Commit 3bd1cee5 authored by Thomas Fetzer's avatar Thomas Fetzer
Browse files

[lowrekepsilon] Add model equations

parent 38add600
......@@ -24,8 +24,57 @@
*
* \copydoc RANSModel
*
* These models calculate the eddy viscosity with two additional PDEs,
* one for the turbulent kinetic energy (k) and for the dissipation (epsilon).
* The low-Reynolds k-epsilon models calculate the eddy viscosity with two additional PDEs,
* one for the turbulent kinetic energy (k) and for the dissipation (\f$ \varepsilon \f$).
* The model uses the one proposed by Chien \cite Chien1982a.
* A good overview and additional models are given in Patel et al. \cite Patel1985a.
*
* The turbulent kinetic energy balance is identical with the one from the k-epsilon model,
* but the dissipation includes a dampening function (\f$ D_\varepsilon \f$):
* \f$ \varepsilon = \tilde{\varepsilon} + D_\varepsilon \f$:
*
* \f[
* \frac{\partial \left( k \right)}{\partial t}
* + \nabla \cdot \left( \textbf{v} k \right)
* - \nabla \cdot \left( \left( \nu + \frac{\nu_\text{t}}{\sigma_\text{k}} \right) \nabla k \right)
* - 2 \nu_\text{t} \textbf{S} \cdot \textbf{S}
* + \tilde{\varepsilon}
* + D_\varepsilon
* = 0
* \f].
*
* The dissipation balance is changed by introducing additional functions
* (\f$ E_\text{k}\f$, \f$ f_1 \f$, and \f$ f_2 \f$) to account for a dampening towards the wall:
* \f[
* \frac{\partial \left( \tilde{\varepsilon} \right)}{\partial t}
* + \nabla \cdot \left( \textbf{v} \tilde{\varepsilon} \right)
* - \nabla \cdot \left( \left( \nu + \frac{\nu_\text{t}}{\sigma_{\varepsilon}} \right) \nabla \tilde{\varepsilon} \right)
* - C_{1\tilde{\varepsilon}} f_1 \frac{\tilde{\varepsilon}}{k} 2 \nu_\text{t} \textbf{S} \cdot \textbf{S}
* + C_{2\tilde{\varepsilon}} f_2 \frac{\tilde{\varepsilon}^2}{k}
* - E_\text{k}
* = 0
* \f].
*
* The kinematic eddy viscosity \f$ \nu_\text{t} \f$ is dampened by \f$ f_\mu \f$:
* \f[
* \nu_\text{t} = C_\mu f_\mu \frac{k^2}{\tilde{\varepsilon}}
* \f].
*
* The auxiliary and dampening functions are defined as:
* \f[ D_\varepsilon = 2 \nu \nicefrac{k}{y^2} \f]
* \f[ E_\text{k} = -2 \nu \frac{\tilde{\varepsilon}}{y^2} \exp \left( -0.5 y^+ \right) \f]
* \f[ f_1 = 1 \f]
* \f[ f_2 = 1 - 0.22 \exp \left( - \left( \frac{\mathit{Re}_\text{t}}{6} \right)^2 \right) \f]
* \f[ f_\mu = 1 - \exp \left( -0.0115 y^+ \right) \f]
* \f[ \mathit{Re}_\text{t} = \frac{k^2}{\nu \tilde{\varepsilon}} \f]
* .
*
* Finally, the model is closed with the following constants:
* \f[ \sigma_\text{k} = 1.00 \f]
* \f[ \sigma_\varepsilon =1.30 \f]
* \f[ C_{1\tilde{\varepsilon}} = 1.35 \f]
* \f[ C_{2\tilde{\varepsilon}} = 1.80 \f]
* \f[ C_\mu = 0.09 \f]
*/
#ifndef DUMUX_LOWREKEPSILON_MODEL_HH
......
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment