Zeta-f model

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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.

1 Turbulent viscosity
2 Turbulent kinetic energy
3 Turbulent kinetic energy dissipation rate
4 Normalized velocity scale
5 Elliptic relaxation function
6 Production of the turbulent kinetic energy
7 Modulus of the mean rate-of-strain tensor
8 Turbulence time scale
9 Turbulence length scale
10 Model coefficients
11 References

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.

This article is a stub, a short article which needs to be improved. You can help by expanding it.

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-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. For the incompressible Newtonian fluid the final set of equations constituting the <math>\zeta-f</math> model is given below.
+== Turbulent viscosity <math>\nu_t</math> ==
+<math>\nu_t = C_\mu \, \zeta \, k \, T</math>
+== 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>
+== 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>
+== Normalized velocity scale <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>
+== 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>
+== 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>
+== Modulus of the mean rate-of-strain tensor <math>S</math> ==
+<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math>
+== 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}{\varepsilon} \right)^{1/2} \right]</math>
+== 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>
+== Model 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>.
+== References ==
+*<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.
+[[Category:Turbulence models]]
+{{stub}}