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

Abstract

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 Mat0=0.5 and Taylor microscale Reynold number Reλ0=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 κmaxη0 ≥ 2.71 and the maximum temporal resolution parameter Δtini/τ0 ≤ 27.08/1000 are adequate for HGKS to resolve the compressible isotropic turbulence, where κmax is the maximum resolved wave number, Δtini is the initial time step, η0 and τ0 are the initial Kolmogorov length scale and time scale, respectively. Guided by such criterion, the compressible isotropic turbulence from subsonic regime Mat0=0.8 to supersonic one Mat0=1.2, and the Taylor microscale Reynolds number Reλ0 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 Mat0 and Reλ0. Meanwhile, the dilational dissipation rate increases with the increasing of Mat0, 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.