CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Wiki > Zeta-f model

Zeta-f model

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
Line 1: Line 1:
-
This is a robust modification of the elliptic relaxation model ([[v2-f]]).
+
The ''zeta-f'' model is a robust modification of the elliptic relaxation model. The set of equations constituting the <math>\zeta-f</math> model reads:
 +
 
 +
 
 +
The turbulent viscosity
 +
 
 +
<math>\nu_t = C_\mu \, \zeta \, k \, T</math>
 +
 
 +
 
 +
The turbulent kinetic energy <math>k</math>
 +
 
 +
<math>\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]</math>
 +
 
 +
 
 +
The turbulent kinetic energy dissipation rate <math>\varepsilon</math>
 +
 
 +
<math>\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]</math>
 +
 
 +
 
 +
The normalized fluctuating velocity normal to the streamlines <math>\zeta</math>
 +
 
 +
<math>\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]</math>
 +
 
 +
 
 +
The elliptic relaxation function <math>f</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 turbulence time scale <math>T</math>
 +
 
 +
<math>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]</math>
 +
 
 +
 
 +
The turbulence length scale <math>L</math>
 +
 
 +
<math>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]</math>
 +
 
 +
 
 +
The coefficients
 +
 
 +
<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>.

Revision as of 11:06, 22 January 2007

The zeta-f model is a robust modification of the elliptic relaxation model. The set of equations constituting the \zeta-f model reads:


The turbulent viscosity

\nu_t = C_\mu \, \zeta \, k \, T


The turbulent kinetic energy k

\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]


The turbulent kinetic energy dissipation rate \varepsilon

\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]


The normalized fluctuating velocity normal to the streamlines \zeta

\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]


The elliptic relaxation function f

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)


The turbulence time scale T

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]


The turbulence length scale L

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]


The coefficients

C_\mu = 0.22, \sigma_{k} = 1, \sigma_{\varepsilon} = 1.3, \sigma_{\zeta} = 1.2, C_{\varepsilon 1} = 1.4 (1 + 0.012 / \zeta), C_{\varepsilon 2} = 1.9, C_1 = 1.4, C_2' = 0.65, C_T = 6, C_L = 0.36 and C_{\eta} = 85.

My wiki