For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the gas-kinetic kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around each cell interface. With the use 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 [18]. In this paper, based on the same time-stepping method and the second-order GKS flux function [34], a fourth-order gas-kinetic scheme is constructed for the Euler and Navier-Stokes equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [21], the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme, even though the third-and fourth-order schemes have similar computation cost. Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. Perfect numerical solutions can be obtained from the high Reynolds number boundary layer solutions to the hypersonic viscous heat conducting flow computations. Many numerical tests, including many difficult ones for the Navier-Stokes solvers, have been used to validate the current fourth-order method. Following the two-stage time-stepping framework, the one-stage third-order GKS can be easily extended to 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 impact on the development of higher-order CFD methods.

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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 highorder 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 evolution solution at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the GKS can get the cell averaged slopes inside each control volume as well through the differences of flow variables at the cell interfaces. So, with the updated flow variables and their slopes inside each cell, the HWENO reconstruction can be naturally implemented for the compact high-order reconstruction at the beginning of next step. Therefore, a compact higher-order GKS, such as the two-stages fourth-order compact scheme can be constructed. This scheme is as robust as second-order one, but more accurate solution can be obtained. In comparison with compact fourth-order DG method, the current scheme has only two stages instead of four within each time step for the fourth-order temporal accuracy, and the CFL number used here can be on the order of 0.5 instead of 0.11 for the DG method. Through this research, it concludes that the use of high-order time evolution model rather than the first order Riemann solution is extremely important for the design of robust, accurate, and efficient higher-order schemes for the compressible flows.2 the second-order or third-order GKS fluxes with the multi-stage multi-derivative technique again, a family of high order gas-kinetic methods has been constructed [17]. The above higher-order GKS uses the higher-order WENO reconstruction for spatial accuracy. These schemes are not compact and have room for further improvement.The GKS time dependent gas-distribution function at a cell interface provides not only the flux evaluation and its time derivative, but also time accurate flow variables at a cell interface. The design of compact GKS based on the cell averaged and cell interface values has been conducted before [44,31,32]. In the previous approach, the cell interface values are strictly enforced in the reconstruction, which may not be an appropriate approach. In this paper, inspired by the Hermite WENO (HWENO) reconstruction and compact fourth order GRP scheme [11], instead of using the interface values we are going to get the slopes inside each control volume first, then based on the cell averaged values and slopes inside each control volume the HWENO reconstruction is implemented for the compact highorder reconstruction. The higher-order compact GKS developed in this paper is basically a unified combination of three ingredients, which are the two-stage fourth-order framework for temporal discretization [33], the higher-order gas evolution model for...

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 turbulent Mach number M a t = 0.5 and Taylor microscale Reynold number Re λ = 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 ∆t ini /τ t 0 ≤ 5.58/1000 are adequate for HGKS to resolve the compressible isotropic turbulence, where κ max is the maximum resolved wave number, ∆t ini is the initial time step, η 0 and τ t 0 are the initial Kolmogorov length scale and the large-eddyturnover time. Guided by such criterion, the compressible isotropic turbulence from subsonic regime M a t = 0.8 to supersonic one M a t = 1.2, and the Taylor microscale Reynolds number Re λ 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 turbulence 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 M a t and Re λ . Meanwhile, the dilational dissipation rate increases with the increasing of M a t , 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.

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.