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S. Scheichl, A. Kluwick:
"On the effects of compressibility in marginally separated flows.";
Talk: GAMM 2008, Bremen, Germany; 03-31-2008 - 04-04-2008; in: "Proceedings of the annual GAMM meeting", PAMM, 8/1 (2008), ISSN: 1617-7061; 10639 - 10640.

If the angle of attack α of a slender airfoil reaches a critical value α_s flow separation is known to occur at the upper surface. Further increase of α initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field - a phenomenon known as marginal separation - but then rather rapidly causes a severe change of the flow behaviour (leading edge stall). According to the asymptotic theory of marginal separation holding in the limit of large Reynolds numbers Re → ∞, the flow in the neighbourhood of the separation bubble is governed by an integrodifferential equation exhibiting a single controlling parameter Γ which relates the angle of attack to the Reynolds number, with a value Γ_s corresponding to α_s. Numerical investigations have shown that solutions to this equation exist up to a critical value Γ_c > Γ_s only and this has been taken as an indication that a substantial change of the flow field must take place if Γ exceeds Γ_c. In the case of unsteady three-dimensional perturbations of a two-dimensional steady marginally separated boundary layer in the limit Γ → Γ_c, the (generalized form of the) integro-differential equation reduces to a nonlinear diffusion equation of Fisher type. This asymptotic description serves as a basis for the investigations concerning compressibility effects, also in respect of the mechanism through which the hydromechanic motion inside the boundary layer could be transformed into aerodynamic sound. A modified interaction equation is obtained, which in addition to Γ, contains the Mach number at the point of separation as a second controlling parameter. The unsteady changes of the boundary layer displacement thickness in the limit ΔΓ = Γ_c − Γ → 0, again governed by Fisher´s equation, then are identified to act as a quadrupole source for sound radiation. The most surprising outcome is the fact that in the three-dimensional case, the second derivative with respect to the lateral direction contained in the Fisher equation changes its sign and becomes negative if the Mach number exceeds a certain value.

http://dx.doi.org/10.1002/pamm.200810639