DOI: 10.1017/s0022112075001814. Progress in the development of a Reynolds-stress turbulence closure

Progress in the development of a Reynolds-stress turbulence closure

10.1017/s0022112075001814

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

Abstract

The paper develops proposals for a model of turbulence in which the Reynolds stresses are determined from the solution of transport equations for these variables and for the turbulence energy dissipation rate ε. Particular attention is given to the approximation of the pressure-strain correlations; the forms adopted appear to give reasonably satisfactory partitioning of the stresses both near walls and in free shear flows.Numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows including the jet, the wake, the mixing layer and plane channel flow. In addition, it is shown that the closure does predict a very strong influence of secondary strain terms for flow over curved surfaces.

Bibliography

Launder, B. E., Reece, G. J., & Rodi, W. (1975). Progress in the development of a Reynolds-stress turbulence closure. Journal of Fluid Mechanics, 68(3), 537â566.

Authors 3

B. E. Launder (first)
G. J. Reece (additional)
W. Rodi (additional)

References 42 Referenced 3,086

Uberoi M. S. 1957 Equipartition of energy and local isotropy in turbulent flows.J. Appl. Phys. 28,1165. (10.1063/1.1722600)
Townsend A. A. 1954 The uniform distortion of homogeneous turbulence.Quart. J. Mech. Appl. Math. 7,104. (10.1093/qjmam/7.1.104)
Rotta J. C. 1962 Turbulent boundary layers in incompressible flow.Prog. Aero. Sci. 2,1. (10.1016/0376-0421(62)90014-3)
Rotta J. C. 1951 Statistische Theorie nichthomogener Turbulent.J. Phys. 129,547.
Rodi W. 1972 The prediction of free turbulent boundary layers by use of a 2-equation model of turbulence. Ph.D. thesis,University of London.
Robins A. 1971 The structure and development of a plane turbulent free jet. Ph.D. thesis,University of London.
Patankar, S. V. & Spalding D. B. 1970 Heat and Mass Transfer on Boundary Layers. London:Intertext Books.
Naot D. , Shavit, A. & Wolfshtein M. 1973 Two-point correlation model and the redistribution of Reynolds stress. Phys. Fluids,16,738. (10.1063/1.1694422)
Naot D. , Shavit, A. & Wolfshtein M. 1970 Interactions between components of the turbulent velocity correlation tensor.Israel J. Tech. 8,259.
Laufer J. 1951 Investigation of turbulent flow in a two-dimensional channel.N.A.C.A. Rep. no. 1053.
Irwin H. P. A. 1973 Measurements in a self-preserving plane wall jet in a positive pressure gradient.J. Fluid Mech. 61,33. (10.1017/S0022112073000558)
Hanjalić, K. & Launder B. E. 1972b A Reynolds stress model of turbulence and its application to thin shear flows.J. Fluid Mech. 52,609. (10.1017/S002211207200268X)
Tucker, H. J. & Reynolds A. J. 1968 The distortion of turbulence by irrotational plane strain.J. Fluid Mech. 32,657. (10.1017/S0022112068000947)
Ubeboi M. S. 1963 Energy transfer in isotropic turbulence. Phys. Fluids,6,1048. (10.1063/1.1706861)
Naot D. , Shavit, A. & Wolfshtein M. 1972 Prediction of flow in square section ducts.Mech. Engng Dept., Technion, Haifa, Intern. Rep. no. 154.
Crou P. Y. 1945 On velocity correlations and the solutions of the equations of turbulent fluctuation.Quart. Appl. Math. 3,38. (10.1090/qam/11999)
Comte-Bellot O. 1965 Ecoulement turbulent entre deux parois parallèes.Publ. Sci. Tech. Ministére de l'Air, no. 419.
Lumley, J. L. & Khajeh-Nouri B. 1974 Computational modelling of turbulent transport. Proc. 2nd IUGG-IUTAM Symp. on Atmos. Diffwion in Environmental Pollution ,Academic. (10.1016/S0065-2687(08)60460-4)
Klebanoff P. S. 1955 Characteristics of turbulence in a boundary layer with zero pressure gradient.N.A.C.A. Rep. no. 1247.
Uberoi M. S. 1956 Effect of wind-tunnel contraction on free-stream turbulence.J. Aerospace Sci. 23,754.
Lumley J. L. 1972 A model for computation of stratified turbulent flows.Int. Symp. on Stratified Floui, Novisibirsk.
Daly, B. J. & Harlow F.H. 1970 Transport equations of turbulence. Phys. Fluids,13,2634. (10.1063/1.1692845)
Reynolds W. C. 1970 Computation of turbulent flows-state-of-the-art, 1970.Stanford University Mech. Engng Dept. Rep. MD-27.
Donaldson C. DUP. 1968 A computer study of an analytical model of boundary layer transition.A.I.A.A. Paper, no. 68–38.
Tucker H. J. 1970 The distortion of turbulence by irrotational strain.McGill University, Mech. Engng Dept. Rep. no. 70-7.
Chevray, R. & Kovasznay L. S. G. 1969 Turbulence measurements in the wake of a thin flat plate.A.I.A.A. J. 7,1641. (10.2514/3.5461)
Castro I. 1973 A highly distorted free shear layer. Ph.D. thesis,University of London.
Hanjalić, K. & Launder B. E. 1972a Asymmetric flow in a plane channel.J. Fluid Mech. 51,301. (10.1017/S0022112072001211)
Tennekes, H. & Lumley J. L. 1972 A First Course in Turbulence .M.I.T. Press. (10.7551/mitpress/3014.001.0001)
Patankar, S. V. & Spalding D. B. 1972 A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer,15,1787. (10.1016/0017-9310(72)90054-3)
Bradshaw P. , Ferriss, D. H. & Johnson R. F. 1964 Turbulence in the noise-producing region of a circular jet.J. Fluid Mech. 19,591. (10.1017/S0022112064000945)
Batchelor, G. K. & Proudman I. 1954 The effect of rapid distortion of a fluid in turbulent motion.Quart. J. Mech. Appl. Math. 7,83. (10.1093/qjmam/7.1.83)
Crow S. C. 1968 Viscoelastic properties of fine-grained incompressible turbulence.J. Fluid Mech. 41,81. (10.1017/S0022112068002314)
Batchelor, G. K. & Townsend A. A. 1948 Decay of isotrophic turbulence in the initial period.Proc. Roy. Soc. A193,539.
Bradbury, L. J. S. 1965 The structure of a self-preserving turbulent plane jet. J. Fluid Mech. 23,31. (10.1017/S0022112065001222)
Patel R. P. 1970 A study of two-dimensional symmetric and asymmetric turbulent shear flows. Ph.D. thesis, McGill University.
Bradshaw P. 1972 The understanding and prediction of turbulent flow.Aero. J. 76,403.
Bradshaw P. 1973a Effects of streamline curvature on turbulent flow.Agardograph, no. 169.
Bradshaw P. 1973b The strategy of calculation methods for complex turbulent flows.Imperial College Aero. Rep. no. 73–05.
Champagne F. H. , Harris, V. G. & Corrsin S. 1970 Experiments on nearly homogeneous shear flow.J. Fluid Mech. 41,81. (10.1017/S0022112070000538)
Donaldson C. DUP. 1971 A progress report on an attempt to construct an invariant model of turbulent shear flows.Proc. Agard Conf. on Turbulent Shear Flows, London, paper B-1.
Hanjalić K. 1970 Two-dimensional flow in an axisymmetric channel. Ph.D. thesis,University of London.

Dates

Type	When
Created	19 years, 4 months ago (March 29, 2006, 8:58 a.m.)
Deposited	4 years ago (July 25, 2021, 3:05 a.m.)
Indexed	59 minutes ago (Aug. 21, 2025, 2:25 a.m.)
Issued	50 years, 4 months ago (April 15, 1975)
Published	50 years, 4 months ago (April 15, 1975)
Published Online	19 years, 4 months ago (March 29, 2006)
Published Print	50 years, 4 months ago (April 15, 1975)

Funders 0

None

BibTeX

@article{Launder_1975, title={Progress in the development of a Reynolds-stress turbulence closure}, volume={68}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112075001814}, DOI={10.1017/s0022112075001814}, number={3}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Launder, B. E. and Reece, G. J. and Rodi, W.}, year={1975}, month=apr, pages={537–566} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        21
      ]
    ],
    "date-time": "2025-08-21T06:25:26Z",
    "timestamp": 1755757526826
  },
  "reference-count": 42,
  "publisher": "Cambridge University Press (CUP)",
  "issue": "3",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            2006,
            3,
            29
          ]
        ],
        "date-time": "2006-03-29T00:00:00Z",
        "timestamp": 1143590400000
      },
      "content-version": "unspecified",
      "delay-in-days": 11306,
      "URL": "https://www.cambridge.org/core/terms"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1975,
        4,
        15
      ]
    ]
  },
  "abstract": "<jats:p>The paper develops proposals for a model of turbulence in which the Reynolds stresses are determined from the solution of transport equations for these variables and for the turbulence energy dissipation rate \u03b5. Particular attention is given to the approximation of the pressure-strain correlations; the forms adopted appear to give reasonably satisfactory partitioning of the stresses both near walls and in free shear flows.</jats:p><jats:p>Numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows including the jet, the wake, the mixing layer and plane channel flow. In addition, it is shown that the closure does predict a very strong influence of secondary strain terms for flow over curved surfaces.</jats:p>",
  "DOI": "10.1017/s0022112075001814",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2006,
        3,
        29
      ]
    ],
    "date-time": "2006-03-29T13:58:31Z",
    "timestamp": 1143640711000
  },
  "page": "537-566",
  "source": "Crossref",
  "is-referenced-by-count": 3086,
  "title": "Progress in the development of a Reynolds-stress turbulence closure",
  "prefix": "10.1017",
  "volume": "68",
  "author": [
    {
      "given": "B. E.",
      "family": "Launder",
      "sequence": "first",
      "affiliation": []
    },
    {
      "given": "G. J.",
      "family": "Reece",
      "sequence": "additional",
      "affiliation": []
    },
    {
      "given": "W.",
      "family": "Rodi",
      "sequence": "additional",
      "affiliation": []
    }
  ],
  "member": "56",
  "published-online": {
    "date-parts": [
      [
        2006,
        3,
        29
      ]
    ]
  },
  "reference": [
    {
      "key": "S0022112075001814_ref041",
      "doi-asserted-by": "crossref",
      "unstructured": "Uberoi M. S. 1957 Equipartition of energy and local isotropy in turbulent flows.J. Appl. Phys. 28,1165.",
      "DOI": "10.1063/1.1722600"
    },
    {
      "key": "S0022112075001814_ref037",
      "doi-asserted-by": "crossref",
      "unstructured": "Townsend A. A. 1954 The uniform distortion of homogeneous turbulence.Quart. J. Mech. Appl. Math. 7,104.",
      "DOI": "10.1093/qjmam/7.1.104"
    },
    {
      "key": "S0022112075001814_ref035",
      "doi-asserted-by": "crossref",
      "unstructured": "Rotta J. C. 1962 Turbulent boundary layers in incompressible flow.Prog. Aero. Sci. 2,1.",
      "DOI": "10.1016/0376-0421(62)90014-3"
    },
    {
      "key": "S0022112075001814_ref034",
      "unstructured": "Rotta J. C. 1951 Statistische Theorie nichthomogener Turbulent.J. Phys. 129,547."
    },
    {
      "key": "S0022112075001814_ref033",
      "unstructured": "Rodi W. 1972 The prediction of free turbulent boundary layers by use of a 2-equation model of turbulence. Ph.D. thesis,University of London."
    },
    {
      "key": "S0022112075001814_ref032",
      "unstructured": "Robins A. 1971 The structure and development of a plane turbulent free jet. Ph.D. thesis,University of London."
    },
    {
      "key": "S0022112075001814_ref028",
      "unstructured": "Patankar, S. V. & Spalding D. B. 1970 Heat and Mass Transfer on Boundary Layers. London:Intertext Books."
    },
    {
      "key": "S0022112075001814_ref027",
      "doi-asserted-by": "crossref",
      "unstructured": "Naot D. , Shavit, A. & Wolfshtein M. 1973 Two-point correlation model and the redistribution of Reynolds stress. Phys. Fluids,16,738.",
      "DOI": "10.1063/1.1694422"
    },
    {
      "key": "S0022112075001814_ref025",
      "unstructured": "Naot D. , Shavit, A. & Wolfshtein M. 1970 Interactions between components of the turbulent velocity correlation tensor.Israel J. Tech. 8,259."
    },
    {
      "key": "S0022112075001814_ref022",
      "unstructured": "Laufer J. 1951 Investigation of turbulent flow in a two-dimensional channel.N.A.C.A. Rep. no. 1053."
    },
    {
      "key": "S0022112075001814_ref020",
      "doi-asserted-by": "crossref",
      "unstructured": "Irwin H. P. A. 1973 Measurements in a self-preserving plane wall jet in a positive pressure gradient.J. Fluid Mech. 61,33.",
      "DOI": "10.1017/S0022112073000558"
    },
    {
      "key": "S0022112075001814_ref019",
      "doi-asserted-by": "crossref",
      "unstructured": "Hanjali\u0107, K. & Launder B. E. 1972b A Reynolds stress model of turbulence and its application to thin shear flows.J. Fluid Mech. 52,609.",
      "DOI": "10.1017/S002211207200268X"
    },
    {
      "key": "S0022112075001814_ref039",
      "doi-asserted-by": "crossref",
      "unstructured": "Tucker, H. J. & Reynolds A. J. 1968 The distortion of turbulence by irrotational plane strain.J. Fluid Mech. 32,657.",
      "DOI": "10.1017/S0022112068000947"
    },
    {
      "key": "S0022112075001814_ref042",
      "doi-asserted-by": "crossref",
      "unstructured": "Ubeboi M. S. 1963 Energy transfer in isotropic turbulence. Phys. Fluids,6,1048.",
      "DOI": "10.1063/1.1706861"
    },
    {
      "key": "S0022112075001814_ref026",
      "unstructured": "Naot D. , Shavit, A. & Wolfshtein M. 1972 Prediction of flow in square section ducts.Mech. Engng Dept., Technion, Haifa, Intern. Rep. no. 154."
    },
    {
      "key": "S0022112075001814_ref011",
      "doi-asserted-by": "crossref",
      "unstructured": "Crou P. Y. 1945 On velocity correlations and the solutions of the equations of turbulent fluctuation.Quart. Appl. Math. 3,38.",
      "DOI": "10.1090/qam/11999"
    },
    {
      "key": "S0022112075001814_ref012",
      "unstructured": "Comte-Bellot O. 1965 Ecoulement turbulent entre deux parois parall\u00e8es.Publ. Sci. Tech. Minist\u00e9re de l'Air, no. 419."
    },
    {
      "key": "S0022112075001814_ref024",
      "doi-asserted-by": "crossref",
      "unstructured": "Lumley, J. L. & Khajeh-Nouri B. 1974 Computational modelling of turbulent transport. Proc. 2nd IUGG-IUTAM Symp. on Atmos. Diffwion in Environmental Pollution ,Academic.",
      "DOI": "10.1016/S0065-2687(08)60460-4"
    },
    {
      "key": "S0022112075001814_ref021",
      "unstructured": "Klebanoff P. S. 1955 Characteristics of turbulence in a boundary layer with zero pressure gradient.N.A.C.A. Rep. no. 1247."
    },
    {
      "key": "S0022112075001814_ref040",
      "unstructured": "Uberoi M. S. 1956 Effect of wind-tunnel contraction on free-stream turbulence.J. Aerospace Sci. 23,754."
    },
    {
      "key": "S0022112075001814_ref023",
      "unstructured": "Lumley J. L. 1972 A model for computation of stratified turbulent flows.Int. Symp. on Stratified Floui, Novisibirsk."
    },
    {
      "key": "S0022112075001814_ref014",
      "doi-asserted-by": "crossref",
      "unstructured": "Daly, B. J. & Harlow F.H. 1970 Transport equations of turbulence. Phys. Fluids,13,2634.",
      "DOI": "10.1063/1.1692845"
    },
    {
      "key": "S0022112075001814_ref031",
      "unstructured": "Reynolds W. C. 1970 Computation of turbulent flows-state-of-the-art, 1970.Stanford University Mech. Engng Dept. Rep. MD-27."
    },
    {
      "key": "S0022112075001814_ref015",
      "unstructured": "Donaldson C. DUP. 1968 A computer study of an analytical model of boundary layer transition.A.I.A.A. Paper, no. 68\u201338."
    },
    {
      "key": "S0022112075001814_ref038",
      "unstructured": "Tucker H. J. 1970 The distortion of turbulence by irrotational strain.McGill University, Mech. Engng Dept. Rep. no. 70-7."
    },
    {
      "key": "S0022112075001814_ref010",
      "doi-asserted-by": "crossref",
      "unstructured": "Chevray, R. & Kovasznay L. S. G. 1969 Turbulence measurements in the wake of a thin flat plate.A.I.A.A. J. 7,1641.",
      "DOI": "10.2514/3.5461"
    },
    {
      "key": "S0022112075001814_ref008",
      "unstructured": "Castro I. 1973 A highly distorted free shear layer. Ph.D. thesis,University of London."
    },
    {
      "key": "S0022112075001814_ref018",
      "doi-asserted-by": "crossref",
      "unstructured": "Hanjali\u0107, K. & Launder B. E. 1972a Asymmetric flow in a plane channel.J. Fluid Mech. 51,301.",
      "DOI": "10.1017/S0022112072001211"
    },
    {
      "key": "S0022112075001814_ref036",
      "doi-asserted-by": "crossref",
      "unstructured": "Tennekes, H. & Lumley J. L. 1972 A First Course in Turbulence .M.I.T. Press.",
      "DOI": "10.7551/mitpress/3014.001.0001"
    },
    {
      "key": "S0022112075001814_ref029",
      "doi-asserted-by": "crossref",
      "unstructured": "Patankar, S. V. & Spalding D. B. 1972 A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer,15,1787.",
      "DOI": "10.1016/0017-9310(72)90054-3"
    },
    {
      "key": "S0022112075001814_ref007",
      "doi-asserted-by": "crossref",
      "unstructured": "Bradshaw P. , Ferriss, D. H. & Johnson R. F. 1964 Turbulence in the noise-producing region of a circular jet.J. Fluid Mech. 19,591.",
      "DOI": "10.1017/S0022112064000945"
    },
    {
      "key": "S0022112075001814_ref001",
      "doi-asserted-by": "crossref",
      "unstructured": "Batchelor, G. K. & Proudman I. 1954 The effect of rapid distortion of a fluid in turbulent motion.Quart. J. Mech. Appl. Math. 7,83.",
      "DOI": "10.1093/qjmam/7.1.83"
    },
    {
      "key": "S0022112075001814_ref013",
      "doi-asserted-by": "crossref",
      "unstructured": "Crow S. C. 1968 Viscoelastic properties of fine-grained incompressible turbulence.J. Fluid Mech. 41,81.",
      "DOI": "10.1017/S0022112068002314"
    },
    {
      "key": "S0022112075001814_ref002",
      "unstructured": "Batchelor, G. K. & Townsend A. A. 1948 Decay of isotrophic turbulence in the initial period.Proc. Roy. Soc. A193,539."
    },
    {
      "key": "S0022112075001814_ref003",
      "doi-asserted-by": "crossref",
      "unstructured": "Bradbury, L. J. S. 1965 The structure of a self-preserving turbulent plane jet. J. Fluid Mech. 23,31.",
      "DOI": "10.1017/S0022112065001222"
    },
    {
      "key": "S0022112075001814_ref030",
      "unstructured": "Patel R. P. 1970 A study of two-dimensional symmetric and asymmetric turbulent shear flows. Ph.D. thesis, McGill University."
    },
    {
      "key": "S0022112075001814_ref004",
      "unstructured": "Bradshaw P. 1972 The understanding and prediction of turbulent flow.Aero. J. 76,403."
    },
    {
      "key": "S0022112075001814_ref005",
      "unstructured": "Bradshaw P. 1973a Effects of streamline curvature on turbulent flow.Agardograph, no. 169."
    },
    {
      "key": "S0022112075001814_ref006",
      "unstructured": "Bradshaw P. 1973b The strategy of calculation methods for complex turbulent flows.Imperial College Aero. Rep. no. 73\u201305."
    },
    {
      "key": "S0022112075001814_ref009",
      "doi-asserted-by": "crossref",
      "unstructured": "Champagne F. H. , Harris, V. G. & Corrsin S. 1970 Experiments on nearly homogeneous shear flow.J. Fluid Mech. 41,81.",
      "DOI": "10.1017/S0022112070000538"
    },
    {
      "key": "S0022112075001814_ref016",
      "unstructured": "Donaldson C. DUP. 1971 A progress report on an attempt to construct an invariant model of turbulent shear flows.Proc. Agard Conf. on Turbulent Shear Flows, London, paper B-1."
    },
    {
      "key": "S0022112075001814_ref017",
      "unstructured": "Hanjali\u0107 K. 1970 Two-dimensional flow in an axisymmetric channel. Ph.D. thesis,University of London."
    }
  ],
  "container-title": "Journal of Fluid Mechanics",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075001814",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2021,
        7,
        25
      ]
    ],
    "date-time": "2021-07-25T07:05:41Z",
    "timestamp": 1627196741000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://www.cambridge.org/core/product/identifier/S0022112075001814/type/journal_article"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1975,
        4,
        15
      ]
    ]
  },
  "references-count": 42,
  "journal-issue": {
    "issue": "3",
    "published-print": {
      "date-parts": [
        [
          1975,
          4,
          15
        ]
      ]
    }
  },
  "alternative-id": [
    "S0022112075001814"
  ],
  "URL": "http://dx.doi.org/10.1017/S0022112075001814",
  "relation": {},
  "ISSN": [
    "0022-1120",
    "1469-7645"
  ],
  "subject": [],
  "container-title-short": "J. Fluid Mech.",
  "published": {
    "date-parts": [
      [
        1975,
        4,
        15
      ]
    ]
  }
}