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Here we introduce new coarse-grained variables for a turbulent flow in theform of moments of its Reynolds-averaged Boltzmann equation. With the exceptionof the collision moments, the transport equations for the new variables areidentical to the usual moment equations, and thus naturally lend themselves tothe variety of already existing closure methods. Under the anelastic turbulenceapproximation, we derive equations for the Reynolds-averaged turbulentfluctuations around the coarse-grained state. We show that the global relativeentropy of the coarse-grained state is bounded from above by the Reynoldsaverage of the fine-grained global relative entropy, and thus obeys the timedecay bound of Desvillettes and Villani. This is similar to what is observed inthe rarefied gas dynamics, which makes the Grad moment closure a good candidatefor truncating the hierarchy of the coarse-grained moment equations. We alsoshow that, under additional assumptions on the form of the coarse-grainedcollision terms, one arrives at the Navier-Stokes closure, which can benaturally extended to the Burnett and super-Burnett orders. Finally, we suggestcrude parameterizations of the coarse-grained collision terms for use asstarting points in numerical simulation and modeling.

Coarse-grained transport of a turbulent flow via moments of the Reynolds-averaged Boltzmann equation