Resumen de Nonlinear subgrid finite element models for low Mach number flows coupled with radiative heat transfer
Matías Oscar Avila Salinas
The general description of a fluid flow involves the solution of the compressible Navier-Stokes equations, a very complex problem whose mathematical structure is not well understood. It is widely accepted that these equations provide an accurate description of any problem in fluid mechanics which may present many different nonlinear physical mechanisms. Depending on the physics of the problem under consideration, different simplified models neglecting some physical mechanisms can be derived from asymptotic analysis. On the other hand, radiative heat transfer can strongly interact with convection in high temperature flows, and neglecting its effects may have significant consequences in the overall predictions. Problems as fire scenarios emphasized the need for an evaluation of the effect of radiative heat transfer. This work is directed to strongly thermally coupled low Mach number flows with radiative heat transfer. The complexity of these mathematical problem makes their numerical solution very difficult. Despite the important difference in the treatment of the incompressibility, the low Mach number equations present the same mathematical structure as the incompressible Navier-Stokes equations, in the sense that the mechanical pressure is determined from the mass conservation constraint. Consequently the same type of numerical instabilities can be found, namely, the problem of compatibility conditions between the velocity and pressure finite element spaces, and the instabilities due to convection dominated flows. These instabilities can be avoided by the use of stabilization techniques. Many stabilization techniques used nowadays are based on the variational multiscale method, in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed. The modeling of the subgrid scale and its influence leads to a modified coarse scale problem providing stability. The quality of the final approximation (accuracy, efficiency) depends on the particular model. The extension of these techniques to nonlinear and coupled problems is presented. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms appearing in the finite element equations and in the subgrid scale model. The first ingredient permits to obtain an improved time discretization scheme(higher accuracy, better stability). The second ingredient permits to prove global conservation properties, being also responsible of the higher accuracy of the method. This ingredient is intimately related to the problem of thermal turbulence modeling from a strictly numerical point of view. The capability for the simulation of turbulent flows is a measure of the ability of modeling the effect of the subgrid flow structures over the coarser ones. The performance of the model in predicting the behavior of turbulent flows is demonstrated. The radiation transport equation has been also approximated within the variational multiscale framework, the design and analysis of stabilized finite element methods is presented.
