Inertial- and Dissipation-Range Asymptotics in Fluid Turbulence

Abstract

We propose and verify a wave-vector-space version of generalized extended self-similarity [R. Benzi et al., Europhys. Lett. 32, 709 (1995)] and broaden its applicability to uncover intriguing, universal scaling in the far dissipation range by computing high-order ( $\ensuremath{\le}20$) structure functions numerically for (1) the three-dimensional, incompressible Navier-Stokes equation (with and without hyperviscosity) and (2) the Gledzer-Ohkitani-Yamada shell model for turbulence. Also, in case (2), with Taylor-microscale Reynolds numbers $4\ifmmode\times\else\texttimes\fi{}{10}^{4}\ensuremath{\le}{\mathrm{Re}}_{\ensuremath{\lambda}}\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{10}^{6}$, we find that the inertial-range exponents ( ${\ensuremath{\zeta}}_{p}$) of the order- $p$ structure functions do not approach their Kolmogorov value $p/3$ as ${\mathrm{Re}}_{\ensuremath{\lambda}}$ increases.