Scaling theory of hydrodynamic turbulence

Abstract

A phenomenological scaling theory for incompressible fluid turbulence in the limit of infinite Reynolds number is proposed. The local vorticity and local dissipation are taken as scaling variables with scaling dimensions $\frac{2}{3}\ensuremath{-}\frac{\ensuremath{\zeta}}{2}$ and $\frac{\ensuremath{\mu}}{2}$, respectively. The 1941 Kolmogorov theory corresponds to $\ensuremath{\mu}=\ensuremath{\zeta}=0$. Experimentally, $\ensuremath{\zeta}$ is small and $\ensuremath{\mu}\ensuremath{\approx}\frac{1}{2}$. This choice of scaling variables gives immediate and simple predictions about measured or readily measurable scaling exponents. An additional dimensionality-dependent scaling relation, $\ensuremath{\mu}=d\ensuremath{-}\frac{8}{3}+2\ensuremath{\zeta}$, is proposed and supported by a physically plausible argument. This relation, which is consistent with experiment, suggests that the 1941 Kolmogorov theory is exact for $2ldl\frac{8}{3}$ and has small corrections for $d=3$. Dynamical reasons for this behavior are suggested. The relation of scaling behavior to intermittency of the dissipation is briefly discussed.