Convergence of approximate deconvolution models to the filtered Navier-Stokes Equations

TL;DR

This paper proves that a class of Approximate Deconvolution Models (ADMs) for 3D fluid flow converge to the filtered Navier-Stokes equations as the deconvolution order increases, validating their approximation effectiveness.

Contribution

It establishes existence, uniqueness, and convergence of ADM solutions to the filtered Navier-Stokes equations for the first time.

Findings

Existence and uniqueness of regular weak solutions for fixed deconvolution order.

Convergence of ADM solutions to the filtered Navier-Stokes equations as deconvolution order tends to infinity.

Rigorous mathematical validation of ADM as an effective approximation method.

Abstract

We consider a 3D Approximate Deconvolution Model (ADM) which belongs to the class of Large Eddy Simulation (LES) models. We work with periodic boundary conditions and the filter that is used to average the fluid equations is the Helmholtz one. We prove existence and uniqueness of what we call a "regular weak" solution $(\wit_N,q_N)$ to the model, for any fixed order $N\in\N$ of deconvolution. Then, we prove that the sequence $\{(\wit_N,q_N)\}_{N \in \N}$ converges -in some sense- to a solution of the filtered Navier-Stokes equations, as $N$ goes to infinity. This rigorously shows that the class of ADM models we consider have the most meaningful approximation property for averages of solutions of the Navier-Stokes equations.