Alignments of Triad Phases in 1D Burgers and 3D Navier-Stokes Flows

The goal of this study is to analyze the fine structure of nonlinear modal interactions in different 1D Burgers and 3D Navier-Stokes flows. This analysis is focused on preferential alignments characterizing the phases of Fourier modes participating in triadic interactions, which are key to determining the nature of energy fluxes between different scales. We develop novel diagnostic tools designed to probe the level of coherence among triadic interactions realizing different flow scenarios. We consider extreme 1D viscous Burgers flows and 3D Navier-Stokes flows which are complemented by singularity-forming inviscid Burgers flows as well as viscous Burgers flows and Navier-Stokes flows corresponding to generic turbulent and simple unimodal initial data, such as the Taylor-Green vortex. The main finding is that while the extreme viscous Burgers and Navier-Stokes flows reveal the same relative level of enstrophy amplification by nonlinear effects, this behaviour is realized via modal interactions with vastly different levels of coherence. In the viscous Burgers flows the flux-carrying triads have phase values which saturate the nonlinearity thereby maximizing the energy flux towards small scales. On the other hand, in 3D Navier-Stokes flows with the extreme initial data the energy flux to small scales is realized by a very small subset of helical triads. The second main finding concerns the role of initial coherence. Comparison of the flows resulting from the extreme and generic initial conditions shows striking similarities between these two types of flows, for the 1D viscous Burgers equation as well as the 3D Navier-Stokes equation.