Universal scaling laws in fully developed turbulence

Abstract

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 ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{l}}$ coarse-grained at inertial-range scale l. These moment ratios ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{l}}^{(\mathit{p})}$=〈${\mathrm{\ensuremath{\epsilon}}}_{\mathit{l}}^{\mathit{p}+1}$〉/〈${\mathrm{\ensuremath{\epsilon}}}_{\mathit{l}}^{\mathit{p}}$〉(p=0, 1, 2,...,) form a hierarchy of structures. The most singular structures ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{l}}^{(\mathrm{\ensuremath{\infty}})}$ are assumed to be filaments, and it is argued that ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{l}}^{(\mathrm{\ensuremath{\infty}})}$\ensuremath{\sim}${\mathit{l}}^{\mathrm{\ensuremath{-}}2/3}$. Furthermore, a universal relation between scalings of successive structures is postulated, which leads to a prediction of the entire set of the scaling exponents: 〈${\mathrm{\ensuremath{\epsilon}}}_{\mathit{l}}^{\mathit{p}}$〉\ensuremath{\sim}${\mathit{l}}_{\mathit{p}}^{\mathrm{\ensuremath{\tau}}}$, ${\mathrm{\ensuremath{\tau}}}_{\mathit{p}}$=-2/3p+2[1-( 2) / 3 ${)}^{\mathit{p}}$] and 〈\ensuremath{\delta}${\mathit{v}}_{\mathit{l}}^{\mathit{p}}$〉\ensuremath{\sim}${\mathit{l}}_{\mathit{p}}^{\mathrm{\ensuremath{\zeta}}}$, ${\mathrm{\ensuremath{\zeta}}}_{\mathit{p}}$=p/9+2[1-(2/3${)}^{\mathit{p}/3}$].