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. Filtered kinetic energy equation In addition to the filtered mass and momentum equations, filtering the kinetic energy equation can provide additional insight. The kinetic energy field can be filtered to yield the total filtered kinetic energy: and the total filtered kinetic energy can be decomposed into two terms: the kinetic energy of the filtered velocity field , and the residual kinetic energy , such that . The conservation equation for can be obtained by multiplying the filtered momentum transport equation by to yield: where is the dissipation of kinetic energy of the filtered velocity field by viscous stress, and represents the sub-filter scale (SFS) dissipation of kinetic energy. The terms on the left-hand side represent transport, and the terms on the right-hand side are sink terms that dissipate kinetic energy. The SFS dissipation term is of particular interest, since it represents the transfer of energy from large resolved scales to small unresolved scales. On average, transfers energy from large to small scales. However, instantaneously can be positive or negative, meaning it can also act as a source term for , the kinetic energy of the filtered velocity field. The transfer of energy from