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Natural convection with mixed insulating and conducting boundary conditions: low- and high-Rayleigh-number regimes

Published online by Cambridge University Press:  24 February 2014

P. Ripesi ,
L. Biferale ,
M. Sbragaglia  and
A. Wirth
P. Ripesi
Affiliation:
Department of Physics and INFN, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy
L. Biferale*
Affiliation:
Department of Physics and INFN, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy
M. Sbragaglia
Affiliation:
Department of Physics and INFN, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy
A. Wirth
Affiliation:
Laboratoire LEGI (UMR 5519, CNRS), Université de Grenoble, 38041 Grenoble CEDEX 9, France
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the stability and dynamics of natural convection in two dimensions, subject to inhomogeneous boundary conditions. In particular, we consider a Rayleigh–Bénard (RB) cell, where the horizontal top boundary contains a periodic sequence of alternating thermal insulating and conducting patches, and we study the effects of the heterogeneous pattern on the global heat exchange, at both low and high Rayleigh numbers. At low Rayleigh numbers, we determine numerically the transition from a regime characterized by the presence of small convective cells localized at the inhomogeneous boundary to the onset of ‘bulk’ convective rolls spanning the entire domain. Such a transition is also controlled analytically in the limit when the boundary pattern length is small compared with the cell vertical size. At higher Rayleigh number, we use numerical simulations based on a lattice Boltzmann method to assess the impact of boundary inhomogeneities on the fully turbulent regime up to $\mathit{Ra} \sim 10^{10}$.

JFM classification

Convection: Convection Instability: Instability Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 742 , 10 March 2014 , pp. 636 - 663
Copyright
© 2014 Cambridge University Press 

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References

Aargard, K. & Carmack, E. C. 1989 The role of sea ice and other fresh water in Arctic circulation. J. Geophys. Res. 94, 14 485–14 498.CrossRefGoogle Scholar
Ahlers, G., Funfschilling, D. & Bodenschatz, E. 2009 Transition in heat transport by turbulent convection at Rayleigh numbers up to $10^{15}$ . New J. Phys. 11, 123001.Google Scholar
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.Google Scholar
Benzi, R., Succi, S. & Vergassola, M. 1992 The lattice Boltzmann equation: theory and applications. Phys. Rep. 222, 145–197.Google Scholar
Benzi, R., Toschi, F. & Tripiccione, R. 1998 On the heat transfer in Rayleigh–Bénard systems. J. Stat. Phys. 93, 901–918.Google Scholar
Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525.Google Scholar
Biferale, L., Perlekar, P., Sbragaglia, M & Toschi, F. 2013 Simulations of boiling systems using a lattice Boltzmann method. Commun. Comput. Phys. 13, 696–705.Google Scholar
Biferale, L. & Procaccia, I. 2005 Anisotropy in turbulent flows and in turbulent transport. Phys. Rep. 414, 43–164.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.Google Scholar
Buick, J. M. & Greated, C. A. 2000 Gravity in a lattice Boltzmann model. Phys. Rev. E 61, 5307–5320.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Chen, S. & Doolen, G. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364.Google Scholar
Chilla, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35, 58–82.Google Scholar
Choudhuri, A. R. 1998 The Physics of Fluids and Plasmas: An Introduction for Astrophysics. Cambridge University Press.Google Scholar
Cieszelski, R. 1998 A case study of Rayleigh–Bénard convection with clouds. Boundary-Layer Meteorol. 88, 211–237.CrossRefGoogle Scholar
Cortet, P.-P., Chiffaudel, A., Daviaud, F. & Dubrulle, B. 2010 Experimental evidences of a phase transition in closed turbulence. Phys. Rev. Lett. 105, 214501.Google Scholar
Freund, G., Pesch, W. & Zimmermann, W. 2011 Rayleigh–Bénard convection in the presence of spatial temperature modulations. J. Fluid Mech. 673, 318–348.CrossRefGoogle Scholar
Gladrow, W. 2000 Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer.Google Scholar
Gonnella, G., Lamura, A. & Sofonea, V. 2007 Lattice Boltzmann simulation of thermal nonideal fluids. Phys. Rev. E 76, 036703.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 2000 Table of Integrals, Series, and Products, 6th edn. Academic Press.Google Scholar
Guillou, L. & Jaupart, C. 1995 On the effect of continents on mantle convection. J. Geophys. Res. 100, 24 217–24 238.CrossRefGoogle Scholar
He, X., Funfschilling, D., Nobach, N., Bodenschatz, E. & Ahlers, G. 2012 Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 024502.Google Scholar
He, X. & Luo, L. S. 1997 Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56, 6811–6817.Google Scholar
He, X., Shan, X. & Doolen, G. 1998 Discrete Boltzmann equation model for nonideal gases. Phys. Rev. E 57, R13–R16.Google Scholar
von der Heydt, A., Grossmann, S. & Lohse, D. 2003 Response maxima in modulated turbulence. Phys. Rev. E 67, 046308.Google Scholar
Holland, M., Bitz, C., Eby, M. & Weaver, A. 2001 The role of ice–ocean interactions in the variability of the North Atlantic thermohailine circulation. J. Clim. 14, 656–675.Google Scholar
Hossain, M. H. & Floryan, J. F. 2013 Instabilities of natural convection in a periodically heated layer. J. Fluid Mech. 733, 33–67.Google Scholar
Jellinek, A. & Lenardic, A. 2009 Effects of spatially varying roof cooling on Rayleigh–Bénard convection in a fluid with a strongly temperature-dependent viscosity. J. Fluid Mech. 692, 109–137.CrossRefGoogle Scholar
Jin, X.-L. & Xia, K.-Q. 2008 An experimental study of kicked thermal turbulence. J. Fluid Mech. 606, 133–151.Google Scholar
Latt, J.2007 Hydrodynamic limit of lattice Boltzmann equations. PhD thesis, University of Geneva.Google Scholar
Lauga, E. & Stone, H. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.Google Scholar
Lenardic, A. & Moresi, L. 2003 Thermal convection below a conducting lid of variable extent: heat flow scalings and two-dimensional, infinite Prandtl number numerical simulations. Phys. Fluids 15, 455–466.Google Scholar
Lenardic, A., Moresi, L., Jellinek, A. M. & Manga, M. 2005 Continental insulation, mantle cooling, and the surface area of oceans and continents. Earth Planet. Sci. Lett. 234, 317–333.Google Scholar
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid. Mech. 42, 335–364.Google Scholar
Marcq, S. & Weiss, J. 2012 Influence of sea ice lead-width distribution on turbulent heat transfer between the ocean and the atmosphere. Cryosphere 6, 143–156.Google Scholar
Martinson, DG. 1990 Evolution of the Southern Ocean winter mixed layer and sea ice: open ocean deep water formation and ventilation. J. Geophys. Res. 95, 11 641–11 654.CrossRefGoogle Scholar
Martys, N., Shan, X. & Chen, H. 1998 Evaluation of the external force term in the discrete Boltzmann equation. Phys. Rev. E 58, 6855–6857.Google Scholar
Philip, J. R. 1972 Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23, 353–370.Google Scholar
Philippi, P. C., Hegele, L. A., Dos Santos, L. O. E. & Surmas, R. 2006 From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models. Phys. Rev. E 73, 056702.CrossRefGoogle Scholar
Prasianakis, N. & Karlin, I. V. 2007 Lattice Boltzmann method for thermal flow simulation on standard lattices. Phys. Rev. E 76, 016702.Google Scholar
Rayleigh, Lord 1916 On the convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 529–546.Google Scholar
Sbragaglia, M., Benzi, R., Biferale, L., Succi, S., Sugiyama, K. & Toschi, F. 2007 Generalized lattice Boltzmann method with multirange pseudopotential. Phys. Rev. E 75, 026702.Google Scholar
Sbragaglia, M. & Prosperetti, A. 2007 A note on the effective slip properties for microchannel flows with ultra-hydrophobic surfaces. Phys. Fluids. 19, 043603.Google Scholar
Scagliarini, A., Biferale, L., Sbragaglia, M., Sugiyama, K. & Toschi, F. 2010 Lattice Boltzmann methods for thermal flows: continuum limit and applications to compressible Rayleigh–Taylor systems. Phys. Fluids 22, 055101.Google Scholar
Seiden, G., Weiss, S., McCoy, J., Pesch, W. & Bodenschatz, E. 2008 Pattern forming system in the presence of different symmetry-breaking mechanisms. Phys. Rev. Lett. 101, 214503.Google Scholar
Shan, X. 1997 Simulation of Rayleigh–Bénard convection using a lattice Boltzmann method. Phys. Rev. E 55, 2780–2788.CrossRefGoogle Scholar
Shan, X. & Chen, H. 1993 Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 47, 1815–1819.Google Scholar
Shan, X. & Doolen, G. 1996 Diffusion in a multicomponent lattice Boltzmann equation model. Phys. Rev. E 54, 3614–3620.Google Scholar
Shan, X., Yuan, X.-F. & Chen, H. 2006 Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation. J. Fluid Mech. 550, 413–441.CrossRefGoogle Scholar
Shishkina, O. & Wagner, C. 2011 Modelling the influence of wall roughness on heat transfer in thermal convection. J. Fluid Mech. 686, 568–582.Google Scholar
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. North-Holland.Google Scholar
Solomatov, V. S. & Moresi, L. N. 2000 Scaling of time-dependent stagnant lid convection: application to small-scale convection on Earth and other terrestrial planets. J. Geophys. Res. 105, 21 795–21 818.Google Scholar
Soloviev, A. & Klinger, B. 2001 Open ocean convection. In Encyclopedia of Ocean Sciences (ed. Steele, J. H., Thorpe, S. A. & Turekian, K. K.), vol. 4, pp. 2015–2022. Academic Press.Google Scholar
Stossel, A., Yang, K. & Kim, S-J. 2002 On the role of sea ice and convection in a global ocean model. J. Phys. Oceanogr. 32, 1194–1208.Google Scholar
Succi, S. 2005 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press.Google Scholar
Tisserand, J. C., Creyssels, M., Gasteuil, Y., Pabiou, H., Gibert, M., Castaing, B. & Chilla, F. 2011 Comparison between rough and smooth plates within the same Rayleigh–Bénard cell. Phys. Fluids 23, 015105.CrossRefGoogle Scholar
Watari, M. 2009 Velocity slip and temperature jump simulations by the three-dimensional thermal finite-difference lattice Boltzmann method. Phys. Rev. E 79, 066706.Google Scholar
Weiss, S., Seiden, G. & Bodenschatz, E. 2011 Pattern formation in spatially forced thermal convection. New J. Phys. 14, 053010.Google Scholar
Wirth, A. & Barnier, B. 2006 Tilted plumes in numerical convection experiments. Ocean Model. 12, 101–111.Google Scholar
Zhang, J. & Tian, F. 2008 A bottom-up approach to non-ideal fluids in the lattice Boltzmann method. Europhys. Lett. 81, 66005.Google Scholar