A direct method to calculate the energy evolution of a turbulent flow

The classical configuration of the power spectrum of a homogeneous turbulent flow provides a functional relation between energy and enstrophy which may be used to find the evolution of energy in the intermediate stages of finite time and viscosity to which the usual asymptotic arguments do not apply. The method has a minimal computational cost compared with the direct numerical integration of the whole Navier–Stokes system. It is applied here to study the energy evolution both theoretically and numerically, obtaining the minima of energy compatible with the power spectra as well as the rate of decay of the energy towards them. The results may be useful to study the compatibility of the Kolmogorov and Kraichnan spectra with the observed energy evolution of a turbulent flow.