Dynamic Triad Interactions and Evolving Turbulence -- Part 1: 4D Modal Interactions

TL;DR

The paper extends turbulence analysis by formulating the Navier–Stokes equations in a 4D Fourier space that includes time, thereby treating velocity fluctuations as four-dimensional modal interactions. It derives the 4D transform, shows that the nonlinear term governs interactions across both wavenumber and frequency, and introduces a generalized phase-match condition that admits delayed and advanced interactions. This framework explains time-dependent energy transfer, the Richardson cascade, and significant non-local interactions in high-intensity and non-Taylor-invariant turbulence, highlighting the impact of finite temporal overlap on turbulence dynamics. The results set the stage for Part2, which discusses practical implications for experiments and simulations with finite resolution and domain size.

Abstract

We investigate the effect of a four-dimensional Fourier transform on the formulation of the Navier-Stokes equation in Fourier space and the way the energy is transferred between Fourier components. Since time in a sampled high intensity turbulence must be considered a stochastic variable in the energy exchange between scales, we refer to these dynamic triad interactions as modal interactions, rather than the commonly referred to triad interactions in the classical 3-dimensional analysis. The inclusion of time as a parameter broadens the phase match condition from the classical one, $Δ\bm{k} \cdot \bm{r} = \left [ \bm{k} - (\bm{k}_1 + \bm{k}_2 ) \right ] \cdot \bm{r}$, to the more general formulation that also includes temporal frequencies: $Δ\bm{k} \cdot \bm{r} - Δωt = \left [ \bm{k} - (\bm{k}_1 + \bm{k}_2 ) \right ] \cdot \bm{r} - \left [ ω- \left (ω_1 + ω_2 \right ) \right ] t$. This renders possible the occurrence of `delayed' and `advanced' interactions. The observation that mismatches in the wavevector triadic interactions may be compensated by a corresponding mismatch in the frequencies supports the empirically deduced delayed interactions reported in [Josserand \textit{et al.}, \textit{J. Stat. Phys.} (2017)]. These results explain the occurrence and inherent time development of the so-called Richardson cascade and also how finite temporal overlap of wave components can result in significant non-local interactions and consequently non-equilibrium turbulence, e.g., fractal grid generated turbulence. The consequences of including time as a parameter in practical experiments or simulations in terms of limited resolution, domain size etc. are treated in the companion paper (Part 2) of the present work.