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Numerical simulation of thermal convection in a two-dimensional finite box

Published online by Cambridge University Press:  26 April 2006

Isaac Goldhirsch ,
Richard B. Pelz  and
Steven A. Orszag
Isaac Goldhirsch
Affiliation:
Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University Ramat-Aviv, Tel-Aviv 69978, Israel
Richard B. Pelz
Affiliation:
Department of Mechanical and Aerospace Engineering, Rutgers University, PO Box 909, Piscataway, NJ 08855, USA
Steven A. Orszag
Affiliation:
Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
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Abstract

The problems of dynamical onset of convection, textural transitions and chaotic dynamics in a two-dimensional, rectangular Rayleigh-Bénard system have been investigated using well-resolved, pseudo-spectral simulations. All boundary conditions are taken to be no-slip. It is shown that the process of creating the temperature gradient in the system, is responsible for roll creation at the side boundaries. These rolls either induce new rolls or move into the interior of the cell, depending on the rate of heating. Complicated flow patterns and textural transitions are observed in both non-chaotic and chaotic flow regimes. Multistability is frequently observed. Intermediate-Prandtl-number fluids (e.g. 0.71) have a quasiperiodic time dependence up to Rayleigh numbers of order 106. When the Prandtl number is raised to 6.8, one observes aperiodic (chaotic) flows of non-integer dimension. In this case roll merging and separation is observed to be an important feature of the dynamics. In some cases corner rolls are observed to migrate into the interior of the cell and to grow into regular rolls; the large rolls may shrink and retreat into corners. The basic flow patterns observed do not change qualitatively when the chaotic regime is entered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 199 , February 1989 , pp. 1 - 28
Copyright
© 1989 Cambridge University Press

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References

Ahlers, G. & Behringer, R. P., 1978 The Rayleigh–Bénard instability and the evolution of turbulence. Prog. Theor. Phys. Supp. 64, 186–201.Google Scholar
Ahlers, G., Cross, M. C., Hohenberg, P. C. & Safran, S., 1981 The amplitude equation near the convective threshold: application to time-dependent heating experiments. J. Fluid Mech. 110, 297–334.Google Scholar
Bolton, E. W., Busse, F. H. & Clever, R. M., 1986 Oscillatory instabilities of convective rolls at intermediate Prandtl numbers. J. Fluid Mech. 164, 469–485.Google Scholar
Boussinesq, J.: 1903 Théorie Analytique de Chaleur, vol. 2, p. 172. Paris: Gauthier-Villars.
Busse, F. F.: 1972 The oscillatory instability of convection rolls in a low Prandtl number fluid. J. Fluid Mech. 52, 97–112.Google Scholar
Busse, F. H.: 1978 Nonlinear properties of thermal convection. Rep. Prog. Phys. 41, 1929.Google 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.Google Scholar
Busse, F. H. & Frick, H., 1985 Square pattern convection in fluids with strongly temperature dependent viscosity. J. Fluid Mech. 150, 451–465.Google Scholar
Chandrasekhar, S.: 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Ciliberto, S. & Gollub, J. P., 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922.Google Scholar
Cross, M. C.: 1982a Ingredients of a theory of convective textures close to onset. Phys. Rev. A A25, 1065–1076.Google Scholar
Cross, M. C.: 1982b Wave-number selection by soft boundaries near threshold. Phys. Rev. A A29, 391–392.Google Scholar
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D., 1980 Effect of distant sidewalls on wave-number selection in Rayleigh–Bénard convection. Phys. Rev. Lett. 45, 898–901.Google Scholar
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D., 1983a Phase-winding solutions in a finite container above the convective threshold. J. Fluid Mech. 127, 155–183.Google Scholar
Cross, M. C., Hohenberg, P. C. & Lücke, M. 1983b Forcing of convection due to time-dependent heating near threshold. J. Fluid Mech. 136, 155–183.Google Scholar
Cross, M. C. & Newell, A. C., 1984 Convective patterns in large aspect ratio systems. Physica 10D, 299–328.Google Scholar
Curry, J. H.: 1978 A generalized Lorenz system. Comm. Math. Phys. 60, 193–204.Google Scholar
Curry, J. C., Herring, J. R., Loncaric, J. & Orszag, S. A., 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38.Google Scholar
Cvitanovic, P.: 1984 Universality in Chaos. Bristol: Adam Hilger.
Frick, H. & Müller, U. 1983 Oscillatory Hele-Shaw convection. J. Fluid Mech. 126, 521–532.Google Scholar
Gershuni, G. Z. & Zhukhovitskii 1976 Convective Stability of Incompressible Fluids. Jerusalem: Keter.
Giglio, M., Musazzi, S. & Perini, U., 1981 Transition to chaotic behaviour via a reproducible sequence of period doubling bifurcations. Phys. Rev. Lett. 47, 243–246.Google Scholar
Gollub, J. P. & Benson, S. V., 1980 Many routes to turbulent convection. J. Fluid Mech. 100, 449–470.Google Scholar
Gollub, J. P. & McCarriar, A. R., 1982 Convection patterns in Fourier space. Phys. Rev. A A26, 3470–3476.Google Scholar
Gollub, J. P., McCarriar, A. R. & Steinman, J. F., 1982 Convective pattern evolution and secondary instabilities. J. Fluid Mech. 125, 259–281.Google Scholar
Gotlieb, D. & Orszag, S. A., 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia.
Grassberger, P. & Procaccia, I., 1983 Characterization of strange attractors. Phys. Rev. Lett. 50, 346.Google Scholar
Greenside, H. S., Ahlers, G., Hohenberg, P. C. & Walden, R. W., 1982 A simple stochastic model for the onset of turbulence in Rayleigh–Bénard convection. Physica 5D, 322–334.Google Scholar
Greenside, H. S., Coughran Jr., W. M. & Schryer, N. L. 1982 Nonlinear pattern formation near the onset of Rayleigh–Bénard convection. Phys. Rev. Lett. 49, 726–729.Google Scholar
Grötzbach, G.: 1982 Direct numerical simulation of laminar and turbulent Bénard convection. J. Fluid Mech. 119, 27–53.Google Scholar
Grötzbach, G.: 1983 Spatial resolution requirement for direct numerical simulation of Rayleigh–Bénard convection. J. Comp. Phys. 49, 241–264.Google Scholar
Guckenheimer, J. & Holmes, P., 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Haidvogel, D. B. & Zang, T., 1979 The accurate solution of Poisson's equation by expansion in Chebyshev polynomials. J. Comp. Phys. 30, 167–180.Google Scholar
Knobloch, E., Moore, D. R., Toomre, J. & Weiss, N. O., 1986 Transitions to chaos in two-dimensional double-diffusive convection. J. Fluid Mech. 166, 409–448.Google Scholar
Korpela, S. A., Gozum, D. & Baxi, C. B., 1973 Intl J. Heat Mass Transfer 16, 1683.
Koster, J. N. & Müller, U. 1984 Oscillatory convection in vertical slots. J. Fluid Mech. 139, 363–390.Google Scholar
Libchaber, A. & Maurer, J., 1980 Une expérience de Rayleigh–Bénard de géométric réduite; multiplication, accrochage et démultiplication de fréquences. J. Phys. (Paris) 41 Colloque C3, 51–56.Google Scholar
Lorenz, E. N.: 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130.Google Scholar
McLaughlin, J. B. & Orszag, S. A., 1982 Transition from periodic to chaotic thermal convection. J. Fluid Mech. 122, 123–142.Google Scholar
Malraison, B., Atten, P., Berge, P. & Dubois, M., 1983 Dimension of strange attractors: an experimental determination for the chaotic regime of two convective systems. J. Physique Lett. 44, L-897–L-902.Google Scholar
Moore, D. R. & Weiss, N. O., 1973 Two-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 58, 289–312.Google Scholar
Newell, A. C. & Whitehead, J. A., 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–303.Google Scholar
Orszag, S. A., Israeli, M. & Deville, M. O., 1986 Boundary conditions for incompressible flows. J. Sci. Comp. 1, 75–111.Google Scholar
Orszag, S. A. & Patera, P. T., 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Schlüter, A., Lortz, D. & Busse, F., 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 35, 129–144.Google Scholar
Segel, L.: 1969 Distant side walls cause slow modulation of cellular convection. J. Fluid Mech. 38, 203–224.Google Scholar
Siggia, E. D. & Zippelius, A., 1981a Dynamics of defects in Rayleigh–Bénard convection. Phys. Rev. A A24, 1036–1049.Google Scholar
Siggia, E. D. & Zippelius, A., 1981b Pattern selection in Rayleigh–Bénard convection near threshold. Phys. Rev. Lett. 47, 835.Google Scholar
Takens, F.: 1981a In Proc. Symp. on Dynamical Systems and Turbulence, University of Warwick, 1979–80 (ed. D. A. Rand & L. S. Young). Springer.
Takens, F.: 1981b Invariants related to dimension and entropy. Proc. 13th Brazilian Colloquium on Mathematics (unpublished).Google Scholar
Yahata, H.: 1984 Onset of chaos in the Rayleigh–Bénard convection. Prog. Theor. Phys. Supp. 79, 26–74.Google Scholar
Zippelius, A. & Siggia, E. D., 1982 Disappearance of stable convection between free-slip boundaries. Phys. Rev. A A26, 1788–1790.Google Scholar