Optimal airfoils in the intermediate Reynolds number range

Abstract

We revisit a classical airfoil design problem: the search for shapes that maximize aerodynamic performance metrics, targeting the underexplored intermediate Reynolds-number regime between 1 and 3000, relevant to small animals and miniature vehicles. The problem is formally stated as the glide ratio or the endurance factor maximization for Joukowski airfoil profiles and for more general airfoil shapes with adjustable position of the maximum camber, under steady inflow. It is solved numerically by a hybrid approach combining stochastic search and direct parameter sweep, and using a steady laminar Navier--Stokes solver based on conformal mapping and second-order finite-difference discretization. Zero-thickness cambered airfoils are found to be globally optimal, within the Joukowski family, across the entire Reynolds-number range considered. The optimal angle of attack decreases monotonically with $Re$, whereas the optimal camber varies non-monotonically, reaching a pronounced maximum near $Re \approx 40-50$ before declining at higher $Re$. At low Reynolds numbers ($Re \lesssim 100$), a broad family of cambered shapes performs within a few per cent of the optimum, indicating weak sensitivity to geometrical parameters. In contrast, for $Re \gtrsim 1000$, the performance landscape becomes sharply localized around a single preferred design, for which geometric refinement is critical.