Os. Ryzhov et Ed. Terentev, A COMPOSITE ASYMPTOTIC MODEL FOR THE WAVE MOTION IN A STEADY 3-DIMENSIONAL SUBSONIC BOUNDARY-LAYER, Journal of Fluid Mechanics, 337, 1997, pp. 103-128
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.