Skip to main content Accessibility help
×
Home

Why, how and when MHD turbulence at low $\mathit{Rm}$ becomes three-dimensional

  • Alban Pothérat (a1) and Rico Klein (a2)

Abstract

Magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number is experimentally investigated by studying a liquid metal flow in a cubic domain. We focus on the mechanisms that determine whether the flow is quasi-two-dimensional, three-dimensional or in any intermediate state. To this end, forcing is applied by injecting a DC current $I$ through one wall of the cube only, to drive vortices spinning along the magnetic field. Depending on the intensity of the externally applied magnetic field, these vortices extend part or all of the way through the cube. Driving the flow in this way allows us to precisely control not only the forcing intensity but also its dimensionality. A comparison with the theoretical analysis of this configuration singles out the influences of the walls and of the forcing on the flow dimensionality. Flow dimensionality is characterised in several ways. First, we show that when inertia drives three-dimensionality, the velocity near the wall where current is injected scales as $U_{b}\sim I^{2/3}$ . Second, we show that when the distance $l_{z}$ over which momentum diffuses under the action of the Lorentz force (Sommeria & Moreau, J. Fluid Mech., vol. 118, 1982, pp. 507–518) reaches the channel width $h$ , the velocity near the opposite wall $U_{t}$ follows a similar law with a correction factor $(1-h/l_{z})$ that measures three-dimensionality. When $l_{z}<h$ , by contrast, the opposite wall has less influence on the flow and $U_{t}\sim I^{1/2}$ . The central role played by the ratio $l_{z}/h$ is confirmed by experimentally verifying the scaling $l_{z}\sim N^{1/2}$ put forward by Sommeria & Moreau ( $N$ is the interaction parameter) and, finally, the nature of the three-dimensionality involved is further clarified by distinguishing weak and strong three-dimensionalities previously introduced by Klein & Pothérat (Phys. Rev. Lett., vol. 104 (3), 2010, 034502). It is found that both types vanish only asymptotically in the limit $N\rightarrow \infty$ . This provides evidence that because of the no-slip walls, (i) the transition between quasi-two-dimensional and three-dimensional turbulence does not result from a global instability of the flow, unlike in domains with non-dissipative boundaries (Boeck et al. Phys. Rev. Lett., vol. 101, 2008, 244501), and (ii) it does not occur simultaneously at all scales.

Copyright

Corresponding author

Email address for correspondence: alban.potherat@coventry.ac.uk

References

Hide All
Akkermans, R. A. D., Kamp, L. P. J., Clercx, H. J. H. & Van Heijst, G. H. F. 2008 Intrinsic three-dimensionality in electromagnetically driven shallow flows. Europhys. Lett. 83 (2), 24001.
Alboussière, T., Uspenski, V. & Moreau, R. 1999 Quasi-2D MHD turbulent shear layers. Exp. Therm. Fluid Sci. 20 (20), 1924.
Alpher, R. A., Hurwitz, H., Johnson, R. H. & White, D. R. 1960 Some studies of free-surface mercury magnetohydrodynamics. Rev. Mod. Phys. 4 (32), 758774.
Andreev, O., Kolesnikov, Y. & Thess, A. 2013 Visualization of the Ludford column. J. Fluid Mech. 721, 438453.
Boeck, T., Krasnov, D. & Thess, A. 2008 Large-scale intermittency of liquid–metal channel flow in a magnetic field. Phys. Rev. Lett. 101, 244501.
Davidson, P. & Pothérat, A. 2002 A note on Bodewädt–Hartmann layer. Eur. J. Mech. (B/Fluids) 21 (5), 541559.
Dousset, V. & Pothérat, A 2012 Characterisation of the flow around a truncated cylinder in a duct under a spanwise magnetic field. J. Fluid Mech. 691, 341367.
Duran-Matute, M., Trieling, R. R. & van Heijst, G. J. F. 2010 Scaling and asymmetry in an electromagnetically forced dipolar flow structure. Phys. Rev. E 83, 016306.
Greenspan, H. P. 1969 Theory of Rotating Fluids. Cambridge University Press.
Hunt, J. C. R., Ludford, G. S. S. & Hunt, J. C. R. 1968 Three dimensional MHD duct flow with strong transverse magnetic field. Part 1. Obstacles in a constant area duct. J. Fluid Mech. 33, 693714.
Kanaris, N., Albets, X., Grigoriadis, D. & Kassinos, K. 2013 3-d numerical simulations of MHD flow around a confined circular cylinder under low, moderate and strong magnetic fields. Phys. Fluids 074102.
Klein, R. & Pothérat, A. 2010 Appearance of three-dimensionality in wall bounded MHD flows. Phys. Rev. Lett. 104 (3), 034502.
Klein, R., Pothérat, A. & Alferjonok, A. 2009 Experiment on a confined electrically driven vortex pair. Phys. Rev. E 79 (1), 016304.
Kljukin, A. & Thess, A. 1998 Direct measurement of the stream-function in a quasi-two-dimensional liquid metal flow. Exp. Fluids 25, 298304.
Ludford, G. S. S. 1961 Effect of a very strong magnetic crossfield on steady motion through a slightly conducting fluid. J. Fluid Mech. 10, 141155.
Moreau, R. 1990 Magnetohydrodynamics. Kluwer Academic Publisher.
Mück, B., Günter, C. & Bühler, L. 2000 Buoyant three-dimensional MHD flows in rectangular ducts with internal obstacles. J. Fluid Mech. 418, 265295.
Paret, J., Marteau, D., Paireau, O. & Tabeling, P. 1997 Are flows electromagnetically forced in thin stratified layers two dimensional? Phys. Fluids 9 (10), 31023104.
Pothérat, A. 2012 Three-dimensionality in quasi-two dimensional flows: recirculations and Barrel effects. Europhys. Lett. 98 (6), 64003.
Pothérat, A. & Dymkou, V. 2010 Direct numerical simulations of low-Rm MHD turbulence based on the least dissipative modes. J. Fluid Mech. 655, 174197.
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.
Pothérat, A., Sommeria, J. & Moreau, R. 2000 An effective two-dimensional model for MHD flows with transverse magnetic field. J. Fluid Mech. 424, 75100.
Pothérat, A., Sommeria, J. & Moreau, R. 2005 Numerical simulations of an effective two-dimensional model for flows with a transverse magnetic field. J. Fluid Mech. 534, 115143.
Roberts, P. H. 1967 Introduction to Magnetohydrodynamics. Longmans.
Schumann, U. 1976 Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field. J. Fluid Mech. 35, 3158.
Shats, M., Byrne, D. & Xia, H. 2010 Turbulence decay rate as a measure of flow dimensionality. Phys. Rev. Lett. 105, 264501.
Sommeria, J. 1986 Experimental study of the two-dimensionnal inverse energy cascade in a square box. J. Fluid Mech. 170, 139168.
Sommeria, J. 1988 Electrically driven vortices in a strong magnetic field. J. Fluid Mech. 189, 553569.
Sommeria, J. & Moreau, R. 1982 Why, how and when MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507518.
Sreenivasan, B. & Alboussière, T. 2002 Experimental study of a vortex in a magnetic field. J. Fluid Mech. 464, 287309.
Thess, A. & Zikanov, O. 2007 Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 579, 383412.
Vetcha, N., Smolentsev, S., Abdou, M. & Moreau, R. 2013 Study of instabilities and quasi-two-dimensional turbulence in volumetrically heated magnetohydrodynamic flows in a vertical rectangular duct. Phys. Fluids 25 (2), 024102.
Zikanov, O. & Thess, A. 1998 Direct simulation of forced MHD turbulence at low magnetic Reynolds number. J. Fluid Mech. 358, 299333.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Why, how and when MHD turbulence at low $\mathit{Rm}$ becomes three-dimensional

  • Alban Pothérat (a1) and Rico Klein (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed