  Models of turbulent flows require the resolution of a vast range of scales,
from large eddies to small-scale features directly associated with dissipation.
As the required resolution is not within reach of large scale numerical
simulations, standard strategies involve a smoothing of the fluid dynamics,
either through time averaging or spatial filtering. These strategies raise
formal issues in general relativity, where the split between space and time is
observer dependent. To make progress, we develop a new covariant framework for
filtering/averaging based on the fibration of spacetime associated with fluid
elements and the use of Fermi coordinates to facilitate a meaningful local
analysis. We derive the resolved equations of motion, demonstrating how
"effective" dissipative terms arise because of the coarse-graining, and paying
particular attention to the thermodynamical interpretation of the resolved
quantities. Finally, as the smoothing of the fluid-dynamics inevitably leads to
a closure problem, we propose a new closure scheme inspired by recent progress
in the modelling of dissipative relativistic fluids, and crucially, demonstrate
the linear stability of the proposed model.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena;Fluid Dynamics