Shape Optimization of Helicopter Rotors in Forward Flight

This article will develop a framework for performing optimal shape design for periodic, unsteady flows. Specifically, the time-spectral method is used to solve the unsteady, Euler equations in conjunction with a continuous-adjoint methodology that returns surface sensitivities. A brief derivation of the continuous-adjoint formulation is provided along with a discussion on implementing the time-spectral method in an existing steady flow solver. Adjoint-based gradients are compared against finite differencing for two separate two-dimensional test cases. Moreover, the method's applicability to aerodynamic shape optimization is demonstrated by means of a simple drag-minimization study.