Three-dimensional acoustic-roughness receptivity of a boundary layer on an airfoil:: experiment and direct numerical simulations

The paper is devoted to an experimental and numerical investigation of the problem of excitation of three-dimensional Tollmien-Schlichting (TS) waves in a boundary layer on an airfoil owing to scattering of an acoustic wave on localized microscopic surface non-uniformities. The experiments were performed at controlled disturbance conditions on a symmetric airfoil section at zero angle of attack. In each set of measurements, the acoustic wave had a fixed frequency f(ac), in the range of unstable TS-waves. The three-dimensional surface non-uniformity was positioned close to the neutral stability point at branch I for the two-dimensional perturbations. To avoid experimental difficulties in the distinction of the hot-wire signals measured at the same (acoustic) frequency but having a different physical nature, the surface roughness was simulated by a quasi-stationary surface non-uniformity (a vibrator) oscillating with a low frequency f(v). This led to the generation of TS-wavetrains at combination frequencies f(1,2) = f(ac) -/+ f(v). The spatial behaviour of these wavetrains has been studied in detail for three different values of the acoustic frequency. The disturbances were decomposed into normal oblique TS-modes. The initial amplitudes and phases of these modes (i.e. at the position of the vibrator) were determined by means of an upstream extrapolation of the experimental data. The shape of the vibrator oscillations was measured by means of a laser triangulation device and mapped onto the Fourier space. The direct numerical simulations (DNS) are based on the vorticity-velocity formulation of the complete Navier-Stokes equations using a uniformly spaced grid in the streamwise and wall-normal direction and a spectral representation in the spanwise direction. For the present investigation, the sound wave in the free stream is prescribed as a solution of the second Stokes' problem at inflow, and a novel wall model has been implemented. The three-dimensional simulations were performed for a stationary surface non-uniformity. As a result of the present study, the acoustic receptivity coefficients are obtained both in experiment and DNS as functions of the spanwise wavenumber and acoustic frequency. The receptivity amplitudes and phases obtained numerically are in very good agreement with those obtained experimentally in the studied range of parameters. The scattering of acoustic waves on three-dimensional surface non-uniformities is found to be significantly stronger than that on two-dimensional surface nonuniformities. The obtained data are independent of the specific shape of the surface non-uniformity and can be used for estimation of the initial amplitudes of the three-dimensional TS-waves, as well as for the validation of other three-dimensional receptivity theories.