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The decay of wall-bounded MHD turbulence at low $Rm$

Published online by Cambridge University Press:  26 October 2015

Alban Pothérat  and
Kacper Kornet
Alban Pothérat*
Affiliation:
Applied Mathematics Research Centre, Coventry University, Coventry CV51FB, UK
Kacper Kornet
Affiliation:
Applied Mathematics Research Centre, Coventry University, Coventry CV51FB, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: alban.potherat@coventry.ac.uk
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Abstract

We present direct numerical simulations of decaying magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number. The domain considered is bounded by periodic boundary conditions in the two directions perpendicular to the magnetic field and by two plane Hartmann walls in the third direction. Regimes of high magnetic fields (Hartmann number of up to 896) are reached thanks to a new spectral method using the eigenvectors of the dissipation operator. The decay is found to proceed through two phases: first, energy and integral length scales vary rapidly during a two-dimensionalisation phase extending over approximately a Hartmann friction time. During this phase, the evolution of the former appears significantly more impeded by the presence of walls than that of the latter. Once the large scales are nearly quasi-two-dimensional, the decay results from the competition of a two-dimensional dynamics driven by dissipation in the Hartmann boundary layers and the three-dimensional dynamics of smaller scales. In the later stages of the decay, three-dimensionality subsists under the form of barrel-shaped structures. A purely quasi-two dimensional decay entirely dominated by friction in the Hartmann layers is not reached because of residual dissipation in the bulk. However, this dissipation is not generated by the three-dimensionality that subsists, but by residual viscous friction due to horizontal velocity gradients. In the process, the energy in the velocity component aligned with the magnetic field is found to be strongly suppressed, as is skewness. This result reproduces the experimental findings of Kolesnikov & Tsinober (Fluid Dyn., vol. 9, 1974, pp. 621–624), where, as in the present simulations, Hartmann walls were present.

JFM classification

MHD and Electrohydrodynamics: High-Hartmann-number flows MHD and Electrohydrodynamics: MHD turbulence Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 783 , 25 November 2015 , pp. 605 - 636
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Copyright
© 2015 Cambridge University Press 

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References

Alemany, A., Moreau, R., Sulem, P. & Frish, U. 1979 Influence of an external magnetic field on homogeneous MHD turbulence. J. Méc. 18 (2), 277–313.Google Scholar
Andreev, O., Kolesnikov, Y. & Thess, A. 2006 Experimental study of liquid metal channel flow under the influence of a non-uniform magnetic field. Phys. Fluids 18, 065108.CrossRefGoogle Scholar
Andreev, O., Kolesnikov, Y. & Thess, A. 2013 Visualization of the Ludford column. J. Fluid Mech. 721, 438–453.CrossRefGoogle Scholar
Baker, N., Pothérat, A. & Davoust, L. 2015 Dimensionality, secondary flows and helicity in low- $Rm$ MHD vortices. J. Fluid Mech. 779, 325–350.Google Scholar
Burattini, P., Zikanov, O. & Knaepen, B. 2010 Decay of magnetohydrodynamic turbulence at low magnetic Reynolds number. J. Fluid Mech. 657, 502–538.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods: Fundamentals in Single Domains. Springer.Google Scholar
Caperan, P. & Alemany, A. 1985 Turbulence homogène MHD à faible nombre de Reynolds magnétique. Étude de la transition vers la phase quasi bidimensionelle et caractérisation de son anisotropie. J. Méc. Theor. Appl. 4 (2), 175–200.Google Scholar
Dymkou, V. & Pothérat, A. 2009 Spectral methods based on the least dissipative modes for wall-bounded MHD turbulence. J. Theor. Comput. Fluid Dyn. 23 (6), 535–555.Google Scholar
Favier, B., Godeferd, F. S., Cambon, C. & Delache, A. 2010 On the two-dimensionalization of quasistatic magnetohydrodynamic turbulence. Phys. Fluids 22, 075104.Google Scholar
Favier, B., Godeferd, F. S., Cambon, C., Delache, A. & Bos, W. J. T. 2011 Quasi-static magnetohydrodynamic turbulence at high Reynolds number. J. Fluid Mech. 681, 434–461.Google Scholar
Klein, R. & Pothérat, A. 2010 Appearance of three-dimensionality in wall bounded MHD flows. Phys. Rev. Lett. 104 (3), 034502.Google Scholar
Knaepen, B. & Moin, P. 2004 Large-eddy simulation of conductive flows at low magnetic Reynolds number. Phys. Fluids 16 (5), 1255–1261.Google Scholar
Kolesnikov, Y. B. & Tsinober, A. B. 1974 Experimental investigation of two-dimensional turbulence behind a grid. Fluid Dyn. 9 (4), 621–624.Google Scholar
Kornet, K. & Pothérat, A. 2015 Spectral methods based on the least dissipative modes for wall bounded MHD flows. J. Comput. Phys. 298, 266–279.Google Scholar
Low, R. & Pothérat, A. 2015 Bounds on the attractor dimension for Low- $Rm$ magnetohydrodynamic channel flow with parallel magnetic field. Phys. Rev. E 5, 053022.Google Scholar
Moffatt, H. K. 1967 On the suppression of turbulence by a uniform magnetic field. J. Fluid Mech. 28, 571–592.Google Scholar
Moreau, R. 1990 Magnetohydrodynamics. Kluwer Academic.Google Scholar
Mück, B., Günter, C. & Bühler, L. 2000 Buoyant three-dimensional MHD flows in rectangular ducts with internal obstacles. J. Fluid Mech. 418, 265–295.Google Scholar
Okamoto, N., Davidson, P. A. & Kaneda, Y. 2010 On the decay of low-magnetic-Reynolds-number turbulence in an imposed magnetic field. J. Fluid Mech. 651, 295–318.Google Scholar
Pothérat, A. 2012 Three-dimensionality in quasi-two dimensional flows: recirculations and ‘barrel’ effects. Europhys. Lett. 98 (6), 64003.Google Scholar
Pothérat, A. & Alboussière, T. 2006 Bounds on the attractor dimension for low- $Rm$ wall-bound MHD turbulence. Phys. Fluids 18 (12), 125102.Google Scholar
Pothérat, A. & Dymkou, V. 2010 Direct numerical simulations of low-Rm MHD turbulence based on the least dissipative modes. J. Fluid Mech. 655, 174–197.Google Scholar
Pothérat, A. & Klein, R. 2014 Why, how and when MHD turbulence at low Rm becomes three-dimensional. J. Fluid Mech. 761, 168–205.Google Scholar
Pothérat, A., Rubiconi, F., Charles, Y. & Dousset, V. 2013 Direct and inverse pumping in flows with homogenenous and non-homogenous swirl. Eur. Phys. J. E 36 (8), 94.Google Scholar
Pothérat, A., Sommeria, J. & Moreau, R. 2000 An effective two-dimensional model for MHD flows with transverse magnetic field. J. Fluid Mech. 424, 75–100.Google Scholar
Roberts, P. H. 1967 Introduction to Magnetohydrodynamics. Longmans.Google Scholar
Schumann, U. 1976 Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field. J. Fluid Mech. 74, 31–58.CrossRefGoogle Scholar
Shebalin, J. V., Matthaeus, W. H. & Montgomery, D. 1983 Anisotropy in MHD turbulence due to a mean magnetic field. J. Plasma Phys. 29 (3), 525–547.Google Scholar
Sommeria, J. 1988 Electrically driven vortices in a strong magnetic field. J. Fluid Mech. 189, 553–569.CrossRefGoogle Scholar
Sommeria, J. & Moreau, R. 1982 Why, how and when MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507–518.CrossRefGoogle Scholar
Verma, M. K., Chatterjee, A., Reddy, S., Yadav, R., Paul, S., Chandra, M. & Samtaney, R. 2013 Benchmarking and scaling studies of pseudo-spectral code tarang for turbulence simulations. Pramana J. Phys. 81 (4), 617–629.CrossRefGoogle Scholar
Vorobev, A., Zikanov, O., Davidson, P. A. & Knaepen, B. 2005 Anisotropy of magnetohydrodynamic turbulence at low magnetic Reynolds number. Phys. Fluids 17, 125105.Google Scholar