On the Modulational Instability of O(1) Amplitude Waves in Supersonic Boundary Layers

The evolution of large amplitude Tollmien-Schlichting waves in a supersonic boundary layer is investigated. Disturbances which have their wavenumber and frequency slowly varying in time and space are described using a phase equation type of approach. Unlike the incompressible case we nd that the initial bifurcation to a nite amplitude Tollmien-Schlichting wave is subcritical for most Mach numbers. In fact the bifurcation is only supercritical for a small range of Mach numbers and even then for only a nite range of wave propagation angles. The modulational instability of large amplitude wavetrains is considered and is shown to be governed by an equation similar to Burgers equation but with the viscous term replaced by a fractional derivative. A numerical investigation of the solution of this equation is described. It is shown that uniform wavetrains are unstable. This research was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 236810001. This research was also supported by the Air Force O ce for Scienti c Research under Grant No. F49620-94-1-0242.