Development of high vorticity structures and geometrical properties of the vortex line representation

TL;DR

This paper investigates the development of high vorticity structures in 3D Euler flows using vortex line representation, revealing the geometric properties and compressibility effects responsible for observed vorticity scaling.

Contribution

It introduces high-accuracy numerical simulations of vortex line representation equations to analyze vorticity growth and geometric properties in Euler flows.

Findings

Vorticity growth linked to vortex line compressibility

Identification of geometric properties responsible for vorticity scaling

Successful simulation on adaptive grids up to 1536^3 nodes

Abstract

The incompressible three-dimensional Euler equations develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling $\omega_{max}\sim\ell^{-2/3}$ between the vorticity maximum and the pancake thickness, as was observed in the recent numerical experiments [D.S. Agafontsev et al, Phys. Fluids 27, 085102 (2015)]. We study the process of pancakes' development in terms of the vortex line representation (VLR), which represents a partial integration of the Euler equations with respect to conservation of the Cauchy invariants and describes compressible dynamics of continuously distributed vortex lines. We present, for the first time, the numerical simulations of the VLR equations with high accuracy, which we perform in adaptive anisotropic grids of up to $1536^3$ nodes. With these simulations, we show that the vorticity growth is connected with the compressibility of the vortex lines and find geometric properties responsible for the observed scaling $\omega_{max}\sim\ell^{-2/3}$.