Free decay of turbulence and breakdown of self-similarity

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