Zhen-Su She and Emmanuel Leveque (1994) – Universal scaling laws in fully developed turbulence

The inertial-range scaling laws of fully developed turbulence are described in terms of scalings of a sequence of moment ratios of the energy dissipation field ${\epsilon }_l$ coarse-grained at inertial-range scale l. These moment ratios ${\epsilon }_l^{\left(p\right)}$=〈${\epsilon }_l^{p+1}$〉/〈${\epsilon }_l^p$〉(p=0, 1, 2,…,) form a hierarchy of structures. The most singular structures ${\epsilon }_l^{\left(\infty \right)}$ are assumed to be filaments, and it is argued that ${\epsilon }_l^{\left(\infty \right)}$∼$l^{-2/3}$. Furthermore, a universal relation between scalings of successive structures is postulated, which leads to a prediction of the entire set of the scaling exponents: 〈${\epsilon }_l^p$〉∼$l_p^{\tau }$, ${\tau }_p$=-2/3p+2[1-( 2) / 3 ${)}^p$] and 〈δ$v_l^p$〉∼$l_p^{\zeta }$, ${\zeta }_p$=p/9+2[1-(2/3${)}^{p/3}$].

Zhen-Su She and Emmanuel Leveque
Phys. Rev. Lett. 72, 336 – Published 17 January 1994