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314 10. Inviscid Compressible Flow
a shock wave travelling at speed c and leaving behind a contact discontinuity
travelling at speed Vc, as shown in Fig. 10.13:
The problem therefore contains three distinct regions (2, 3, 4) in addition
to the regions at rest (1 and 5), of which two represents discontinuous regions
(region 3-4 and 4-5). The closed form solution of this problem is given in [6].
10.10.1 Initial Conditions
The shock tube is of length 1000, and the diaphragm is located at x = 500. The
initial conditions at t = 0 are:
u(x, 0) = 0 for 0 < x < 1000
Q(X, 0) = 1, p(x, 0) = 1 for x < 500
Q(x, 0) = 4, p{x, 0) = 4 for x > 500
10.10.2 Boundary Conditions
The shock tube is considered of infinite length, which avoids waves being re-
flected at its end. Therefore, extrapolation from the interior of the domain is
used on both boundaries:
g(i = 0, t) = g{i = 1, t) g(i = N,t) = g(i = N - 1, t)
u(i = 0,t) = u(i = 1,t) u(i = N,t) = u(i = N - l,t)
p(i = 0, t) = p(i = 1, t) p(i = N,t) =p(i = N - 1, t)
Note that we make use of halo cells to introduce the boundary conditions, as
discussed in subsection 10.5.2.
10.10.3 Solution Procedure and Sample Calculations
The computer program is given in Appendix B. First, a 1-D grid with equidistant
spacing is generated (subroutine generate_grid). The flux variables are then
initialized according to the specified boundary condition of subection 10.9.2 for
i
a diatomic gas (subroutine n i t i a l - c o n d i t i o n s ) . Then the algorithm marches
in time, with increments satisfying the stability restrictions of Section 10.8 (sub-
t
routine imestep). The MacCormack predictor-corrector steps of Section 10.9
are computed (subroutine Maccormack) to yield the updated flux values. The
program ends when the time has reached 250. Note that a subroutine has been
devised to compute the Euler fluxes / given the primitive variables (subroutine
flux). This way, the predictor and corrector steps are simply done by calling
the flux subroutine for each of the steps of Eqs. (10.6.5) and (10.6.6). The nu-
merical boundary schemes are used in both steps, but at the opposite end of
the tube.