Time correlations and random sweeping in isotropic turbulence

The Lagrangian and Eulerian frequency spectrum in isotropic turbulence is considered without a mean flow, concentrating on its second moment, the mean-square acceleration. The pressure and viscous contributions are reviewed and the scaling properties of the advective term 〈[(v·∇)v] ²〉 are examined. Random sweeping from this term is shown to be dominant at large Reynolds numbers if the fluctuation spectrum of the kinetic energy v²(x) scales as k ^-5/3. If on the other hand it satisfies the same Kolmogorov scaling as the pressure going as k ^-7/3, then the recent renormalization group prediction of no sweeping is recovered. This question is subject to direct experimental resolution. The experiments of Van Atta and Wyngaard [J. Fluid Mech. 72, 673 (1975)] strongly indicate that the spectrum of v² goes as k ^-5/3 at high Reynolds numbers, thereby supporting the sweeping hypothesis.