Green's functions for a source in a boundary layer: direct waves, channelled waves and diffracted waves

Green's functions for a source embedded in an isothermal transversely sheared boundary layer are compared with direct numerical simulation (DNS) at various frequencies and free-stream Mach numbers. The procedures developed for a mixing layer in Part 1 (Suzuki & Lele 2003) are applied to derive the low- and high-frequency Green's functions for direct waves, i.e. the third-order convective wave equation is solved using asymptotic matching. In addition, channelled waves propagating downstream along the wall are analysed using the normal mode decomposition. By introducing an adjoint operator of the convective wave equation with a mixed-type boundary condition on the wall, the corresponding Hilbert space is defined and eigenfunctions of channelled waves are normalized. Furthermore, diffracted waves in the shadow zone are formulated in the high-frequency limit. These theoretical predictions are compared with numerical simulations in two dimensions: DNS are performed based on the full Navier-Stokes equations (the ratios between the acoustic wavelength and the boundary layer thickness are lambda/delta(BL) = 4.0, 1.0, and 0.25 at a free-stream Mach number of M-infinity = 0.8; and lambda/delta(BL) = 1.0 at M-infinity = 0.3 and 1.2). The DNS results generally agree with the theories: the pressure amplitudes of direct waves and diffracted waves follow the high-frequency limit with a reasonable degree of accuracy in the intermediate- and high-frequency cases (lambda/delta(BL) = 1.0 and 0.25). The DNS results for channelled waves also agree with the theoretical predictions fairly well. In addition, the acoustic impedance on the wall under a strongly sheared viscous boundary layer is derived asymptotically based on the modal analysis.