Page 327 - Computational Fluid Dynamics for Engineers

10.11 Model Problem for the MacCormack Method: Quasi 1-D Nozzle 317

10.11.2 Boundary Conditions

The incorporation of the source term poses no difficulty but the non-reflecting
boundary conditions must now be changed to subsonic Riemman type boundary
conditions when subsonic flow is present and to supersonic boundary conditions
when supersonic flow is present. The supersonic boundary conditions are (with
right running characteristics)
Qi = Qinf
u
l = ^inf (10.11.5)
Pi = Pint
t
at he inlet, and
QN = QN-i
UN = UN-I (10.11.6)
PN =PN-I
at the outlet. Equations (10.7.1) must be used when subsonic flow is present.
Since in our model problem the nozzle entrance will be supersonic and the
nozzle exit subsonic, we will develop the appropriate boundary condition for
the exit station i = N. Referring to Fig. 10.15, the two Riemann invariants
meeting at the boundary point J are emanating from the point i = N — 1 and
from downstream infinity, where the pressure is given.
Along the right running characteristic, we have

QR =QN-i
PR =PN-i

from which the entropy SR and the speed of sound c# are computed. It is known
that entropy in an inviscid subsonic flow is constant so that the far-field entropy
SL carried by a left running characteristic is thus equal to the entropy carried
by the right wave: si = SR. Since the back pressure p^ is given as a boundary
condition, the density QL and the speed of sound CL are found at once.
The right running Riemann invariant R\ meets the left running Riemann
invariant i?2 at point i = N and, by substracting the two invariants,
= R 1- - 1)
R 2 4 c L / ( 7

Fig. 10.15. Far-field boundary conditions using Riemann
N-l N oo . . .
invariants