# Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

132

analysis of the mentioned problem due to the essential computer time saving.

Flow analysis in a divergent nozzle, whose cross-section area varies along the x axis of the nozzle according to the law [15]

S ( x ) = 1 . 398 + 0 . 347 tanh ( 0 . 8 x + 0 . 4 ) ,

is the second task in which the results obtained by applying the block-diagonal procedure and the implicit LU factorization will be compared. The diagram in Figure 4.7 shows the pressure and density distribution in case of supersonic flow in entire flow domain, while the diagram in Figure 4.8 illustrates the mentioned distribution in the presence of a strong shock wave.

Without shock wave

1.2

1.1

pr ROE (BD) ess. dens. ROE (BD) pr ROE (LU) ess. dens. ROE (LU) pr analyt. ess. dens. analyt.

1

0.9

0.8

0.7

p/p 1 ρ ρ/ 1

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.25

0.5 x/l

0.75

1

Figure 4.7: Flow in divergent nozzle – supersonic flow.

Qualitative and quantitative matching of numerical solutions with theoretically ones is remarkable in the case of a entirely supersonic flow, which can be concluded by analyzing the diagram on Figure 4.7. When a strong shock wave is formed in the nozzle, as shown on diagram in Figure 4.8, the numerical solutions are almost identical with theoretical in the zone of continuous change, except in the zone of abrupt discontinuity of flow quantities through the shock wave, where the block-diagonal procedure shows some advantage. It is obvious that this quality is paid with the longer computational time.

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