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Inertial regimes in a curved electromagnetically forced flow

Published online by Cambridge University Press:  26 January 2017

J. Boisson ,
R. Monchaux  and
S. Aumaître
J. Boisson*
Affiliation:
IMSIA, ENSTA ParisTech, CNRS, CEA, EDF, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau CEDEX, France
R. Monchaux
Affiliation:
IMSIA, ENSTA ParisTech, CNRS, CEA, EDF, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau CEDEX, France
S. Aumaître
Affiliation:
SPHYNX, Service de Physique de l’Etat Condensé, CNRS UMR 3680, Université Paris-Saclay, CEA Saclay, F-91191 Gif-sur-Yvette CEDEX, France Laboratoire de Physique de l’ École Normale Supérieure de Lyon, CNRS and Université de Lyon, 46 allée d’Italie, F-69364 Lyon CEDEX 07, France
*
†Email address for correspondence: jean.boisson@ensta-paristech.fr
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Abstract

We investigated experimentally the flow driven by a Lorentz force induced by an axial magnetic field $\boldsymbol{B}$ and a radial electric current $I$ applied between two fixed concentric copper cylinders. The gap geometry corresponds to a rectangular section with an aspect ratio of $\unicode[STIX]{x1D702}=4$ and we probe the azimuthal and axial velocity profiles of the flow along the vertical axis by using ultrasonic Doppler velocimetry. We have performed several runs at moderate magnetic field strengths, corresponding to moderate Hartmann numbers $M\leqslant 300$. At these forcing parameters and because of the geometry of our experimental device, we show that the inertial terms are not negligible and an azimuthal velocity that depends on both $I$ and $B$ is induced. From measurements of the vertical velocity we focus on the characteristics of the secondary flow: the time-averaged velocity profiles are compatible with a secondary flow presenting two pairs of stable vortices, as pointed out by previous numerical studies. The flow exhibited a transition between two dynamical modes, a high- and a low-frequency one. The high-frequency mode, which emerges at low magnetic field forcing, corresponds to the propagation in the radial $r$-direction of tilted vortices. This mode is consistent with our previous experiments and with the instability described in Zhao et al. (Phys. Fluids, vol. 23 (8), 2011, 084103) taking place in an elongated duct geometry. The low-frequency mode, observed for high magnetic field forcing, consists of large excursions of the vortices. The dynamics of these modes matches the first axisymmetric instability described in Zhao & Zikanov (J. Fluid Mech., vol. 692, 2012, pp. 288–316) taking place in an square duct geometry. We demonstrated that this transition is controlled by the inertial magnetic thickness $H^{\prime }$ which is the characteristic length we introduce as a balance between the advection and the Lorentz force. The key point here is that when the inertial magnetic thickness $H^{\prime }$ is comparable to one geometric characteristic length ($H/2$ in the vertical or $\unicode[STIX]{x0394}r$ in the radial direction) the corresponding mode is favoured. Therefore, when $H^{\prime }/(H/2)\approx 1$ we observe the high-frequency mode taking place in an elongated duct geometry, and when $H^{\prime }/\unicode[STIX]{x0394}r\approx 1$ we observe the low-frequency mode taking place in square duct geometry and high magnetic field.

JFM classification

Boundary Layers: Boundary Layers Instability: Instability MHD and Electrohydrodynamics: MHD and Electrohydrodynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 813 , 25 February 2017 , pp. 860 - 881
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Copyright
© 2017 Cambridge University Press 

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