Optimal Aerodynamic design of airfoils in unsteady viscous flows

Abstract

A continuous adjoint formulation is used to determine optimal airfoil shapes in unsteady viscous flows at Re = 1 × 10⁴. The Reynolds number is based on the free-stream speed and the chord length of the airfoil. A finite element method based on streamline-upwind Petrov/Galerkin (SUPG) and pressure-stabilized Petrov/Galerkin (PSPG) stabilizations is used to solve both the flow and adjoint equations. The airfoil is parametrized via a Non-Uniform Rational B-Splines (NURBS) curve. Three different objective functions are used to obtain optimal shapes: maximize lift, minimize drag and minimize ratio of drag to lift. The objective functions are formulated on the basis of time-averaged aerodynamic coefficients. The three objective functions result in diverse airfoil geometries. The resulting airfoils are thin, with the largest thickness to chord ratio being only 5.4%. The shapes obtained are further investigated for their aerodynamic performance. Maximization of time-averaged lift leads to an airfoil that produces more than six times more lift compared to the NACA 0012 airfoil. The excess lift is a consequence of the large peak and extended region of high suction on the upper surface and high pressure on the lower surface. Minimization of drag results in an airfoil with a sharp leading edge. The flow remains attached for close to 70% of the chord length. Minimization of the ratio of drag to lift results in an airfoil with a shallow dimple on the upper surface. It leads to a fairly large value of the time-averaged ratio of lift to drag (~ 17.8). The high value is mostly achieved by a 447% increase in lift and 16% reduction in drag, compared to a NACA 0012 airfoil. Imposition of volume constraint, for the cases studied, is found to result in airfoils that have lower aerodynamic performance.