3D stability analysis of Rayleigh-Benard convection of a liquid metal layer in the presence of a magnetic field-effect of wall electrical conductivity

Abstract

Rayleigh-Benard stability of a liquid metal layer of rectangular cross section is examined in the presence of a strong magnetic field that is aligned with the horizontal direction of the cross section. The latter is much longer than the vertical direction and the cross section assumes a large aspect ratio. The side walls are treated as highly conducting. Linear stability analysis is performed allowing for three-dimensional instabilities that develop along the longitudinal direction. The finite element methodology is employed for the discretization of the stability analysis formulation while accounting for the electrical conductivity of the cavity walls. The Arnoldi method provides the dominant eigenvalues and eigenvectors of the problem. In order to facilitate parallel implementation of the numerical solution at large Hartmann numbers, Ha, domain decomposition is employed along the horizontal direction of the cross section. As the Hartmann number increases a real eigenvalue emerges as the dominant unstable eigenmode, signifying the onset of thermal convection, whose major vorticity component in the core of the layer is aligned with the direction of the magnetic field. Its wavelength along the longitudinal direction of the layer is on the order of twice its height and increases as Ha increases. The critical Grashof was obtained for large Ha and it was seen to scale like Ha(2) signifying the balance between buoyancy and Lorentz forces. For well conducting side walls, the nature of the emerging flow pattern is determined by the combined conductivity of Hartmann walls and Hartmann layers, c(H) + Ha(-1). When poor conducting Hartmann walls are considered, c(H) << 1, the critical eigensolution is characterized by well defined Hartmann and side layers. The side layers are characterized by fast fluid motion in the magnetic field direction as a result of the electromagnetic pumping in the vicinity of the Hartmann walls. Increasing the electrical conductivity of the Hartmann walls was seen to delay the onset of thermal convection, while retaining the above scaling at criticality. Furthermore, for both conducting and insulating Hartmann walls and the entire range of Ha numbers that was examined, there was no tendency for a well defined quasi two-dimensional structure to develop owing to the convective motion in the core. A connection is made between the above findings and previous experimental investigations indicating the onset of standing waves followed by travelling waves as Gr is further increased beyond its critical value.