# Long-Time Reynolds Averaging of Reduced Order Models for Fluid Flows: Preliminary Results

## TL;DR

This paper investigates the long-term average energy transfer among modes in reduced order models of fluid flows, combining theoretical proofs and numerical experiments using the Burgers equation.

## Contribution

It provides the first analytical results on energy exchange in ROM modes and compares eigenfunction and POD modes, supported by numerical evidence.

## Key findings

- Time-average energy transfer from low to high POD modes is positive over long intervals.
- Analytical differences between eigenfunction and POD modes are highlighted.
- Numerical results support theoretical predictions about energy transfer.

## Abstract

We perform a theoretical and numerical investigation of the time-average of energy exchange among modes of Reduced Order Models (ROMs) of fluid flows. We are interested in the statistical equilibrium problem, and especially in the possible forward and backward average transfer of energy among ROM basis functions (modes). We consider two types of ROM modes: eigenfunctions of the Stokes operator and Proper Orthogonal Decomposition (POD) modes. We prove analytical results for both types of ROM modes and we highlight the differences between them. We also investigate numerically whether the time-average energy exchange between POD modes is positive. To this end, we utilize the one-dimensional Burgers equation as a simplified mathematical model, which is commonly used in ROM tests. The main conclusion of our numerical study is that, for long enough time intervals, the time-average energy exchange from low index POD modes to high index POD modes is positive, as predicted by our theoretical results.