Editing Large eddy simulation (section)

====Derivation====
Using [[Einstein notation]], the Navier–Stokes equations for an incompressible fluid in Cartesian coordinates are
: <math> \frac{\partial u_i}{\partial x_i} = 0 </math>
: <math> \frac{\partial u_i}{\partial t} + \frac{\partial u_iu_j}{\partial x_j}
= - \frac{1}{\rho} \frac{\partial p}{\partial x_i}
+ \nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}.
</math>

Filtering the momentum equation results in
: <math> \overline{\frac{\partial u_i}{\partial t}} + \overline{\frac{\partial u_iu_j}{\partial x_j}}
= - \overline{\frac{1}{\rho} \frac{\partial p}{\partial x_i}}
+ \overline{\nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}}.
</math>

If we assume that filtering and differentiation commute, then
: <math> \frac{\partial \bar{u_i}}{\partial t} + \overline{\frac{\partial u_iu_j}{\partial x_j}}
= - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i}
+ \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j}.
</math>

This equation models the changes in time of the filtered variables <math>\bar{u_i}</math>.  Since the unfiltered variables <math>u_i</math> are not known, it is impossible to directly calculate <math>\overline{\frac{\partial u_iu_j}{\partial x_j}}</math>.  However, the quantity <math> \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j}</math> is known.  A substitution is made:
: <math> \frac{\partial \bar{u_i}}{\partial t} +  \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j}
= - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i}
+ \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j}
- \left(\overline{ \frac{\partial u_iu_j}{\partial x_j}} -  \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j}\right).
</math>

Let <math>\tau_{ij} = \overline{u_i u_j} - \bar{u}_{i} \bar{u}_{j}</math>. The resulting set of equations are the LES equations:
:<math> \frac{\partial \bar{u_i}}{\partial t} + \bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j}
= - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i}
+ \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j}
- \frac{\partial\tau_{ij}}{\partial x_j}.
</math>