Application of deconvolution on the turbulent boundary layer wall pressure spectrum

Abstract

Turbulent boundary layer cross-spectral data has recently been measured for different velocities at zero pressure gradient. The obtained wavenumber-frequency spectra were not sufficiently accurate to detect and characterize the acoustic region. Therefore applicability of source localization techniques is explored in an attempt to increase image quality and sharpness. The lack of comprehensive literature coping with physical background and limitations of beamforming algorithms inspired a ground-up step-by-step explanation including infinite and nearfield Delay-and-Sum procedures as well as the κ − ω transformation and the subsequent power algorithms. The idea of deconvolution is additionally explained and derived for the different techniques. It is shown that infinite beamforming is an unbiased estimate for the κ − ω transformation and as a result a non-equidistantly spaced Fourier deconvolution approach is implemented to account for spatial weighting of arbitrary microphone arrays. A range of common deconvolution procedures (DAMAS,DAMAS2,Richardson-Lucy,Lawson-Hanson) is explained and applied for infinite beamforming and the κ − ω transformation to analyze their potential using a mixed Corcos-Diffuse model for the turbulent boundary layer wavenumber frequency spectrum. The Richardson-Lucy algorithm applied on a κ − ω transformed map shows best results for the highly polluted acoustic region while DAMAS2 deconvolution is more efficient in removing lobes within the aerodynamic region. The deconvolution procedure is applied on experimental data at two different low machnumber cases (M = 0.13, M = 0.23). The edge of the acoustic field can for the first time be visualized for a low Machnumber case (M = 0.23). At M = 0.13 strong orthogonally impinging waves (κ = 0) are detected. Isolated acoustic cross-power spectral density matrices are obtained from the deconvolved κ − ω-spectra by inverse Fourier transform.