https://www.cfd-online.com/W/index.php?title=Special:Contributions/R.absi&feed=atom&limit=50&target=R.absi&year=&month=
CFD-Wiki - User contributions [en]
2024-03-28T22:04:59Z
From CFD-Wiki
MediaWiki 1.16.5
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-22T12:47:40Z
<p>R.absi: /* The mean velocity profile */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
[http://www.cfd-online.com/Wiki/Standard_k-epsilon_model <math>k</math>-<math>\epsilon</math> model]<br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
[http://www.cfd-online.com/Wiki/Prandtl%27s_one-equation_model One-equation model]<br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled <math>k</math>-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation.<br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
For local equilibrium, we are able to find the mean velocity profile <math>u</math> from the turbulent kinetic energy <math>k</math> (equation 4) and the mixing length <math>l_m</math> (equation 5), by: <br />
<table width="70%"><tr><td><br />
<math> <br />
{{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
<br />
Figure (1) shows that the velocity profile obtained from equations (8), (4) and (5) (solid line) is more accurate than the logarithmic velocity profile (dash-dotted line). <br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
'''Figure 1''', Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data ([[#References|[Sukhodolov ''et al.'' (1998)]]]). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; (values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov ''et al.'' (1998)]]]); Figure from [[#References|[Absi (2006)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=''Journal of Hydraulic, Coastal and Environmental Engineering, (Doboku Gakkai Ronbunshuu B),'' Japan Society of Civil Engineers, Vol. '''62''', No. 4, pp.437-446}}<br />
<br />
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}<br />
<br />
* {{reference-paper|author=Sukhodolov, A., Thiele, M. and Bungartz, H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=''Water Resour. Res.'', Vol. '''34''', pp. 1317-1334}}<br />
<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-22T11:54:45Z
<p>R.absi: /* References */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
[http://www.cfd-online.com/Wiki/Standard_k-epsilon_model <math>k</math>-<math>\epsilon</math> model]<br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
[http://www.cfd-online.com/Wiki/Prandtl%27s_one-equation_model One-equation model]<br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled <math>k</math>-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation.<br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
In local equilibrium region, we are able to find the mean velocity <math>u</math> profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by: <br />
<br />
<table width="70%"><tr><td><br />
<math> <br />
{{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
With equations (4) and (5), we obtain [[#References|[Absi (2006)]]]:<br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
Fig. Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov et al. (1998)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=''Journal of Hydraulic, Coastal and Environmental Engineering, (Doboku Gakkai Ronbunshuu B),'' Japan Society of Civil Engineers, Vol. '''62''', No. 4, pp.437-446}}<br />
<br />
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}<br />
<br />
* {{reference-paper|author=Sukhodolov, A., Thiele, M. and Bungartz, H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=''Water Resour. Res.'', Vol. '''34''', pp. 1317-1334}}<br />
<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-21T15:19:31Z
<p>R.absi: /* Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
[http://www.cfd-online.com/Wiki/Standard_k-epsilon_model <math>k</math>-<math>\epsilon</math> model]<br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
[http://www.cfd-online.com/Wiki/Prandtl%27s_one-equation_model One-equation model]<br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled <math>k</math>-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation.<br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
In local equilibrium region, we are able to find the mean velocity <math>u</math> profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by: <br />
<br />
<table width="70%"><tr><td><br />
<math> <br />
{{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
With equations (4) and (5), we obtain [[#References|[Absi (2006)]]]:<br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
Fig. Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov et al. (1998)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=''Journal of Hydraulic, Coastal and Environmental Engineering, (Doboku Gakkai Ronbunshuu B),'' Japan Society of Civil Engineers, Vol. 62, No. 4, pp.437-446}}<br />
<br />
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}<br />
<br />
* {{reference-paper|author=Sukhodolov A., Thiele M. and Bungartz H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=''Water Resour. Res.'', 34, pp. 1317-1334}}<br />
<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/Standard_k-epsilon_model
Standard k-epsilon model
2007-06-21T15:16:53Z
<p>R.absi: /* Transport equations for standard <math>k</math>-<math>\epsilon</math> model */</p>
<hr />
<div>{{Turbulence modeling}}<br />
<br />
== Transport equations for standard k-epsilon model ==<br />
<br />
For turbulent kinetic energy <math> k </math> <br><br />
:<math> \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k </math><br />
<br />
<br><br />
For dissipation <math> \epsilon </math><br />
<br><br />
<br />
:<math> <br />
\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon}<br />
</math><br />
<br />
== Modeling turbulent viscosity ==<br />
Turbulent viscosity is modelled as: <br><br />
:<math><br />
\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}<br />
</math><br />
<br />
== Production of k ==<br />
<br />
:<math><br />
P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \mu_t S^2 </math><br />
<br />
Where <math> S </math> is the modulus of the mean rate-of-strain tensor, defined as : <br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
== Effect of buoyancy ==<br />
<br />
:<math><br />
P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i} <br />
</math><br />
<br />
<br />
<br><br />
where Pr<sub>t</sub> is the turbulent [[Prandtl number]] for energy and g<sub>i</sub> is the component of the gravitational vector in the ith direction. For the standard and realizable - models, the default value of Pr<sub>t</sub> is 0.85.<br />
<br />
The coefficient of thermal expansion, <math> \beta </math> , is defined as <br><br />
:<math> <br />
\beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p <br />
</math><br />
<br />
== Model constants ==<br />
<br />
:<math><br />
C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3 <br />
</math><br />
<br />
<br />
[[Category:Turbulence models]]</div>
R.absi
https://www.cfd-online.com/Wiki/Standard_k-epsilon_model
Standard k-epsilon model
2007-06-21T15:16:19Z
<p>R.absi: /* Transport equations for standard k-epsilon model */</p>
<hr />
<div>{{Turbulence modeling}}<br />
<br />
== Transport equations for standard <math>k</math>-<math>\epsilon</math> model ==<br />
<br />
For turbulent kinetic energy <math> k </math> <br><br />
:<math> \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k </math><br />
<br />
<br><br />
For dissipation <math> \epsilon </math><br />
<br><br />
<br />
:<math> <br />
\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon}<br />
</math><br />
<br />
== Modeling turbulent viscosity ==<br />
Turbulent viscosity is modelled as: <br><br />
:<math><br />
\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}<br />
</math><br />
<br />
== Production of k ==<br />
<br />
:<math><br />
P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \mu_t S^2 </math><br />
<br />
Where <math> S </math> is the modulus of the mean rate-of-strain tensor, defined as : <br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
== Effect of buoyancy ==<br />
<br />
:<math><br />
P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i} <br />
</math><br />
<br />
<br />
<br><br />
where Pr<sub>t</sub> is the turbulent [[Prandtl number]] for energy and g<sub>i</sub> is the component of the gravitational vector in the ith direction. For the standard and realizable - models, the default value of Pr<sub>t</sub> is 0.85.<br />
<br />
The coefficient of thermal expansion, <math> \beta </math> , is defined as <br><br />
:<math> <br />
\beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p <br />
</math><br />
<br />
== Model constants ==<br />
<br />
:<math><br />
C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3 <br />
</math><br />
<br />
<br />
[[Category:Turbulence models]]</div>
R.absi
https://www.cfd-online.com/Wiki/Standard_k-epsilon_model
Standard k-epsilon model
2007-06-21T15:15:36Z
<p>R.absi: /* Transport equations for standard k-epsilon model */</p>
<hr />
<div>{{Turbulence modeling}}<br />
<br />
== Transport equations for standard k-epsilon model ==<br />
<br />
For turbulent kinetic energy <math> k </math> <br><br />
:<math> \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k </math><br />
<br />
<br><br />
For dissipation <math> \epsilon </math><br />
<br><br />
<br />
:<math> <br />
\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon}<br />
</math><br />
<br />
== Modeling turbulent viscosity ==<br />
Turbulent viscosity is modelled as: <br><br />
:<math><br />
\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}<br />
</math><br />
<br />
== Production of k ==<br />
<br />
:<math><br />
P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i} <br />
</math><br />
<br><br />
:<math> P_k = \mu_t S^2 </math><br />
<br />
Where <math> S </math> is the modulus of the mean rate-of-strain tensor, defined as : <br><br />
:<math><br />
S \equiv \sqrt{2S_{ij} S_{ij}} <br />
</math><br />
<br />
== Effect of buoyancy ==<br />
<br />
:<math><br />
P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i} <br />
</math><br />
<br />
<br />
<br><br />
where Pr<sub>t</sub> is the turbulent [[Prandtl number]] for energy and g<sub>i</sub> is the component of the gravitational vector in the ith direction. For the standard and realizable - models, the default value of Pr<sub>t</sub> is 0.85.<br />
<br />
The coefficient of thermal expansion, <math> \beta </math> , is defined as <br><br />
:<math> <br />
\beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p <br />
</math><br />
<br />
== Model constants ==<br />
<br />
:<math><br />
C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3 <br />
</math><br />
<br />
<br />
[[Category:Turbulence models]]</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-21T15:12:21Z
<p>R.absi: /* One-equation eddy viscosity model */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
[http://www.cfd-online.com/Wiki/Prandtl%27s_one-equation_model One-equation model]<br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled <math>k</math>-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation.<br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
In local equilibrium region, we are able to find the mean velocity <math>u</math> profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by: <br />
<br />
<table width="70%"><tr><td><br />
<math> <br />
{{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
With equations (4) and (5), we obtain [[#References|[Absi (2006)]]]:<br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
Fig. Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov et al. (1998)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=''Journal of Hydraulic, Coastal and Environmental Engineering, (Doboku Gakkai Ronbunshuu B),'' Japan Society of Civil Engineers, Vol. 62, No. 4, pp.437-446}}<br />
<br />
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}<br />
<br />
* {{reference-paper|author=Sukhodolov A., Thiele M. and Bungartz H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=''Water Resour. Res.'', 34, pp. 1317-1334}}<br />
<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-21T15:09:40Z
<p>R.absi: /* Algebraic model for the turbulent kinetic energy */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled <math>k</math>-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation.<br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
In local equilibrium region, we are able to find the mean velocity <math>u</math> profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by: <br />
<br />
<table width="70%"><tr><td><br />
<math> <br />
{{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
With equations (4) and (5), we obtain [[#References|[Absi (2006)]]]:<br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
Fig. Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov et al. (1998)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=''Journal of Hydraulic, Coastal and Environmental Engineering, (Doboku Gakkai Ronbunshuu B),'' Japan Society of Civil Engineers, Vol. 62, No. 4, pp.437-446}}<br />
<br />
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}<br />
<br />
* {{reference-paper|author=Sukhodolov A., Thiele M. and Bungartz H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=''Water Resour. Res.'', 34, pp. 1317-1334}}<br />
<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-21T13:50:50Z
<p>R.absi: /* References */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled k-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation. <br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
In local equilibrium region, we are able to find the mean velocity <math>u</math> profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by: <br />
<br />
<table width="70%"><tr><td><br />
<math> <br />
{{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
With equations (4) and (5), we obtain [[#References|[Absi (2006)]]]:<br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
Fig. Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov et al. (1998)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=''Journal of Hydraulic, Coastal and Environmental Engineering, (Doboku Gakkai Ronbunshuu B),'' Japan Society of Civil Engineers, Vol. 62, No. 4, pp.437-446}}<br />
<br />
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}<br />
<br />
* {{reference-paper|author=Sukhodolov A., Thiele M. and Bungartz H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=''Water Resour. Res.'', 34, pp. 1317-1334}}<br />
<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-21T12:21:19Z
<p>R.absi: /* The mean velocity profile */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled k-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation. <br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
In local equilibrium region, we are able to find the mean velocity <math>u</math> profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by: <br />
<br />
<table width="70%"><tr><td><br />
<math> <br />
{{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
With equations (4) and (5), we obtain [[#References|[Absi (2006)]]]:<br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
Fig. Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov et al. (1998)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}<br />
<br />
* {{reference-paper|author=Sukhodolov A., Thiele M. and Bungartz H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=Water Resour. Res., 34, pp. 1317-1334}}<br />
<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-21T12:18:11Z
<p>R.absi: /* References */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled k-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation. <br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
In local equilibrium region, we are able to find the mean velocity profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by: <br />
<br />
<table width="70%"><tr><td><br />
<math> <br />
{{d U} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
With equations (4) and (5), we obtain:<br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
Fig. Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>U_f = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>U_f = 3.97 cm/s</math> ; values of <math>y_0 , h, U_f</math> are from [[#References|[Sukhodolov et al. (1998)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}<br />
<br />
* {{reference-paper|author=Sukhodolov A., Thiele M. and Bungartz H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=Water Resour. Res., 34, pp. 1317-1334}}<br />
<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-21T12:12:51Z
<p>R.absi: /* Algebraic eddy viscosity model */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled k-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation. <br />
<br />
===Algebraic model for the mixing length=== <br />
<br />
For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]: <br />
<br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness. <br />
For a smooth wall (<math>y_0 = 0</math>): <br />
<table width="70%"><tr><td><br />
<math> <br />
l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<br />
==The mean velocity profile==<br />
<br />
In local equilibrium region, we are able to find the mean velocity profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by: <br />
<br />
<table width="70%"><tr><td><br />
<math> <br />
{{d U} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} <br />
</math></td><td width="5%">(8)</td></tr></table> <br />
With equations (4) and (5), we obtain:<br />
<br />
[[Image:fig7a.jpg]]<br />
[[Image:fig7b.jpg]]<br />
<br />
Fig. Vertical distribution of mean flow velocity. <br />
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; <br />
Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>U_f = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>U_f = 3.97 cm/s</math> ; values of <math>y_0 , h, U_f</math> are from [[#References|[Sukhodolov et al. (1998)]]].<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/File:Fig7b.jpg
File:Fig7b.jpg
2007-06-21T12:05:18Z
<p>R.absi: </p>
<hr />
<div></div>
R.absi
https://www.cfd-online.com/Wiki/File:Fig7a.jpg
File:Fig7a.jpg
2007-06-21T12:00:34Z
<p>R.absi: </p>
<hr />
<div></div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T20:31:52Z
<p>R.absi: /* Two-equation eddy viscosity model */</p>
<hr />
<div>==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \epsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
for a smooth wall (<math>y_0 = 0</math>):<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:30:11Z
<p>R.absi: /* Algebraic model for the turbulent kinetic Energy */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.<br />
<br />
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
for a smooth wall (<math>y_0 = 0</math>):<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:29:29Z
<p>R.absi: /* the algebraic eddy viscosity model is therefore */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic Energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter. <br />
<br />
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
for a smooth wall (<math>y_0 = 0</math>):<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:28:55Z
<p>R.absi: /* the algebraic eddy viscosity model is therefore */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic Energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter. <br />
<br />
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
for a smooth wall (<math>y_0 = 0</math>):<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:26:36Z
<p>R.absi: /* References */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic Energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter. <br />
<br />
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:20:37Z
<p>R.absi: /* Algebraic eddy viscosity model [Absi (2006)] */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic Energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter. <br />
<br />
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=A roughness and time dependent mixing length equation|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:19:48Z
<p>R.absi: /* Algebraic eddy viscosity model */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model [[#References|[Absi (2006)]]]==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic Energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter. <br />
<br />
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=A roughness and time dependent mixing length equation|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:18:47Z
<p>R.absi: /* Algebraic model for the mixing length, based on (4) */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic Energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter. <br />
<br />
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=A roughness and time dependent mixing length equation|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:10:57Z
<p>R.absi: /* Algebraic eddy viscosity model */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
===Algebraic model for the turbulent kinetic Energy===<br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter. <br />
<br />
===Algebraic model for the mixing length, based on (4)=== <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
===the algebraic eddy viscosity model is therefore=== <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=A roughness and time dependent mixing length equation|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T15:07:42Z
<p>R.absi: /* Algebraic eddy viscosity model */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
Algebraic model for the turbulent kinetic Energy: <br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity <br />
<br />
Algebraic model for the mixing length, based on (4) : <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
the algebraic eddy viscosity model is therefore: <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=A roughness and time dependent mixing length equation|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T14:57:40Z
<p>R.absi: /* Kinematic Eddy Viscosity */</p>
<hr />
<div>==Two-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where: <br />
<math> C_{\mu} = 0.09 </math><br />
<br />
==One-equation eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
==Algebraic eddy viscosity model==<br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) <br />
</math></td><td width="5%">(3)</td></tr></table><br />
<math>l_m</math> is the mixing length. <br />
<br />
where: <br />
<table width="70%"><tr><td><br />
<math><br />
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(4)</td></tr></table><br />
<math>u_\tau </math> is the shear velocity <br />
<br />
and: <br />
<table width="70%"><tr><td><br />
<math><br />
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness<br />
<br />
therefore: <br />
<table width="70%"><tr><td><br />
<math> <br />
\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)<br />
u_\tau e^{\frac{-y}{A}} <br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=A roughness and time dependent mixing length equation|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T14:37:19Z
<p>R.absi: /* Kinematic Eddy Viscosity */</p>
<hr />
<div>==Kinematic Eddy Viscosity==<br />
Two-equation model: <br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math><br />
where: <math> C_{\mu} = 0.09 </math><br />
<br />
One-equation model: <br />
<math> <br />
\nu _t = l k^{{1 \over 2}} = {C_{\mu}}^{1/4} l_m k^{{1 \over 2}} <br />
</math><br />
<br />
Algebraic model: <br />
<math><br />
k^{{1 \over 2}} = {1 \over {C_{\mu}}^{1/4}}<br />
</math><br />
<br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Absi, R. |year=2006|title=A roughness and time dependent mixing length equation|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T14:32:13Z
<p>R.absi: /* Kinematic Eddy Viscosity */</p>
<hr />
<div>==Kinematic Eddy Viscosity==<br />
Two-equation model: <br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math><br />
where: <math> C_{\mu} = 0.09 </math><br />
<br />
One-equation model: <br />
<math> <br />
\nu _t = l k^{{1 \over 2}} = {C_{\mu}}^{1/4} l_m k^{{1 \over 2}} <br />
</math><br />
<br />
Algebraic model: <br />
<math><br />
k^{{1 \over 2}} = {1 \over {C_{\mu}}^{1/4}}<br />
</math></div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T14:31:35Z
<p>R.absi: /* Kinematic Eddy Viscosity */</p>
<hr />
<div>==Kinematic Eddy Viscosity==<br />
Two-equation model: <br />
<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math><br />
where: <math> C_{\mu} = 0.09 </math><br />
<br />
One-equation model: <br />
:<math> <br />
\nu _t = l k^{{1 \over 2}} = {C_{\mu}}^{1/4} l_m k^{{1 \over 2}} <br />
</math><br />
<br />
Algebraic model: <br />
k^{{1 \over 2}} = {1 \over {C_{\mu}}^{1/4}}</div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T14:27:59Z
<p>R.absi: /* Kinematic Eddy Viscosity */</p>
<hr />
<div>==Kinematic Eddy Viscosity==<br />
Two-equation model : <br />
:<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math><br />
where : :<math> C_{\mu} = 0.09 </math><br />
<br />
One-equation model : <br />
:<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></div>
R.absi
https://www.cfd-online.com/Wiki/A_roughness-dependent_model
A roughness-dependent model
2007-06-19T14:26:52Z
<p>R.absi: New page: ==Kinematic Eddy Viscosity== Two-equation model : :<math> \nu _t = C_{\mu} {{k^2 } \over \varepsilon } </math> One-equation model : :<math> \nu _t = k^{{1 \over 2}} l </math></p>
<hr />
<div>==Kinematic Eddy Viscosity==<br />
Two-equation model : <br />
:<math> <br />
\nu _t = C_{\mu} {{k^2 } \over \varepsilon }<br />
</math><br />
<br />
One-equation model : <br />
:<math> <br />
\nu _t = k^{{1 \over 2}} l <br />
</math></div>
R.absi
https://www.cfd-online.com/Wiki/Turbulence_modeling
Turbulence modeling
2007-06-19T14:23:24Z
<p>R.absi: /* Content of turbulence modeling section */</p>
<hr />
<div>Turbulence modeling is a key issue in most CFD simulations. Virtually all engineering applications are turbulent and hence require a turbulence model.<br />
<br />
==Classes of turbulence models==<br />
<br />
*Algebraic models<br />
*Eddy viscosity transport models, one and two equation models<br />
*Non-linear eddy viscosity models and algebraic stress models<br />
*Reynolds stress transport models<br />
*Detached eddy simulations and other hybrid models<br />
*Large eddy simulations<br />
*Direct numerical simulations<br />
<br />
==Content of turbulence modeling section==<br />
<br />
# [[Turbulence]] <br />
# [[Algebraic turbulence models|Algebraic models]]<br />
##[[Cebeci-Smith model]]<br />
##[[Baldwin-Lomax model]]<br />
## [[Johnson-King model]]<br />
## [[A roughness-dependent model]]<br />
# [[One equation turbulence models|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 />
### [[Realisability and Schwarz' inequality]]<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>
R.absi