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A computational study of Rayleigh–Bénard convection. Part 1. Rayleigh-number scaling

Published online by Cambridge University Press:  26 April 2006

Anil E. Deane  and
Lawrence Sirovich
Anil E. Deane
Affiliation:
Center for Fluid Mechanics and The Division of Applied Mathematics, Brown University, Providence, RI 02912, USA Present address: Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
Lawrence Sirovich
Affiliation:
Center for Fluid Mechanics and The Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
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Abstract

A parametric study is made of chaotic Rayleigh–Bénard convection over moderate Rayleigh numbers. As a basis for comparison over the Rayleigh number (Ra) range we consider mean quantities, r.m.s. fluctuations, Reynolds number, probability distributions and power spectra. As a further means of investigating the flow we use the Karhunen–Loéve procedure (empirical eigenfunctions, proper orthogonal decomposition). Thus, we also examine the variation in eigenfunctions with.Ra. This in turn provides an analytical basis for describing the manner in which the chaos is enriched both temporarily and spatially as Ra increases. As Ra decreases, the significant mode count decreases but, in addition, the eigenfunctions tend more nearly to the eigenfunctions of linearized theory. As part of this parametric study a variety of scaling properties are investigated. For example it is found that the empirical eigenfunctions themselves show a simple scaling in Ra.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 231 - 250
Copyright
© 1991 Cambridge University Press

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References

Ash, R. B. & Gardner, M. F., 1975 Topics in Stochastic Processes. Academic.
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E., 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115.Google Scholar
Balachandar, S., Maxey, M. R. & Sirovich, L., 1989 Numerical simulation of high Rayleigh number convection. J. Sci. Comput. 4, 219.Google Scholar
Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J.-M. 1980 Lyapunov characteristic exponents for smooth dynamical systems and Hamiltonian systems: A method for computing all of them. Pts. 1 and 2. Meccanica 15, 9–30.Google Scholar
Busse, F. H.: 1985 In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). Springer.
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libschaber, A., Thomae, S., Wu, X-Z., Zaleski, S. & Zanetti, G., 1989 Scaling of hard thermal turbulence in Rayleigh-Bénard convection. J. Fluid Mech. 204, 1.Google Scholar
Catton, I.: 1966 Natural convection in horizontal liquid layers. Phys. Fluids 19, 252.Google Scholar
Chandrasekharr, S.: 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Constantin, P., Foiaş, C., Manley, O. P. & Temam, R., 1985 Determining modes and fractal dimension of turbulent flows. J. Fluid Mech. 150, 427.Google Scholar
Drazin, P. G. & Reid, W. H., 1981 Hydrodynamic Stability. Cambridge University Press.
Feigenbaum, M. J.: 1978 Quantitative universality for a class of nonlinear transformations. J. Statist. Phys. 19, 25.Google Scholar
Fenstermacher, P. R., Swinney, H. L. & Gollub, J. P., 1979 Dynamical instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94, 103.Google Scholar
Foiaş, C., Manley, O. & Temam, R., 1987 Nonlin. Anal. Theory, Meth. Applies. 11, 939–967.
Garon, A. M. & Goldstein, R. J., 1973 Velocity and heat transfer measurements in thermal convection. Phys. Fluids 16, 1818–1825.Google Scholar
Gilbert, N. & Kleiser, L., 1987 Low-resolution simulations of transitional and turbulent channel flow. In Proc. Intl Conf. on Fluid Mechanics. Peking University Press, Beijing, China.
Glauser, M. N., Lieb, S. J. & George, N. K., 1987 Coherent structure in the axisymmetric jet mixing layer. In Turbulent Shear Flows 5 (ed. F. Durst et al.). Springer.
Grenander, U. & Szego, G., 1958 Toeplitz Forms and Their Application. University of California Press, Berkeley.
Herring, J. H. & Wyngaard, J., 1987 Convection with a simple chemically reactive passive scalar. 10.39–10.43 In Turbulent Shear Flows 5 (ed. F. Dust et al.), p. 328. Springer.
Kaplan, J. & Yorke, J., 1979 Chaotic behavior in multi-dimensional difference equations. In Functional Differential Equations and the Approximation of Fixed Points (ed. H. O. Peitgen & H. O. Walther). In Lecture Notes in Mathematics, vol. 730, p. 228. Springer.
Keefe, L. & Moin, P., 1987 Bull. Am. Phys. Soc. II 32, 2026.
Landau, L. D.: 1944 Dokl. Akad. Nauk. 8SSR 44, 339.
Lorenz, E. N.: 1959 Prospects for statistical weather forecasting. Final Rep., Statistical Forecasting Project, MIT.Google Scholar
Lorenz, E. N.: 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130.Google Scholar
Lorenz, E. N.: 1980 Noisy periodicity and reverse bifurcation. In Nonlinear Dynamics (ed. R. H. G. Helleman). NY Acad. Sci.
Lumley, J. L.: 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. A. M. Yaglom & V. I. Tatarski), pp. 166–178. Moskow: Nauka.
Lumley, J. L.: 1970 Stochastic Tools in Turbulence. Academic.
Lumley, J. L.: 1981 Coherent structures in turbulence. In Transition and Turbulence (ed. R. E. Meyer), pp. 215–242. Academic.
Malkus, W. V. R.: 1954a Discrete transitions in turbulent convection. Proc. R. Soc. Lond. 225, 185.Google Scholar
Malkus, W. V. R.: 1954b The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. 225, 198.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225.Google Scholar
Malraison, D., Atten, P., Bergé, P. & Dubois, M. 1983 Dimension d'attracteurs étranges: une détermination expérimentale en régime chaotique de deux systems convectifs. CR Acad. Sci. Paris C297, 209.Google Scholar
Massaguer, J. M. & Mercader, I., 1988 Instability of swirl in low-Prandtl-number thermal convection. J. Fluid Mech. 189, 367.Google Scholar
Moin, P.: 1984 Probing turbulence via large eddy simulation. AIAA Paper-84–0174.Google Scholar
Moin, P. & Moseh, R. D., 1989 Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech. 200, 471.Google Scholar
Monin, A. S. & Yaglom, A. M., 1975 Statistical Fluid Mechanics Vol. II. MIT Press.
Pillow, A. F.: 1952 The free convection cell in two dimensions. Rep. Aero. Res. Lab. Melbourne A79.Google Scholar
Preisendorfer, R. W.: 1988 Principal component analysis in meteorology and oceanography. In Meteorology and Oceanography. Elsevier.
Ruelle, D. & Takens, F., 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167.Google Scholar
Shimada, I. & Nagashima, T., 1979 A numerical approach to the ergodic problem of dissipative dynamical systems. Prog. Theor. Phys. 61, 1605.Google Scholar
Sirovich, L.: 1987a Turbulence and the dynamics of coherent structures, Pt. I: Coherent Structures. Q. Appl. Maths XLV, 561.Google Scholar
Sirovich, L.: 1987b Turbulence and the dynamics of coherent structures, Pt. II. Symmetries and transformations. Q. Appl. Maths XLV, 573.Google Scholar
Sirovich, L., Balachandar, S. & Maxey, M. R., 1989a Simulations of turbulent thermal convection. Phys. Fluids A 1, 1911–1914.Google Scholar
Sirovich, L. & Deane, A. E., 1991 A computational study of Rayleigh-Bénard convection. Part 2. Dimension considerations. J. Fluid Mech. 222, 251 (referred to herein as II).Google Scholar
Sirovich, L., Maxey, M. & Tarman, H., 1989b Analysis of turbulent thermal convection. In Sixth Symposium on Turbulent Shear Flow, Toulouse, France, p. 68. Springer.
Sirovich, L. & Rodriguez, J. D., 1987 Coherent structures and chaos: A model problem. Phys. Lett A 120, 211–214.Google Scholar
Spiegel, E. A.: 1962 On the Malkus Theory of Turbulence, p. 181. Centre National de la Researche Scientifique, Paris.
Tarman, H.: 1989 Analysis of turbulent thermal convection. Ph.D. thesis, Brown University.
Tarman, H. & Sirovich, L., 1990 Low dimensional dynamics for the turbulent convection problem (to appear).
Theodorsen, T.: 1952 Mechanism of turbulence. In Proc. 2nd Midwestern Conf. on Fluid Mech., Ohio State Univ., Columbus, Ohio.
Threlfall, D. C.: 1975 Free convection in low-temperature gaseous helium. J. Fluid Mech. 67, 17.Google Scholar
Toomre, J., Gough, D. O. & Spiegel, E. A., 1977 Numerical solutions of single-mode convection equations. J. Fluid Mech. 79, 1.Google Scholar
Toomre, J., Gough, D. O. & Spiegel, E. A., 1982 Time-dependent solutions of multimode convection equations. J. Fluid Mech. 125, 99.Google Scholar
Townsend, A. A.: 1956 The Structure of Turbulent Shear flow. Cambridge University Press.
Turner, J. S.: 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Wolf, A., Swiftr, J. B., Swinney, H. L. & Vastano, J. A., 1985 Determining Lyapunov exponents from a time series. Physica 16D, 285.Google Scholar