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.