The authors perform high-resolution numerical simulations of 2D incompressible Euler flows using the Characteristic Mapping Method (CMM) to study the dynamics of thin vortex layers. The key findings are:
For decreasing vortex sheet thickness δ, the initial palinstrophy growth exhibits a t^2 scaling, followed by an exponential growth phase. The steepness of the exponential growth increases with decreasing δ, suggesting palinstrophy divergence in the limit δ → 0.
Tracking the vortex center line reveals singular-like behavior in the curvature and true vortex strength. These quantities are found to scale with δ^-0.9 and δ^-0.31 respectively, supporting the conjecture that in the limit of vanishing thickness, the vortex core coalesces to a point with infinite curvature.
Temporal and spatial rescaling of the vortex dynamics shows self-similar behavior, with the critical time ts and scaling factors depending on the vortex sheet thickness δ. This suggests the presence of flow singularities that can be tracked in the complex plane.
The energy spectra exhibit an initial k^-2 scaling, characteristic of the vortex sheet, which transitions to an exponential decay at high wavenumbers as the flow evolves. The impact of the vortex formation and merging processes on the energy spectra is discussed.
Overall, the high-resolution CMM simulations provide detailed insights into the singular behavior of 2D Euler flows with non-smooth initial data, shedding light on the non-uniqueness of weak solutions in this regime.
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by Julius Bergm... à arxiv.org 04-03-2024
https://arxiv.org/pdf/2404.02008.pdfQuestions plus approfondies