A Phase Equation Approach to Boundary Layer Instability Theory: Tollmien Schlichting Waves
نویسنده
چکیده
Our concern is with the evolution of large amplitude Tollmien-Schlichting waves in boundary layer ows. In fact the disturbances we consider are of a comparable size to the unperturbed state. We shall describe two-dimensional disturbances which are locally periodic in time and space. This is achieved using a phase equation approach of the type discussed by Howard and Kopell (1977) in the context of reaction-di usion equations. We shall consider both large and O(1) Reynolds numbers ows though, in order to keep our asymptotics respectable, our nite Reynolds number calculation will be carried out for the asymptotic suction ow. Our large Reynolds number analysis, though carried out for Blasius ow, is valid for any steady two-dimensional boundary layer. In both cases the phase equation approach shows that the wavenumber and frequency will develop shocks or other discontinuities as the disturbance evolves. As a special case we consider the evolution of constant frequency/wavenumber disturbances and show that their modulational instability is controlled by Burgers equation at nite Reynolds number and by a new integro-di erential evolution equation at large Reynolds numbers. For the large Reynolds number case the evolution equation points to the development of a spatially localized singularity at a nite time. The three-dimensional generalizations of the evolution equations is also given for the case of weak spanwise modulations. This research was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. This work was also supported by SERC and NSF under contract NSF CTS-9123553
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