A resolvent-analysis-based optimization design method for airfoils at low Reynolds number

Control of flow separation at high angles of attack plays a crucial role in enhancing the aerodynamic performance of airfoils at low Reynolds number, as well as in reducing flow-induced structural vibrations and noise. To address the inefficiency of current empirical trial-and-error and a posteriori based flow control methods, we propose a novel approach that integrates resolvent analysis with airfoil optimization design to achieve passive surface deformation control of unsteady flows. First, we construct an input-output dynamical model of the flow system. The excitation with the strongest response (i.e., forcing modes), the states with the highest receptivity (i.e., response modes) and the amplification between them (i.e., resolvent gains) under harmonic inputs at varying frequencies can be identified through resolvent analysis. Second, a quantitative correlation between the resolvent gain and flow stability is established. When the flow system satisfies the rank-1 approximation condition, a reduction in the maximum resolvent gain directly corresponds to an improvement in flow stability. Finally, an efficient shape optimization framework is developed, with the objective function defined as the minimization of the maximum gain, and the optimization process combining a penalty function approach with the nonlinear conjugate gradient method. The NACA0012 airfoil under the design condition Ma=0.1, Re=200, angle of attack 18° is selected as the test case, and the two optimized airfoils are obtained by varying the penalty parameters. The computational results demonstrate that, under the condition of no loss or even an improvement in aerodynamic performance at small angles of attack, the maximum resolvent gain of the two airfoils is reduced by 63.49% and 54.44%, respectively. The flow stability at supercritical angles of attack is significantly enhanced, with the amplitude of lift fluctuations attenuating by an average of 16.20% and 13.79%. Additionally, the time-averaged drag coefficient decreases by 2.44% and 1.84%, respectively. Analysis of the flow field evolution reveals that the alternating generation and shedding of leading-edge and trailing-edge separation vortices at high angles of attack lead to significant lift oscillations. The shape optimization effectively suppresses flow separation, achieving a synergistic improvement in both the aerodynamic performance and flow stability of the optimized airfoils. This study provides new theoretical guidance for airfoil separation flow control and establishes a resolvent-analysis-based shape optimization design method that is anticipated to be applicable to separation flow problems involving various types of flow instabilities.