Normal and anomalous scaling of turbulence

Abstract

A non-Gaussian model of the probability density function (PDF) of $|\ensuremath{\Delta}{u}_{r}|$ is proposed to study how the scaling exponents ${S}_{p}$ of the structure function $〈|\ensuremath{\Delta}{u}_{r}{|}^{p}〉$ of finite Reynolds number turbulence depends upon the Taylor-microscale Reynolds number ${R}_{\ensuremath{\lambda}}.$ Here $\ensuremath{\Delta}{u}_{r}$ is the longitudinal velocity difference across a distance r, and 〈 〉 is the statistical average. The model not only predicts anomalous scaling ${(S}_{p}<p/3$ for $p>3$) observed in experiments at a finite Reynolds number, but also predicts that ${S}_{p}$ approaches normal scaling $p/3$ while ${R}_{\ensuremath{\lambda}}$ is very high. Hence, in contrast to the prevailing multiscaling models, the non-Gaussian PDF model suggests a completely different picture of scaling of isotropic turbulence: the anomalous scaling observed in experiments is a finite Reynolds number effect, and the normal scaling is valid in the real Kolmogorov inertial range corresponding to an infinite Reynolds number.