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

The matrices and forcing vectors are again similar and given as
…
ME ˆ N T …
ENE d CE ˆ N T Er T …
…uNE† d Cp ˆ N T Er T …uNp† d
…
KT ˆ …rNE† T …
krNT d K E ˆ B T …I0 ÿ 2
3 mmT †B d
KuE ˆÿ1 …
2 …r T …uNE†† T …
…rNE† d fe ˆ N
ÿ
T En T …t d u ‡ krT† dÿ
Kup ˆÿ1 …
2 …r T …uNE†† T …rNp† d
…3:62†
The forcing term f es contains source terms. If no source terms are available this term is
equal to zero.
It is of interest to observe that the process of Step 4 can be extended to include in an
identical manner the equations describing the transport of quantities such as turbulence
parameters, 36 chemical concentrations, etc., once the ®rst essential Steps 1±3
have been completed.
Split B
With Split B, the discretization and solution procedures have to be modi®ed slightly.
Leaving the details of the derivation to the reader and using identical discretization
processes, the ®nal steps can be summarized as:
Step 1
~U i ˆÿM ÿ1
u t …C u ~U ‡ K ~u ‡ G T ~p ÿ f†ÿ t K u ~U ‡ f s ‡ t
2 P~p
where all matrices are the same as in Split A except the forcing term f which is
…
f ˆ N T …
u g d ‡
since the pressure term has now been integrated by parts
Step 2
and
Step 3
n
…3:63†
N
ÿ
T u t d dÿ …3:64†
…M p ‡ t 2 1 2H† ~p ˆ t‰G ~ U n ‡ 1G ~ U ÿ f pŠ n
Characteristic-based split (CBS) algorithm 75
…3:65†
~U ˆ ~U ÿ M ÿ1
u t 2G T ~p …3:66†
Step 4, calculation of the energy, is unchanged. The reader can notice the minor
di€erences in the above equations from those of Split A.