CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Wiki > Favre averaged Navier-Stokes equations

Favre averaged Navier-Stokes equations

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
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:
 +
 +
<table width="100%">
 +
<tr><td>
:<math>
:<math>
\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
-
</math>      (1)
+
</math>       
-
 
+
</td><td width="5%">(1)</td></tr>
 +
<tr><td>
:<math>
:<math>
\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
-
</math> (2)
+
</math>
-
 
+
</td><td>(2)</td></tr>
 +
<tr><td>
:<math>
:<math>
\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
-
</math> (3)
+
</math>
 +
</td><td>(3)</td></tr>
 +
</table>
-
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:
+
-
<math>
+
<table width="100%">
 +
<tr><td>
 +
:<math>
\tau_{ij} = 2 \mu S_{ij}^*
\tau_{ij} = 2 \mu S_{ij}^*
</math>
</math>
 +
</td><td width="5%">(4)</td></tr>
 +
</table>
-
Where the trace-less viscous strain-rate is defined
+
Where the trace-less viscous strain-rate is defined by:
-
by:
+
-
<math>
+
<table width="100%">
 +
<tr><td>
 +
:<math>
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}
</math>
</math>
 +
</td><td width="5%">(5)</td></tr>
 +
</table>
The heat-flux, <math>q_j</math>, is given by Fourier's law:
The heat-flux, <math>q_j</math>, is given by Fourier's law:
-
<math>
+
<table width="100%">
 +
<tr><td>
 +
:<math>
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}
</math>
</math>
 +
</td><td width="5%">(6)</td></tr>
 +
</table>
-
Where the laminar Prandtl number <math>Pr</math> is defined
+
Where the laminar Prandtl number <math>Pr</math> is defined by:
-
by:
+
-
<math>
+
<table width="100%">
 +
<tr><td>
 +
:<math>
Pr \equiv \frac{C_p \mu}{\lambda}
Pr \equiv \frac{C_p \mu}{\lambda}
</math>
</math>
 +
</td><td width="5%">(7)</td></tr>
 +
</table>
-
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:
+
-
<math>
+
<table width="100%">
 +
<tr><td>
 +
:<math>
\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
</math>
</math>
 +
</td><td width="5%">(8)</td></tr>
 +
</table>
-
Where <math>\gamma, C_p, C_v</math> and <math>R</math> are constant.
+
Where <math>\gamma</math>, <math>C_p</math>, <math>C_v</math> and <math>R</math> are constant.
The total energy <math>e_0</math> is defined by:
The total energy <math>e_0</math> is defined by:
-
<math>
+
<table width="100%">
 +
<tr><td>
 +
:<math>
e_0 \equiv e + \frac{u_k u_k}{2}
e_0 \equiv e + \frac{u_k u_k}{2}
</math>
</math>
 +
</td><td width="5%">(9)</td></tr>
 +
</table>
-
Note that the
+
Note that the corresponding expression <table><tr><td bgcolor="red">Insert Reference</td></tr></table> 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 <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.
 +
==

Revision as of 08:36, 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.

==



\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

#total_energy

My wiki