This talk will address the issue of sub-grid closure in large eddy simulations, leveraging ideas from non-equilibrium statistical mechanics. The approach is based on the Mori-Zwanzig (M-Z) formalism, which provides a framework to re-cast a high-dimensional dynamical system into an equivalent, lower-dimensional system. In this reduced system, which is in the form of a generalized Langevin equation (GLE), the effect of the unresolved modes on the resolved modes appears as a convolution integral (which is sometimes referred to as memory). The appearance of the memory term in the GLE demonstrates that, coarse-graining non-scale-separated systems (such as turbulence) leads to non-local effects. The M-Z formalism alone does not lead to a reduction in computational complexity as it requires the solution of the orthogonal (unresolved) dynamics equation. A model for the memory is constructed by assuming that memory effects have a finite temporal support and by exploiting the Germano identity. The appeal of the proposed model, is that it is parameter-free and has a structural form imposed by the mathematics of the coarse-graining process (rather than the phenomenological assumptions made by the modeler, such as in classical subgrid scale models). Demonstrations are presented in the context of predictive LES for rotating and non-rotating turbulence, and turbulent channel flow. Recent results on extending these techniques in the context of Discontinuous Galerkin discretizations and will also be presented.
1. Parish, E. and Duraisamy, K., “A Dynamic Sub-grid Scale Model for Large Eddy Simulations based on the Mori-Zwanzig formalism," Journal of Computational Physics, 2017.
2. Gouasmi, A., Parish, E., and Duraisamy, K., “Characterizing Memory Effects in Coarse-Grained Nonlinear Systems Using the Mori-Zwanzig formalism," Under revision, Proceedings of Royal Society Series A, 2017. arXiv:1611.06277.
3. Parish, E. and Duraisamy, K., “Non-local closure models for Large Eddy Simulations based on the Mori-Zwanzig formalism," Physical Review Fluids, 2017.