Application of biorthogonal eigenfunction system for extraction of Tollmien-Schlichting waves in acoustic receptivity simulations

Acoustic receptivity of a two-dimensional boundary layer on a flat plate with elliptic leading edge is studied through direct numerical simulation (DNS). Sound waves are modelled by a uniform oscillation of freestream boundaries in time which results to an infinite-wavelength acoustic wave. Acoustic disturbances interact with strong streamwise gradients at the leading edge or surface nonhomogeneities and create Tollmien-Schlichting (TS) waves inside the boundary layer. Measuring amplitude of TS waves created by sound waves is challenging due to presence of Stokes wave (acoustic boundary layer) with the same temporal frequency of TS waves. In this study biorthogonal eigenfunction system of local linear stability equations has been utilised to extract TS wave amplitudes. This method is based on the concept of using adjoint mode as a projector where the TS amplitude is obtained by projecting the DNS solution onto adjoint TS modes. However, the computed TS wave amplitude employing this method found to be modulated. It is shown that the modulation is due to existence of a small amplitude wave in the DNS data that is not expandable onto the basis of local linear stability equations.