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316 10. Inviscid Compressible Flow

readily be obtained, and the solutions have many branch points, i.e. fully sub-
sonic, subsonic/supersonic, fully supersonic and lastly subsonic/supersonic and
subsonic at the exit (underexpanded jets), it provides a useful way to evalu-
ate the influence of boundary conditions while also yielding information on the
convergence speed and accuracy of the selected schemes.
The compressible Euler equations for a quasi 1-D nozzle are

(10.11.1)
dt dx

where the source term is related to the cross-sectional area of the nozzle. In
this equation, the vectors Q and F are very close to their ID counterpart [Eqs.
(2.1.32) and (2.1.33)], but they contain an additional factor, which corresponds
to the influence of the cross-sectional nozzle area,

QAl QUA 0
Q QUA , E = (gu 2 +p)A and S P dA (10.11.2)
eA \ u(e + p)A dx
0

The MacCormack scheme is modified to take into account the source term but
there are many possibilities, and here the correction will be added at each step:

predictor step:
Qi = Q? - At ( E? Af~ l 1 + 4<ST (10.11.3)

corrector step:

^
4 = Q? - At ( ± 1 ^ ) + AtS* (10.11.4)
Ax

Updating is as described by Eq. (10.6.4). Note that S is the source term evalu-
ated with the predicted value from the predictor step. Since there are only slight
modifications to the ID Euler flow equations, the computer program of Section
10.10 is used here, with the proper modifications for the area and source terms
made.

10.11.1 Initial Conditions

The nozzle is of length 10, with the incoming flow supersonic (M = 1.3) and
the outgoing flow subsonic (M < 1). The entire flowfield is initialized with the
incoming flow values.