# Favre averaged Navier-Stokes equations

(Difference between revisions)
 Revision as of 08:09, 5 September 2005 (view source)Jola (Talk | contribs)← Older edit Revision as of 08:36, 5 September 2005 (view source)Jola (Talk | contribs) Newer edit → Line 1: Line 1: - The instantaneous continuity equation, + == Instantaneuos Equations == - momentum equation and energy equation + - for a compressible fluid can be written as: + + The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as: + + +
:$:[itex] \frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0 \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0 -$      (1) + [/itex] - + (1)
:$:[itex] \frac{\partial}{\partial t}\left( \rho u_i \right) + \frac{\partial}{\partial t}\left( \rho u_i \right) + \frac{\partial}{\partial x_j} \frac{\partial}{\partial x_j} \left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0 \left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0 -$ (2) + - + (2)
:$:[itex] \frac{\partial}{\partial t}\left( \rho e_0 \right) + \frac{\partial}{\partial t}\left( \rho e_0 \right) + \frac{\partial}{\partial x_j} \frac{\partial}{\partial x_j} \left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0 \left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0 -$ (3) + + (3)
+ + + - For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous + For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by: - stress is given by: + - $+ + + :[itex] \tau_{ij} = 2 \mu S_{ij}^* \tau_{ij} = 2 \mu S_{ij}^*$ [/itex] + (4)
+ - Where the trace-less viscous strain-rate is defined + Where the trace-less viscous strain-rate is defined by: - by: + - $+ + + :[itex] S_{ij}^* \equiv S_{ij}^* \equiv \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + Line 36: Line 46: \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$ [/itex] + (5)
+ The heat-flux, $q_j$, is given by Fourier's law: The heat-flux, $q_j$, is given by Fourier's law: - $+ + + :[itex] q_j = -\lambda \frac{\partial T}{\partial x_j} q_j = -\lambda \frac{\partial T}{\partial x_j} \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j} \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}$ [/itex] + (6)
+ - Where the laminar Prandtl number $Pr$ is defined + Where the laminar Prandtl number $Pr$ is defined by: - by: + - $+ + + :[itex] Pr \equiv \frac{C_p \mu}{\lambda} Pr \equiv \frac{C_p \mu}{\lambda}$ [/itex] + (7)
+ - To close these equations it is also necessary to specify an equation of state. + To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid: - Assuming a calorically perfect gas the following relations are valid: + - $+ + + :[itex] \gamma \equiv \frac{C_p}{C_v} ~~,~~ \gamma \equiv \frac{C_p}{C_v} ~~,~~ p = \rho R T ~~,~~ p = \rho R T ~~,~~ Line 60: Line 80: C_p - C_v = R C_p - C_v = R$ [/itex] + (8)
+ - Where $\gamma, C_p, C_v$ and $R$ are constant. + Where $\gamma$, $C_p$, $C_v$ and $R$ are constant. The total energy $e_0$ is defined by: The total energy $e_0$ is defined by: - $+ + + :[itex] e_0 \equiv e + \frac{u_k u_k}{2} e_0 \equiv e + \frac{u_k u_k}{2}$ [/itex] + (9)
+ - Note that the + Note that the corresponding expression
Insert Reference
for Favre averaged turbulent flows contains an extra term related to the turbulent energy. - corresponding expression~\ref{eq:fav_total_energy} + - for Favre averaged turbulent flows contains an + - extra term related to the turbulent energy. + + Equations (1)-(9), supplemented with gas data for $\gamma$, $Pr$, $\mu$ and perhaps $R$, form a closed set of partial differential equations, and need only be complemented with boundary conditions. + ==

## Instantaneuos Equations

The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:

 $\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0$ (1) $\frac{\partial}{\partial t}\left( \rho u_i \right) + \frac{\partial}{\partial x_j} \left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0$ (2) $\frac{\partial}{\partial t}\left( \rho e_0 \right) + \frac{\partial}{\partial x_j} \left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0$ (3)

For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:

 $\tau_{ij} = 2 \mu S_{ij}^*$ (4)

Where the trace-less viscous strain-rate is defined by:

 $S_{ij}^* \equiv \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$ (5)

The heat-flux, $q_j$, is given by Fourier's law:

 $q_j = -\lambda \frac{\partial T}{\partial x_j} \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}$ (6)

Where the laminar Prandtl number $Pr$ is defined by:

 $Pr \equiv \frac{C_p \mu}{\lambda}$ (7)

To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:

 $\gamma \equiv \frac{C_p}{C_v} ~~,~~ p = \rho R T ~~,~~ e = C_v T ~~,~~ C_p - C_v = R$ (8)

Where $\gamma$, $C_p$, $C_v$ and $R$ are constant.

The total energy $e_0$ is defined by:

 $e_0 \equiv e + \frac{u_k u_k}{2}$ (9)
Note that the corresponding expression
 Insert Reference
for Favre averaged turbulent flows contains an extra term related to the turbulent energy.

Equations (1)-(9), supplemented with gas data for $\gamma$, $Pr$, $\mu$ and perhaps $R$, form a closed set of partial differential equations, and need only be complemented with boundary conditions.

==

$\frac{\partial \overline{\rho}}{\partial t} + \frac{\partial}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0$

$\frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{u_i} \right) + \frac{\partial}{\partial x_j} \left[ \overline{\rho} \widetilde{u_j} \widetilde{u_i} + \overline{p} \delta_{ij} - \widetilde{\tau_{ji}^{tot}} \right] = 0$