Transition to turbulence and unstable nonlinear magnetohydrodynamic flow states in ducts with a transverse magnetic field

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Transition to turbulence is important in various flow systems. Upon increasing the Reynolds number Re, it typically occurs sub-critically in wall-bounded flows, i.e. irrespective of an instability of the laminar flow state. The transition is typically interpreted in the framework of dynamical systems theory. A flow state is regarded as a point in the phase space of possible flow states. Laminar and turbulent flow have different basins of attraction that are separated by a hypersurface (edge of chaos). On this surface the flow develops neither towards the laminar nor towards the turbulent state, but to an attractor within the surface that is called the edge state. During transition to turbulence, the flow may first reach the vicinity of the edge state before it eventually becomes turbulent. Edge states correspond to non-trivial time-dependent flows that have simpler dynamics than turbulent solutions. They can therefore serve as models for studyingthe physical mechanisms of transition. Edge states can, e.g., correspond to wave-like, time-periodic or chaotic solutions. They were studied in detail for a number of prototypical shear flows, but have hardly ever been considered for magnetohydrodynamic (MHD) flows although transition in these flows is also sub-critical. In MHD flows, the Hartmann number Ha appears as a second parameter for the magnetic field in addition to the Reynolds number as a dimensionless parameter for the velocity. In the project, the route to turbulence and the mechanisms of transition in MHD duct flows will be determined that are largely unknown. The specific configurations to be studied are flows in ducts with rectangular cross-section and transversal homogeneous magnetic field. These flows are three-dimensional and characterized by boundary layers on the walls parallel and perpendicular to the magnetic field that are calledShercliff and Hartmann layers, respectively. The flows will be investigated by highly resolved direct numerical simulations (DNS) and stability analysis. First, the boundary between laminar and turbulent states will be determined in the Re-Ha-A-parameter space, where A is theperturbation amplitude of the initial state. In the second step, edge states will be determined by DNS and their dynamics and specific properties will be analyzed. In the third step, the bifurcations of flow states upon changing of Re and Ha will be investigated that give rise to edge states. By that one can also obtain a better understanding why transition is determined by a parameter that is based on the thickness of the Hartmann layers although turbulence appears first in the Shercliff layers. Our results are relevant for electromagnetic flow control in metallurgical and heat transfer applications with conducting liquids.