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Maximum palinstrophy growth in 2D incompressible flows

Published online by Cambridge University Press:  21 February 2014

Diego Ayala  and
Bartosz Protas
Diego Ayala
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1
Bartosz Protas*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1
*
†Email address for correspondence: bprotas@mcmaster.ca
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Abstract

In this study we investigate vortex structures which lead to the maximum possible growth of palinstrophy in two-dimensional incompressible flows on a periodic domain. The issue of palinstrophy growth is related to a broader research program focusing on extreme amplification of vorticity-related quantities which may signal singularity formation in different flow models. Such extreme vortex flows are found systematically via numerical solution of suitable variational optimization problems. We identify several families of maximizing solutions parameterized by their palinstrophy, palinstrophy and energy and palinstrophy and enstrophy. Evidence is shown that some of these families saturate estimates for the instantaneous rate of growth of palinstrophy obtained using rigourous methods of mathematical analysis, thereby demonstrating that this analysis is in fact sharp. In the limit of small palinstrophies the optimal vortex structures are found analytically, whereas for large palinstrophies they exhibit a self-similar multipolar structure. It is also shown that the time evolution obtained using the instantaneously optimal states with fixed energy and palinstrophy as the initial data saturates the upper bound for the maximum growth of palinstrophy in finite time. Possible implications of this finding for the questions concerning extreme behaviour of flows are discussed.

JFM classification

Mathematical Foundations: Variational methods Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 742 , 10 March 2014 , pp. 340 - 367
Copyright
© 2014 Cambridge University Press 

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