[doc][cleanup] Use the proposed limited sTilde method, add doxygen

Fix problem with sanitizelinks.sh

doc/doxygen/modules.txt

@@ -194,7 +194,7 @@
        * \ingroup RANSModel
        * \defgroup OneEqModel 1-Eq. Models
        * \brief One-equation turbulence model by Spalart-Allmaras
-       * \copydetails ./freeflow/oneeq/model.hh
+       * \copydetails ./freeflow/rans/oneeq/model.hh
        */
       /*!
        * \ingroup RANSModel

doc/doxygen/sanitizelinks.sh

 #make the modules page default view clearer (toggleLevel(1))
-sed -i 's/\(init_search();\)/\1 toggleLevel(1);/' html/modules.html
+if [ -e html/modules.html ]; then
+  sed -i 's/\(init_search();\)/\1 toggleLevel(1);/' html/modules.html
+fi

doc/handbook/dumux-handbook.bib

@@ -1725,3 +1725,14 @@ url={http://dx.doi.org/10.1007/s11242-015-0599-1}
   address =   {Harlow},
   edition =   {2},
 }
+
+@InCollection{Spalart1992a,
+  author =        {Spalart, P. R. and Allmaras, S. R.},
+  title =         {{A one-equation turbulence model for aerodynamic flows}},
+  booktitle =     {Aerospace Sciences Meetings},
+  publisher =     {American Institute of Aeronautics and Astronautics},
+  year =          {1992},
+  pages =         {--},
+  doi =           {10.2514/6.1992-439},
+  url =           {http://dx.doi.org/10.2514/6.1992-439}
+}

dumux/freeflow/rans/oneeq/model.hh

@@ -24,7 +24,54 @@
  *
  * \copydoc RANSModel
  *
- * \todo please doc me
+ * This model, published by Spalart and Allmaras 1992 \cite Spalart1992a,
+ * uses one additional PDE for a working variable \f$ \tilde{\nu} \f$.
+ * This variable has the units of a viscosity and can be converted to the eddy
+ * viscosity via a model function~(\f$ f_\text{v1} \f$):
+ * \f[
+ *  \nu_\text{t} = \tilde{\nu} f_\text{v1}
+ * \f]
+ *
+ * Here, as proposed by Wilcox \cite Wilcox2008a and Versteeg \cite Versteeg2009a, the correction
+ * term which account for the transition or trip, is dropped from the original equations,
+ * such that the balance equation simplifies to:
+ * \f[
+ *   \frac{\partial \tilde{\nu}}{\partial t}
+ *   + \nabla \cdot \left( \tilde{\nu} \textbf{v} \right)
+ *   - c_\text{b1} \tilde{S} \tilde{\nu}
+ *   - \frac{1}{\sigma_{\tilde{\nu}}} \nabla \cdot \left( \left[ \nu + \tilde{\nu} \right] \nabla \tilde{\nu} \right)
+ *   - \frac{c_\text{b2}}{\sigma_{\tilde{\nu}}} \left| \nabla \tilde{\nu} \right|^2
+ *   + c_\text{w1} f_\text{w} \frac{\tilde{\nu}^2}{y^2}
+ *   = 0
+ * \f]
+ *
+ * Here, a modified mean effective strain rate (\f$ \tilde{S} \f$) based on
+ * the mean rotation rate tensor (\f$ \mathbf{\Omega} \f$) is used:
+ * \f[
+ *  \tilde{S} = \sqrt{2 \mathbf{\Omega} \cdot \mathbf{\Omega}} + \frac{\tilde{\nu}}{\kappa^2 y^2} f_\text{v2}
+ * \f]
+ * \f[
+ *  \mathbf{\Omega} = \frac{1}{2} \left( \nabla \textbf{v}_\text{g}
+ *                                  - \nabla \textbf{v}_\text{g}^\intercal \right)
+ * \f]
+ *
+ * This balance equation is linked to the flow geometry by the distance to the closest wall ($y$).
+ * Further, the model uses the following functions and expressions:
+ * \f[ \chi = \frac{\tilde{\nu}}{\nu} \f]
+ * \f[ f_\text{v1} = \frac{\chi^3}{\chi^3+c_\text{v1}^3} \f]
+ * \f[ f_\text{v2} = 1 - \frac{\chi}{1+f_\text{v1}\chi} \f]
+ * \f[ f_\text{w} = g_\text{w} \left( \frac{1+c_\text{w3}^6}{g^6_\text{w}+c_\text{w3}^6}
+ *                             \right)^\frac{1}{6} \f]
+ * \f[ g_\text{w} = r_\text{w} + c_\text{w2} (r_\text{w}^6 - r_\text{w}) \f]
+ * \f[ r_\text{w} = \min \left[ \frac{\tilde{\nu}}{\tilde{S}\kappa^2 y^2},10\right] \f]
+ * \f[ \sigma_{\tilde{\nu}} = \nicefrac{2}{3} \f]
+ * \f[ c_\text{b1} = 0.1355 \f]
+ * \f[ c_\text{b2} = 0.622 \f]
+ * \f[ c_\text{v1} = 7.1 \f]
+ * \f[ c_\text{w1} = \frac{c_\text{b1}}{\kappa^2}
+ *                   + \frac{1+c_\text{b2}}{\sigma_{\tilde{\nu}}} \f]
+ * \f[ c_\text{w2} = 0.3 \f]
+ * \f[ c_\text{w3} = 2 \f]
  */
 
 #ifndef DUMUX_ONEEQ_MODEL_HH

dumux/freeflow/rans/oneeq/volumevariables.hh

@@ -199,8 +199,9 @@ public:
     Scalar ft2() const
     {
         using std::exp;
+        // the trip correction term is dropped according to Versteeg2009 and Wilcox2006
+        // return ct3() * exp(-ct4() * viscosityRatio()  * viscosityRatio());
         return 0.0;
-//         return ct3() * exp(-ct4() * viscosityRatio()  * viscosityRatio());
     }
 
     //! \brief Returns a model function
@@ -231,36 +232,32 @@ public:
     Scalar viscosityRatio() const
     { return viscosityTilde() / RANSParentType::kinematicViscosity(); }
 
-    //! \brief Returns a modified version of the stress tensor scalar product
+    /*
+     * ! \brief Returns a modified version of the stress tensor scalar product
+     *
+     * According to <a href="https://turbmodels.larc.nasa.gov/spalart.html">NASA</a>
+     * this term should never be zero and different limiters might be used.
+     * The Implementation uses the one proposed in:
+     * Allmaras, S. R., Johnson, F. T., and Spalart, P. R.,
+     * "Modifications and Clarifications for the Implementation of the Spalart-Allmaras Turbulence Model," ICCFD7-1902
+     */
     Scalar stressTensorScalarProductTilde() const
     {
-
-        static const int sTildeTerm = getParamFromGroup<int>("",
-                                                             "OneEq.STildeType", 0);
-        if (sTildeTerm == 1)
-        {
-            return std::max(0.3 * vorticityMagnitude(),
-                   vorticityMagnitude()
-                   + viscosityTilde() * fv2()
-                     / RANSParentType::karmanConstant() / RANSParentType::karmanConstant()
-                     / RANSParentType::wallDistance() / RANSParentType::wallDistance());
-        }
-        else if (sTildeTerm == 2)
-        {
-            Scalar sBar = viscosityTilde() * fv2()
-                          / RANSParentType::karmanConstant() / RANSParentType::karmanConstant()
-                          / RANSParentType::wallDistance() / RANSParentType::wallDistance();
-            return sBar < -c2() * vorticityMagnitude()
-                  ? vorticityMagnitude()
-                    + (vorticityMagnitude() * (c2() * c2() * vorticityMagnitude() + c3() * sBar))
-                      / ((c3() - 2.0 * c2()) * vorticityMagnitude() - sBar)
-                  : vorticityMagnitude() + sBar;
-        }
-
-        return vorticityMagnitude()
-               + viscosityTilde() * fv2()
-                 / RANSParentType::karmanConstant() / RANSParentType::karmanConstant()
-                 / RANSParentType::wallDistance() / RANSParentType::wallDistance();
+        // original form
+        // return vorticityMagnitude()
+        //        + viscosityTilde() * fv2()
+        //          / RANSParentType::karmanConstant() / RANSParentType::karmanConstant()
+        //          / RANSParentType::wallDistance() / RANSParentType::wallDistance();
+
+        // limiter form, literature source see above
+        Scalar sBar = viscosityTilde() * fv2()
+                      / RANSParentType::karmanConstant() / RANSParentType::karmanConstant()
+                      / RANSParentType::wallDistance() / RANSParentType::wallDistance();
+        return sBar < -c2() * vorticityMagnitude()
+               ? vorticityMagnitude()
+                 + (vorticityMagnitude() * (c2() * c2() * vorticityMagnitude() + c3() * sBar))
+                   / ((c3() - 2.0 * c2()) * vorticityMagnitude() - sBar)
+               : vorticityMagnitude() + sBar;
     }
 
     //! \brief Returns the magnitude of the vorticity

test/freeflow/rans/test_pipe_laufer.input

@@ -21,9 +21,6 @@ InletVelocity = 2.5 # [m/s]
 EnableGravity = false
 SandGrainRoughness = 0.0 # [m] # not implemented for EddyViscosityModel = 3
 
-[OneEq]
-STildeType = 2
-
 [RANS]
 EddyViscosityModel = "baldwinLomax"
 UseStoredEddyViscosity = false