Weak Dominant Balance: A robust method for identifying structure in more complex fluid flows

The data-driven dominant balance algorithm is an unsupervised approach to system identification that extracts a low-dimensional representation of the dominant physical processes underlying high-dimensional fluid flow data. The original implementation was able to extract sparse representations from time-averaged DNS data; however, it is not able to adequately identify governing patterns in more challenging conditions, e.g., uncertain measurements and complex mesh geometries which often lead to numerical inaccuracies. We address these issues by leveraging the integral form of the governing equations to cast a "weak form" of the problem which ultimately allows for more robust analysis on data for which obtaining the gradients would typically be challenging. The effectiveness of this formulation is demonstrated primarily on an example of transitional boundary layer flow, where the time-averaged fields (i.e. velocity, pressure, and Reynolds Stress) have been corrupted with Gaussian noise proportional to the magnitude of the wall-normal velocity. Our "weak dominant balance" method is able to decompose the flow into regions of distinct dynamics—identifying laminar, transitional, inertially-dominated, viscous-dominated, and freestream regions across the spatial domain of the flow. Weak dominant balance quantitatively surpasses the original data-driven dominant balance algorithm in correctly identifying the dynamics in each of these regions across many levels of noise. We also extend this framework to two new flows: 1) the secondary flows in turbulent ducts, which involve higher-order gradients, and 2) the adverse/favorable pressure gradients present in wavy channel flow, based on experimental (PIV) data and DNS data on a non-rectilinear mesh geometry.