Finite Element Approximation Of Thermal Models For Lowspeed Flows

Abstract

Thermally coupled low speed flows have been traditionally modeled by the incompressible
Boussinesq approximation. Its relationship with models obtained from the zero Mach limit of
the general compressible Navier-Stokes equations is a subject that still deserves interest, both
from the conceptual and the numerical points of view. On the one hand, the way to justify the
Boussinesq model by using asymptotic expansions is not unique. Several geometrical and/or
thermodynamic assumptions may be used. On the other hand, numerical experiments can serve
as a virtual laboratory to test the validity of the Boussinesq approach in terms of the temperature
gradients present in the flow.
In this work we discuss the relationship between the Boussinesq model and asymptotic models
for thermally coupled low Mach number flows, trying to clarify their connections. Likewise,
we propose a finite element approximation for these models using stabilization to treat cases
dominated by convection and allowing equal interpolation for all the variables. The numerical
formulation is based on the subgrid scale concept.
Both in the description of the thermal models and in the presentation of the stabilized finite
element techniques we employ to approximate them, our intention is to introduce these subjects
rather than to present the latest research results in these fields.