On the conservation of physical properties in operator interpolation of parameterized hydrodynamic systems

TL;DR

This paper evaluates operator interpolation methods for reduced-order models of fluid flows, demonstrating that DMD-based ROMs more accurately predict flow dynamics across parameters than Galerkin-based ROMs, especially when carefully selecting subspaces.

Contribution

It introduces a comprehensive assessment of operator interpolation in ROMs for fluid systems, highlighting the robustness of DMD-based methods over Galerkin projection in parametric settings.

Findings

DMD-based ROM accurately predicts fundamental and harmonic frequencies.

Galerkin ROM captures only the fundamental frequency, missing higher harmonics.

Careful subspace selection improves interpolation accuracy.

Abstract

Reduced-order models (ROMs) that capture changes in fluid systems due to variations in parameters, such as the Reynolds number or the shape of a stationary body placed in the flow, are attracting increasing attention in engineering applications. In this study, we identify linear operators that characterize the behavior of fluid systems across a wide parameter range by using flow field datasets at several representative parameter values. We then comprehensively assess the applicability of ROMs constructed through the interpolation of these operators. Specifically, we consider two intrusive operator-based ROMs: one derived from Galerkin projection and the other based on operator inference using dynamic mode decomposition (DMD). The performance of these ROMs is evaluated for flows around circular and elliptical cylinders over a range of Reynolds numbers and aspect ratios. The Galerkin-based ROM successfully predicts only the eigenvalue and eigenmode corresponding to the fundamental frequency of the K\'arm\'an vortex shedding, while other frequencies are not captured and show a decaying behavior. In contrast, the DMD-based ROM accurately predicts both the fundamental frequency and its higher harmonics. Furthermore, visualization of the linear operator matrix elements reveals that interpolation fails when the subspace includes bases contaminated by numerical errors. However, by carefully selecting the subspace dimension and reference conditions, it is possible to accurately predict eigenvalues and corresponding modes even under conditions where multiple low-frequency modes exist outside the harmonic structure of the fundamental frequency. These findings underscore the robustness of the DMD-based parametric operator ROM approach.