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Rotating magnetic field driven flow: Multiple steady solutions and stability

Grants, I.; Gerbeth, G.
An axially symmetric isothermal liquid metal flow induced by a low frequency low induction rotating magnetic field in a cylindrical cavity is considered. This problem has already been addressed by several investigations but the existing results are still contradicting. We consider the problem numerically employing highly accurate spectral methods. We seek for the steady solution in vorticity-stream function formulation in space of base functions build up from Chebyshev polynomials to satisfy appropriate boundary conditions. The dynamical system is obtained and the spectrum of the linearized problem is found gradually increasing the forcing parameter. We evaluate the critical magnetic Taylor number for the basic solution in 0.1% accuracy as 166300 (for a diameter to height ratio 1). The oscillatory instability is found to be of another type than the expected Taylor-Goertler vortices. Several additional monotonically unstable steady solutions are detected in the range where the basic solution is stable. These additional solutions are marked by an additional couple of secondary vortices counter-rotating in the meridional plane near the side wall at the mid-height. Some of the additional steady solutions are particularly close to the basic steady state and therefore take high spatial resolution to be distinguished. We consider also the time dependent solution that allows much higher spatial resolution. The linear stability results were verified comparing to the dynamical parameters of the time evolution in the near-critical regime. This comparison shows 0.1% agreement. The unstable additional steady solutions close to the basic steady state indicate instability to small finite size perturbations. The closest one gives an estimate from above for the minimum energy of the unstable finite perturbation. This estimate is improved considerably by series of numerical time-dependent solutions developing from additional steady states. The basic state is found to be unstable with respect to a finite size Taylor-Goertler type perturbation of less than 3.0E-8 size (in relative energy terms). Such a small perturbation excites long-lasting non-linear oscillations already at 3/4 of the critical forcing. This excited regime in turn is found to be unstable to another instability that finally reestablishes the basic state. These results predict the possibility of an unpredictably oscillating flow already in the linearly stable regime. The results indicate that an experimental observation of the linear instability is hardly possible. The obtained results also explain the big differences in previous numerical results concerning the stability of the rotating magnetic field driven flow.

  • Lecture (Conference)
    4th International Conference "MHD at dawn of 3rd Millennium", Presqu'ile de Giens, France, September 18-22, 2000

Publ.-Id: 3216