The principle of maximum randomness in the theory of fully developed turbulence. II. Isotropic decaying turbulence

A statistical model for describing the decay of developed isotropic turbulence of an incompressible fluid is proposed. The model uses the distribution function of the velocity pulsations introduced earlier by the authors on the basis of the principle of maximum randomness of the velocity field for a given spectral energy flux. The renormalization-group technique and $\varepsilon'$ expansion are used to calculate the correlation functions of the velocity that occur in the equation of spectral energy balance. This leads to a closed equation for the dependence of the energy spectrum on the integral turbulence scaler $r_c(t)$. In the inertial interval, this equation gives the Kolmogorov asymptotic spectrum, while for the time dependence ofr $r_c(t)$ and the pulsation energy $e(t)$ it predicts the power lawsr $r_c(t)\sim t^{2/5}$ and $e(t)\sim t^{-6/5}$.