Inertial Steady 2D Vector Field Topology

Abstract

Figure 1: Visualization of inertial critical points and a glyph-based visualization of the asympotic behavior of inertial particles. The sink (blue critical point) that is reached by an inertial particle depends on its initial position x 0 and the initial velocity v 0 . The glyph location encodes x 0 and the glyph disc color-codes the sink that is reached for a certain v 0 . The glyph center represents v 0 = 0 and the selected glyph is enlarged in the bottom right corner, showing the seeds of inertial particle trajectories. Left: BORROMEAN RINGS with particle diameter dp = 70 µm and maximal velocity vmax = 10 m/s (occurs at glyph boundary), middle: DUFFING OSCILLATOR with dp = 100 µm, vmax = 5 m/s, and right: NINE CP with dp = 70 µm, vmax = 10 m/s. Inertial critical point types are listed in Table 1. AbstractVector field topology is a powerful and matured tool for the study of the asymptotic behavior of tracer particles in steady flows. Yet, it does not capture the behavior of finite-sized particles, because they develop inertia and do not move tangential to the flow. In this paper, we use the fact that the trajectories of inertial particles can be described as tangent curves of a higher dimensional vector field. Using this, we conduct a full classification of the first-order critical points of this higher dimensional flow, and devise a method to their efficient extraction. Further, we interactively visualize the asymptotic behavior of finite-sized particles by a glyph visualization that encodes the outcome of any initial condition of the governing ODE, i.e., for a varying initial position and/or initial velocity. With this, we present a first approach to extend traditional vector field topology to the inertial case.

Flow-Induced Inertial Steady Vector Field Topology

“…Starting with simple models is also an essential step in the research process, where we later try to extend and generalize. An example is the extension of inertial critical point classification [GT16a] to other equations of motion, as done in this paper, or the extraction of so‐called influence curves [GT16c] in more general inertial particle models [GT17]. Influence curves recover the origin of an inertial particle.…”

“…Another approach is the definition of an abstract inertial flow map, which maps a seed location/velocity to the target location/velocity that is reached after integration for a certain duration. This approach was used for integration‐based concepts, such as inertial integral curves [GKKT13], the determination of stable sets [GT16a], or derived quantities such as multiplicity maps [SJJ* 17] or singular flow map gradients [GT16b]. Based on the flow map, finite‐time Lyapunov exponents (FTLE) can be defined, which measure the spatial separation of nearby‐released inertial particles [SH09,PD09].…”

“…The reduction to lower‐dimensional spaces was also of interest. Günther and Theisel [GT16a] used glyphs to encode stable sets for varying initial positions and velocities in underlying 2D flows. In [GT16c], they also used glyphs to depict the particle velocities that are observed at certain locations in the domain.…”

“…Sagristà et al [SJJ* 17] overlaid quantities measured in the spatial subspace and the velocity subspace, and proposed to use stacked visualizations. In some cases, the extracted features turn out to be structures that can be searched in the domain of the underlying flow, such as vortex corelines [GT14] or critical points [GT16a].…”

“…One way to approach this high‐dimensional problem is to focus on topology, since it is a compact description of asymptotic behavior. Recently, Günther and Theisel [GT16a] have shown that the critical points (in dynamical systems these are also known as fixed points, stationary points or singularities) of a certain simple inertial system can be classified by the eigenvalues of the Jacobian of the underlying 2D air flow. Such a classification, however, depends on the underlying particle model and has to be revisited for every other equation of motion.…”