added: spalart-allmaras turbulence model integrand and test application

git-svn-id: http://svn.sintef.no/trondheim/IFEM/trunk@1076 e10b68d5-8a6e-419e-a041-bce267b0401d

doc/Integrands/Spalart-Allmaras/spalart-allmaras.tex

@@ -0,0 +1,120 @@
+\documentclass[twoside, 11pt, a4paper]{article}
+\usepackage{amsmath,amsfonts,amssymb,graphicx,parskip}
+\usepackage[utf8]{inputenc}
+
+\DeclareMathOperator{\eps}{\epsilon}
+\newcommand{\dee}{\mathrm{d}}
+
+\title{Spalart-Allmaras turbulence model - SplineFEM implementation}
+\author{Arne Morten Kvarving}
+\begin{document}
+\maketitle
+This document describe the derivation of the weak form, and the associated
+Jacobian, for the Spalart-Allmaras turbulence model given in \cite{sa}.
+It is implemented in SplineFEM in the SpalartAllmaras
+Integrand, see src/Integrands/SpalartAllmaras.h/C, while
+you can find a test application (and SIM classes) in Apps/SpalartAllmaras.
+
+Initial expression:
+\begin{equation}
+	\frac{\partial\tilde{\nu}}{\partial t} + \mathbf{u}\cdot\nabla\tilde{\nu} = c_{b1}\tilde{S}\tilde{\nu}+\frac{1}{\sigma}\left(\nabla\cdot\left(\nu + \tilde{\nu}\right)\nabla\tilde{\nu} + c_{b2}\left|\nabla\tilde{\nu}\right|^2\right) - c_{w1}f_w\left(\frac{\tilde{\nu}}{d}\right)^2.
+\end{equation}
+Here $\tilde{S}$ models the production and is given by
+\begin{equation}
+	\tilde{S} = S + \frac{\tilde{\nu}}{k^2d^2}f_{\tilde{\nu}2}
+\end{equation}
+where $S$ is the normal strain tensor
+\begin{equation}
+	\begin{split}
+		S &\equiv \sqrt{\hat{\Omega}_{ij}\hat{\Omega}_{ij}}, \\
+		\hat{\Omega}_{ij} &\equiv \partial_j u_i - \partial_i u_j, \\
+		X &= \frac{\tilde{\nu}}{\nu}, \\
+		f_{\tilde{\nu}1} &= \frac{X^3}{X^3+c_{\tilde{\nu}1}^3}, \\
+		f_{\tilde{\nu}2} &= 1-\frac{X}{1+Xf_{\tilde{\nu}1}}, \\
+		c_{\tilde{\nu}1} &= 7.1, \\
+		c_{b1} &= 0.1355, \\
+		k &= 0.41.
+	\end{split}
+\end{equation}
+
+The center terms models the diffusion. Here
+\begin{equation}
+	\sigma = \frac{2}{3},\ c_{b2} = 0.622
+\end{equation}
+
+Finally, the last term models destruction. Here
+\begin{equation}
+	\begin{split}
+		f_w = g\left[\frac{1+c_{w3}^6}{g^6+c_{w3}^3}\right]^{1/6}; \quad g = r + c_{w2}\left(r^6-r\right); \quad r \equiv \frac{\tilde{\nu}}{\tilde{S}k^2d^2} \\
+		c_{w1} = c_{b1}/k^2+(1+c_{b2})/\sigma,\quad c_{w2} = 0.3,\quad c_{w3} = 2.
+	\end{split}
+\end{equation}
+
+Furthermore, we have that $\hat{\Omega}_{ij} \equiv \partial_j u_i-\partial_i u_j$.
+That is
+\[
+  \hat{\Omega}  = \begin{bmatrix}
+                    0 & \frac{1}{2}\left(\frac{\partial u_1}{\partial x_2}-\frac{\partial u_2}{\partial x_1}\right) & \frac{1}{2}\left(\frac{\partial u_1}{\partial x_3}-\frac{\partial u_3}{\partial x_1}\right) \\
+                    \frac{1}{2}\left(\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}\right) & 0 & \frac{1}{2}\left(\frac{\partial u_2}{\partial x_3}-\frac{\partial u_3}{\partial x_2}\right) \\
+                    \frac{1}{2}\left(\frac{\partial u_3}{\partial x_1}-\frac{\partial u_1}{\partial x_3}\right) & \frac{1}{2}\left(\frac{\partial u_3}{\partial x_2}-\frac{\partial u_2}{\partial x_3}\right) & 0
+                  \end{bmatrix} =
+                  \begin{bmatrix}
+                     0 & a & b \\
+                    -a & 0 & c \\
+                    -b & -c & 0
+                  \end{bmatrix}
+\]
+Thus,
+\[
+  \hat{\Omega}\hat{\Omega}  = 2a^2+2b^2+2c^2
+\]
+and
+\[
+  S = 2\sqrt{a^2+b^2+c^2}.
+\]
+
+\subsection*{Derivation of the Jacobian}
+Temporal term: In this case we consider a BDF1 discretization.
+
+\subsubsection*{Advection term}
+\bf Strong form:\rm
+\begin{equation}
+  \mathbf{u}\cdot\nabla\delta\nu
+\end{equation}
+The weak form is straight forward.
+
+\subsubsection*{Production term}
+\bf Strong from:\rm
+\begin{equation}
+  c_{b1}\hat{S}\delta\nu
+\end{equation}
+The weak form gives a simple mass term.
+
+\subsubsection*{Diffusion terms}
+\bf Strong form:\rm
+\begin{equation}
+  \frac{1}{\sigma}\left(\nabla\cdot\left(\nu+\tilde{\nu}+\delta\nu\right)\nabla\left(\tilde{\nu}+\delta\nu\right)+c_{b2}\left|\nabla\tilde{\nu}+\nabla\delta\nu\right|^2\right)
+\end{equation}
+\bf Linearized, first term:\rm
+\begin{equation}
+  \nabla\cdot\nu\nabla\delta\nu + \nabla\cdot\tilde{\nu}\nabla\delta\nu + \nabla\cdot\delta\nu\nabla\tilde{\nu}.
+\end{equation}
+\bf Linearized, second term:\rm
+\begin{equation}
+  2\nabla\tilde{\nu}\cdot\nabla\delta\nu
+\end{equation}
+
+\bf Weak form, first term:\rm
+\begin{equation}
+  \begin{aligned}
+    \int_\Omega \nabla \cdot \left(\nu\nabla\delta\nu + \tilde{\nu}\nabla\delta\nu + \delta\nu\nabla\tilde{\nu}\right)v\,\dee\Omega &=&
+    \int_{\partial\Omega} \left(\nu\nabla\delta\nu + \tilde{\nu}\nabla\delta\nu + \delta\nu\nabla\tilde{\nu}\right)v\cdot \mathbf{n}\,\dee S \\
+    &-&\int_\Omega \left(\nu\nabla\delta\nu + \tilde{\nu}\nabla\delta\nu + \delta\nu\nabla\tilde{\nu}\right)\cdot \nabla v\,\dee\Omega.
+   \end{aligned}
+\end{equation}
+
+\bf Weak form, second term:\rm \\
+Straight forward, no integration by parts have to be performed.
+
+\end{document}
+

src/SIM/SIMenums.h

@@ -24,7 +24,8 @@ namespace SIM //! Simulation scope
     LINEAR    = 1,
     NONLINEAR = 2,
     LAPLACE   = 10,
-    STRESS    = 11
+    STRESS    = 11,
+    RANS      = 12
   };
 
   //! \brief Enum defining the various solution modes that may occur.