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Zeta-f model

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(The turbulent kinetic energy <math>k</math>)
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The ''zeta-f'' model is a robust modification of the elliptic relaxation model. The set of equations, for the incompressible Newtonian fluid, constituting the <math>\zeta-f</math> model is given below.
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The ''zeta-f'' model is a robust modification of the elliptic relaxation model. For the incompressible Newtonian fluid the final set of equations constituting the <math>\zeta-f</math> model is given below.
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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>
<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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== 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>
<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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== Normalized velocity scale <math>\zeta</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>
<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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== Elliptic relaxation function <math>f</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>
<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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== Production of the turbulent kinetic energy <math>P_k</math> ==
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== The production of the turbulent kinetic energy <math>P_k</math> ==
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:<math>
:<math>
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P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i}   
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P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i}   
</math>
</math>
<br>
<br>
:<math> P_k = \nu_t S^2 </math>
:<math> P_k = \nu_t S^2 </math>
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== Modulus of the mean rate-of-strain tensor <math>S</math> ==
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== The modulus of the mean rate-of-strain tensor <math>S</math> ==
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<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math>
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<br>
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:<math>
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S \equiv \sqrt{2S_{ij} S_{ij}}  
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</math>
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== Turbulence time scale <math>T</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}{\varepsilon} \right)^{1/2} \right]</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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== Turbulence length scale <math>L</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}, \,
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,
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   \left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math>
   \left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math>
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== Model coefficients ==
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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>.
<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>.
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== References ==
== References ==
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*<b>Popovac, M., Hanjalic, K.</b> Compound Wall Treatment for RANS Computation of Complex Turbulent Flows and Heat Transfer, Flow, Turbulence and Combustion, DOI 10.1007/s10494-006-9067-x, 2007.
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*<b>Popovac, M., Hanjalic, K.</b> Compound Wall Treatment for RANS Computation of Complex Turbulent Flows and Heat Transfer, Flow Turbulence and Combustion, 78, 177-202, 2007.
*<b>Hanjalic, K., Popovac, M., Hadziabdic, M.</b> A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD, Int. J. Heat Fluid Flow, 25, 1047–1051, 2004.
*<b>Hanjalic, K., Popovac, M., Hadziabdic, M.</b> A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD, Int. J. Heat Fluid Flow, 25, 1047–1051, 2004.
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[[Category:Turbulence models]]
[[Category:Turbulence models]]
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Latest revision as of 10:06, 17 December 2008

The zeta-f model is a robust modification of the elliptic relaxation model. For the incompressible Newtonian fluid the final set of equations constituting the \zeta-f model is given below.


Contents

Turbulent viscosity \nu_t

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

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]

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]

Normalized velocity scale \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]

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)

Production of the turbulent kinetic energy P_k


P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i}


 P_k = \nu_t S^2

Modulus of the mean rate-of-strain tensor S

S \equiv \sqrt{2S_{ij} S_{ij}}

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}{\varepsilon} \right)^{1/2} \right]

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]

Model 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.

References

  • Popovac, M., Hanjalic, K. Compound Wall Treatment for RANS Computation of Complex Turbulent Flows and Heat Transfer, Flow Turbulence and Combustion, 78, 177-202, 2007.
  • Hanjalic, K., Popovac, M., Hadziabdic, M. A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD, Int. J. Heat Fluid Flow, 25, 1047–1051, 2004.


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