Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-20T07:18:45.140Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Airiau, C. & Castets, M. 2004 On the amplification of small disturbances in a channel flow with a normal magnetic field. Phys. Fluids 16, 2991–3005.CrossRefGoogle Scholar
von Ammon, W., Gelfgat, Y., Gorbunov, L., Muhlbauer, A., Muiznieks, A., Makarov, Y., Virbulis, J. & Muller, G. 2005 Application of magnetic fields in industrial growth of silicon single crystals. In 15th Riga and 6th PAMIR Conf. on Fundamental and Applied MHD Modeling of MHD turbulence, vol. I, pp. 41–54. Riga, Latvia.Google Scholar
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150.CrossRefGoogle Scholar
Barleon, L., Burr, U., Mack, K. J. & Stieglitz, R. 2001 Magnetohydrodynamic heat transfer research related to the design of fusion blankets. Fusion Technol 39, 127–156.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids 4, 1637–1650.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quateroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. SpringerCrossRefGoogle Scholar
Davidson, P. A. 1999 Magnetohydrodynamics in materials processing. Annu. Rev. Fluid Mech. 31, 273–300.CrossRefGoogle Scholar
Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory part I: autonomous operators. J. Atmos. Sci. 53, 2025–2040.2.0.CO;2>CrossRefGoogle Scholar
Gerard-Varet, D. 2002 Amplification of small perturbations in a Hartmann layer. Phys. Fluids 14, 1458–1467.CrossRefGoogle Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods. Philadelphia: SIAM.CrossRefGoogle Scholar
Jimenez, J. 1990 Transition to turbulence in two-dimensional Poiseuille flow. J. Fluid Mech. 218, 265–297.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
Krasnov, D. S., Zienicke, E., Zikanov, O., Boeck, T. & Thess, A. 2004 Numerical study of the instability of the Hartmann layer. J. Fluid Mech. 504, 183–211.CrossRefGoogle Scholar
Lee, D. & Choi, H. 2001 Magnetohydrodynamic turbulent flow in a channel at low magnetic Reynolds number. J. Fluid Mech. 429, 367–394.CrossRefGoogle Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.CrossRefGoogle Scholar
Mansour, N. N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15–44.CrossRefGoogle Scholar
Moreau, R. 1990 Magnetohydrodynamics. Kluwer.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J., Bagget, P. & Henningson, D. S. 1998 On the stability of stream-wise streaks and transition thresholds in plane channel flow. J. Fluid Mech. 365, 269–303.CrossRefGoogle Scholar
Roberts, P. H. 1967 An Introduction to Magnetohydrodynamics. Longmans, Green.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer Verlag.CrossRefGoogle Scholar
Schmid, P. J. & Rossi, M. 2004 Three-dimensional stability of a Burgers vortex. J. Fluid Mech. 500, 103–112.CrossRefGoogle Scholar
Thomas, B. G. & Zhang, L. 2001 Mathematical modeling of fluid flow in continuous casting: a review. ISIJ Intl 41, 1181–1193.CrossRefGoogle Scholar
Zikanov, O. 1996 On the instability of pipe Poiseuille flow. Phys. Fluids 8, 2923–2932.CrossRefGoogle Scholar
Supplementary material: PDF

Krasnov Supplementary material

Appendix.pdf

Download Krasnov Supplementary material(PDF)
PDF 86.2 KB