Low-Re k-epsilon models

{{Turbulence modeling}}
==Introduction==

Good reviews of classical Low-Re k-epsilon models can be found in [[#References|[Patel (1995)]]] and [[#References|[Rodi (1993)]]]. There are hundreds of different low-Re k-epsilon models in the literature. This article tries to summarize and describe the most common and classical models. Feel free to add more models here, but please only add models that have gained a widespread use in the CFD community.

==Overview of models==

The models presented presented here are:

{| cellpadding="5" cellspacing="0" border="1"
|-
|'''Model''' || '''Reference''' || '''Description'''
|- 
|nowrap|Chien Model ||nowrap| [[#References|[Chien (1982}]]] || A very common model in turbomachinery applications. Has nice numerical properties.
|-
|nowrap|Launder-Sharma Model ||nowrap| [[#References|[Launder (1974)]]] || An old classical model which has attracted some attention for its ability to in model cases predict by-pass transition.
|-
|nowrap|Nagano-Tagawa Model ||nowrap|[[#References|[Nagano (1990)]]] || A model originally developed for heat-transfer applications.
|}

==Governing equations==

These models can be written in a general form like:

:<math>
\frac{\partial}{\partial t} \left( \rho k \right) +
\frac{\partial}{\partial x_j} 
\left[
 \rho k u_j - \left( \mu + \frac{\mu_t}{\sigma_k} \right) 
 \frac{\partial k}{\partial x_j}
\right]
=
P - \rho \epsilon - \rho D
</math>

:<math>
\frac{\partial}{\partial t} \left( \rho \epsilon \right) +
\frac{\partial}{\partial x_j} 
\left[
 \rho \epsilon u_j - \left( \mu + \frac{\mu_t}{\sigma_\epsilon} \right) 
 \frac{\partial \epsilon}{\partial x_j}
\right]
=
\left( C_{\epsilon_1} f_1 P - C_{\epsilon_2} f_2 \rho \epsilon \right)
\frac{\epsilon}{k}
+ \rho E
</math>

:<math>
\mu_t = C_\mu f_\mu \rho \frac{k^2}{\epsilon}
</math>

:<math>
P = \tau_{ij}^{turb} \frac{\partial u_i}{\partial x_j}
</math>

Where <math>C_{\epsilon_1}</math>, <math>C_{\epsilon_2}</math>, <math>C_\mu</math>, <math>\sigma_k</math> and <math>\sigma_\epsilon</math> are model constants. The damping functions <math>f_\mu</math>, <math>f_1</math> and <math>f_2</math> and the extra source terms <math>D</math> and <math>E</math> are only active close to solid walls and makes it possible to solve <math>k</math> and <math>\epsilon</math> down to the viscous sublayer. Table below summarizes the constants, damping functions and boundary conditions for all k-epsilon models presented here.

==Model constants, daming functions and boundary conditions==

{| cellpadding="5" cellspacing="0" border="1"
|- 
|||'''Chien''' || '''Launder-Sharma''' || '''Nagano-Tagawa'''
|-
|<math>c_\mu</math> || <math>0.09</math> || <math>0.09</math> || <math>0.09</math>
|-
|<math>\sigma_k</math> || <math>1</math> || <math>1</math> || <math>1.4</math>
|-
|<math>\sigma_\epsilon</math> || <math>1.3</math> || <math>1.3</math> || <math>1.3</math>
|-
|<math>D</math> || <math>2\, \nu \, \frac{k}{y^2}</math> || <math>2 \, \nu \, \left( \frac{\partial \sqrt{k}}{\partial y} \right)^2</math> || <math>0</math>
|-
|<math>E</math> || <math>-\frac{2 \nu \epsilon}{y^2} \exp{-0.5y^+}</math> || <math>2 \, \nu \, \nu_t \, \left(\frac{\partial^2 u}{\partial y^2}\right)^2</math> || <math>0</math>
|-
|<math>\epsilon_{wall}</math> || <math>0</math> || <math>0</math> || <math>\nu \, \left( \frac{\partial \sqrt{k}}{\partial y} \right)^2</math>
|-
|<math>C_{\epsilon_1}</math> || <math>1.35</math> || <math>1.44</math> || <math>1.45</math>
|-
|<math>C_{\epsilon_2}</math> || <math>1.8</math> || <math>1.92</math> || <math>1.9</math>
|-
|<math>f_\mu</math> || <math>1-\exp{-0.0115 y^+}</math> || <math> \exp{\frac{-3.4}{\left( 1 + R_t/50 \right)^2}}</math> A || <math>\left(1-\exp{\frac{-y^+}{26}}\right)^2 \left(1+\frac{4.1}{Re_t^{3/4}}\right)</math>
|-
|<math>f_1</math> || <math>1</math> || <math>1</math> || <math>1</math>
|-
|<math>f_2</math> || <math>1-0.22 \exp{-\left(\frac{Re_t}{6}\right)^2}</math> || <math>1-0.3 \exp{-Re_t^2}</math> || <math>\left(1\!-\!0.3\exp{-\left(\frac{Re_t}{6.5}\right)^2}\right)\!\! \left(1\!-\!\exp{\frac{-y^+}{6}}\right)^2</math>
|}
Where <math>Re_t \equiv \frac{k^2}{\nu \epsilon}</math>, <math>y^+  \equiv \frac{u^*y}{\nu}</math> and <math>k_{wall} = 0</math>.

A According to P. Gardin, M.Brunet, J.F.Domgin and K. Pericleous, "An experimental and numerical CFD study of turbulence in a tundish container", 2nd International Conference on CFD, CSIRO, 1999. There are three options for <math>f_\mu</math>


D1

<math> \min \left( 1.0 , \frac{R_t^{1/3}}{45} \right)</math>


D2

<math> 1 - \exp{\frac{-R_t}{ 30.1862 }}</math>


D3

<math> \exp{\frac{-3.4}{\left( 1 + R_t/50 \right)^2}}</math>

==Performance, applicability and limitations==

''Not written yet''

==Implementation issues==

''Not written yet''

==References==

{{reference-paper|author=Chien, K.-Y.|year=1982|title=Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds Number Turbulence Model|rest=AIAA Journal, Vol. 20, No. 1, pp. 33-38}}

{{reference-paper|author=Nagano, Y. and Tagawa, M.|year=1990|title=An Improved k-epsilon Model for Boundary Layer Flows|rest=Journal of Fluids Engineering, Vol. 112, pp. 33-39}}

{{reference-paper|author=Patel, V. C. and Rodi, W. and Scheuerer, G.|year=1985|title=Turbulence Models for Near-Wall and Low Reynolds Number Flows: A Review|rest=AIAA Journal, Vol. 23, No. 9, pp. 1308-1319}} 

{{reference-paper|author=Rodi, W. and Mansour, N. N.|year=1993|title=Low Reynolds Number k-epsilon Modeling with the Aid of Direct Simulation Data|rest=Journal of Fluid Mechanics, Vol. 250, pp. 509-529}}

{{reference-paper|author=Launder, B. E. and Sharma, B. I.|year=1974|title=Application of the Energy-Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc|rest=Letters in Heat and Mass Transfer, Vol. 1, No. 2, pp. 131-138}}

[[Category:Turbulence models]]