5. Illustrative Examples
5.4. Investigation of a Complex Acoustic System
In previous sections some elementary systems were considered. In this section a complex example is illustrated: the investigated system is composed of several acoustic elements. The virtual construction is illustrated on figure 5.8:
Figure 5.8.: The virtual construction of a composed acoustic system: ¬ to contraction element, to ® flame element, ® to ¯ uniform tube without viscous loss,
¯ to ° expansion element, ° to ± uniform tube with thermo-viscous loss.
¬to: sudden expansion with the cross-sectional area ratioα12= AA2
1 and the transfer matrix for the Riemann invariants, derived fromTpu, which is given in section 5.3:
Tf g12= 1
to®: acoustic element, which represents a flame. Its transfer matrix for the pressure and velocity fluctuations, derived by W. Polifke [7], is given by
Tpu23=
F(ω)is a flame transfer function:
F(ω) = (1 +a)e−iωτ1−ω
® to ¯: uniform tube with non-viscous flow and the length l34. Its transfer matrix for the Riemann invariants is given in section 5.1:
Tf g34=
¯to °: sudden contraction with the cross-sectional area ratioα45= AA5
° to ±: uniform tube with flow and thermo-viscous loss. The transfer matrix for the Riemann invariants is given in section 5.2:
Tf g56=
This example only serves to show, how a composed element can look like, and it has no practical relevance. It would lead too far to consider any realistic system in this thesis.
The computation is made for frequencies between 20 Hz and 400 Hz. The flow and geometry parameters parameters are listed in appendix A.5.
To keep the entire system as simple as possible, a lot of assumptions are made, even not realistic (for exampleu3 =u4 and p3 =p4).
The transfer matrix of the system is composed from the transfer matrices of the elements:
Fig. 5.9 illustrate the potentialities of the generated acoustic power for the current system in comparison to a flame element alone with the same computation parameters. Both of the systems are different: the generated power spectrum of the flame is larger. Furthermore for frequencies between 100 Hz and 200 Hz the flame element generates much more (up to 5 times more) acoustic power than the compound. Apparently, interaction with other system components cause this variation. That was not really expected, because the behaviour of these systems was already investigated in previous sections, and there is no element, which provokes such radical change. As soon as the system contains several acoustic elements, its behaviour can not be really predicted from the behaviour of each separate element. These elements might interact in an unpredictable way. That is illustrated on figure 5.10, where the potentialities of a current system are compared with the product of the potentialities of each separate element.
Furthermore, the elements, which are fully neutral, if they are alone, can also provoke the acoustic power generation or dissipation. In fig. 5.11 λmax and λmin are shown for different lengths of the tube without viscous loss at the middle of the construction. This tube shifts the phase of the waves, which go through the tube. The phase shifting is a function of the wave frequency and the tube length. The larger the length, the higher the difference in phase
Figure 5.9.: The potentialities of a composed system in comparison to a flame element with the same computation parameters. The flame element alone generates more acoustic power than the entire compound.
shifting for neighbouring frequencies. That results in different system behaviours for these neighbouring frequencies and is illustrated in the above-mentioned figure. For elementary systems the phase shifting can change the location (the ratio of the Riemann invariants) of the maximal value of the generated power but not the value itself. Therefore the potentialities are not affected by the phase shifting. This complex example shows that the phase shifting by an element, which is placed not at the ends of the compound, changes the potentialities.
There is one more example that the interaction of elements can have an unpredictable re-sult. From section refSudden Area Change with Incompressible non-Viscous Flow it is known,
Figure 5.10.: The potentialities of the composed system in comparison to the product of potentialities of each separate element of this system. The difference between the potentialities is caused by the correlation between the elements.
that the expansion systems are passive and dissipate more acoustic power with raisingα. The contraction systems are active and produce more acoustic power if α sinks. These effects are not depending on the wave frequency. That doesn’t hold in general for complex systems.
Figures 5.12 and 5.13 show the potentialities of the generated acoustic power for different cross-sectional area ratios of the contraction and expansion elements. The changes in poten-tialities depend on the acoustic wave frequency, therefore it is not enough only to distinguish
Figure 5.11.: The potentialities of the composed system with different lengths of the tube without viscous loss at the middle of the compound. The upper plot shows λmax and the lower plotλmin. The larger the tube the higher the phase shifting.
That leads to the oscillation of the potentialities over the wave frequency.
the sudden area change between the contraction expansion elements to describe its behaviour.
Fig. 5.14 illustrate ηsi as a function of the Riemann invariants f1 and g6 and their phase difference. The plot is made for the frequency f = 135 Hz, where λmax has the maximal value: λmax = 9.139. The maximum and the minimum of ηsi are achieved not for the waves in phase or out of phase in comparison to the elementary systems, considered before:
(ηsi)max = 9.139 for ϕ= 353.0◦ and gf2
1 = 0.899 (ηsi)min = 9.139 for ϕ= 173.0◦ and gf2
1 = 1.250
Figure 5.12.: The potentialities of the composed system with different α12. The upper plot shows λmax and the lower plot λmin. The variation of α12 causes frequency dependent change of the potentialities.
To distinguish between the dissipated and generated acoustic power the custom colormap is used to create a polar plot (see fig. 5.15). According to λmin = 0.145 the system is not always unstable (λmin = 9.139), it also dissipates a lot of acoustic power (95.5%Pinc).
Nevertheless the area, where the acoustic power dissipation is possible, is much smaller than the acoustic power generation area. Therefore there is a low probability that a such system shows passive behaviour.
Figure 5.13.: The potentialities of the composed system with different α45. The upper plot shows λmax and the lower plot λmin. The variation of α45 causes frequency dependent change of the potentialities.
Concluding, the investigation of a composed system shows that its behaviour can not be completely derived from the behaviour of each separate element because of their interactions.
Therefore it is required to compute the scattering or transfer matrix of the entire system to analyse it. All kinds of the ”predictions” should be considered with scepticism and have to be verified on the entire system.
1.3
Figure 5.14.: The plot of ηsi with the default colormap for the frequency 135 Hz, ||g||f6||
1|| is a radius in the range between 0 and 5.
1.3
Figure 5.15.: The plot of ηsi with the custom colormap for the frequency 135 Hz, ||g||f6||
1|| is a radius in the range between 0 and 5. In the red colored area ηsi >2, the green color denotes the transition area, where ηsi ≈1, the blue colored area illustrate the acoustic power dissipation ηsi<1
6. Conclusion
The computation examples, given in previous chapter show that the developed MATLAB tool can be very useful for the investigation of one-dimensional linear acoustic elements or compounds.
At first, it is possible to obtain quickly the potentialities of the generated acoustic power from the scattering or transfer matrix of an element, if flow parameters are known. Therefore Limits of the acoustic power generation can be determined.
This tool also allows the deeper analysis. It can show the distribution of the scattered acoustic power over the incident acoustic power as a function of the incident Riemann invari-ants or the Riemann invariinvari-ants at the inlet of the system. In this way the constellation of the Riemann invariants can be found, which provokes the required acoustic power generation. In the similar way the value of the generated acoustic power can be found for defined constellation of the Riemann invariants.