Homological Analysis of Schlieren Images of Supersonic Elliptical Jets

This research explores the application of computational homology to the analysis of schlieren photographs of supersonic elliptical jets. In this study, computational homology is used to compute topological invariants called the Betti numbers, which are values associated with the homology groups of a topological space that measure the number of connected components and enclosed regions in the space. To compute the Betti numbers, grayscale schlieren images of supersonic elliptical jets are converted into images that have only black and white pixels. Different threshold values defining black and white regions emphasize different flow structures associated with variations in the fluid density. The schlieren images are for supersonic elliptical jets issuing from a nozzle designed to produce a shock-free jet with a Mach number of 1.52. The jet has an aspect ratio of two with a minor axis length of one inch. Schlieren images are analyzed for both an under-expanded jet and an over-expanded jet with Mach numbers of 1.36 and 1.64 respectively. Both of these off-design jet flows produce complex shock wave phenomena. Computed Betti numbers are discussed in terms of jet aerodynamics as well as the methods used to convert the schlieren images to pictures with only black and white regions.