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

Document Type

Article

Publication Date

11-20-2024

Abstract

We consider Euler flows on two-dimensional (2-D) periodic domain and are interested in the stability, both linear and nonlinear, of a simple equilibrium given by the 2-D 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 (Intl Math. Res. Not., vol. 2004, issue 41, 2004, pp. 2147–2178). 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 partial differential equation (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.

Identifier

85210184115 (Scopus)

Publication Title

Journal of Fluid Mechanics

External Full Text Location

https://doi.org/10.1017/jfm.2024.946

e-ISSN

14697645

ISSN

00221120

Volume

999

Grant

DMS–2107956

Fund Ref

Natural Sciences and Engineering Research Council of Canada

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