On the use of two-point velocity correlation in wall-pressure models for turbulent flow past a trailing edge under adverse pressure gradient

Two-point velocity statistics near the trailing edge of a controlled diffusion airfoil are obtained, both experimentally and analytically, by decomposing Poisson's equation for pressure into the mean-shear (MS) and turbulence-turbulence (TT) interaction terms. The study focuses on the modeling of each interaction term, in order to allow for the reconstruction of the wall-pressure spectra from tomographic velocimetry data, without numerically solving for pressure. The two-point correlation of the wall-normal velocity that describes the magnitude of the MS source term is found to be influenced by various competing factors such as blocking, mean-shear, and the adverse mean pressure gradient. The blocking term is found to supersede the other interaction terms close to the wall, making the two-point velocity correlation self-similar. The most dominant TT term that contributes to far-field noise for an observer located perpendicular to the airfoil chord at the mid-span is shown to be the one that quantifies the variation of the wall-normal velocity fluctuations in the longitudinal direction because of the statistical homogeneity of turbulence in planes parallel to the wall. A model to determine the contribution of the TT interaction term is proposed where the fourth-order two-point correlation can be modeled using Lighthill's approximation. However, its contribution toward wall-pressure spectra is found to be substantially lower than the MS term in the present case.