Zeta-f model

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(Difference between revisions)

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 $C_{\eta} = 85$ .

@@ 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>.

Zeta-f model

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Revision as of 11:06, 22 January 2007

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