From linear to nonlinear: Transient growth in confined magnetohydrodynamic flows

Abstract

The underlying flow mechanisms for the destabilisation of an electrically conducting fluid under the influence of a transverse magnetic field in a square duct are investigated. Such flows are applicable to metallurgical processes where magnetic fields are used to dampen disturbances to increase homogeneity in material production, in addition to cooling blankets of nuclear fusion reactors, where flow disturbances can aid in improving convective heat transfer. A preliminary investigation into optimal linear growths at Hartmann numbers 10 _ Ha _ 1000 and Reynolds number Re = 5000 identifies two regimes for the scaling of optimal linear growths; when perturbation structures are dominated by three-dimensional variation in the vertical side-wall boundary layers, and 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 an excellent predictor of fundamental growth mechanisms for Ha > 150. A two-step method incorporating the seeding of an unperturbed base flow with optimal linear perturbations in a high magnetic field strength regime shows that no increase in energy amplification can be achieved via initial seeding energies in the range 10�����6 _ Ep _ 10�����2. The dominant dissipative mechanisms for these different seeding energies are also analysed, where it is shown that strong magnetic damping does not always necessitate the smoothing of the velocity field towards pure anisotropy, which has potentially useful applications for aiding convective heat transfer in magnetically damped flows.