Multiscale finite element method applied to the Spalart–Allmaras turbulence model for 3D detached-eddy simulation

Abstract

Employing the advection–diffusion–reaction equation as a model problem we present a multiscale method that yields a stabilized finite element formulation for Reynolds-Averaged Navier–Stokes (RANS) based turbulence models. The multiscale method arises from a decomposition of the scalar field into coarse (resolved) and fine (unresolved) scales. Modeling of the unresolved scales corrects the lack of stability of the standard Galerkin formulation. The proposed method possesses superior properties like that of the Streamline Upwind/Petrov–Galerkin (SUPG) method and the Galerkin/Least-Squares (GLS) method. The stabilization terms appear naturally and the resulting formulation provides effective stabilization in turbulent computations where reaction-dominated effects strongly influence the boundary layer prediction. A family of 2D and 3D elements is developed and comparison of the proposed method with the SUPG method is presented. The multiscale formulation is then applied to the Spalart–Allmaras turbulence model [1] in FENSAP-ICE [2–7] for detached-eddy simulation (DES). Numerical results obtained by the proposed method are compared with experimental and DNS results for backward-facing step problem at Reynolds number of 5000.