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Take the divergence (instead of rotation) of
and use
(with
) to obtain
the basic equations
Two-dimensional 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 8.67-8.68
(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: 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