Entropy-stable \(p\)-nonconforming discretizations with the summation-by-parts property for the compressible Navier-Stokes equations. (English) Zbl 1521.76326
Summary: The entropy-conservative/stable, curvilinear, nonconforming, \(p\)-refinement algorithm for hyperbolic conservation laws of D. C. Del Rey Fernández et al. [J. Comput. Phys. 392, 161–186 (2019; Zbl 1452.76085)] is extended from the compressible Euler equations to the compressible Navier-Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming elements is used. Curvilinear volume metric terms are numerically approximated via a minimization procedure and satisfy the discrete geometric conservation law conditions. Distinct curvilinear surface metrics are used on the adjoining interfaces to construct the interface coupling terms, thereby localizing the discrete geometric conservation law constraints to each individual element. The resulting scheme is entropy conservative/stable, element-wise conservative, and freestream preserving. Viscous interface dissipation operators that retain the entropy stability of the base scheme are developed. The accuracy and stability of the resulting numerical scheme are shown to be comparable to those of the original conforming scheme in [M. H. Carpenter et al., SIAM J. Sci. Comput. 36, No. 5, B835–B867 (2014; Zbl 1457.65140)] and [M. Parsani et al., SIAM J. Sci. Comput. 38, No. 5, A3129–A3162 (2016; Zbl 1457.65149)], i.e., this scheme achieves \(\sim p + 1 / 2\) convergence on geometrically high-order distorted element grids; this is demonstrated in the context of the viscous shock problem, the Taylor-Green vortex problem at a Reynolds number of \(R e = 1,600\), and a subsonic turbulent flow past a sphere at \(R e = 2,000\).
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76N06 | Compressible Navier-Stokes equations |
Keywords:
compressible Navier-Stokes equations; nonconforming interfaces; nonlinear entropy stability; summation-by-parts and simultaneous approximation terms; curved elements; unstructured gridsReferences:
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