# Characteristic Curves of Vector Fields

## by Tino Weinkauf • June 8, 2012 • Highlights

*Flows of fluids or gases are of great importance in many technical processes. For example, designing a fuel efficient car involves reducing its aerodynamic drag. To this end, the flow around the car is simulated with a computer program. A subsequent flow analysis reveals vortex structures, which usually have a negative influence on the aerodynamic drag. Modern analysis methods aim at raising the physical understanding of a flow in order to allow for conclusions about necessary changes to the car’s design.*

*Here we discuss new visualization methods for flows that became available due to novel mathematical insights. They bridge a longstanding gap between flow visualization in real-world experiments and computer-based flow visualization.*

## Real-World Flow Visualization

The properties of a flow can be explained by considering the motion of particles. A particle moves with the flow on its so-called *trajectory*, which is a line consisting of all point locations visited by the particle over time. Unfortunately, a trajectory is hard to visualize in a real-world flow experiment since a simple photo shows only the current location of the particle. Therefore, other *characteristic curves* are used for visualization: a *streak line* consists of a large number of particles, which have been injected into the flow one after another from the same location. A common way to achieve this effect in a flow lab is to constantly release smoke from a nozzle. It moves with the flow and thereby forms the streak line. A *time line* is another means of visualizing a flow. It is created by an instantaneous release of smoke from a slit: the initially straight line of smoke is transported by the flow and rolls up in vortices. This reveals interesting flow patterns.

## Computer-Based Flow Visualization

While streak and time lines are easier to visualize than particle trajectories in real-world experiments, it is exactly the other way around in computer-based visualizations. It is well-known that particle trajectories can be expressed mathematically using *ordinary differential equations*, which can be solved using standard methods. Furthermore, these equations allow conclusions about important properties of particle trajectories. For example, it is possible to determine their curvature without actually computing the trajectories themselves. Many important methods of computer-based flow analysis rely on the simple, yet powerful, representation of particle trajectories using ordinary differential equations.

For a long time, only comparatively complex algorithms existed for the computation of streak and time lines, and they did not allow any conclusions about the inherent properties of these lines. Together with our collaboration partner, Prof. Dr. Holger Theisel of Magdeburg University, we have succeeded in developing a novel mathematical approach that allows streak and time lines to be described using ordinary differential equations. This work was awarded the *Best Paper Award* of the annual IEEE Visualization conference.

## New Possibilities

This allows for a wide range of new methods for analyzing flows since the inherent properties of streak and time lines can now be expressed in a compact mathematical form. The Figure to the right shows a so-called vortex core line in the center of swirling streak lines. For the first time, such vortex cores can be computed using our novel mathematical approach. They are an important basis for the identification of vortices in flows.

Furthermore, streak and time lines can often be computed significantly faster using the new approach. The 5000 streak lines in the Figure below were computed within a minute using our new method. The classic algorithm requires more than two hours for this.