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Numerical simulation of the von Kármán sodium dynamo experiment

Published online by Cambridge University Press:  03 September 2018

C. Nore Open the ORCID record for C. Nore [Opens in a new window] ,
D. Castanon Quiroz ,
L. Cappanera  and
J.-L. Guermond
C. Nore*
Affiliation:
Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur, LIMSI, CNRS, Univ. Paris-Sud, Université Paris-Saclay, Bâtiment 508, rue John von Neumann, Campus Universitaire, F-91405 Orsay, France
D. Castanon Quiroz
Affiliation:
BCAM – Basque Center for Applied Mathematics, Mazarredo 14, E-48009 Bilbao, Basque Country, Spain
L. Cappanera
Affiliation:
Department of Computational and Applied Mathematics, Rice University, 6100 Main, MS-134, Houston, TX 77005, USA
J.-L. Guermond
Affiliation:
Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843-3368, USA
*
†Email address for correspondence: caroline.nore@limsi.fr
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Abstract

We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flows in the von Kármán sodium (VKS) set-up. The counter-rotating impellers made of soft iron that were used in the successful 2006 experiment are represented by means of a pseudo-penalty method. Hydrodynamic simulations are performed at high kinetic Reynolds numbers using a large eddy simulation technique. The results compare well with the experimental data: the flow is laminar and steady or slightly fluctuating at small angular frequencies; small scales fill the bulk and a Kolmogorov-like spectrum is obtained at large angular frequencies. Near the tips of the blades the flow is expelled and takes the form of intense helical vortices. The equatorial shear layer acquires a wavy shape due to three coherent co-rotating radial vortices as observed in hydrodynamic experiments. MHD computations are performed: at fixed kinetic Reynolds number, increasing the magnetic permeability of the impellers reduces the critical magnetic Reynolds number for dynamo action; at fixed magnetic permeability, increasing the kinetic Reynolds number also decreases the dynamo threshold. Our results support the conjecture that the critical magnetic Reynolds number tends to a constant as the kinetic Reynolds number tends to infinity. The resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated near the impellers as observed in the VKS experiment. A speculative mechanism for dynamo action in the VKS experiment is proposed.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory MHD and Electrohydrodynamics: MHD and Electrohydrodynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 854 , 10 November 2018 , pp. 164 - 195
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Copyright
© 2018 Cambridge University Press 

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References

Balay, S., Abhyankar, S., Adams, M. F., Brown, J., Brune, P., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Rupp, K., Smith, B. F. & Zhang, H.2014 PETSc users manual. Tech. Rep. ANL-95/11 – Revision 3.5, Argonne National Laboratory.Google Scholar
Boisson, J., Aumaitre, S., Bonnefoy, N., Bourgoin, M., Daviaud, F., Dubrulle, B., Odier, P., Pinton, J.-F., Plihon, N. & Verhille, G. 2012 Symmetry and couplings in stationary von Kármán sodium dynamos. New J. Phys. 14 (1), 013044.Google Scholar
Boisson, J. & Dubrulle, B. 2011 Three-dimensional magnetic field reconstruction in the VKS experiment through Galerkin transforms. New J. Phys. 13 (2), 023037.Google Scholar
Bonito, A. & Guermond, J.-L. 2011 Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements. Math. Comput. 80 (276), 1887–1910.Google Scholar
Bonito, A., Guermond, J.-L. & Luddens, F. 2013 Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains. J. Math. Anal. Appl. 408 (2), 498–512.Google Scholar
Bullard, E. C. 1955 The stability of a homopolar dynamo. Proc. Camb. Phil. Soc. 51, 744–760.Google Scholar
Caffarelli, L., Kohn, R. & Nirenberg, L. 1982 Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Maths 35 (6), 771–831.Google Scholar
Castanon Quiroz, D.2015 Solution of the MHD equations in the presence of non-axisymmetric conductors using the Fourier-finite element method. PhD thesis, Texas A&M College Station.Google Scholar
Colgate, S. A., Beckley, H., Si, J., Martinic, J., Westpfahl, D., Slutz, J., Westrom, C., Klein, B., Schendel, P., Scharle, C., McKinney, T., Ginanni, R., Bentley, I., Mickey, T., Ferrel, R., Li, H., Pariev, V. & Finn, J. 2011 High magnetic shear gain in a liquid sodium stable couette flow experiment: a prelude to an 𝛼-𝛺 dynamo. Phys. Rev. Lett. 106, 175003.Google Scholar
Cortet, P.-P., Diribarne, P., Monchaux, R., Chiffaudel, A., Daviaud, F. & Dubrulle, B. 2009 Normalized kinetic energy as a hydrodynamical global quantity for inhomogeneous anisotropic turbulence. Phys. Fluids 21 (2), 025104.Google Scholar
Frick, P., Noskov, V., Denisov, S. & Stepanov, R. 2010 Direct measurement of effective magnetic diffusivity in turbulent flow of liquid sodium. Phys. Rev. Lett. 105, 184502.Google Scholar
Frisch, U. 1995 Turbulence: the Legacy of AN Kolmogorov. Cambridge University Press.Google Scholar
Gailitis, A., Lielausis, O., Dement’ev, S., Platacis, E. & Cifersons, A. 2000 Detection of a flow induced magnetic field eigenmode in the Riga dynamo facility. Phys. Rev. Lett. 84, 4365–4368.Google Scholar
Gailitis, A., Lielausis, O., Platacis, E., Gerbeth, G. & Stefani, F. 2003 The Riga dynamo experiment. Surv. Geophys. 24 (3), 247–267.Google Scholar
Giesecke, A., Nore, C., Stefani, F., Gerbeth, G., Léorat, J., Herreman, W., Luddens, F. & Guermond, J.-L. 2012 Influence of high-permeability discs in an axisymmetric model of the Cadarache dynamo experiment. New J. Phys. 14 (5), 053005.Google Scholar
Giesecke, A., Nore, C., Stefani, F., Gerbeth, G., Léorat, J., Luddens, F. & Guermond, J.-L. 2010 Electromagnetic induction in non-uniform domains. Geophys. Astrophys. Fluid Dyn. 104 (5), 505–529.Google Scholar
Gissinger, C. 2009 A numerical model of the VKS experiment. Europhys. Lett. 87, 39002.Google Scholar
Gissinger, C., Iskakov, A., Fauve, S. & Dormy, E. 2008 Effect of magnetic boundary conditions on the dynamo threshold of von Kármán swirling flows. Europhys. Lett. 82, 29001.Google Scholar
Guermond, J.-L., Laguerre, R., Léorat, J. & Nore, C. 2007 An interior penalty Galerkin method for the MHD equations in heterogeneous domains. J. Comput. Phys. 221 (1), 349–369.Google Scholar
Guermond, J.-L., Laguerre, R., Léorat, J. & Nore, C. 2009 Nonlinear magnetohydrodynamics in axisymmetric heterogeneous domains using a Fourier/finite element technique and an interior penalty method. J. Comput. Phys. 228, 2739–2757.Google Scholar
Guermond, J.-L., Léorat, J., Luddens, F., Nore, C. & Ribeiro, A. 2011a Effects of discontinuous magnetic permeability on magnetodynamic problems. J. Comput. Phys. 230, 6299–6319.Google Scholar
Guermond, J.-L., Marra, A. & Quartapelle, L. 2006 Subgrid stabilized projection method for 2d unsteady flows at high Reynolds number. Comput. Methods Appl. Mech. Engng 195, 5857–5876.Google Scholar
Guermond, J.-L., Pasquetti, R. & Popov, B. 2011b Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230 (11), 4248–4267.Google Scholar
Guermond, J.-L., Pasquetti, R. & Popov, B. 2011c From suitable weak solutions to entropy viscosity. J. Sci. Comput. 49 (1), 35–50.Google Scholar
Guermond, J.-L. & Shen, J. 2004 On the error estimates for the rotational pressure-correction projection methods. Maths Comput. 73 (248), 1719–1737; (electronic).Google Scholar
Herbert, E., Cortet, P.-P., Daviaud, F. & Dubrulle, B. 2014 Eckhaus-like instability of large scale coherent structures in a fully turbulent von Kármán flow. Phys. Fluids 26 (1), 015103.Google Scholar
Iskakov, A. B., Schekochihin, A. A., Cowley, S. C., McWilliams, J. C. & Proctor, M. R. E. 2007 Numerical demonstration of fluctuation dynamo at low magnetic Prandtl numbers. Phys. Rev. Lett. 98, 208501.Google Scholar
Karypis, G. & Kumar, V. 1998 A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20 (1), 359–392.Google Scholar
Kreuzahler, S., Ponty, Y., Plihon, N., Homann, H. & Grauer, R. 2017 Dynamo enhancement and mode selection triggered by high magnetic permeability. Phys. Rev. Lett. 119, 234501.Google Scholar
Kreuzahler, S., Schulz, D., Homann, H., Ponty, Y. & Grauer, R. 2014 Numerical study of impeller-driven von Kármán flows via a volume penalization method. New J. Phys. 16 (10), 103001.Google Scholar
Laguerre, R., Nore, C., Léorat, J. & Guermond, J.-L. 2006 Effects of conductivity jumps in the envelope of a kinematic dynamo flow. C. R. Méc. 334, 593.Google Scholar
Laguerre, R., Nore, C., Ribeiro, A., Léorat, J., Guermond, J.-L. & Plunian, F. 2008 Impact of impellers on the axisymmetric magnetic mode in the VKS2 dynamo experiment. Phys. Rev. Lett. 101 (10), 104501.Google Scholar
Marié, L., Normand, C. & Daviaud, F. 2006 Galerkin analysis of kinematic dynamos in the von Kármán geometry. Phys. Fluids 18, 017102.Google Scholar
Miralles, S., Bonnefoy, N., Bourgoin, M., Odier, P., Pinton, J.-F., Plihon, N., Verhille, G., Boisson, J., Daviaud, F. & Dubrulle, B. 2013 Dynamo threshold detection in the von Kármán sodium experiment. Phys. Rev. E 88, 013002.Google Scholar
Miralles, S., Plihon, N. & Pinton, J.-F. 2015 Lorentz force effects in the Bullard–von Kármán dynamo: saturation, energy balance and subcriticality. J. Fluid Mech. 775, 501–523.Google Scholar
Moffatt, H. 1978 Magnetic Field Generation in Electrically Conducting Fluids, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press.Google Scholar
Monchaux, R., Berhanu, M., Bourgoin, M., Odier, P., Moulin, M., Pinton, J.-F., Volk, R., Fauve, S., Mordant, N., Pétrélis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Gasquet, C., Marié, L. & Ravelet, F. 2007 Generation of magnetic field by a turbulent flow of liquid sodium. Phys. Rev. Lett. 98, 044502.Google Scholar
Müller, U., Stieglitz, R. & Horanyi, S. 2004 A two-scale hydromagnetic dynamo experiment. J. Fluid Mech. 498, 31–71.Google Scholar
Nore, C., Tuckerman, L. S., Daube, O. & Xin, S. 2003 The 1:2 mode interaction in exactly counter-rotating von Kármán swirling flow. J. Fluid Mech. 477, 51–88.Google Scholar
Nore, C., Zaidi, H., Bouillault, F., Bossavit, A. & Guermond, J.-L. 2016 Approximation of the time-dependent induction equation with advection using whitney elements: application to dynamo action. COMPEL – Intl J. Comput. Math. Electr. Electron. Engng 35 (1), 326–338.Google Scholar
Nore, C., Castanon Quiroz, D., Cappanera, L. & Guermond, J.-L. 2016 Direct numerical simulation of the axial dipolar dynamo in the von Kármán sodium experiment. Europhys. Lett. 114 (6), 65002.Google Scholar
Nornberg, M. D., Spence, E. J., Kendrick, R. D., Jacobson, C. M. & Forest, C. B. 2006 Measurements of the magnetic field induced by a turbulent flow of liquid metal. Phys. Plasmas 13 (5), 055901.Google Scholar
Pasquetti, R., Bwemba, R. & Cousin, L. 2008 A pseudo-penalization method for high Reynolds number unsteady flows. Appl. Numer. Maths 58 (7), 946–954.Google Scholar
Peffley, N. L., Cawthorne, A. B. & Lathrop, D. P. 2000 Toward a self-generating magnetic dynamo: the role of turbulence. Phys. Rev. E 61, 5287–5294.Google Scholar
Ponty, Y. & Plunian, F. 2011 Transition from large-scale to small-scale dynamo. Phys. Rev. Lett. 106, 154502.Google Scholar
Ponty, Y., Mininni, P., Pinton, J.-F., Politano, H. & Pouquet, A. 2007 Dynamo action at low magnetic Prandtl numbers: mean flow versus fully turbulent motion. New J. Phys. 9, 296.Google Scholar
Ravelet, F.2005 Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de von Kármán turbulent. PhD thesis, Ecole Polytechnique X.Google Scholar
Ravelet, F., Chiffaudel, A. & Daviaud, F. 2008 Supercritical transition to turbulence in an inertially driven von Kármán closed flow. J. Fluid Mech. 601, 339–364.Google Scholar
Ravelet, F., Chiffaudel, A., Daviaud, F. & Léorat, J. 2005 Towards an experimental von Kármán dynamo: numerical studies for an optimized design. Phys. Fluids 17, 117104.Google Scholar
Ravelet, F., Dubrulle, B., Daviaud, F. & Ratié, P.-A. 2012 Kinematic alpha tensors and dynamo mechanisms in a von Kármán swirling flow. Phys. Rev. Lett. 109, 024503.Google Scholar
Reuter, K., Jenko, F. & Forest, C. B. 2011 Turbulent magnetohydrodynamic dynamo action in a spherically bounded von Kármán flow at small magnetic Prandtl numbers. New J. Phys. 13 (7), 073019.Google Scholar
Scheffer, V. 1987 Nearly one-dimensional singularities of solutions to the Navier–Stokes inequality. Commun. Math. Phys. 110 (4), 525–551.Google Scholar
Sisan, D. R., Shew, W. L. & Lathrop, D. P. 2003 Lorentz force effects in magneto-turbulence. Phys. Earth Planet. Inter. 135 (2–3), 137–159.Google Scholar
Stefani, F., Xu, M., Gerbeth, G., Ravelet, F., Chiffaudel, A., Daviaud, F. & Léorat, J. 2006 Ambivalent effects of added layers on steady kinematic dynamos in cylindrical geometry: application to the VKS experiment. Eur. J. Mech. (B/Fluids) 25, 894.Google Scholar
Stieglitz, R. & Müller, U. 2001 Experimental demonstration of a homogeneous two-scale dynamo. Phys. Fluids 13, 561.Google Scholar
Varela, J., Brun, S., Dubrulle, B. & Nore, C. 2015 Role of boundary conditions in helicoidal flow collimation: consequences for the von Kármán sodium dynamo experiment. Phys. Rev. E 92, 063015.Google Scholar
Verhille, G., Plihon, N., Bourgoin, M., Odier, P. & Pinton, J.-F. 2010 Induction in a von Kármán flow driven by ferromagnetic impellers. New J. Phys. 12 (3), 033006.Google Scholar