Clustering Properties in Turbulent Signals

Abstract

We consider the telegraph approximation (TA) of turbulent signals by ignoring their amplitude variability and retaining only their ‘zero’-crossing information. We establish a unique relationship between the spectral exponent of a signal and that of its TA, whenever the signal possesses a Gaussian PDF and a spectral shape in which the high-frequency cut-off is sufficiently sharp. The velocity signals in most turbulent flows away from the wall satisfy these conditions adequately, so that the Kolmogorov spectral exponent of −5/3 for the turbulent velocity spectrum corresponds to a −4/3 spectral exponent for its TA. By introducing a new scaling exponent to characterize the tendency of small-scale fluctuations to cluster, we show that the velocity and passive scalar signals display a finite tendency to cluster even in the limit of Re $\rightarrow \infty$. We advance the notion, on the basis of the properties of the TA, that turbulent processes belong to one of two classes—either the ‘white noise’ type or the ‘Markov-Lorentzian’ type.