Aerodynamic shape optimization of airfoils in unsteady flow

Abstract

Aerodynamic shape optimization of airfoils is carried out for two values of Reynolds numbers: 10³ and 10⁴, for an angle of attack of 5° . The objective functions used are (a) maximization of lift (b) minimization of drag and (c) minimization of drag to lift ratio. The surface of the airfoil is parametrized by a 4^th order non-uniform rational B-Spline (NURBS) curve with 61 control points. Unlike the efforts in the past, the relatively large number of control points used in this study offer a rich design shape with the possibility of local bumps and valleys on the airfoil surface. The airfoils obtained via the optimization exhibit significantly improved aerodynamic performance compared to the NACA 0012 airfoil. Maximization of lift results in airfoils with corrugations on the upper surface. These corrugations are accentuated in the airfoils that are optimal at Re=10⁴. The improved performance is due to the formation of small vortices for the optimal airfoil. These vortices lead to an increase in suction on the airfoil surface. The time-averaged flow field shows that these vortices occupy the valleys in the corrugated upper surface of the optimal airfoil. Compared to a NACA 0012 airfoil, the time-averaged lift coefficient increases by 36.58% at Re=10³ and 220.72% at Re=10⁴, over the NACA 0012 airfoil. Minimization of drag results in airfoils with flatter upper surface. Compared to the NACA 0012 airfoil, the time-averaged drag coefficient for the optimal airfoil is 2.1% smaller at Re=10³. The reduction at Re=104 is 15.36% compared to the NACA 0012 airfoil. The effect of the time period for time averaging of the flow, on the convergence of the optimization cycle, is investigated. It is found that time period for which the unsteady data is sampled and time-averaged should be sufficiently long time to achieve good convergence in the optimization cycle. A continuous adjoint based method is used for shape optimization. The flow as well as the adjoint equations are solved via a stabilized finite element method. The limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm is used to minimize the objective function.