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Optimal growth and transition to turbulence in channel flow with spanwise magnetic field

Published online by Cambridge University Press:  17 January 2008

DMITRY KRASNOV ,
MAURICE ROSSI ,
OLEG ZIKANOV  and
THOMAS BOECK
DMITRY KRASNOV
Affiliation:
Fakultät Maschinenbau, Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany
MAURICE ROSSI
Affiliation:
Institut Jean Le Rond D'Alembert, Université Pierre et Marie Curie, 4 place Jussieu, F-75252 Paris Cedex 05, France
OLEG ZIKANOV
Affiliation:
Department of Mechanical Engineering, University of Michigan - Dearborn, 4901 Evergreen Road, Dearborn, MI 48128-1491, USA
THOMAS BOECK
Affiliation:
Fakultät Maschinenbau, Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany
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Abstract

Instability and transition to turbulence in a magnetohydrodynamic channel flow are studied numerically for the case of a uniform magnetic field imposed along the spanwise direction. Optimal perturbations and their maximum amplifications over finite time intervals are computed in the framework of the linear problem using an iterative scheme based on direct and adjoint governing equations. It is shown that, at sufficiently strong magnetic field, the maximum amplification is no longer provided by classical streamwise rolls, but rather by rolls oriented at an oblique angle to the basic flow direction. The angle grows with the Hartmann number Ha and reaches the limit corresponding to purely spanwise rolls at Ha between 50 and 100 depending on the Reynolds number. Direct numerical simulations are applied to investigate the transition to turbulence at a single subcritical Reynolds number Re = 5000 and various Hartmann numbers. The transition is caused by the transient growth and subsequent breakdown of optimal perturbations, which take the form of one or two symmetric optimal modes (streamwise, oblique or spanwise modes depending on Ha) with low-amplitude three-dimensional noise added at the moment of strongest energy amplification. A sufficiently strong magnetic field (Ha larger than approximately 30) is found to completely suppress the instability. At smaller Hartmann numbers, the transition is observed but it is modified in comparison with the pure hydrodynamic case.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 596 , 25 January 2008 , pp. 73 - 101
Copyright
Copyright © Cambridge University Press 2008

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