Abstract
This paper presents a comprehensive analytical approach to the modelling of wall-pressure fluctuations under a turbulent boundary layer, unifying and expanding the analytical models that have been proposed over many decades. The Poisson equation governing pressure fluctuations is Fourier transformed in the wavenumber domain to obtain a modified Helmholtz equation, which is solved with a Green’s function technique. The source term of the differential equations is composed of turbulence–mean shear and turbulence–turbulence interaction terms, which are modelled separately within the hypothesis of a joint normal probability distribution of the turbulent field. The functional expression of the turbulence statistics is shown to be the most critical point for a correct representation of the wall-pressure spectrum. The effect of various assumptions on the shape of the longitudinal correlation function of turbulence is assessed in the first place with purely analytical considerations using an idealised flow model. Then, the effect of the hypothesis on the spectral distribution of boundary-layer turbulence on the resulting wall-pressure spectrum is compared with the results of direct numerical simulation computations and pressure measurements on a controlled-diffusion aerofoil. The boundary layer developing over the suction side of this aerofoil in test conditions is characterised by an adverse pressure gradient. The final part of the paper discusses the numerical aspect of wall-pressure spectrum computation. A Monte Carlo technique is used for a fast evaluation of the multi-dimensional integral formulation developed in the theoretical part.
Highlights
The statistical characterisation of wall-pressure fluctuations under a turbulent boundary layer is necessary in many industrial applications
A characteristic of boundary-layer turbulence that is not included in this model is the 45o orientation of the vorticity vectors due to mean straining, the effect of which was deemed by Kraichnan (1956b) negligible with respect to that of length scale anisotropy
The results show that all turbulence models considered in this study are able to represent the quadratic rise of the wall-pressure power spectral density (PSD) at low frequencies
Summary
= (k1, k3), planar wavenumber vector = k12 + k32, planar wavenumber vector magnitude = aerodynamic wavenumbers. = coordinate system for boundary-layer description = fluctuating velocity component uτ
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