filtered Navier–Stokes equations, where is the filtered pressure field and is the rate-of-strain tensor evaluated using the filtered velocity. The nonlinear filtered advection term is the chief cause of difficulty in LES modeling. It requires knowledge of the unfiltered velocity field, which is unknown, so it must be modeled. The analysis that follows illustrates the difficulty caused by the nonlinearity, namely, that it causes interaction between large and small scales, preventing separation of scales. The filtered advection term can be split up, following Leonard (1975), as: where is the residual stress tensor, so that the filtered Navier-Stokes equations become with the residual stress tensor grouping all unclosed terms. Leonard decomposed this stress tensor as and provided physical interpretations for each term. , the Leonard tensor, represents interactions among large scales, , the Reynolds stress-like term, represents interactions among the sub-filter scales (SFS), and , the Clark tensor, represents cross-scale interactions between large and small scales. Modeling the unclosed term is the task of sub-grid scale (SGS) models. This is made challenging by the fact that the subgrid stress tensor must account for interactions among all scales, including filtered scales with unfiltered scales. The filtered governing equation for a passive scalar , such as mixture fraction or temperature, can be written as where is the diffusive flux of , and is the sub-filter flux for the scalar . The filtered diffusive flux is unclosed, unless a particular form is assumed for it, such as a gradient diffusion model . is defined analogously to , and can similarly be split up into contributions from interactions between various scales. This sub-filter flux also requires a sub-filter model. Derivation Using Einstein notation, the Navier–Stokes equations for an incompressible fluid in Cartesian coordinates are Filtering the momentum equation results in If we assume that filtering and differentiation commute, then This equation models the changes in time of the filtered variables . Since the unfiltered variables are not known, it is impossible to directly calculate . However, the quantity is known. A substitution is made: Let . The resulting set of equations are the LES equations: Compressible governing equations For the governing equations of compressible flow, each equation, starting with the conservation of mass, is filtered. This gives: which results in an additional sub-filter term. However, it is desirable to avoid having to model the sub-filter scales of the mass conservation equation. For this reason, Favre