Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-22T02:27:24.558Z Has data issue: false hasContentIssue false
  • English
  • Français

A ν-fluid model of homogeneous turbulence subjected to uniform mean distortion

Published online by Cambridge University Press:  29 March 2006

J. M. Dowden
J. M. Dowden
Affiliation:
Department of Mathematics, University of Essex, Colchester, England
Get access Rights & Permissions [Opens in a new window]

Abstract

In two previous papers (Proudman 1970; Dowden 1972) it has been shown that some of the phenomena of turbulence a t high Reynolds numbers can be modelled by a suitable chosen member of the class of ν-fluids. These are non-Newtonian fluids all of whose properties depend only on a single dimensional constant whose dimensions are those of viscosity. The purpose of this paper is to construct an equation to model homogeneous turbulence in the presence of a spatially constant rate of deformation in the limit of infinite Reynolds number.

The equation employed is that of a doubly degenerate third-order v-fluid (in Proudman's classification) in the limit ν → 0. In such a fluid the stress tensor S is governed by an equation of the form \[ A\dot{S}\ddot{S}+B\dot{S}^2+Cu^{\prime}S\dot{S}+D\dot{u}^{\prime}S^2 + Eu^{\prime 2}S^2 = 0, \] where A, B,…, E are isotropic tensor constants of the fluid, u′ is the total rate of deformation tensor and dots denote time derivatives. A list of properties required of the equation and its solution is proposed, and the most general form of A, B,…, E is given consistent with these requirements. Computed solutions of this equation are compared with the results of experiments on homogeneous turbulence, and are found to agree well with them.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 63 , Issue 1 , 18 March 1974 , pp. 33 - 50
Copyright
© 1974 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

Batckelor, G.K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Quart. J. Mech. Appl. Math. 7, 83.Google Scholar
Chanpagne, F. H., Harris, V. G. & Corrsin, S. 1970 Experiments on nearly homogeneous turbulent shear flow. J. Fluid Mech. 41, 81.Google Scholar
Dowden, J.M. 1972 The relaxation of stress in a v-fluid with reference to the decay of homogeneous turbulence. J. Fluid Mech. 56, 641.Google Scholar
Lumley, J. L. 1970 Toward a turbulent constitutive relation. J. Fluid Mech. 41, 413.Google Scholar
Maréchal, J. 1967 Anisotropie d'une turbulence de grille déformée par un champ de vitesse moyenne homogitne. C. R. Acad. Sci., Paris, A 265, 478.Google Scholar
Proudman, I. 1970 On the motion of v-fluids. J. Fluid Mech. 44, 563.Google Scholar
Rose, W.G. 1966 Results of an attempt to generate a homogeneous turbulent shear flow. J. Fluid Mech. 25, 97.Google Scholar
Smith, G.F. 1965 On isotropic integrity bases. Arch. Rat. Mech. Anal. 68, 282.Google Scholar
Spencer, A. J. M. & Rivlin, R.S. 1962 Isotropic integrity bases for vectors and second order tensors, Part I. Arch. Rat. Mech. Anal. 9, 45.Google Scholar
Taylor, G.I. 1938 The spectrum of turbulence. Proc. Roy. Soc. A 164, 476.Google Scholar
Townsend, A.A. 1954 The uniform distortion of homogeneous turbulence. Quart. J. Mech. Appl. Math. 7, 104.Google Scholar
Traugott, S.C. 1958 Influence of solid-body rotation on screen-produced turbulence. N.A.C.A. Tech. Note, no. 4135, p. 1.Google Scholar
Tucker, H. J. & Reynolds, A.J. 1968 The distortion of turbulence by irrotational and plane strain. J. Fluid Mech. 32, 657.Google Scholar