The consequences of finite-time proper orthogonal decomposition for an extensively chaotic flow field

TL;DR

This paper investigates how finite-time proper orthogonal decomposition (POD) affects the analysis of chaotic fluid flows, revealing slow convergence and significant errors, and compares two methods for estimating the flow's dimension.

Contribution

It demonstrates the slow convergence of POD in chaotic flows and compares the effectiveness of the snapshot and Fourier methods for dimension estimation.

Findings

Convergence of POD eigenvalues and eigenfunctions is very slow in chaotic flows.

The Fourier method significantly improves convergence over the snapshot method.

The flow's Karhunen-Loève dimension is much larger than the Lyapunov dimension.

Abstract

The use of proper orthogonal decomposition (POD) to explore the complex fluid flows that are common in engineering applications is increasing and has yielded new physical insights. However, for most engineering systems the dimension of the dynamics is expected to be very large yet the flow field data is available only for a finite time. In this context, it is important to establish the convergence of the POD in order to accurately estimate such quantities as the Karhunen-Lo\`{e}ve dimension. Using direct numerical simulations of Rayleigh-B\'{e}nard convection in a finite cylindrical geometry we explore a regime exhibiting extensive chaos and demonstrate the consequences of performing a POD with a finite amount of data. In particular, we show that the convergence in time of the eigenvalue spectrum, the eigenfunctions, and the dimension are very slow in comparison with the time scale of the convection rolls and that the errors incurred by not using the asymptotic values can be significant. We compute the dimension using two approaches, the method of snapshots and a Fourier method that exploits the azimuthal symmetry. We find that the convergence rate of the Fourier method is vastly improved over the method of snapshots. The dimension is found to be extensive as the system size is increased and for a dimension measurement that captures 90% of the variance in the data the Karhunen-Lo\`{e}ve dimension is about 20 times larger than the Lyapunov dimension.