Theoretical Refinements of the Smagorinsky Model for Turbulence Simulations

Abstract: This paper presents a rigorous theoretical extension of the Smagorinsky model for turbulence simulations. The author builds on its fundamental framework, addressing known limitations, and making new mathematical advances. Specifically, this work introduces new theorems on the existence and uniqueness of weak solutions and establishes the stability of the model under small perturbations. The approach leverages advanced tools in functional analysis, particularly Sobolev and Banach spaces, increasing the theoretical robustness of the model. Each theorem is accompanied by detailed proofs employing variational techniques and energy estimations. Furthermore, the author proposes a variational formulation that demonstrates improvements in the mathematical behavior of the model's subgrid-scale stress representation. This work aims to provide a deeper understanding of the Smagorinsky model, setting the stage for further theoretical and computational developments. The obtained results are expected to impact turbulence modeling by offering a more robust theoretical basis for Large Eddy Simulations (LES).