On the inviscid instability of the 2D Taylor-Green vortex

TL;DR

This paper investigates the stability of the 2D Taylor-Green vortex in inviscid Euler flows, revealing unstable eigenvalues and non-modal growth mechanisms that differ from viscous flow behaviors, highlighting unique inviscid stability properties.

Contribution

It provides numerical evidence of embedded unstable eigenvalues and introduces a non-modal growth mechanism, advancing understanding of inviscid flow instabilities beyond prior modal analyses.

Findings

Unstable eigenvalues are embedded in the essential spectrum.

A non-modal growth mechanism involving a continuous family of functions is identified.

Inviscid flows exhibit different stability properties compared to viscous flows.

Abstract

We consider Euler flows on two-dimensional (2D) periodic domain and are interested in the stability, both linear and nonlinear, of a simple equilibrium given by the 2D Taylor-Green vortex. As the first main result, numerical evidence is provided for the fact that such flows possess unstable eigenvalues embedded in the band of the essential spectrum of the linearized operator. However, the unstable eigenfunction is discontinuous at the hyperbolic stagnation points of the base flow and its regularity is consistent with the prediction of Lin (2004). This eigenfunction gives rise to an exponential transient growth with the rate given by the real part of the eigenvalue followed by passage to a nonlinear instability. As the second main result, we illustrate a fundamentally different, non-modal, growth mechanism involving a continuous family of uncorrelated functions, instead of an eigenfunction of the linearized operator. Constructed by solving a suitable PDE optimization problem, the resulting flows saturate the known estimates on the growth of the semigroup related to the essential spectrum of the linearized Euler operator as the numerical resolution is refined. These findings are contrasted with the results of earlier studies of a similar problem conducted in a slightly viscous setting where only the modal growth of instabilities was observed. This highlights the special stability properties of equilibria in inviscid flows.