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

32 Convection dominated problems
2. Speci®cation of no boundary condition at the outlet edge in the case when k > 0,
which is equivalent to imposing a zero conduction ¯ux there, generally results in
quite acceptable solutions with standard Galerkin weighting even for quite high
Peclet numbers.
Part II: Transients
2.5 Transients ± introductory remarks
2.5.1 Mathematical background
The objective of this section is to develop procedures of general applicability for the
solution by direct time-stepping methods of Eq. (2.1) written for scalar values of , F i
and G i:
@
@t ‡ @Fi ‡
@xi @Gi ‡ Q ˆ 0 …2:60†
@xi though consideration of the procedure for dealing with a vector-valued function will
be included in Part III. However, to allow a simple interpretation of the various
methods and of behaviour patterns the scalar equation in one dimension [see
Eq. (2.10)], i.e.
@ @ @
‡ U ÿ
@t @x @x
k @
@x
‡ Q ˆ 0 …2:61a†
will be considered. This of course is a particular case of Eq. (2.60) in which F ˆ F… †,
U ˆ @F=@ and Q ˆ Q… ; x† and therefore
@F @F @ @
ˆ ˆ U
@x @ @x @x
…2:61b†
The problem so de®ned is non-linear unless U is constant. However, the non-conservative
equations (2.61) admit a spatial variation of U and are quite general.
The main behaviour patterns of the above equations can be determined by a change
of the independent variable x to x 0 such that
Noting that for ˆ …x 0 i; t† we have
@
ˆ
@t x const
@
@x0 @x
i
0 i @
‡
@t @t x0 const
dx 0 i ˆ dx i ÿ U i dt …2:62†
@
ˆÿUi @x 0 i
The one-dimensional equation (2.61a) now becomes simply
@
@t
‡ @
@t x 0 const
…2:63†
@ @
ÿ 0 k
@x @x0 ‡ Q…x0 †ˆ0 …2:64†