Intermittency as a consequence of a stationarity constraint on the energy flux

Abstract

In his seminal work on turbulence, Kolmogorov made use of the stationary hypothesis to determine the Power Density Spectrum of the velocity field in turbulent flows. However to our knowledge, the constraints that stationary processes impose on the fluctuations of the energy flux have never been used in the context of turbulence. Here we recall that the Power Density Spectra of the fluctuations of the injected power, the dissipated power and the energy flux have to converge to a common value at vanishing frequency. Hence, we show that the intermittent GOY--shell model fulfills these constraints. We argue that they can be related to intermittency. Indeed, we find that the constraint on the fluctuations of the energy flux implies a relation between the scaling exponents that characterize intermittency, which is verified by the GOY--shell model and in agreement with the She-Leveque formula. It also fixes the intermittency parameter of the log-normal model at a realistic value. The relevance of these results for real turbulence is drawn in the concluding remarks.