Editing Large eddy simulation (section)

=== Compressible governing equations ===

For the governing equations of compressible flow, each equation, starting with the conservation of mass, is filtered.  This gives:

:<math>
\frac{\partial \overline{\rho}}{\partial t} + \frac{ \partial \overline{u_i \rho} }{\partial x_i} = 0
</math>

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<ref name="Favre_1983">{{cite journal
|author=Favre, Alexandre
|title=Turbulence: space-time statistical properties and behavior in supersonic flows
|year=1983
|journal=Physics of Fluids A
|volume=23
|issue=10
|pages=2851–2863
|doi=10.1063/1.864049|bibcode = 1983PhFl...26.2851F }}</ref> proposed a density-weighted filtering operation, called Favre filtering, defined for an arbitrary quantity <math>\phi</math> as:

:<math>
\tilde{\phi} = \frac{ \overline{\rho \phi} }{ \overline{\rho} }
</math>

which, in the limit of incompressibility, becomes the normal filtering operation.  This makes the conservation of mass equation:

:<math>
\frac{\partial \overline{\rho}}{\partial t} + \frac{ \partial \overline{\rho} \tilde{u_i} }{ \partial x_i } = 0.
</math>

This concept can then be extended to write the Favre-filtered momentum equation for compressible flow.  Following Vreman:<ref name="Vreman_1995">{{cite journal
|author1=Vreman, Bert
|author2=Geurts, Bernard
|author3=Kuerten, Hans
|journal=[[Applied Scientific Research]]
|year=1995
|volume=45
|issue=3
|doi=10.1007/BF00849116
|title=Subgrid-modelling in LES of compressible flow
|pages=191–203|bibcode=1995FTC....54..191V
|url=https://research.utwente.nl/en/publications/subgridmodelling-in-les-of-compressible-flow(7c54958d-ebdd-4422-bf9d-c17052984a68).html
}}</ref>

:<math>
\frac{ \partial \overline{\rho} \tilde{u_i} }{ \partial t }
+ \frac{ \partial \overline{\rho} \tilde{u_i} \tilde{u_j} }{ \partial x_j }
+ \frac{ \partial \overline{p} }{ \partial x_i }
- \frac{ \partial \tilde{\sigma}_{ij} }{ \partial x_j }
= - \frac{ \partial \overline{\rho} \tau_{ij}^{r} }{ \partial x_j }
+ \frac{ \partial }{ \partial x_j } \left( \overline{\sigma}_{ij} - \tilde{\sigma}_{ij} \right)
</math>

where <math>\sigma_{ij}</math> is the [[shear stress]] tensor, given for a Newtonian fluid by:

:<math>
\sigma_{ij} = 2 \mu(T) S_{ij} - \frac{2}{3} \mu(T) \delta_{ij} S_{kk}
</math>

and the term <math>\frac{ \partial }{\partial x_j} \left( \overline{\sigma}_{ij} - \tilde{\sigma}_{ij} \right)</math> represents a sub-filter viscous contribution from evaluating the viscosity <math>\mu(T)</math> using the Favre-filtered temperature <math>\tilde{T}</math>.  The subgrid stress tensor for the Favre-filtered momentum field is given by

:<math>
\tau_{ij}^{r} = \widetilde{ u_i \cdot u_j } - \tilde{u_i} \tilde{u_j}
</math>

By analogy, the Leonard decomposition may also be written for the residual stress tensor for a filtered triple product <math>\overline{\rho \phi \psi}</math>.  The triple product can be rewritten using the Favre filtering operator as <math>\overline{\rho} \widetilde{\phi \psi}</math>, which is an unclosed term (it requires knowledge of the fields <math>\phi</math> and <math>\psi</math>, when only the fields <math>\tilde{\phi}</math> and <math>\tilde{\psi}</math> are known).  It can be broken up in a manner analogous to <math>\overline{u_i u_j}</math> above, which results in a sub-filter stress tensor <math>\overline{\rho} \left( \widetilde{\phi \psi} - \tilde{\phi} \tilde{\psi} \right)</math>.  This sub-filter term can be split up into contributions from three types of interactions: the Leondard tensor <math>L_{ij}</math>, representing interactions among resolved scales; the Clark tensor <math>C_{ij}</math>, representing interactions between resolved and unresolved scales; and the Reynolds tensor <math>R_{ij}</math>, which represents interactions among unresolved scales.<ref name="Sagaut_2009">{{cite book
|author1=Garnier, E.
|author2=Adams, N.
|author3=Sagaut, P.
|title=Large eddy simulation for compressible flows
|year=2009
|publisher=Springer
|isbn=978-90-481-2818-1
|doi=10.1007/978-90-481-2819-8|url=https://cds.cern.ch/record/1339029
}}</ref>