### [handbook] Modify section on staggered grid discretization.

parent aeacd3d6
 ... ... @@ -210,64 +210,12 @@ volume centers). \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} \includegraphics[width=.8\linewidth]{./pdf/staggered_grid.pdf} \caption{\label{pc:staggered} Discretization of the staggered-grid method. The figure shows the different control volume arrangements, which are staggered with respect to each other. There are the control volumes centered around the scalar primary variables in black, the control volumes located around the $x$-component of the velocity in blue and the control volumes located around the $y$-components of the velocity in red. The control volume boundaries are given by lines. Additionally, there is one shaded example control volume each.\\ In the two-dimensional free-flow models, the continuity equation is discretized using the black control volumes, the $x$-component of the momentum equation is discretized using the blue control volumes and the $y$-component is discretized using the red control volumes. In three dimensions this works analogously.} \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 or marker-and-cell method uses a finite volume method with different control volumes for different equations. There are control volumes centered around the scalar primary variables. They correspond to the finite volume mesh. Additionally, there are control volumes located around the $x,y$ and (in 3D) $z$ velocity components which are shifted in the $x,y$ and $z$ 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}.\\ 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. Currently, all free-flow models in \Dumux use the staggered-grid discretization.
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