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3D-case

  First we show in Figure 4.2 the staggered grid arrangements of the unknowns together with the control volumes.

  
Figure: Arrangement of the unknowns for a staggered grid

If we integrate the continuity equation over a p-cell get

 

this is a simple extension of formula (4.4).
Just as the 2D-case we splitup the momentum equation. The formulae (4.5) - (4.14) are all most the same in 3D. The main difference between 2D and 3D follows from the fact that not a discretization of formula (4.2) is used but from:

 

This formula (4.16) follows directly from equation (4.2), (2.16), (2.21) and .
The discretization of the time derivative is given by:

 

where (0, 0, 0) is the centre of a -cell.
The discretization of the right-hand-side term is given by

 

The discretization of the convective terms is given by a straight forward extension of formulae (7.7) and (7.8a)-(7.8b) of Van Kan et al. (1991):

 

For all non-linear terms in (4.19) we used the Newton linearization given by

 

where is taken at the new time level and at the old level.
Unknowns not present at points where they are required, are computed by linear interpolation using the fewest number of interpolation points.
The discretization of the deviatoric stress tensor is carried out according to:

 

with given by formula (4.16).
The derivatives in (4.16) are computed by central differences, thus:

 

where

 

is replaced by . We make here also the remark that it might be better to replace by . So use

 

instead of formula (4.16) in (4.21).
Finally the discretization of the pressure term is carried out by a generalization of formula (3.14) in [25]:

 



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