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.
Measurement in Unsteady Boundary Layer of Propagating Pressure Wave by Laser Differential Interferometry