Vortices, Maximum Growth and the Problem of Finite-Time Singularity Formation

Abstract

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 & Protas (2013) 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.

15 pages, 7 figures; to appear in "Fluid Dynamics Research" (Special Issue for IUTAM Symposium on Vortex Dynamics)

References in corpus (2)

• Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem
• Maximum Palinstrophy Growth in 2D Incompressible Flows

Cited by in corpus (10)

• On reference solutions and the sensitivity of the 2D Kelvin-Helmholtz instability problem
• Maximum Palinstrophy Growth in 2D Incompressible Flows
• Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows
• On Singular Vortex Patches, I: Well-posedness Issues
• Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics
• Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation
• Maximum palinstrophy amplification in the two-dimensional Navier-Stokes equations
• Transient Growth in Stochastic Burgers Flows
• Searching for Singularities in Navier-Stokes Flows Based on the Ladyzhenskaya-Prodi-Serrin Conditions
• On Maximum Enstrophy Dissipation in 2D Navier-Stokes Flows in the Limit of Vanishing Viscosity