Discrete Models of Fluids: Spatial Averaging, Closure, and Model Reduction

The main question addressed in the paper is how to obtain closed form continuum equations governing spatially averaged dynamics of semi-discrete ODE models of fluid flow. In the presence of multiple small scale heterogeneities, the size of these ODE systems can be very large. Spatial averaging is then a useful tool for reducing computational complexity of the problem. The averages satisfy balance equations of mass, momentum and energy. These equations are exact, but they do not form a continuum model in the true sense of the word because calculation of stress and heat flux requires solving the underlying ODE system. To produce continuum equations that can be simulated without resolving micro-scale dynamics, we developed a closure method based on the use of regularized deconvolutions. We mostly deal with non-linear averaging suitable for Lagrangian particle solvers, but consider Eulerian linear averaging where appropriate. The results of numerical experiments show good agreement between our closed form flux approximations and their exact counterparts.