- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Convectons in unbalanced natural doubly diffusive convection

J. Tumelty, C. Beaume*, and A. M. Rucklidge†

  • *Contact author: c.m.l.beaume@leeds.ac.uk
  • †Contact author: a.m.rucklidge@leeds.ac.uk
PDF

Phys. Rev. Fluids 10, 044401 – Published 10 April, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.044401

Abstract

Fluids subject to both thermal and compositional variations can undergo doubly diffusive convection when these properties both affect the fluid density and diffuse at different rates. In natural doubly diffusive convection, the gradients of temperature and salinity are aligned with each other and orthogonal to gravity. The resulting buoyancy-driven flows are known to lead to the formation of a variety of patterns, including spatially localized states of convection surrounded by quiescent fluid. These states are known as convectons. Localized pattern formation in natural doubly diffusive convection has been studied under a specific balance where the effects of temperature and salinity changes are opposite but of equal intensity on the fluid density. In this case, a steady conduction state exists and convectons bifurcate from it. The aforementioned buoyancy balance underpins our knowledge of this pattern formation but it is an ideal case that can hardly be met experimentally or in nature. This article addresses how localized pattern formation in natural doubly diffusive convection is affected by departures from the balanced case. In particular, the absence of a conduction state leads to the unfolding of the bifurcations to convectons. In thermally dominated regimes, the background flow promotes localized states with convection rolls attached to the end walls, known as anticonvectons, and the existence of these states is found to be related to the emergence of convectons. Convectons are found to exist over a range of buoyancy ratio values, disappearing at saddle nodes of anticonvecton branches in the thermally dominated regime and smoothly transitioning into domain-filling states in the solutally dominated regime. These observations extend the scope of our understanding of localized pattern formation in fluid systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (49)

  1. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, Garden City, 1981).
  2. E. L. Koschmieder, Bénard Cells and Taylor Vortices (Cambridge University Press, Cambridge, 1993).
  3. M. Golubitsky, J. W. Swift, and E. Knobloch, Symmetries and pattern selection in Rayleigh–Bénard convection, Physica D 10, 249 (1984).
  4. J. Pallares, M. P. Arroyo, F. X. Grau, and F. Giralt, Experimental laminar Rayleigh–Bénard convection in a cubical cavity at moderate Rayleigh and Prandtl numbers, Exp. Fluids 31, 208 (2001).
  5. D. Puigjaner, J. Herrero, F. Giralt, and C. Simó, Bifurcation analysis of multiple steady flow patterns for Rayleigh–Bénard convection in a cubical cavity at Pr=130, Phys. Rev. E 73, 046304 (2006).
  6. K. Borońska and L. S. Tuckerman, Extreme multiplicity in cylindrical Rayleigh–Bénard convection. I. Time dependence and oscillations, Phys. Rev. E 81, 036320 (2010).
  7. K. Borońska, L. S. Tuckerman, L. M. Witkowski, M. C. Navarro, D. Barkley, and R. Bessaih, Extreme multiplicity in cylindrical Rayleigh–Bénard convection. II. Bifurcation diagram and symmetry classification, Phys. Rev. E 81, 036321 (2010).
  8. A. Alonso, O. Batiste, A. Meseguer, and I. Mercader, Complex dynamical states in binary mixture convection with weak negative Soret coupling, Phys. Rev. E 75, 026310 (2007).
  9. M. Stern, Convective instability of salt fingers, J. Fluid Mech. 35, 209 (1969).
  10. D. E. Kelley, H. J. S. Fernando, A. E. Gargett, J. Tanny, and E. Özsoy, The diffusive regime of double diffusive convection, Prog. Oceanogr. 56, 461 (2003).
  11. H. E. Huppert and J. S. Turner, Double-diffusive convection, J. Fluid Mech. 106, 299 (1981).
  12. K. Ghorayeb and A. Mojtabi, Double diffusive convection in a vertical rectangular cavity, Phys. Fluids 9, 2339 (1997).
  13. S. Xin, P. Le Quéré, and L. S. Tuckerman, Bifurcation analysis of double-diffusive convection with opposing horizontal thermal and solutal gradients, Phys. Fluids 10, 850 (1998).
  14. C. Beaume, A. M. Rucklidge, and J. Tumelty, Near-onset dynamics in natural doubly diffusive convection, J. Fluid Mech. 934, A42 (2022).
  15. A. Bergeon and E. Knobloch, Natural doubly diffusive convection in three-dimensional enclosures, Phys. Fluids 14, 3233 (2002).
  16. J. Burke and E. Knobloch, Localized states in the generalized Swift–Hohenberg equation, Phys. Rev. E 73, 056211 (2006).
  17. P. D. Woods and A. R. Champneys, Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate Hamiltonian–Hopf bifurcation, Physica D 129, 147 (1999).
  18. A. Bergeon and E. Knobloch, Spatially localized states in natural doubly diffusive convection, Phys. Fluids 20, 034102 (2008).
  19. C. Beaume, A. Bergeon, and E. Knobloch, Convectons and secondary snaking in three-dimensional natural doubly diffusive convection, Phys. Fluids 25, 024105 (2013).
  20. C. Beaume, A. Bergeon, and E. Knobloch, Three-dimensional doubly diffusive convectons: Instability and transition to complex dynamics, J. Fluid Mech. 840, 74 (2018).
  21. C. Beaume, E. Knobloch, and A. Bergeon, Nonsnaking doubly diffusive convectons and the twist instability, Phys. Fluids 25, 114102 (2013).
  22. C. Beaume, Transition to doubly diffusive chaos, Phys. Rev. Fluids 5, 103903 (2020).
  23. I. Mercader, O. Batiste, A. Alonso, and E. Knobloch, Localized pinning states in closed containers: Homoclinic snaking without bistability, Phys. Rev. E 80, 025201(R) (2009).
  24. I. Mercader, O. Batiste, A. Alonso, and E. Knobloch, Convectons, anticonvectons and multiconvectons in binary fluid convection, J. Fluid Mech. 667, 586 (2011).
  25. T. Watanabe, K. Toyabe, M. Iima, and Y. Nishiura, Time-periodic traveling solutions of localized convection cells in binary fluid mixture, Theor. Appl. 59, 211 (2010).
  26. T. Watanabe, M. Iima, and Y. Nishiura, Spontaneous formation of travelling localized structures and their asymptotic behaviour in binary fluid convection, J. Fluid Mech. 712, 219 (2012).
  27. S. Blanchflower, Magnetohydrodynamic convectons, Phys. Lett. A 261, 74 (1999).
  28. P. Assemat, A. Bergeon, and E. Knobloch, Spatially localized states in Marangoni convection in binary mixtures, Fluid Dyn. Res. 40, 852 (2008).
  29. D. Lo Jacono, A. Bergeon, and E. Knobloch, Spatially localized binary fluid convection in a porous medium, Phys. Fluids 22, 073601 (2010).
  30. C. Beaume, A. Bergeon, H.-C. Kao, and E. Knobloch, Convectons in a rotating fluid layer, J. Fluid Mech. 717, 417 (2013).
  31. J. Tumelty, C. Beaume, and A. M. Rucklidge, Toward convectons in the supercritical regime: Homoclinic snaking in natural doubly diffusive convection, SIAM J. Appl. Dyn. Syst. 22, 1710 (2023).
  32. C. Beaume, A. Bergeon, and E. Knobloch, Homoclinic snaking of localized states in doubly diffusive convection, Phys. Fluids 23, 094102 (2011).
  33. C. K. Mamun and L. S. Tuckerman, Asymmetry and Hopf bifurcation in spherical Couette flow, Phys. Fluids 7, 80 (1995).
  34. C. Beaume, Adaptive Stokes preconditioning for steady incompressible flows, Commun. Comput. Phys. 22, 494 (2017).
  35. S. M. Houghton and E. Knobloch, Homoclinic snaking in bounded domains, Phys. Rev. E 80, 026210 (2009).
  36. A. Bergeon, J. Burke, E. Knobloch, and I. Mercader, Eckhaus instability and homoclinic snaking, Phys. Rev. E 78, 046201 (2008).
  37. G. K. Batchelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures, Q. Appl. Math. 12, 209 (1954).
  38. A. Bahloul, N. Boutana, and P. Vasseur, Double-diffusive and Soret-induced convection in a shallow horizontal porous layer, J. Fluid Mech. 491, 325 (2003).
  39. J. Tumelty, Localised states in natural doubly diffusive convection, Ph.D. thesis, University of Leeds, 2022, https://etheses.whiterose.ac.uk/id/eprint/31748/.
  40. C. M. Vest and V. S. Arpaci, Stability of natural convection in a vertical slot, J. Fluid Mech. 36, 1 (1969).
  41. A. V. Yulin and A. R. Champneys, Discrete snaking: multiple cavity solitons in saturable media, SIAM J. Appl. Dyn. Syst. 9, 391 (2010).
  42. A. R. Champneys, E. Knobloch, Y.-P. Ma, and T. Wagenknecht, Homoclinic snakes bounded by a saddle-center periodic orbit, SIAM J. Appl. Dyn. Syst. 11, 1583 (2012).
  43. D. Lo Jacono, A. Bergeon, and E. Knobloch, Localized traveling pulses in natural doubly diffusive convection, Phys. Rev. Fluids 2, 093501 (2017).
  44. I. Mercader, O. Batiste, A. Alonso, and E. Knobloch, Effect of small inclination on binary convection in elongated rectangular cells, Phys. Rev. E 99, 023113 (2019).
  45. J. Lee, M. T. Hyun, and Y. S. Kang, Confined natural convection due to lateral heating in a stably stratified solution, Int. J. Heat Mass Transf. 33, 869 (1990).
  46. J. W. Lee and J. M. Hyun, Double diffusive convection in a cavity under a vertical solutal gradient and a horizontal temperature gradient, Int. J. Heat Mass Transf. 34, 2423 (1991).
  47. H. A. Dijkstra and E. J. Kranenborg, A bifurcation study of double diffusive flows in a laterally heated stably stratified liquid layer, Int. J. Heat Mass Transf. 39, 2699 (1996).
  48. S. Thangam, A. Zebib, and C. F. Chen, Double-diffusive convection in an inclined fluid layer, J. Fluid Mech. 116, 363 (1982).
  49. G. Bardan, A. Bergeon, E. Knobloch, and A. Mojtabi, Nonlinear doubly diffusive convection in vertical enclosures, Physica D 138, 91 (2000).

Outline

Information

Phys. Rev. Fluids 10, 044401– Published 10 April, 2025
  • Received 7 November 2024
  • Accepted 12 March 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation