Unveiling Singularity Formation in 2D Euler Equations through High-Resolution Vortex Sheet Simulations

Centrala begrepp

High-resolution numerical simulations of 2D incompressible Euler flows with vortex sheet initial conditions reveal the formation of singular structures, including unbounded vorticity and curvature in the vanishing thickness limit.

Sammanfattning

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:

1. 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.

2. 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.

3. 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.

4. 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.

Statistik

The initial enstrophy Z(t=0) scales with δ^-1.
The initial palinstrophy P(t=0) scales with δ^-3.

Citat

"The curvature was found to empirically scale with δ^-0.9 and the true vortex strength with δ^-0.31, for which an explanation was not found."
"Interestingly, over time both quantities show the same behavior with an initial steep increase to an overall maximum, decreasing again with oscillations."

Viktiga insikter från

Singularity formation of vortex sheets in 2D Euler equations using the characteristic mapping method

by Julius Bergm... på arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.02008.pdf

Djupare frågor

How do the observed scaling laws and singular behavior of the 2D Euler flows compare to the 3D case, where finite-time singularity formation is a long-standing open problem

The observed scaling laws and singular behavior in 2D Euler flows differ from the 3D case, where finite-time singularity formation remains a challenging problem. In 2D Euler flows, the scaling laws and singular behavior indicate that as the vortex sheet thickness approaches zero, the curvature and true vortex strength tend towards infinity, leading to a singularity in finite time. This behavior is distinct from the 3D case, where the dynamics are more complex and the formation of singularities is not well understood. In 3D Euler flows, the existence of finite-time singularities is a long-standing open problem, with the Navier-Stokes equations exhibiting a different behavior compared to the 2D case. The insights gained from studying the scaling laws and singular behavior in 2D Euler flows provide valuable information for understanding the dynamics of vorticity in two dimensions, but the transition to 3D systems introduces additional complexities and challenges in predicting and analyzing singularity formation.

What are the implications of the non-uniqueness of weak solutions for the predictability and modeling of high-Reynolds number turbulent flows

The non-uniqueness of weak solutions in 2D Euler flows has significant implications for the predictability and modeling of high-Reynolds number turbulent flows. In turbulent flows, where vorticity plays a crucial role in the dynamics, the presence of non-smooth initial data can lead to the formation of singularities and the emergence of complex flow structures. The existence of infinitely many non-stationary weak solutions for vortex sheet initial data in 2D Euler equations highlights the challenges in accurately predicting the evolution of turbulent flows with non-smooth features. This non-uniqueness complicates the modeling of turbulent phenomena and emphasizes the need for robust numerical methods and models that can capture the intricate dynamics of high-Reynolds number flows. Understanding the implications of non-uniqueness is essential for improving the accuracy and reliability of turbulent flow simulations and enhancing our ability to predict complex flow behavior.

Can the insights from this study be leveraged to develop more accurate numerical methods or subgrid-scale models for simulating turbulent flows with non-smooth features

The insights gained from the study of singular behavior and scaling laws in 2D Euler flows can be leveraged to develop more accurate numerical methods and subgrid-scale models for simulating turbulent flows with non-smooth features. By understanding the dynamics of vortex sheets and the formation of singularities in 2D flows, researchers can refine existing numerical techniques and develop new approaches to capture the intricate flow structures that arise in high-Reynolds number turbulence. The characteristic mapping method (CMM) used in the study, with its ability to resolve fine-scale flow structures and exponential resolution in linear time, offers a promising framework for simulating turbulent flows with non-smooth initial data. By incorporating the insights from this study into numerical simulations and modeling approaches, researchers can enhance the predictive capabilities of turbulent flow simulations and improve the understanding of complex flow phenomena in high-Reynolds number flows.