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

1. Solid boundaries with no slip conditions: On such boundaries the ¯uid is assumed to
stick or attach itself to the boundary and thus all velocity components become
zero. Obviously this condition is only possible for viscous ¯ows.
2. Solid boundaries in inviscid ¯ow (slip conditions): When the ¯ow is inviscid we will
always encounter slipping boundary conditions where only the normal velocity
component is speci®ed and is in general equal to zero in steady-state motion.
Such boundary conditions will invariably be imposed for problems of Euler
¯ow whether it is compressible or incompressible.
3. Prescribed traction boundary conditions: The last category is that on which tractions
are prescribed. This includes zero traction in the case of free surfaces of ¯uids or any
prescribed tractions such as those caused by wind being imposed on the surface.
These three basic kinds of boundary conditions have to be imposed on the ¯uid
and special consideration has to be given to these when split operator schemes
are used.
3.6.3 Application of real boundary conditions in the
discretization using the CBS split
We shall ®rst consider the treatment of boundaries described under (1) or (2) of the
previous section. On such boundaries
and either
u n ˆ 0; normal velocity zero …3:84†
ts ˆ 0; tangential traction zero for inviscid flow
or (3.85)
us ˆ 0; tangential velocity zero for viscous flow
In early applications of the CBS algorithm it appeared correct that when computing
~U i no velocity boundary conditions be imposed and to use instead the value of
boundary tractions which corresponds to the deviatoric stresses and pressures computed
at time tn. We note that if the pressure is removed as in Split A these pressures
could also be removed from the boundary traction component. However in Split B no
such pressure removal is necessary. This requires, in viscous problems, evaluation of
the boundary ij's and this point is explained further later.
When computing or p we integrate by parts obtaining (Eq. 3.53)
…
N k 1
p
c2 …
p d ˆÿ t N k @
p U n i ‡ 1 Ui ÿ 1 t @pn ‡ !
2
d
… k
@Np ˆ t
…
ÿ t
@x i
ÿ
@x i
U n i ‡ 1 U i ÿ t @pn ‡ 2
@x i
@x i
N k p n i U n i ‡ 1 U b ÿ t @pn ‡ 2
@x i
n i dÿ
Boundary conditions 83
dÿ …3:86†