A COMPOSITE ASYMPTOTIC MODEL FOR THE WAVE MOTION IN A STEADY 3-DIMENSIONAL SUBSONIC BOUNDARY-LAYER

The problem for a thin near-wall region is reduced, within the triple- deck approach, to unsteady three-dimensional nonlinear boundary-layer equations subject to an interaction law. A linear version of the bound ary-value problem describes eigenmodes of different nature (including crossflow vortices) coupled together. The frequency omega of the eigen modes is connected with the components k and m of the wavenumber vecto r through a dispersion relation. This relation exhibits two singular p roperties. One of them is of basic importance since it makes the imagi nary part Im(omega) of the frequency increase without bound as k and m tend to infinity along some curves in the real (k,m)-plane. The singu larity turns out to be strong, rendering the Cauchy problem ill posed for linear equations. Accounting for the second-order approximation in asymptotic expansions for the upper and main decks brings about signi ficant alterations in the interaction law. A new mathematical model le ans upon a set of composite equations without rescaling the original i ndependent variables and desired functions. As a result, the right-han d side of a modified dispersion relation involves an additional term m ultiplied by a small parameter epsilon = R-1/8, R being the reference Reynolds number. The aforementioned strong singularity is missing from solutions of the modified dispersion relation. Thus, the range of val idity of a linear approximation becomes far more extended in omega, k and m, but the incorporation of the higher-order term into the interac tion law means in essence that the Reynolds number is retained in the formulation of a key problem for the lower deck.