J. math. fluid mech. 9 (2007) 44–74 1422-6928/07/010044-31 c ° 2006 Birkh¨auser Verlag, Basel DOI 10.1007/s00021-006-0191-0 Journal of Mathematical Fluid Mechanics Analytical and Numerical Results for the Rational Large Eddy Simulation Model David Barbato, Luigi C. Berselli* and Carlo R. Grisanti Communicated by G. P. Galdi Abstract. In this paper we analyze the Rational Large Eddy Simulation model. We start by introducing the system ofpartial differentialequations we shallconsider,together with its derivation. Then, we prove a result of fullregularity for strong solutions in the space periodic setting.We also construct some exact solutions useful for the numerical benchmarking and finally we provide the results of some numerical experiments we performed. Mathematics Subject Classification (2000). Primary 76 D03; Secondary 35 Q30, 76 F 65. Keywords. Large Eddy Simulation, regularity, exact solutions, numerical simulations. 1. Introduction In this paper we analyze the Rational Large Eddy Simulation (RLES in the sequel) model, introduced by Galdi and Layton [17]:                    ∂w ∂t + ∇q + ∇ · (w ⊗ w) − 1 Re ∆w + ∇ · µ I − δ 2 24 ∆ ¶ −1 · δ 2 12 ∇w∇w T ¸ = f , ∇ · w = 0, w(x, 0) = w 0 (x). (1.1) Here, Re > 0 is the Reynolds number, while the vector field w : Ω × [0, T ] → R 3 is an approximation, formally of order O(δ 4 ), of u that comes out by filtering the ∗ Corresponding author.