Abstract
Abstract In this paper we study a phase change problem for non-isothermal incompressible viscous flows. The underlying continuum is modelled as a viscous Newtonian fluid where the change of phase is either encoded in the viscosity itself, or in the Brinkman–Boussinesq approximation where the solidification process influences the drag directly. We address these and other modelling assumptions and their consequences in the simulation of differentially heated cavity flows of diverse type. A second order finite element method for the primal formulation of the problem in terms of velocity, temperature, and pressure is constructed, and we provide conditions for its stability. We finally present several numerical tests in 2D and 3D, corroborating the accuracy of the numerical scheme as well as illustrating key properties of the model.
Highlights
The phenomenon of natural convection driven by variations in temperature distribution has been extensively studied from the viewpoint of physical properties and using computational methods
Recent numerical methods dedicated for phase change Boussinesq models include a class of stabilised discontinuous Galerkin and finite volume methods proposed for porosity-based models in [2] and [6], respectively; and the primal finite element scheme for viscosity-based models, introduced in [5]
We have addressed the modelling of phase change in Boussinesq models within porous media
Summary
The phenomenon of natural convection driven by variations in temperature distribution has been extensively studied from the viewpoint of physical properties and using computational methods. Recent numerical methods dedicated for phase change Boussinesq models include a class of stabilised discontinuous Galerkin and finite volume methods proposed for porosity-based models in [2] and [6], respectively; and the primal finite element scheme for viscosity-based models, introduced in [5]. These contributions do not address the stability of the discrete or continuous problems. These tests include a qualitative analysis on the micro-structure and its relationship with our modelling assumptions, and we close with some remarks and discussions on alternative models
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