Optimal Transport Aligned Latent Embeddings for Fluid Flow Analysis

Quantifying differences between flow fields is a key challenge in fluid mechanics, particularly when evaluating the impact of actuation. Traditional vector metrics, such as the one induced by the $L^2$ norm, offer straightforward pointwise comparisons. However, they fail to utilize the underlying geometric structure of flow field distributions, such as when they are disjointly supported. To address this limitation, we employ optimal transport (OT) theory, which is a mathematical framework built on probability and measure theory. Rather than only considering pointwise differences, OT distances effectively quantify a cost of transporting some resource or measure between distributions. As a demonstration, this study analyzes the effect of unsteady thermal actuation on separated flows past a NACA 0012 airfoil at a Reynolds number of 23,000 for angles of attack of $6^\circ$ and $9^\circ$ using a modified OT autoencoder approach. With the inclusion of an additional loss term in the autoencoder training, we simultaneously learn to reconstruct flow fields from a low-dimensional latent space while aligning Euclidean distances between flow fields in the latent space with the length of corresponding OT geodesics. We identify a two-dimensional embedding that succinctly describes the time-average behavior of the separation bubble as well as additional features representing the laminarization of the flow. We found that the overall flow response strongly depended on the actuation frequency, with secondary dependence on the spanwise wavenumber outside of two-dimensional actuation. The present results demonstrate the potential of OT in the analysis of fluid flows.