Black and White Manifolds
Melissa Green GS
Department of Mechanical and Aerospace Engineering
In the area of fluid mechanics, one particular source of controversy is an analytical definition of a vortex. A tornado ripping through the atmosphere is easy to identify, but in a mathematical sense, what velocity information is needed and how can we identify where the vortex begins and where it ends? A new method to do this involves following particle trajectories and investigating where the distances between trajectories stretch. This image shows some results of that work. We look at a very chaotic, periodic velocity field, named the ABC flow after its developers. We integrate trajectories both forward and backward in time, and observe the maximum stretching that occurs. The panels in the picture represent two-dimensional cuts through the three-dimensional velocity field. The top left and bottom right panels show a contour plot of the stretching parameter calculated in backward time, with the bright white curves illustrating the maxima. The plot in the top right and bottom left is the same type of contour, but for trajectories integrated in forward time. The most interesting aspect of this plot is the symmetry of the two calculations. This is suspected to be due to the periodicity of the flow, but a complete and satisfying conclusion has not yet been obtained.