Long-time Reynolds averaging of reduced order models for fluid flows: Preliminary results

Luigi C. Berselli; Traian Iliescu; Birgul Koc; Roger Lewandowski; Luigi C. Berselli; Traian Iliescu; Birgul Koc; Roger Lewandowski

doi:10.3934/mine.2020001

Mathematics in Engineering

2020, Volume 2, Issue 1: 1-25. doi: 10.3934/mine.2020001

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Long-time Reynolds averaging of reduced order models for fluid flows: Preliminary results

1.
Dipartimento di Matematica, Università di Pisa, Pisa, I-56127, Italy
2.
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
3.
Univ Rennes & INRIA, CNRS–IRMAR UMR 6625 & Fluminance team, Rennes, F-35042, France

Received: 15 January 2019 Accepted: 29 August 2019 Published: 22 October 2019

We perform a preliminary 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.
- reduced order model,
- long-time behavior,
- Reynolds equations,
- statistical equilibrium
Citation: Luigi C. Berselli, Traian Iliescu, Birgul Koc, Roger Lewandowski. Long-time Reynolds averaging of reduced order models for fluid flows: Preliminary results[J]. Mathematics in Engineering, 2020, 2(1): 1-25. doi: 10.3934/mine.2020001

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Abstract

We perform a preliminary 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.

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