Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

142

10 -2

3

CFL=5 CFL=10 CFL=20 CFL=25

2.5

10 -3

2

10 -4

1.5

10 -5

residual term

CFL=5 CFL=10 CFL=20 CFL=25

1

v

mass flux error (%)

10 -6

0.5

10 -7

0

0

0.25

0.5 x/l

0.75

1

250

500

750

1000

number of iterations

Figure 4.11: Convergence rate - supersonic outflow.

Figure 4.12: Mass flux error – supersonic outflow.

In the case of a supersonic outflow, a shock wave does not occur in the flow field inside the nozzle and nonlinear effects do not appear significantly. Stability analysis of system of linear equations does not introduce a step integration constraints in the application of implicit numerical schemes, and allows the implementation of relatively large integration steps and corresponding numbers CFL to obtain convergent solution. The above conclusions are confirmed by the Figure 4.11. Distribution of relative mass flux error in the continuity equation is given along the nozzle axis in the Figure 4.12. Absolute value of the relative error increases slightly in the zones of more intensive variations of flow quantities, while close to inflow and outflow boundaries is almost negligible, since the change of the nozzle cross section area in that zone is less evident.

10 -2

20

18

CFL=5 CFL=10 CFL=20 CFL=25

10 -3

16

14

10 -4

12

10

10 -5

8

re idu s al term

CFL=5 CFL=10 CFL=20 CFL=25

6

v

mass flux error (%)

10 -6

4

2

10 -7

0

250

500

750

1000

0

0.25

0.5 x/l

0.75

1

number of iteratons

Figure 4.13: Convergence rate – subsonic outflow.

Figure 4.14: Mass flux error – subsonic outflow.