The 16th FDR prize

Abstract

Objective identification of complex and apparently chaotic structures appearing in turbulent flows without arbitrariness is a crucial step toward understanding turbulent phenomena. Various identification criteria have been proposed, particularly, for vortical structures, which undergo less non-local effects of pressure (compared to rate of strain) so that their temporal evolution may be discussed locally. Typical kinematic criteria are based on the invariants of the velocity gradient tensor. Among them, the identificationmethod proposed byHunt et al (1988) is one of themost widely known. In theirmethod, vortical structures are extracted from incompressible turbulent flows as regions with a positive value of the second invariant of the velocity gradient tensor (meaning the dominance of rotation over deformation). For two-dimensional flows, the same criterion has independently been proposed by Okubo (1970) and Weiss (1991) long before Hunt et al (1988). This criterion, currently called the Okubo-Weiss criterion, is often used for the identification of vortical structures in two-dimensional turbulent flows. The above prize-winning paper has provided an interesting theoretical discussion on the Okubo-Weiss criterion. The Okubo-Weiss criterion is expressed in terms of the square of the eigenvalue of the (two-dimensional) velocity gradient tensor,Q referred to as the Okubo-Weiss parameter (a positive value of which corresponds to a negative value of the second invariant). If the Okubo-Weiss parameter Q is negative (or positive), rotation (or deformation) is dominant, i.e., streamlines are elliptic (or hyperbolic). For two-dimensional incompressible Euler flows, the Lagrangian derivative of the divorticity vector is equal to the product of the velocity gradient tensor and the divorticity vector, implying that the divorticity vector will be frozen in the fluid, where the divorticity vector is defined as the rotation of the vorticity and is tangential to iso-contours of the vorticity. This paper has discussed the relation betweenQ and the Gaussian curvature of the vorticity distribution under the condition (a kind of the Beltrami condition) that the divorticity vector and the velocity vector are parallel to each other, and has demonstrated that the Gaussian curvature of the vorticity distribution is negative at points where deformation is dominant and thus Q is positive, indicating that the vorticity field has a saddle point. This is an interesting result that can characterise the two-dimensional flow field under the Beltrami condition in terms of the Gaussian curvature of the vorticity field. The authors have also given an expression of the Okubo-Weiss parameter in a plane polar coordinate system, and using the given expression they have identified the flow fields associated with the Lamb-Oseen vortex and the (three-dimensional axisymmetric) Burgers vortex as elliptic (or hyperbolic) near (or far from) the vortex. Furthermore, the authors have extended the Okubo-Weiss criterion to quasi-geostrophic flows. Assuming the β-plane approximation to the Coriolis parameter under the Beltrami condition of the potential divorticity and the velocity, the above relation between the positive Okubo-Weiss parameter and the negative Gaussian curvature of the vorticity distribution in