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 proposed a density-weighted filtering operation, called Favre filtering, defined for an arbitrary quantity  as:
which, in the limit of incompressibility, becomes the normal filtering operation.  This makes the conservation of mass equation:
This concept can then be extended to write the Favre-filtered momentum equation for compressible flow.  Following Vreman:
where  is the shear stress tensor, given for a Newtonian fluid by:
and the term  represents a sub-filter viscous contribution from evaluating the viscosity  using the Favre-filtered temperature .  The subgrid stress tensor for the Favre-filtered momentum field is given by
By analogy, the Leonard decomposition may also be written for the residual stress tensor for a filtered triple product .  The triple product can be rewritten using the Favre filtering operator as , which is an unclosed term (it requires knowledge of the fields  and , when only the fields  and  are known).  It can be broken up in a manner analogous to  above, which results in a sub-filter stress tensor .  This sub-filter term can be split up into contributions from three types of interactions: the Leondard tensor , representing interactions among resolved scales; the Clark tensor , representing interactions between resolved and unresolved scales; and the Reynolds tensor , which represents interactions among unresolved scales.