Generic objective vortices for flow visualization

Abstract

In flow visualization, vortex extraction is a long-standing and unsolved problem. For decades, scientists developed numerous definitions that characterize vortex regions and their corelines in different ways, but none emerged as ultimate solution. One reason is that almost all techniques have a fundamental weakness: they are not invariant under changes of the reference frame, i.e., they are not objective. This has two severe implications: First, the result depends on the movement of the observer, and second, they cannot track vortices that are moving on arbitrary paths, which limits their reliability and usefulness in practice. Objective measures are rare, but recently gained more attention in the literature. Instead of only introducing a new objective measure, we show in this paper how all existing measures that are based on velocity and its derivatives can be made objective. We achieve this by observing the vector field in optimal local reference frames, in which the temporal derivative of the flow vanishes, i.e., reference frames in which the flow appears steady. The central contribution of our paper is to show that these optimal local reference frames can be found by a simple and elegant linear optimization. We prove that in the optimal frame, all local vortex extraction methods that are based on velocity and its derivatives become objective. We demonstrate our approach with objective counterparts to λ 2 , vorticity and Sujudi-Haimes.

The State of the Art in Vortex Extraction

“…For these reasons, Günther et al . [GGT17] did not seek for a global (spatially constant) reference frame, but for a local one. They estimated an optimal local reference frame for every point , in which the transformed velocity field is as steady as possible in a local neighbourhood U around .…”

“…Here, a line integral convolution (time slice) and pathlines (black) are shown for three different reference frame movements: standing still, linearly translating and swinging along a sine curve—unfortunately, all give different results. The illustration is courtesy from [GGT17].…”

“…Current state‐of‐the‐art feature extractors aspire to be objective [Hal15, HHFH16, GGT17]. In continuum mechanics, objectivity refers to the invariance of a measure under a change of the reference frame that transforms a point in space‐time to a new point by where is a rotation matrix, c is a translation vector and a is a constant.…”

“…Generally, v , , J , a and Ω are not objective, as they do not fulfil Definition , cf. [GGT17]. The divergence and the strain rate tensor S are some of the few objective first‐order differential properties of a vector field, which may be used to characterize a vortex in an objective way, cf.…”

“…The divergence and the strain rate tensor S are some of the few objective first‐order differential properties of a vector field, which may be used to characterize a vortex in an objective way, cf. [Hal05, GGT17]. The objectivity of divergence is easily seen, since it can be computed from S as .…”