In this study, flow phenomena associated with inflectional and boundary-layer instabilities, as well as a mixed instability mode are accessed with the help of a parametrical model, which describes a family of quasi-two-dimensional (Q2D) magnetohydrodynamic (MHD) flows in a rectangular duct, where the near-wall jet is produced by imposing an external flow-opposing force. By varying this force, various instability modes and transition scenarios are reproduced via changes in the basic velocity profile. First, linear stability analysis is performed and then nonlinear effects are studied using DNS for Hartmann numbers 100 and 200 and Reynolds numbers from 1800 to 5000. A special attention is paid to the location of the inflection point with respect to the duct wall. A more complex flow dynamics, including various vortex-wall and vortex-vortex interactions, and even negative turbulence production are observed and analyzed as the inflection point approaches the wall. The obtained results as well as their qualitative comparisons with previous experimental and numerical data for the flows with the "M-shaped" velocity profile give a deeper look into what is usually called "jet instability", which, in fact, appears to be a complex integrated phenomenon that involves both linear and nonlinear mechanisms.

We also consider MHD rectangular duct flows with volumetric heating. The flows are upward, subject to a strong transverse magnetic field perpendicular to the temperature gradient, such that the flow dynamics is Q2D. The internal volumetric heating imitates conditions of a blanket of a fusion power reactor, where a buoyancy-driven flow is imposed on the forced flow. Studies of this mixed-convection flow include analysis for the basic flow, linear stability analysis and DNS-type computations. The parameter range covers the Hartmann number (Ha) up to 500, the Reynolds number (Re) from 1000 to 10,000 and the Grashof number (Gr) from 105 to 109. The linear stability analysis predicts two primary instability modes: (i) bulk instability associated with the inflection point in the velocity profile near the "hot" wall and (ii) side-wall boundary layer instability. A mixed instability mode is also possible. An equation for the critical Hartmann number has been obtained as a function of Re and Gr. Effects of Ha, Re and Gr on turbulent flows are addressed via non-linear computations that demonstrate two characteristic turbulence regimes. In the "weak" turbulence regime, the induced vortices are localized near the inflection point of the basic velocity profile, while the boundary layer at the wall parallel to the magnetic field is slightly disturbed. In the "strong" regime, the bulk vortices interact with the boundary layer causing its destabilization and formation of secondary vortices that may travel across the flow, even reaching the opposite wall. In this regime, the key phenomena are vortex-wall and various vortex-vortex interactions.