Does deterministic chaos imply intermittency in fully developed turbulence?

Abstract

A Fourier–Weierstrass decomposition of the velocity field is introduced. The admitted number of real amplitudes is 572 or 836. They are determined numerically from the Navier–Stokes equation including viscosity, driven by constant energy input stirring the largest eddies only. In another calculation, the energy input is provided by an external shear. The Reynolds number Re is about 106, the inertial range comprises about 2 decades, and there are 11 levels of successively decaying eddies. The hierarchical mode ansatz thus allows for a state of high turbulence, which usually is inaccessible numerically. Deterministic chaos is found on all scales. The mean values of the amplitudes scale with the eddy size r as r ζ with ζ very near 1/3. Expected deviations δζ=ζ−1/3, as one typical signature of intermittency, are very small only, well compatible with none at all. So, despite stochasticity (chaos) in the Fourier–Weierstrass decomposition with a tractably restricted set of plane waves, intermittency in the velocity scaling cannot be found. This changes if, in addition to temporal chaos, a spatial branching of the eddy decay process is also admitted.