Initial evolution of 3D turbulence en route to the Kolmogorov state: emergence and transformations of coherent structures, self-similarity and instabilities

TL;DR

This study numerically investigates the evolution of 3D turbulence from initial random conditions, revealing stages of vortex formation, instability, and a final turbulent state with Kolmogorov spectrum, highlighting structural transformations and self-similarity.

Contribution

It provides a detailed numerical analysis of the stages leading to turbulence, including vortex pancake formation, instability, and the emergence of Kolmogorov scaling, advancing understanding of turbulence development.

Findings

Self-similar spectrum evolution with exponential wave number front.

Formation and instability of vortex pancakes and filaments.

Development of a Kolmogorov energy spectrum in the turbulent state.

Abstract

In this work, we study numerically the temporal evolution of an initially random large-scale velocity field under governed by the hyperviscous incompressible Navier-Stoke equations. Three stages are clearly observed during the evolution. First, the initial condition development is characterized by a spectrum evolving in a self-similar way with a wave number front $k_*(t)$ propagating toward high values exponentially in time. This evolution corresponds to the formation and shrinking of think vortex pancakes exponentially in time, as it has been previously reported in simulations of the incompressible Euler equations. At the second stage, pancakes become unstable, rolling up on the edges and breaking up in the middle, leading to the emergence of vortex ribs -- quasi-periodic arrangements of vortex filaments. Those filaments then twist creating structures akin to ropes. At the last stage, a fully developed turbulent state is observed, characterized by a Kolmogorov energy spectrum and exhibiting a decay law compatible with standard turbulent predictions in the case of an integral scale saturated at the scale of the box.