Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T00:30:12.175Z Has data issue: false hasContentIssue false
  • English
  • Français

A Reynolds stress model of turbulence and its application to thin shear flows

Published online by Cambridge University Press:  29 March 2006

K. Hanjalić  and
B. E. Launder
K. Hanjalić
Affiliation:
Mechanical Engineering Department, Imperial College, London Present address: Mašinski Fakultet, Sarajevo, Yugoslavia.
B. E. Launder
Affiliation:
Mechanical Engineering Department, Imperial College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper provides a model of turbulence which effects closure through approximated transport equations for the Reynolds stress tensor $\overline{u_iu_j}$ and for the turbulence energy-dissipation rate ε. In its most general form the model thus entails the solution of seven transport equations for turbulence quantities but contains only six constants to be determined by experiment. It is demonstrated that the proposed approximation to the pressure-rate-of-strain correlations leads to satisfactory predictions of the component stress levels in plane homogeneous turbulence, including the non-equality of the lateral and transverse normal-stress components.

For boundary-layer flows a simpler version of the model is derived wherein transport equations are solved only for the shear stress $-\overline{u_1u_2}$ the turbulence energy κ and ε. This model has been incorporated in the numerical solution procedure of Patankar & Spalding (1970) and applied to the prediction of a number of boundary-layer flows including examples of flow remote from walls, those developing along one wall and those confined within ducts. Three of the flows are strongly asymmetric with respect to the surface of zero shear stress and here the turbulent shear stress does not vanish where the mean rate of strain goes to zero. In most cases the predicted profiles and other quantities accord with the data within the probable accuracy of the measurements.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 52 , Issue 4 , 25 April 1972 , pp. 609 - 638
Copyright
© 1972 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. & Townsend, A. A.1948 Decay of isotropic turbulence in the initial period. Proc. Roy. Soc. A193, 539ndash;558.
Bradbury, L. J. S.1965 The structure of a self-preserving turbulent plane jet. J. Fluid Mech. 23, 31.Google Scholar
Brbdshaw, P.1967 The turbulence structure of equilibrium boundary layers. J. Fluid Mech, 29, 625.Google Scholar
Bradshaw, P., Ferriss, D. H. & Atwell, N. P.1967 Calculation of boundary-layer development using the turbulent energy equation. J. Fluid Mech. 28, 593.Google Scholar
Champagne, F. H., Harris, V. G. & Corrsin, S.1970 Experiments on nearly homogeneous turbulent shear flow. J. Fluid Mech. 41, 81–141Google Scholar
Chou, P. Y.1945 On velocity correlations and the solution of the equations of turbulent fluctuation. Quart. Appl. Math. 3, 31.Google Scholar
Comte-Bellot, G.1965 Écoulement turbulent entre deux parois parallèles. Publ. Sceientifiques et Techniques du Ministere de l'Air, no 419.
Daly, B. J. & Harlow, F. H.1970 Transport equations in turbulence. Phys. Fluids, 13, 2634.Google Scholar
Davidov, B. I.1961 On the statistical dynamics of an incompressible turbulent fluid. Dokl. Akad. Nauk S.S.S.R. 136, 47–50.Google Scholar
Donaldson, C. Du P.1968 A computer study of an analytical model of boundary layer transition. A.I.A.A. Paper, no. 68-38.
Hanjalić, K. 1970 Two-dimensional asymmetric turbulent flow in ducts. Ph.D. thesis, University of London.
Hanjalić, K. & Launder, B. E. 1972 Fully developed asymmetric flow in a plane channel. J. Fluid Mech. 51, 301.Google Scholar
Harlow, F. H. & Hirt, C. W.1969 Generalized transport theory of anisotropic turbulence. Los Alamos Sci. Lab. University of California Rep. LA 4086.
Harlow, F. H. & Nakayama, P. I.1968 Transport of turbulence energy decay rate Los Alamos Sci. Lab. University of California Rep. LA 3854.
Hinze, J. O.1959 Turbulence. McGraw-Hill.
Jones, W. P. & Launder, B. E.1972 The prediction of laminarization with a 2-equation model of turbulence. Int. J. Heat. Mass Transfer, 15, 301.Google Scholar
Klebanoff, P. S.1955 Characteristics of turbulence in a boundary layer with zero pressure gradient. N.A.C.A. Rep. no. 1247.Google Scholar
Kolavandin, B. A. & Vatutin, I. A.1969 On statistical theory of non-uniform turbulence. Int. Seminar on Heat and Mass Transfer, Herceg Novi, Yugoslavia,
Laufer, J.1951 Investigation of turbulent flow in a two-dimensional channel. N.A.C.A. Rep. no. 1053.Google Scholar
Launder, B. E. & Ying, W. M.1971 Fully-developed turbulent flow in ducts of square-cross section. Mech. Engng. Dept. Imperial College. Rep. TM/TN/A/11.
Lawn, C. J.1970 Application of the turbulence energy equation to fully developed flow in simple ducts. C.E.G.B. Rep. RD/B 1575.
Lawn, C. J. & Hamlin, M. J.1969 Velocity measurements in roughened annuli. C.E.G.B. Rep. RD/B/N 1278.
Liepmann, H. N. & Laufer, J.1957 Investigation of free turbulent mixing. N.A.C.A. Tech. Work, no. 1257.
Millionshtchikov, M. D.1941 On the theory of homogeneous isotropic turbulence. C.R. Acad. Sci. S.S.S.R. 32, 615–619.Google Scholar
Nee, V. W. & Kovasznay, L. S. G.1968 The calculation of the incompressible turbulent boundary layers by a simple theory. Conference on Conzputation of Turbulent Boundary Layers. Stanford University.
Ng, K. H. & Spalding, D. B.1969 Some applications of a model of turbulence for boundary layers near walls. Mech. Engng. Dept. Imperial College. Rep. BL/TNA/14.
Patankar, S. V. & Spaldinc, D. B.1970 Heat and Mass Transfer in Boundary Layers, 2nd edn. Intertext Books.
Patel, V. C.1965 Calibration of the Preston tube and limitations on its use in pressure gradient. J. Fluid Mech. 23, 185–208.Google Scholar
Prandtl, L.1925 Bericht uber Untersuchungen zur ausgebildeten Turbulenz. Z. angew. Math. Mech. 5, 136.Google Scholar
Rodi, W.1971 On the equation governing the rate of turbulent energy dissipation. Mech. Engng. Dept. Imperial College. Rep. TM/TN/A/14.
Rodi, W. & Spalding, D. B.1970 A two-parameter model of turbulence and its application to free jets. Wärmeund Stoffübertragung 3, 85–95.Google Scholar
Rotta, J.1951 Statistische Theorie nichthomogener Turbulenz. Z. Phys. 129, 547–572.Google Scholar
Rotta, J.1962 Turbulent boundary layers in incompressible flow. In Progress in Aeronautical Sciences, vol. 2. (ed. A. Ferri, D. Kuchemann & L. H. G. Sterne), pp. 1ndash;221. Macmillan.
Spalding, D. B.1970 The prediction of two-dimensional steady turbulent flows. Mech. Engng Dept., Rep. EF/TN/A/16.Google Scholar
Tailland, A. & Mathieu, J.1967 Jet Parietal. J. Mécan. 6, 103–131.Google Scholar
Townsend, A. A.1951 The structure of the turbulent boundary layer. Proc. Camb. Phil. Soc. 47, 375.Google Scholar
Tucker, H. J. & Reynolds, A. J.1968 The distortion of turbulence by irrotational plane strain. J. Fluid Mech. 32, 657.Google Scholar
Uberoi, M. S.1957 Equipartition o f energy and local isotropy in turbulent flows. J. Fluid Mech. J. App1. Phys. 28, 1165ndash;1170.Google Scholar
Wygnanski, I. & Fiedler, H. E.1970 Two-dimensional mixing region. J. Fluid Mech. 41, 327–363.Google Scholar