  This work uses the Mori-Zwanzig (M-Z) formalism, a concept originating from
non-equilibrium statistical mechanics, as a basis for the development of
coarse-grained models of turbulence. The mechanics of the generalized Langevin
equation (GLE) are considered and insight gained from the orthogonal dynamics
equation is used as a starting point for model development. A class of sub-grid
models is considered which represent non-local behavior via a finite memory
approximation (Stinis, P., "Mori-Zwanzig reduced models for uncertainty
quantification I: Parametric uncertainty," \textit{arXiv:1211.4285}, 2012.),
the length of which is determined using a heuristic that is related to the
spectral radius of the Jacobian of the resolved variables. The resulting models
are intimately tied to the underlying numerical resolution and are capable of
approximating non-Markovian effects. Numerical experiments on the Burgers
equation demonstrate that the M-Z-based models can accurately predict the
temporal evolution of the total kinetic energy and the total dissipation rate
at varying mesh resolutions. The trajectory of each resolved mode in
phase-space is accurately predicted for cases where the coarse-graining is
moderate. LES of homogeneous isotropic turbulence and the Taylor Green Vortex
show that the M-Z-based models are able to provide excellent predictions,
accurately capturing the sub-grid contribution to energy transfer. Lastly, LES
of fully developed channel flow demonstrate the applicability of M-Z-based
models to non-decaying problems. It is notable that the form of the closure is
not imposed by the modeler, but is rather derived from the mathematics of the
coarse-graining, highlighting the potential of M-Z-based techniques to define
LES closures.
