By Barnett I. A., Nathan D.

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The key step in the derivation of the LES or RANS equations is averaging over small scales by the aforementioned procedures applied to the Navier– Stokes equations. ) What distinguishes LES from RANS is the deﬁnition of the small scales. LES assumes the small scales to be smaller than the mesh size ∆, and RANS assumes them to be smaller than the largest eddy scale L. 32) and write where u ¯ is the ﬁltered (or averaged) value of u, and u is the ﬂuctuating component. 32) and assuming that G is chosen so that the ﬁltering and diﬀerentiation operators commute, we have ∂ ∂u ¯ + (uu) = 0 .

23) and the calorically perfect gas equation of state p= ρe . 24) Cp . 3 Historical Perspective Before commencing our summary of the basic equations of ﬂuid dynamics, we provide a brief historical perspective on the evolution of the mathematical description of ﬂuid motion. Navier (1827) must be credited with the ﬁrst attempt at deriving the equations for homogeneous incompressible viscous ﬂuids on the basis of considerations involving the action of intermolecular forces. Poisson (1831) derived the equations for compressible ﬂuids from a similar molecular model.

Various hybrid RANS/LES approaches have been developed. These use RANS models in part of the ﬂow and LES models in other parts. One example is the detached-eddy simulation (DES) approach (see Spalart (2000)) that uses RANS models in attached boundary layers and LES models in regions of separated ﬂow. Sagaut (2006) provides extensive coverage of LES formulations and models, albeit only for incompressible ﬂow. Pope (2000) discusses LES for compressible (and reacting) ﬂows. Gatski, Hussaini and Lumley (1996) provide an introduction to DNS, LES and RANS modeling of turbulent ﬂows.