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A. Kluwick, S. Braun:
"Response of marginally separated boundary layers to three-dimensional unsteady perturbations";
Talk: 5th European Fluid Mechanics Conference (EFMC), Toulouse/FR; 08-24-2003 - 08-28-2003.

Two-dimensional marginally separated laminar boundary layers have been
investigated first by Ruban 1981 and independently by Stewartson, Smith
and Kaups 1982 who showed
that the nondimensional wall shear (or equivalently the nondimensional
perturbation displacement thickness) A is governed by a nonlinear
integro-differential equation. This equation was found to have solutions up to
a critical value Gamma_c of the controlling parameter Gamma
characterizing, for example, the angle of attack of a slender airfoil.

Here we investigate three-dimensional unsteady perturbations of a
two-dimensional marginally separated laminar boundary layer with special
emphasis on the flow behaviour near Gamma_c. Specifically, it is shown that
the integro-differential equation which governs these disturbances if
Gamma_c-Gamma=O(1) reduces to a nonlinear partial equation - known as
Fisher equation, Fisher 1937 - as Gamma approaches the critical value
Gamma_c. This in
turn leads to a significant simplification of the problem allowing, among
others, a systematic study of devices used in boundary layer control and
an analytical analysis of the conditions leading to the formation of
finite time singularities which have been observed in earlier numerical studies
of two-dimensional, Smith 1982, and three-dimensional flows in the vicinity of
a line of symmetry, Duck 1990. Also it
is found possible to construct exact solutions which describe waves of constant
form travelling in the spanwise direction. These waves may contain
singularities which can be interpreted as vortex sheets. The existence of these
solutions strongly suggests that solutions of the Fisher equation which lead to
finite time blow up may be extended beyond the blow-up time thereby generating
vortical structures qualitatively similar to those emerging in direct numerical
simulations of near critical (i.e. transitional) laminar separation bubbles.
This is supported by asymptotic analysis.