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Cascades of owls: singular integral equations in aerodynamics

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Porous aerofoils have excellent aeroacoustic properties, albeit at the expense of aerodynamic performance. In this talk, we will investigate the aerodynamic performance of a variety of aerofoil configurations through the analysis of singular integral equations. We will begin by studying the basic single rigid aerofoil problem and introduce two methods of solution: inversion via a Riemann-Hilbert problem and expansion in weighted Chebyshev polynomials. We shall show how the former method can be extended to porous aerofoils (which satisfy a Darcy condition along their chord), but breaks down when they are undergoing unsteady motions. Consequently, we extend the Chebyshev method to porous aerofoils by using asymptotic analysis to determine the parameters of a weighted Jacobi polynomial expansion. We will also apply the Riemann-Hilbert method to cascades which may be rigid and stationary, porous and stationary (i.e. a cascade of owls), or rigid and moving. The latter allows many results for single aerofoils to be generalised to cascade geometries, such as the Theodorsen function and Sears gust response function. Some preliminary experimental results into steady ground effect will also be presented.

This talk is part of the Waves Group (DAMTP) series.

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