A Phase Equation Approach to Boundary Layer Instability Theory: Tollmien Schlichting Waves
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING HAMPTON VA
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Our concern is with the evolution of large amplitude Tollmien-Schlichting waves in boundary layer flows. 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-diffusion equations. We shall consider both large and 01 Reynolds numbers flows though, in order to keep our asymptotics respectable, our finite Reynolds number calculation will be carried out for the asymptotic suction flow. Our large Reynolds number analysis, though carried out for Blasius flow, 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 frequencywavenumber disturbances and show that their modulational instability is controlled by Burgers equation at finite Reynolds number and by a new integro-differential 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 finite time. The three-dimensional generalizations of the evolution equations is also given for the case of weak spanwise modulations. Boundary layer, Phase equation.
- Fluid Mechanics