Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T01:47:41.619Z Has data issue: false hasContentIssue false
  • English
  • Français

Measuring the departures from the Boussinesq approximation in Rayleigh–Bénard convection experiments

Published online by Cambridge University Press:  22 July 2011

H. KURTULDU ,
K. MISCHAIKOW  and
M. F. SCHATZ
H. KURTULDU
Affiliation:
Center for Nonlinear Science and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
K. MISCHAIKOW
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
M. F. SCHATZ*
Affiliation:
Center for Nonlinear Science and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
*
†Email address for correspondence: michael.schatz@physics.gatech.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Algebraic topology (homology) is used to characterize quantitatively non-Oberbeck–Boussinesq (NOB) effects in chaotic Rayleigh–Bénard convection patterns from laboratory experiments. For fixed parameter values, homology analysis yields a set of Betti numbers that can be assigned to hot upflow and, separately, to cold downflow in a convection pattern. An analysis of data acquired under a range of experimental conditions where NOB effects are systematically varied indicates that the difference between time-averaged Betti numbers for hot and cold flows can be used as an order parameter to measure the strength of NOB-induced pattern asymmetries. This homology-based measure not only reveals NOB effects that Fourier methods and measurements of pattern curvature fail to detect, but also permits distinguishing pattern changes caused by modified lateral boundary conditions from NOB pattern changes. These results suggest a new approach to characterizing data from either experiments or simulations where NOB effects are expected to play an important role.

JFM classification

Convection: Buoyancy-driven instability Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 682 , 10 September 2011 , pp. 543 - 557
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

Ahlers, G., Dressel, B., Oh, J. & Pesch, W. 2009 Strong non-Boussinesq effects near the onset of convection in a fluid near its critical point. J. Fluid Mech. 642, 15–48.CrossRefGoogle Scholar
Bodenschatz, E., de Bruyn, J. R., Ahlers, G. & Cannell, D. S. 1991 Transition between patterns in thermal convection. Phys. Rev. Lett. 67, 3078–3081.CrossRefGoogle ScholarPubMed
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.Google Scholar
Boussinesq, J. 1903 Théorie Analytique de la Chaleur, vol. 2. Gauthier-Villars.Google Scholar
de Bruyn, J. R., Bodenschatz, E., Morris, S. W., Trainoff, S. P., Hu, Y., Cannell, D. S. & Ahlers, G. 1996 Apparatus for the study of Rayleigh–Bénard convection in gases under pressure. Rev. Sci. Instrum. 67, 2043–2067.CrossRefGoogle Scholar
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.CrossRefGoogle Scholar
Cakmur, R. V., Egolf, D. A., Plapp, B. B. & Bodenschatz, E. 1997 Bistability and competition of spatiotemporal chaotic and fixed point attractors in Rayleigh–Bénard convection. Phys. Rev. Lett. 79, 1853–1856.Google Scholar
CHomP 2010 Computational Homology Project. Available at: http://chomp.rutgers.edu.Google Scholar
Ciliberto, S., Pampaloni, E. & Peréz-García, C. 1988 Competition between different symmetries in convective patterns. Phys. Rev. Lett. 61, 1198–1201.CrossRefGoogle ScholarPubMed
Getling, A. V. 1998 Rayleigh–Bénard Convection: Structures and Dynamics. World Scientific.Google Scholar
Heutmaker, M. S. & Gollub, J. P. 1987 Wave-vector field of convective flow patterns. Phys. Rev. A 35, 242–260.CrossRefGoogle ScholarPubMed
Hu, Y., Ecke, R. & Ahlers, G. 1993 Convection near threshold for Prandtl numbers near 1. Phys. Rev. E 48, 4399–4413.CrossRefGoogle ScholarPubMed
Hu, Y., Ecke, R. E. & Ahlers, G. 1995 Convection for Prandtl numbers near 1: Dynamics of textured patterns. Phys. Rev. E 51, 3263–3279.Google Scholar
Kaczynski, T., Mischaikow, K. & Mrozek, M. 2004 Computational Homology. Springer.CrossRefGoogle Scholar
Krishan, K., Kurtuldu, H., Schatz, M. F., Madruga, S., Gameiro, M. & Mischaikow, K. 2007 Homology and symmetry breaking in Rayleigh–Bénard convection. Phys. Fluids 19, 117105.Google Scholar
Madruga, S. & Riecke, H. 2007 a Hexagons and spiral defect chaos in non-Boussinesq convection at low Prandtl numbers. Phys. Rev. E 75, 026210.Google Scholar
Madruga, S. & Riecke, H. 2007 b Reentrant and whirling hexagons in non-Boussinesq convection. Eur. Phys. J. Special Topics 146, 279–290.CrossRefGoogle Scholar
Madruga, S., Riecke, H. & Pesch, W. 2006 Re-entrant hexagons in non-Boussinesq convection. J. Fluid Mech. 548, 341–360.CrossRefGoogle Scholar
Mischaikow, K. & Wanner, T. 2007 Probabilistic validation of homology computations for nodal domains. Ann. Appl. Prob. 17, 980–1018.CrossRefGoogle Scholar
Morris, S. W., Bodenschatz, E., Cannell, D. S. & Ahlers, G. 1993 Spiral defect chaos in large aspect ratio Rayleigh–Bénard convection. Phys. Rev. Lett. 71, 2026–2029.CrossRefGoogle ScholarPubMed
Morris, S. W., Bodenschatz, E., Cannell, D. S. & Ahlers, G. 1996 The spatio-temporal structure of spiral-defect chaos. Physica D 97, 164–179.CrossRefGoogle Scholar
Niemela, J. & Sreenivasen, K. R. 2003 Confined turbulent convection. J. Fluid Mech. 481, 355–384.CrossRefGoogle Scholar
Oberbeck, A. 1879 Über die Wärmeleitung der Flüssigkeiten bei der Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem. 7, 271–292.CrossRefGoogle Scholar
Schluter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Semwogerere, D. & Schatz, M. F. 2004 Secondary instabilities of hexagonal patterns in a Bénard–Marangoni convection experiment. Phys. Rev. Lett. 93, 124502.Google Scholar
Xi, H. & Gunton, J. D. 1995 Spatiotemporal chaos in a model of Rayleigh–Bénard convection. Phys. Rev. E 52, 4963–4975.CrossRefGoogle Scholar
Supplementary material: PDF

Kurtuldo et al. supplementary material

Supplementary information

Download Kurtuldo et al. supplementary material(PDF)
PDF 76.6 KB