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

TL;DR

This paper introduces stabilised discontinuous Galerkin finite element methods that effectively handle boundary and moving interior layers in incompressible natural convection flows on non-adapted meshes, improving accuracy and robustness.

Contribution

It presents a novel combination of grad-div and pressure jump stabilisation in dG-FEM for non-isothermal flows, enabling accurate solutions on coarse, non-adapted meshes.

Findings

Enhanced mass conservation with stabilised dG-FEM

Accurate handling of boundary and interior layers

Robustness and efficiency on coarse meshes

Abstract

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.