Energy transfer in an isotropic turbulent flow

This paper examines 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 for isotropic turbulence on the basis of this hypothesis are integrated numerically as an initial value problem for an inviscid fluid. The most remarkable feature revealed by the computation is that the energy spectrum function becomes negative during the course of time in certain regions of wave-number space. This situation is similar to the result obtained previously for two-dimensional turbulence. Truncation errors that arise from finite-difference approximations in numerical integration are examined. It is tentatively concluded that this unphysical negative energy is not generated by the truncation errors but is the consequence of the quasi-normality hypothesis.