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2D-case

  The discretization of the continuity equation is straightforward. We use a staggered grid arrangement as plotted in Figure 4.1.

  
Figure 4.2: Arrangement of the unknowns for a staggered grid


The continuity equation is integrated over a so-called pressure-cell. This yields:

 

where the local numbering of Figure 3.1 is used.
With respect to the discretization of the momentum equations we distinguish between the time-derivative, the convection term, the pressure gradient, the deviatoric stress tensor and the right-hand-side term.
The discretization of the time-derivative is given by formula (5.35) of Van Kan et al. (1991)):

 

where (0, 0) is the center of a -cell.
The discretization of the right-hand-side term is given by formula (5.34) of that report:

 

In order to solve the so-called no flow problem, the discretization of the right-hand side has slightly been improved by taking

See [26].
The discretization of the convective terms requires a linearization. At this moment only one type of linearization is available, the Newton linearization given by

 

where is taken at the new time level and at the preceding one.
Apart from the linearization, the discretization of the convective terms is given by formulae (5.8) and (5.9) of Van Kan et al. (1991):

-cell:

 


-cell:

 


Unknowns not present at points where they are required, are computed by linear interpolation using the least number of neighbouring points possible.
The discretization of the deviatoric stress tensor is carried out according to formulae (5.23) to (5.25) in Van Kan et al. (1991):

-cell:

 


-cell:

 


with given by formula (2.22) and by formula (4.2).
The derivatives are computed by central differences, hence

  

where for the local numbering is used.
The same type of interpolation is used as for the convective terms. is replaced by in the points where is evaluated, although a better method might be to replace by , since .
Finally, the discretization of the pressure gradient is carried by formula (3.14) in [25]:

 



Tatiana Tijanova
Wed Mar 26 10:36:42 MET 1997