ADJOINT SYSTEMS AND THEIR ROLE IN THE RECEPTIVITY PROBLEM FOR BOUNDARY-LAYERS

The effectiveness with which various sources excite convective instabi lities in a boundary layer is found by a simple method. Chosen field v alues of the adjoint to the Tollmien-Schlichting eigensolution, normal ized appropriately, indicate the amplitude of the unstable disturbance which will result for direct time-harmonic forcing by sources of mome ntum, mass and vorticity, as well as by boundary motions. For the Blas ius boundary layer, forcing in the vicinity of the critical layer indu ces the largest response. At this position, the response to forcing in the wall-normal direction is typically 5% of that resulting from stre amwise forcing of the same magnitude. At the wall, normal motions elic it a much stronger response than streamwise motions. Forcing close to the lower branch of the neutral stability curve leads to the largest r esponse. The adjoint field values are equivalent to the residues of Fo urier-inversion integrals. This equivalence is discussed for two probl ems; the vibrating ribbon problem and excitation of an inviscid free s hear layer by a vorticity source. The efficiency factor is calculated for the scattering of 'acoustic' waves into Tollmien-Schlichting waves in the presence of small surface roughness, at a finite Reynolds numb er, based on the Orr-Sommerfeld operator. This is achieved by using th e solution of an inhomogeneous adjoint problem. The results are compar ed with the asymptotic solutions obtained from triple-deck theory, and agree with previous finite-Reynolds-number calculations.