Techniques for the optimization and control of large-scale systems with application to jet noise

The present work focuses on the analysis and optimization of jet noise and some of the related challenges that high-dimensional approximations of infinite-dimensional, chaotic, multiscale systems present to the gradient-based optimization framework.; We first present the optimization of flow/acoustic interactions in jets. This optimization is based on a gradient obtained from an adjoint field. The ability to modify favorably the high-frequency acoustic field via low-frequency modulation of the hydrodynamic field near the jet exit is confirmed with this method. Due to the complexity of the system under consideration, some simplifications have been applied in the derivation of equations governing the adjoint field. In order to evaluate the correctness of the adjoint code and the adequacy of these simplifications, complex-step derivative (CSD) method has been used to calculate the gradient directly from the flow solver. Until now, the CSD method has been applied only to physical space simulation codes. In the present work, a non-trivial extension of this technique to pseudospectral codes has been developed, as many of the numerical codes used for turbulent flows problems leverage pseudospectral techniques to calculate spatial derivatives in one or more directions.; Once the gradient information obtained with the adjoint field has been so validated, optimizations may be performed. Note that the control schedule that we have designed for the jet system is periodic in time, due both to the quasi-periodic nature of the jet system and reasons of practical implementation. We have encountered certain fundamental challenges related to the fact that, when applied to time evolving systems, the adjoint field grows exponentially backward in time, resulting in a gradient dominated by effects in a narrow time window no matter what time horizon is used for the optimization. In such a situation, one can not leverage the ergodicity of the controlled chaotic system (by integrating over a sufficiently long time horizon) in order to achieve a general applicability of the control design. New methods to treat such control problems directly in the time-periodic framework have been proposed and are currently under investigation. A difficult step in such time-periodic control optimizations is the development of efficient techniques to obtain the time-periodic orbits of a chaotic system, as will be presented.