The linear stability of a fully developed liquid–metal magnetohydrodynamic pipe flow subject to a transverse magnetic field is studied numerically. Because of the lack of axial symmetry in the mean velocity profile, we need to perform a BiGlobal stability analysis. For that purpose, we develop a two-dimensional complex eigenvalue solver relying on a Chebyshev–Fourier collocation method in physical space. By performing an extensive parametric study, we show that in contrast to the Hagen–Poiseuille flow known to be linearly stable for all Reynolds numbers, the magnetohydrodynamic pipe flow with transverse magnetic field is unstable to three-dimensional disturbances at sufficiently high values of the Hartmann number and wall conductance ratio. The instability observed in this regime is attributed to the presence of velocity overspeed in the so-called Roberts layers and the corresponding inflection points in the mean velocity profile. The nature and characteristics of the most unstable modes are investigated, and we show that they vary significantly depending on the wall conductance ratio. A major result of this paper is that the global critical Reynolds number for the magnetohydrodynamic pipe flow with transverse magnetic field is Re = 45 230, and it occurs for a perfectly conducting pipe wall and the Hartmann number Ha = 19.7.
Many shear flows exhibit laminar to turbulent transitions at subcritical Reynolds numbers. In this context, the computation of the perturbations that exhibit the largest possible transient growth is of central interest as it often sheds light on the actual bypass transition route. In this work, we consider the effect on the transient growth of a spanwise magnetic field in a boundary layer flow of a liquid metal over an electrically insulating flat plate. We compute the optimal perturbations using non-modal theory and perform a parametric study to measure the influence of the magnetic field on their amplification and orientation with respect to the flow direction. We also perform direct numerical simulations to examine how the optimal perturbations evolve in the nonlinear regime. To assess the influence of the boundary layer development, we consider both a constant Blasius base flow profile and a growing base profile. We show that the properties of the optimal perturbations are not significantly affected by the choice of the base profile, whereas it has a more important impact on their evolution in the nonlinear regime.
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