Using Optimal Transport Aligned Latent Embeddings for Separated Flow Analysis

TL;DR

The paper addresses the challenge of comparing high-dimensional flow fields by moving beyond pointwise metrics to an unbalanced optimal transport (OT) framework. It introduces an OT-aligned autoencoder that includes an embedding loss to align latent-space distances with OT-based dissimilarities, yielding a low-dimensional, physically interpretable latent space. Applied to separated flow over a NACA0012 under periodic heat-flux actuation, the method produces a two-dimensional latent representation where the first axis tracks separation-bubble size and lift-to-drag performance, while the second captures laminarization and trailing-edge effects; this structure is consistent across AoA values. The approach offers a geometry-aware tool for flow control analysis with potential applications in interpolation, design optimization, and uncertainty quantification.

Abstract

Quantifying differences between flow fields is a key challenge in fluid mechanics, particularly when evaluating the effectiveness of flow control. 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 utilize this OT-based analysis on controlled, separated flows past a NACA 0012 airfoil with a chord-based Reynolds number of 23,000 and a freestream Mach number of 0.3 for two angles of attack 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, characterized by the degree of suppression of the separation bubble and secondary effects from laminarization and trailing-edge separation. The interpretation of the latent representation was found to be consistent across the two angles of attack, 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.

Figures (14)

• Figure 1: Pedagogical schematic of optimal transport between a supply and demand distribution representative of two different flow fields. The optimal transport distance is the minimum total cost to move the supply to the demand distribution.
• Figure 2: Comparison of unbalanced optimal transport distance $d_{\text{UOT}}$ with the $L^2$ metric. (a) Gaussian pulse undergoing advection and diffusion. (b) Comparison of distances between the evolving pulse and the initial condition.
• Figure 3: Examples of baseline flow and various responses of a separated wake past a NACA 0012 airfoil at $\alpha=6^\circ$ to a heat flux actuator input at the leading edge yeh2019resolvent. The $Q L_c^2/u_\infty^2=50$ isosurface colored by the normalized streamwise velocity is shown.
• Figure 4: Illustration of the augmented autoencoder problem setup. During training, the model learns to associate Euclidean distances in the latent space (red line) with flow field dissimilarities computed with optimal transport.
• Figure 5: Example parameter study of the OT-based autoencoder for $\alpha=9^\circ$. (a) L-curve showing the trade-off between $\mathcal{L}_1$ and $\mathcal{L}_2$ for the test set with $10^{-4} \le \lambda \le 10^4$. (b) Variation of the total loss $\mathcal{L}_1 + \lambda \mathcal{L}_2$ with respect to the latent space dimension for $\lambda = 0.1$. Standard deviation for the last 500 epochs is colored in gray.