Very high-order accurate finite volume scheme for the steady-state incompressible Navier–Stokes equations with polygonal meshes on arbitrary curved boundaries

• Published in: Computer methods in applied mechanics and engineering Vol. 396; p. 115064
• Main Authors: Costa, Ricardo, Clain, Stéphane, Machado, Gaspar J., Nóbrega, João M.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2022; Elsevier BV
• Subjects: Algorithms; Arbitrary curved boundaries; Boundary conditions; Convergence; Discretization; Finite volume method; Fluid dynamics; Fluid flow; Incompressible flow; Incompressible fluid flows; Linear transformations; Mathematical analysis; Mesh generation; Navier-Stokes equations; Parameterization; Polygons; Polynomials; Quadratures; Reconstruction for off-site data method; Robustness (mathematics); Staggered polygonal meshes; Very high-order of convergence; Navier–Stokes equations; Reconstruction for off-site data method; Incompressible fluid flows; Arbitrary curved boundaries; Staggered polygonal meshes; Very high-order of convergence
• ISSN: 0045-7825
• DOI: 10.1016/j.cma.2022.115064

The numerical solution of the incompressible Navier–Stokes equations raises challenging numerical issues on the development of accurate and robust discretisation techniques due to the div-grad duality. Significant advances have been achieved in the context of high-order of convergence methods, but many questions remain unsolved in the quest of practical, accurate, and efficient approaches. In particular, the discretisation of problems with arbitrary curved boundaries has been the subject of intensive research, where the conventional practice consists in employing curved meshes to avoid accuracy deterioration. The present work proposes a novel approach, the reconstruction for off-site data method, which employs polygonal meshes to approximate arbitrary smooth curved boundaries with linear piecewise elements. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed on the polygonal mesh elements. Boundary conditions are taken into account via polynomial reconstructions with specific linear constraints defined for a set of points on the physical boundary. The very-high order of convergence is preserved without relying on the full curve parametrisation, avoiding the limitations of curved mesh approaches, such as sophisticated mesh generation algorithms, cumbersome quadrature rules, and complex non-linear transformations. A comprehensive verification benchmark is provided, with numerical test cases for several fluid flow problems, to demonstrate the capability of the proposed approach to achieve very high-orders of convergence. •Steady-state incompressible Navier–Stokes equations in curved domains.•Very high-order accurate finite volume scheme.•Arbitrary curved boundaries approximated with unstructured polygonal meshes.•Verification benchmark comprising several fluid flows problems.•Sixth-order of convergence for non-trivial arbitrary curved boundaries.