Superfast amplification and superfast nonlinear saturation of perturbations as the mechanism of turbulence

Abstract

Ruelle predicted that the maximal amplification of perturbations in homogeneous isotropic turbulence is exponential $e^{σ\sqrt{Re} t}$ (where $σ\sqrt{Re}$ is the maximal Liapunov exponent). In our earlier works, we predicted that the maximal amplification of perturbations in fully developed turbulence is faster than exponential $e^{σ\sqrt{Re} \sqrt{t} +σ_1 t}$. That is, we predicted superfast initial amplification of perturbations. Built upon our earlier numerical verification of our prediction, here we conduct a large numerical verification with resolution up to $2048^3$ and Reynolds number up to $6210$. Our direct numerical simulation here confirms our analytical prediction. Our numerical simulation also demonstrates that such superfast amplification of perturbations leads to superfast nonlinear saturation. We conclude that such superfast amplification and superfast nonlinear saturation of ever existing perturbations serve as the mechanism for the generation, development and persistence of fully developed turbulence.