Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows

TL;DR

This paper investigates the maximum possible growth of enstrophy in 3D Navier-Stokes flows using computational optimization, finding that enstrophy growth is finite and scales with initial enstrophy, with no evidence of finite-time singularities.

Contribution

It introduces a PDE optimization framework to quantify maximum enstrophy growth in 3D Navier-Stokes flows, demonstrating bounded growth and elucidating flow behaviors leading to extreme enstrophy.

Findings

Maximum enstrophy growth scales as rac{3}{2} with initial enstrophy.

Enstrophy remains bounded, showing no finite-time singularity evidence.

Extreme enstrophy events involve vortex reconnection phenomena.

Abstract

This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system --- as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy $\mathcal{E}_0$ are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time $T$. Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of $\mathcal{E}_0$ and $T$, we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to $\mathcal{E}_0^{3/2}$ as $\mathcal{E}_0$ becomes large. Thus, in such worst-case scenario the enstrophy still remains bounded for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.