Stabilised dG-FEM for incompressible natural convection flows with boundary and moving interior layers on non-adapted meshes

• Published in: Journal of computational physics Vol. 335; pp. 760 - 779
• Main Authors: Schroeder, Philipp W., Lube, Gert
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 15.04.2017; Elsevier Science Ltd
• Subjects: Approximation; Boundary layer; Boundary layers; Boussinesq approximation; Computational fluid dynamics; Computational physics; Differential equations; Differentially heated square cavity; Discontinuous Galerkin method; Enthalpy; Finite element analysis; Finite element method; Fluid dynamics; Fluid flow; Free convection; Galerkin method; Gallium; Grad-div stabilisation; Incompressible flow; Mathematical analysis; Mathematical models; Melting of pure gallium; Multiphase flow; Numerical methods; Porosity; Pressure jump; Pressure jump stabilisation; Robustness (mathematics); Solidification; Stabilization; Studies; Weakly non-isothermal incompressible flow; Discontinuous Galerkin method; Pressure jump stabilisation; Melting of pure gallium; Grad-div stabilisation; Differentially heated square cavity; Weakly non-isothermal incompressible flow
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2017.01.055

This paper presents heavily grad-div and pressure jump stabilised, equal- and mixed-order discontinuous Galerkin finite element methods for non-isothermal incompressible flows based on the Oberbeck–Boussinesq approximation. In this framework, the enthalpy-porosity model for multiphase flow in melting and solidification problems can be employed. By considering the differentially heated cavity and the melting of pure gallium in a rectangular enclosure, it is shown that both boundary layers and sharp moving interior layers can be handled naturally by the proposed class of non-conforming methods. Due to the stabilising effect of the grad-div term and the robustness of discontinuous Galerkin methods, it is possible to solve the underlying problems accurately on coarse, non-adapted meshes. The interaction of heavy grad-div stabilisation and discontinuous Galerkin methods significantly improves the mass conservation properties and the overall accuracy of the numerical scheme which is observed for the first time. Hence, it is inferred that stabilised discontinuous Galerkin methods are highly robust as well as computationally efficient numerical methods to deal with natural convection problems arising in incompressible computational thermo-fluid dynamics. •We model incompressible natural convection via the Boussinesq approximation.•Equal-/mixed-order stabilised dG-FEM with improved mass conservation are used.•Grad-div stabilisation improves overall accuracy of pressure stabilised dG-FEM.•Our methods are superior to standard FEM and H(div)-conforming methods.•No adaptive mesh refinement is necessary for resolving boundary and interior layers.