DOI: 10.1017/s0022112080001243. Many routes to turbulent convection

10.1017/s0022112080001243

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

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.

Bibliography

Gollub, J. P., & Benson, S. V. (1980). Many routes to turbulent convection. Journal of Fluid Mechanics, 100(3), 449â470.

Authors 2

J. P. Gollub (first)
S. V. Benson (additional)

References 41 Referenced 576

Dates

Type	When
Created	19 years, 4 months ago (April 19, 2006, 10:55 a.m.)
Deposited	2 months ago (June 20, 2025, 9:38 p.m.)
Indexed	2 weeks, 2 days ago (Aug. 6, 2025, 9:35 a.m.)
Issued	44 years, 10 months ago (Oct. 16, 1980)
Published	44 years, 10 months ago (Oct. 16, 1980)
Published Online	19 years, 4 months ago (April 19, 2006)
Published Print	44 years, 10 months ago (Oct. 16, 1980)

Funders 0

None

BibTeX

@article{Gollub_1980, title={Many routes to turbulent convection}, volume={100}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112080001243}, DOI={10.1017/s0022112080001243}, number={3}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Gollub, J. P. and Benson, S. V.}, year={1980}, month=oct, pages={449–470} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        6
      ]
    ],
    "date-time": "2025-08-06T13:35:23Z",
    "timestamp": 1754487323375,
    "version": "3.41.0"
  },
  "reference-count": 41,
  "publisher": "Cambridge University Press (CUP)",
  "issue": "3",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            2006,
            4,
            19
          ]
        ],
        "date-time": "2006-04-19T00:00:00Z",
        "timestamp": 1145404800000
      },
      "content-version": "unspecified",
      "delay-in-days": 9316,
      "URL": "https://www.cambridge.org/core/terms"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1980,
        10,
        16
      ]
    ]
  },
  "abstract": "<jats:p>Using automated laser-Doppler methods we have identified four distinct sequences of instabilities leading to turbulent convection at low Prandtl number (2\u00b75\u20135\u00b70), 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 <jats:italic>R</jats:italic>, 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 <jats:italic>R</jats:italic> 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 10<jats:sup>4</jats:sup> to 10<jats:sup>5</jats:sup>. Finally, we observe a process of intermittent non-periodicity first identified by Libchaber &amp; Maurer at lower Prandtl number. In this case the fluid alternates between quasi-periodic and non-periodic states over a finite range in <jats:italic>R</jats:italic>. 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.</jats:p>",
  "DOI": "10.1017/s0022112080001243",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2006,
        4,
        19
      ]
    ],
    "date-time": "2006-04-19T14:55:21Z",
    "timestamp": 1145458521000
  },
  "page": "449-470",
  "source": "Crossref",
  "is-referenced-by-count": 576,
  "title": "Many routes to turbulent convection",
  "prefix": "10.1017",
  "volume": "100",
  "author": [
    {
      "given": "J. P.",
      "family": "Gollub",
      "sequence": "first",
      "affiliation": []
    },
    {
      "given": "S. V.",
      "family": "Benson",
      "sequence": "additional",
      "affiliation": []
    }
  ],
  "member": "56",
  "published-online": {
    "date-parts": [
      [
        2006,
        4,
        19
      ]
    ]
  },
  "reference": [
    {
      "key": "S0022112080001243_ref021",
      "doi-asserted-by": "crossref",
      "unstructured": "Gollub, J. P. , Brunner, T. O. , & Danly, B. G. 1978 Periodicity and chaos in coupled nonlinear oscillators.Science 200,48\u201350.",
      "DOI": "10.1126/science.200.4337.48"
    },
    {
      "key": "S0022112080001243_ref025",
      "doi-asserted-by": "crossref",
      "unstructured": "Krishnamurti, R. 1970 On the transition to turbulent convection. Part 2. The transition to time-dependent flow.J. Fluid Mech. 42,309\u2013320.",
      "DOI": "10.1017/S0022112070001283"
    },
    {
      "key": "S0022112080001243_ref014",
      "doi-asserted-by": "crossref",
      "unstructured": "Feigenbaum, M. J. 1980 The onset spectrum of turbulence.Phys. Lett. A74,375\u2013378.",
      "DOI": "10.1016/0375-9601(79)90227-5"
    },
    {
      "key": "S0022112080001243_ref038",
      "doi-asserted-by": "crossref",
      "unstructured": "Ruelle, D. & Takens, F. 1971 On the nature of turbulence.Comm. Math. Phys. 20,167\u2013192.",
      "DOI": "10.1007/BF01646553"
    },
    {
      "key": "S0022112080001243_ref035",
      "doi-asserted-by": "crossref",
      "unstructured": "Normand, C. Y. , Pomeau, Y. & Velarde, M. G. 1977 Convective instability: a physicist's approach.Rev. Mod. Phys. 49,581\u2013624.",
      "DOI": "10.1103/RevModPhys.49.581"
    },
    {
      "key": "S0022112080001243_ref028",
      "unstructured": "Libchaber, A. & Maurer, J. 1979 Une exp\u00e9rience de Rayleigh-B\u00e9nard de g\u00e9om\u00e9trie r\u00e9duite: multiplication, accrochage et d\u00e9multiplication de fr\u00e9quences.J. Phys. (to appear)."
    },
    {
      "key": "S0022112080001243_ref016",
      "doi-asserted-by": "crossref",
      "unstructured": "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\u2013128.",
      "DOI": "10.1017/S0022112079000963"
    },
    {
      "key": "S0022112080001243_ref017",
      "doi-asserted-by": "crossref",
      "unstructured": "Flaherty, J. E. & Hoppensteadt, F. C. 1978 Frequency entrainment of a forced Van der Pol oscillator.Stud. Appl. Math. 58,5\u201315.",
      "DOI": "10.1002/sapm19785815"
    },
    {
      "key": "S0022112080001243_ref031",
      "doi-asserted-by": "crossref",
      "unstructured": "Maurer, J. & Libchaber, A. 1979 Rayleigh-B\u00e9nard experiment in liquid helium: frequency locking and the onset of turbulence.J. Phys. Lett. 40,L419\u2013L423.",
      "DOI": "10.1051/jphyslet:019790040016041900"
    },
    {
      "key": "S0022112080001243_ref020",
      "doi-asserted-by": "crossref",
      "unstructured": "Gollub, J. P. , Benson, S. V. & Steinman, J. 1980 A subharmonic route to turbulent convection.Ann. N. Y. Acad. Sci. (to appear).",
      "DOI": "10.1111/j.1749-6632.1980.tb29671.x"
    },
    {
      "key": "S0022112080001243_ref003",
      "doi-asserted-by": "crossref",
      "unstructured": "Ahlers, G. & Behringer, R. P. 1979 The Rayleigh-B\u00e9nard instability and the evolution of turbulence.Prog. Theor. Phys. Suppl. 64,186\u2013201.",
      "DOI": "10.1143/PTPS.64.186"
    },
    {
      "key": "S0022112080001243_ref019",
      "doi-asserted-by": "crossref",
      "unstructured": "Gollub, J. P. & Benson, S. V. 1979 Phase locking in the oscillations leading to turbulence. In Pattern Formation (ed. H. Haken ),pp.74\u201380.Springer.",
      "DOI": "10.1007/978-3-642-67480-8_7"
    },
    {
      "key": "S0022112080001243_ref001",
      "doi-asserted-by": "crossref",
      "unstructured": "Ahlers, G. 1974 Low-temperature studies of the Rayleigh-B\u00e9nard instability and turbulence.Phys. Rev. Lett. 33,1185.",
      "DOI": "10.1103/PhysRevLett.33.1185"
    },
    {
      "key": "S0022112080001243_ref009",
      "doi-asserted-by": "crossref",
      "unstructured": "Busse, F. H. & Whitehead, J. A. 1974 Oscillatory and collective instabilities in large Prandtl number convection.J. fluid Mech. 66,67\u201379.",
      "DOI": "10.1017/S0022112074000061"
    },
    {
      "key": "S0022112080001243_ref008",
      "doi-asserted-by": "crossref",
      "unstructured": "Busse, F. H. & Clever, R. M. 1979.J. Fluid Mech. 91,319\u2013335.",
      "DOI": "10.1017/S002211207900015X"
    },
    {
      "key": "S0022112080001243_ref005",
      "doi-asserted-by": "crossref",
      "unstructured": "Berg\u00e9, P. & Dubois, M. 1976 Time-dependent velocity in Rayleigh-B\u00e9nard convection: a transition to turbulence.Opt. Comm. 19,129\u2013133.",
      "DOI": "10.1016/0030-4018(76)90403-X"
    },
    {
      "key": "S0022112080001243_ref022",
      "doi-asserted-by": "crossref",
      "unstructured": "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\u2013439.Plenum.",
      "DOI": "10.1007/978-1-4757-1668-9_12"
    },
    {
      "key": "S0022112080001243_ref039",
      "doi-asserted-by": "crossref",
      "unstructured": "Saltzman, B. 1962 Finite amplitude free convection as an initial value problem.J. Atmos. Sci. 19,329\u2013341.",
      "DOI": "10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2"
    },
    {
      "key": "S0022112080001243_ref002",
      "doi-asserted-by": "crossref",
      "unstructured": "Ahlers, G. & Behringer, R. 1978 Evolution of turbulence from the Rayleigh-B\u00e9nard instability.Phys. Rev. Lett. 40,712\u2013716.",
      "DOI": "10.1103/PhysRevLett.40.712"
    },
    {
      "key": "S0022112080001243_ref036",
      "doi-asserted-by": "crossref",
      "unstructured": "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\u2013629.",
      "DOI": "10.1017/S0022112079000781"
    },
    {
      "key": "S0022112080001243_ref026",
      "doi-asserted-by": "crossref",
      "unstructured": "Krishnamurti, R. 1973 Some further studies on the transition to turbulent convection.J. Fluid Mech. 60,285\u2013303.",
      "DOI": "10.1017/S0022112073000170"
    },
    {
      "key": "S0022112080001243_ref023",
      "doi-asserted-by": "crossref",
      "unstructured": "Gollub, J. P. , Romer, E. J. & Socolar, J. E. 1980 Trajectory divergence for coupled relaxation oscillators: measurements and models.J. Stat. Phys. 23,321\u2013333.",
      "DOI": "10.1007/BF01011372"
    },
    {
      "key": "S0022112080001243_ref018",
      "doi-asserted-by": "crossref",
      "unstructured": "Gollub, J. P. & Benson, S. V. 1978 Chaotic response to periodic perturbation of a convecting fluid.Phys. Rev. Lett. 41,948\u2013951.",
      "DOI": "10.1103/PhysRevLett.41.948"
    },
    {
      "key": "S0022112080001243_ref011",
      "doi-asserted-by": "crossref",
      "unstructured": "Curry, J. H. 1978 A generalized Lorenz system.Comm. Math. Phys. 60,193\u2013204.",
      "DOI": "10.1007/BF01612888"
    },
    {
      "key": "S0022112080001243_ref012",
      "doi-asserted-by": "crossref",
      "unstructured": "Davis, S. H. 1967 Convection in a box: linear theory.J. Fluid Mech. 30,467\u2013478.",
      "DOI": "10.1017/S0022112067001545"
    },
    {
      "key": "S0022112080001243_ref004",
      "unstructured": "Berg\u00e9, P. 1979 Experiments on hydrodynamic instabilities and the transition to turbulence. In Dynamical Critical Phenomena and Related Topics (ed. H. P. Enz ).Springer."
    },
    {
      "key": "S0022112080001243_ref033",
      "doi-asserted-by": "crossref",
      "unstructured": "Monin, A. S. 1979 On the nature of turbulence.Sov. Phys. Usp. 21,429\u2013442.",
      "DOI": "10.1070/PU1978v021n05ABEH005554"
    },
    {
      "key": "S0022112080001243_ref032",
      "doi-asserted-by": "crossref",
      "unstructured": "McLaughlin, J. B. & Martin, P. C. 1975 Transition to turbulence in a statically stressed fluid system.Phys. Rev. A 12,186\u2013203.",
      "DOI": "10.1103/PhysRevA.12.186"
    },
    {
      "key": "S0022112080001243_ref029",
      "doi-asserted-by": "crossref",
      "unstructured": "Lipps, F. B. 1976 Numerical simulation of three-dimensional B\u00e9nard convection in air.J. Fluid Mech. 75,113\u2013148.",
      "DOI": "10.1017/S0022112076000141"
    },
    {
      "key": "S0022112080001243_ref041",
      "doi-asserted-by": "crossref",
      "unstructured": "Willis, G. E. & Deardorff, J. W. 1970 The oscillatory motions of Rayleigh convection.J. Fluid Mech. 44,661\u2013672.",
      "DOI": "10.1017/S0022112070002070"
    },
    {
      "key": "S0022112080001243_ref030",
      "doi-asserted-by": "crossref",
      "unstructured": "Lorenz, E. N. 1963 Deterministic nonperiodic flow.J. Atmos. Sci. 20,130\u2013141.",
      "DOI": "10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2"
    },
    {
      "key": "S0022112080001243_ref010",
      "doi-asserted-by": "crossref",
      "unstructured": "Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection.J. Fluid Mech. 65,625\u2013645.",
      "DOI": "10.1017/S0022112074001571"
    },
    {
      "key": "S0022112080001243_ref007",
      "doi-asserted-by": "crossref",
      "unstructured": "Busse, F. H. 1980 Transition to turbulence in Rayleigh-B\u00e9nard convection. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub ).Springer.",
      "DOI": "10.1007/978-3-662-02330-3_5"
    },
    {
      "key": "S0022112080001243_ref027",
      "doi-asserted-by": "crossref",
      "unstructured": "Libchaber, A. & Maurer, J. 1978 Local probe in a Rayleigh-B\u00e9nard experiment in liquid helium.J. Phys. Lett. 39,L369\u2013L372.",
      "DOI": "10.1051/jphyslet:019780039021036900"
    },
    {
      "key": "S0022112080001243_ref006",
      "doi-asserted-by": "crossref",
      "unstructured": "Busse, F. H. 1978 Non-linear properties of thermal convection.Rep. Prog. Phys. 41,1929\u20131967.",
      "DOI": "10.1088/0034-4885/41/12/003"
    },
    {
      "key": "S0022112080001243_ref034",
      "doi-asserted-by": "crossref",
      "unstructured": "Newhouse, S. , Ruelle, D. & Takens, F. 1978 Occurrence of strange axiom A attractors near quasi-periodic flows on Tm, \u03bc3.Comm. Math. Phys. 64,35\u201340.",
      "DOI": "10.1007/BF01940759"
    },
    {
      "key": "S0022112080001243_ref015",
      "doi-asserted-by": "crossref",
      "unstructured": "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\u2013666.",
      "DOI": "10.1111/j.1749-6632.1979.tb29505.x"
    },
    {
      "key": "S0022112080001243_ref037",
      "unstructured": "Otnes, R. K. & Enochson, L. 1972 Digital Time Series Analysis ,p.286.Wiley."
    },
    {
      "key": "S0022112080001243_ref040",
      "doi-asserted-by": "crossref",
      "unstructured": "Stork, K. & M\u00fcller, U. 1972 Convection in boxes: experiments.J. Fluid Mech. 54,599\u2013611.",
      "DOI": "10.1017/S0022112072000898"
    },
    {
      "key": "S0022112080001243_ref024",
      "doi-asserted-by": "crossref",
      "unstructured": "Koschmieder, E. L. 1974 B\u00e9nard convection.Adv. Chem. Phys. 26,177\u2013212.",
      "DOI": "10.1002/9780470143780.ch4"
    },
    {
      "key": "S0022112080001243_ref013",
      "doi-asserted-by": "crossref",
      "unstructured": "Dubois, M. & Berg\u00e9, P. 1978 Experimental study of the velocity field in Rayleigh-B\u00e9nard convection.J. Fluid Mech. 85,641\u2013653.",
      "DOI": "10.1017/S002211207800083X"
    }
  ],
  "container-title": "Journal of Fluid Mechanics",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112080001243",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2025,
        6,
        21
      ]
    ],
    "date-time": "2025-06-21T01:38:22Z",
    "timestamp": 1750469902000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://www.cambridge.org/core/product/identifier/S0022112080001243/type/journal_article"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1980,
        10,
        16
      ]
    ]
  },
  "references-count": 41,
  "journal-issue": {
    "issue": "3",
    "published-print": {
      "date-parts": [
        [
          1980,
          9,
          11
        ]
      ]
    }
  },
  "alternative-id": [
    "S0022112080001243"
  ],
  "URL": "http://dx.doi.org/10.1017/S0022112080001243",
  "relation": {},
  "ISSN": [
    "0022-1120",
    "1469-7645"
  ],
  "subject": [],
  "container-title-short": "J. Fluid Mech.",
  "published": {
    "date-parts": [
      [
        1980,
        10,
        16
      ]
    ]
  }
}