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The FTLE field of the double gyre with parameters t 0 = 0 and T = 15 (i.e., 1.5 periods). (a) The unfiltered field. (b) The filtered by setting FTLE values to 0 for sep(x) ≤ 0.5. Path lines confirm that the persistent ridge is indeed due to spatial separation.
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In many cases, feature detection for flow visualization is structured in two phases: first candidate identification, and then filtering. With this paper, we propose to use the directional information contained in the finite-time Lyapunov exponents (FTLE) computation, in order to filter the FTLE field. Thereby we can focus on those separation struct...
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... × [0, 1] × R. All com- putations for this example have been carried out using the MAPLE software pack- age. The flow map was computed using MAPLEs seventh-eighth order continuous Runge-Kutta method dverk78, for estimation of the deformation gradient tensor we used central finite differences in the coordinate directions with spacing h = 0.01. Fig. 4(a) shows the FTLE field with parameters t 0 = 0 and T = 15, i.e., 1.5 periods. The filtering is emulated by setting the FTLE value of points with sep(x) ≤ 0.5 to 0. We see that the filtering produces sharper ridges as the original FTLE field, high- lighting in particular one ridge associated with rather low FTLE values. Seeding path lines ...
Citations
... A detailed description of the analysis steps is given in Sec.4.3. Figure 11: Final result of the interactive flow analysis based on the her proposed feature set. We are able to identify the recirculation area in front of the of the obstacle described by Pobitzer et al. [23] different types of path line behavior present. Comparing the seed points of the path lines found by our analysis to the seed points of the reference path lines, we see a clear correspondence of the two sets ( Fig. 9, in contrast to the situation in Fig. 6(a) bottom). ...
... This data set has been investigated by Pobitzer et al. [23] in the context of finite-time Lyapunov exponents (FTLE). One of the interesting features is a separation structure in front of the obstacle, separating particles passing at the two sides of the obstacle. ...
... In Fig. 11 we see the selection and the resulting path lines. We conclude that we found the recirculation zone Pobitzer et al. found the boundary of in their paper [23]. Investigating the remaining attributes, we see a clear split in the second quadratic statistical invariant, color coding the path lines according to this attribute, yields Fig. 12, revealing that the left-right separation structure is also present inside the recirculation, an insight the FTLE-based analysis of Pobitzer et al. failed to convey. ...
Recent work has shown the great potential of interactive flow analysis by the analysis of path lines. The choice of suitable attributes, describing the path lines, is, however, still an open question. This paper addresses this question performing a statistical analysis of the path line attribute space. In this way we are able to balance the usage of computing power and storage with the necessity to not loose relevant information. We demonstrate how a carefully chosen attribute set can improve the benefits of state-of-the art interactive flow analysis. The results obtained are compared to previously published work.
... The currently most common approach to detect such behavior makes use of the FTLE, which is, informally speaking, the maximal local separation rate. We developed a filter that distinguishes between separation due to different flow directions and from separation due to different flow speeds [4]. The filter follows the geometric intuition behind the original definition of FTLE. ...
The model of a double gyre flow by Shadden et al. is a standard benchmark data set for the computation of hyperbolic Lagrangian Coherent Structures (LCS) in flow data. While structurally extremely simple, it generates hyperbolic LCS of arbitrary complexity. Unfortunately, the double gyre does not come with a well‐defined ground truth: the location of hyperbolic LCS boundaries can only be approximated by numerical methods that usually involve the gradient of the flow map. We present a new benchmark data set that is a small but carefully designed modification of the double gyre, which comes with ground truth closed‐form hyperbolic trajectories. This allows for computing hyperbolic LCS boundaries by a simple particle integration without the consideration of the flow map gradient. We use these hyperbolic LCS as a ground truth solution for testing an existing numerical approach for extracting hyperbolic trajectories. In addition, we are able to construct hyperbolic LCS curves that are significantly longer than in existing numerical methods.
Vector Field Topology describes the asymptotic behavior of a flow in a vector field, i.e., the behavior for an integration time converging to infinity. For some applications, a segmentation of the flow in areas of similar behavior for a finite integration time is desired. We introduce an approach for a finite-time segmentation of a steady 2D vector field which avoids the systematic evaluation of the flow map in the whole flow domain. Instead, we consider the separatrices of the topological skeleton and provide them with additional information on how the separation evolves at each point with ongoing integration time. We analyze this behavior and its distribution along a separatrix, and we provide a visual encoding for it. The result is an augmented topological skeleton. We demonstrate the approach on several artificial and simulated vector fields.
Integration-based flow visualization provides important visual cues about fluid transport. Analyzing the behavior of infinitesimal volumes as opposed to the behavior of rigid particles provides additional details valuable to flow visualization research. Our work concentrates on examining the local velocity gradient tensor along the path of a particle seeded within time-varying flow to produce a visualization highlighting temporal characteristics of particle behaviors, such as deformation. We present a framework for the analysis and visualization of such characteristics, focused on providing concise representations of physically meaningful flow features such as separation regions and vorticity. We apply the derived techniques to two data sets, highlighting the importance of such higher order Lagrangian analysis techniques to time-varying flow analysis.
It is a wide-spread convention to identify repelling Lagrangian Coherent Structures (LCSs) with ridges of the forward finite-time Lyapunov exponent (FTLE) field and to identify attracting LCSs with ridges of the backward FTLE. However, we show that, in two-dimensional incompressible flows, also attracting LCSs appear as ridges of the forward FTLE field. This raises the issue of the characterization of attracting LCSs using a forward finite-time Lyapunov analysis. To this end, we extend recent results regarding the relationship between forward and backward maximal and minimal FTLEs, to both the whole finite-time Lyapunov spectrum and to stretch directions. This is accomplished by considering the singular value decomposition (SVD) of the linearized flow map. By virtue of geometrical insights from the SVD, we provide characterizations of attracting LCSs in forward time for two geometric approaches to hyperbolic
The Finite Time Lyapunov Exponent (FTLE) has become a widespread tool for analyzing unsteady flow behavior. For its computation, several numerical methods have been introduced, which provide trade-offs between performance and accuracy. In order to decide which methods and parameter settings are suitable for a particular application, an evaluation of the different FTLE methods is necessary. We propose a general benchmark for FTLE computation, which consists of a number of 2D time-dependent flow fields and error measures. Evaluating the accuracy of a numerically computed FTLE field requires a ground truth, which is not available for realistic flow data sets, since such fields can generally not be described in a closed form. To overcome this, we introduce approaches to create non-trivial vector fields with a closed-form formulation of the FTLE field. Using this, we introduce a set of benchmark flow data sets that resemble relevant geometric aspects of Lagrangian structures, but have an analytic solution for FTLE. Based on this ground truth, we perform a comparative evaluation of three standard FTLE concepts. We suggest error measures based on the variance of both, the fields and the extracted ridge structures.
