# Favre averaged Navier-Stokes equations

### From CFD-Wiki

(Difference between revisions)

Line 1: | Line 1: | ||

+ | The instantaneous continuity equation, | ||

+ | momentum equation and energy equation | ||

+ | for a compressible fluid can be written as: | ||

+ | |||

+ | <math> | ||

+ | \frac{\partial \rho}{\partial t} + | ||

+ | \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0 | ||

+ | </math> | ||

+ | |||

+ | <math> | ||

+ | \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 | ||

+ | </math> | ||

+ | |||

+ | <math> | ||

+ | \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 | ||

+ | </math> | ||

+ | |||

+ | For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous | ||

+ | stress is given by: | ||

+ | |||

+ | <math> | ||

+ | \tau_{ij} = 2 \mu S_{ij}^* | ||

+ | </math> | ||

+ | |||

+ | Where the trace-less viscous strain-rate is defined | ||

+ | by: | ||

+ | |||

+ | <math> | ||

+ | S_{ij}^* \equiv | ||

+ | \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + | ||

+ | \frac{\partial u_j}{\partial x_i} \right) - | ||

+ | \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} | ||

+ | </math> | ||

+ | |||

+ | The heat-flux, <math>q_j</math>, is given by Fourier's law: | ||

+ | |||

+ | <math> | ||

+ | q_j = -\lambda \frac{\partial T}{\partial x_j} | ||

+ | \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j} | ||

+ | </math> | ||

+ | |||

+ | Where the laminar Prandtl number <math>Pr</math> is defined | ||

+ | by: | ||

+ | |||

+ | <math> | ||

+ | Pr \equiv \frac{C_p \mu}{\lambda} | ||

+ | </math> | ||

+ | |||

+ | To close these equations it is also necessary to specify an equation of state. | ||

+ | Assuming a calorically perfect gas the following relations are valid: | ||

+ | |||

+ | <math> | ||

+ | \gamma \equiv \frac{C_p}{C_v} ~~,~~ | ||

+ | p = \rho R T ~~,~~ | ||

+ | e = C_v T ~~,~~ | ||

+ | C_p - C_v = R | ||

+ | </math> | ||

+ | |||

+ | Where <math>\gamma, C_p, C_v</math> and <math>R</math> are constant. | ||

+ | |||

+ | The total energy <math>e_0</math> is defined by: | ||

+ | |||

+ | <math> | ||

+ | e_0 \equiv e + \frac{u_k u_k}{2} | ||

+ | </math> | ||

+ | |||

+ | Note that the | ||

+ | corresponding expression~\ref{eq:fav_total_energy} | ||

+ | for Favre averaged turbulent flows contains an | ||

+ | extra term related to the turbulent energy. | ||

+ | |||

+ | |||

+ | |||

+ | |||

<math> | <math> | ||

\frac{\partial \overline{\rho}}{\partial t} + | \frac{\partial \overline{\rho}}{\partial t} + | ||

Line 14: | Line 92: | ||

= 0 | = 0 | ||

</math> | </math> | ||

+ | |||

+ | [[#total_energy]] | ||

+ | |||

+ | [[Category:Fluid Dynamics]][[Category:Equations]] |

## Revision as of 07:40, 5 September 2005

The instantaneous continuity equation, momentum equation and energy equation for a compressible fluid can be written as:

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.