arXiv subjects

Galen Wilcox

Publications and source records attributed to Galen Wilcox.

3 recordsLinked to original sources

arXiv2024

A modern approach to the Kelvin-Helmholtz instability on circular vortex sheets

We represent the outermost shear interface of an eddy by a circular vortex sheet in two dimensions, and provide a new proof of linear instability via the Birkhoff-Rott equation. Like planar vortex sheets, circular sheets are found to be susceptible to a violent short-wave instability known as the Kelvin-Helmholtz instability, with some modifications due to vortex sheet geometry. This result is in agreement with the classical derivation of (Moore 1974, Saffman 1992), but our modern approach provides greater clarity. We go on to show that the linear evolution problem can develop a singularity from analytic initial data in a time proportional to the square of the vortex sheet radius. Numerical evidence is presented that suggests this linear instability captures the wave-breaking mechanism observed in nonlinear point vortex simulations. Based on these results, we hypothesize that the Kelvin-Helmholtz instability can contribute to the development of secondary instability for eddies in two-dimensional turbulent flow.

physics.flu-dyn↗

arXiv2023

The Influence of Vortex Sheet Geometry on the Kelvin-Helmholtz Instability

This article revisits the instability of sharp shear interfaces, also called vortex sheets, in incompressible fluid flows. We study the Birkhoff-Rott equation, which describes the motion of vortex sheets according to the incompressible Euler equations in two dimensions. The classical Kelvin-Helmholtz instability demonstrates that an infinite, flat vortex sheet has a strong linear instability. We show that this is not the case for circular vortex sheets: such a configuration has a delicate linear stability, and is the first example of a linearly stable solution to the Birkhoff-Rott equation. We subsequently derive a sufficient condition for linear instability of a circular vortex sheet for a family of generalized Birkhoff-Rott kernels, and prove that a common regularized kernel used in numerical simulation and analysis destabilizes the circular vortex sheet. Absent a destabilizing kernel regularization, our work suggests that the nonlinear dynamics are critical for understanding circular vortex sheet instability, and so the essential mechanism of the Kelvin-Helmholtz instability is dependent on global vortex sheet geometry. As expected, nonlinear numerical simulations utilizing the regularized kernel exhibit unstable behavior. Finally, we show experimental results which qualitatively match the types of instabilities that are observed numerically, demonstrating the persistence of the Kelvin-Helmholtz instability in real circular shear flows.

physics.flu-dyn↗

arXiv2023

Non-Linear Singularity Formation for Circular Vortex Sheets

We study the evolution of vortex sheets according to the Birkhoff-Rott equation, which describe the motion of sharp shear interfaces governed by the incompressible Euler equation in two dimensions. In a recent work, the authors demonstrated within this context a marginal linear stability of circular vortex sheets, standing in sharp contrast with classical instability of the flat vortex sheet, which is known as the Kelvin-Helmholtz instability. This article continues that analysis by investigating how non-linear effects induce singularity formation near the circular vortex sheet. In high-frequency regimes, the singularity formation is primarily driven by a complex-valued, conjugated Burgers equation, which we study by modifying a classical argument from hyperbolic conservation laws. This provides a deeper understanding of the mechanisms driving the breakdown of circular vortex sheets, which are observed both numerically and experimentally.

physics.flu-dyn↗