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8.2.3 Free Surfaces: Marker-and-Cell Method (MAC)

Introduce appropriate boundary conditions to handle such an open surface: ``marker'' particles distinguish between liquid-filled and empty Euler cells.

The hydrodynamic equations with gravity,
$\displaystyle \frac{\partial \mbox{$\bf v$}}{\partial t}$ $\textstyle =$ $\displaystyle -\nabla \bar{p}-(\mbox{$\bf v$}\cdot
\nabla) \mbox{$\bf v$} + \nu \nabla^{2}\mbox{$\bf v$}+\mbox{$\bf g$}$ (8.79)
$\displaystyle \nabla \cdot \mbox{$\bf v$}$ $\textstyle =$ $\displaystyle 0$ (8.80)

are integrated by any of the foregoing techniques (usually the pressure method). The marker particles in each cell move along according to $\mbox{$\bf r$}^{n+1}=\mbox{$\bf r$}^{n}+\mbox{$\bf v$}^{n}\Delta t$, where $\mbox{$\bf v$}^{n}$ is a particle velocity calculated by interpolation (with suitable weights) from the velocities $v_{x}, v_{y}$ in the adjacent Euler cells [HARLOW 65].

Treatment of boundary cells: Four possible types - see Figure 8.2 for the respective velocity boundary conditions. The pressure boundary conditions are the same in all cases: $p=p_{vac}$, where $p_{vac}$ is the ``external'' pressure in the empty Euler cells.
Figure 8.2: MAC method: the 4 types of surface cells and the appropriate boundary conditions for $v_{x}, v_{y}$ (see [POTTER 80]).
\begin{figure}\includegraphics[width=300pt]{figures/f8mac.ps}\end{figure}

next up previous
Next: 8.3 Lattice Gas Models Up: 8.2 Incompressible Flow with Previous: 8.2.2 Pressure Method
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001