstructured meshes [42] is adopted. In the classical finite volume type WENO schemes, the linear weights for candidate stencils are obtained by solving linear systems at Gaussian quadrature points of cell interface. For the three-dimensional scheme, such operations at twenty-four Gaussian quadrature points of a hexahedron would reduce the efficiency greatly, especially for the moving-mesh computation. Another drawback of the classical WENO schemes is the appearance of negative weights with irregular local topology, which affect the robustness of spatial reconstruction. For the three-dimensional WENO-AO scheme, a simple strategy of selecting big stencil and sub-stencils for reconstruction is proposed. With the reconstructed quadratic polynomial from big stencil and linear polynomials from sub-stencils, the linear weights are chosen as positive numbers with the requirement that their summation equals to one and be independent of local mesh topology. With such WENO reconstruction, a high-order gas-kinetic scheme (HGKS) is developed for both three-dimensional inviscid and viscous flows. Taken the grid velocity into account, the scheme is extended into the moving-mesh computation as well. Numerical results are provided to illustrate the good performance of such new finite volume WENO scheme. In the future, such WENO reconstruction will be extended to the unstructured meshes. [ABSTRACT FROM AUTHOR]

In this paper, a third-order gas-kinetic scheme is developed on the three-dimensional hybrid unstructured meshes for the compressible inviscid and viscous flows. In the classical weighted essentially non-oscillatory (WENO) scheme, the high-order spatial accuracy is achieved by the non-linear combination of lower order polynomials. However, for the hybrid unstructured meshes, the procedures, including the selection of candidate stencils and calculation of linear weights, become extremely complicated, especially for three-dimensional problems. To overcome the drawbacks, a third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A general strategy is given for the selection of big stencil and sub-stencils for hybrid meshes, and the topologically independent linear weights are used in the spatial reconstruction. A unified interpolation is also used for the volume integral of different types of control volumes, as well as the flux integration over different cell interfaces. Incorporate with the two-stage fourth-order temporal discretization, the explicit high-order gas-kinetic schemes are developed for unsteady problems. With the lower–upper symmetric Gauss–Seidel (LU-SGS) methods and Jacobi iteration, the implicit high-order gas-kinetic schemes are developed for steady problems. To accelerate the computation, both explicit and implicit scheme is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). Especially, for implicit parallel computation, Jacobi iteration is used. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows. • A third-order GKS is developed on 3D hybrid unstructured meshes for compressible flows. • A general selection of stencil is given for 3D hybrid meshes. • Based on the selected stencils, the topologically independent linear weights are used. • A unified method is given for geometrical computation and flux integration for different cells. • To accelerate the computation, the current scheme is implemented with GPU. [ABSTRACT FROM AUTHOR]

The recently proposed discrete unified gas kinetic scheme (DUGKS) is a finite volume method for multiscale flow computations with asymptotic preserving property. The solution of the Boltzmann model equation is directly used for the construction of numerical flux and makes the scheme applicable in all flow regimes. In previous applications of the DUGKS, structured meshes have been mostly employed, which may have difficulties for problems with complex geometries. In this paper we will extend the DUGKS to unstructured meshes, with the implementation of computational fluid dynamics techniques to the DUGKS. Several test cases, i.e., the cavity flow ranging from continuum to free molecular regimes, a multiscale flow problem between two connected cavities with large pressure and density variations, high speed flows past multiple cylinders in slip and transitional regimes, and an impulsive start problem are performed. The results are compared with the well-defined Direct Simulation Monte Carlo (DSMC) or Navier–Stokes (NS) solutions in their applicable regimes. The numerical results demonstrate the effectiveness of the proposed DUGKS for the study of multiscale flow problems. [ABSTRACT FROM AUTHOR]

In this paper, the high-order compact gas-kinetic scheme (CGKS) on three-dimensional hybrid unstructured mesh is further developed with the p-multigrid technique for steady-state solution acceleration. The p-multigrid strategy is a two-level algorithm. On the high-order level, the CGKS with high-order compact initial reconstruction is used to evolve both cell-averaged conservative flow variables and their gradients at the beginning of next time step. On the low-order level, starting from the high-order level solution the cell-averaged conservative flow variables is evolved by a first-order scheme, where implicit backward Euler smoother is adopted for accelerating the convergence of steady-state solution. The final iterative updating scheme becomes numerically simple and computationally efficient. The effectiveness of the p-multigrid method is validated in both subsonic and supersonic flow simulations in two- and three-dimensional space with hybrid unstructured mesh. One order of magnitude speedup in the convergence rate has been achieved by the approach in comparison with the explicit counterpart. • The p-multigrid technique is firstly adopted to the compact gas-kinetic scheme. • One order of magnitude speedup is achieved compared with the explicit counterpart. • The resulting two-level p-multigrid compact gas-kinetic scheme is simple and robust. • The effectiveness of the proposed method is validated in 2-D and 3-D flow simulations. [ABSTRACT FROM AUTHOR]

• A fourth-order multi-dimensional WENO reconstruction is developed, in which a simple strategy of selecting stencils is adopted and the topology independent linear weights are used. • In corporate with two-stage fourth-order temporal discretization and the fourth-order WENO reconstruction, a high-order gas-kinetic scheme is proposed for DNS and LES of turbulence. • The fourth-order reconstruction improves the order of accuracy of the previous thirdorder WENO reconstruction, which is too dissipative for the simulation of turbulent flows. • Numerical examples, including the classical turbulence cases, are provided to illustrate the good performance of such WENO scheme for turbulence simulation. In this paper, a fourth-order multi-dimensional weighted essentially non-oscillatory (WENO) reconstruction is developed, which is aiming at the complicated flows. In corporate with two-stage fourth-order temporal discretization, a high-order gas-kinetic scheme is proposed for the direct numerical simulation (DNS) and large eddy simulation (LES) of turbulence. In such WENO scheme, a simple strategy of selecting stencils is adopted and the topology independent linear weights are used. The fourth-order reconstruction improves the order of accuracy of the previous third-order WENO reconstruction [Computers and Fluids 198 (2020) 104401], which is too dissipative for the simulation of turbulent flows. Compared with the classical dimension-by-dimension WENO reconstruction, the order of accuracy can be also achieved for any non-uniform and curvilinear meshes in the finite volume framework. Numerical examples, including the classical turbulence cases, are provided to illustrate the good performance of such WENO scheme for turbulence simulation. More importantly, the fourth-order WENO scheme is robust and works well from the subsonic to hypersonic flows. In the future, the current scheme will be extended to the genuinely unstructured meshes for more complicated turbulent flows. [ABSTRACT FROM AUTHOR]

• Isotropic compressible turbulence at supersonic regime has been studied by the high-order gas-kinetic scheme. • Detailed statistical solutions are provided as the benchmark solutions. • The DNS solutions provide useful data for the construction of compressible LES model in supersonic regime. • All these results and the analysis of physical mechanism are useful for the future study of isotropic compressible turbulence. In this paper, we intend to address the high-order gas-kinetic scheme (HGKS) in the direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime. To validate the performance of HGKS, the compressible isotropic turbulence with initial turbulent Mach number M a t 0 = 0.5 and Taylor microscale Reynold number R e λ 0 = 72 is simulated as a benchmark. With the consideration of robustness and accuracy, the WENO-Z scheme is adopted for spatial reconstruction in the current higher-order scheme. Statistical quantities are compared with the high-order compact finite difference scheme to determine the spatial and temporal criterion for DNS. According to the grid and time convergence study, it can be concluded that the minimum spatial resolution parameter κ max η 0 ≥ 2.71 and the maximum temporal resolution parameter Δ t ini / τ 0 ≤ 27.08/1000 are adequate for HGKS to resolve the compressible isotropic turbulence, where κ max is the maximum resolved wave number, Δ t ini is the initial time step, η 0 and τ 0 are the initial Kolmogorov length scale and time scale, respectively. Guided by such criterion, the compressible isotropic turbulence from subsonic regime M a t 0 = 0.8 to supersonic one M a t 0 = 1.2 , and the Taylor microscale Reynolds number Re λ 0 ranging from 10 to 72 are simulated. With the high initial turbulent Mach number, the strong random shocklets and high expansion regions are identified, as well as the wide range of probability density function over local turbulent Mach number. All those impose great challenge for high-order schemes. In order to construct compressible large eddy simulation models at high turbulent Mach number, the ensemble budget of turbulent kinetic energy is fully analyzed. The solenoidal dissipation rate decreases with the increasing of Ma t 0 and Re λ 0. Meanwhile, the dilational dissipation rate increases with the increasing of Ma t 0 , which cannot be neglected for constructing supersonic turbulence model. The current work shows that HGKS provides a valid tool for the numerical and physical studies of isotropic compressible turbulence in supersonic regime, which is much less reported in the current turbulent flow study. [ABSTRACT FROM AUTHOR]

• Proposed fifth order finite volume WENO-C in orthogonally-curvilinear coordinates. • WENO-Curvilinear reconstruction and averaging weights provided for standard grids. • Analytical values of weights for Radius → ∞ case conform to conventional WENO-JS. • Tests performed in curvilinear coordinates testify for fifth order & ENO properties. A fifth order finite volume WENO reconstruction scheme is proposed in the framework of orthogonally - curvilinear coordinates for solving hyperbolic conservation equations. The derivation employs a piecewise parabolic polynomial approximation to the zone averaged values (Q ¯ i) to reconstruct the right (q i +), middle (q i M), and left (q i −) interface values. The grid dependent linear weights of the WENO are recovered by inverting a Vandermonde - like linear system of equations with spatially varying coefficients. A scheme for calculating the linear weights, optimal weights, and smoothness indicator on a regularly - /irregularly - spaced grid in orthogonally - curvilinear coordinates is proposed. A grid independent relation for evaluating the smoothness indicator is derived from the basic definition. Finally, a computationally efficient extension to multi - dimensions is proposed along with the procedures for flux and source term integrations. Analytical values of the linear weights, optimal weights, and weights for flux and source term integrations are provided for a regularly - spaced grid in Cartesian, cylindrical, and spherical coordinates. Conventional fifth order WENO - JS can be fully recovered in the case of limiting curvature (R → ∞). The fifth order finite volume WENO - C (orthogonally - curvilinear version of WENO) reconstruction scheme is tested for several 1D and 2D benchmark tests involving smooth and discontinuous flows in cylindrical and spherical coordinates. [ABSTRACT FROM AUTHOR]

High-order gas-kinetic scheme (HGKS) with 5th-order non-compact reconstruction has been well implemented for implicit large eddy simulation (ILES) in nearly incompressible turbulent channel flows. In this study, the HGKS with higher-order non-compact reconstruction and compact reconstruction will be validated in turbulence simulation. For higher-order non-compact reconstruction, 7th-order reconstruction in both normal and tangential directions are implemented. On the other hand, the compact HGKS with 5th-order compact reconstruction in normal direction is developed. Current work aims to show the benefits of high-order non-compact reconstruction and compact reconstruction for ILES. The accuracy of HGKS is verified by numerical simulation of three-dimensional advection of density perturbation. For the non-compact 7th-order scheme, 16 Gaussian points are required on the cell-interface to preserve the order of accuracy. Then, HGKS with non-compact and compact reconstruction is used in the three-dimensional Taylor–Green vortex (TGV) problem and turbulent channel flows. Accurate ILES solutions have been obtained from HGKS. In terms of the physical modeling underlying the numerical algorithms, the compact reconstruction has the consistent physical and numerical domains of dependence without employing additional information from cells which have no any direct physical connection with the targeted cell. The compact GKS shows a favorable performance for turbulence simulation in resolving the multi-scale structures. • The HGKS with non-compact and compact reconstruction are developed for ILES. • Compact reconstruction shows consistent physical and numerical domains of dependence. • The results with compact reconstruction outweigh those of non-compact reconstruction. • In the same order, compact reconstruction resolves much smaller turbulent structures. • The HGKS with compact reconstruction shows great strength in multi-scale turbulence. [ABSTRACT FROM AUTHOR]