Favre averaged Navier-Stokes equations

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Revision as of 09:02, 5 September 2005

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.

Favre Averaged Equations

It is not possible to solve the instantaneous equations directly for the applications of interest here. At the Reynolds numbers typically present in a turbine these equations have very chaotic turbulent solutions, and it is necessary to model the influence of the smallest scales. All turbulence models used in this work are based on one-point averaging of the instantaneous equations. The averaging procedure used is described in the next sections.

Averaging

Let $\Phi$ be any dependent variable. It is convenient to define two different types of averaging of $\Phi$ :

Classical time average (Reynolds average):

$\overline{\Phi} \equiv \frac{1}{T} \int_T \Phi(t) dt$	(10)
$\Phi' \equiv \Phi - \overline{\Phi}$

Density weighted time average (Favre average):

$\widetilde{\Phi} \equiv \frac{\overline{\rho \Phi}}{\overline{\rho}}$	(11)
$\Phi'' \equiv \Phi - \widetilde{\Phi}$

Note that with the above definitions $\overline{\Phi'} = 0$ , but $\overline{\Phi''} \neq 0$ .

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

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

@@ Line 99: / Line 99: @@
 Equations (1)-(9), supplemented with gas data for <math>\gamma</math>, <math>Pr</math>, <math>\mu</math> and perhaps <math>R</math>, form a closed set of partial differential equations, and need only be complemented with boundary conditions.
-==
+== Favre Averaged Equations ==
+It is not possible to solve the instantaneous equations directly for the applications of interest here. At the Reynolds numbers typically present in a turbine these equations have very chaotic turbulent solutions, and it is necessary to model the influence of the smallest scales. All turbulence models used in this work are based on one-point averaging of the instantaneous equations. The averaging procedure used is described in the next sections.
+== Averaging ==
+Let <math>\Phi</math> be any dependent variable. It is convenient to define
+two different types of averaging of <math>\Phi</math>:
+* Classical time average (Reynolds average):
+<table width="100%">
+<tr><td>
+:<math>\overline{\Phi} \equiv \frac{1}{T} \int_T \Phi(t) dt</math>
+</td>
+<td rowspan="2" width="5%">(10)</td></tr>
+<tr><td>
+:<math>\Phi' \equiv \Phi - \overline{\Phi}</math>
+</td></tr>
+</table>
+* Density weighted time average (Favre average):
+<table width="100%">
+<tr><td>
+:<math>\widetilde{\Phi} \equiv  \frac{\overline{\rho \Phi}}{\overline{\rho}}</math>
+</td><td rowspan="2" width="5%">(11)</td></tr>
+<tr><td>
+:<math>\Phi'' \equiv \Phi - \widetilde{\Phi}</math>
+</td></tr>
+</table>
+Note that with the above definitions <math>\overline{\Phi'} = 0</math>, but <math>\overline{\Phi''} \neq 0</math>.
 <math>
@@ Line 118: / Line 147: @@
 </math>
-[[#total_energy]]
 [[Category:Fluid Dynamics]][[Category:Equations]]

Favre averaged Navier-Stokes equations

From CFD-Wiki

Revision as of 09:02, 5 September 2005

Instantaneuos Equations

Favre Averaged Equations

Averaging

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