Receptivity in parallel flows: An adjoint approach

Linear receptivity studies in parallel flows are aimed at understanding how external forcing couples to the natural unstable motions which a flow can support. The vibrating ribbon problem models the original Schubauer and Skramstad boundary layer experiment and represents the classic boundary layer receptivity problem. The process by which disturbances are initiated in convectively-unstable jets and shear layers has also received attention. Gaster was the first to handle the boundary layer analysis with the recognition that spatial modes, rather than temporal modes, were relevant when studying convectively-unstable flows that are driven by a time-harmonic source. The amplitude of the least stable spatial mode, far downstream of the source, is related to the source strength by a coupling coefficient. The determination of this coefficient is at the heart of this type of linear receptivity study. The first objective of the present study was to determine whether the various wave number derivative factors, appearing in the coupling coefficients for linear receptivity problems, could be reexpressed in a simpler form involving adjoint eigensolutions. Secondly, it was hoped that the general nature of this simplification could be shown; indeed, a rather elegant characterization of the receptivity properties of spatial instabilities does emerge. The analysis is quite distinct from the usual Fourier-inversion procedures, although a detailed knowledge of the spectrum of the Orr-Sommerfeld equation is still required. Since the cylinder wake analysis proved very useful in addressing control considerations, the final objective was to provide a foundation upon which boundary layer control theory may be developed.