A Definition of Vortex Vector and Vortex

TL;DR

This paper introduces a precise mathematical definition of vortex and vortex vector in fluid dynamics, enabling better visualization and understanding of turbulent vortical structures.

Contribution

It proposes a new vortex vector concept based on fluid rotation, addressing the lack of a clear mathematical vortex definition in turbulence research.

Findings

Vortex vector is defined as a local fluid rotation descriptor.

The vortex is characterized as a connected flow region with positive vortex vector magnitude.

The new definitions are validated using DNS and LES data.

Abstract

Vortex is ubiquitous in nature. However, there is not a consensus on the vortex definition in fluid dynamics. Lack of mathematical definition has caused considerable confusions in visualizing and understanding the coherent vortical structures in turbulence. According to previous study, it is realized that vortex is not the vorticity tube and vorticity should be decomposed into a rotational part which is the vortex vector and a non-rotational part which is the shear. In this paper, several new concepts such as fluid rotation of local point, the direction of fluid rotation axis and the strength of fluid rotation are proposed by investigating the kinematics of fluid element in the 2D and 3D flows. A definition of a new vector quantity called vortex vector is proposed to describe the local fluid rotation. The direction of the vortex vector is defined as the direction of local fluid rotation axis. The velocity components in the plane orthogonal to the vortex vector have zero derivatives along the vortex vector direction. The magnitude of the vortex vector is defined as the rotational part of vorticity in the direction of the vortex vector, which is the twice of the minimum angular velocity of fluid around the point among all azimuth in the plane perpendicular to vortex vector. According to the definition of the vortex vector, vortex is defined as a connected flow region where the magnitude of the vortex vector at each point is larger than zero. The new definition for the vortex vector and vortex follows three principles: 1. Local in quantity, 2. Galilean invariant, 3. Unique. The definitions are carefully checked by DNS and LES examples which clearly show that the new defined vortex vector and vortex can fully represent the complex structures of vortices in turbulence.