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Adjoint systems and their role in the receptivity problem for boundary layers. (English) Zbl 0866.76029
The author introduces an alternative treatment of the receptivity problem by extensively using the properties of adjoint solutions. He demonstrates that the adjoint eigensolution field defines the efficiency with which a particular forcing excites the eigensolution. The amplitude of the convectively unstable eigensolution induced by a harmonic point forcing is shown to be simply the value of its adjoint eigensolution at the forcing location and is equivalent to the residue of the Fourier inversion integral. This is demonstrated for the examples of a vibrating ribbon and the excitation of an inviscid free shear layer by a vorticity source. For the scattering of acoustic waves into Tollmien-Schlichting waves in the presence of a small surface roughness, an inhomogeneous adjoint problem is solved. The solution approaches the triple-deck solution of M. E. Goldstein [J. Fluid Mech. 154, 509-529 (1985; Zbl 0576.76064)] close to the lower branch for high Reynolds numbers and agrees directly with the finite-Reynolds number computations of J. D. Crouch [Phys. Fluids, A, 4 1408-1414 (1992; Zbl 0825.76159)]and M. Choudhari and C. L. Strett [Phys. Fluids, A, 4, No. 11, 2495-2514 (1992; Zbl 0762.76024)].

MSC:
76E05 Parallel shear flows in hydrodynamic stability
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
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