Flow-Induced Inertial Steady Vector Field Topology

Abstract

Model I: dp = 100 µm Model II: R = 1, St = 0.2 Model I: dp = 100 µm Model II: R = 1, St = 0.2 Figure 1: Critical points in 2D and 3D flows for two different particle models. In contrast to massless flows, inertial particles might oscillate. AbstractTraditionally, vector field visualization is concerned with 2D and 3D flows. Yet, many concepts can be extended to general dynamical systems, including the higher-dimensional problem of modeling the motion of finite-sized objects in fluids. In the steady case, the trajectories of these so-called inertial particles appear as tangent curves of a 4D or 6D vector field. These higher-dimensional flows are difficult to map to lower-dimensional spaces, which makes their visualization a challenging problem. We focus on vector field topology, which allows scientists to study asymptotic particle behavior. As recent work on the 2D case has shown, both extraction and classification of isolated critical points depend on the underlying particle model. In this paper, we aim for a model-independent classification technique, which we apply to two different particle models in not only 2D, but also 3D cases. We show that the classification can be done by performing an eigenanalysis of the spatial derivatives' velocity subspace of the higher-dimensional 4D or 6D flow. We construct glyphs that depict not only the types of critical points, but also encode the directional information given by the eigenvectors. We show that the eigenvalues and eigenvectors of the inertial phase space have sufficient symmetries and structure so that they can be depicted in 2D or 3D, instead of 4D or 6D.

Visualization of 4D Vector Field Topology

“…Second, the physical phase space of 2D steady inertial dynamics is a four‐dimensional vector field, where two components correspond to space and two components to momentum. While, as presented by Günther and Gross [GG17], dynamics of flow‐induced inertial dynamics can be reduced to the spatial dimension, general inertial dynamics need to be considered in the full 2 n ‐dimensional phase space. Further, being restricted to the n ‐dimensional spatial domain, Günther and Gross cannot show separatrices in the 2 n ‐dimensional phase space.…”

“…Two‐parameter‐dependent 2D vector fields [WTHS06] only consider vectors in 2‐space, while the tracking of 3D critical points [GTS04] over time only considers 3D vector fields at different instances in time. Flow‐induced inertial dynamics of 2D systems [GG17], on the other hand, has an underlying 4D phase space, but the underlying flow allows the reduction of the analysis to the two‐dimensional spatial domain. Our work focuses on general four‐dimensional vector fields.…”

Phase Space Projection of Dynamical Systems

“…In scientific visualization, several feature extraction algorithms have been extended to higher‐dimensional flows [LMGP97], including the analysis of finite‐sized particles in fluids. For instance, Günther et al extracted critical points in the high‐dimensional phase space [GG17] and extracted attracting manifolds via backward integration of inertial particles [GT17]. Due to the structure of the inertial phase space, a globally attracting manifold [MBZ06] exists that was visualized by Baeza Rojo et al [BRGG18].…”

The State of the Art in Vortex Extraction

“…In divergence‐free flows for instance, sources and sinks never occur. Higher‐dimensional flows that describe the motion of finite‐sized objects do not contain sources [GT16, GG17] and time‐dependent flows do not contain any critical points in space‐time, since particles always move forward in time, cf. Equation .…”