This thesis is an interdisciplinary work in the field of scientific visualization as part of computer science and the field of fluid mechanics. It is focussed on the analysis of time-dependent, two-dimensional flow fields. In this set- ting, the search for relevant structures – often called features – is one of the main topics. In my thesis, I am concerned with the extraction of Lagrangian coherent structures (LCSs). While this concept is one of the most discussed in the literature, there exists no commonly accepted definition. For instance, some researchers associate LCSs with vortices and others with distinguished manifolds of particle divergence and convergence. Due to the vague notion of LCSs, their extraction is split into different domain-specific and algorithmic challenges: What quantities are useful for identifying these features? How can these structures be robustly extracted? How can they be tracked efficiently over time? What are appropriate measures that enable a spatiotemporal filtering of the extracted features? I contribute to the above-mentioned questions by investigating the finite-time Lyapunov exponent (FTLE) and the acceleration.
The FLTE measures the separation and convergence of particles. Structures visible in the FTLE field are a popular realization of LCSs. In this thesis, an alternative algorithm for computing the FTLE field is given based on the Jacobian of the flow field. In addition, a critical review of the FTLE approach shows problems regarding the applicability to complex flow configurations.
Using the acceleration, I begin with transferring the concept of critical points of velocity field topology to time-dependent flow fields. This concept does only reveal significant structures for stationary flow fields. I show that features defined as minima of the acceleration magnitude serve as time-dependent counterparts of these points. These minima are introduced in this thesis as Lagrangian equilibrium points (LEPs). Similar to the centers of standard velocity field topology, a subset of the LEPs represents vortices. Within the concept of LEPs, I present three major contributions. At first, I introduce a hierarchy that is based on a spatiotemporal importance measure. It consists of the lifetime of the features and will be later on combined with homological persistence. The second contribution is the robust extraction of the LEPs and their evolution. I present an approach to extract a vortex merge graph. An existing tracking approach is adapted to the underlying physics while staying compatible with homological persistence, which enables a noise resilient extraction. Last, I present an approach to robustly extract vortex regions and their evolution. Employing the same robust combinatorial tools as for the vortex merge graph, I show how vortex regions can also be based on the acceleration magnitude. I also investigate the resulting vortex merge graph and the associated vortex regions based on the acceleration magnitude and compare the acceleration to other vortex related quantities.
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