https://cfd-online.com/W/index.php?title=Special:Contributions/Mirzapopovac&feed=atom&limit=50&target=Mirzapopovac&year=&month=CFD-Wiki - User contributions [en]2024-03-29T07:34:33ZFrom CFD-WikiMediaWiki 1.16.5https://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-03-16T18:02:32Z<p>Mirzapopovac: /* References */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== Turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
== Turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
== Model coefficients ==<br />
<br />
<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>.<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-03-16T18:02:06Z<p>Mirzapopovac: /* References */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
Instead of [[turbulent kinetic energy]] <math>k</math>, the <math>\overline{v^2}-f</math> model uses a velocity scale <math>\overline {v^2}</math> (hence the name <math>\overline{v^2}-f</math> or the ''v2-f model'') for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_j} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<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><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-03-16T18:01:24Z<p>Mirzapopovac: /* References */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
Instead of [[turbulent kinetic energy]] <math>k</math>, the <math>\overline{v^2}-f</math> model uses a velocity scale <math>\overline {v^2}</math> (hence the name <math>\overline{v^2}-f</math> or the ''v2-f model'') for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_j} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<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><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-03-16T18:00:38Z<p>Mirzapopovac: /* References */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== Turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
== Turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
== Model coefficients ==<br />
<br />
<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>.<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/CFD-Wiki:Community_portalCFD-Wiki:Community portal2007-01-26T14:28:59Z<p>Mirzapopovac: /* CFD-Wikians - Who we are */</p>
<hr />
<div>This section is intended for people who work on adding content to the Wiki. So fellow CFD-Wikians, this is your page, private hideout, coffee room, coordination center, after-hours bar or whatever you want to use it for. If you still haven't contributed to the Wiki [[CFD-Wiki:Contribute something today|please do so today]]! We need your help and everyone is welcome to join our team of Wiki authors. <br />
<br />
==What's in the works==<br />
<br />
You who do significant additions to the Wiki, please add some information about your work, plans and progress here so that others can see what you are working on and perhaps help, monitor, come with suggestions and most importantly, be inspired by.<br />
<br />
* In May 2006 the focus area of CFD-Wiki is [[Turbulence modeling|turbulence modeling]] - read [[Focus area May 2006: Turbulence modeling|here]]. --[[User:Jola|Jola]] 12:11, 6 May 2006 (MDT)<br />
<br />
== What needs to be done ==<br />
<br />
''Anything that you want!'' Be bold and just pick something that you feel that you can improve! If you need some help with good ideas on things to work on here are a few suggestions:<br />
<br />
* We have many turbulence models listed in the [[turbulence modeling]] section which still lack any description. Feel free to pick a model that you are familiar with and write a description of it. --[[User:Jola|Jola]] 01:50, 13 September 2005 (MDT)<br />
<br />
* The [[FAQ's | FAQ]] section is still very thin. If you are familiar with one of the larger CFD codes please consider adding a few questions and answers to the FAQ. --[[User:Jola|Jola]] 08:28, 13 September 2005 (MDT)<br />
<br />
* If you are an experienced CFD engineer and an expert in a special application area you are very welcome to start a [[Best practise guidelines|best practise guideline]] for your speciality. --[[User:Jola|Jola]] 10:44, 18 September 2005 (MDT)<br />
<br />
* ''... add your suggestions on what should be done here''<br />
<br />
* As commercial CFD codes become more and more capable, CFD design agencies are more and more pressed to reconsider the cost vs. benefit of developing and maintaing in house codes. I have had many conversations on this topic with other CFD managers and practitioners - there is a lot to think about: We should have a discussion forum on this topic.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==Other resources of interest==<br />
<br />
Here are a few links to pages that are of special interest for us CFD-Wikians:<br />
<br />
*[http://www.cfd-online.com/Forum/wiki.cgi Wiki Discussion Forum]<br />
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* [[User:mirzapopovac]] - Mirza Popovac<br />
* [[User:bajjal]] - Bajjal Raghavendra<br />
* [[User:anurag]] - Anurag Sharma<br />
* [[User:Suneesh]] - Suneesh S.S.<br />
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* [[User:Tsaad]] - [http://jedi.knows.it Tony Saad]<br />
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* [[User:Anand]] - www.iitm.ac.in<br />
* [[User:prokol]] - www.my-area.net</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-26T10:24:28Z<p>Mirzapopovac: /* The <math>\zeta - f</math> equations */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
Instead of [[turbulent kinetic energy]] <math>k</math>, the <math>\overline{v^2}-f</math> model uses a velocity scale <math>\overline {v^2}</math> (hence the name <math>\overline{v^2}-f</math> or the ''v2-f model'') for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_j} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<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><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-26T10:23:19Z<p>Mirzapopovac: /* The <math>\overline{\upsilon^2} - f</math> equations */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
Instead of [[turbulent kinetic energy]] <math>k</math>, the <math>\overline{v^2}-f</math> model uses a velocity scale <math>\overline {v^2}</math> (hence the name <math>\overline{v^2}-f</math> or the ''v2-f model'') for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_j} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} P_k + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-26T10:18:36Z<p>Mirzapopovac: /* Turbulence time scale <math>T</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== Turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
== Turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
== Model coefficients ==<br />
<br />
<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>.<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-25T15:11:19Z<p>Mirzapopovac: </p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== Turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
== Turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
== Model coefficients ==<br />
<br />
<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>.<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-22T12:47:11Z<p>Mirzapopovac: /* The <math>\zeta - f</math> equations */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
Instead of [[turbulent kinetic energy]] <math>k</math>, the <math>\overline{v^2}-f</math> model uses a velocity scale <math>\overline {v^2}</math> (hence the name <math>\overline{v^2}-f</math> or the ''v2-f model'') for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} P_k + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-22T12:44:36Z<p>Mirzapopovac: /* Introduction */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
Instead of [[turbulent kinetic energy]] <math>k</math>, the <math>\overline{v^2}-f</math> model uses a velocity scale <math>\overline {v^2}</math> (hence the name <math>\overline{v^2}-f</math> or the ''v2-f model'') for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} P_k + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-22T12:43:38Z<p>Mirzapopovac: /* Introduction */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
Instead of [[turbulent kinetic energy]], <math>k</math>, the <math>\overline{v^2}-f</math> model uses a velocity scale <math>\overline {v^2}</math> (hence the name <math>\overline{v^2}-f</math> or the ''v2-f model'') for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} P_k + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Turbulence_modelingTurbulence modeling2007-01-22T12:38:43Z<p>Mirzapopovac: </p>
<hr />
<div># [[Turbulence]] <br />
# [[Zero equation models]]<br />
##[[Cebeci-Smith model]]<br />
##[[Baldwin-Lomax model]]<br />
# [[Half equation models]]<br />
## [[Johnson-King model]]<br />
# [[One equation models]]<br />
## [[Prandtl's one-equation model]]<br />
## [[Baldwin-Barth model]]<br />
## [[Spalart-Allmaras model]]<br />
# [[Two equation models]]<br />
## [[k-epsilon models]]<br />
### [[Standard k-epsilon model]]<br />
### [[Realisable k-epsilon model]]<br />
### [[RNG k-epsilon model]]<br />
### [[Near-wall treatment for k-epsilon models]]<br />
## [[k-omega models]]<br />
### [[Wilcox's k-omega model]]<br />
### [[Wilcox's modified k-omega model]]<br />
### [[SST k-omega model]]<br />
### [[Near-wall treatment for k-omega models]]<br />
## [[Two equation turbulence model constraints and limiters]]<br />
### [[Kato-Launder modification]]<br />
### [[Durbin's realizability constraint]]<br />
### [[Yap correction]]<br />
# [[v2-f models]]<br />
## <math>\overline{\upsilon^2}-f</math> model<br />
## <math>\zeta-f</math> model<br />
# [[Reynolds stress model (RSM) ]]<br />
# [[Large eddy simulation (LES) ]]<br />
## [[Smagorinsky-Lilly model]]<br />
## [[Dynamic subgrid-scale model]]<br />
## [[RNG-LES model]]<br />
## [[Wall-adapting local eddy-viscosity (WALE) model]]<br />
## [[Kinetic energy subgrid-scale model]]<br />
## [[Near-wall treatment for LES models]]<br />
# [[Detached eddy simulation (DES) ]]<br />
# [[Direct numerical simulation (DNS) ]]<br />
# [[Turbulence near-wall modeling]]<br />
# [[Turbulence free-stream boundary conditions]]<br />
## [[Turbulence intensity]]<br />
## [[Turbulent length scale]]<br />
<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:31:29Z<p>Mirzapopovac: /* The coefficients */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== Turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
== Turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
== Model coefficients ==<br />
<br />
<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>.<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:31:05Z<p>Mirzapopovac: /* The turbulence length scale <math>L</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== Turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
== Turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:30:41Z<p>Mirzapopovac: /* The turbulence time scale <math>T</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== Turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:30:02Z<p>Mirzapopovac: /* Production of the turbulent kinetic energy <math>P_k</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial U_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:29:14Z<p>Mirzapopovac: /* The modulus of the mean rate-of-strain tensor <math>S</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== Modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br />
<math>S \equiv \sqrt{2S_{ij} S_{ij}}</math><br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:28:24Z<p>Mirzapopovac: /* The production of the turbulent kinetic energy <math>P_k</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== Production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
== The modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:28:04Z<p>Mirzapopovac: /* The elliptic relaxation function <math>f</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== Elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
== The production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
== The modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:27:41Z<p>Mirzapopovac: /* The normalized fluctuating velocity normal to the streamlines <math>\zeta</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== Normalized velocity scale <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
== The elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
<br />
== The production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
== The modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:26:47Z<p>Mirzapopovac: /* The turbulent kinetic energy dissipation rate <math>\varepsilon</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== Turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
== The normalized fluctuating velocity normal to the streamlines <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
<br />
== The elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
<br />
== The production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
== The modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:26:28Z<p>Mirzapopovac: /* The turbulent kinetic energy <math>k</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== Turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
== The turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
<br />
== The normalized fluctuating velocity normal to the streamlines <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
<br />
== The elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
<br />
== The production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
== The modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:26:10Z<p>Mirzapopovac: /* The turbulent viscosity <math>\nu_t</math> */</p>
<hr />
<div>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.<br />
<br />
<br />
== Turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
== The turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
<br />
== The normalized fluctuating velocity normal to the streamlines <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
<br />
== The elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
<br />
== The production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
== The modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:21:30Z<p>Mirzapopovac: </p>
<hr />
<div>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.<br />
<br />
<br />
== The turbulent viscosity <math>\nu_t</math> ==<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
<br />
== The turbulent kinetic energy <math>k</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulent kinetic energy dissipation rate <math>\varepsilon</math> ==<br />
<br />
<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><br />
<br />
<br />
== The normalized fluctuating velocity normal to the streamlines <math>\zeta</math> ==<br />
<br />
<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><br />
<br />
<br />
== The elliptic relaxation function <math>f</math> ==<br />
<br />
<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><br />
<br />
<br />
== The production of the turbulent kinetic energy <math>P_k</math> ==<br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
== The modulus of the mean rate-of-strain tensor <math>S</math> ==<br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
== The turbulence time scale <math>T</math> ==<br />
<br />
<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><br />
<br />
<br />
== The turbulence length scale <math>L</math> ==<br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
== The coefficients ==<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:15:04Z<p>Mirzapopovac: </p>
<hr />
<div>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 reads:<br />
<br />
<br />
The turbulent viscosity<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
<br />
The turbulent kinetic energy <math>k</math><br />
<br />
<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><br />
<br />
<br />
The turbulent kinetic energy dissipation rate <math>\varepsilon</math><br />
<br />
<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><br />
<br />
<br />
The normalized fluctuating velocity normal to the streamlines <math>\zeta</math><br />
<br />
<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><br />
<br />
<br />
The elliptic relaxation function <math>f</math><br />
<br />
<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><br />
<br />
<br />
The production of the turbulent kinetic energy <math>P_k</math><br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
The modulus of the mean rate-of-strain tensor <math>S</math><br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
The turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
<br />
The turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
The coefficients<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T12:13:22Z<p>Mirzapopovac: /* References */</p>
<hr />
<div>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:<br />
<br />
<br />
The turbulent viscosity<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
<br />
The turbulent kinetic energy <math>k</math><br />
<br />
<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><br />
<br />
<br />
The turbulent kinetic energy dissipation rate <math>\varepsilon</math><br />
<br />
<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><br />
<br />
<br />
The normalized fluctuating velocity normal to the streamlines <math>\zeta</math><br />
<br />
<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><br />
<br />
<br />
The elliptic relaxation function <math>f</math><br />
<br />
<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><br />
<br />
<br />
The production of the turbulent kinetic energy <math>P_k</math><br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
The modulus of the mean rate-of-strain tensor <math>S</math><br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
The turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
<br />
The turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
The coefficients<br />
<br />
<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>.<br />
<br />
<br />
== References ==<br />
<br />
*<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.<br />
<br />
*<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.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T11:13:03Z<p>Mirzapopovac: </p>
<hr />
<div>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:<br />
<br />
<br />
The turbulent viscosity<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
<br />
The turbulent kinetic energy <math>k</math><br />
<br />
<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><br />
<br />
<br />
The turbulent kinetic energy dissipation rate <math>\varepsilon</math><br />
<br />
<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><br />
<br />
<br />
The normalized fluctuating velocity normal to the streamlines <math>\zeta</math><br />
<br />
<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><br />
<br />
<br />
The elliptic relaxation function <math>f</math><br />
<br />
<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><br />
<br />
<br />
The production of the turbulent kinetic energy <math>P_k</math><br />
<br />
:<math><br />
P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \nu_t S^2 </math><br />
<br />
<br />
The modulus of the mean rate-of-strain tensor <math>S</math><br />
<br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
<br />
The turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
<br />
The turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
The coefficients<br />
<br />
<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>.<br />
<br />
<br />
==References==<br />
M.P</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T11:06:44Z<p>Mirzapopovac: </p>
<hr />
<div>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:<br />
<br />
<br />
The turbulent viscosity<br />
<br />
<math>\nu_t = C_\mu \, \zeta \, k \, T</math><br />
<br />
<br />
The turbulent kinetic energy <math>k</math><br />
<br />
<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><br />
<br />
<br />
The turbulent kinetic energy dissipation rate <math>\varepsilon</math><br />
<br />
<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><br />
<br />
<br />
The normalized fluctuating velocity normal to the streamlines <math>\zeta</math><br />
<br />
<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><br />
<br />
<br />
The elliptic relaxation function <math>f</math><br />
<br />
<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><br />
<br />
<br />
The turbulence time scale <math>T</math><br />
<br />
<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><br />
<br />
<br />
The turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
<br />
The coefficients<br />
<br />
<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>.</div>Mirzapopovachttps://cfd-online.com/Wiki/Zeta-f_modelZeta-f model2007-01-22T10:38:22Z<p>Mirzapopovac: </p>
<hr />
<div>This is a robust modification of the elliptic relaxation model ([[v2-f]]).</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-22T10:32:44Z<p>Mirzapopovac: /* The <math>\overline{\upsilon^2} - f</math> equations */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} P_k + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-22T10:29:56Z<p>Mirzapopovac: /* The <math>\zeta - f</math> equations */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \varepsilon \frac{\overline{\upsilon^2}}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} P_k + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<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><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-22T10:27:38Z<p>Mirzapopovac: /* The <math>\overline{\upsilon^2} - f</math> equations */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \varepsilon \frac{\overline{\upsilon^2}}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-22T10:24:45Z<p>Mirzapopovac: /* Introduction */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \varepsilon \frac{\overline{\upsilon^2}}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-22T10:21:14Z<p>Mirzapopovac: /* Notes */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal fluctuating velocity <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name the <math>\zeta-f</math> model).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \varepsilon \frac{\overline{\upsilon^2}}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-20T17:44:55Z<p>Mirzapopovac: /* The <math>\upsilon^2 - f</math> equations */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal fluctuating velocity <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name the <math>\zeta-f</math> model).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\overline{\upsilon^2} - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math><br />
<br />
<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \varepsilon \frac{\overline{\upsilon^2}}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\upsilon^2-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-20T17:33:44Z<p>Mirzapopovac: /* Introduction */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal fluctuating velocity <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name the <math>\zeta-f</math> model).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\upsilon^2 - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\upsilon^2} = C_\mu \, \upsilon^2 \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\upsilon^2</math><br />
<br />
<math>\frac{\partial \upsilon^2}{\partial t} + U_j \frac{\partial \upsilon^2}{\partial x_j} = k f - \varepsilon \frac{\upsilon^2}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\upsilon^2}} \right) \frac{\partial \upsilon^2}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\upsilon^2}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\upsilon^2} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\upsilon^2-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-20T17:08:59Z<p>Mirzapopovac: /* Introduction */</p>
<hr />
<div>==Introduction==<br />
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.<br />
<br />
The modification of the original model of \shortciteN{Durbin1991a}<br />
is done by introducing as a new variable .<br />
<br />
In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.<br />
Instead of using the wall-normal fluctuating velocity <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name the <math>\zeta-f</math> model).<br />
This turbulence variable can be regarded as the ratio of the two time<br />
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).<br />
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.<br />
<br />
==The <math>\upsilon^2 - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\upsilon^2} = C_\mu \, \upsilon^2 \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\upsilon^2</math><br />
<br />
<math>\frac{\partial \upsilon^2}{\partial t} + U_j \frac{\partial \upsilon^2}{\partial x_j} = k f - \varepsilon \frac{\upsilon^2}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\upsilon^2}} \right) \frac{\partial \upsilon^2}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\upsilon^2}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\upsilon^2} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\upsilon^2-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-20T16:44:38Z<p>Mirzapopovac: /* References */</p>
<hr />
<div>==Introduction==<br />
The <math>v^2-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>v^2-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall.<br />
<br />
==The <math>\upsilon^2 - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\upsilon^2} = C_\mu \, \upsilon^2 \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\upsilon^2</math><br />
<br />
<math>\frac{\partial \upsilon^2}{\partial t} + U_j \frac{\partial \upsilon^2}{\partial x_j} = k f - \varepsilon \frac{\upsilon^2}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\upsilon^2}} \right) \frac{\partial \upsilon^2}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\upsilon^2}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\upsilon^2} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\upsilon^2-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
*<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.<br />
<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovachttps://cfd-online.com/Wiki/V2-f_modelsV2-f models2007-01-20T16:40:52Z<p>Mirzapopovac: /* Limitations */</p>
<hr />
<div>==Introduction==<br />
The <math>v^2-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.<br />
The <math>v^2-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall.<br />
<br />
==The <math>\upsilon^2 - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^{\upsilon^2} = C_\mu \, \upsilon^2 \, T</math><br />
<br />
and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\upsilon^2</math><br />
<br />
<math>\frac{\partial \upsilon^2}{\partial t} + U_j \frac{\partial \upsilon^2}{\partial x_j} = k f - \varepsilon \frac{\upsilon^2}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\upsilon^2}} \right) \frac{\partial \upsilon^2}{\partial x_k} \right]<br />
</math><br />
<br />
and the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\upsilon^2}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}</math><br />
<br />
where the turbulence length scale <math>L</math><br />
<br />
<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math><br />
<br />
and the turbulence time scale <math>T</math><br />
<br />
<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</math><br />
<br />
are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\upsilon^2} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.<br />
<br />
<br />
==The <math>\zeta - f</math> equations==<br />
The turbulent viscosity is defined as<br />
<br />
<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math><br />
<br />
and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads<br />
<br />
<math>\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math><br />
<br />
and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math><br />
<br />
<math>L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</math><br />
<br />
where the turbulence time scale <math>T</math><br />
<br />
<math>T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</math><br />
<br />
and the turbulence length scale <math>L</math><br />
<br />
<math>L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \,<br />
\frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta}<br />
\left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]</math><br />
<br />
are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.<br />
<br />
The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</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>.<br />
<br />
<br />
==Notes==<br />
This model can not be used to solve Eulerian multiphase problems.<br />
<br />
Mathematically and physically the <math>\upsilon^2-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.<br />
<br />
== References ==<br />
<br />
*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.<br />
<br />
{{stub}}<br />
[[Category:Turbulence models]]</div>Mirzapopovac