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

172 Compressible high-speed gas ¯ow
6.3.1 Euler equation
Here only ®rst-order derivatives occur and the number of boundary conditions is less
than that for the full Navier±Stokes problem.
For a solid wall boundary, ÿ u, only the normal component of velocity u n needs to be
speci®ed (zero if the wall is stationary). Further, with lack of conductivity the energy
¯ux across the boundary is zero and hence E (and ) remain unspeci®ed.
In general the analysis domain will be limited by some arbitrarily chosen external
boundaries, ÿ s, for exterior or internal ¯ows, as shown in Fig. 6.1 (see also Sec. 3.6,
Chapter 3).
Here, as discussed in Sec. 2.10.3, it will in general be necessary to perform a
linearized Riemann analysis in the direction of the outward normal to the boundary
n to determine the speeds of wave propagation of the equations. For this linearization
of the Euler equations three values of propagation speeds will be found
1 ˆ u n
2 ˆ un ‡ c
3 ˆ un ÿ c
…6:8†
where un is the normal velocity component and c is the compressible wave celerity
(speed of sound) given by
r
c ˆ
p
…6:9†
As of course no disturbances can propagate at velocities greater than those of Eqs
(6.8) and in the case of supersonic ¯ow, i.e. when the local Mach number is
M ˆ junj 5 1 …6:10†
c
we shall have to distinguish two possibilities:
(a) supersonic in¯ow boundary where
un < ÿc
and the analysis domain cannot in¯uence the exterior, for such boundaries all
components of the vector must be speci®ed; and
Γ u
Γ s
External flow Internal flow
Fig. 6.1 Boundaries of a computation domain. ÿ u, wall boundary; ÿ s, ®ctitious boundary.
Γ s
Γ u
Γ s