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<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
Main page: [[Two equation models#Near-wall treatments| Two equation near-wall treatments]]<br />
<br />
For <math>k</math> the boundary conditions imposed at the solid boundary are:<br />
:<math><br />
\begin{matrix}<br />
\frac{\partial k}{\partial n} = 0 & & \frac{\partial \omega}{\partial n} = 0<br />
\end{matrix}<br />
</math><br />
where <math>n</math> is the normal to the boundary.<br />
<br />
Moreover the centroid values in cells adjacent to solid wall are specified as<br />
:<math><br />
\begin{matrix}<br />
k_p = \frac{u^2_\tau}{\sqrt{C_\mu}y_p},<br />
&&<br />
\omega_p = \frac{u_\tau}{\sqrt{C_\mu}\kappa y_p} = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p}.<br />
\end{matrix}<br />
</math><br />
In the alternative approach <math>k</math> production terms is modified.<br />
==Automatic wall treatments==<br />
The purpose of automatic wall treatments is to make results insensitive with respect to wall mesh refinement. Many blending approaches have been proposed. The one by Menter takes advantage of the fact that the solution to <math>\omega</math> equations is known for both viscous and log layer<br />
:<math><br />
\begin{matrix}<br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2} & \omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
\end{matrix}<br />
</math><br />
where <math>y</math> is the cell centroid distance from the wall. Using this a blending can take the following form:<br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
Note that for low <math>y</math> values the <math>1/y^2</math> will dominate and therefore viscous value of <math>\omega</math> will be reproduced. Conversely, for larger values of <math>y</math>, <math>1/y</math> will be dominant and logarithmic value will be recovered.<br />
<br />
Subsequently Menter proposes also blending for friction velocity. Friction velocity for viscous and logarithmic region are:<br />
:<math><br />
\begin{matrix}<br />
u^\text{vis}_\tau = \frac{U}{y^{+}} & & u_\tau^\text{log} = \frac{U}{\log E y^{+}}<br />
\end{matrix}<br />
</math><br />
And the blending suggested:<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-03T20:55:26Z<p>AlmostSurelyRob: /* Near-wall treatments */</p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|320px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows| introduction to wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) <br />
* High Reynolds number treatment (HRN) <br />
<br />
LRN integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
<br />
HRN also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
There are two<ref>{{Citation<br />
| last = Bredberg | first = J. <br />
| title = On the Wall Boundary Condition for Turbulence Models<br />
| publisher = Chalmers University of Technology Goteborg, Sweden<br />
| url = http://www.tfd.chalmers.se/~lada/postscript_files/jonas_report_WF.pdf<br />
| year = 2000<br />
}}</ref> possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard wall functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in the [[Law of the wall|logarithmic law of the wall]]. Now using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> resulting from asymptotic analysis<ref>{{Citation<br />
| last = Wilcox | first = D.C.<br />
| title = Turbulence Modeling for CFD<br />
| publisher = DCW Industries, Inc.<br />
| year = 1993<br />
}}</ref> of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding wall functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure<ref>{{Citation<br />
| last =Launder | first = B.E. | last2=Spalding |first2= D.B.<br />
| issn = 00457825<br />
| journal = Computer Methods in Applied Mechanics and Engineering<br />
| number = 2<br />
| pages = 269-289<br />
| publisher = Elsevier<br />
| title = The numerical computation of turbulent flows<br />
| url = http://linkinghub.elsevier.com/retrieve/pii/0045782574900292<br />
| volume = 3<br />
| year = 1974<br />
}}</ref> is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
==See Also==<br />
* [[Boussinesq eddy viscosity assumption]]<br />
== References ==<br />
{{reflist}}<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-03T20:50:33Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|320px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) <br />
* High Reynolds number treatment (HRN) <br />
<br />
LRN integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
<br />
HRN also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
There are two<ref>{{Citation<br />
| last = Bredberg | first = J. <br />
| title = On the Wall Boundary Condition for Turbulence Models<br />
| publisher = Chalmers University of Technology Goteborg, Sweden<br />
| url = http://www.tfd.chalmers.se/~lada/postscript_files/jonas_report_WF.pdf<br />
| year = 2000<br />
}}</ref> possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in the [[Law of the wall|logarithmic law of the wall]]. Now using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> resulting from asymptotic analysis<ref>{{Citation<br />
| last = Wilcox | first = D.C.<br />
| title = Turbulence Modeling for CFD<br />
| publisher = DCW Industries, Inc.<br />
| year = 1993<br />
}}</ref> of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure<ref>{{Citation<br />
| last =Launder | first = B.E. | last2=Spalding |first2= D.B.<br />
| issn = 00457825<br />
| journal = Computer Methods in Applied Mechanics and Engineering<br />
| number = 2<br />
| pages = 269-289<br />
| publisher = Elsevier<br />
| title = The numerical computation of turbulent flows<br />
| url = http://linkinghub.elsevier.com/retrieve/pii/0045782574900292<br />
| volume = 3<br />
| year = 1974<br />
}}</ref> is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
==See Also==<br />
* [[Boussinesq eddy viscosity assumption]]<br />
== References ==<br />
{{reflist}}<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-03T20:43:11Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|320px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) <br />
* High Reynolds number treatment (HRN) <br />
<br />
LRN integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
<br />
HRN also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
There are two<ref>{{Citation<br />
| last = Bredberg | first = J. <br />
| title = On the Wall Boundary Condition for Turbulence Models<br />
| publisher = Chalmers University of Technology Goteborg, Sweden<br />
| url = http://www.tfd.chalmers.se/~lada/postscript_files/jonas_report_WF.pdf<br />
| year = 2000<br />
}}</ref> possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in the [[Law of the wall|logarithmic law of the wall]]. Now using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> resulting from asymptotic analysis<ref>{{Citation<br />
| last = Wilcox | first = D.C.<br />
| title = Turbulence Modeling for CFD<br />
| publisher = DCW Industries, Inc.<br />
| year = 1993<br />
}}</ref> of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure<ref>{{Citation<br />
| last =Launder | first = B.E. | last2=Spalding |first2= D.B.<br />
| issn = 00457825<br />
| journal = Computer Methods in Applied Mechanics and Engineering<br />
| number = 2<br />
| pages = 269-289<br />
| publisher = Elsevier<br />
| title = The numerical computation of turbulent flows<br />
| url = http://linkinghub.elsevier.com/retrieve/pii/0045782574900292<br />
| volume = 3<br />
| year = 1974<br />
}}</ref> is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
== References ==<br />
{{reflist}}<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-02T12:02:13Z<p>AlmostSurelyRob: /* Near-wall treatments */</p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
There are two<ref>{{Citation<br />
| last = Bredberg | first = J. <br />
| title = On the Wall Boundary Condition for Turbulence Models<br />
| publisher = Chalmers University of Technology Goteborg, Sweden<br />
| url = http://www.tfd.chalmers.se/~lada/postscript_files/jonas_report_WF.pdf<br />
| year = 2000<br />
}}</ref> possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in the [[Law of the wall|logarithmic law of the wall]]. Now using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> resulting from asymptotic analysis<ref>{{Citation<br />
| last = Wilcox | first = D.C.<br />
| title = Turbulence Modeling for CFD<br />
| publisher = DCW Industries, Inc.<br />
| year = 1993<br />
}}</ref> of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure<ref>{{Citation<br />
| last =Launder | first = B.E. | last2=Spalding |first2= D.B.<br />
| issn = 00457825<br />
| journal = Computer Methods in Applied Mechanics and Engineering<br />
| number = 2<br />
| pages = 269-289<br />
| publisher = Elsevier<br />
| title = The numerical computation of turbulent flows<br />
| url = http://linkinghub.elsevier.com/retrieve/pii/0045782574900292<br />
| volume = 3<br />
| year = 1974<br />
}}</ref> is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
== References ==<br />
{{reflist}}<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-11-02T11:59:33Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
Main page: [[Two equation models#Near-wall treatments| Two equation near-wall treatments]]<br />
<br />
For <math>k</math> the boundary conditions imposed at the solid boundary are:<br />
:<math><br />
\begin{matrix}<br />
\frac{\partial k}{\partial n} = 0 & & \frac{\partial \omega}{\partial n} = 0<br />
\end{matrix}<br />
</math><br />
where <math>n</math> is the normal to the boundary.<br />
<br />
Moreover the centroid values in cells adjacent to solid wall are specified as<br />
:<math><br />
\begin{matrix}<br />
k_p = \frac{u^2_\tau}{\sqrt{C_\mu}y_p},<br />
&&<br />
\omega_p = \frac{u_\tau}{\sqrt{C_\mu}\kappa y_p} = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p}.<br />
\end{matrix}<br />
</math><br />
In the alternative approach <math>k</math> production terms is modified.<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Boussinesq_eddy_viscosity_assumptionBoussinesq eddy viscosity assumption2011-11-02T08:44:43Z<p>AlmostSurelyRob: </p>
<hr />
<div>In 1877 Boussinesq postulated<ref>{{Citation<br />
| last =Boussinesq | first = J.<br />
| journal = Mémoires présentés par divers savants à l'Académie des Sciences <br />
| number = 1<br />
| pages = 1-680<br />
| title = Essai sur la théorie des eaux courantes<br />
| volume = 23<br />
| address = Paris<br />
| year = 19877<br />
}}</ref><ref><br />
{{Citation<br />
| last =Schmitt | first = F.G.<br />
| journal = Comptes Rendus Mécanique<br />
| number = (9-10)<br />
| pages = 617-627<br />
| title = About Boussinesq’s turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity<br />
| volume = 335<br />
| url = http://hal.archives-ouvertes.fr/hal-00264386/fr/<br />
| year = 2007<br />
}}<br />
</ref> that the momentum transfer caused by turbulent eddies can be modeled with an eddy viscosity. This is in analogy with how the momentum transfer caused by the molecular motion in a gas can be described by a molecular viscosity. The Boussinesq assumption states that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the trace-less mean strain rate tensor, <math>S_{ij}^*</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]]. The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho \overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} - \frac{2}{3} \frac{\partial U_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}</math><br />
<br />
Note that for incompressible flow:<br />
<br />
:<math>\frac{\partial U_k}{\partial x_k} = 0</math><br />
<br />
The Boussinesq eddy viscosity assumption is also often called the Boussinesq hypothesis or the Boussinesq approximation.<br />
== See also ==<br />
*[[Two equation models]]<br />
*[[Introduction to turbulence/Reynolds averaged equations#Turbulence closure problem and eddy viscosity|Turbulence closure problem and eddy viscosity]]<br />
==References==<br />
{{reflist}}<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-01T22:28:33Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
There are two<ref>{{Citation<br />
| last = Bredberg | first = J. <br />
| title = On the Wall Boundary Condition for Turbulence Models<br />
| publisher = Chalmers University of Technology Goteborg, Sweden<br />
| url = http://www.tfd.chalmers.se/~lada/postscript_files/jonas_report_WF.pdf<br />
| year = 2000<br />
}}</ref> possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> resulting from asymptotic analysis<ref>{{Citation<br />
| last = Wilcox | first = D.C.<br />
| title = Turbulence Modeling for CFD<br />
| publisher = DCW Industries, Inc.<br />
| year = 1993<br />
}}</ref> of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure<ref>{{Citation<br />
| last =Launder | first = B.E. | last2=Spalding |first2= D.B.<br />
| issn = 00457825<br />
| journal = Computer Methods in Applied Mechanics and Engineering<br />
| number = 2<br />
| pages = 269-289<br />
| publisher = Elsevier<br />
| title = The numerical computation of turbulent flows<br />
| url = http://linkinghub.elsevier.com/retrieve/pii/0045782574900292<br />
| volume = 3<br />
| year = 1974<br />
}}</ref> is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
<br />
== References ==<br />
{{reflist}}<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-11-01T22:26:49Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
Main page: [[Two equation models#Near-wall treatments| Two equation near-wall treatments]]<br />
<br />
For <math>k</math> the boundary conditions imposed are<br />
:<math><br />
\frac{\partial k}{\partial y} = 0<br />
</math><br />
Moreover the centroid values in cells adjacent to solid wall are specified as<br />
:<math><br />
k_p = \frac{u^2_\tau}{\sqrt{C_\mu}y_p}<br />
</math><br />
:<math><br />
\omega_p = \frac{u_\tau}{\sqrt{C_\mu}\kappa y_p} = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
In the alternative approach <math>k</math> production terms is modified.<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-01T22:25:06Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
<br />
There are two<ref>{{Citation<br />
| last = Bredberg | first = J. <br />
| title = On the Wall Boundary Condition for Turbulence Models<br />
| publisher = Chalmers University of Technology Goteborg, Sweden<br />
| url = http://www.tfd.chalmers.se/~lada/postscript_files/jonas_report_WF.pdf<br />
| year = 2000<br />
}}</ref> possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> resulting from asymptotic analysis<ref>{{Citation<br />
| last = Wilcox | first = D.C.<br />
| title = Turbulence Modeling for CFD<br />
| publisher = DCW Industries, Inc.<br />
| year = 1993<br />
}}</ref> of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure<ref>{{Citation<br />
| last =Launder | first = B.E. | last2=Spalding |first2= D.B.<br />
| issn = 00457825<br />
| journal = Computer Methods in Applied Mechanics and Engineering<br />
| number = 2<br />
| pages = 269-289<br />
| publisher = Elsevier<br />
| title = The numerical computation of turbulent flows<br />
| url = http://linkinghub.elsevier.com/retrieve/pii/0045782574900292<br />
| volume = 3<br />
| year = 1974<br />
}}</ref> is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
<br />
== References ==<br />
{{reflist}}<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-01T22:22:40Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
<br />
There are two<ref>{{Citation<br />
| last = Bredberg | first = J. <br />
| title = On the Wall Boundary Condition for Turbulence Models<br />
| publisher = Chalmers University of Technology Goteborg, Sweden<br />
| url = http://www.tfd.chalmers.se/~lada/postscript_files/jonas_report_WF.pdf<br />
| year = 2000<br />
}}</ref> possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> asymptotic<ref>{{Citation<br />
| last = Wilcox | first = D.C.<br />
| title = Turbulence Modeling for CFD<br />
| publisher = DCW Industries, Inc.<br />
| year = 1993<br />
}}</ref> analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure<ref>{{Citation<br />
| last =Launder | first = B.E. | last2=Spalding |first2= D.B.<br />
| issn = 00457825<br />
| journal = Computer Methods in Applied Mechanics and Engineering<br />
| number = 2<br />
| pages = 269-289<br />
| publisher = Elsevier<br />
| title = The numerical computation of turbulent flows<br />
| url = http://linkinghub.elsevier.com/retrieve/pii/0045782574900292<br />
| volume = 3<br />
| year = 1974<br />
}}</ref> is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
<br />
== References ==<br />
{{reflist}}<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-11-01T22:06:05Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
Main page: [[Two equation models#Near-wall treatments| Two equation near-wall treatments]]<br />
<br />
For <math>k</math> the boundary conditions imposed are<br />
:<math><br />
\frac{\partial k}{\partial y} = 0<br />
</math><br />
Moreover the centroid values in cells adjacent to solid wall are specified as<br />
:<math><br />
k_p = \frac{u^2_\tau}{\sqrt{C_\mu}y_p}<br />
</math><br />
:<math><br />
\omega_p = \frac{u_\tau}{\sqrt{C_\mu}\kappa y_p} = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
In the alternative approach <math>k</math> production terms is modified.<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-01T18:36:35Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure<ref>{{Citation<br />
| last =Launder | first = B.E. | last2=Spalding |first2= D.B.<br />
| issn = 00457825<br />
| journal = Computer Methods in Applied Mechanics and Engineering<br />
| number = 2<br />
| pages = 269-289<br />
| publisher = Elsevier<br />
| title = The numerical computation of turbulent flows<br />
| url = http://linkinghub.elsevier.com/retrieve/pii/0045782574900292<br />
| volume = 3<br />
| year = 1974<br />
}}</ref> is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
<br />
== References ==<br />
{{reflist}}<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-01T18:22:31Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k}.<br />
</math></td><td width="5%">(4)</td></tr></table><br />
A three step procedure is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-11-01T18:21:19Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
===Standard Wall Functions===<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
and for e.g. <math>k</math>-<math>\omega</math> model:<br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
===Launder-Spalding Wall Functions===<br />
This approaches utilises the relation<br />
<table width="70%"><tr><td><br />
:<math><br />
u_\tau = C_\mu^{1/4} \sqrt{k},<br />
</math></td><td width="5%">(4)</td></tr></table><br />
then the three step procedure is adopted:<br />
* Calculate wall viscosity.<br />
* Modify the production term in <math>k</math> equation in the wall adjacent cell.<br />
* Specify the value of the second variable in the wall adjacent cell.<br />
<br />
After substituing (4) to log law relation we obtain:<br />
:<math><br />
U = \frac{C_\mu^{1/4}\sqrt{k}}{\kappa}\ln E y^*,<br />
</math><br />
where<br />
:<math><br />
y^* = \frac{C_\mu^{1/4}\sqrt{k} y}{ \nu},<br />
</math><br />
<br />
The production term (just in the first cell!) can be approximated as follows:<br />
:<math><br />
{P}_p = -\overline{u_1^'u_2^'}\frac{\partial U_1}{\partial x_2} = <br />
\underbrace{(\nu_t + \nu)\frac{\partial U_1}{\partial x_2}}_{\text{Boussinesq assumption}}<br />
\overbrace{\frac{C_\mu^{1/4} \sqrt{k_p}}{\kappa y_p}}^{\text{Logarithmic velocity}}<br />
</math><br />
<br />
Lastly, we specify the remaining turbulence quantities in the same manner as in the previous approach.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T15:59:48Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}\sim 1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T15:59:14Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in <math>y^{+}~1</math>. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T15:22:12Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in viscous sublayer. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T15:14:49Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, remembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p},<br />
</math><br />
where <math>U_p</math> and <math>y_p</math> are the velocity and the position of the cell centroid in the first cell adjacent to the wall.<br />
<br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity in the log-law region:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T15:01:34Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, remembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p},<br />
</math><br />
where <math>U_p</math> and <math>y_p</math> are the velocity and the position of the cell centroid in the first cell adjacent to the wall.<br />
<br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T15:01:20Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p},<br />
</math><br />
where <math>U_p</math> and <math>y_p</math> are the velocity and the position of the cell centroid in the first cell adjacent to the wall.<br />
<br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T15:00:55Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p},<br />
</math><br />
where <math>U_p</math> and <math>y_p</math> are the velocity and the position of the cell centroid in the first cell adjacent to the wall.<br />
<br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T14:59:09Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.<br />
==Standard wall functions==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T14:57:22Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
==Standard wall functions==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T13:32:34Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T13:31:04Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T13:24:43Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(4right). HRN uses log law in order to estimate gradient in the cell.]]<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in viscous sublayer. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T13:24:04Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
[[File:HRNvsLRN.png|left|thumb|350px|HRN (left) vs LRN(4right). HRN uses log law in order to estimate gradient in the cell.]]<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in viscous sublayer. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an ''ad hoc'' solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T13:19:47Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. See [[Introduction to turbulence/Wall bounded turbulent flows]] for more detail. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in viscous sublayer. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
[[File:HRNvsLRN.png|right|thumb|390px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an \textit{ad hoc} solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T13:16:09Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in viscous sublayer. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
[[File:HRNvsLRN.png|right|thumb|390px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an \textit{ad hoc} solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T13:15:52Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in viscous sublayer. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
[[File:HRNvsLRN.png|right|thumb|370px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an \textit{ad hoc} solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T13:15:20Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in viscous sublayer. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
[[File:HRNvsLRN.png|right|thumb|350px|HRN (left) vs LRN(right). HRN uses log law in order to estimate gradient in the cell.]]<br />
<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an \textit{ad hoc} solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/File:HRNvsLRN.pngFile:HRNvsLRN.png2011-10-31T13:11:02Z<p>AlmostSurelyRob: Near Wall Treatment - High Reynolds Number approach vs Low Reynolds Number approach. HRN requires different gradient estimation.</p>
<hr />
<div>Near Wall Treatment - High Reynolds Number approach vs Low Reynolds Number approach. HRN requires different gradient estimation.</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Two_equation_turbulence_modelsTwo equation turbulence models2011-10-31T13:08:30Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
Two equation turbulence models are one of the most common type of turbulence models. Models like the [[K-epsilon models|k-epsilon model]] and the [[K-omega models|k-omega model]] have become industry standard models and are commonly used for most types of engineering problems. Two equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.<br />
<br />
By definition, two equation models include two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. <br />
<br />
Most often one of the transported variables is the [[Turbulent kinetic energy|turbulent kinetic energy]], <math>k</math>. The second transported variable varies depending on what type of two-equation model it is. Common choices are the turbulent [[Dissipation|dissipation]], <math>\epsilon</math>, or the [[Specific dissipation|specific dissipation]], <math>\omega</math>. The second variable can be thought of as the variable that determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, <math>k</math>, determines the energy in the turbulence.<br />
<br />
==Boussinesq eddy viscosity assumption==<br />
<br />
The basis for all two equation models is the [[Boussinesq eddy viscosity assumption]], which postulates that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way:<br />
<br />
:<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]] which is normally computed from the two transported variables. The last term is included for modelling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed:<br />
<br />
:<math>k=\frac{\overline{u'_i u'_i}}{2}</math><br />
<br />
The same equation can be written more explicitly as:<br />
<br />
:<math> -\rho\overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3}\rho k \delta_{ij}</math><br />
<br />
The Boussinesq assumption is both the strength and the weakness of two equation models. This assumption is a huge simplification which allows one to think of the effect of turbulence on the mean flow in the same way as molecular viscosity affects a laminar flow. The assumption also makes it possible to introduce intuitive scalar turbulence variables like the turbulent energy and dissipation and to relate these variables to even more intuitive variables like [[Turbulence intensity|turbulence intensity]] and [[Turbulence length scale|turbulence length scale]]. <br />
<br />
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing which says that the Reynolds stress tensor must be proportional to the strain rate tensor. It is true in simple flows like straight boundary layers and wakes, but in complex flows, like flows with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption is simply not valid. This give two equation models inherent problems to predict strongly rotating flows and other flows where curvature effects are significant. Two equation models also often have problems to predict strongly decellerated flows like stagnation flows.<br />
<br />
==Near-wall treatments==<br />
The structure of turbulent boundary layer exhibits large, compared with the flow in the core region, gradients of velocity and quantities characterising turbulence. In a collocated grid these gradients will be approximated using discretisation procedures which are not suitable for such high variation since they usually assume linear interpolation of values between cell centres. <br />
<br />
Moreover, the additional quantities appearing in two-equation models require specification of their own boundary conditions that on purely physical grounds cannot be specified ''a priori''.<br />
<br />
This situation gave rise to a plethora of near-wall treatments. Generally speaking two approaches can be distinguished:<br />
* Low Reynolds number treatment (LRN) integrates every equation up to the viscous sublayer and therefore the first computational computational cell must have its centroid in viscous sublayer. This results in very fine meshes close to the wall. Additionally, for some models additional treatment (damping functions) of equations is required to guarantee asymptotic consistency with the turbulent boundary layer behaviour. This often makes the equations stiff and further increases computation time.<br />
* High Reynolds number treatment (HRN) also known as wall functions approach relies on log-law velocity profile and therefore the first computational cell must have its centroid in the log-layer. Use of HRN enhances convergence rate and often numerical stability.<br />
<br />
Interestingly, none of the current approaches can deal with buffer layer i.e. the layer in which both viscous and Reynolds stresses are significant. The first computational cell should be either in viscous sublayer or in log-layer -- not in-between. Automatic wall treatments, available in some codes, are an \textit{ad hoc} solution but the blending techniques employed there are usually arbitrary and though they can achieve the switching between HRN and LRN treatments they cannot be regarded as the correct representation of the buffer layer.<br />
<br />
<br />
[[Category: Turbulence models]]<br />
<br />
{{stub}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T13:04:05Z<p>AlmostSurelyRob: </p>
<hr />
<div>{{Turbulence modeling}}<br />
==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox2006 asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T12:51:10Z<p>AlmostSurelyRob: </p>
<hr />
<div>==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox2006 asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
Menter suggested a mechanism that switches automatically between HRN and LRN treatments.<br />
<br />
''The full description to appear soon. The idea is based on blending:''<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T12:48:04Z<p>AlmostSurelyRob: </p>
<hr />
<div>==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox2006 asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
==Automatic wall treatments==<br />
<br />
:<math><br />
\omega_\text{vis} = \frac{6\nu}{\beta y^2}<br />
</math><br />
<br />
:<math><br />
\omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}<br />
</math><br />
:<math><br />
\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},<br />
</math><br />
<br />
<br />
:<math><br />
u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},<br />
</math><br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T12:29:21Z<p>AlmostSurelyRob: </p>
<hr />
<div>==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox2006 asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T12:25:42Z<p>AlmostSurelyRob: </p>
<hr />
<div>==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox2006 asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
== References ==<br />
<br />
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation}</div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T12:23:31Z<p>AlmostSurelyRob: </p>
<hr />
<div>==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox2006 asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.<br />
<br />
<br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
Reference: <br />
FLUENT 6.2 Documentation, 2006 <br></div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T12:23:04Z<p>AlmostSurelyRob: </p>
<hr />
<div>==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(2)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox2006 asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math>. Many extension of this approach has been proposed.<br />
<br />
<br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
Reference: <br />
FLUENT 6.2 Documentation, 2006 <br></div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Near-wall_treatment_for_k-omega_modelsNear-wall treatment for k-omega models2011-10-31T12:22:20Z<p>AlmostSurelyRob: </p>
<hr />
<div>==Standard wall function approach==<br />
There are two possible ways of implementing wall functions in a finite volume code:<br />
* Additional source term in the momentum equations.<br />
* Modification of turbulent viscosity in cells adjacent to solid walls.<br />
<br />
The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress. <br />
<br />
Using the compact version of log-law<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
where <math>E=9.8</math> is equivalent to additive constants, and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:<br />
:<math><br />
\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},<br />
</math><br />
On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:<br />
:<math><br />
\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.<br />
</math><br />
<br />
Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:<br />
<table width="70%"><tr><td><br />
<math><br />
\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).<br />
</math> </td><td width="5%">(3)</td></tr></table><br />
Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining <math>y^{+}</math> has to be employed. <br />
<br />
After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:<br />
<table width="70%"><tr><td><br />
<math><br />
\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.<br />
</math></td><td width="5%">(3)</td></tr></table><br />
This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is <br />
:<math><br />
y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.<br />
</math><br />
Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:<br />
:<math><br />
k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}<br />
</math><br />
:<math><br />
\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},<br />
</math><br />
which are the values for <math>k</math> and <math>\omega</math> according to Wilcox2006 asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.<br />
<br />
The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math>. Many extension of this approach has been proposed.<br />
<br />
<br />
<br />
==FLUENT==<br />
Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models. <br><br />
<br />
The wall boundary conditions for the k equation in the k- omega models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k- epsilon models. <br><br />
<br />
This means that all boundary conditions for <br><br />
- '''wall-function meshes''' will correspond to the wall function approach, while for the <br><br />
- '''fine meshes''', the appropriate low-Reynolds-number boundary conditions will be applied. <br><br />
<br />
In Fluent, that means:<br />
<br />
If the '''Transitional Flows option is enabled '''in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment <br><br />
''(y+ at the wall-adjacent cell should be on the order of '''y+ = 1'''. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)''<br><br />
<br />
If '''Transitional Flows option is not active''', then the mesh guidelines should be the same as for the wall functions.<br><br />
''(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound '''y+ = 30''' is most desirable.)''<br />
<br />
<br><br />
<br />
Reference: <br />
FLUENT 6.2 Documentation, 2006 <br></div>AlmostSurelyRobhttp://www.cfd-online.com/Wiki/Introduction_to_turbulence/Wall_bounded_turbulent_flowsIntroduction to turbulence/Wall bounded turbulent flows2011-10-29T15:37:44Z<p>AlmostSurelyRob: </p>
<hr />
<div>== Introduction ==<br />
Without the presence of walls or surfaces, turbulence in the absence of density<br />
fluctuations could not exist. This is because it is only at surfaces that vorticity can actually be generated by an on-coming flow is suddenly brought to rest to satisfy the no-slip condition. The vorticity generated at the leading edge can then be diffused, transported and amplified. But it can only be generated at the wall, and then only at the leading edge at that. Once the vorticity has been generated, some flows go on to develop in the absence of walls, like the free shear flows we considered earlier. Other flows remained “attached” to the surface and evolve entirely under the influence of it. These are generally referred to as “wall-bounded flows” or “boundary layer flows”.<br />
The most obvious causes for the effects of the wall on the flow arise from the<br />
wall-boundary conditions. In particular, <br />
<br />
* The '''kinematic''' boundary condition demands that the normal velocity of<br />
the fluid on the surface be equal to the normal velocity of the surface. This<br />
means there can be no-flow through the surface. Since the velocity normal<br />
to the surface cannot just suddenly vanish, the kinematic boundary condition<br />
ensures that the normal velocity components in wall-bounded flows are<br />
usually much less than in free shear flows. Thus the presence of the wall<br />
reduces the entrainment rate. Note that viscosity is not necessary in the<br />
equations to satisfy this condition, and it can be met even by solutions to<br />
to the inviscid Euler’s equations.<br />
<br />
<br />
* The '''no-slip''' boundary condition demands that the velocity component tangential to the wall be the same as the tangential velocity of the wall. If the<br />
wall is at rest relative, then the no-slip condition demands the tangential flow<br />
velocity be identically zero at the surface.<br />
<br />
<br />
Figure 8.1: Flow around a simple airfoil without separation.<br />
<br />
<br />
It is the no-slip condition, of course, that led Ludwig Prandtl1 to the whole<br />
idea of a boundary layer in the first place. Professor Prandt literally saved fluid mechanics from d’Alembert’s paradox: the fact that there seemed to be no drag in an inviscid fluid (not counting form drag). Prior to Prandtl, everyone thought that as the Reynolds number increased, the flow should behave more and more like an inviscid fluid. But when there were surfaces, it clearly didn’t. Instead of behaving like those nice potential flow solutions (like around cylinders, for example), the flow not only produced drag, but often separated and produced wakes and other free shear flows. Clearly something was very wrong, and as a result fluid mechanics didn’t get much respect from engineers in the 19th century.<br />
<br />
<br />
And with good reason: how useful could a bunch of equations be if they couldn’t<br />
find viscous drag, much less predict how much. But Prandtl’s idea of the boundary<br />
layer saved everything.<br />
<br />
Prandtl’s great idea was the recognition that the viscous no-slip condition<br />
could not be met without somehow retaining at least one viscous stress term in<br />
the equations. As we shall see below, this implies that there must be at least two<br />
length scales in the flow, unlike the free shear flows we considered in the previous<br />
chapter for which the mean flow could be characterized by only a single length<br />
scale. The second length scale characterizes changes normal to the wall, and make<br />
it clear precisely which viscous term in the instantaneous equations is important.<br />
<br />
== Review of laminar boundary layers ==<br />
<br />
Let’s work this all out for ourselves by considering what happens if we try to<br />
apply the kinematic and no-slip boundary conditions to obtain solutions of the<br />
Navier-Stokes equations in the infinite Reynolds number limit. Let’s restrict our<br />
attention for the moment to the laminar flow of a uniform stream of speed, <math>U_o</math>,<br />
around a body of characteristic dimension, D, as shown in Figure 8.1. It is easy<br />
to see the problem if we non-dimensionalize our equations using the free stream<br />
boundary condition and body dimension. The result is:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{D \tilde{\tilde{u_{i}}}}{D \tilde{\tilde{t}}} = - \frac{\partial \tilde{\tilde{p}}}{\partial \tilde{\tilde{x_{i}}}} + \frac{1}{Re} \frac{\partial^{2} \tilde{\tilde{u_{i}}}}{\partial \tilde{\tilde{x_{i}}}}<br />
</math><br />
</td><td width="5%">(1)</td></tr></table><br />
<br />
where <math>\tilde{\tilde{u_{i}}} \equiv u_{i}/U_{0}</math>, <math>\tilde{\tilde{x_{i}}} \equiv x_{i} / D</math>, <math>\tilde{\tilde{t}} \equiv U_{0} t / D</math> and <math>\tilde{\tilde{p}} \equiv p / \left( \rho U^{2}_{0} \right) </math>. The kinematic viscosity, <math>\nu</math> has disappeared entirely, and is included in the '''Reynolds number''' defined by:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
Re \equiv \frac{U_{o} D}{\nu}<br />
</math><br />
</td><td width="5%">(2)</td></tr></table><br />
<br />
Now consider what happens as the Reynolds number increases, due to the increase of <math>U_{o}</math> or <math>L</math>, or even a decrease in the viscosity. Obviously the viscous terms become relatively less important. In fact, if the Reynolds number is large enough it is hard to see at first glance why any viscous term should be retained at all. Certainly in the limit as <math>Re \rightarrow \infty</math>, our equations must reduce to Euler’s equations which have no viscous terms at all; i.e., in dimensionless form,<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{D \tilde{u_{i}} }{ D \tilde{t} } = - \frac{\partial \tilde{p} }{\partial \tilde{x_{i}}}<br />
</math><br />
</td><td width="5%">(3)</td></tr></table><br />
<br />
Now if we replace the Navier-Stokes equations by Euler’s equations, this presents<br />
no problem at all in satisfying the kinematic boundary condition on the body’s<br />
surface. We simply solve for the inviscid flow by replacing the boundary of the<br />
body by a streamline. This automatically satisfies the kinematic boundary condition. If the flow can be assumed irrotational, then the problem reduces to a<br />
solution of Laplace’s equation, and powerful potential flow methods can be used.<br />
<br />
<br />
In fact, for potential flow, it is possible to show that the flow is entirely determined by the ''normal'' velocity at the surface. And this is, of course, the source of our problem. There is no role left for the ''viscous'' no-slip boundary condition. And indeed, the potential flow has a tangential velocity along the surface streamline that is not zero. The problem, of course, is the absence of viscous terms in the Euler equations we used. Without viscous stresses acting near the wall to retard the flow, the solution cannot adjust itself to zero velocity at the wall. But how can viscosity enter the equations when our order-of-magnitude analysis says they are negligible at large Reynolds number, and exactly zero in the infinite Reynolds number limit. At the end of the nineteenth century, this was arguably the most serious problem confronting fluid mechanics.<br />
<br />
<br />
Prandtl was the first to realize that there must be at least one viscous term<br />
in the problem to satisfy the no-slip condition. Therefore he postulated that the<br />
strain rate very near the surface would become as large as necessary to compensate for the vanishing effect of viscosity, so that at least one viscous term remained. This very thin region near the wall became known as Prandtl’s boundary layer, and the length scale characterizing the necessary gradient in velocity became known as the boundary layer “thickness”.<br />
<br />
Prandtl’s argument for a laminar boundary layer can be quickly summarized<br />
using the same kind of order-of-magnitude analysis we used in Chapter 7 '''(here necessary make navigation !!!)'''. For the leading viscous term:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\nu \frac{\partial^{2} u}{ \partial y^{2}} \sim \nu \frac{U_{s}}{ \delta^{2}}<br />
</math><br />
</td><td width="5%">(4)</td></tr></table><br />
<br />
where <math>\delta</math> is the new length scale characterizing changes normal to the plate near the wall and we have assumed <math>\Delta U_{s} = U_{s}</math>. In fact, for a boundary layer next to walls driven by an external stream speed <math>U_{\infty}</math>. For the leading convection term:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
U \frac{\partial U}{ \partial x} \sim \frac{ U^{2}_{s}}{L}<br />
</math><br />
</td><td width="5%">(5)</td></tr></table><br />
<br />
The viscous term can survive only if it is the same order of magnitude as the convection term. Hence it follows that we must have:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\nu \frac{U_{s}}{ \delta^{2}} \sim \frac{ U^{2}_{s}}{L} <br />
</math><br />
</td><td width="5%">(6)</td></tr></table><br />
<br />
This in turn requires that new length scale <math>\delta</math> must satisfy:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\delta \sim \left[ \frac{\nu L}{U_{s}} \right]^{1/2}<br />
</math><br />
</td><td width="5%">(7)</td></tr></table><br />
<br />
or <br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{\delta}{L} \sim \left[ \frac{\nu }{U_{s} L} \right]^{1/2} <br />
</math><br />
</td><td width="5%">(8)</td></tr></table><br />
<br />
Thus, for a ''laminar flow'' <math>\delta</math> grows like <math>L^{1/2}</math>. Now if you go back to your fluid mechanics texts and look at the similarity solution for a Blasius boundary layer, you will see this is exactly right if you take <math>L \propto x</math>, which is what we might have guessed anyway.<br />
<br />
It is very important to remember that the momentum equation is a vector equation, and we therefore have to scale all components of this vector equation the same way to preserve its direction. Therefore we must carry out the same kind of estimates for the cross-stream momentum equations as well. For a laminar<br />
boundary layer this can be easily be shown to reduce to:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\left[ \frac{\delta}{L} \right] \left\{ \frac{ \partial \tilde{\tilde{ u}}}{ \partial \tilde{\tilde{t}}} + \tilde{\tilde{u}} \frac{\partial \tilde{\tilde{ u}} }{\tilde{\tilde{ x}}}+ \tilde{\tilde{v}} \frac{\partial \tilde{\tilde{ v}} }{\tilde{\tilde{ y}}} \right\} = - \frac{ \partial \tilde{\tilde{p}}_{\infty}}{ \partial \tilde{\tilde{ y}} } + \frac{1}{Re} \left[ \frac{\delta}{L} \right] \left\{ \frac{ \partial^{2} \tilde{\tilde{v}} }{ \partial \tilde{\tilde{ y}}^{2}} \right\} + \frac{1}{Re} \left\{ \frac{ \partial^{2} \tilde{\tilde{v}} }{ \partial \tilde{\tilde{ x}}^{2}} \right\} <br />
</math><br />
</td><td width="5%">(9)</td></tr></table><br />
<br />
<br />
Note that a only single term survives in the limit as the Reynolds number goes to<br />
infinity, the cross-stream pressure gradient. Hence, for very large Reynolds number, the pressure gradient across the boundary layer equation is very small. Thus the pressure ''in the boundary'' is imposed on it by the flow ''outside'' the boundary layer. And this flow '''outside''' the boundary layer is governed to first order by Euler’s equation.<br />
<br />
<br />
The fact that the pressure is ''imposed'' on the boundary layer provides us an<br />
easy way to calculate such a flow. First calculate the inviscid flow along the<br />
surface using Euler’s equation. Then use the pressure along the surface from this ''inviscid'' solution together with the boundary layer equation to calculate the boundary layer flow. If you wish, you can even use an iterative procedure where you recalculate the outside flow over a streamline which was been displaced from the body by the boundary layer displacement thickness, and then re-calculate the boundary layer, etc. Before modern computers, this was the only way to calculate the flow around an airfoil, for example.<br />
<br />
== The "outer" turbulent boundary layer ==<br />
<br />
The understanding of turbulent boundary layers begins with exactly the same averaged equations we used for the free shear layers of Chapter 7'''(here necessary to add link!!!)'''; namely, <br />
<br />
<math>x</math>-component:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
U \frac{\partial U}{ \partial x} + V \frac{\partial U}{ \partial y} = - \frac{1}{\rho} \frac{\partial P}{ \partial x} - \frac{\partial \left\langle u^{2} \right\rangle}{ \partial x} - \frac{\partial \left\langle uv \right\rangle}{ \partial y} + \nu \frac{ \partial^{2} U }{ \partial x^{2}} + \nu \frac{ \partial^{2} U }{ \partial y^{2}}<br />
</math><br />
</td><td width="5%">(10)</td></tr></table><br />
<br />
<math>y</math>-component:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
U \frac{\partial V}{ \partial x} + V \frac{\partial V}{ \partial y} = - \frac{1}{\rho} \frac{\partial P}{ \partial y} - \frac{\partial \left\langle uv \right\rangle}{ \partial x} - \frac{\partial \left\langle u^{2} \right\rangle}{ \partial y} + \nu \frac{ \partial^{2} V }{ \partial x^{2}} + \nu \frac{ \partial^{2} V }{ \partial y^{2}}<br />
</math><br />
</td><td width="5%">(11)</td></tr></table><br />
<br />
'''two-dimensional mean continuity'''<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{\partial U}{ \partial x} + \frac{\partial V}{ \partial y} = 0<br />
</math><br />
</td><td width="5%">(12)</td></tr></table><br />
<br />
In fact, the order of magnitude analysis of the terms in this equation proceeds exactly the same as for free shear flows. If we take <math>U_{s} = \Delta U_{s} = U_{\infty}</math>, then the ultimate problem again reduces to how to keep a turbulence term. And, as before this requires:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{\delta}{L} \sim \frac{u^{2}}{U_{s} \Delta U_{s}} <br />
</math><br />
</td><td width="5%">(00)</td></tr></table><br />
<br />
<br />
No problem, you say, we expected this. But the problem is that by requiring this be true, we also end up concluding that even the ''leading'' viscous term is also negligible! Recall that the leading viscous term is of order:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\frac{\nu}{U_{s} \delta} \frac{L}{\delta}<br />
</math><br />
</td><td width="5%">(13)</td></tr></table><br />
<br />
compared to the unity.<br />
<br />
In fact to leading order, there are no viscous terms in either component of the<br />
momentum equation. In the limit as <math>U_{\infty} \delta / \nu \rightarrow \infty</math>, they are exactly the same as for the free shear flows we considered earlier; namely,<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
U \frac{\partial U}{ \partial x} + \left\{ V \frac{\partial U}{ \partial y} \right\} = - \frac{1}{\rho} \frac{\partial P}{ \partial x} - \frac{\partial }{ \partial y} \left\langle uv \right\rangle - \left\{ \frac{\partial }{ \partial x} \left\langle u^{2} \right\rangle \right\} <br />
</math><br />
</td><td width="5%">(14)</td></tr></table><br />
<br />
and<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
0 = - \frac{1}{\rho} \frac{\partial P}{ \partial y} - \frac{\partial }{ \partial y} \left\langle v^{2} \right\rangle<br />
</math><br />
</td><td width="5%">(15)</td></tr></table><br />
<br />
<br />
And like the free shear flows these can be integrated from the free stream to a<br />
given value of <math>y</math> to obtain a single equation:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
U \frac{\partial U}{ \partial x} + V \frac{\partial U}{ \partial y} = - \frac{d P_{\infty}}{dx} - \frac{\partial }{ \partial y} \left\langle uv \right\rangle - \left\{ \frac{\partial}{\partial x} \left[ \left\langle u^{2} \right\rangle - \left\langle v^{2} \right\rangle \right] \right\}<br />
</math><br />
</td><td width="5%">(16)</td></tr></table><br />
<br />
The last term in brackets is the gradient of the difference in the normal Reynolds stresses, and is of order <math>u^2 / U^2</math> compared to the others, so is usually just ignored. The bottom line here is that even though we have attempted to carry out an order of magnitude analysis for a ''boundary layer'', we have ended up with exactly the equations for a free shear layer. Only the boundary conditions are different— most notably the kinematic and no-slip conditions at the wall. Obviously, even though we have equations that describe a turbulent boundary layer, we cannot satisfy the no-slip condition without a viscous term. In other words, we are right back where we were ''before'' Prandtl invented the boundary layer for laminar flow! We need a boundary layer within the boundary layer to satisfy the no-slip condition. In the next section we shall in fact show that such an ''inner boundary layer'' exists. And that everything we analysed in this section applies only to the ''outer boundary layer'' — which is NOT to be confused with the ''outer flow'' which is non-turbulent and still governed by Euler’s equation. Note that the presence of <math>P_{\infty}</math> in our outer boundary equations means that (to first order in the turbulence intensity), the pressure is still imposed on the boundary layer by the flow outside it, exactly as for laminar boundary layers (and all free shear flows, for that matter).<br />
<br />
== The “inner” turbulent boundary layer ==<br />
<br />
We know that we cannot satisfy the no-slip condition ''unless'' we can figure out<br />
how to keep a viscous term in the governing equations. And we know there can<br />
be such a term only if the mean velocity near the wall changes rapidly enough so<br />
that it remains, no matter how small the viscosity becomes. In other words, we<br />
need a length scale for changes in the <math>y</math>-direction very near the wall which enables us keep a viscous term in our equations. This new length scale, let’s call it <math>\eta</math>, is going to be much smaller than <math>\delta</math>, the boundary layer thickness. But how much smaller?<br />
<br />
<br />
Obviously we need to go back and look at the full equations again, and re-scale<br />
them for the near wall region. To do this, we need to first decide how the mean<br />
and turbulence velocities scale near the wall scale. We are clearly so close to the wall and the velocity has dropped so much (because of the no-slip condition) that it makes no sense to characterize anything by <math>U_{\infty}</math>. But we don’t have anyway of knowing yet what this scale should be, so let’s just call it <math>u_{w}</math> and define it later. Also, we do know from experiment that the turbulence intensity near the wall is relatively high (30% or more). So there is no point in distinguishing between a turbulence scale and the mean velocity, we can just use <math>u_{w}</math> for both. Finally we will still use <math>L</math> to characterize changes in the <math>x</math>-direction, since these will vary no more rapidly than in the outer boundary layer above this very near wall region we are interested in.<br />
<br />
For the complete <math>x</math>-momentum equation we estimate<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
U \frac{\partial U}{ \partial x} & + & V \frac{ \partial U}{ \partial y} \\<br />
u_{w} \frac{u_{w}}{L} & & \left( u_{w} \frac{\eta}{L} \right) \frac{u_{w}}{\eta} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(00)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
= & - \frac{1}{\rho} \frac{\partial P}{ \partial x} & - & \frac{\partial \left\langle u^{2} \right\rangle}{\partial x} & - & \frac{\partial \left\langle uv \right\rangle}{\partial y} & + & \nu \frac{ \partial^{2} U }{ \partial x^{2}} & + & \nu \frac{ \partial^{2} U }{ \partial y^{2}} \\<br />
& ? & & \frac{u^{2}_{w}}{\eta} & & \frac{u^{2}_{w}}{L} & & \nu \frac{u_{w}}{L^{2}} & & \nu \frac{u_{w}}{\eta^{2}} \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(00)</td></tr></table><br />
<br />
<br />
where we have used the continuity equation to estimate <math>V \sim u_{w} \eta / L</math> near the wall.<br />
<br />
Now as always, we have to decide which terms we have to keep so we know what to compare the others with. But that is easy here, we MUST insist that at least one viscous term survive. Since the largest is of order <math>\nu u_{w} / \eta^{2}</math> , we can divide by it to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
U \frac{\partial U}{ \partial x} & + & V \frac{ \partial U}{ \partial y} \\<br />
\left( \frac{u_{w} \eta }{ \nu } \right) \frac{\eta}{L} & & \left( \frac{u_{w} \eta }{ \nu } \right) \frac{\eta}{L}<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(00)</td></tr></table><br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
\begin{matrix}<br />
= & - \frac{1}{\rho} \frac{\partial P}{ \partial x} & - & \frac{\partial \left\langle u^{2} \right\rangle}{\partial x} & - & \frac{\partial \left\langle uv \right\rangle}{\partial y} & + & \nu \frac{ \partial^{2} U }{ \partial x^{2}} & + & \nu \frac{ \partial^{2} U }{ \partial y^{2}} \\<br />
& ? & & \left( \frac{u_{w} \eta }{ \nu } \right) \frac{\eta}{L} & & \frac{u_{u} \eta}{\nu} & & \frac{\eta^{2}}{L^{2}} & & 1 \\<br />
\end{matrix}<br />
</math><br />
</td><td width="5%">(00)</td></tr></table><br />
<br />
Now we have an interesting problem. We have the power to whether the Reynolds shear stress term survives or not by our choice of <math>\eta</math>; i.e. we can pick <math>u_{w} \eta / \nu \sim 1</math> or <math>u_{w} \eta / \nu \rightarrow 0</math>. Note that we can not choose it so this term blows up, or else our viscous term will not be at least equal to the leading term. The most general choice is to pick <math>\eta \sim \nu / u_{w} </math> so the Reynolds shear stress remains too. (This is called the distinguished limit in asymptotic analysis.) By making this choice we eliminate the necessity of having to go back and find there is another layer in which only the Reynolds stress survives — as we shall see below. Obviously if we choose <math>\eta \sim \nu / u_{w} </math>, then all the other terms vanish, except for the viscous one. In fact, if we apply the same kind of analysis to the <math>y</math>-mean momentum equation, we can show that the pressure in our near wall layer is also imposed from the outside. Moreover, even the streamwise pressure gradient disappears in the limit as <math>u_{w} \eta / \nu \rightarrow \infty</math>. These are relatively easy to show and left as exercises.<br />
<br />
So to first order in <math>\eta / L \sim \nu / \left( u_{w} L \right)</math>, our mean momentum equation for the ''near wall region'' reduces to:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
0 \approx \frac{\partial}{\partial y} \left[ - \left\langle uv \right\rangle + \nu \frac{\partial U}{ \partial y} \right]<br />
</math><br />
</td><td width="5%">(17)</td></tr></table><br />
<br />
In fact, this equation is exact in the limit as <math>u_{w} \delta / \nu \rightarrow \infty</math>, '''but only for the very near wall region!'''<br />
<br />
Equation (17) can be integrated from the wall to location <math>y</math> to obtain:<br />
<br />
<table width="70%"><tr><td><br />
:<math> <br />
0 = - \left\{ \left\langle uv \right\rangle - \left\langle uv \right\rangle |_{y=0} \right\} + \nu \left\{ \frac{\partial U}{ \partial y} - \frac{\partial U}{ \partial y} \bigg|_{y=0} \right\}<br />
</math><br />
</td><td width="5%">(18)</td></tr></table></div>AlmostSurelyRob