Three-dimensional stability of free convection vortices in the presence of a magnetic field

Abstract

Three-dimensional (3D) stability of 2D vortical flow of a liquid metal in a cavity of square cross section is examined. Vortices are produced as a result of free convection and internal heating in the cavity in the presence of a magnetic field. Low-magnetic-Reynolds-number equations are used for the base flow and stability formulation. Finite element methodology is used to discretize the problem. Efficient calculation of the dominant eigenvalues is afforded by the Arnoldi method, while neutral stability diagrams are constructed using continuation techniques. The number of vortices exhibited by the base flow switches from one to two as the internal heating crosses a threshold value. The dominant instability mechanism is the Gortler instability in the case of a single vortex and elliptical instability in the case of two vortices. In elliptic instability, axial vorticity is symmetric, it is characterized by two lobed structures aligned with one of the two principal directions of strain and the dominant eigenmode assumes the form of a traveling wave. The magnetic field opposes buoyancy, alters the direction of maximal strain by accentuating wall shear layers in comparison with the vortex pair in the core and leads to smaller frequencies at criticality.