On the identification of a vortex

TLDR

The definition of a vortex remains unclear, yet coherent structures in turbulent flows are increasingly regarded as vortices, making the question highly relevant. The authors aim to provide an objective vortex definition that enables vortex‑dynamics analysis of coherent structures, their formation, evolution, and role in turbulence, and supports turbulence modeling and control. They define a vortex in incompressible flow via the eigenvalues of the tensor S²+Ω², and benchmark this against prior schemes using exact and numerical solutions of the Euler and Navier–Stokes equations for various laminar and turbulent flows. This definition correctly captures pressure minima perpendicular to the vortex axis at high Reynolds numbers, accurately delineates vortex cores at low Reynolds numbers, and reliably identifies vortex cores in flows with clear geometry, outperforming criteria based on the second invariant or complex eigenvalues of the velocity gradient.

Abstract

Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$ ; here ${\bm {\cal S}}$ and ${\bm \Omega}$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$ . This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$ , our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.