Effects of a Nonlinear Boundary Condition on the Steady Aerodynamics of Porous Airfoils

Abstract

This paper considers the lift coefficient of an airfoil with nonzero prescribed thickness and camber, along with a prescribed porosity distribution in a steady incompressible flow. In similar prior work, considering a Darcy-type porosity condition on the airfoil surface provides a Fredholm integral equation that can be solved exactly for Hölder-continuous porosity distributions. However, the generated lift predicted by the model diverges from the experimental data for porous airfoils with large values of the porosity parameter. This indicates a fundamental characteristic is missing in the mathematical model. Consequently, in this paper, the (linear) Darcy porosity condition is replaced by the (nonlinear) Forchheimer porosity condition. The Forchheimer porosity condition is decomposed into linear sections and furnishes a modified system of Fredholm integral equations, enabling an approximate solution by superposition for Hölder-continuous porosity distributions. The solution is then verified against the SD7003 airfoil results, and the comparison shows better agreement than considering the Darcy porosity condition. The methodology and results presented here may be combined with previous work on the aeroacoustics of porous airfoils with a Forchheimer boundary condition to address the conflicting aims of improving aerodynamic performance while reducing unwanted aeroacoustic emissions.