Linear global stability of a downward flow of liquid metal in a vertical duct under strong wall heating and transverse magnetic field

The linear global stability of a downward flow of liquid metal in a vertical duct under strong wall heating and a transverse magnetic field is examined numerically. The two-dimensional steady-state magnetohydrodynamic (MHD) mixed convection with an upward reverse flow is first computed by the finite-element method. Then linear global stability equations of the MHD mixed convection are derived and solved by the spatial discretization of the Taylor-Hood finite element and an implicitly restarted Arnoldi algorithm for the resulted generalized eigenvalue problem. Elevator and oscillatory unstable modes are revealed through the eigenspetrum computation of linear global stability equations. The elevator mode is found to be always unstable and independent of the basic flow profile, though a large magnetic field may suppress its growth rate. The unstable oscillatory mode is directly related to the basic upward reverse flow and first occurs at the specific flow structure which has an upward reverse flow near the heating wall and a downward flow near the opposite wall. The critical curves of the Grashof number with respect to the Hartmann number for the three-dimensional oscillatory mode are plotted and reveal that larger Hartmann numbers have larger critical Grashof numbers. Energy budget analyses are also performed and show that the shear Kelvin-Helmholtz instability due to the existence of an inflection point is the key instability mechanism of the three-dimensional oscillatory mode. The appearance of the unstable oscillatory mode may be regarded as an alternative physical explanation of the high-amplitude, low-frequency pulsations of temperature in the experiments and related numerical simulations.

Automated summary

This study examines the linear global stability of a downward flow of liquid metal in a vertical duct under strong wall heating and a transverse magnetic field through numerical methods. The results show that the elevator mode is consistently unstable, while the unstable oscillatory mode is directly related to the specific flow structure with upward reverse flow.