Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence I: criterion for direct numerical simulation

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. 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 $\kappa_{max} \eta_0 \ge 2.71$ and the maximum temporal resolution parameter $\Delta t_{ini}/\tau_{t_0} \leq 5.58/1000$ are adequate for HGKS to resolve the compressible isotropic turbulence, where $\kappa_{max}$ is the maximum resolved wave number, $\Delta t_{ini}$ is the initial time step, $\eta_0$ and $\tau_{t_0}$ are the initial Kolmogorov length scale and the large-eddy-turnover time. Guided by such criterion, the compressible isotropic turbulence from subsonic regime $Ma_t=0.8$ to supersonic one $Ma_t=1.2$, and the Taylor microscale Reynolds number $Re_{\lambda}$ 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 $Ma_t$ and $Re_{\lambda}$. Meanwhile, the dilational dissipation rate increases with the increasing of $Ma_t$, which cannot be neglected for constructing supersonic turbulence model.