Zeta-f model
From CFD-Wiki
(Difference between revisions)
Mirzapopovac (Talk | contribs) |
Mirzapopovac (Talk | contribs) |
||
| Line 25: | Line 25: | ||
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{P_k}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math> | <math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{P_k}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math> | ||
| + | |||
| + | |||
| + | The production of the turbulent kinetic energy <math>P_k</math> | ||
| + | |||
| + | :<math> | ||
| + | P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} | ||
| + | </math> | ||
| + | <br> | ||
| + | :<math> P_k = \nu_t S^2 </math> | ||
| + | |||
| + | |||
| + | The modulus of the mean rate-of-strain tensor <math>S</math> | ||
| + | <br> | ||
| + | :<math> | ||
| + | S \equiv \sqrt{2S_{ij} S_{ij}} | ||
| + | </math> | ||
| Line 42: | Line 58: | ||
<math>C_\mu = 0.22</math>, <math>\sigma_{k} = 1</math>, <math>\sigma_{\varepsilon} = 1.3</math>, <math>\sigma_{\zeta} = 1.2</math>, <math>C_{\varepsilon 1} = 1.4 (1 + 0.012 / \zeta)</math>, <math>C_{\varepsilon 2} = 1.9</math>, <math>C_1 = 1.4</math>, <math>C_2' = 0.65</math>, <math>C_T = 6</math>, <math>C_L = 0.36</math> and <math>C_{\eta} = 85</math>. | <math>C_\mu = 0.22</math>, <math>\sigma_{k} = 1</math>, <math>\sigma_{\varepsilon} = 1.3</math>, <math>\sigma_{\zeta} = 1.2</math>, <math>C_{\varepsilon 1} = 1.4 (1 + 0.012 / \zeta)</math>, <math>C_{\varepsilon 2} = 1.9</math>, <math>C_1 = 1.4</math>, <math>C_2' = 0.65</math>, <math>C_T = 6</math>, <math>C_L = 0.36</math> and <math>C_{\eta} = 85</math>. | ||
| + | |||
| + | |||
| + | ==References== | ||
| + | M.P | ||
Revision as of 11:13, 22 January 2007
The zeta-f model is a robust modification of the elliptic relaxation model. The set of equations constituting the 
model reads:
The turbulent viscosity
The turbulent kinetic energy 
![\frac{\partial k}{\partial t} + U_j \frac{\partial k}{\partial x_j} = P_k - \varepsilon + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{k}} \right) \frac{\partial k}{\partial x_j} \right]](/W/images/math/4/e/a/4eac951b79fdf4d26cac2b6cca1540bd.png)
The turbulent kinetic energy dissipation rate 
![\frac{\partial \varepsilon}{\partial t} + U_j \frac{\partial \varepsilon}{\partial x_j} = \frac{C_{\varepsilon 1} P_k - C_{\varepsilon 2} \varepsilon}{T} + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\varepsilon}} \right) \frac{\partial \varepsilon}{\partial x_j} \right]](/W/images/math/5/b/e/5be411c44d64810a0a4d341ada81137c.png)
The normalized fluctuating velocity normal to the streamlines 
![\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} P_k + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_j} \right]](/W/images/math/1/4/3/143979f772a60676f05d24a868fa7f47.png)
The elliptic relaxation function 
The production of the turbulent kinetic energy 
The modulus of the mean rate-of-strain tensor 
The turbulence time scale 
![T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{0.6}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]](/W/images/math/6/1/7/617b18437b40d1988d1ca62041a1bca0.png)
The turbulence length scale 
![L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]](/W/images/math/7/0/7/70730f002515f1c28758512281d69cd2.png)
The coefficients
, 
, 
, 
, 
, 
, 
, 
, 
, 
and 
.
References
M.P
