Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

68 A general algorithm for compressible and incompressible ¯ows
(2.11). This term can however be treated as a known (source type) quantity providing
we have an independent way of evaluating the pressure. Before proceeding with the
algorithm, we rewrite Eq. (3.4) in the form given below to which the characteristic±Galerkin
method can be applied
@U i
@t
@
ˆÿ …u
@x
jUi†‡ j
@ ij
‡ gi ‡ Q
@xj n ‡ 2
i
…3:17†
with Q n ‡ 2 being treated as a known quantity evaluated at t ˆ t n ‡ 2 t in a time
increment t. In the above equation
with
or
In this
@p n ‡ 2
@x i
Q n ‡ 2
i
ˆ 2
@p n ‡ 2
@x i
ˆÿ @pn ‡ 2
@x i
n ‡ 1
@p
‡…1ÿ @x
2†
i
@pn
@xi ˆ @pn @ p
‡ 2
@xi @xi p ˆ p n ‡ 1 ÿ p n
Using Eq. (2.91) of the previous chapter and replacing by U i, we can write
n ‡ 1
Ui ÿ U n i ˆ t ÿ @
…u
@x
jUi† j
n ‡ @ n ij
‡ Q
@xj n ‡ 2
i ÿ… gi† n
‡
t
2 uk @
@x k
@
@x j
…u jU i†ÿQ i ‡ g i
n
…3:18†
…3:19†
…3:20†
…3:21†
…3:22†
At this stage we have to introduce the `split' in which we substitute a suitable
approximation for Q which allows the calculation to proceed before p n ‡ 1 is
evaluated. Two alternative approximations are useful and we shall describe these as
Split A and Split B respectively. In the ®rst we remove all the pressure gradient
terms from Eq. (3.22); in the second we retain in that equation the pressure gradient
corresponding to the beginning of the step, i.e. @p n =@x i. Though it appears that the
second split might be more accurate, there are other reasons for the success of the
®rst split which we shall refer to later. Indeed Split A is the one which we shall
universally recommend.
Split A
In this we introduce an auxiliary variable U i such that
U i ˆ U i ÿ U n i
ˆ t ÿ @
…u
@x
jUi†‡ j
@ ij
ÿ gi ‡
@xj t
2 uk @
@x k
@
@x j
…u jU i†‡ g i
n
…3:23†