Dynamic Triad Interactions and Evolving Turbulence -- Part 2: Implications for Practical Signals

TL;DR

This work shows that real turbulence measurements, constrained by finite spatial and temporal extents and digital sampling, broaden the traditional triad interaction framework by replacing ideal delta-phase matching with windowed sinc responses. The resulting spectral windows enable both local and non-local energy transfers and introduce cascaded delays that shape the time evolution of spectra, as demonstrated by experiments (hot-wire, LDA) and a 1D Navier–Stokes model. The findings explain how finite measurement domains influence spectrum formation, coherence of structures, and the development of non-equilibrium turbulence, including fractal-grid cases. Overall, the study provides a practical, measurement-centric view of triad interactions that bridges theory with observable turbulence dynamics.

Abstract

We investigate how momentum and kinetic energy is transferred between Fourier components (the so-called triad interactions) in measured turbulent flow fields, i.e. in practical, discretely sampled signals with limited temporal and spatial domains. We empirically observe that the finite resolution in experimental investigations causes a decoupling between time and space, which broadens the phase match condition to include both spatial and temporal frequencies as predicted in Part 1. It is also empirically observed that the Fourier components may interact with a finite time delay and within a broadened frequency window (finite overlap widths) in both time and space, as compared to the usual integrals over infinite ranges where Fourier components interact by overlapping Dirac delta functions. Furthermore, it is empirically observed how the finite temporal and spatial measurement domains of velocity records can have a significant effect on the efficiency of the triad interactions and thereby on the shape and development of measured velocity power spectra. These finite spatial/temporal domains thus influence the measured spatial and temporal development of turbulence, the possibility for non-local interactions and hence also non-equilibrium turbulence, e.g. fractal grid generated turbulence.

Figures (9)

• Figure 1: Wave vectors for three interacting Fourier components (plane waves) in the spatial domain $L \times L$.
• Figure 2: Overlap integral as a function of $\bm{k}_2$ for a fixed value of $\bm{k}_1$.
• Figure 3: Computer simulation of the development of the spectrum of an initial signal consisting of a single spatial frequency of $50 \, m^{-1}$. Figure 4(a): Large interaction volume, high spatial bandwidth which does not interfere with the conversion efficiency. Figure 4(b): Small interaction volume, the sinc-function spectral window influences the conversion efficiency.
• Figure 4: (a) Frequency doubling. (b) Frequency tripling. (c) Frequency quadrupling.
• Figure 5: (Left) Picture and (Right) sketch of the simple experimental setup used to isolate the interactions resulting from a distinct frequency shed off from a rectangular rod. The jet was large compared to the rod, such that the laminar core of the jet could create an environment similar to an open wind tunnel test section. A single component hot-wire (as seen in the left figure) was used to trace the downstream development from the initial single frequency injected by the rectangular rod.