This paper presents an analytic solution for the sound generated by rotor–stator interaction for aerofoils with small camber and thickness subject to a background flow with small angle of attack. The interaction is modelled as a convected, unsteady vortical or entropic gust incident on an infinite rectilinear cascade of staggered aerofoils in a background flow that is uniform far away from the cascade. Applying rapid distortion theory (RDT) and transforming to an orthogonal coordinate system reduces the cascade of aerofoils to a cascade of flat plates. By seeking a perturbation expansion in terms of the disturbance of the background flow from uniform flow, leading- and first-order governing equations and boundary conditions are obtained for the acoustic potential. The system is then solved analytically using the Wiener–Hopf method. The resulting expression is inverted to give the acoustic potential function in the entire domain, i.e. a solution to the inhomogeneous convected Helmholtz equation with inhomogeneous boundary conditions in a cascade geometry. The solution significantly extends previous analytical work that is restricted to flat plates or only calculates the far-upstream radiation, and as such can give insight into the role played by blade geometry on the acoustic field upstream, downstream and in the important inter-blade region of the cascade. This new solution is validated against solutions that only account for flat plates at zero angle of attack. Various aeroacoustic results, including the scattered pressure, unsteady lift and sound power output, are discussed for a range of geometries and angles of attack.

‘Ground effect’ refers to the enhanced performance enjoyed by fliers or swimmers operating close to the ground. We derive a number of exact solutions for this phenomenon, thereby elucidating the underlying physical mechanisms involved in ground effect. Unlike previous analytic studies, our solutions are not restricted to particular parameter regimes, such as ‘weak’ or ‘extreme’ ground effect, and do not even require thin aerofoil theory. Moreover, the solutions are valid for a hitherto intractable range of flow phenomena, including point vortices, uniform and straining flows, unsteady motions of the wing, and the Kutta condition. We model the ground effect as the potential flow past a wing inclined above a flat wall. The solution of the model requires two steps: firstly, a coordinate transformation between the physical domain and a concentric annulus; and secondly, the solution of the potential flow problem inside the annulus. We show that both steps can be solved by introducing a new special function which is straightforward to compute. Moreover, the ensuing solutions are simple to express and offer new insight into the mathematical structure of ground effect. In order to identify the missing physics in our potential flow model, we compare our solutions against new experimental data. The experiments show that boundary layer separation on the wing and wall occurs at small angles of attack, and we suggest ways in which our model could be extended to account for these effects.

We present a solution for the scattered field caused by an incident wave interacting with an infinite cascade of blades with complex boundary conditions. This extends previous studies by allowing the blades to be compliant, porous or satisfy a generalised impedance condition. Beginning with the convected wave equation, we employ Fourier transforms to obtain an integral equation amenable to the Wiener–Hopf method. This Wiener–Hopf system is solved using a method that avoids the factorisation of matrix functions. The Fourier transform is inverted to obtain an expression for the acoustic potential function that is valid throughout the entire domain. We observe that the principal effect of complex boundary conditions is to perturb the zeros of the Wiener–Hopf kernel, which correspond to the duct modes in the inter-blade region. We focus efforts on understanding the role of porosity, and present a range of results on sound transmission and generation. The behaviour of the duct modes is discussed in detail in order to explain the physical mechanisms behind the associated noise reductions. In particular, we show that cut-on duct modes do not exist for arbitrary porosity coefficients. Conversely, the acoustic far-field modes are unchanged by modifications to the boundary conditions. We apply our solution to a cascade of perforated plates and see that a fractional open area of 1 % is sufficient to significantly attenuate backscattering. The solution is essentially analytic (the only numerical requirements are matrix inversion and root finding) and is therefore extremely rapid to compute.