Non-normal nonlinear transition to turbulence in a magnetically driven swirling flow

It is well-known that several shear flows become turbulent far below their linear stability limit. The transition to turbulence is triggered by unavoidable finite-amplitude disturbances. Although all infinitesimal disturbances eventually decay, their amplitude can grow temporarily. The lower limit of the control parameter when the amplitude of disturbances can grow initially is called limit of energetical instability [1]. As a rule, this limit considerably underestimates the threshold when the transition to turbulence is observed experimentally in various shear flows. Only recently certain clues of theoretical prediction of the phenomenon of non-linear non-normal transition have been found [2]. However, a routine method to predict this so-called non-normal non-linear transition is still a challenge.
We consider the flow of an electrically conducting melt driven by a rotating magnetic field (RMF) in a cylindrical cavity. Our numerical analysis evidenced that the intermittency route may take place also for this example of a rotating flow [3]. A pronounced flow sensitivity and additional unstable saddle type steady solutions have been detected at control parameters several times below the critical value for the onset of linear axisymmetric instability. The presence of such additional solutions is a sign of a possible non-linear non-normal transition due to finite perturbations despite the linear stability of the basic flow.
We present experimental and numerical results for the transition to turbulence in the RMF driven flow in a cylindrical cavity. The experimental approach is based on a non-invasive examination of the sensitive flow field by hanging the cavity on a torsion wire [4]. Under carefully prepared, clean conditions the occurrence of first flow oscillations was observed at a control parameter close to the linear stability threshold. If an artificial perturbation like a thermocouple was installed at the inner cavity wall, the transition was observed at a considerably lower forcing in the linearly stable parameter range. Examples of the typically intermittent onset of transition will be given. The numerical results will comprise the linear stability analysis based on a highly accurate spectral method and, in particular, the investigation of additional unstable steady solutions. Current evidence suggests that the lowest possible control parameter for the non-linear transition may be reasonably associated with the first bifurcation of such solutions. Indeed, a good quantitative agreement was found between experiments and numeric for the value of this global stability threshold.
Further results will be presented for the linear as well as global stability limits if the RMF driven flow in the cylinder is additionally exposed to a heating from below or a superimposed steady magnetic field. The latter typically delays the transition. The stability analysis of such flows is of particular interest in crystal growth technologies from the melt, for which the occurrence of a turbulent flow should often be prevented.

[1] D. D. Joseph, Stability of Fluid Motions Vol. I, Springer, Berlin, 1976.
[2] S. Grossmann, The onset of shear flow turbulence, Rev. Mod. Phys. 72, 603, 2000.
[3] I. Grants, G. Gerbeth, Stability of axially symmetric flow driven by a rotating magnetic field in a cylindrical cavity, J. Fluid Mech. 431, 407, 2000.
[4] I. Grants, G. Gerbeth, Experimetal study of non-normal nonlinear transition to turbulence in a rotating magnetic field driven flow, Phys. Fluids 15, 2803, 2003.