Monolithic One-Shot Optimization for Time-Periodic Flows Using Harmonic Balance

Abstract

Adjoint methods for aerodynamic design optimization have been a considerable area of research in computational fluid dynamics. Several approaches have been developed to fully couple the flow, adjoint, and design equations for steady flows. However, only limited attempts have been made to apply those to unsteady flows using a monolithic approach. This study aims to fully couple the primal, adjoint, and design equations based on a harmonic balance technique allowing the unsteady flow optimization to be performed in a single monolithic computation. The harmonic balance is the appropriate method for such problems as it reformulates unsteady time-periodic problems as “mathematically steady,” where one does not need to store the entire time history of the primal solution. The novel unsteady technique will be applied to a time-periodic lid-driven cavity. Here, the Reynolds number is taken to be the design variable so as to match a target vorticity distribution at the first subtime level of the time-spectral (harmonic balance) solution. Finally, mean drag minimization of a rotationally oscillating cylinder in crossflow is performed. In all cases, it is shown that the proposed method is quite efficient, requiring less than a couple of hundred total iterations to identify the “optimum” conditions.