Schémas compacts basés sur le résidu d'ordre élevé pour des écoulements compressibles instationnaires. Application à de la capture de fines échelles.

Abstract : Computational Fluid Dynamics (CFD) solvers have reached maturity in terms of solution accuracy as well as computational efficiency. However, progress remains to be done for unsteady flows especially when governed by large, coherent structures. For these flows, current CFD solvers do not provide accurate solutions unless very fine mesh are used. Moreover, high-accuracy is a crucial feature for the application of advanced turbulence simulation strategies, like Large Eddy Simulation (LES). In order to apply high-order methods to complex unsteady flows several issues needs to be addressed among which numerical robustness and the capability of handling complex geometries.In the present work, we study a family of compact approximations that provide high accuracy not for each space derivative treated apart but for the complete residual r, i.e. the sum of all of the terms in the governing equations. For steady problems solved by time marching, r is the residual at steady state and it involves space derivatives only; for unsteady problems, r also includes the time derivative. Schemes of this type are referred-to as Residual-Based Compact (RBC). Precisely, we design high-order finite difference RBC schemes for unsteady compressible flows, and provide a comprehensive study of their dissipation properties. The dissipation and dispersion errors introduced by RBC schemes are investigated to quantify their capability of resolving a given wave length using a minimal number of grid-points. The capabilities of RBC dissipation to drain energy only at small, ill-resolved scales are also discussed in view of the application of RBC schemes to implicit LES (ILES) simulations. Finally, RBC schemes are extended to the Finite Volume (FV) framework in order to handle complex geometries. A high-order accuracy preserving FV formulation of the third-order RBC scheme for general irregular grids is presented and analysed. Numerical applications, including complex Reynolds-Averaged Navier-Stokes unsteady simulation of turbomachinery flows and ILES simulations of turbulent flows dominated by coherent structure dynamics or decay, support the theoretical results.
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Karim Grimich. Schémas compacts basés sur le résidu d'ordre élevé pour des écoulements compressibles instationnaires. Application à de la capture de fines échelles.. Mécanique des fluides [physics.class-ph]. Ecole nationale supérieure d'arts et métiers - ENSAM, 2013. Français. ⟨NNT : 2013ENAM0033⟩. ⟨tel-01178904⟩

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