Acoustic Receptivity of a Boundary-Layer to Tollmien-Schlichting Waves Resulting From a Finite-Height Hump At Finite Reynolds-Numbers

The acoustic receptivity of a boundary layer to Tollmien-Schlichting (T-S) waves resulting from a finite-height hump at finite Reynolds numbers is investigated. The steady flow is calculated using an interacting boundary-layer (IBL) scheme that accounts for viscous/inviscid interactions. The unsteady flow is written as the sum of a Stokes wave and a traveling wave generated due to the interaction of the Stokes flow with the steady disturbance resulting from the hump. The traveling wave is governed by a set of nonhomogeneous equations, which is a generalization of the Orr-Sommerfeld equation. The solution of these nonhomogeneous equations is projected onto the quasiparallel eigenmode using the quasiparallel adjoint. This leads to a nonhomogeneous equation with variable coefficients governing the amplitude and phase of the T-S wave. Results are presented for the amplitude variation and the receptivity at finite Reynolds numbers. The results are in good agreement with the experimental results of Saric, Hoos, and Radeztsky [Boundary Layer Stability and Transition to Turbulence (ASME, New York, 1991), FED No. 114, pp. 17-22] for all tested hump heights at the two-tested sound pressure levels. Application of this paper's theory to small humps yields results that agree with those of Choudhari and Streett [Phys. Fluids A 4, 2495 (1992)]; and Crouch [Phys. Fluids A 4, 1408 (1992)]. Application of suction is shown to reduce the receptivity resulting from the hump.