-- DominiqueLaurence - 2010-03-29
  \begin{center} {\large MSc. Advanced CFD (2010)} \end{center}  \section{Finite Volume Methods for Unstructured Grids}  \subsection{Introduction} The \emph{finite volume} (FV) method starts with the integral form of a conservation equation:%  \begin{equation} \frac{\partial}{\partial t}\int_{\Omega}\rho\,\phi\,d\Omega+\int_{S}\rho \,\phi\,\underline{v}.\underline{n}\,dS=\int_{S}\rho\,\gamma\,(\underline {grad\,}(\phi)\,.\underline{n}\,)dS+\int_{\Omega}q_{\phi}\,d\Omega \label{IntCE}% \end{equation} and is obtained by integrating: \begin{equation} \frac{\partial\rho\phi}{\partial t}+div(\underline{v\,}\rho\,\phi )=div(\rho\,\gamma\underline{grad\,}(\phi))+q_{\phi}% \end{equation} over any \emph{control volume} $\Omega$\ bounded by the surface $S,$ on which $\underline{n}$ is the normal directed toward the outside of $\Omega$. The integral conservation equation (\ref{IntCE}) is valid for any control volume $\Omega$. It is applied to every cell of the mesh. These cells usually consist of rectangles or triangles in 2D, and hexaedra, prisms, or tetrahedra in 3D. More generally, any type of volume, bounded by $N_{s}$ planar surfaces, is possible.%  %TCIMACRO{\FRAME{dthFU}{271.4375pt}{72.1875pt}{0pt}{\Qcb{Hexahedra, Prisms, and %Tetrahedra}}{}{Figure }{\special{ language "Scientific Word"; %type "GRAPHIC"; maintain-aspect-ratio TRUE; display "PICT"; %valid_file "T"; width 271.4375pt; height 72.1875pt; depth 0pt; %original-width 251.75pt; original-height 65.5625pt; cropleft "0"; %croptop "1"; cropright "1"; cropbottom "0"; %tempfilename 'L01IXT00.wmf';tempfile-properties "XPR";}}}% %BeginExpansion \begin{center} \includegraphics[ natheight=65.562500pt, natwidth=251.750000pt, height=72.1875pt, width=271.4375pt ]% {L01IXT00.wmf}% \\ Hexahedra, Prisms, and Tetrahedra \end{center} %EndExpansion The control volumes (CV) are noted $T_{i}$ (considering Triangles as generic examples), the union of all CV $(i=1,..,N_{CV})$ forming a partition of the entire flow domain. The surface integrals appearing in (\ref{IntCE}) when applied to the CV $T_{i}$ is simply the sum of fluxes over the number $Ns$ of planar surfaces separating $T_{i}$\ from the neighbor cells: \[ \int_{T_{i}}\rho\,\phi\,\underline{v}.\underline{n}\,dS=\sum_{j=1,Ns}% \int_{S_{ij}}\rho\,\phi\,\underline{v}.\underline{n}\,dS \]

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