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Instability and transition to turbulence in a free shear layer affected by a parallel magnetic field

Published online by Cambridge University Press:  15 February 2007

A. VOROBEV  and
O. ZIKANOV
A. VOROBEV
Affiliation:
Department of Mechanical Engineering, University of Michigan – Dearborn, 4901 Evergreen Road, Dearborn, MI 48128-1491, USA
O. ZIKANOV*
Affiliation:
Department of Mechanical Engineering, University of Michigan – Dearborn, 4901 Evergreen Road, Dearborn, MI 48128-1491, USA
*
†Author to whom correspondence should be addressed: zikanov@umd.umich.edu
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Abstract

Instability and transition to turbulence in a temporally evolving free shear layer of an electrically conducting fluid affected by an imposed parallel magnetic field is investigated numerically. The case of low magnetic Reynolds number is considered. It has long been known that the neutral disturbances of the linear problem are three-dimensional at sufficiently strong magnetic fields. We analyse the details of this instability solving the generalized Orr–Sommerfeld equation to determine the wavenumbers, growth rates and spatial shapes of the eigenmodes. The three-dimensional perturbations are identified as oblique waves and their properties are described. In particular, we find that at high hydrodynamic Reynolds number, the effect of the strength of the magnetic field on the fastest growing perturbations is limited to an increase of their oblique angle. The dimensions and spatial shape of the waves remain unchanged. The transition to turbulence triggered by the growing oblique waves is investigated in direct numerical simulations. It is shown that initial perturbations in the form of superposition of two symmetric waves are particularly effective in inducing three-dimensionality and turbulence in the flow.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 574 , 10 March 2007 , pp. 131 - 154
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Abas, S. 1969 The effect of a parallel magnetic field on the stability of free boundary-layer type flows of low magnetic Reynolds number. J. Fluid Mech. 38, 243–253.CrossRefGoogle Scholar
Betchov, R. & Szewczyk, A. 1963 Stability of a shear layer between parallel streams. Phys. Fluid. 6 (10), 1391.CrossRefGoogle Scholar
Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.CrossRefGoogle Scholar
Drazin, P. G. 1960 Stability of parallel flow in a parallel magetic field at small magnetic Reynolds numbers. J. Fluid Mech. 8, 130.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Gotoh, K. 1971 Hydromagnetic instability of a free shear layer at small magnetic Reynolds numbers. J. Fluid Mech. 49, 21.CrossRefGoogle Scholar
Hunt, J. C. R. 1966 On the stability of parallel flows with parallel magnetic fields. Proc. R. Soc. Lond. A 293, 342.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Maslowe, S. A. 1981 Shear Flows Instabilities and Transition. In Hydrodynamic Instabilities and the Transition to Turbulence, (ed. Swinney, H. L. & Gollub, J. P.), p. 181.Google Scholar
Metcalfe, R. W., Orszag, S. A., Brachet, M. E., Menon, S. & Riley, J. J. 1987 Secondary instability of a temporally growing mixing layer. J. Fluid Mech. 184, 207–243.CrossRefGoogle Scholar
Michael, D. H. 1953 The stability of plane parallel flows of electrically conducting fluids. Proc. Camb. Phil. Soc. 49, 166.CrossRefGoogle Scholar
Moreau, R. 1990 Magnetohydrodynamics, Kluwer.CrossRefGoogle Scholar
Müller, U. & Bühler, L. 2001 Magnetohydrodynamics in Channels and Containers. Springer.CrossRefGoogle Scholar
Rogers, M. M. & Moser, R. D. 1992 The three-dimensional evolution of a plane mixing layer: the Kelvin–Helmholtz rollup. J. Fluid Mech. 243, 183–226.CrossRefGoogle Scholar
Rogers, M. M. & Moser, R. D. 1993 Spanwise scale selection in plane mixing layers. J. Fluid Mech. 247, 321–337.CrossRefGoogle Scholar
Squire, H. B. 1933 On the stability of three-dimensional disturbances of viscous fluid flow between parallel walls Proc. R. Soc. Lond. A 142, 621.Google Scholar
Stuart, J. T. 1954 On the stability of viscous flow between parallel planes in the presence of a coplanar magnetic field. Proc. R. Soc. Lond. A 221, 189.Google Scholar
Thess, A. & Zikanov, O. 2005 On the transition from two-dimensional to three-dimensional MHD turbulence. Proc. of 2004 CTR Summer Program, Stanford University, pp. 63–74.Google Scholar
Tsinober, A. 1990 MHD flow drag reduction. In Viscous Drag Reduction in Boundary Layers. (ed. Bushnell, D. M. & Hefner, J. N.), pp. 327–349.Google Scholar
Vorobev, A., Zikanov, O., Davidson, P. & Knaepen, B. 2005 Anisotropy of MHD turbulence of low magnetic Reynolds number. Phys. Fluid. 17 (12), 125105.CrossRefGoogle Scholar
Zikanov, O. & Thess, A. 1998 Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds number. J. Fluid Mech. 358, 299–333.CrossRefGoogle Scholar
Zikanov, O., Slinn, D. N., & Dhanak, M. R. 2004 Turbulent convection driven by surface cooling in shallow water J. Fluid Mech. 464, 81–111.CrossRefGoogle Scholar