Decay of three-dimensional turbulence at high Reynolds numbers

Abstract

A modification of the Loitsyanskii integral is proposed as an invariant of three-dimensional decaying homogeneous and isotropic turbulence. As a result, the kinetic energy of a flow, generated in an infinitely large volume by the initial energy spectrum $E(k,t\,{=}\,0)$, which peaks in the vicinity of the wavenumber $k_{0}\,{=}\,1/{\cal L}(0)\,{=}\,O(1)$, $\cal L$ being an integral scale, decays with time in accordance with Kolmogorov's (1941) prediction: ${\cal K}\,{=}\,\frac{1}{2}\overline {u^{2}}\,{\propto}\, t^{-\gamma}$ with $\gamma\,{=}\,10/7$.