Visualization of Intricate Flow Structures for Vortex Breakdown Analysis
University of Utah
University of Kaiserslautern
University of Utah
University of Kaiserslautern
University of Leipzig
University of Utah
Figure 1: Vortex breakdown bubble in numerical simulation of a cylindrical container. Flow topology is illustrated with stagnation points (red),singularity paths (yellow), and streamlines (blue) on three axially oriented cutting planes. Volume rendering illustrates additional aspects of flowstructure, using a two-dimensional transfer function (widget, right) of a Jacobian-related invariant (horizontal axis) and vorticity (vertical axis).
Vortex breakdowns and flow recirculation are essential phenomenain aeronautics where they appear as a limiting factor in the designof modern aircrafts. Because of the inherent intricacy of these fea-tures, standard flow visualization techniques typically yield clut-tered depictions. The paper addresses the challenges raised by thevisual exploration and validation of two CFD simulations involvingvortex breakdown. To permit accurate and insightful visualizationwe propose a new approach that unfolds the geometry of the break-down region by letting a plane travel through the structure alonga curve. We track the continuous evolution of the associated pro-jected vector field using the theoretical framework of parametrictopology. To improve the understanding of the spatial relationshipbetween the resulting curves and lines we use direct volume ren-dering and multi-dimensional transfer functions for the display offlow-derived scalar quantities. This enriches the visualization andprovides an intuitive context for the extracted topological informa-tion. Our results offer clear, synthetic depictions that permit newinsight into the structural properties of vortex breakdowns.
CR Categories: I.4.7 [Image Processing and Computer Vision]:Feature Measurement [I.6.6]: Simulation And ModelingSimulation Output Analysis J.2 [Physical Sciences and Engineer-ing]: Engineering.
Keywords: flow visualization, vortex analysis, parametric topol-ogy, cutting planes, volume rendering
Computational Fluid Dynamics (CFD) has become an essential toolin various engineering fields. In aeronautics it is a key element
in the design of modern aircrafts. The speed of todays comput-ers combined with the increasing complexity of physical modelsyields numerical simulations that accurately reproduce the subtleflow structures observed in practical experiments and permit tostudy their impact on flight stability. Yet, to fully exploit the hugeamount of information contained in typical data sets, engineers re-quire post-processing techniques providing insight into the resultsof their computation.
In order to meet these needs the research in Flow Visualizationhas designed various methods aimed at efficiently exploring fluidflow data and automatically characterizing their essential proper-ties. Unfortunately, vortex breakdowns and their associated flowrecirculation patterns remain challenging structures and none of theexisting visualization techniques can offer satisfying depictions ofthese features. Their truly three-dimensional nature is poorly visu-alized by conventional methods, such as e.g. streamlines and iso-surfaces. Stream surfaces  improve on these basic techniques butstill obscure the intricate internal structures of recirculation bub-bles.
The work described in this paper has its origins in the col-laboration between engineering and visualization. The problemposed was to find efficient methods to validate two large simula-tion datasets and analyze the contained features, with an emphasison vortices and vortex breakdowns. The approach presented in thefollowing consists in compounding two kinds of visualization tech-niques that had little interaction so far. More precisely, we asso-ciate topology-based flow visualization methods and volume ren-dering. This provides depictions that both convey the subtle struc-tures present in vortical flows, especially during vortex breakdown,and provide an intuitive understanding of their spatial context andassociated physical properties.
The central idea of our flow visualization method is to extendthe basic and widely used cutting plane technique to make it a flex-ible and powerful tool for exploring flow volumes in a continuousway. These moving cutting planes, as we term them in this pa-per, smoothly travel along trajectories that can be either obtainedautomatically by standard feature extraction schemes or providedby the user to explore a particular region. Building on an existing
technique we accurately track the vector field topology observed onthe cutting planes. This allows us to detect and visualize essentialproperties of the flow, especially for recirculation bubbles. Our ap-plication of the volume rendering technique is based on the conceptof multidimensional transfer functions. In the processing of ourdata sets this methodology proves extremely useful in permittingthe simultaneous and coherent depiction of multiple flow-derivedscalar fields, traditionally used to analyze vortical structures. Com-bined with the topological information gathered by our moving cut-ting plane this enhances the visualization and facilitates the under-standing of both the geometry and the physical properties of ourfluid flow data. Observe that although the data at hand are time-dependent we chose to restrict our visualization to the analysis ofthe structural features contained in individual time steps.
2 RELATED WORK
The study of vortex breakdown is a field of its own in the fluidmechanics community. The corresponding literature mostly usesstreamlines or particles advection to observe and analyze the prop-erties of breakdown bubbles [22, 26]. From the viewpoint of Scien-tific Visualization the phenomenon has not received much attentionso far and Kenwright and Haimes  published one of the fewpapers explicitly considering this problem. Vortices, however, arean essential topic of Flow Visualization and have been treated quiteextensively. In the absence of a formal characterization of vorticalstructures, swirling motion around some central region is used as aworking definition [20, 24]. Depending on the approach taken, thisleads to features that are either lines, surfaces or volumes. Vortexcore lines appear to be the most prominent feature type. Banks andSinger  extracted them by looking for points with low pressureand high absolute vorticity. Sujudi and Haimes  applied a cell-wise linear pattern matching strategy to find vortex core lines overtetrahedral grids. This fast method is probably the most widely usedin practice. Peikert and Roth  showed that this and most otherline-based feature detection methods can be reformulated using theconcept of Parallel Operator, leading to continuous features. Theyalso proposed a second order method . A famous region-basedmethod is the 2-criterion proposed by Jeong and Hussein . Itconnects a vortical region to the negative eigenvalues of a symmet-ric matrix derived from the Jacobian. Alternative techniques wereintroduced recently [10, 6]. Aside from this line of research, vor-tices are usually characterized by certain physical quantities likepressure, vorticity or helicity. Finding the corresponding regionscan then be formulated as a level set problem and reduced to iso-surface extraction. Unfortunately, none of the methods mentionedabove is able to deal with the very complex flow behaviors associ-ated with vortex breakdowns.
Although topology-based methods have been widely and suc-cessfully applied to the visualization of planar vector fields, the ex-tension of this technique to three-dimensional flows is still incom-plete. Early contributions [8, 7] were restricted to the extraction andidentification of first-order critical points and the integration of line-type separatrices. Recently, Theisel et al. presented a method forthe visualization of so-called saddle connectors . These line-type features avoid occlusion but provide an incomplete structuralpicture. Mahrous et al.  described an approach for the topologi-cal segmentation of 3D data sets in which separatrices are obtainedas implicit stream surfaces . As a consequence, the accuracy istypically limited in regions of intricate flow. Concerning unsteadydata, Tricoche et al. proposed a technique for tracking the topologyof planar flows over time . A different approach was introducedlater by Theisel and Seidel . The method presented in this paperbuilds on the original idea of .
Direct volume rendering is a powerful tool for the visualizationof scalar volumetric data because of its simplicity and flexibility. Its
simplicity facilitates interactive implementation on graphics hard-ware while its flexibility is grounded in its reliance on the trans-fer function, which maps from data values to the colors and opac-ities. Additional flexibility comes from transfer functions with amulti-dimensional domain, allowing the rendering to display notjust (soft) isosurfaces of individual data values, but the relation-ships between them. Multi-dimensional transfer functions were pi-oneered by Levoy , generalized by Kindlmann and Durkin ,and more recently advanced by Kniss et al. for the visualization ofscalar datasets , as well as for color cryosection data and meteo-rological simulations . Note that, starting from scalar quantitiesprovided by CFD simulations, Ebert et al. already proposed to usevolume rendering for the visualization of gases in . We use a dif-ferent ap