diff --git a/doc/handbook/4_structure.tex b/doc/handbook/4_structure.tex index 228c71fe44eb865fa70fddb43364af9d9d647f47..649eb2f79a08ee53ce60c43b7fa35f94d90c6dcc 100644 --- a/doc/handbook/4_structure.tex +++ b/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}; diff --git a/doc/handbook/5_spatialdiscretizations.tex b/doc/handbook/5_spatialdiscretizations.tex index b8861d74797093ba2faaf8ffce509b705c9c8936..516d1bfe8405256ef26e3acf08d9a0ac38e768f3 100644 --- a/doc/handbook/5_spatialdiscretizations.tex +++ b/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.