Unsteady aerodynamics of porous aerofoils

Abstract: We extend unsteady thin aerofoil theory to model aerofoils with generalised chordwise porosity distributions. The analysis considers a linearised porosity boundary condition where the seepage velocity through the aerofoil is related to the local pressure jump across the aerofoil surface and to the unsteady characteristics of the porous medium. Application of the Plemelj formulae to the resulting boundary value problem yields a singular Fredholm--Volterra integral equation which does not admit an analytic solution. Accordingly, we develop a numerical solution scheme by expanding the bound vorticity distribution in terms of appropriate basis functions. Asymptotic analysis at the leading- and trailing-edges reveals that the appropriate basis functions are weighted Jacobi polynomials whose parameters are related to the porosity distribution. The Jacobi polynomial basis enables the construction of a numerical scheme that is accurate and rapid, in contrast to the standard choice of Chebyshev basis functions that are shown to {be unsuitable} for porous aerofoils. Applications of the numerical solution scheme to discontinuous porosity profiles, quasi-static problems, and the separation of circulatory and non-circulatory contributions are presented. Analogues to the classical Theodorsen and Sears functions are computed numerically, which show that an effect of trailing-edge porosity is to reduce the amount of vorticity shed into the wake, thereby reducing the magnitude of the circulatory lift. {Fourier transform inversion of these frequency-domain functions produces porous extensions to the Wagner and K\"{u}ssner functions for transient aerofoil motions or gust encounters, respectively.}