Favre averaged Navier-Stokes equations
From CFD-Wiki
The instantaneous continuity equation, momentum equation and energy equation for a compressible fluid can be written as:
![\frac{\partial \rho}{\partial t} +
\frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0](/W/images/math/0/7/1/071e4f5508fd336ddad848551ce3188e.png)
![\frac{\partial}{\partial t}\left( \rho u_i \right) +
\frac{\partial}{\partial x_j}
\left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0](/W/images/math/6/f/c/6fc8041faa4be98ee72ec1e670fb22c7.png)
![\frac{\partial}{\partial t}\left( \rho e_0 \right) +
\frac{\partial}{\partial x_j}
\left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0](/W/images/math/8/1/7/8176cdf87a72617542883cbbeafc50cc.png)
For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:
Where the trace-less viscous strain-rate is defined by:
The heat-flux, 
, is given by Fourier's law:
Where the laminar Prandtl number 
is defined
by:
To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:
Where 
and 
are constant.
The total energy 
is defined by:
Note that the corresponding expression~\ref{eq:fav_total_energy} for Favre averaged turbulent flows contains an extra term related to the turbulent energy.
![\frac{\partial \overline{\rho}}{\partial t} +
\frac{\partial}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0](/W/images/math/d/c/d/dcd521718edfb790fcf5915dfff51dde.png)
![\frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{u_i} \right) +
\frac{\partial}{\partial x_j}
\left[
\overline{\rho} \widetilde{u_j} \widetilde{u_i}
+ \overline{p} \delta_{ij}
- \widetilde{\tau_{ji}^{tot}}
\right]
= 0](/W/images/math/c/9/f/c9fb98d7716cc9a91f7e78b0ed597728.png)
