A composite asymptotic model for the wave motion in a steady three-dimensional subsonic boundary layer

Abstract

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 boundary-value problem describes eigenmodes of different nature (including crossflow vortices) coupled together. The frequency ω of the eigenmodes is connected with the components k and m of the wavenumber vector through a dispersion relation. This relation exhibits two singular properties. One of them is of basic importance since it makes the imaginary part Im(ω) of the frequency increase without bound as k and m tend to infnity along some curves in the real (k, m)-plane. The singularity 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 significant alterations in the interaction law. A new mathematical model leans upon a set of composite equations without rescaling the original independent variables and desired functions. As a result, the right-hand side of a modified dispersion relation involves an additional term multiplied by a small parameter ε=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 validity of a linear approximation becomes far more extended in ω, k and m, but the incorporation of the higher-order term into the interaction law means in essence that the Reynolds number is retained in the formulation of a key problem for the lower deck.