Feature-Based Analysis and Comparison of Coherent Structures in Parameter-Dependent Flows
In this project methods are developed for extraction of coherent structures or dominant features in complex turbulent, time-dependent flows. Since flow fields are inherently noisy, robust methods are needed. A special focus is put on parameter-dependent fields. These parameters can be flow parameters such as actuation, or model parameters like the shape of the boundaries that influence the flow. New feature based methods for comparing flow fields shall also be developed. For this measures to compare the different structures are investigated. Those techniques will be integrated into a virtual lab.
Active flow control involves setting a high number of parameters in order to achieve the goal of, e.g., higher lift. Using the methods developed in this project we want to ease the exploration of this high-dimensional parameter space. For this, it is important to compare flow fields, which have been simulated using different parameters. We have chosen a comparison approach based on flow features, since they capture the most important structures of the flow. The following features will be considered in this project:
- Critical points
- Vortex core lines
- Vortex regions
- Attachment and detachment lines
After developing methods and tools to extract those features in a robust manner, we are going to develop comparison techniques for them. This means to develop solutions for the following tasks:
- Relate a feature of flow A with a similar feature of flow B.
- Compute their distance in feature space using a metric.
- Determine the influence of a feature on the flow.
- Weight several distance measures based on the specific application and combine them to one metric.
- Visualize the feature based difference of flow A and flow B.
This approach of visual analysis helps us to optimize actuation and to validate the used simulation methods. Since computation times are long and expensive, a tightly focused optimization is needed. This is only possible by gaining a thorough understanding of the dominant processes inherent to the flow.
Furthermore, feature based comparison bridges the gap between simulation, experiment and low-dimensional modelling. Those three techniques generate data with very different properties, i.e., data from simulations has a much higher resolution than from an experiment and modelled flows are often much smoother than their simulation input. However, all three techniques aim to resolve the same flow processes in the same setting. By looking at the most important features instead of the data itself, we build up some kind of abstraction and offer more meaningful comparisons of those different types of data.
Localized Finite-time Lyapunov Exponent
A promising way to better understand the dynamics of time-dependent flows is to extract Lagrangian coherent structures. According to S. K. Robinson ("Coherent motions in the turbulent boundary layer", Annu. Rev. Fluid Mech., 23:601-639, 1991) a coherent motion is defined as "a region over which at least one fundamental flow variable exhibits significant correlation with itself or with another variable over a range of space and/or time that is significantly larger than the smallest local scales". Such structures can be extracted for instance by visualizing the spatio-temporal distribution of the finite-time Lyapunov exponent (FTLE) introduced by Haller ("Distinguished material surfaces and coherent structures in three-dimensional fluid flows", Physica D 149, 248–277, 2001). The FTLE field characterizes at each space-time point the rate of separation of infinitesimally close trajectories when moving with the flow for some given time interval T. Thereby coherency in flows is characterized in terms of repelling and attracting manifolds.
Commonly it is defined based on the flow map, analyzing the separation of trajectories of nearby particles over a finite-time span. Another approach to compute the FTLE is by integrating a local separation measure. This measure is based on the Jacobian matrix as it is the generator of separation. This definition is called Localized FTLE and suggests an fast algorithm for computating this measure by reusing computed values along the pathline.
Finite Time Topology
Standard vector field topology separates a flow field into areas of similar streamline behavior in terms of limit sets. This is represented by a skeleton consisting of critical points and separatrices. A critical point is a point where the flow field vanishes; a separatrix is a stream line or stream surface that separates regions of different flow behavior. This approach is limited by the fact that this is only defined for steady vector fields. A basic drawback is that critical points and thus also vector field topology are Galilean-variant, i.e. they depend on the frame of reference. In a turbulent flow, however, there is no preferred frame of reference - at least sufficiently far from the boundaries. Furthermore, for unsteady flow fields the vector field topology can be extracted only for snapshots in time. These do not reveal important aspects of the unsteady flow. In particular so-called Lagrangian properties do not become visible. Lagrangian analysis follows the trajectories of individual fluid particles and observes the changes they experience, whereas Eulerian analysis captures what happens at fixed points in space.
As noted above, the analysis of the FTLE field is a widely used approach to analyze time-dependent fields. Unfortunately, it only considers separating and diverging structures. Other structures are not extracted by analysis of the FTLE field.
The goal of this project is to overcome these limitations by developing a more general concept than vector field topology that reveals Lagrangian coherent structures. Since we also consider flows for certain periods of time T and again aim at specific point-like features and a partitioning of space-time in subdomains of similar flow behaviour, we call it "Finite-Time Topology".
The first building block of a Galilean invariant flow topology is a new set of distinguished points, which we call Langrangian Equilibrium Points (LEP). The definition of LEPs is based on the acceleration field and therefore is Galilean invariant. In stationary flow fields the magnitude of the acceleration ||a|| vanishes at locations that are a superset of the critical points of the velocity field. In unsteady flows, however, additional terms contribute to ||a||. Thus, the acceleration magnitude does not vanish at such locations, but in general takes a local minimum. Therefore, we define Lagrangian Equilibrium Points as the points where the acceleration magnitude has a local minimum.
In order to extract only long living features based upon the acceleration, we define a 'feature lifetime' and use it for filtering. Moreover, this lifetime serves as an interval for averaging the acceleration magnitude. These features can be seen in Figure 2 for a mixing of Oseen vortices. The lifetime is depicted as illuminated pathline segments in Figure 3.
Derivative-free Methods for Flow Analysis
The analysis of flow fields is often done by extracting geometrical features from the field. These features can be lines or points or - in a three-dimensional setting - also surfaces. Since all flow fields resulting from simulation or experiments contain noise, robust feature extraction algorithms are needed. One approach is to use derivative-free methods that work directly on the underlying discrete field, do not require interpolation, and include topological controlled smoothing.
For analysis of two-dimensional time-dependent scalar fields, there are two approaches: using methods for three-dimensional data or using two-dimensional methods for each time step and track the extracted features over time. In this project, both approaches are investigated.
Vortex merge graphs
Instead of extracting vortex core lines in two-dimensional time-dependent flow fields as a set of three lines, we now consider the temporal domain explicitly. In this part of the project, we employ robust methods for extracting scalar field topology including a combinatorial tracking approach for critical points. Unfortunately, this approach is not able to extract mergers of critical points. However, vortex mergers are common events in time-dependent flow fields. We therefore adapt this approach to the underlying physics. It thereby stays compatible with homological persistence, which enables the extraction of vortex merge graphs even for complex flow configurations. The general approach is applicable to several different quantities and in this project we compare the results -- in particular the exact merge events resulting from the different quantities.
Time-dependent Vortex Regions
Since the acceleration field captures Galilean invariant properties of particle motion, it is a fundamental quantity describing fluid flows. Interestingly, vortex-like minima of the magnitude of the acceleration are enclosed by particularly pronounced ridges. This makes it possible to define boundaries of vortex regions in a parameter-free way. Utilizing scalar field topology, the robust algorithms presented above can be used to extract these boundaries. This novel definition of vortex regions combines the following three advantages:
1. It is Galilean invariant. The definition of the vortex regions is solely based on the acceleration, which is Galilean invariant. This is an essential property when dealing with time-dependent flows.
2. It allows for vortex regions of arbitrary shape. Using scalar field topology as the basis for the definition of vortex regions, the extraction algorithm is not restricted to star- or convex-shaped geometries.
3. It does not require any threshold like, e.g., an iso value. This enables an unsupervised extraction of vortex regions in complex time-dependent datasets. This is important, e.g., for very large data sets.
- J. Kasten, A. Zoufahl, H.-C. Hege, and I. Hotz. Analysis of vortex merge graphs. VMV 2012: Vision, Modeling and Visualization, Magdeburg, Germany, pp. 111-118.
- J.Kasten, I.Hotz, B.R. Noack, H.-C. Hege. Vortex merge graphs in two-dimensional unsteady flow fields. Proceedings Joint IEEE/EG Conference on Visualization -- EuroVis 2012, Short Paper Proceedings, pp. 1-5, Eurographics Association.
- H.-C. Hege, J. Kasten, I. Hotz. Distillation and Visualization of Spatiotemporal Structures in Turbulent Flow Fields. J. Phys.: Conf. Ser., Vol. 318, pp. 062009, 2011.
- J. Kasten, J. Reininghaus, I. Hotz, H.-C. Hege. Two-Dimensional Time-Dependent Vortex Regions Based on the Acceleration Magnitude. IEEE Transactions on Visualization and Computer Graphics, Vol. 17, No. 12, pp. 2080-2087, 2011.
- J. Kasten, I. Hotz, H.-C. Hege. On the Elusive Concept of Lagrangian Coherent Structures. Topological Methods in Data Analysis and Visualization II, Mathematics and Visualization Ronny Peikert, Helwig Hauser, Hamish Carr (Ed.), pp. 207 - 220, Springer, Berlin, 2011.
- J. Kasten, J. Reininghaus, K. Oberleithner, I. Hotz, B. R. Noack, H.-C. Hege. Flow over a Cavity -- Evolution of the Vortex Skeleton. Visualization at 28th Annual Gallery of Fluid Motion exhibit, held at the 63th Annual Meeting of the American Physical Society, Division of Fluid Dynamics (Long Beach, CA, USA, November 21-23, 2010)., 2010.
- J. Kasten, C. Petz, I. Hotz, G. Tadmor, B. R. Noack, H.-C. Hege. Lagrangian Feature Extraction of the Cyliner Wake. Physics of Fluids, Vol. 22, 9, Gallery of Fluid Motion Award 2009, 2010.
- J. Kasten, T. Weinkauf, C. Petz, I. Hotz, B. R. Noack, H.-C. Hege. Extraction of Coherent Structures from Natural and Actuated Flows. Active Flow Control II, 108:373-387, 2010.
- J. Kasten, I. Hotz, B. Noack, H.-C. Hege. On the Extraction of Long-living Features in Unsteady Fluid Flows. V. Pascucci et al. (ed.), pp. 115-126, Springer, Berlin, 2010.
- J. Kasten, C. Petz, I. Hotz, B. R. Noack, H.-C. Hege. Localized Finite-time Lyapunov Exponent for Unsteady Flow Analysis. In: Marcus Magnor, Bodo Rosenhahn and Holger Theisel, editors, Vision, Modeling and Visualization, vol. 1, pp. 265-274. Universität Magdeburg, Institut für Simulation und Graphik, 2009.
- T. Weinkauf, D. Guenther. Separatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods. Computer Graphics Forum, 28(5):1519–1528, 2009.
- K. Shi, H. Theisel, H. Hauser, T. Weinkauf, K. Matkovic, H.-C. Hege, H.-P. Seidel. Path Line Attributes - an Information Visualization Approach to Analyzing the Dynamic Behavior of 3D Time-Dependent Flow Fields. Topology-Based Methods in Visualization II, p. 75–88, 2009.
- W. von Funck, T. Weinkauf, H. Theisel, H.-P. Seidel. Smoke Surfaces: An Interactive Flow Visualization Technique Inspired by Real-World Flow Experiments. IEEE Transactions on Visualization and Computer Graphics, 14(6):1396–1403, 2008.
- K. Shi, H. Theisel, T. Weinkauf, H.-C. Hege, H.-P. Seidel. Visualizing Transport Structures of Time-Dependent Flow Fields. IEEE Computer Graphics and Applications, 28(5):24 – 36, 2008.
- T. Weinkauf. Extraction of Topological Structures in 2D and 3D Vector Fields, PhD Thesis. 2008.
- K. Shi, H. Theisel, T. Weinkauf, H.-C. Hege, H.-P. Seidel. Finite-Time Transport Structures of Flow Fields. Proc. IEEE Pacific Visualization 2008, p. 63–70, Kyoto, Japan, 2008.
- T. Weinkauf, J. Sahner, B. Gunther, H. Theisel, H.-C. Hege, F. Thiele. Feature-based Analysis of a Multi-Parameter Flow Simulation. Proc. SimVis 2008, p. 237–252, Magdeburg, Germany, 2008.
- R. S. Laramee, G. Erlebacher, C. Garth, T. Schafhitzel, H. Theisel, X. Tricoche, T. Weinkauf, D. Weiskopf. Applications of Texture-Based Flow Visualization. Engineering Applications of Computational Fluid Mechanics (EACFM), 2(3):264–274, 2008.
- H. Theisel, C. Rossl, T. Weinkauf. Morphological Representations of Vector Fields. Shape Analysis and Structuring, p. 215–240, 2008.