Modeling error in Approximate Deconvolution Models

TL;DR

This paper analyzes the asymptotic behavior of modeling errors in approximate deconvolution models for 3D periodic flows, comparing Helmholz and Gaussian filters as the deconvolution order increases.

Contribution

It provides convergence rate estimates for Helmholz filters and discusses limitations in analyzing Gaussian filters, advancing understanding of error behavior in these models.

Findings

Helmholz filters show convergence to zero error with increasing deconvolution order.

Energy budgets and inequalities are used to estimate convergence rates.

The analysis for Gaussian filters remains inconclusive, highlighting open research issues.

Abstract

We investigate the assymptotic behaviour of the modeling error in approximate deconvolution model in the 3D periodic case, when the order $N$ of deconvolution goes to $\infty$. We consider successively the generalised Helmholz filters of order $p$ and the Gaussian filter. For Helmholz filters, we estimate the rate of convergence to zero thanks to energy budgets, Gronwall's Lemma and sharp inequalities about Fouriers coefficients of the residual stress. We next show why the same analysis does not allow to conclude convergence to zero of the error modeling in the case of Gaussian filter, leaving open issues.