  Large amplitude gust encounters exhibit a range of separated flow phenomena,
making them difficult to characterize using the traditional tools of
aerodynamics. In this work, we propose a dynamical systems approach to gust
encounters, viewing the flow as a cycle (or a closed trajectory) in state
space. We posit that the topology of this cycle, or its shape and structure,
provides a compact description of the flow, and can be used to identify
coordinates in which the dynamics evolve in a simple, intuitive way. To
demonstrate this idea, we consider flowfield measurements of a transverse gust
encounter. For each case in the dataset, we characterize the full-state
dynamics of the flow using persistent homology, a tool that identifies holes in
point cloud data, and transform the dynamics to a reduced-order space using a
nonlinear autoencoder. Critically, we constrain the autoencoder such that it
preserves topologically relevant features of the original dynamics, or those
features identified by persistent homology. Using this approach, we are able to
transform six separate gust encounters to a three-dimensional latent space, in
which each gust encounter reduces to a simple circle, and from which the
original flow can be reconstructed. This result shows that topology can guide
the creation of low-dimensional state representations for strong transverse
gust encounters, a crucial step toward the modeling and control of airfoil-gust
interactions.
