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Favre averaged Navier-Stokes equations

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The instantaneous continuity equation,
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== Instantaneuos Equations ==
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momentum equation and energy equation
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for a compressible fluid can be written as:
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 +
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
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</math>      (1)
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</math>       
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</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
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</math> (2)
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</math>
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</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
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</math> (3)
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</math>
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</td><td>(3)</td></tr>
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</table>
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For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous
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For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:
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stress is given by:
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<math>
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<table width="100%">
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<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>
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Where the trace-less viscous strain-rate is defined
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Where the trace-less viscous strain-rate is defined by:
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by:
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<math>
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<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} +
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                 \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>
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</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:
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<math>
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<table width="100%">
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<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>
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</td><td width="5%">(6)</td></tr>
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</table>
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Where the laminar Prandtl number <math>Pr</math> is defined
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Where the laminar Prandtl number <math>Pr</math> is defined by:
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by:
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<math>
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<table width="100%">
 +
<tr><td>
 +
:<math>
Pr \equiv \frac{C_p \mu}{\lambda}
Pr \equiv \frac{C_p \mu}{\lambda}
</math>
</math>
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</td><td width="5%">(7)</td></tr>
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</table>
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To close these equations it is also necessary to specify an equation of state.
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To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:
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Assuming a calorically perfect gas the following relations are valid:
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<math>
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<table width="100%">
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<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 ~~,~~
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C_p - C_v = R
C_p - C_v = R
</math>
</math>
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</td><td width="5%">(8)</td></tr>
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</table>
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Where <math>\gamma, C_p, C_v</math> and <math>R</math> are constant.
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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:
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<math>
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<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>
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</td><td width="5%">(9)</td></tr>
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</table>
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Note that the
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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.
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corresponding expression~\ref{eq:fav_total_energy}
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for Favre averaged turbulent flows contains an
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extra term related to the turbulent energy.
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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.
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==

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

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