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

Luigi C Berselli,Roger Lewandowski,Birgul Koc,Traian Iliescu

doi:10.3934/mine.2020001

Abstract

group GNAMPA of INdAM; University of Pisa [PRA 2018 52]; NSFNational Science Foundation (NSF) [DMS-1821145]

Highlights

In this note we combine some results on the long-time averaging of fluid equations with the recently developed techniques for reduced order model (ROM) development
In this preliminary work we start proving some analytical results that characterize the timeaveraged effect of the exchange of energy between various modes, both in the case of the computable decomposition made with proper orthogonal decomposition (POD) type basis functions and with the abstract basis made with eigenfunctions
We investigated theoretically and numerically the time-average of the exchange of energy among modes of reduced order models (ROMs) of fluid flows

Summary

Introduction

In this note we combine some results on the long-time averaging of fluid equations with the recently developed techniques for reduced order model (ROM) development In this preliminary work we start proving some analytical results that characterize the timeaveraged effect of the exchange of energy between various modes, both in the case of the computable decomposition made with proper orthogonal decomposition (POD) type basis functions and with the abstract basis made with eigenfunctions. The properties of a turbulent flow are computable (and relevant) only in an average sense In this respect, we want to follow the most classical approach dating back to Stokes, Reynolds, and Prandtl of considering long-time averages of the solution as the key quantity to be computed or observed. We do not need to consider statistical averages and link them with time averaging by means of (unproved) ergodic hypotheses

Objectives

Results

Conclusion