Steady Aerodynamic Theory for Porous Membrane Wings

The growing interest in small-scale uncrewed air vehicles (UAVs) for surveillance, reconnaissance, and urban mobility purposes continues to spur scientific interest in novel aerodynamic design solutions inspired by biological fliers. A special focus has been given to compliant membrane wings, which passively adapt their shape to accommodate the flow. The coupled fluid-structure interaction inherent to these flexible structures has been shown to improve aerodynamic performance in steady flow by delaying the onset of stall and enhancing lift. However, these benefits were identified for impermeable membranes, whereas in nature similar compliant structures commonly include porosity. Such permeability enables flow seepage through the airfoil surface, which alters the pressure distribution and reduces the aerodynamic load on the airfoil in steady flow conditions. For a flexible membrane wing, such flow seepage also alters the membrane camber line through its tight coupling with the surrounding flow. Therefore, the aim of the current study is to explore the potentially contradicting effects of membrane compliance and porosity on the aeroelastic stability and aerodynamic performance of porous membrane airfoils. To this end, we develop a theoretical framework to study the steady aerodynamic properties of a porous membrane airfoil of uniform porosity, immersed in inviscid incompressible flow. We utilize Darcy's law to derive a closed-form expression for the aerodynamic load on a porous (thin) cambered airfoil of small camber at small angles of attack. This aerodynamic model is embedded in the membrane equilibrium equation to study the aeroelastic response of porous membrane airfoils and its aerodynamic properties in the stable regime. The stability threshold of the porous membrane airfoil is identified at zero angle of attack, for which solution of the eigenvalue problem provides the minimum tension coefficient that is required to obtain a stable membrane profile. For tension coefficients above this threshold, we analyze the stable equilibrium profiles and aerodynamic characteristics, including the lift slope, pitch moment slope, and aerodynamic center, obtained for varying porosity and tension coefficient values. The results demonstrate the role of porosity in the static aeroelastic stability and aerodynamic performance of porous membrane wings.