Extrapolative Potential of Turbulent ROMs

Introduction

A significant limitation of any statistically driven reduced-order model is that its applicability is constrained by the representativeness of its data set; in other words, the ROM is only at best as representative of a phenomenon’s general behavior as is the data used to generate it. A turbulent flow regime is high-dimensional, non-conservative, seemingly chaotic, and nonlinear. Thus, most turbulent ROMs are limited in accuracy to, almost exclusively, the flow regime used to generate them.

The question is thus begged: how applicable is a reduced-order model if it can only be used to recreate, at best, the flow behavior used to generate it in the first place? There are two significant limitations to the applicability of a ROM that can only recreate the original data, regardless of whether this data was generated using experimental or computational methods:

1)      Dynamics perspective: A ROM that does not have any extrapolative potential is not likely to improve the dynamical understanding of the system

2)      Engineering perspective: A ROM that does not have any extrapolative potential will not likely be used for research or design when the exact solution has already been generated

It is important to note that ROMs can still be functional if they can only ever recreate the initial data, though in the case of turbulence these two limitations are significant inhibitors of the widespread application of ROMs. Data compression of any kind uses model-order reduction mathematics; it is the seemingly chaotic nature of turbulence that complicates this particular application.

Proper Orthogonal Decomposition

Though called many names, and though rediscovered in different fields at different times, the proper orthogonal decomposition (POD) was arguably first introduced to turbulence by J.L. Lumley in 1967. The reader is directed to an Annual Review on this topic for a brief mathematical and historical overview. The general principle is to reproduce a signal (in this case the flow behavior) u(x) using a sum of orthogonal eigenvalue-eigenvector pairs,

such that the signal is reproduced using only its dominant coherent structures. In turbulence, it is believed that these coherent structures are the key to understanding the dynamics of the system. The challenge, as is the challenge with any eigenvalue problem, is finding the appropriate eigenvalues and eigenvectors for the system. Then, in the case of model-order reduction, one must identify which eigenfunctions are required to adequately replicate the original function. (As the goal is to reduce the order of the system, a series of eigenfunctions in length equal to the original number of degrees of freedom within the system would not be very valuable.) Identifying these eigenvalues, eigenvectors, and modal combinations are at the heart of contemporary ROM research.

Extrapolative Potential

Model-order reduction in the context of turbulence is an active field of research with a rich history dating back almost a century. However, much of the work being done aims to minimize the number of modes required to replicate a given flow regime. While this modal optimization has its benefits, it also means that the system has been optimized exactly for that one flow regime. This research hypothesizes that a modal framework optimized for not one flow regime but of many similar flow regimes will improve the extrapolative potential of the resulting reduced-order model.