Hard wall - soft wall - vorticity scattering in shear flow

An analytically exact solution, for the problem of low Mach number incident vorticity scattering at a hard-soft wall transition, is obtained in the form of Fourier integrals by using the Wiener-Hopf method. Harmonic vortical perturbations of inviscid linear shear flow are scattered at the wall transition. This results in a far field which is qualitatively different for low shear and high shear cases. In particular, for high shear the pressure (apparently driven by the mean flow) does not decay and its Fourier representation involves a diverging integral which is to be interpreted in generalised sense. Then the incompressible hydrodynamic (Wiener-Hopf) "inner" solution is matched asymptotically to an acoustic outer field in order to determine the sound associated to the scattering. The qualitative difference between low and high shear is also apparent here. The low shear case matches successfully. In the high shear case only a partial matching was possible.