Seminar: Transition between turbulent and laminar flow in magnetohydrodynamic channels and ducts
Date & Time: Monday, October 23rd at 10:00.
Speaker: Mattias Brynjell-Rahkola (TU Ilmenau)
Topic: Transition between turbulent and laminar flow in magnetohydrodynamic channels and ducts
Abstract: During the last two decades, the understanding of subcritical transition in shear flows has significantly progressed thanks to the dynamical systems perspective [1]. In hydrodynamic systems, the appearance of turbulent flow emerges due to a saddle-node bifurcation, which appears as the Reynolds number (Re) is increased. In this bifurcation, two solution branches emerge additional to the laminar solution. While the upper branch typically is associated with turbulent dynamics, the lower branch involve limiting perturbations such as the edge state, whose stable manifold may be interpreted as the basin boundary of the laminar solution [2].
In magnetohydrodynamic (MHD) flows, energy may be dissipated not only through viscosity but also through the Lorentz force. Within the quasi-static MHD approximation considered in our work, the resulting Joule dissipation is characterized by a second non-dimensional parameter known as the Hartmann number (Ha), proportional to the magnetic field strength. In this setting, it is now possible to hypothesize a second saddle-node bifurcation with respect to Ha instead of Re. However, due to the different nature of the viscous and Joule dissipation, the behavior of the flow near the two bifurcation points will be significantly different.
During the seminar, the concept of edge states will be presented, and the change in edge dynamics with increasing Ha for a doubly periodic channel with a spanwise magnetic field and electrically insulating walls [3] will be shown. In particular, the above described saddle-node bifurcation for sufficiently large Ha will be illustrated. Afterwards, the transition process in a MHD square duct with electrically insulating walls and a transverse magnetic field will be presented. Due to the presence of the magnetic field, this flow features two types of boundary layer, namely Hartmann and Shercliff layers [4]. Given their different stability properties, two transition scenarios can be hypothesized that involve either of these boundary layers. Using edge state calculations and direct numerical simulations, the interaction between them will be outlined, along with possible transition routes for MHD square ducts. The results presented have been obtained with the open source spectral element code NEK5000 [5], which offers high accuracy and geometric flexibility. We have recently extended the capabilities of this code to quasi-static MHD flows. Therefore, a brief overview of the spectral element method along with the discretization of the electromagnetic quantities will be provided as well.
[1] Eckhardt et al., Annu. Rev. Fluid Mech. 39, 447-468 (2007).
[2] Skufca et al., Phys. Rev. Lett., 96(17), 174101 (2006).
[3] Krasnov et al., J. Fluid Mech., 596, 73-101 (2008).
[4] Mueller & Buehler, Magnetofluiddynamics in Channels and Containers, Springer (2001).
[5] Fischer et al., https://nek5000.mcs.anl.gov (2008).