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Optimal linear growth in magnetohydrodynamic duct flow

Published online by Cambridge University Press:  16 April 2010

DMITRY KRASNOV ,
OLEG ZIKANOV ,
MAURICE ROSSI  and
THOMAS BOECK
DMITRY KRASNOV*
Affiliation:
Institute of Thermodynamics and Fluid Mechanics, Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany
OLEG ZIKANOV
Affiliation:
Department of Mechanical Engineering, University of Michigan–Dearborn, 4901 Evergreen Road, Dearborn, MI 48128-1491, USA
MAURICE ROSSI
Affiliation:
UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
THOMAS BOECK
Affiliation:
Institute of Thermodynamics and Fluid Mechanics, Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany
*
†Email address for correspondence: dmitry.krasnov@tu-ilmenau.de
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Abstract

Transient linear growth in laminar magnetohydrodynamic duct flow is analysed. The duct is straight with rectangular cross-section and electrically insulating walls. The applied uniform magnetic field is oriented perpendicular to the mean flow direction and parallel to one of the walls. Optimal perturbations and their maximum amplifications over finite time intervals are computed. The optimal perturbations are increasingly damped by the magnetic field, localized in the boundary layers parallel to the magnetic field irrespective of the duct aspect ratio. Typically, the optimal perturbations have non-vanishing streamwise wavenumber as found in magnetohydrodynamic channel flow with spanwise magnetic field. The Hartmann boundary layers perpendicular to the magnetic field do not contribute to the transient growth.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 653 , 25 June 2010 , pp. 273 - 299
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Adams, J. C. 1999 MUDPACK: multigrid software for elliptic partial differential equations. http://www.cisl.ucar.edu/css/software/mudpack/.Google Scholar
Adams, J. C., Swarztrauber, P. & Sweet, R. 1999 Efficient fortran subprograms for the solution of separable elliptic partial differential equations. http://www.cisl.ucar.edu/css/software/fishpack/.Google Scholar
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 The 15th Riga and 6th PAMIR Conference on Fundamental and Applied MHD Modeling of MHD Turbulence, vol. I, pp. 41–54, Riga, Latvia.Google Scholar
Biau, D., Soueid, H. & Bottaro, A. 2008 Transition to turbulence in duct flow. J. Fluid Mech. 596, 133–142.CrossRefGoogle Scholar
Boeck, T., Krasnov, D., Thess, A. & Zikanov, O. 2008 Large-scale intermittency of liquid-metal channel flow in a magnetic field. Phys. Rev. Lett. 101 (24), 244501-1–244501-4.CrossRefGoogle ScholarPubMed
Branover, H. 1978 Magnetohydrodynamic Flow in Ducts. John Wiley.Google Scholar
Cukierski, K. & Thomas, B. G. 2008 Flow control with local electromagnetic braking in continuous casting of steel slabs. Metall. Mater. Trans. B 39 (1), 94–107.CrossRefGoogle Scholar
Galletti, B. & Bottaro, A. 2004 Large-scale secondary structures in duct flow. J. Fluid Mech. 512, 85–94.CrossRefGoogle Scholar
Gerard-Varet, D. 2002 Amplifcation of small perturbations in a Hartmann layer. Phys. Fluids 14, 1458–1467.CrossRefGoogle Scholar
Hartmann, J. & Lazarus, F. 1937 Hg-dynamics. Part II. Experimental investigations on the flow of mercury in a homogeneous magnetic field. Mat.-Fys. Medd. K. Dan. Vidensk. Selsk. 15 (7), 1–45.Google Scholar
Hof, B., van Doorne, C., Westerweel, J. & Nieuwstadt, F. 2005 Turbulence regeneration in pipe flow at moderate Reynolds numbers. Phys. Rev. Lett. 95, 214502.CrossRefGoogle ScholarPubMed
Kobayashi, H. 2008 Large eddy simulation of magnetohydrodynamic turbulent duct flows. Phys. Fluids 20, 015102.CrossRefGoogle Scholar
Krasnov, D., Rossi, M., Zikanov, O. & Boeck, T. 2008 Optimal growth and transition to turbulence in channel flow with spanwise magnetic field. J. Fluid Mech. 596, 73–101.CrossRefGoogle Scholar
Krasnov, D. S., Zienicke, E., Zikanov, O., Boeck, T. & Thess, A. 2004 Numerical study of instability and transition to turbulence in the Hartmann flow. J. Fluid Mech. 504, 183–211.CrossRefGoogle Scholar
Landahl, M. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Lingwood, R. J. & Alboussière, T. 1999 On the stability of the Hartmann layer. Phys. Fluids 11, 2058–2068.CrossRefGoogle Scholar
Moresco, P. & Alboussière, T. 2004 Experimental study of the instability of the Hartmann layer. J. Fluid Mech. 504, 167–181.CrossRefGoogle Scholar
Müller, U. & Bühler, L. 2001 Magnetohydrodynamics in Channels and Containers. Springer.CrossRefGoogle Scholar
Potherat, A. 2007 Quasi-two-dimensional perturbations in duct flows under transverse magnetic field. Phys. Fluids 19, 074104.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J., Bagget, P. & Henningson, D. S. 1998 On the stability of streamwise streaks and and transition thresholds in plane channel flow. J. Fluid Mech. 365, 269–303.CrossRefGoogle Scholar
Reed, C. B. & Picologlou, B. F. 1989 Side wall flow instabilities in liquid metal MHD flow under blanket relevant conditions. Fusion Technol. 15, 705–715.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Smolentsev, S. & Dajeh, D. 1996 The effect of a conducting inclusion on the heat transfer in a fully developed MHD flow in a rectangular channel. Magnetohydrodynamics 32, 331–335.Google Scholar
Smolentsev, S., Moreau, R. & Abdou, M. 2008 Characterization of key magnetohydrodynamic phenomena for PbLi flows for the US DCLL blanket. Fusion Engng Design 83, 771–783.CrossRefGoogle Scholar
Tatsumi, T. & Yoshimura, T. 1990 Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212, 437–449.CrossRefGoogle Scholar
Ting, A. L., Walker, J. S., Moon, T. J., Reed, C. B. & Picologlou, B. F. 1991 Linear stability analysis for high-velocity boundary layers in liquid-metal magnetohydrodynamic flows. Intl J. Engng Sci. 29 (8), 939–948.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.CrossRefGoogle Scholar
Zikanov, O. 1996 On the instability of pipe Poiseuille flow. Phys. Fluids 8, 2923–2932.CrossRefGoogle Scholar