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Up: 8.2 Incompressible Flow with
Previous: 8.2.1 Vorticity Method
Take the divergence (instead of rotation) of
and use
(with
) to obtain
the basic equations
Twodimensional case, explicit:
Apply finite difference scheme.
Note: Divergence condition
must
stay intact in the course of the calculation.
Harlow and Welch procedure([HARLOW 65], see also [POTTER 80]):
 Let the pressure be localized at the
centers of the Euler cells
 Velocity components and are localized
at the right and upper box sides, respectively
Figure 8.1:
Grid structure in the pressure method

 Now approximate the divergence of the velocity by

(8.70) 
or, in ``geographical'' notation,

(8.71) 
for vanishing divergence.
 Apply the Lax scheme to 8.678.68
(all terms on the r.h.s. are taken at time ):
 Insert the new velocity components in 8.71 to find
with
 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
and deviate from zero.
The simple ansatz

(8.76) 
for the pressures is therefore not usable. Rather, we write
Stability:
Again, the conditions are

(8.78) 
Next: 8.2.3 Free Surfaces: MarkerandCell
Up: 8.2 Incompressible Flow with
Previous: 8.2.1 Vorticity Method
Franz J. Vesely Oct 2005
See also: "Computational Physics  An Introduction," KluwerPlenum 2001