Potential flow through a cascade of aerofoils: direct and inverse problems

The potential flow through an infinite cascade of aerofoils is considered as both a direct and inverse problem. In each case, a perturbation expansion about a background uniform flow is assumed where the size of the perturbation is comparable to the aspect ratio of the aerofoils. This perturbation must decay far upstream and also satisfy particular edge conditions, including the Kutta condition at each trailing edge. In the direct problem, the flow field through a cascade of aerofoils of known geometry is calculated. This is solved analytically by recasting the situation as a Riemann-Hilbert problem with only imaginary values prescribed on the chords. As the distance between aerofoils is taken to infinity, the solution is seen to converge to a known analytic expression for a single aerofoil. Analytic expressions for the surface velocity, lift and deflection angle are presented as functions of aerofoil geometry, angle of attack and stagger angle; these show good agreement with numerical results. In the inverse problem, the aerofoil geometry is calculated from a prescribed tangential surface velocity along the chords and upstream angle of attack. This is found via the solution of a singular integral equation prescribed on the chords of the aerofoils.