Scaling exponents for turbulence and other random processes and their relationships with multifractal structure

Abstract

In the recent literature on high-Reynolds-number turbulence, several different types of scaling exponents—such as multifractal exponents for velocity increments, for energy and scalar dissipation, for the square of the local vorticity, and so forth—have been introduced. More recently, a new exponent called the cancellation exponent has been introduced for characterizing rapidly oscillating quantities. Not all of these exponents are independent; some of them are simply related to more familiar scaling for velocity and temperature structure functions either exactly or through plausible hypotheses familiar for turbulence. A primary purpose of this paper is to establish the interrelationships among the various exponents. In doing so, we obtain several additional relations. Much of the paper is relevant to general stochastic processes, although the discussion is heavily influenced by the turbulent context. We first examine the case of one-dimensional random processes and subsequently consider two- and three-dimensional processes. Special consideration is given to characteristic values appropriate to the geometry of turbulence, as well as the lifetimes of eddies of various scales. Finally, we discuss some properties of the tails of the probability density function to which the scaling properties of high-order structure functions are related and discuss the implications of multifractality on their structure.