Efficient second-order analytical solutions for airfoils in subsonic flows

This paper presents simple and efficient analytical solutions for the velocity and pressure distributions on airfoils of arbitrary shapes, which are obtained considering the rigorous boundary conditions, without the use of the small perturbation assumption. A second-order accurate method using special singularities in the expression of the fluid velocity is first developed for airfoils in inviscid incompressible flows, by simultaneously solving the symmetric and anti-symmetric flow components defined by coupled boundary conditions. Accurate analytical solutions in closed form are thus derived and then successfully validated by comparison with the exact solutions for special airfoils obtained by conformal transformation and with numerical inviscid results. The method is then extended to model the main viscous effects on the velocity and pressure distributions, by considering the real velocity behavior at the airfoil trailing edge in viscous flow and the displacement thickness of the boundary layer developed on the airfoil and the wake. The resulting analytical solutions including viscous effects, derived for attached flows past airfoils at moderate angles of attack, were found in good agreement with experimental and numerical viscous results for various incidences, Reynolds and Mach numbers (Karman-Tsien compressibility correction was used to extend the solutions to compressible subsonic flows). In all cases studied (symmetric and cambered thick airfoils of arbitrary shapes, with rounded or pointed leading edges), these analytical solutions in closed form were found to be very efficient and accurate.