Data Assimilation in Large Eddy Simulation: Addressing Model-Observation Mismatch from Navier-Stokes Data

TL;DR

This paper develops and analyzes a continuous data assimilation algorithm for Large Eddy Simulation models, demonstrating its effectiveness in synchronizing with Navier-Stokes data and providing theoretical and numerical validation.

Contribution

It introduces a new CDA approach for LES models with Navier-Stokes data, proving convergence and demonstrating effectiveness through simulations.

Findings

Proved exponential convergence of the CDA algorithm in 2D.

Validated synchronization through numerical simulations in 2D and 3D.

Quantified the error bound related to turbulence viscosity.

Abstract

In atmospheric and turbulent flow modeling, Large Eddy Simulation (LES) is often used to reduce computational cost, while observational data typically originates from the underlying physical system. Motivated by this setting, we study a continuous data assimilation (CDA) algorithm applied to a Smagorinsky/Ladyzhenskaya-type LES model, in which the observational data is generated from the full Navier--Stokes equations (NSE). In the two-dimensional setting, we establish global well-posedness of the assimilated system and prove exponential convergence to the true solution, up to an error of order $\bar{\nu}^{1/2}$, where $\bar{\nu}$ is the turbulence viscosity parameter. In addition to rigorous analysis in 2D, we provide numerical simulations in both 2D domains with physical boundary conditions and 3D periodic domains, demonstrating effective synchronization in these cases, and corroborating our theoretical predictions.