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

Abstract

This paper is a continuation of previous work (Ogura 1962 a, 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 R λ = 28·8 and 14·4. This first occurs at a time approximately 2·8 for R λ = 28·8 and 4·2 for R λ = 14·4 The time-scale here is $(E_0 k^3_0)^-{\frac{1}{2}}$ , where k 0 is a wave-number scale typical of the energy-containing velocity component and E1 0 , 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 R λ = 7·2 and 1·8. It is observed that inertial effects are relatively weak at R λ = 7·2 and the decay process is largely controlled by viscous effects. For R λ = 1·8 a purely viscous calculation is found to be adequate to account for the numerically integrated results.