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Rayleigh–Bénard instability in the presence of phase boundary and shear

Published online by Cambridge University Press:  15 September 2022

Cailei Lu ,
Mengqi Zhang Open the ORCID record for Mengqi Zhang [Opens in a new window] ,
Kang Luo Open the ORCID record for Kang Luo [Opens in a new window] ,
Jian Wu Open the ORCID record for Jian Wu [Opens in a new window]  and
Hongliang Yi Open the ORCID record for Hongliang Yi [Opens in a new window]
Cailei Lu
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China Key Laboratory of Aerospace Thermophysics, Harbin Institute of Technology, Harbin 150001, PR China
Mengqi Zhang
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575 Singapore
Kang Luo*
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Jian Wu
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Hongliang Yi*
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China Key Laboratory of Aerospace Thermophysics, Harbin Institute of Technology, Harbin 150001, PR China
*
† Email addresses for correspondence: [email protected], [email protected]
† Email addresses for correspondence: [email protected], [email protected]
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Abstract

We study the two-dimensional Rayleigh–Bénard instability subject to the combined effects of a solid–liquid phase boundary and shear using linear stability theory and energy analysis. We consider two thermal states of the solid (isothermal and conducting), and two types of shear that can arise in different contexts. When the melting temperature is equal to that held at the top boundary, three instability modes can arise with the increase of Reynolds number Re that characterizes the shear intensity: a boundary mode, a mixed boundary–bulk mode and a bulk flow mode. When the melting temperature lies between the top and the bottom boundaries, the introduction of Couette flow, independent of its intensity, always leads to the mixed mode, whereas the instability with Poiseuille flow is dominated by the bulk flow mode once Re exceeds a critical value, below which the mixed mode dominates. The energy analysis suggests that there exist two mechanisms by which the shear flow affects the system: one is by inhibiting the upward heat flux and another is by absorbing energy from the perturbed hydrodynamic field. These two mechanisms can play totally different roles in different cases. Results in the high-Re regime indicate that, when Re exceeds its classical threshold, i.e. Re = 5772.2 for Poiseuille flow, the Tollmien–Schlichting instability will be dominant in the present system.

JFM classification

Instability: Absolute/convective instability Phase change: Solidification/melting
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 948 , 10 October 2022 , A46
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

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The online version of this article has been updated since original publication. A notice detailing the change has also been published.

References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys. 81, 503–537.CrossRefGoogle Scholar
Anderson, D.M. & Guba, P. 2020 Convective phenomena in mushy layers. Annu. Rev. Fluid Mech. 52, 93–119.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Nield, D.A. 2011 On the onset of dissipation thermal instability for the Poiseuille flow of a highly viscous fluid in a horizontal channel. J. Fluid Mech. 681, 499–514.CrossRefGoogle Scholar
Barletta, A. & Nield, D.A. 2010 Convection-dissipation instability in the horizontal plane Couette flow of a highly viscous fluid. J. Fluid Mech. 662, 475–492.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent development in Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Bushuk, M., Holland, M.D., Stanton, T.P., Stern, A. & Gray, C. 2019 Ice scallops: a laboratory investigation of the ice-water interface. J. Fluid Mech. 873, 942–976.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chung, C.A. & Chen, F. 2001 Morphological instability in a directionally solidifying binary solution with an imposed shear flow. J. Fluid Mech. 436, 85–106.CrossRefGoogle Scholar
Couston, L.-A., Hester, E., Favier, B., Taylor, J.R., Holland, P.R. & Jenkins, A. 2021 Topography generation by melting and freezing in a turbulent shear flow. J. Fluid Mech. 911, A44.CrossRefGoogle Scholar
Dallaston, M.C., Hewitt, I.J. & Wells, A.J. 2015 Channelization of plumes beneath ice shelves. J. Fluid Mech. 785, 109–134.CrossRefGoogle Scholar
Davis, S.H., Müller, U. & Dietsche, C. 1984 Pattern selection in single-component system coupling Bénard convection and solidification. J. Fluid Mech. 144, 133–151.CrossRefGoogle Scholar
Dutrieux, P., Vaughan, D.G., Corr, H.F.J., Jenkins, A., Holland, P.R., Joughin, I. & Fleming, A.H. 2013 Pine Island Glacier ice shelf melt distributed at kilometer scales. Cryosphere 7 (5), 1543–1555.CrossRefGoogle Scholar
Esfahani, B.R., Hirata, S.C., Berti, S. & Calzavarini, E. 2018 Basal melting driven by turbulent thermal convection. Phys. Rev. Fluids 3, 053501.CrossRefGoogle Scholar
Favier, B., Purseed, J. & Duchemin, L. 2019 Rayleigh-Bénard convection with a melting boundary. J. Fluid Mech. 858, 437–473.CrossRefGoogle Scholar
FitzMaurice, A., Straneo, F., Cendese, C. & Andres, M. 2016 Effect of sheared flow on iceberg motion and melting. Geophys. Res. Lett. 43 (12), 12520–12527.CrossRefGoogle Scholar
Fujimura, K. & Kelly, R.E. 1988 Stability of unstably stratified shear flow between parallel plates. Fluid Dyn. Res. 2 (4), 281–292.CrossRefGoogle Scholar
Gage, K.S. & Reid, W.H. 1968 The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33, 21–32.CrossRefGoogle Scholar
Gallagher, A.P. & Mercer, A.McD. 1965 On the behavior of small disturbances in plane Couette flow with a temperature gradient. Proc. R. Soc. Lond. A 286, 117–128.Google Scholar
Gilpin, R.R., Hirata, T. & Cheng, K.C. 1980 Wave formation and heat transfer at an ice-water interface in the presence of a turbulent flow. J. Fluid Mech. 99, 619–640.CrossRefGoogle Scholar
Grue, J., Gjevik, B. & Weber, J.E. 1996 Waves and Nonlinear Processes in Hydrodynamics. Springer.CrossRefGoogle Scholar
Hewitt, I.J. 2020 Subglacial plumes. Annu. Rev. Fluid Mech. 52, 145–169.CrossRefGoogle Scholar
Kato, Y. & Fujimura, K. 2000 Prediction of pattern selection due to an interaction between longitudinal rolls and transverse modes in a flow thorugh a rectangular channel heated from below. Phys. Rev. E 62, 1.CrossRefGoogle Scholar
Kim, M.C., Lee, D.W. & Choi, C.K. 2008 Onset of buoyancy-driven convection in melting from below. Korean J. Chem. Engng 25 (6), 1239–1244.CrossRefGoogle 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.CrossRefGoogle Scholar
Lebrocq, A.M., et al. 2013 Evidence from ice shelves for channelized meltwater flow beneath the Antarctic Ice Sheet. Nature Geosci. 6, 945–948.CrossRefGoogle Scholar
Luijkx, J.-M., Platten, J.K. & Legros, J.C. 1981 On the existence of thermoconvective rolls, transverse to a superimposed mean Poiseuille flow. Intl J. Heat Mass Transfer 24, 1287–1291.CrossRefGoogle Scholar
Monteux, J., Andrault, D., Guitreau, M., Samuel, H. & Demouchy, S. 2020 A mushy Earth's mantle for more than 500 Myr after the magma ocean solidification. Geophys. J. Intl 221, 1165–1181.CrossRefGoogle Scholar
Müller, H.W., Lücke, M. & Kamps, M. 1992 Transverse convection patterns in horizontal shear flow. Phys. Rev. A 45, 6.CrossRefGoogle ScholarPubMed
Naughten, K.A., De Rydt, J., Rosier, S.H.R., Jenkins, A., Holland, P.R. & Ridley, J.K. 2021 Two-timescale response of a large Antarctic ice shelf to climate change. Nat. Commun. 12, 1991.CrossRefGoogle ScholarPubMed
Neufeld, J.A. & Wettlaufer, J.S. 2008 Shear-enhanced convection in a mushy layer. J. Fluid Mech. 612, 339–361.CrossRefGoogle Scholar
Nield, D.A. 1968 The Rayleigh-Jeffreys problem with boundary slab of finite conductivity. J. Fluid Mech. 32, 393–398.CrossRefGoogle Scholar
Polashenski, C., Golden, K.M., Perovich, D.K., Skyllingstad, E., Arnsten, A., Stwertka, C. & Wright, N. 2017 Percolation blockage: a process that enables melt pond formation on first year Arctic sea ice. J. Geophys. Res. Oceans 122, 413–440.CrossRefGoogle Scholar
Polashenski, C., Perovich, D.K. & Courville, Z. 2012 The mechanisms of sea ice melt pond formation and evolution. J. Geophys. Res. 117, C01001.CrossRefGoogle Scholar
Purseed, J., Favier, B. & Duchemin, L. 2020 Bistability in Rayleigh-Bénard convection with a melting boundary. Phys. Rev. Fluids 5, 023501.CrossRefGoogle Scholar
Ravichandran, S. & Wettlaufer, J.S. 2021 Melting driven by rotating Rayleigh-Bénard convection. J. Fluid Mech. 916, A28.CrossRefGoogle Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32, 529–546.CrossRefGoogle Scholar
Requilé, Y., Hirata, S.C. & Quarzazi, M.N. 2020 a Viscous dissipation effects on the linear stability of Rayleigh-Bénard-Poiseuille/Couette convection. Intl J. Heat Mass Transfer 146, 118834.CrossRefGoogle Scholar
Requilé, Y., Hirata, S.C., Quarzazi, M.N. & Barletta, A. 2020 b Weakly nonlinear analysis of viscous dissipation thermal instability in plane Poiseuille and plane Couette flow. J. Fluid Mech. 886, A26.CrossRefGoogle Scholar
Ristroph, L. 2018 Sculpting with flow. J. Fluid Mech. 838, 1–4.CrossRefGoogle Scholar
Roberts, P.H. & King, E.M. 2013 On the genesis of the Earth's magnetism. Rep. Prod. Phys. 76 (9), 6801.Google ScholarPubMed
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Squire, V.A. 2020 Ocean wave interactions with sea ice: a reappraisal. Annu. Rev. Fluid Mech. 52, 37–60.CrossRefGoogle Scholar
Toppaladoddi, S. 2021 Nonlinear interactions between an unstably stratified shear flow and a phase boundary. J. Fluid Mech. 919, A28.CrossRefGoogle Scholar
Toppaladoddi, S. & Wettlaufer, J.S. 2019 Combined effects of shear and buoyancy on phase boundary stability. J. Fluid Mech. 868, 648–665.CrossRefGoogle Scholar
Ulvrová, M., Labrosse, S., Coltice, N., Råback, P. & Tackley, P. 2012 Numerical modelling of convection interacting with a melting and solidification front: application to the thermal evolution of basal magma ocean. Phys. Earth Planet. Inter. 206–207, 21–66.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.CrossRefGoogle Scholar
Wang, Z.Q., Calzavarini, E., Sun, C. & Toschi, F. 2021 How the growth of ice depends on the fluid dynamics underneath. Proc. Natl Acad. Sci. USA 118, e2012870118.CrossRefGoogle ScholarPubMed
Weideman, J.A. & Reddy, S.C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519.CrossRefGoogle Scholar
Worster, M.G. 1991 Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359.CrossRefGoogle Scholar
Worster, M.G. 2000 Solidification of fluids. In Perspective in Fluid Dynamics (ed. G.K. Batchelor, H.K. Moffatt & M.G. Worster), pp. 393–446. Cambridge University Press.Google Scholar
Zhang, M., Martinelli, F., Wu, J., Schmid, P.J. & Quadrio, M. 2015 Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow. J. Fluid Mech. 770, 319–349.CrossRefGoogle Scholar