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
• Separatrices
• Vortices
• 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.

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.