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Abstract Recent interest in urban and regional air mobility and the need to improve the aviation industry’s emissions has motivated research and development of novel propeller-driven vehicles. These vehicles range in configuration from conventional takeoff and landing designs to complex rotorcraft that transition between vertical and horizontal flight. These designs must be optimized to ensure optimal efficiency throughout their missions, leveraging the tightly coupled nature of propeller-wing interaction. In this work, we study the NASA tiltwing concept vehicle wing with varying numbers of propellers, ranging from no propellers to five propellers evenly spaced along the wing. Using aerodynamic shape optimization, we optimize the wing shapes for each propeller-wing configuration, minimizing the wing drag. These optimizations are carried out with DAFoam, a discrete adjoint implementation of OpenFOAM, embedded within OpenMDAO and the MPhys optimization framework. The optimizations show that the lowest drag configuration is a single propeller mounted at the wing tip. Increasing the number of propellers slightly increases drag compared to the single propeller configuration. However, aerodynamic shape optimization considering propeller-wing interaction yields a negligible benefit compared to aerodynamic optimization of an isolated wing that is subsequently trimmed to a desired flight condition in the presence of a propeller. 

Koopman decomposition is a nonlinear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman-invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on a multi-task feature learning to extract the most informative Koopman-invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity-promoting extensions of EDMD/KDMD. Furthermore, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship-airwake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analysed, with an emphasis on characterizing transient dynamics. The results are compared with existing theoretical expositions and numerical approximations.