Large Eddy Simulation - Derivation

Derivation

Using Einstein notation, the Navier-Stokes equations for an incompressible fluid in Cartesian coordinates are

\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}.

Filtering the momentum equation results in

\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}}.

If we assume that filtering and differentiation commute, then

\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}.

This equation models the changes in time of the filtered variables . Since the unfiltered variables are not known, it is impossible to directly calculate . However, the quantity is known. A substitution is made:

\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).

Let . The resulting set of equations are the LES equations:

\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}.

Read more about this topic:  Large Eddy Simulation