On the leading-edge suction and stagnation-point location in unsteady flows past thin aerofoils

Abstract

Unsteady thin-aerofoil theory is a low-order method for calculating the forces and moment developed on a camber line undergoing arbitrary motion, based on potential-flow theory. The vorticity distribution is approximated by a Fourier series, with a special ‘A0’ term that is infinite at the leading edge representing the ‘suction peak’. Though the integrated loads are finite, the pressure and velocity at the leading edge in this method are singular owing to the term. In this article, the principle of matched asymptotic expansions is used to resolve the singularity and obtain a uniformly valid first-order solution. This is performed by considering the unsteady thin-aerofoil theory as an outer solution, unsteady potential flow past a parabola as an inner solution, and by matching them in an intermediate region where both are asymptotically valid. Resolution of the leading-edge singularity allows for derivation of the velocity at the leading edge and location of the stagnation point, which are of physical and theoretical interest. These quantities are seen to depend on only the A0 term in the unsteady vorticity distribution, which may be interpreted as an ‘effective unsteady angle of attack’. The leading-edge velocity is proportional to A0 and inversely proportional to the square root of leading-edge radius, while the chordwise stagnation-point location is proportional to the square of A0 and independent of the leading-edge radius. Closed-form expressions for these in simplified scenarios such as quasi-steady flow and small-amplitude harmonic oscillations are derived.