5. Illustrative Examples
5.3. Sudden Area Change with Incompressible Non-Viscous FlowNon-Viscous Flow
The dissipated acoustic power of a sudden area change with non-viscous flow The transfer matrix for sudden area change is given by W. Polifke [3]:
Tpu =
1. It holds under following assumptions: absence of mean flow, adiabatic process without viscous loses. In this section the sudden area change is considered in presence of mean flow. To use the above-mentioned transfer matrix as an approximation, only small Mach numbers are considered2. According to these assumptions, the density is constant ρ1 = ρ2 = ρ. To simplify the calculation, it is assumed that the sound of speed remains approximately constant c1 ≈c2 ≈c. As consequence,
p02
The continuity equation delivers a correlation between upstream and downstream velocities:
A1ρ1u1 =A2ρ2u2 =⇒ u2 = u1
α. (5.19)
To compute the dissipated acoustic power eq. (3.2) and eq. (3.4) are used:
Pdiss=P1−P2 =
2There are also further models for sudden area change with mean flow, which can be found in [8]. Nevertheless the analytical computation ofηsi for these models is very complex.
< 02m02 >=<
With this knowledge the computation of dissipated energy can go on:
Pdiss=A1
If there is no mean flow (u1 = 0), the acoustic element is fully neutral (Pdiss= 0). Therefore the computed maximum and minimum of potentiality should be both equal one.
In presence of flow the system behaviour depends on the cross-sectional area ratio α, the intensities of f1 and g2 and their phase difference, because of the mixed time average
< f1g2 >. They determine a sigh ofPdiss, i.e. the behaviour of the system (active or passive).
If the cross-sectional area ratioαis held constantPdisscan still change the sign depending on
f1 and g2. Therefore for eachα there are always potentialities λmax and λmin, where one of them is smaller than 1and another, which is greater than 1.
If αand the intensities off1 and g2 are held constant,Pdissis a function only of the phase difference off1 and g2. < f1g2 > has the maximal (positive) value if the waves are in phase and the minimal (negative) value if the waves are out of phase. This knowledge leads to the following conclusions:
a) If α >1 and the incoming waves are out of phase (phase difference 180), the maximal value of the dissipation is achieved. For the waves in phase the dissipation is minimal.
b) Ifα <1 and the incoming waves are in phase the system produce the maximal amount of acoustic power. For the incoming waves out of phase the acoustic power generation is minimal.
All of these statements are illustrated by a following example.
For the computation with the MATLAB tool, air is chosen as flow medium with following parameters:
Name Variable Value
density ρ 1.225 kg/m3
temperature T 288.15K
specific gas constant R 287.15J/(kgK)
adiabatic index γ 1.4
sound of speed c √
γRT [m/s]
Table 5.2.: Computation parameters.
In the first case with absence of flow potentiality computation.m delivers λmin = λmax = 1 for α = 2. As expected, both of the potentialities are equal 1, so the system is neutral. There is no need to use the function potentiality plot.m, because the transfer matrix is not a function of frequency. That meansλmax and λmin have the same value for all of the frequencies.
Now, the same system with α= 2 is investigated in presence of flow. The Mach number at the inlet is set to 0.01. The flow velocity at the inlet is u1 = M1c and at the outlet u2 =u1/α. This leads to:
λmax ≈1.00000025 λmin ≈0.9802
The system is not neutral. It can be passive or active, as expected after considering eq. (5.25). As already mentioned, the results are for all acoustic wave frequencies the same.
To investigate the system, a plot of the potentiality versus the frequency is not really useful.
Figure 5.4.: The potentialities versus the cross-sectional area ratio α for M1 = 0.01. The system is active for α <1 and passive forα <1.
Hence a plot of the potentiality versus the cross-sectional area ratio α is more meaningful, which is illustrated in fig. 5.4. For α < 1 λmax >1 and for α > 1 λmin < 1. The remaining potentialities are approximately one3. According to eq. (5.25) the mean flow velocity doesn’t change the sign of Pdiss, it serves only as an amplification factor. Therefore fig 5.4 enables statements about the system in general and not only forM1 = 0.01. Concluding, the behaviour of the system depends deeply on α. According to the mean flow direction, the contraction process (α <1) is active and expansion process is passive. Furthermore, with the increasing contraction or expansion, the dominating behaviour of the system is more and more amplified.
The mean flow velocity, as already mentioned, is an amplification factor. In fig. 5.5 there is a plot of the dominating potentialities (different from 1) versus the Mach number at the inlet for cross-sectional area ratios α = 0.5 and α = 2. The recessive potentialities are not illustrated, because they are weak influenced by any change in the mean flow and remain equal to 1. The figure shows that the mean flow amplifies the dominating potentialities. Therefore contracting systems generate more energy and expanding systems dissipate more energy, if the Mach number raises.
Using the function pgen plot f1g2.m, ηsi is visualised as a function of the amplitudes of the incoming Riemann invariants ||g||f2||
1|| and their phase difference ϕ. The polar plot is created for M1 = 0.01 and α = 2 (see Figure 5.6). In contrast to the uniform tube with flow and thermo-viscous loses from section 5.2, the behaviour of this system depends on the phase difference of the incoming waves. If the waves are in phase, the system is almost neutral.
3For α < 1 λmin ≈ 1− and for α > 1 λmax ≈ 1+. Therefore, theoretically, the system has for each frequency one potentiality bigger than one and another smaller than one, which is consistent with the analytical solution.
Figure 5.5.: The dominating potentialities (different from 1) versus the Mach number at the inlet for two different cross-sectional area ratios. the mean flow amplifies the potentialities.
1.3 2.5
3.8 5
30°
210°
60°
240°
90°
270° 120°
300° 150°
330°
180° 0°
0.982 0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 Pscat / P
inc
Figure 5.6.: ηsi of an expanding element with α = 2 and M1 = 0.01. The radius is ||g||f2||
1|| in the range between 0 and 5, the angle is the phase difference ϕ between g2 and f1. The system is almost neutral for the acoustic waves in phase and passive for the waves out of phase.
If the waves are out of phase and the amplitudes of the incident Riemann invariants are approximately equal, the system dissipates the maximal value of acoustic power.
The plot of ηsi of an expansion system with α = 0.5 and M1 = 0.005 is illustrated on figure 5.7. The system is the active for the waves out of phase. The maximum of the generated
1.3 2.5
3.8 5
30°
210°
60°
240°
90°
270° 120°
300° 150°
330°
180° 0°
1 1.002 1.004 1.006 1.008 1.01 1.012 1.014 1.016 1.018 1.02 Pscat / P
inc
Figure 5.7.: ηsi of a contracting element with α = 0.5 and M1 = 0.005. The radius is ||g||f2||
1||
with the range between 0 and 5, the angle is the phase difference ϕ between g2 and f1. The system is almost neutral for the acoustic waves in phase and active for the waves out of phase.
acoustic power is achieved, if the amplitude of the upstream propagating wave g2 is about two times higher than the amplitude of the downstream propagating wavef1. Forϕ= 0◦ the system is almost neutral.
The investigation of the sudden area change with non-viscous mean flow shows that the contraction systems (α < 1) are active and expansion systems (α > 1) are passive. The dominating behaviour of the system is amplified with the raising Mach number and with further contraction or expansion, depending on the case. Finally, the contraction system generates the most of the acoustic power and the expansion system dissipates the most of the acoustic power, if the incoming waves are out of phase.