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.
Real-world flow visualization
The properties of a flow can be explained by considering the motion of particles. A particle moves with the fl ow 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 fl ow. 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.
Figure 1: The time lines of the fl ow behind a cylinder reveal – in contrast to particle trajectories – the swirling motion in the fl ow.
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 fl ow 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.
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 com- pact mathematical form. Figure 2 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 time lines in Figure 1 were computed within 14 seconds using our new method. The classic algorithm requires 45 minutes for this.