Diego Ayala

Maximum palinstrophy growth in 2D incompressible flows

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 rigorous 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 behavior of flows are discussed.

Vortices, maximum growth and the problem of finite-time singularity formation

In this work we are interested in extreme vortex states leading to the maximum possible growth of palinstrophy in 2D viscous incompressible flows on periodic domains. This study is a part of a broader research effort motivated by the question about the finite-time singularity formation in the 3D Navier?Stokes system and aims at a systematic identification of the most singular flow behaviors. We extend the results reported in Ayala and Protas (2014 J. Fluid Mech. 742 340?67) where extreme vortex states were found leading to the growth of palinstrophy, both instantaneously and in finite time, which saturates the estimates obtained with rigorous methods of mathematical analysis. Here we uncover the vortex dynamics mechanisms responsible for such extreme behavior in time-dependent 2D flows. While the maximum palinstrophy growth is achieved at short times, the corresponding long-time evolution is characterized by some nontrivial features, such as vortex scattering events.

Extreme vortex states and the growth of enstrophy in three-dimensional incompressible flows

In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy

On maximum enstrophy growth in a hydrodynamic system

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Maximum Palinstrophy Growth in 2D Incompressible Flows: Instantaneous Case

which maximize the instantaneous rate of growth of enstrophy

Maximum Production of Enstrophy in Swirling Viscous Flows

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On the growth of enstrophy in axisymmetric 3D Euler flows with swirl

. We provide {an analytic} characterization of these extreme vortex states in the limit of vanishing enstrophy

Extreme Vortex States and the Hydrodynamic Blow-Up Problem (Probing Fundamental Bounds in Hydrodynamics Using Variational Optimization Methods)

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Extreme Vortex States and the Growth of Palinstrophy in Two Dimensions

and, in particular, show that the Taylor-Green vortex is in fact a local maximizer of

On Optimal Vortex Structures for Palinstrophy Generation

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