Low-dissipative Numerical Schemes And The Applications In Direct Numerical Simulations

High speed flows are almost invariably involved with discontinuities and complicated flow structures, which put strict requirements on numerical methods. Comprehensive investigations are thus conducted into high-order shock-capturing schemes, a class of schemes possessing high-order accuracy, non-oscillation property and high computational efficiency is developed, these several competitive advantages make the schemes quite suitable for DNS of supersonic flows.It's pointed out in the present paper that the classical third-order WENO scheme loses order of accuracy at critical points where the third derivative doesn't vanish together with the first derivative, and instead of achieving third-order, it only obtains once-order accuracy. A new smooth indicator of sufficient high-order is proposed, combined with the framework of WENO-Z, the new WENO scheme has recovered third-order accuracy, which is verified by theoretical analyses and numerical results. The optimized WENO scheme has a large enhancement in the dissipation property and representation of discontinuities, further improvements have been made by exploiting the hybridization framework.Based on central schemes and WENO schemes, a systematic method of hybridization is proposed, and a new class of hybrid schemes have been formulated, which preserves the feature of FVS schemes and possesses a neat and concise intrinsic form. The competitive advantages of the resulting schemes allows the introduction of monotonicity preserving(MP) bounds and discontinuity sharpening technic. The MP bounds are utilized to improvement the robustness of the hybrid scheme, since the hybrid schemes themselves may incur some slight oscillations. Numerical results demonstrate that the monotonicity-preserving hybrid schemes not only perform excellently in resolving the complicated flow structures, but also are high efficient in terms of computational cost, therefore, this hybrid scheme is quite suitable for DNS of supersonic flows.Finally, tentative simulations of supersonic turbulent boundary layer are conducted by utilizing the numerical schemes presented in this paper and with the appropriate inflow condition for boundary layer transition.