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.