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

82 A general algorithm for compressible and incompressible ¯ows
INLET
(Free-stream)
X 2
X 1
Fig. 3.2 Fictitious and real boundaries.
2. Whereas for supersonic ¯ows all the variables can be prescribed at the inlet, at the
exit however no boundary conditions are imposed simply because by de®nition the
disturbances caused by the boundary conditions cannot travel faster than the
speed of sound.
With subsonic exit conditions the situation is somewhat more complex and here
various possibilities exist. We again illustrate such conditions in Fig. 3.2.
Condition A: Denoting the most obvious assumptions with regard to the traction and
velocities.
Condition B: A more sophisticated condition of zero gradient of traction and stresses
existing there. Such conditions will of course always apply to the exit domains for
incompressible ¯ow. Condition B was ®rst introduced by Zienkiewicz et al. 47 and is
discussed fully by Papanastasiou et al. 55 This condition is of some importance as it
gives remarkably good answers.
We shall refer to these open boundary conditions in various classes of problems
dealt with later in this book and shall discuss them in detail. In particular the kind
of di€erences that may occur in incompressible ¯ows in conduits under di€erent
exit conditions are considered.
Of considerable importance, especially in view of the new schemes, are however
conditions which we will encounter on real boundaries.
3.6.2 Real boundaries
SIDE
(Free-stream)
SLIP or NO-SLIP
SIDE
(Free-stream)
a b
By real boundaries we mean limits of ¯uid domains which are physically de®ned and
here three di€erent possibilities exist.
EXIT
u 2 = 0
t 1 = 0
(parallel flow)
t b = (τ 11, τ 12) a
(A)
(B)