Shear Stress Transport model.

The different advantages of two very popular models have been combined in the Shear Stress Transport (SST) model of Menter \cite{Men94}. As a first step, Menter proposed a blending Baseline Model (BSL) model between the $k-\omega$ and the $k-\varepsilon$. By rewriting the $k-\varepsilon$ equations in terms of the rate of dissipation per unit of kinetic energy $\omega$, then the extra cross gradient term that emerges is multiplied by a blending function, allowing a return to the original $k-\omega$ near wall boundaries. The blending function is also used in the calculation of all the closure coefficients. The equations of the BSL model are:

 \begin{gather*} \frac{\partial k}{\partial t}+\left\langle U_j\right\rangle \frac{\partial k}{\partial x_j} =P_{k}-\beta^* k \omega +\frac{\partial }{\partial x_{j}}\left[ \left( \nu +\dfrac{\nu _{t}}{\sigma _{k}}\right) \frac{\partial k}{\partial x_{j}}\right] \label{eq:kBSL}\\[2mm] \frac{\partial \omega}{\partial t}+\left\langle U_j\right\rangle \frac{\partial \omega}{\partial x_j} = \alpha \frac{\omega}{k}P_k -\beta \omega^2 +\frac{\partial}{\partial x_j}\left[ \left(\nu + \frac{\nu_t}{\sigma_{\omega}}\right)  \frac{\partial \omega}{\partial x_j}\right]+2\frac{(1-F_1)}{\sigma_{\omega 2}}\frac{1}{\omega} \frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}\label{eq:omBSL} \end{gather*}

The model uses a value of any coefficient $c$ computed from:

 \begin{equation*} c = F_1 c_1 + (1-F_1)c_2 \end{equation}

where the subscript 1 represents the original coefficient of the $k-\omega$ model and the subscript 2 represents the coefficients of the transformed $k-\varepsilon$ model. The blending function is defined such that it makes a smooth transition from $k-\omega$ at the wall to $k-\varepsilon$ far from it. In this way the sensitivity of $\omega$ in the free stream is reduced and the problem in the viscous sublayer of the $k-\varepsilon$ does not play a part in the solution. The blending function is given by:

 \begin{gather*} F_1 = \tanh (arg_1^4)\\[2mm]\label{eq:f1_sst} arg_1 = min \left[ max \left( \frac{\sqrt{k}}{0.09 \omega y};\frac{500     \nu}{y^2 \omega } \right); \frac{4 \rho k}{\sigma_{\omega       2}CD_{k \omega}y^2 }\right]\\[2mm] CD_{k \omega} = max \left( 2 \rho \frac{1}{\sigma_{\omega2}\omega } \frac {\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}; 10^{-20}\right) \end{gather*}

here, $y$ represents the distance from the wall, which can be difficult to define in complex geometries. This wall distance dependence is undesirable but not uncommon.

The second step presented by Menter \cite{Men94} is the Shear Stress Transport model (SST), in which a function similar to the blending function $F_1$ of the BSL is used to limit the value of the eddy viscosity, therefore maintaining the shear stress proportional to the turbulent kinetic energy in the wake of the boundary layer.

 \begin{gather*} \nu_t = \frac{a_1 k}{max(a_1 \omega ; S F_2)}\\[2mm] F_2 = \tanh(arg_2^2)\\[2mm] arg_2 = max \left( \frac{2 \sqrt{k}}{0.09 \omega y }; \frac{500\nu}{y^2 \omega}\right) \end{gather*}

where $S = \sqrt{2S_{ij} S_{ij}}$.

The closure coefficients for the SST model are shown in Table \ref{tab:SSTcnt}.

 \begin{table}[h] \centering \begin{tabular}{||c|c|c|c|c|c||} \hline $\alpha_1$&$\beta_1$&$\sigma_{k1}$&$\sigma_{\omega1}$&   $a_1$& $\kappa$\\   \hline $\beta_1/\beta^* -\kappa^2/(\sigma_{\omega_1}   \sqrt{\beta^*})$& 0.075& 1.176& 2& 0.31& 0.41\\ \hline \hline  $\alpha_2$&$\beta_2$&$\sigma_{k2}$&$\sigma_{\omega2}$&$\beta^*$& \\ \hline $\beta_2/\beta^*   -\kappa^2/(\sigma_{\omega_2}\sqrt{\beta^*})$&0.0828 &1.0 &1.1682& 0.09&  \\ \hline \end{tabular} \caption{Coefficients of the SST model} \label{tab:SSTcnt} \end{table}
The boundary condition on $\omega$ proposed by Menter is given by:
 \begin{equation*} \omega_w = 10\frac{6\nu}{\beta_1y^2} \end{equation*}

Results

The following table shows the cases that have results available with the SST model in the CfdTM database:

Test Case Case Results Author Model
Channel Flow TestCase001Res001 Juan Uribe Phi-f, SST, EBM (Elliptic Blending Model)
Channel Flow TestCase001Res002 Juan Uribe k-eps, SST, v2F, SSG, LES
Channel Flow TestCase001Res004 J. Uribe
Flow past a heated circular cylinder TestCase002Res001 Juan Uribe SST
Vertical Heated Pipe TestCase005Res008 Stefano Rolfo k-omega-SST
3D Diffuser TestCase010Res000 Juan Uribe, Flavien Billard
Asymmetric plane diffuser TestCase013Res000 Juan Uribe, Flavien Billard SST, V2f models
Asymmetric plane diffuser TestCase013Res002 J .Uribe
Flow over 2D periodic hills TestCase014Res000 Juan Uribe
  TestCase015Res000 J. Uribe
  TestCase015Res001 E. Moreau
Flow over a 2D hill TestCase019Res001 J. Uribe
  TestCase025Res000 E. Moreau
Number of topics: 13

Performance

The SST model is in reality a combination of two eddy viscosity models plus a limiter on the turbulent viscosity and as such it inherits their disadvantages but not necessarily all their advantages, since it is based on purely empirical blending functions.

Because it uses the Boussinesq approximation, it is unable to predict anisotropy. This limits its performance in flows with strong curvature, important near wall effects and flows with secondary flows.

The great advantage of the SST model is its robustness and ease of use. This comes from the use of $\omega$ rather than $\varepsilon$ which allows a direct integration over the viscus sub layer without the need of damping functions. It is important to note that, although the model can be used in the viscous sublayer, it has not been designed to take into account any of the wall effects. In fact the actual turbulent variables are wrongly predicted in this region.

In a channel flow simulation, the turbulent kinetic energy is under predicted, but the rate of dissipation is overpredicted and since the the turbulent viscosity is defined as $\nu_t=k/\omega$ the errors compensate each other.

The SST model is good in attached boundary layers and some separated flows where the separation point is dictated by the geometry rather than by and adverse pressure gradient. On the second type of flow, many authors have shown that the SST model predicts recirculation better than the traditional $k-\varepsilon$ but these comparisons are always difficult to read since one model solves the viscous sublayer and the other uses a wall function approach. One of the most popular low-Re version of $k-\varepsilon$ is the Launder-Sharma but this model was designed for very low Re numbers with transition in mind.

Non-linear models or elliptic relaxation based models tend to do better than the SST in adverse pressure gradients situations, but the great advantage of the SST is its popularity which has led to a high number of publications and cases where its performance can be evaluated.

Channel Flow

In the channel flow case, the blending functions are negligible and the model acts as the standard $k-\omega$. Here the dissipation (computed as $\varepsilon = C_{\mu} k \omega$) has the wrong behaviour near the wall. This is compensated by an under-prediction of the turbulent kinetic energy which results in a good turbulent viscosity profile that leads to a good velocity prediction.

Even though Wilcox devised a series of damping functions to correct the behaviour of the turbulent variables in the viscous sub-layer, this are never used on the SST model.

For more comparisons of model behaviour on channel flow, check here .

U.png nut.png k.png eps.png
Velocity profiles  Eddy viscosity (-uv/dU/dy for EBM and DNS)  Kinetic energy profiles  Epsilon profile 

Heated cylinder

In this case the SST model gives good predictions of Nusselt number and skin friction inmost of the domain, but in other similar experiments the recirculation length has been reported to be too large. This can be seen in the region of $80 <\theta <130$ where the Nusselt number is over-predicted.

Nu_36000_lowTu_ke_sst_nonames.png      
Nusselt Number distribution.     

Vertical heated pipe

This is a very bad case for the SST model since it is not capable to predict the correct relaminarisation. Here the model predicts the wrong turbulent length scale and therefore stays turbulent. This in turn makes the prediction of velocity profiles and temperature completely wrong.

slide0006_image020.png slide0012_image032.png slide0012_image033.png slide0012_image034.png
Nussel Number versu Buoyancy  Velocity  Temperature  Kinetic energy 

Heated cavity

For the long tall cavity the SST model does a good job in predicting values for the vertical velocity and the temperature.
In this case it is interesting to note that the model acts in $k-\omega$ mode in most of the domain as shown in the figure. Therefore the predictions are very similar to the original $k-\omega$ model.

Error: no images found

Mixed convection in vertical channel.

This is another case where the SST model cannot reproduce the correct turbulent parameters, as in the channel flow. Given that, the velocity profiles are in good agreement with the DNS data due to error cancelling.

Kasagi_K.jpg Kasagi_Phi.jpg Kasagi_Velocity.jpg latexbf44c2e303fa6bda92cd49752e71461a.png
Kasagi_K  Kasagi_Phi  Kasagi_Velocity  latexbf44c2e303fa6bda92cd49752e71461a 

Diffuser flow

This case is particularly challenging for Eddy viscocity models due to the nature of the separation which is not dictated by the geometry but by the effect of the adverse pressure gradient. The correct prediction of the evolution of the shear stress in the inclined wall is critical.

Unfortunately the SST model does a poor job here and the separation is way too early, almost as soon as the diffuser starts. The recirculation predicted by the model is too large, giving a reattachment point too far downstream.

Periodic hill flow

In this case, as with the diffuser, the recirculation predicted by the SST model is too large. The models tends to give a large recirculation zone which in turns creates a under-developed profile for the next hill causing a smaller shear stress at the top of the hill.

stl2.png streamlinesEBM.jpg streamlinesPhi.png streamlinessst.png
Streamlines LES  streamlines EBM  Streamlines Phi-f  Streamlines SST 

"

3D Diffuser

This case is a very challenging case for all EVMs since the inlet is a fully developed flow on a rectangular duct. The Boussinesqu approximation does not allow for anisotropy and therefore the models cannot predict the secondary motion at the inlet.

In this case the SST (and most of the EVMs) predict a separation zone in the wrong wall. This might be due to the fact that the separation is predicted too early and too strong. Because is too early the recirculation zone reaches the lower corner when the height is still too small and then it stays on the diverging wall with the lower angle.

Velocity_Begin.png Velocity_End.png    
Velocity profiles at z/H=2  Velocity profiles at z/H=2 (second half)    


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Topic revision: r17 - 2011-07-15 - 08:22:01 - AlexSkillen
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