Rotation Invariant Vortices for Flow Visualization

Interactive Exploration of Physically-Observable Objective Vortices in Unsteady 2D Flow

State-of-the-art computation and visualization of vortices in unsteady fluid flow employ objective vortex criteria, which makes them independent of reference frames or observers. However, objectivity by itself, although crucial, is not sufficient to guarantee that one can identify physically-realizable observers that would perceive or detect the same vortices. Moreover, a significant challenge is that a single reference frame is often not sufficient to accurately observe multiple vortices that follow different motions. This paper presents a novel framework for the exploration and use of an interactively-chosen set of observers, of the resulting relative velocity fields, and of objective vortex structures. We show that our approach facilitates the objective detection and visualization of vortices relative to well-adapted reference frame motions, while at the same time guaranteeing that these observers are in fact physically realizable. In order to represent and manipulate observers efficiently, we make use of the low-dimensional vector space structure of the Lie algebra of physically-realizable observer motions. We illustrate that our framework facilitates the efficient choice and guided exploration of objective vortices in unsteady 2D flow, on planar as well as on spherical domains, using well-adapted reference frames.

Vector Field Decompositions Using Multiscale Poisson Kernel

Extraction of multiscale features using scale-space is one of the fundamental approaches to analyze scalar fields. However, similar techniques for vector fields are much less common, even though it is well known that, for example, turbulent flows contain cascades of nested vortices at different scales. The challenge is that the ideas related to scale-space are based upon iteratively smoothing the data to extract features at progressively larger scale, making it difficult to extract overlapping features. Instead, we consider spatial regions of influence in vector fields as scale, and introduce a new approach for the multiscale analysis of vector fields. Rather than smoothing the flow, we use the natural Helmholtz-Hodge decomposition to split it into small-scale and large-scale components using progressively larger neighborhoods. Our approach creates a natural separation of features by extracting local flow behavior, for example, a small vortex, from large-scale effects, for example, a background flow. We demonstrate our technique on large-scale, turbulent flows, and show multiscale features that cannot be extracted using state-of-the-art techniques.

Rotation Invariant Predictor-Corrector for Smoothed Particle Hydrodynamics Data Visualization

In order to extract the vortex features more accurately, a new method of vortex feature extraction on the Smoothed Particle Hydrodynamics data is proposed in the current study by combining rotation invariance and predictor-corrector method. There is a limitation in the original rotation invariance, which can only extract the vortex features that perform equal-speed rotations. The limitation is slightly weakened to a situation that the rotation invariance can be used, given that a specific axis is existed in the fluid to replace the axis needed for it. Therefore, as long as the axis exists, the modified rotation invariant method can be used. Meanwhile, the vortex features are extracted by predictor-corrector method. By calculating the cross product of the parallel vector field, the seed candidates of vortex core lines can be obtained, and the real seed points can be gained from the rotation invariant Jacobian. Finally, the seed point and a series of candidates based on the predictor-corrector method are connected to draw the vortex core lines. Compared with the original method, the rotation invariant predictor-corrector method not only expands the application scope, but also ensures the accuracy of extraction. Our method adds the steps of calculating the rotation invariant Jacobian, the performance is slightly lower, but with the increase of the particle number, the performance gradually tends to the original method.

Objective Observer-Relative Flow Visualization in Curved Spaces for Unsteady 2D Geophysical Flows

Computing and visualizing features in fluid flow often depends on the observer, or reference frame, relative to which the input velocity field is given. A desired property of feature detectors is therefore that they are objective, meaning independent of the input reference frame. However, the standard definition of objectivity is only given for Euclidean domains and cannot be applied in curved spaces. We build on methods from mathematical physics and Riemannian geometry to generalize objectivity to curved spaces, using the powerful notion of symmetry groups as the basis for definition. From this, we develop a general mathematical framework for the objective computation of observer fields for curved spaces, relative to which other computed measures become objective. An important property of our framework is that it works intrinsically in 2D, instead of in the 3D ambient space. This enables a direct generalization of the 2D computation via optimization of observer fields in flat space to curved domains, without having to perform optimization in 3D. We specifically develop the case of unsteady 2D geophysical flows given on spheres, such as the Earth. Our observer fields in curved spaces then enable objective feature computation as well as the visualization of the time evolution of scalar and vector fields, such that the automatically computed reference frames follow moving structures like vortices in a way that makes them appear to be steady.

Visibility, Topology, and Inertia: New Methods in Flow Visualization

In this article, we address three different topics in scientific visualization. The first part introduces optimization strategies that determine the visibility of line and surface geometry, such that a balance between occlusion avoidance and preservation of context is found. The second part proposes new methods for the visualization of time-dependent fluid flows, including the accurate depiction of Lagrangian scalar fields, as well as a new category of vortex identification methods. The third part introduces finite-sized particles as new application area for flow visualization, covering geometry-based methods, particle separation, topology, vortex corelines, and the determination of the origin of finite-sized particles.

Hyper-Objective Vortices

Almost all properties of vector fields, including magnitude, direction, λ₂ and vorticity change under arbitrary movements of the observer. This is undesirable since measurements of physical properties should ideally not depend on the way the (virtual) measurement device moves. There are some properties that are invariant under certain types of reference frame transformations: Galilean invariance (invariance under equal-speed translation) and objectivity (invariance under any smooth rotation and translation of the reference frame). In this paper, we introduce even harder conditions than objectivity: we demand invariance under any smooth similarity transformation (rotation, translation and uniform scale) as well as invariance under any smooth affine transformation of the reference frame. We show that these new hyper-objective measures allow the extraction of vortices that change their volume or deform. Further, we present a generic approach that transforms almost any vortex measure into a hyper-objective one. We apply our methods to vortex extraction in 2D and 3D vector fields, and analyze the numerical robustness, extraction time and the minimization residuals for the Galilean invariant, objective, and the two new hyper-objective approaches.

Vector Field Topology of Time-Dependent Flows in a Steady Reference Frame

The topological analysis of unsteady vector fields remains to this day one of the largest challenges in flow visualization. We build up on recent work on vortex extraction to define a time-dependent vector field topology for 2D and 3D flows. In our work, we split the vector field into two components: a vector field in which the flow becomes steady, and the remaining ambient flow that describes the motion of topological elements (such as sinks, sources and saddles) and feature curves (vortex corelines and bifurcation lines). To this end, we expand on recent local optimization approaches by modeling spatially-varying deformations through displacement transformations from continuum mechanics. We compare and discuss the relationships with existing local and integration-based topology extraction methods, showing for instance that separatrices seeded from saddles in the optimal frame align with the integration-based streakline vector field topology. In contrast to the streakline-based approach, our method gives a complete picture of the topology for every time slice, including the steps near the temporal domain boundaries. With our work it now becomes possible to extract topological information even when only few time slices are available. We demonstrate the method in several analytical and numerically-simulated flows and discuss practical aspects, limitations and opportunities for future work.

Extraction and Visual Analysis of Potential Vorticity Banners around the Alps

Potential vorticity is among the most important scalar quantities in atmospheric dynamics. For instance, potential vorticity plays a key role in particularly strong wind peaks in extratropical cyclones and it is able to explain the occurrence of frontal rain bands. Potential vorticity combines the key quantities of atmospheric dynamics, namely rotation and stratification. Under suitable wind conditions elongated banners of potential vorticity appear in the lee of mountains. Their role in atmospheric dynamics has recently raised considerable interest in the meteorological community for instance due to their influence in aviation wind hazards and maritime transport. In order to support meteorologists and climatologists in the analysis of these structures, we developed an extraction algorithm and a visual exploration framework consisting of multiple linked views. For the extraction we apply a predictor-corrector algorithm that follows streamlines and realigns them with extremal lines of potential vorticity. Using the agglomerative hierarchical clustering algorithm, we group banners from different sources based on their proximity. To visually analyze the time-dependent banner geometry, we provide interactive overviews and enable the query for detail on demand, including the analysis of different time steps, potentially correlated scalar quantities, and the wind vector field. In particular, we study the relationship between relative humidity and the banners for their potential in indicating the development of precipitation. Working with our method, the collaborating meteorologists gained a deeper understanding of the three-dimensional processes, which may spur follow-up research in the future.

Visual Analysis of Spatio-temporal Relations of Pairwise Attributes in Unsteady Flow

Despite significant advances in the analysis and visualization of unsteady flow, the interpretation of it's behavior still remains a challenge. In this work, we focus on the linear correlation and non-linear dependency of different physical attributes of unsteady flows to aid their study from a new perspective. Specifically, we extend the existing spatial correlation quantification, i.e. the Local Correlation Coefficient (LCC), to the spatio-temporal domain to study the correlation of attribute-pairs from both the Eulerian and Lagrangian views. To study the dependency among attributes, which need not be linear, we extend and compute the mutual information (MI) among attributes over time. To help visualize and interpret the derived correlation and dependency among attributes associated with a particle, we encode the correlation and dependency values on individual pathlines. Finally, to utilize the correlation and MI computation results to identify regions with interesting flow behavior, we propose a segmentation strategy of the flow domain based on the ranking of the strength of the attributes relations. We have applied our correlation and dependency metrics to a number of 2D and 3D unsteady flows with varying spatio-temporal kernel sizes to demonstrate and assess their effectiveness.

Time-Dependent Flow seen through Approximate Observer Killing Fields

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.

Objective Vortex Corelines of Finite-sized Objects in Fluid Flows

Vortices are one of the most-frequently studied phenomena in fluid flows. The center of the rotating motion is called the vortex coreline and its successful detection strongly depends on the choice of the reference frame. The optimal frame moves with the center of the vortex, which incidentally makes the observed fluid flow steady and thus standard vortex coreline extractors such as Sujudi-Haimes become applicable. Recently, an objective optimization framework was proposed that determines a near-steady reference frame for tracer particles. In this paper, we extend this technique to the detection of vortex corelines of inertial particles. An inertial particle is a finite-sized object that is carried by a fluid flow. In contrast to the usual tracer particles, they do not move tangentially with the flow, since they are subject to gravity and exhibit mass-dependent inertia. Their particle state is determined by their position and own velocity, which makes the search for the optimal frame a high-dimensional problem. We demonstrate in this paper that the objective detection of an inertial vortex coreline can be reduced in 2D to a critical point search in 2D. For 3D flows, however, the vortex coreline criterion remains a parallel vectors condition in 6D. To detect the vortex corelines we propose a recursive subdivision approach that is tailored to the underlying structure of the 6D vectors. The resulting algorithm is objective, and we demonstrate the vortex coreline extraction in a number of 2D and 3D vector fields.