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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.

A weakly compressible flow direct numerical simulation of a controlled-diffusion aerofoil at $8^{\circ }$ geometrical angle of attack, a chord-based Reynolds number of $Re_{c}=150\,000$ and a Mach number of $M=0.25$ based on the free-stream velocity relevant to many industrial applications was conducted to improve the understanding of the impact of the pressure gradient on the development of turbulent structures. The evolution equations for the two invariants $Q$ and $R$ of the velocity-gradient tensor have been studied at various locations along the aerofoil chord on its suction side. The shape of the mean evolution of the velocity-gradient tensor invariants were found to vary strongly when the flow encounters favourable, zero and adverse pressure gradients and as well for different wall-normal locations. The coupling between the pressure-Hessian tensor and the velocity-gradient tensor was found to be the major factor that causes these changes and is greatly influenced by the mean pressure-gradient condition and the wall-normal distance. Striking differences exist from the mean trajectories of this coupling at least in the log layer and outer layer subject to different mean pressure gradients. The nonlinearity and viscous diffusion effects keep their respective invariant characters regardless of the pressure-gradient effects and wall-normal locations. The wall and the mean adverse pressure gradient were both found to suppress the vortical stretching features of the flow. These features are of great importance for the development of future turbulence models on wall-bounded flows, especially on surfaces with significant curvature such as cambered aerofoils and blades for which significant mean pressure gradients exist.

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Direct numerical simulations (DNSs) of an incompressible turbulent boundary layer on an airfoil (suction side) and that on a flat plate are compared to characterize the non-equilibrium turbulence and the effect of wall curvature on the flow. The two simulations effectively impose matching streamwise distributions of adverse pressure gradient (APG) quantified by the acceleration parameter ( K). For the airfoil flow, an existing compressible DNS carried out by Wu et al. [“Effects of pressure gradient on the evolution of velocity-gradient tensor invariant dynamics on a controlled-diffusion aerofoil at Rec = 150,000,” J. Fluid Mech. 868, 584–610 (2019)] of the flow around a controlled-diffusion airfoil is used. For the flat-plate flow, a separate simulation is carried out with the aim to reproduce the flow in the region of the airfoil boundary layer with zero to adverse pressure gradients. Comparison between the two cases extracts the effect of a mild convex wall curvature on velocity and wall-pressure statistics in the presence of APG. In the majority part of the boundary layer development, curvature effect on the flow is masked by that of the APG, except for the region with weak pressure gradients or a thick boundary layer where the effect of wall curvature appears to interact with that of APG. High-frequency wall-pressure fluctuations are also augmented by the wall curvature. Overall, the boundary layers are qualitatively similar with and without the wall curvature. This indicates that a flat-plate boundary layer DNS may serve as a low-cost surrogate of a boundary layer over the airfoil or other objects with mild curvatures to capture important flow features to aid modeling efforts.