Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-01T19:20:10.875Z Has data issue: false hasContentIssue false

A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence

Published online by Cambridge University Press:  28 March 2006

Yoshimitsu Ogura
Affiliation:
Department of Meteorology, Massachusetts Institute of Technology. Cambridge, Massachusetts

Abstract

This paper is a continuation of previous work (Ogura 1962a, b) on the dynamical consequence of the hypothesis that fourth-order mean values of the fluctuating velocity components are related to second-order mean values as they would be for a normal joint-probability distribution. The equations derived by Tatsumi (1957) for isotropic turbulence on the basis of this hypothesis are integrated numerically for specific intitial conditions. The initial values of the Reynolds number, $R_ \lambda = (u^{\overline{2}})^{\frac {1}{2}} \lambda|v$, assigned in this investigation are 28·8, 14·4, 7·2 and 1·8, where $(u^{\overline{2}})^{\frac {1}{2}}$ is the root-mean-square turbulent velocity, λ the dissipation length and v the kinematic viscosity coefficient.

The result of such computations is that the energy spectrum does develop negative values for Rλ = 28·8 and 14·4. This first occurs at a time approximately 2·8 for Rλ = 28·8 and 4·2 for Rλ = 14·4 The time-scale here is $(E_0 k^3_0)^-{\frac{1}{2}}$, where k0 is a wave-number scale typical of the energy-containing velocity component and E10, a typical value of the energy spectrum, is given by $4 \pi ^-{\frac{1}{2}}k_0^{-1}\overline{u^2}$

There is no evidence of the energy distribution tending to become negative for Rλ = 7·2 and 1·8. It is observed that inertial effects are relatively weak at Rλ = 7·2 and the decay process is largely controlled by viscous effects. For Rλ = 1·8 a purely viscous calculation is found to be adequate to account for the numerically integrated results.

Type
Research Article
Copyright
© 1963 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

Kraichnan, R. H. 1962 The closure problem of turbulence theory. Proc. 13th Symp. in Appl. Math., Amer. Math. Soc., 199.Google Scholar
Millionshtchikov, M. 1941a On the theory of homogeneous isotropic turbulence. C.R. Acad. Sci. U.R.S.S. 32, 615.Google Scholar
Millionshtchikov, M. 1941b On the role of the third moments in isotropic turbulence. C.R. Acad. Sci. U.R.S.S. 32, 619.Google Scholar
O'Brien, E. E. & Francis, G. C. 1962 A consequence of the zero fourth cumulant approximation. J. Fluid Mech. 13, 369.Google Scholar
Ogura, Y. 1962a Energy transfer in a normally distributed and isotropic turbulent velocity field in two dimensions. Phys. Fluids, 5, 395.Google Scholar
Ogura, Y. 1962b Energy transfer in an isotropic turbulent flow. J. Geophys. Res. 67, 3143.Google Scholar
Proudman, I. & Reid, W. H. 1954 On the decay of a normally distributed and homogeneous turbulent velocity field. Phil. Trans. A, 247, 163.Google Scholar
Tatsumi, T. 1957 The theory of decay process of incompressible, isotropic turbulence. Proc. Roy. Soc. A, 239, 16.Google Scholar