Optimal airfoils in the intermediate-Reynolds-number range

We revisit a classical airfoil design problem: the search for shapes that maximise 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 maximisation 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 discretisation. 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 the Reynolds number ( italic Re <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:math> $\textit{Re}$ ), whereas the optimal camber varies non-monotonically, reaching a pronounced maximum near italic Re almost equals 40 minus 50 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mo>≈</mml:mo> <mml:mn>40</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> <mml:mn>50</mml:mn> </mml:math> $\textit{Re} \approx 40{-}50$ before declining at higher italic Re <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:math> $\textit{Re}$ . At low Reynolds numbers ( italic Re less than or equivalent to 100 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mo>≲</mml:mo> <mml:mn>100</mml:mn> </mml:math> $\textit{Re} \lesssim 100$ ), a broad family of cambered shapes performs within a few per cent of the optimum, indicating weak sensitivity to geometrical parameters. In contrast, for italic Re greater than or equivalent to 1000 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> <mml:mo>≳</mml:mo> <mml:mn>1000</mml:mn> </mml:math> $\textit{Re} \gtrsim 1000$ , the performance landscape becomes sharply localised around a single preferred design, for which geometric refinement is critical.