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• 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]

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]