Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-28T01:55:56.010Z Has data issue: false hasContentIssue false
  • English
  • Français

Rayleigh–Bénard convection in the presence of spatial temperature modulations

Published online by Cambridge University Press:  24 February 2011

G. FREUND ,
W. PESCH  and
W. ZIMMERMANN
G. FREUND
Affiliation:
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany
W. PESCH*
Affiliation:
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany
W. ZIMMERMANN
Affiliation:
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany
*
†Email address for correspondence: werner.pesch@uni-bayreuth.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by recent experiments, we study a rich variation of the familiar Rayleigh–Bénard convection (RBC), where the temperature at the lower boundary varies sinusoidally about a mean value. As usual the Rayleigh number R measures the average temperature gradient, while the additional spatial modulation is characterized by a (small) amplitude δm and a wavevector qm. Our analysis relies on precise numerical solutions of suitably adapted Oberbeck–Boussinesq equations (OBE). In the absence of forcing (δm = 0), convection rolls with wavenumber qc bifurcate only for R above the critical Rayleigh number Rc. In contrast, for δm≠0, convection is unavoidable for any finite R; in the most simple case in the form of ‘forced rolls’ with wavevector qm. According to our first comprehensive stability diagram of these forced rolls in the qm – R plane, they develop instabilities against resonant oblique modes at R ≲ Rc in a wide range of qm/qc. Only for qm in the vicinity of qc, the forced rolls remain stable up to fairly large R > Rc. Direct numerical simulations of the OBE support and extend the findings of the stability analysis. Moreover, we are in line with the experimental results and also with some earlier theoretical results on this problem, based on asymptotic expansions in the limit δm → 0 and R → Rc. It is satisfying that in many cases the numerical results can be directly interpreted in terms of suitably constructed amplitude and generalized Swift–Hohenberg equations.

JFM classification

Instability: Nonlinear instability Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 673 , 25 April 2011 , pp. 318 - 348
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Buka, A., Dressel, B., Kramer, L. & Pesch, W. 2004 Direct transition to electroconvection in a homeotropic nematic liquid crystal. Chaos 14, 793–802.CrossRefGoogle Scholar
Busse, F. H. & Clever, R. M. 1979 Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319–335.CrossRefGoogle Scholar
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305–320.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chiam, K. H., Paul, M. R., Cross, M. C. & Greenside, H. S. 2003 Mean flow and spiral defect chaos in Rayleigh–Bénard convection. Phys. Rev. E 67, 056206.CrossRefGoogle ScholarPubMed
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625–645.CrossRefGoogle Scholar
Coullet, P. 1986 Commensurate–incommensurate transition in nonequilibrium systems. Phys. Rev. Lett. 56, 724–727.CrossRefGoogle ScholarPubMed
Coullet, P. & Huerre, P. 1986 Resonance and phase solitons in spatially forced thermal convection. Physica D 23, 27–44.CrossRefGoogle Scholar
Cross, M. C. 1980 Derivation of the amplitude equation at the Rayleigh–Bénard instability. Phys. Fluids 23, 1727–1731.CrossRefGoogle Scholar
Cross, M. C. & Greenside, H. 2009 Pattern Formation and Dynamics in Nonequilibrium Systems. Cambridge University Press.CrossRefGoogle Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
Decker, W. & Pesch, W. 1994 Order parameter and amplitude equations for the Rayleigh–Bénard convection. J. Phys. II 4, 419–438.Google Scholar
Hall, P. & Walton, I. C. 1978 The smooth transition to a convective regime in a two-dimensional box. Proc. R. Soc. Lond. A 358, 199–221.Google Scholar
Henriot, M., Burguete, J. & Ribotta, R. 2003 Entrainment of a spatially extended nonlinear structure under selective forcing. Phys. Rev. Lett. 91, 104501.CrossRefGoogle ScholarPubMed
Kelly, R. E. 1993 Thermal convection in fluid layers with nonuniform thickness or surface temperature. In IUTAM Symp. on Nonlinear Instability of Nonparallel Flows (ed. Lin, S. P., Phillips, W. R. C. & Valentine, D. T.), pp. 1–12. Springer.Google Scholar
Kelly, R. E. & Pal, D. 1976 Thermal convection induced between non-uniformly heated horizontal surfaces. In Proc. 1976 Heat Transfer and Fluid Mechanics Institute (ed. McKillop, A. A., Baughn, J. W. & Dwyer, H. A.), pp. 1–17. Stanford University Press.Google Scholar
Kelly, R. E. & Pal, D. 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid Mech. 86, 433–456.CrossRefGoogle Scholar
Lowe, M. & Gollub, J. P. 1985 a Pattern selection near the onset of convection: the Eckhaus instability. Phys. Rev. Lett. 55, 2575–2578.CrossRefGoogle ScholarPubMed
Lowe, M. & Gollub, J. P. 1985 b Solitons and the commensurate–incommensurate transition in a convecting nematic fluid. Phys. Rev. A 31, 3893–3897.CrossRefGoogle Scholar
Manneville, P. & Piquemal, J. M. 1983 Zigzag instability and axisymmetric rolls in Rayleigh–Bénard convection: the effects of curvature. Phys. Rev. A 28, 1774–1790.CrossRefGoogle Scholar
Manor, R., Hagberg, A. & Meron, E. 2008 Wavenumber locking in spatially forced pattern-forming systems. Europhys. Lett. 83, 10005.CrossRefGoogle Scholar
Manor, R., Hagberg, A. & Meron, E. 2009 Wavenumber locking and pattern formation in spatially forced systems. New J. Phys. 11, 063016.CrossRefGoogle Scholar
McCoy, J. 2008 Pattern formation due to spatially periodic forcing. PhD thesis, Cornell University.Google Scholar
McCoy, J. H., Brunner, W., Pesch, W. & Bodenschatz, E. 2008 Self-organization of topological defects due to applied constraints. Phys. Rev. Lett. 101, 254102.CrossRefGoogle ScholarPubMed
Obé, M. & Khayat, R. E. 2010 Spatially modulated thermal convection. Intl J. Numer. Meth. Heat Fluid Flow 20, 17–36.CrossRefGoogle Scholar
Pal, D. & Kelly, R. E. 1979 Three-dimensional thermal convection produced by two-dimensional thermal forcing. In Proc. Joint ASME/AIChE 18th Natl Heat Transfer Conf., San Diego, California, 1979, pp. 1–8. ASME 79-HT-109.Google Scholar
Pesch, W. 1996 Complex spatiotemporal convection patterns. Chaos 6, 348–357.CrossRefGoogle ScholarPubMed
Plaut, E. & Pesch, W. 1999 Extended weakly nonlinear theory of planar nematic convection. Phys. Rev. E 59, 1747–1769.CrossRefGoogle Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.CrossRefGoogle Scholar
Schmitz, R. & Zimmermann, W. 1996 Spatially periodic modulated Rayleigh–Bénard convection. Phys. Rev. E 53, 5993–6011.CrossRefGoogle ScholarPubMed
Seiden, G., Weiss, S. & Bodenschatz, E. 2009 Superlattice patterns in forced thermal convection. Chaos 19, 041102.CrossRefGoogle ScholarPubMed
Seiden, G., Weiss, S., McCoy, J. H., Pesch, W. & Bodenschatz, E. 2008 Pattern forming system in the presence of different symmetry-breaking mechanisms. Phys. Rev. Lett. 101, 214503.CrossRefGoogle ScholarPubMed
Semwogerere, D. & Schatz, M. F. 2002 Evolution of hexagonal patterns from controlled initial conditions in a Bénard–Marangoni convection experiment. Phys. Rev. Lett. 88, 054501.CrossRefGoogle Scholar
Trainoff, S. P. & Cannell, D. S. 2002 Physical optics treatment of the shadowgraph. Phys. Fluids 14, 1340–1363.CrossRefGoogle Scholar
Vozovoi, L. P. & Nepomnyashchy, A. A. 1974 Convection in a horizontal layer in the presence of spatial modulation of temperature at the boundaries. Gidrodinamika 7, 105–117 (published by the Governmental Pedagogical Institute of the University of Perm, Russia).Google Scholar
Weiss, S. 2009 On forcing in thermal convection experiments. PhD thesis, Universität Göttingen.Google Scholar
Xi, H. W., Gunton, J. D. & Vinals, J. 1993 Spiral defect chaos in a model of Rayleigh–Bénard convection. Phys. Rev. Lett. 71 (13), 2030–2033.CrossRefGoogle Scholar
Zimmermann, W., Ogawa, A., Kai, S., Kawasaki, K. & Kawakatsu, T. 1993 Wavelength competition in convective systems. Europhys. Lett. 24, 217–222.CrossRefGoogle Scholar