Merge branch 'handbook/staggeredDiscretization' into 'master'

[doc][handbook] Add first draft for staggered-grid discretization

See merge request !1060

doc/handbook/4_structure.tex

@@ -90,7 +90,7 @@
   [.\node[FirstLevel] {freeflow};
     [.\node[SecondLevel] {\emph{models}};
       \node[ThirdLevel] {Single-phase free flow models using Navier-Stokes
-                         and algebraic turbulence models.};
+                         and eddy-viscosity based Reynolds-averaged Navier-Stokes turbulence models.};
     ]
   ]
   [.\node[FirstLevel] {io};

doc/handbook/5_spatialdiscretizations.tex

@@ -202,11 +202,73 @@ should only be applied for structured grids
 direction of the gradient between the two element/control
 volume centers).
 
-% \subsubsection{MPFA}\label{staggered}
+% \subsubsection{MPFA}\label{cc_mpfa}
 % TODO
-% \subsubsection{NLTPFA}\label{staggered}
+% \subsubsection{NLTPFA}\label{cc_nltpfa}
 % TODO
 
-% \subsection{Staggered Grid -- A Short Introduction}\label{staggered}
-% TODO
+\subsection{Staggered Grid -- A Short Introduction}\label{staggered}
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[scale=3.0,font=\normalsize]
+\begin{scope}[shift={(0,2.5)}]
+% control volume pressure
+\fill[gray!40] (1,1) rectangle (2,2);
+% control volume for vertical velocity
+\fill[dumuxBlue!30] (0.0,0.5) rectangle (1.0,1.5);
+% control volume for horizontal velocity
+\fill[dumuxYellow!30] (0.5,0) rectangle (1.5,1);
+
+% grid cells
+\draw [thick] (0,0) grid (2,2);
+
+% cell centers
+\foreach \x in {0.5,...,1.5}
+  \foreach \y in {0.5,...,1.5}
+    \fill[gray](\x,\y) circle(0.06);
+% velocity x
+\foreach \x in {0,1,2}
+  \foreach \y in {0.5,...,1.5}
+    \draw[dumuxYellow,->,ultra thick](\x-0.15,\y) -- (\x+0.15,\y);
+% velocity y
+\foreach \x in {0.5,...,1.5}
+  \foreach \y in {0,...,2.0}
+    \draw[dumuxBlue,->,ultra thick](\x,\y-0.15) -- (\x,\y+0.15);
+\end{scope}
+
+% annotations
+\begin{scope}[shift={(1.0,2.7)}]
+\fill[gray](2.3,1.8) circle(0.06);
+\draw(2.5,1.8) node[right, align=left] {cell-centered primary variables\\\color{gray}($p_\alpha$, $x^\kappa_\alpha$, $T$, $k$, $\varepsilon$, ...)};
+\draw[dumuxYellow,->,ultra thick](2.2,1.4) -- (2.4,1.4);
+\draw[dumuxYellow](2.5,1.4) node[right, align=left] {$v_{\alpha\textrm{,x}}$};
+\draw[dumuxBlue,->,ultra thick](2.3,0.9) -- (2.3,1.1);
+\draw[dumuxBlue](2.5,1.0) node[right, align=left] {$v_{\alpha\textrm{,y}}$};
+
+\draw[thick](2.2,0.5) rectangle (2.4,0.7);
+\draw(2.5,0.6) node[right, align=left] {finite volume mesh};
+\draw[draw=none,fill=gray!40](2.2,0.1) rectangle (2.4,0.3);
+\draw(2.5,0.2) node[right, align=left] {control volumes\\(cell-centered primary variables)};
+\draw[draw=none,fill=dumuxYellow!30](1.9,-0.1) rectangle (2.1,-0.3);
+\draw[draw=none,fill=dumuxBlue!30](2.2,-0.1) rectangle (2.4,-0.3);
+\draw(2.5,-0.2) node[right, align=left] {staggered control volumes\\(velocity components)};
+\end{scope}
+\end{tikzpicture}
+\caption{\label{pc:staggered} Discretization of the staggered-grid method}
+\end{figure}
 
+The staggered-grid or marker-and-cell method uses a cell-centered finite volume method
+for the scalar primary variables.
+The control volumes for the velocity components are shifted half-a-cell in each direction,
+such that the velocity components are located on the edges of the
+cell-centered finite volume mesh (see Figure~\ref{pc:staggered}).
+As for the cell-centered method, the fluxes are evaluated at the edges
+of each control volume with a two-point flux approximation, cf. \ref{cc}.\\
+For cell-centered variables, the boundary handling is as for the cell-centered method.
+For the velocity components, Dirichlet values for the component normal to the boundary
+face can directly be applied.
+For the tangential components the boundary values are treated with contribution of the boundary flux. \\
+The staggered-grid method is robust, mass conservative, and free of pressure oscillations
+but should, as the cell-centered TPFA method, only be applied for structured grids.
+At the moment the staggered-grid is the base discretization for all free-flow models.