Zeta-f model
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- | + | 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: | |
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+ | The turbulent viscosity | ||
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+ | <math>\nu_t = C_\mu \, \zeta \, k \, T</math> | ||
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+ | The turbulent kinetic energy <math>k</math> | ||
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+ | <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> | ||
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+ | The turbulent kinetic energy dissipation rate <math>\varepsilon</math> | ||
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+ | <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> | ||
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+ | The normalized fluctuating velocity normal to the streamlines <math>\zeta</math> | ||
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+ | <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> | ||
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+ | The elliptic relaxation function <math>f</math> | ||
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+ | <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> | ||
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+ | The turbulence time scale <math>T</math> | ||
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+ | <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> | ||
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+ | The turbulence length scale <math>L</math> | ||
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+ | <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> | ||
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+ | The coefficients | ||
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+ | <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 model reads:
The turbulent viscosity
The turbulent kinetic energy
The turbulent kinetic energy dissipation rate
The normalized fluctuating velocity normal to the streamlines
The elliptic relaxation function
The turbulence time scale
The turbulence length scale
The coefficients
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