Time-Dependent Flow seen through Approximate Observer Killing Fields

Abstract

Flow fields are usually visualized relative to a global observer, i.e., a single frame of reference. However, often no global frame can depict all flow features equally well. Likewise, objective criteria for detecting features such as vortices often use either a global reference frame, or compute a separate frame for each point in space and time. We propose the first general framework that enables choosing a smooth trade-off between these two extremes. Using global optimization to minimize specific differential geometric properties, we compute a time-dependent observer velocity field that describes the motion of a continuous field of observers adapted to the input flow. This requires developing the novel notion of an observed time derivative. While individual observers are restricted to rigid motions, overall we compute an approximate Killing field, corresponding to almost-rigid motion. This enables continuous transitions between different observers. Instead of focusing only on flow features, we furthermore develop a novel general notion of visualizing how all observers jointly perceive the input field. This in fact requires introducing the concept of an observation time, with respect to which a visualization is computed. We develop the corresponding notions of observed stream, path, streak, and time lines. For efficiency, these characteristic curves can be computed using standard approaches, by first transforming the input field accordingly. Finally, we prove that the input flow perceived by the observer field is objective. This makes derived flow features, such as vortices, objective as well.

Can vortex criteria be objectivized?

“…As we show in appendix A, however, unsteadiness-minimizing transforms of v (under any self-consistent measure of unsteadiness, including J t in (4.12)) are never objective. Therefore, contrary to the assertions of Günther et al (2017), as well as those of Hadwiger et al (2018), Günther & Theisel (2020) and Rojo & Günther (2020), applications of local vortex criteria to minimally unsteady transforms of v do not constitute objectivizations of those criteria. In addition to the fundamental non-objectivity of unsteadiness-minimizing frame changes, further issues arise with the proposed numerical implementations of the original optimization problem (4.12).…”

“…We have not required the local observer changes defined in (3.2) to be necessarily Euclidean (distance preserving). Therefore, theorem 3.2 is also applicable to the non-Euclidean local observer changes considered in Hadwiger et al (2018) and Günther & Theisel (2020). We note, however, that all local vortex criteria surveyed in the Introduction assume that the flow is incompressible.…”

“…For instance, the prescribed Jacobian field G(x; t) field may be a time-dependent rotation tensor field that makes v locally as steady as possible (Günther et al 2017;Hadwiger et al 2018;Günther & Theisel 2020;Rojo & Günther 2020). As another example, G(x; t) may align the coordinate axes locally with the eigenvectors of S(x, t) (Astarita 1979;Tabor & Klapper 1994;Lapeyre et al 1999).…”

“…Different rotating observers applying these criteria in their own frames will identify different physical regions as vortices using the fluid velocity field of experiment in figure 1 in their own frames. This discrepancy between frame-dependent results from local vortex criteria and frame-indifferent results from experimental observations has prompted several authors to propose objective modifications to these criteria (Martins et al 2016;Günther, Gross & Theisel 2017;Hadwiger et al 2018;Liu, Gao & Liu 2019a;Liu et al 2019b;Günther & Theisel 2020;Rojo & Günther 2020). Some of these objectivization procedures involve the formal replacement of the spin tensor with other tensors in the original criteria.…”

Robust Reference Frame Extraction from Unsteady 2D Vector Fields with Convolutional Neural Networks

“…Our CNN-based approach removes the noise implicitly. The global optimization of Hadwiger et al [HMTR19] also requires the time partial vt . It remains to be tested, whether a high noise level on vt also influences their result.…”