Quasi-two-dimensional convection in a three-dimensional laterally heated box in a strong magnetic field normal to main circulation

Abstract

Convection in a laterally heated three-dimensional box affected by a strong magnetic field is considered in the quasi-two-dimensional (Q2D) formulation. It is assumed that the magnetic field is strong and is normal to the main convective circulation. The stability of the resulting Q2D flow is studied for two values of the Hartmann number scaled by half of the width ratio, 100 and 1000, and for either thermally insulating or perfectly conducting horizontal boundaries. The aspect length-to-height ratio of the box is varied continuously between 4 and 10. It is shown that the magnetic field damps the bulk flow and creates thermal and Shercliff boundary layers at the boundaries, which become the main source of instabilities. In spite of the general tendency of the flow stabilization by the magnetic field, the flow instability takes place in different ways depending on the boundary conditions and the aspect ratio. Similarities with other magnetic field directions and flows with larger Prandtl numbers are discussed.

Computational modeling of magnetoconvection: Effects of discretization method, grid refinement and grid stretching

“…The Q2D model was originally derived for an isothermal flow along a duct with electrically insulated walls and imposed transverse magnetic field, but applied to many other configurations since then (see, e.g. [9,16,38] is the Reynolds number, are both much larger to one. In essence, the model assumes that under the action of a strong magnetic field, the flow acquires a state with velocity components virtually uniform along the field lines except for the exponential distributions within the thin Hartmann boundary layers.…”

“…It has been recently understood that the suppression of turbulence by the magnetic field does not necessarily mean that the flow acquires a simple laminar steady-state form. On the contrary, growth of the MHD-specific convection instability modes that have weak or zero variations along the magnetic field lines and, thus, are not suppressed, may lead to unsteady, essentially nonlinear and complex flow dynamics [7][8][9][10][11][12][13][14][15][16].…”

The linear stability of Hunt-Rayleigh-Bénard flow

“…Their stability analysis demonstrated that the Hopf bifurcation was a periodic branch and internal heat generation enhanced flow instability by decreasing the critical value of the Grashof number. Through the quasi-two-dimensional (Q2D) formulation, Gelfgat and Molokov 23 indicated that the magnetic field damped the bulk flow and created thermal and Shercliff boundary layers, which became the main source of instabilities. However, the stability of a duct flow with throughflow including the temperature and magnetic fields is complicated, with the unique M-shape velocity distribution, which is the most important phenomenon of the MHD effects and can cause different instability modes.…”

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

“…More recently, the quasi 2D assumption has been employed in order to study the stability of free and mixed convection flows at large Hartmann values. Gelfgat and Molokov (2011) studied the stability of recirculation vortices when the magnetic field is perpendicular to the plane of circulation using the quasi 2D flow assumption, and showed the dependence of the stability pattern on the box aspect ratio and boundary conditions. Smolentsev et al (2012) studied the stability of the M-shaped profile in an upward flow in order to assess the relative importance of jet formation and the appearance of inflection points in the base flow velocity profile, in destabilizing the flow.…”