Abstract

In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD problem couples the Navier–Stokes equations and a Darcy-type system for the electric potential via Lorentz’s force in the momentum equation of the Navier–Stokes equations and the currents generated by the moving fluid in Ohm’s law. The key feature of the FE formulation resides in the design of the stabilization terms, which serve several purposes. First, the formulation is suitable for convection dominated flows. Second, there is no need to use interpolation spaces constrained to a compatibility condition in both sub-problems and therefore, equal-order interpolation spaces can be used for all the unknowns. Finally, this formulation leads to a coupled linear system; this monolithic approach is effective, since the coupling can be dealt by effective preconditioning and iterative solvers that allows to deal with high Hartmann numbers.

Abstract

Objective: In this article we study the approximation to thermal turbulence from a strictly numerical point of view, without the use of any physical model. The main goal is to analyze the behavior of our numerical method in the large eddy simulation (LES) of thermally coupled turbulent flows at low Mach number.

Methods: Our numerical method is a stabilized finite element approximation 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. Modeling the subscale and taking its effect on the coarse scale problem into account results in a stable formulation. The quality of the final approximation (accuracy, efficiency as LES model) depends on the particular subscale model. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms. Another important contribution of this work is the extension of the orthogonal subgrid scale method widely tested for incompressible flows to variable density flows, using a density-weighted L 2 product to define the orthogonality of the subscales and the finite element spaces.

Results: Referring to numerical testing, we present numerical results for a laminar testcase validation that shows the dissipative behavior of the different stabilized methods. Then, we present results of the numerical simulation of two turbulent flow problems, the turbulent channel flow with large temperature differences in the wall normal direction at Res ¼ 180, and the turbulent thermally driven cavity with aspect ratio 4. The behavior of the method is evaluated by comparison against results available in the literature obtained using LES and direct numerical simulation (DNS). They are explained based on a careful analysis of the dissipative structure of the method, showing the physical interpretation of the subgrid scale method presented.

Conclusion: The material presented here is a clear indication of the potential of the method to model all kinds of turbulent thermally coupled flows. The formulation is the same in laminar and turbulent regimes