An entropy-stable discontinuous Galerkin approximation of the Spalart-Allmaras turbulence model for the compressible Reynolds averaged Navier-Stokes equations. (English) Zbl 07518083
Summary: We present an entropy-stable formulation for the compressible Reynolds Averaged Navier-Stokes (RANS) Discontinuous Galerkin (DG) equations and the Spalart-Allmaras one-equation closure. The model is designed to satisfy an entropy law, which includes free- and no-slip wall boundary conditions. We then construct a high-order DG approximation of the model that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. With the help of a discrete stability analysis, we construct two approximations: a kinetic energy preserving scheme based on Pirozzoli’s two-point flux and a thermodynamic entropy conserving one based on Chandrashekar’s split-form. The schemes are applicable to, and the stability proofs hold for, three-dimensional unstructured meshes with curvilinear hexahedral elements. We test the convergence of the schemes on a manufactured solution for increasing polynomial orders and mesh refinement levels, to then assess their numerical stability by propagating a flow from a random initial condition, and finally solve the flow around a two-dimensional flat plate and a NACA 0012 airfoil, comparing numerical results with those available in the literature. The proposed schemes are entropy-stable, and provide accurate solutions for the selected test cases.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76Mxx | Basic methods in fluid mechanics |
| 76Fxx | Turbulence |
Keywords:
Reynolds averaged Navier-Stokes; Spalart-Allmaras; entropy-stable; discontinuous Galerkin; high-order methodsReferences:
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