Mathematical Study on Local Fluid Rotation Axis: Vorticity is Not the Rotation Axis

Abstract

A vortex can be intuitively recognized as the rotational/swirling motion of the fluids. The fascination of the phenomenon brought about many years of research to classify and identify the vortical structure. Vorticity was one of the first theories developed to identify the vortex. Although the vorticity theory was mathematically sound, it did not line up with experimental results. This setback brought forth new eigenvalue-based methods such as Q, λ_ci, and λ₂. However, since these eigenvalue-based methods are scalar-valued, many researchers and textbooks accepted that vorticity is vortex. In recent years, Dr. Liu developed a new vortex identification method called Luitex. Liutex is a vector quantity with a clear physical meaning that overcomes the drawbacks of the previous methods. The physics behind a vortex reveals that there exists a local fluid rotation axis. In this paper, we conducted a mathematical study on five local fluid rotation axis candidates. Namely, the symmetrical tensor’s three eigenvectors, the vorticity vector, and the Liutex directional vector. The results show that the vorticity vector satisfied the local fluid rotation axis requirements only in particular cases. While the Liutex directional vector unconditionally satisfied the requirements to be considered the local fluid rotation axis.