Optimal Closures in a Simple Model for Turbulent Flows

In this work we introduce a computational framework for determining optimal closures of the eddy-viscosity type for large-eddy simulations (LESs) of a broad class of PDE models, such as the Navier--Stokes equation. This problem is cast in terms of PDE-constrained optimization where an error functional representing the misfit between the target and predicted observations is minimized with respect to the functional form of the eddy viscosity in the closure relation. Since this leads to a PDE optimization problem with a nonstandard structure, the solution is obtained computationally with a flexible and efficient gradient approach relying on a combination of modified adjoint-based analysis and Sobolev gradients. By formulating this problem in the continuous setting we are able to determine the optimal closure relations in a very general form subject only to some minimal assumptions. The proposed framework is thoroughly tested on a model problem involving the LES of the one-dimensional Kuramoto--Sivashinsky equation, where optimal forms of the eddy viscosity are obtained as generalizations of the standard Smagorinsky model. It is demonstrated that while the solution trajectories corresponding to the direct numerical simulation and LES still diverge exponentially, with such optimal eddy viscosities the rate of divergence is significantly reduced as compared to the Smagorinsky model. By systematically finding optimal forms of the eddy viscosity within a certain general class of closure models, this framework can thus provide insights about the fundamental performance limitations of these models.