Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier

Abstract

Kolmogorov’s “third hypothesis” asserts that in intermittent turbulence the average $\bar \varepsilon$ of the dissipation ε, taken over any domain D, is ruled by the lognormal probability distribution. This hypothesis will be shown to be logically inconsistent, save under assumptions that are extreme and unlikely. A widely used justification of lognormality due to Yaglom and based on probabilistic argument involving a self-similar cascade, will also be discussed. In this model, lognormality indeed applies strictly when D is “an eddy,” typically a three-dimensional box embedded in a self-similar hierarchy, and may perhaps remain a reasonable approximation when D consists of a few such eddies. On the other hand, the experimental situation is better described by considering averages taken over essentially one-dimensional domains D.