Optimal mode decomposition for unsteady flows

TLDR

The OMD method generalizes dynamic mode decomposition by optimizing the subspace rather than fixing it to POD modes. The study introduces optimal mode decomposition (OMD) as a method to derive a linear model that captures the evolution of fluid flow and can be applied to experimental and numerical data to compute an optimal low‑order model with a user‑defined rank for unsteady and turbulent flows. The method estimates the linear dynamics of a high‑dimensional system projected onto a user‑defined fixed rank subspace, iteratively optimizes the linear model and subspace to minimize residual error, and compares OMD to DMD using synthetic and experimental data. OMD provides an approximation to the Koopman modes and eigenvalues, yields lower residual errors than DMD, and delivers more accurate eigenvalue estimates on a synthetic waveform.

Abstract

Abstract A new method, herein referred to as optimal mode decomposition (OMD), of finding a linear model to describe the evolution of a fluid flow is presented. The method estimates the linear dynamics of a high-dimensional system which is first projected onto a subspace of a user-defined fixed rank. An iterative procedure is used to find the optimal combination of linear model and subspace that minimizes the system residual error. The OMD method is shown to be a generalization of dynamic mode decomposition (DMD), in which the subspace is not optimized but rather fixed to be the proper orthogonal decomposition (POD) modes. Furthermore, OMD is shown to provide an approximation to the Koopman modes and eigenvalues of the underlying system. A comparison between OMD and DMD is made using both a synthetic waveform and an experimental data set. The OMD technique is shown to have lower residual errors than DMD and is shown on a synthetic waveform to provide more accurate estimates of the system eigenvalues. This new method can be used with experimental and numerical data to calculate the ‘optimal’ low-order model with a user-defined rank that best captures the system dynamics of unsteady and turbulent flows.