Free Decay of Turbulence and Breakdown of Self-Similarity

It has been generally assumed, since the work of von Karman and Howarth in 1938, that free decay of fully-developed turbulence is self-similar. We present here a simple phenomenological model of the decay of 3D incompressible turbulence, which predicts breakdown of self-similarity for low-wavenumber spectral exponents $n$ in the range $n_c<n<4$, where $n_c$ is some threshold wavenumber. Calculations with the eddy-damped quasi-normal Markovian approximation give the value as $n_c\approx 3.45$. The energy spectrum for this range of exponents develops two length-scales, separating three distinct wavenumber ranges.