Linear stability analysis and dynamic simulations of free convection in a differentially heated cavity in the presence of a horizontal magnetic field and a uniform heat source

Abstract

The steady state and stability of two-dimensional free convection flow in a square cavity is examined, in the presence of a uniform internal heat source and a uniform magnetic field that is perpendicular to gravity and parallel to an imposed temperature gradient. The finite element method is used for calculating the steady and dynamic state of the system in the parameter space defined by the dimensionless numbers, Gr, Ha, Pr, and S. The trapezoidal rule is used for time integration. Linear stability analysis is performed by solving a generalized eigenvalue problem. The Arnoldi method is used for the calculation of eigenvalues with significant savings in storage and CPU time requirements. The base solution normally exhibits two recirculation regions when the heat production term is large enough. Stability analysis predicts a Hopf bifurcation to a periodic branch. A neutral stability diagram is constructed for a range of values of Ha, Gr, and S for liquid lithium, Pr=0.0321. Internal heat generation, i.e,. increasing S, enhances instability by decreasing the critical value of Grashof, Grcr, determining the onset of the Hopf branch, whereas intensifying the magnetic field, i.e., increasing Ha, stabilizes the flow by increasing Grcr. Dynamic simulations confirm the above structure, identify the oscillatory solution branch as a supercritical Hopf bifurcation for the entire parameter range that was examined, and recover the time constants predicted by stability analysis. As Gr increases or as Ha decreases symmetric arrangement of the two rolls is eliminated and the steady flow configuration loses stability when Gr>Grcr. Subsequently, time periodicity sets in leading to more or less efficient heat removal in terms of lowering or increasing the average cavity temperature, in comparison with the steady flow configuration for the same Gr, depending on whether Ha lies below or above a critical value, respectively, for fixed S and Pr.