OPTIMAL CLOSURES IN HYDRODYNAMIC MODELS

Abstract

In this work, we investigate the performance limitations characterizing certain common closure models for nonlinear models of fluid flow. The need for closures arises when for computational reasons first-principles models, such as the Navier-Stokes equations, are replaced with their simplified (filtered) versions such as the Large-Eddy Simulation (LES). In the present work, we focus on a simple model problem based on the 1D Kuramoto-Sivashinsky equation with a Smagorinsky-type eddy-viscosity closure model. The eddy viscosity is assumed to be a function of the state (flow) variable whose optimal functional form is determined in a very general form in the continuous setting. It is found by solving a PDE-constrained optimization problem in which the least-squares error between the output of the LES and the true flow evolution is minimized with respect to the functional form of the eddy viscosity. This problem is solved using a gradient-based technique utilizing a suitable adjoint-based variational data-assimilation approach implemented in the optimize-then-discretize setting using state-of-the-art techniques. The numerical computations are thoroughly validated. The obtained results indicate how the standard Smagorinsky closure model can be refined such that the corresponding LES evolution approximates more accurately the evolution of the original (unfiltered) flow.