Adjoint systems and their role in the receptivity problem for boundary layers

Abstract

The effectiveness with which various sources excite convective instabilities in a boundary layer is found by a simple method. Chosen field values of the adjoint to the Tollmien–Schlichting eigensolution, normalized appropriately, indicate the amplitude of the unstable disturbance which will result for direct time-harmonic forcing by sources of momentum, mass and vorticity, as well as by boundary motions. For the Blasius boundary layer, forcing in the vicinity of the critical layer induces the largest response. At this position, the response to forcing in the wall-normal direction is typically 5% of that resulting from streamwise forcing of the same magnitude. At the wall, normal motions elicit a much stronger response than streamwise motions. Forcing close to the lower branch of the neutral stability curve leads to the largest response. The adjoint field values are equivalent to the residues of Fourier-inversion integrals. This equivalence is discussed for two problems; the vibrating ribbon problem and excitation of an inviscid free shear 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 number, based on the Orr–Sommerfeld operator. This is achieved by using the solution of an inhomogeneous adjoint problem. The results are compared with the asymptotic solutions obtained from triple-deck theory, and agree with previous finite-Reynolds-number calculations.