Using optimal transport aligned latent embeddings for separated flow analysis

Quantifying differences between flow fields is a key challenge in fluid mechanics, particularly when evaluating the effectiveness of flow control or other problem parameters. Traditional vector metrics, such as the Euclidean distance, provide straightforward pointwise comparisons but can fail to distinguish distributional changes in flow fields. To address this limitation, we employ optimal transport (OT) theory, which is a mathematical framework built on probability and measure theory. By aligning Euclidean distances between flow fields in a latent space learned by an autoencoder with the corresponding OT geodesics, we seek to learn low-dimensional representations of flow fields that are interpretable from the perspective of unbalanced OT. As a demonstration, we utilise this OT-based analysis on separated flows past a NACA 0012 airfoil with periodic heat flux actuation near the leading edge. The cases considered are at a chord-based Reynolds number of 23 000 and a free-stream Mach number of 0.3 for two angles of attack (AoA) of $6^\circ$ and $9^\circ$ . For each angle of attack, we identify a two-dimensional embedding that succinctly captures the different effective regimes of flow responses and control performance, characterised by the degree of suppression of the separation bubble and secondary effects from laminarisation and trailing-edge separation. The interpretation of the latent representation was found to be consistent across the two AoA, suggesting that the OT-based latent encoding was capable of extracting physical relationships that are common across the different suites of cases. This study demonstrates the potential utility of optimal transport in the analysis and interpretation of complex flow fields.