Transient analysis of global dominant modes in quasi-static magnetohydrodynamic flows

The global maximum transient amplifications of an electrically conducting fluid under the influence of a transverse magnetic field in square duct were investigated. A range of Hartmann numbers for 10 ≤ Ha ≤ 1000 were tested at a fixed Re = 5000 to elucidate the processes through which transition from three-dimensional perturbation states at low magnetic field strengths give way to predominantly anisotropic two-dimensional structures at higher field strengths. Such flows are applicable to metallurgical processes where magnetic fields are used to dampen disturbances to increase homogeneity in material production, as well as in the cooling blankets of nuclear fusion reactors where instabilities can aid in improving convective heat transfer. Two regimes are identified for the scaling of maximum transient energy amplification; when perturbation structures are dominated by three dimensional variation in the vertical side-wall boundary layers a scaling of G_max ∝ Ha^−1.6 for 10 ≤ Ha ≤ 100 was found, while scaling of G_max ∝ Ha^−0.37 over 150 ≤ Ha ≤ 1000 occurs for when quasi-two-dimensional (Q2D) disturbances are prevalent. Through comparison with existing literature, the Q2D model of Sommeria & Moreau (1982) is shown to be valid for Ha > 150.