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Many routes to turbulent convection

Published online by Cambridge University Press:  19 April 2006

J. P. Gollub  and
S. V. Benson
J. P. Gollub
Affiliation:
Physics Department, Haverford College, Haverford, Pa 19041, U.S.A.
S. V. Benson
Affiliation:
Physics Department, Haverford College, Haverford, Pa 19041, U.S.A. Present address: Physics Department, Stanford University, Stanford, Ca 94305.
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Abstract

Using automated laser-Doppler methods we have identified four distinct sequences of instabilities leading to turbulent convection at low Prandtl number (2·5–5·0), in fluid layers of small horizontal extent. Contour maps of the structure of the time-averaged velocity field, in conjunction with high-resolution power spectral analysis, demonstrate that several mean flows are stable over a wide range in the Rayleigh number R, and that the sequence of time-dependent instabilities depends on the mean flow. A number of routes to non-periodic motion have been identified by varying the geometrical aspect ratio, Prandtl number, and mean flow. Quasi-periodic motion at two frequencies leads to phase locking or entrainment, as identified by a step in a graph of the ratio of the two frequencies. The onset of non-periodicity in this case is associated with the loss of entrainment as R is increased. Another route to turbulence involves successive subharmonic (or period doubling) bifurcations of a periodic flow. A third route contains a well-defined regime with three generally incommensurate frequencies and no broadband noise. The spectral analysis used to demonstrate the presence of three frequencies has a precision of about one part in 104 to 105. Finally, we observe a process of intermittent non-periodicity first identified by Libchaber & Maurer at lower Prandtl number. In this case the fluid alternates between quasi-periodic and non-periodic states over a finite range in R. Several of these processes are also manifested by rather simple mathematical models, but the complicated dependence on geometrical parameters, Prandtl number, and mean flow structure has not been explained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 100 , Issue 3 , 16 October 1980 , pp. 449 - 470
Copyright
© 1980 Cambridge University Press

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References

Ahlers, G. 1974 Low-temperature studies of the Rayleigh-Bénard instability and turbulence. Phys. Rev. Lett. 33, 1185.Google Scholar
Ahlers, G. & Behringer, R. 1978 Evolution of turbulence from the Rayleigh-Bénard instability. Phys. Rev. Lett. 40, 712–716.Google Scholar
Ahlers, G. & Behringer, R. P. 1979 The Rayleigh-Bénard instability and the evolution of turbulence. Prog. Theor. Phys. Suppl. 64, 186–201.Google Scholar
Bergé, P. 1979 Experiments on hydrodynamic instabilities and the transition to turbulence. In Dynamical Critical Phenomena and Related Topics (ed. H. P. Enz). Springer.
Bergé, P. & Dubois, M. 1976 Time-dependent velocity in Rayleigh-Bénard convection: a transition to turbulence. Opt. Comm. 19, 129–133.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.Google Scholar
Busse, F. H. 1980 Transition to turbulence in Rayleigh-Bénard convection. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). Springer.
Busse, F. H. & Clever, R. M. 1979. J. Fluid Mech. 91, 319–335.
Busse, F. H. & Whitehead, J. A. 1974 Oscillatory and collective instabilities in large Prandtl number convection. J. fluid Mech. 66, 67–79.Google Scholar
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625–645.Google Scholar
Curry, J. H. 1978 A generalized Lorenz system. Comm. Math. Phys. 60, 193–204.Google Scholar
Davis, S. H. 1967 Convection in a box: linear theory. J. Fluid Mech. 30, 467–478.Google Scholar
Dubois, M. & Bergé, P. 1978 Experimental study of the velocity field in Rayleigh-Bénard convection. J. Fluid Mech. 85, 641–653.Google Scholar
Feigenbaum, M. J. 1980 The onset spectrum of turbulence. Phys. Lett. A74, 375–378.Google Scholar
Fenstermacher, P. R., Swinney, H. L., Benson, S. A. & Gollub, J. P. 1979 Bifurcations to periodic, quasiperiodic, and chaotic regimes in rotating and convecting fluids. Ann. N. Y. Acad. Sci. 316, 652–666.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, 103–128.Google Scholar
Flaherty, J. E. & Hoppensteadt, F. C. 1978 Frequency entrainment of a forced Van der Pol oscillator. Stud. Appl. Math. 58, 5–15.Google Scholar
Gollub, J. P. & Benson, S. V. 1978 Chaotic response to periodic perturbation of a convecting fluid. Phys. Rev. Lett. 41, 948–951.Google Scholar
Gollub, J. P. & Benson, S. V. 1979 Phase locking in the oscillations leading to turbulence. In Pattern Formation (ed. H. Haken), pp. 74–80. Springer.
Gollub, J. P., Benson, S. V. & Steinman, J. 1980 A subharmonic route to turbulent convection. Ann. N. Y. Acad. Sci. (to appear).Google Scholar
Gollub, J. P., Brunner, T. O., & Danly, B. G. 1978 Periodicity and chaos in coupled nonlinear oscillators. Science 200, 48–50.Google Scholar
Gollub, J. P., Hulbert, S. L., Dolny, G. M. & Swinney, H. L. 1977 Laser Doppler study of the onset of turbulent convection at low Prandtl number. In Photon Correlation Spectroscopy and Velocimetry (ed. H. Z. Cummins & E. R. Pike), pp. 425–439. Plenum.
Gollub, J. P., Romer, E. J. & Socolar, J. E. 1980 Trajectory divergence for coupled relaxation oscillators: measurements and models. J. Stat. Phys. 23, 321–333.Google Scholar
Koschmieder, E. L. 1974 Bénard convection. Adv. Chem. Phys. 26, 177–212.Google Scholar
Krishnamurti, R. 1970 On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309–320.Google Scholar
Krishnamurti, R. 1973 Some further studies on the transition to turbulent convection. J. Fluid Mech. 60, 285–303.Google Scholar
Libchaber, A. & Maurer, J. 1978 Local probe in a Rayleigh-Bénard experiment in liquid helium. J. Phys. Lett. 39, L369–L372.Google Scholar
Libchaber, A. & Maurer, J. 1979 Une expérience de Rayleigh-Bénard de géométrie réduite: multiplication, accrochage et démultiplication de fréquences. J. Phys. (to appear).Google Scholar
Lipps, F. B. 1976 Numerical simulation of three-dimensional Bénard convection in air. J. Fluid Mech. 75, 113–148.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Maurer, J. & Libchaber, A. 1979 Rayleigh-Bénard experiment in liquid helium: frequency locking and the onset of turbulence. J. Phys. Lett. 40, L419–L423.Google Scholar
McLaughlin, J. B. & Martin, P. C. 1975 Transition to turbulence in a statically stressed fluid system. Phys. Rev. A 12, 186–203.Google Scholar
Monin, A. S. 1979 On the nature of turbulence. Sov. Phys. Usp. 21, 429–442.Google Scholar
Newhouse, S., Ruelle, D. & Takens, F. 1978 Occurrence of strange axiom A attractors near quasi-periodic flows on Tm, μ3. Comm. Math. Phys. 64, 35–40.Google Scholar
Normand, C. Y., Pomeau, Y. & Velarde, M. G. 1977 Convective instability: a physicist's approach. Rev. Mod. Phys. 49, 581–624.Google Scholar
Olson, J. M. & Rosenberger, F. 1979 Convective instabilities in a closed vertical cylinder heated from below. Part 1. Mono-component gases. J. Fluid Mech. 92, 609–629.Google Scholar
Otnes, R. K. & Enochson, L. 1972 Digital Time Series Analysis, p. 286. Wiley.
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Comm. Math. Phys. 20, 167–192.Google Scholar
Saltzman, B. 1962 Finite amplitude free convection as an initial value problem. J. Atmos. Sci. 19, 329–341.Google Scholar
Stork, K. & Müller, U. 1972 Convection in boxes: experiments. J. Fluid Mech. 54, 599–611.Google Scholar
Willis, G. E. & Deardorff, J. W. 1970 The oscillatory motions of Rayleigh convection. J. Fluid Mech. 44, 661–672.Google Scholar