Modelling wall pressure fluctuations under a turbulent boundary layer

The derivation of the wave vector-frequency (w-f) spectrum of wall pressure fluctuations below a turbulent boundary layer developed over a rigid flat plate is re-considered. The Lighthill's equation for pressure fluctuations is derived in a frame of reference fix with respect to the plate, at low Mach numbers, and transformed into the convected frame moving with the flow. To model the source terms of the Lighthill equation, it is assumed that in the inertial range, the turbulence is locally isotropic in the convected frame. The w-f spectrum of isotropic turbulence is obtained from symmetry considerations by extending the isotropy to space time, based on the concept of sweeping velocity. The resulting solution for the pressure w-f spectrum contains a term (the mean shear-turbulence term) which does not fulfill the Kraichnan Philipps theorem, due to the form of the selected turbulent velocity spectrum. The viscous effects are accounted for by a cut-off depending on wall distance; this procedure allows extending the model beyond the inertial range contribution. The w-f pressure spectrum is derived and compared to the experimental low wavenumber data of Farabee and Geib (1991) [8] and Bonness et al. (2010) [5], for which a good agreement is obtained. The derived expression is also compared to Chase theoretical model Chase (1987) [6] and found to agree well in the vicinity of the convective ridge of the subsonic domain and to differ significantly both in supersonic and subsonic low wavenumber limits. The pressure spectrum derived from the model and its scaling are discussed and compared to experimental data and to the empirical model of Goody (2002) [23], which results from the compilation of a large set of experimental data. Very good agreement is obtained, except at vanishing frequencies where it is claimed that the experimental results lack of significance due to the limited size of the experimental facilities. This hypothesis supported by the results obtained from numerical simulations.