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Next: 8.2.3 Free Surfaces: Marker-and-Cell Up: 8.2 Incompressible Flow with Previous: 8.2.1 Vorticity Method

8.2.2 Pressure Method

Take the divergence (instead of rotation) of

\begin{displaymath}
\frac{\partial \mbox{$\bf v$}}{\partial t} + (\mbox{$\bf v$}...
...box{$\bf v$}=
- \nabla \bar{p} + \nu \nabla^{2}\mbox{$\bf v$}
\end{displaymath}

and use

\begin{displaymath}
\nabla \cdot \left( \mbox{$\bf v$} \cdot \nabla \right)\mbox{$\bf v$} =
(\nabla \mbox{$\bf v$}):(\nabla \mbox{$\bf v$})
\end{displaymath}

(with $\mbox{${\bf A}$}:\mbox{${\bf B}$} \equiv \sum_{i}\sum_{j}A_{ij}B_{ji}$) to obtain the basic equations


$\displaystyle \frac{\partial \mbox{$\bf v$}}{\partial t} + (\mbox{$\bf v$} \cdot \nabla )\mbox{$\bf v$}$ $\textstyle =$ $\displaystyle - \nabla \bar{p} + \nu \nabla^{2}\mbox{$\bf v$}$ (8.65)
$\displaystyle \nabla^{2} \bar{p}$ $\textstyle =$ $\displaystyle -(\nabla \mbox{$\bf v$}):(\nabla \mbox{$\bf v$})$ (8.66)

Two-dimensional case, explicit:

$\displaystyle \frac{\partial v_{x}}{\partial t}$ $\textstyle =$ $\displaystyle -\frac{\partial \bar{p}}{\partial x}+\nu
\left[\frac{\partial^{2}...
...
-\frac{\partial v_{x}^{2}}{\partial x}
-\frac{\partial v_{x}v_{y}}{\partial y}$ (8.67)
$\displaystyle \frac{\partial v_{y}}{\partial t}$ $\textstyle =$ $\displaystyle -\frac{\partial \bar{p}}{\partial y}+\nu
\left[\frac{\partial^{2}...
...
-\frac{\partial v_{y}^{2}}{\partial y}
-\frac{\partial v_{x}v_{y}}{\partial x}$ (8.68)
$\displaystyle \frac{\partial^{2}\bar{p}}{\partial x^{2}}
+\frac{\partial^{2}\bar{p}}{\partial y^{2}}$ $\textstyle =$ $\displaystyle -\left[\left(\frac{\partial v_{x}}{\partial x}\right)^{2}
+2\left...
...{\partial x}\right)
+\left(\frac{\partial v_{y}}{\partial y}\right)^{2} \right]$ (8.69)

$\Longrightarrow$Apply finite difference scheme.

Note: Divergence condition $\nabla \cdot \mbox{$\bf v$}=0$ must stay intact in the course of the calculation.
$\Longrightarrow$Harlow and Welch procedure([HARLOW 65], see also [POTTER 80]):
- Let the pressure $p_{i,j}$ be localized at the centers of the Euler cells
- Velocity components $v_{x,i,j}$ and $v_{y,i,j}$ are localized at the right and upper box sides, respectively
Figure 8.1: Grid structure in the pressure method
\begin{figure}\includegraphics[width=150pt]{figures/f8dm1.ps}\end{figure}

- Now approximate the divergence of the velocity by
\begin{displaymath}
D_{i,j} \equiv \frac{1}{\Delta l}\left[v_{x,i,j}-v_{x,i-1,j} \right]
+ \frac{1}{\Delta l}\left[v_{y,i,j}-v_{y,i,j-1} \right]
\end{displaymath} (8.70)

or, in ``geographical'' notation,
\begin{displaymath}
D_{C} \equiv \frac{1}{\Delta l}\left[v_{x,C}-v_{x,W} \right]
+ \frac{1}{\Delta l}\left[v_{y,C}-v_{y,S} \right]
\end{displaymath} (8.71)

$\Longrightarrow$$D_{C}=0$ for vanishing divergence.

- Apply the Lax scheme to 8.67-8.68 (all terms on the r.h.s. are taken at time $t_{n}$):
$\displaystyle v_{x,C}^{n+1}$ $\textstyle =$ $\displaystyle \frac{1}{4}\left[v_{x,N}+v_{x,E}+v_{x,S}+v_{x,W} \right]
-\frac{\Delta t}{2\Delta l}\left[v_{x,E}^{2}-v_{x,W}^{2} \right]$  
    $\displaystyle -\frac{\Delta t}{2\Delta l}\left[\frac{1}{2}\left(v_{y,E}+v_{y,C}...
...t)-\frac{1}{2}\left(v_{y,S}+v_{y,SE}\right)
\left(v_{x,S}+v_{x,C}\right)\right]$  
    $\displaystyle -\frac{\Delta t}{\Delta l}\left(\bar{p}_{E}-\bar{p}_{C} \right)
+...
...Delta t}{(\Delta l)^{2}}
\left(v_{x,N}+v_{x,E}+v_{x,S}+v_{x,W}-4v_{x,C} \right)$ (8.72)
       
$\displaystyle v_{y,C}^{n+1}$ $\textstyle =$ $\displaystyle \frac{1}{4}\left[v_{y,N}+v_{y,E}+v_{y,S}+v_{y,W} \right]
-\frac{\Delta t}{2\Delta l}\left[v_{y,N}^{2}-v_{y,S}^{2} \right]$  
    $\displaystyle -\frac{\Delta t}{2\Delta l}\left[\frac{1}{2}\left(v_{x,N}+v_{x,C}...
...t)
-\frac{1}{2}\left(v_{x,NW}+v_{x,W}\right)\left(v_{y,W}+v_{y,C}\right)\right]$  
    $\displaystyle -\frac{\Delta t}{\Delta l}\left(\bar{p}_{N}-\bar{p}_{C} \right)
+...
...Delta t}{(\Delta l)^{2}}
\left(v_{y,N}+v_{y,E}+v_{y,S}+v_{y,W}-4v_{y,C} \right)$ (8.73)

- Insert the new velocity components in 8.71 to find
$\displaystyle D_{C}^{n+1}$ $\textstyle =$ $\displaystyle \frac{1}{4}\left(D_{N}^{n}+D_{E}^{n}+D_{S}^{n}+D_{W}^{n} \right)
-\frac{\Delta t}{2(\Delta l)^{2}}S_{C}^{n}$  
    $\displaystyle -\frac{\Delta t}{(\Delta l)^{2}}
\left(\bar{p}_{N}^{\,n}
+\bar{p}_{E}^{\,n}+\bar{p}_{S}^{\,n}+\bar{p}_{W}^{\,n}
-4\bar{p}_{C}^{\,n} \right)$  
    $\displaystyle +\frac{\nu \Delta t}{(\Delta l)^{2}}
\left(D_{N}^{n}+D_{E}^{n}+D_{S}^{n}+D_{W}^{n}-4D_{C}^{n} \right)$ (8.74)

with
$\displaystyle S_{C}$ $\textstyle \equiv$ $\displaystyle \left(v_{x,E}^{2}-v_{x,C}^{2}-v_{x,W}^{2}+v_{x,WW}^{2} \right)
+\left(v_{y,N}^{2}-v_{y,C}^{2}-v_{y,S}^{2}+v_{y,SS}^{2} \right)$  
    $\displaystyle +\frac{1}{2}\left(v_{y,E}+v_{y,C}\right)\left(v_{x,N}+v_{x,C}\right)
-\frac{1}{2}\left(v_{y,S}+v_{y,SE}\right)\left(v_{x,S}+v_{x,C}\right)$  
    $\displaystyle -\frac{1}{2}\left(v_{x,NW}+v_{x,W}\right)\left(v_{y,C}+v_{y,W}\right)
+\frac{1}{2}\left(v_{x,W}+v_{x,SW}\right)\left(v_{y,ES}+v_{y,SW}\right)$ (8.75)


- Now solve the Poisson equation 8.69 to obtain the pressures.
Problem: After applying the Poisson solver the pressures contain small errors which cause the central values $D_{i,j}^{n}$ and $D_{i,j}^{n+1}$ deviate from zero. The simple ansatz
\begin{displaymath}
\bar{p}_{N}+\bar{p}_{E}+\bar{p}_{S}+\bar{p}_{W}-4\bar{p}=-S_{C}
\end{displaymath} (8.76)

for the pressures is therefore not usable. Rather, we write
$\displaystyle \bar{p}_{N}+\bar{p}_{E}+\bar{p}_{S}+\bar{p}_{W}-4\bar{p}_{C}$ $\textstyle =$ $\displaystyle -S_{C}
+\frac{(\Delta l)^{2}}{4\Delta t}\left(D_{N}+D_{E}+D_{S}+D_{W} \right)$  
    $\displaystyle +\nu \left(D_{N}+D_{E}+D_{S}+D_{W}-4D_{C}\right)$ (8.77)



Stability: Again, the conditions are
\begin{displaymath}
\Delta t \leq \frac{\Delta l}{\sqrt{2}\vert v\vert _{max}}\,...
...\;\;\;
\Delta t \leq \frac{1}{2} \, \frac{(\Delta l)^{2}}{\nu}
\end{displaymath} (8.78)


next up previous
Next: 8.2.3 Free Surfaces: Marker-and-Cell Up: 8.2 Incompressible Flow with Previous: 8.2.1 Vorticity Method
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001