Analytical Framework for Unsteady Aerodynamics of Airfoil with Chord-wise Varying Porosity and Nonlinear Boundary Condition.

The application of porous airfoils offers a significant solution for noise mitigation on aircraft wings. However, because of the nature of flow through porous surfaces, predicting the aerodynamic performance of these porous air foils becomes challenging. This study addresses the challenge by developing an analytical framework for the unsteady aerodynamics of porous airfoils by considering non-linear boundary conditions and vortex formation. Traditional Darcy-based linear boundary conditions, which assume negligible inertial contributions within the porous medium, become unreliable for higher Reynolds numbers. Hence, we incorporate the linear models with non-linear "Forchheimer boundary condition" to account for the presence of nonlinear effects, and represent flow mechanics in high-porosity and high Reynolds number regimes. In our methodology, we linearize the nonlinear Forchheimer condition through a piecewise approximation approach. Harmonic assumptions are then applied to transform the time-dependent variables into spatial ones, yielding a Singular Fredholm-Volterra Integral Equation. This model is then subsequently used to numerically analyze the aerodynamic impact of various bio-inspired chord-wise varying porosity distributions. Ultimately, this work presents the vorticity distribution for unsteady aerodynamics of airfoils with non-linear porous boundary conditions. This analytical approach provides the insights that will enable finely optimized porous airfoil design and performance in noise mitigation applications.