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Rayleigh–Bénard convection with a melting boundary

Published online by Cambridge University Press:  06 November 2018

B. Favier Open the ORCID record for B. Favier [Opens in a new window] ,
J. Purseed  and
L. Duchemin
B. Favier*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
J. Purseed
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
L. Duchemin
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
*
†Email address for correspondence: favier@irphe.univ-mrs.fr
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Abstract

We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane-layer geometry, this can be seen as classical Rayleigh–Bénard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier–Stokes equations in the Boussinesq approximation. The advantage of this approach is that it requires only moderate modifications of classical numerical methods. We focus on the case where the solid is initially nearly isothermal, so that the evolution of the topography is related to the inhomogeneous heat flux from thermal convection, and does not depend on the conduction problem in the solid. From a very thin stable layer of fluid, convection cells appear as the depth – and therefore the effective Rayleigh number – of the layer increases. The continuous melting of the solid leads to dynamical transitions between different convection cell sizes and topography amplitudes. The Nusselt number can be larger than its value for a planar upper boundary, due to the feedback of the topography on the flow, which can stabilize large-scale laminar convection cells.

JFM classification

Phase change: Solidification/melting Geophysical and Geological Flows: Topographic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 858 , 10 January 2019 , pp. 437 - 473
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Copyright
© 2018 Cambridge University Press 

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References

Alboussière, T., Deguen, R. & Melzani, M. 2010 Melting-induced stratification above the Earth’s inner core due to convective translation. Nature 466, 744–747.Google Scholar
Almgren, R. F. 1999 Second-order phase field asymptotics for unequal conductivities. SIAM J. Appl. Maths 59 (6), 2086–2107.Google Scholar
Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 2000 A phase-field model of solidification with convection. Physica D 135 (1), 175–194.Google Scholar
Andersson, C.2002 Phase-field simulation of dendritic solidification. PhD thesis, Royal Institute of Technology KTH, Department of Numerical Analysis and Computer Science.Google Scholar
Angot, P., Bruneau, C.-H. & Fabrie, P. 1999 A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81 (4), 497–520.Google Scholar
Beckermann, C., Diepers, H.-J., Steinbach, I., Karma, A. & Tong, X. 1999 Modeling melt convection in phase-field simulations of solidification. J. Comput. Phys. 154 (2), 468–496.Google Scholar
Bhattacharjee, K. J. 1991 Parametric resonance in Rayleigh–Bénard convection with corrugated geometry. Phys. Rev. A 43, 819–821.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32 (1), 709–778.Google Scholar
Boettinger, W. J., Warren, J. A., Beckermann, C. & Karma, A. 2002 Phase-field simulation of solidification. Annu. Rev. Mater. Res. 32 (1), 163–194.Google Scholar
Busse, F. H. 1983 Generation of mean flows by thermal convection. Physica D 9 (3), 287–299.Google Scholar
Caginalp, G. 1989 Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. Phys. Rev. A 39, 5887–5896.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Ciliberto, S. & Laroche, C. 1999 Random roughness of boundary increases the turbulent convection scaling exponent. Phys. Rev. Lett. 82, 3998–4001.Google Scholar
Claudin, P., Durán, O. & Andreotti, B. 2017 Dissolution instability and roughening transition. J. Fluid Mech. 832, R2.Google Scholar
Coullet, P. & Huerre, P. 1986 Resonance and phase solitons in spatially-forced thermal convection. Physica D 23 (1), 27–44.Google Scholar
Couston, L.-A., Lecoanet, D., Favier, B. & Le Bars, M. 2017 Dynamics of mixed convective–stably-stratified fluids. Phys. Rev. Fluids 2, 094804.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.Google Scholar
Curry, J. H., Herring, J. R., Loncaric, J. & Orszag, S. A. 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38.Google Scholar
Davaille, A. 1999 Two-layer thermal convection in miscible viscous fluids. J. Fluid Mech. 379, 223–253.Google Scholar
Davis, S. H., Müller, U. & Dietsche, C. 1984 Pattern selection in single-component systems coupling Bénard convection and solidification. J. Fluid Mech. 144, 133–151.Google Scholar
Du, Y.-B. & Tong, P. 2000 Turbulent thermal convection in a cell with ordered rough boundaries. J. Fluid Mech. 407, 57–84.Google Scholar
Favier, B. & Bushby, P. J. 2012 Small-scale dynamo action in rotating compressible convection. J. Fluid. Mech. 690, 262–287.Google Scholar
Favier, B., Silvers, L. J. & Proctor, M. R. E. 2014 Inverse cascade and symmetry breaking in rapidly rotating Boussinesq convection. Phys. Fluids 26 (9), 096605.Google Scholar
Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G.2008 NEK5000 v17.0: open source spectral element CFD solver. Argonne National Laboratory, Illinois. Available at: http://nek5000.mcs.anl.gov.Google Scholar
Fitzgerald, J. G. & Farrell, B. F. 2014 Mechanisms of mean flow formation and suppression in two-dimensional Rayleigh–Bénard convection. Phys. Fluids 26 (5), 054104.Google Scholar
Gastine, T., Wicht, J. & Aurnou, J. M. 2015 Turbulent Rayleigh–Bénard convection in spherical shells. J. Fluid Mech. 778, 721–764.Google Scholar
Gibou, F., Chen, L., Nguyen, D. & Banerjee, S. 2007 A level set based sharp interface method for the multiphase incompressible Navier–Stokes equations with phase change. J. Comput. Phys. 222 (2), 536–555.Google Scholar
Goldhirsch, I., Pelz, R. B. & Orszag, S. A. 1989 Numerical simulation of thermal convection in a two-dimensional finite box. J. Fluid Mech. 199, 1–28.Google Scholar
Gollub, J. P. & Benson, S. V. 1980 Many routes to turbulent convection. J. Fluid Mech. 100 (3), 449–470.Google Scholar
Goluskin, D., Johnston, H., Flierl, G. R. & Spiegel, E. A. 2014 Convectively driven shear and decreased heat flux. J. Fluid Mech. 759, 360–385.Google Scholar
Grannan, A. M., Favier, B., Le Bars, M. & Aurnou, J. M. 2017 Tidally forced turbulence in planetary interiors. Geophys. J. Intl 208 (3), 1690–1703.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.Google Scholar
Jiaung, W.-S., Ho, J.-R. & Kuo, C.-P. 2001 Lattice Boltzmann method for the heat conduction problem with phase change. Numer. Heat Transfer 39 (2), 167–187.Google Scholar
Karma, Alain & Rappel, Wouter-Jan 1996 Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys. Rev. E 53, R3017–R3020.Google Scholar
Keitzl, T., Mellado, J. P. & Notz, D. 2016 Impact of thermally driven turbulence on the bottom melting of ice. J. Phys. Oceanogr. 46 (4), 1171–1187.Google Scholar
Kelly, R. E. & Pal, D. 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid Mech. 86 (3), 433–456.Google Scholar
Killworth, P. D. & Manins, P. C. 1980 A model of confined thermal convection driven by non-uniform heating from below. J. Fluid Mech. 98 (3), 587–607.Google Scholar
Kogan, A. B., Murphy, D. & Meyer, H. 1999 Rayleigh–Bénard convection onset in a compressible fluid: 3He near T C . Phys. Rev. Lett. 82, 4635–4638.Google Scholar
Kolomenskiy, D. & Schneider, K. 2009 A fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys. 228 (16), 5687–5709.Google Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5 (11), 1374–1389.Google Scholar
Labrosse, S., Morison, A., Deguen, R. & Alboussière, T. 2018 Rayleigh–Bénard convection in a creeping solid with melting and freezing at either or both its horizontal boundaries. J. Fluid Mech. 846, 5–36.Google Scholar
Le Bars, M. & Worster, M. G. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173.Google Scholar
Mackenzie, J. A. & Robertson, M. L. 2002 A moving mesh method for the solution of the one-dimensional phase-field equations. J. Comput. Phys. 181 (2), 526–544.Google Scholar
Manneville, P. 2006 Rayleigh–Bénard Convection: Thirty Years of Experimental, Theoretical, and Modeling Work, pp. 41–65. Springer.Google Scholar
Martin, S. & Kauffman, P. 1977 An experimental and theoretical study of the turbulent and laminar convection generated under a horizontal ice sheet floating on warm salty water. J. Phys. Oceanogr. 7 (2), 272–283.Google Scholar
Matthews, P. C., Proctor, M. R. E. & Weiss, N. O. 1995 Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour. J. Fluid Mech. 305, 281–305.Google Scholar
Meakin, P. & Jamtveit, B. 2010 Geological pattern formation by growth and dissolution in aqueous systems. Proc R. Soc. Lond. A 466 (2115), 659–694.Google Scholar
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37 (1), 239–261.Google Scholar
Moore, D. R. & Weiss, N. O. 1973 Nonlinear penetrative convection. J. Fluid Mech. 61 (3), 553–581.Google Scholar
Moore, M. N. J., Ristroph, L., Childress, S., Zhang, J. & Shelley, M. J. 2013 Self-similar evolution of a body eroding in a fluid flow. Phys. Fluids 25 (11), 116602.Google Scholar
Penrose, O. & Fife, P. C. 1990 Thermodynamically consistent models of phase-field type for the kinetic of phase transitions. Physica D 43 (1), 44–62.Google Scholar
Prat, J., Massaguer, J. M. & Mercader, I. 1995 Large scale flows and resonances in 2D thermal convection. Phys. Fluids 7 (1), 121–134.Google Scholar
Rabbanipour Esfahani, B., Hirata, S. C., Berti, S. & Calzavarini, E. 2018 Basal melting driven by turbulent thermal convection. Phys. Rev. Fluids 3, 053501.Google Scholar
Ristroph, L. 2018 Sculpting with flow. J. Fluid Mech. 838, 1–4.Google Scholar
Roberts, P. H. 2015 Theory of the geodynamo. In Treatise on Geophysics, 2nd edn (ed. Schubert, G.), pp. 57–90. Elsevier.Google Scholar
Roche, P.-E., Gauthier, F., Kaiser, R. & Salort, J. 2010 On the triggering of the ultimate regime of convection. New J. Phys. 12 (8), 085014.Google Scholar
Roppo, M. N., Davis, S. H. & Rosenblat, S. 1984 Bénard convection with time periodic heating. Phys. Fluids 27 (4), 796–803.Google Scholar
Rossby, H. T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep Sea Res. 12 (1), 9–16.Google Scholar
Rusaouën, E., Liot, O., Castaing, B., Salort, J. & Chillà, F. 2018 Thermal transfer in Rayleigh–Bénard cell with smooth or rough boundaries. J. Fluid Mech. 837, 443–460.Google Scholar
Singh, J., Bajaj, R. & Kaur, P. 2015 Bicritical states in temperature-modulated Rayleigh–Bénard convection. Phys. Rev. E 92, 013005.Google Scholar
Sondak, D., Smith, L. M. & Waleffe, F. 2015 Optimal heat transport solutions for Rayleigh–Bénard convection. J. Fluid Mech. 784, 565–595.Google Scholar
Stevens, B. 2005 Atmospheric moist convection. Annu. Rev. Earth Planet. Sci. 33 (1), 605–643.Google Scholar
Tackley, P. J. 1996 Effects of strongly variable viscosity on three-dimensional compressible convection in planetary mantles. J. Geophys. Res. 101 (B2), 3311–3332.Google Scholar
Toppaladoddi, S., Succi, S. & Wettlaufer, J. S. 2015 Tailoring boundary geometry to optimize heat transport in turbulent convection. Europhys. Lett. 111 (4), 44005.Google Scholar
Ulvrová, M., Labrosse, S., Coltice, N., Råback, P. & Tackley, P. J. 2012 Numerical modelling of convection interacting with a melting and solidification front: application to the thermal evolution of the basal magma ocean. Phys. Earth Planet. Inter. 206–207, 51–66.Google Scholar
Vasil, G. M. & Proctor, M. R. E. 2011 Dynamic bifurcations and pattern formation in melting-boundary convection. J. Fluid Mech. 686, 77–108.Google Scholar
Venezian, G. 1969 Effect of modulation on the onset of thermal convection. J. Fluid Mech. 35 (2), 243–254.Google Scholar
Verhoeven, J., Wiesehöfer, T. & Stellmach, S. 2015 Anelastic versus fully compressible turbulent Rayleigh–Bénard convection. Astrophys. J. 805 (1), 62.Google Scholar
Voller, V. R., Swaminathan, C. R. & Thomas, B. G. 1990 Fixed grid techniques for phase change problems: a review. Intl J. Numer. Meth. Engng 30 (4), 875–898.Google Scholar
Walton, I. C. 1982 On the onset of Rayleigh–Bénard convection in a fluid layer of slowly increasing depth. Stud. Appl. Maths 67 (3), 199–216.Google Scholar
Wang, S.-L., Sekerka, R. F., Wheeler, A. A., Murray, B. T., Coriell, S. R., Braun, R. J. & McFadden, G. B. 1993 Thermodynamically-consistent phase-field models for solidification. Physica D 69 (1), 189–200.Google Scholar
Weiss, S., Seiden, G. & Bodenschatz, E. 2014 Resonance patterns in spatially forced Rayleigh–Bénard convection. J. Fluid Mech. 756, 293–308.Google Scholar
Woods, A. W. 1992 Melting and dissolving. J. Fluid Mech. 239, 429–448.Google Scholar
Worster, M. G. 2000 Solidification of fluids. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), Cambridge University Press.Google Scholar
Zhang, Y.-Z., Sun, C., Bao, Y. & Zhou, Q. 2018 How surface roughness reduces heat transport for small roughness heights in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 836, R2.Google Scholar

Favier et al. supplementary movie

Visualizations of the total numerical domain for case C in Table 1. The temperature is shown on the left (dark red corresponds to θ=1 while dark blue corresponds to θ=θM) while vorticity is shown on the right (blue and red colors correspond to ±0.25ωmax respectively). The grey line corresponds to the interface defined by the isosurface φ=1/2.

Download Favier et al. supplementary movie(Video)
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