Dynamic triad interactions and evolving turbulence spectra

We investigate the effect of a four-dimensional Fourier transform on the formulation of Navier-Stokes equation in Fourier space and the way the energy is transferred between Fourier components (the so-called triad interactions). We consider the effect of a finite, digitally sampled velocity record on the triad interactions and find that Fourier components may interact within a broadened frequency window as compared to the usual integrals over infinite ranges. We also see how finite velocity records have a significant effect on the efficiency of the different triad interactions and thereby on the shape and development of velocity power spectra. These results explain the occurrence and time development of the so-called Richardson cascade and also why deviations from the classical Richardson cascade may occur. Finally, we quote results from companion papers that deal with laboratory and computer measurements of the time development of velocity power spectra in a turbulent jet flow into which a single Fourier mode is injected.