On the behavior of high order one-step monotonicity-preserving scheme for direct numerical simulation of turbulent flows
Résumé
High Reynolds number supersonic flows require accurate predictions, in the one hand, for the design of industrial devices, and on the other hand, for the physical understanding and modeling of these high speed flows that still raise numerous scientific issues. To this end, Direct Numerical Simulations (DNS) are nowadays an adequate tool since all characteristic time and space scales are computed without using any modeling. Among the great variety of high-order numerical schemes using shock-capture features developed for DNS, high-order One-Step (OS) Monotonicity-Preserving (MP) schemes have been developed [1,2] that control the total truncation error involving both errors due to time integration and spatial discretization. The objective of this paper is to check the ability of the high-order OSMP scheme to accurately compute turbulent compressible flows with a special focus on the effect of the MP constraints on solutions of turbulent flows without shock wave. The classical3-D Taylor-Green vortex test case [3] is first used for evaluating the accuracy of the solver to compute continuous turbulent solutions. Secondly, the spatial development of a supersonic turbulent boundary layer is simulated. A compressible version of the Synthetic Eddy Method is developed for prescribing inflow conditions. We showed that these specific conditions greatly lower the adaptation length compared to more classical inflow conditions. As diffusion in the near wall region is one of the key phenomena, the influence of the viscous flux discretization is reviewed with 2nd-order and 4th-order centered approximations. Compared to reference solutions from the literature, we showed that the OSMP scheme produces reliable solutions that are in very good agreements with references. Even for bounded flows, the use of an order of accuracy higher than the 2nd-order for approximating the diffusive fluxes was shown to have a negligible influence on solutions. We conclude that the OSMP scheme coupled with a centered 2nd-order approximation for the diffusive fluxes constitutes a reliable numerical tool for the DNS of compressible turbulent flows, that is completely competitive compared to other approaches since the overall errors are lowered of about one order of magnitude for the same grid resolution.
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