Numerical stability analysis and flow simulation of lid-driven cavity subjected to high magnetic field

Abstract

To cite this version:Luca Marioni, François Bay, Elie Hachem. Numerical stability analysis and flow simulation of liddriven cavity subjected to high magnetic field. SUMMARYIn this work we study the flow of a conducting fluid inside a two-dimensional square domain. The problem is solved by using a variational multiscale finite element approach. The study focuses on a high magnetic interaction parameter range and high Reynolds number. Under the imposition of a high magnetic field, the flow gets regularized, but fast transient phenomena take place, which could lead to numerical errors. An expression to compute the maximum time-step which guarantees convergence in explicit schemes is proposed and validated through numerical tests.

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

“…The present results are in excellent agreement with the numerical result of Marioni et al in their Figure 5A, even though the magnetic field produced by the conducting fluid motion is neglected in their simulations assuming that Re m ≪1 (inductionless MHD). The effect of the horizontal magnetic field to the flow structure within the square lid‐driven cavity is illustrated in Figure and the Reynolds number and the Stuart (magnetic interaction) number are taken from the work of Marioni et al where the authors investigated the flow behavior at relatively high Stuart numbers. The present streamlines are in very good agreement with those of Marioni et al (please see their Figure 5).…”

“…Finally the multiple high‐aspect‐ratio cellular structures form within the cavity and these eddies perfectly align with the externally applied magnetic field with nearly uniform sizes. This phenomena known as the braking effect of the magnetic field and it can be seen clearly in Figure A at Re =1000, Re m =1, and N =5 with an external magnetic field of B =(1,0) ⊤ . The simulations indicate that the number of cells increases with the increase in the Stuart number.…”

“…The effect of the horizontal magnetic field to the flow structure within the square lid‐driven cavity is illustrated in Figure and the Reynolds number and the Stuart (magnetic interaction) number are taken from the work of Marioni et al where the authors investigated the flow behavior at relatively high Stuart numbers. The present streamlines are in very good agreement with those of Marioni et al (please see their Figure 5). As the Stuart number increases, the large primary vortex moves upwards due to the increased Lorentz force and its strength is decreased significantly.…”

“…Although this modification leads to a conservative finite volume approach with the exact mass conservation, it is not clear how to extend to three dimensions, where the streamfunction vector can be simply integrated as in two dimensions. The second modification to the original algorithm is made by replacing the Lorentz force term in the momentum equation by employing σ [ E + u × B ]× B and neglecting the electric field E term in two dimensions since it leads the Laplacian of the electric scalar potential with Neumann boundary condition due to the electrically insulating wall. Therefore, the remaining term can be treated in a similar manner to the mass matrix if the magnetic field vector B is taken from the previous step.…”

“…Van Opstal et al proposed a pointwise satisfaction of the zero‐divergence constraint on the discrete velocity field for the incompressible Navier‐Stokes equations using divergence‐conforming B‐splines. The second modification is made in two dimensions to the Lorentz force term in the momentum equation by employing σ [ E + u × B ]× B and neglecting the electric field E term in two dimensions since it leads to the Laplacian of the electric scalar potential for any in‐plane magnetic field vector superimposed to a two‐dimensional flow . The modification significantly improves the preconditioning of the monolithic approach as well as the numerical accuracy at high Hartmann numbers.…”