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

86 A general algorithm for compressible and incompressible ¯ows
The inlet Mach number is assumed to be equal to 0.5 and all variables except the
density are prescribed at the inlet. The density is imposed at the exit of the domain.
Both the top and bottom sides are assumed to be symmetric with normal component
of velocity equal to zero. A slipping boundary is assumed on the surface of the
aerofoil. No additional viscosity in any form is used in this problem when we use
the CBS algorithm. However other schemes do need additional di€usions to get a
reasonable solution.
Figure 3.3(c) shows the comparison of the density evolution at the stagnation point
of the aerofoil. It is observed that the di€erence between the single- and two-step
schemes is negligibly small. Further tests on these schemes are carried out at a
higher inlet Mach number of 1.2 with the ¯ow being supersonic, and a di€erent
mesh with a higher number of nodes (3753) and elements (7351). Here all the variables
at the inlet are speci®ed and the exit is free. As we can see from Fig. 3.3(d) and (e), the
(a) (b)
Density
1.250
1.000
0.750
0.500
T–G scheme (Cs = 0.0)
T–G scheme (Cs = 0.5)
Present schemes (Cs = 0.0)
0.250
–50.00 –40.00 –30.00 –20.00 –10.00 0
Distance X
(c) M = 0.5
(d)
Fig. 3.4 Subsonic inviscid ¯ow past a NACA0012 aerofoil with ˆ 0 and M ˆ 0:5: (a) Density contours
with TG scheme with no additional viscosity; (b) Density contours with TG scheme with additional viscosity;
(c) Density contours with CBS scheme with no additional viscosity; (d) Comparison of density along the
stagnation line.