New Formulation for Chaotic Intermittency without Noise

Abstract

The proper description of turbulent flows presents difficulty for researchers in fluid mechanics.
One feature of some of these flows is intermittency. The intermittency phenomenon in Chaotic Dynamics theory is understood as a specific route to the deterministic chaos when spontaneous transitions between laminar and chaotic dynamics occur. A correct characterization of the intermittency is important, principally, to study those problems having partially unknown governing equations or there are experimental or numerical data series. This paper presents a review of a new methodology to investigate systems showing chaotic intermittency phenomenon without noise is presented. To evaluate the statistical properties of the chaotic intermittency a theoretical RPD is obtained. This function depends on two parameters, the lower bound of reinjection and an exponent which describes the non-linear reinjection processes. Once evaluated the RPD function, other properties such as the probability of the laminar length and the characteristic relation are obtained. The key of the new formulation is the introduction of a new function, called M(x), which is utilized to calculate the RPD function in place to consider the huge number of numerical or experimental data. The function M(x) depends on two integrals, this characteristic reduces the influence on the statistical fluctuations in the data series. Also, the function M(x) is easy to evaluate. With this new approach, more accurate analytical expressions for the intermittency statistical properties are obtained. And, it is shown that the behavior of the intermittency phenomena is more rich and complex that those given by the classical theory used until now. On the other hand, the new analytical RPD function is more general than the previous ones; and the classical uniform reinjection is only a particular case.