On the nonlinear growth of two-dimensional Tollmien-Schlichting waves in aflat-plate boundary layer

This paper studies the nonlinear development of two-dimensional Tollmien-Sc hlichting waves in an incompressible flat-plate boundary layer at asymptoti cally large values of the Reynolds number. Attention is restricted to the ' far-downstream lower-branch' regime where a multiple-scales analysis is pos sible. It is supposed that to leading-order the waves are inviscid and neut ral, and governed by the [Davis-Acrivos-] Benjamin-Ono equation. This has a three-parameter family of periodic solutions, the large-amplitude (soliton ) limit of which bears a qualitative resemblance to the 'spikes' observed i n certain 'K-type' transition experiments. The variation of the parameters over slow length- and timescales is controlled by a viscous sublayer. For t he case of a purely temporal evolution, it is shown that a solution for thi s sublayer ceases to exist when the amplitude reaches a certain finite valu e. For a purely spatial evolution, it appears that an initially linear dist urbance does not evolve to a fully nonlinear stage of the envisaged form. T he implications of these results for the 'soliton' theory of spike formatio n are discussed.