Editing Large eddy simulation (section)

=== 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:

:<math>
\overline{E} = \frac{1}{2} \overline{ u_i u_i }
</math>

and the total filtered kinetic energy can be decomposed into two terms: the kinetic energy of the filtered velocity field <math>E_f</math>,

:<math>
E_f = \frac{1}{2} \overline{u_i}  \, \overline{u_i}
</math>

and the residual kinetic energy <math>k_r</math>,

:<math>
k_r = \frac{1}{2} \overline{ u_i u_i } - \frac{1}{2} \overline{u_i} \, \overline{u_i} = \frac{1}{2} \tau_{ii}^{r}
</math>

such that <math>\overline{E} = E_f + k_r</math>.

The conservation equation for <math>E_f</math> can be obtained by multiplying the filtered momentum transport equation by <math>\overline{u_i}</math> to yield:

:<math>
\frac{\partial E_f}{\partial t} 
+ \overline{u_j} \frac{\partial E_f}{\partial x_j} 
+ \frac{1}{\rho} \frac{\partial \overline{u_i} \bar{p} }{ \partial x_i }
+ \frac{\partial \overline{u_i} \tau_{ij}^{r}}{\partial x_j} 
- 2 \nu \frac{ \partial \overline{u_i} \bar{S_{ij}} }{ \partial x_j }
= 
- \epsilon_{f} 
- \Pi
</math>

where <math>\epsilon_{f} = 2 \nu \bar{S_{ij}} \bar{S_{ij}}</math> is the dissipation of kinetic energy of the filtered velocity field by viscous stress, and <math>\Pi = -\tau_{ij}^{r} \bar{S_{ij}}</math> 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.<ref name="Pope_2000" />

The <math>\Pi</math> SFS dissipation term is of particular interest, since it represents the transfer of energy from large resolved scales to small unresolved scales.  On average, <math>\Pi</math> transfers energy from large to small scales.  However, instantaneously <math>\Pi</math> can be positive ''or'' negative, meaning it can also act as a source term for <math>E_f</math>, the kinetic energy of the filtered velocity field.  The transfer of energy from unresolved to resolved scales is called '''backscatter''' (and likewise the transfer of energy from resolved to unresolved scales is called '''forward-scatter''').<ref name="Piomelli_1991">{{cite journal
|author1=Piomelli, U.
|author2=Cabot, W.
|author3=Moin, P.
|author4=Lee, S.
|title=Subgrid-scale backscatter in turbulent and transitional flows
|journal=Physics of Fluids A
|year=1991
|volume=3
|issue=7
|pages=1766–1771
|doi=10.1063/1.857956|bibcode = 1991PhFlA...3.1766P
|s2cid=54904570
|authorlink3=Parviz Moin }}</ref>