JET NOISE MODELS FOR FORCED MIXER NOISE PREDICTIONS A Thesis Submitted to the Faculty of Purdue University by Loren A. Garrison In Partial Fulfillment of the Requirements for the Degree of Masters of Science in Aeronautics and Astronautics December 2003
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JET NOISE MODELS FOR FORCED MIXER NOISE PREDICTIONS
A Thesis
Submitted to the Faculty
of
Purdue University
by
Loren A. Garrison
In Partial Fulfillment of the
Requirements for the Degree
of
Masters of Science in Aeronautics and Astronautics
December 2003
ii
ACKNOWLEDGMENTS
I would like to thank Professor Tasos Lyrintzis and Professor Greg Blasidell for
giving me the opportunity to work on this project and for their leadership and
guidance. The work summarized in this thesis is part of a joint effort with the
Rolls-Royce Corporation, Indianapolis and has been sponsored by the Indiana 21st
Century Research and Technology Fund. I would also like to thank Bill Dalton at
the Rolls-Royce Corporation, Indianapolis for his many valuable discussions and for
providing the technical data and the experimental acoustic data used in this research.
I would like to thank Professor Stuart Bolton for serving on my advisory committee.
I would like to thank Dr. Rod Self, Dr. Brian Tester, and Prof. Mike Fisher at the
Institute of Sound and Vibration Research for both their guidance while I studied
there, and for their valuable advice and suggestions throughout my research. I would
like to thank my colleague Ali Uzun for his help and assistance.
where SPLsu refers to the noise from the upstream secondary jet source and SPL
refers to a single jet prediction based on the secondary flow values, Vs, Ts, and
Ds. Similar to the previous source model, a spectral filter is applied to eliminate
the low frequency part of the single jet prediction, and the overall source levels are
augmented by the source strength term ∆dBsu.
For simplicity, it is assumed that the cut-off Strouhal numbers of the low fre-
quency and high frequency sources are equal. The cut-off frequency, fc, can be
calculated from the cut-off Strouhal number, Stc, through the relation
fc = StcV
D(5.7)
40
It is expected that Model 1, which uses a secondary jet to represent the upstream
portion of the actual jet plume, will produce better predictions for the low penetra-
tion mixer. Likewise, it is expected that Model 2, which uses a mixed jet to represent
the upstream portion of the actual jet plume, will produce better predictions for the
high penetration mixer. These trends are expected due to the fact that for the case
of the high penetration mixer the stronger stream wise vortices result in increased
mixing inside the nozzle. As a result as the penetration is increased, the flow at
the final nozzle exit will be more characteristic of a fully mixed. Similarly, as the
penetration decreases the impact of the stream wise vortices decreases resulting in a
flow that will be more characteristic of a secondary jet.
5.3 Two-Source Model Parameter Optimization
The proposed Two-Source model has three variable parameters, the low fre-
quency source reduction, ∆dBmd, the high frequency source augmentation, ∆dBmu
or ∆dBsu, and the common cut-off Strouhal number, Stc. The optimum values of
these variable parameters for a given mixer and nozzle geometry are determined em-
pirically through the use of a non-linear least squares optimization method. In this
method the best set of variable parameters are found which minimize the weighted
errors between the model prediction and the experimental data. This process essen-
tially curve-fits the experimental data using the Two-Source model. The optimized
parameters that result from this analysis can then be correlated with the changes in
the mixer geometry, namely the amount of penetration.
The non-linear least squares optimization is performed using MATLAB’s lsqnonlin
function. This routine uses a Levenberg-Marquardt method for minimizing the er-
rors between the model prediction and the experimental data. This non-linear least
squares optimization routine is used to find the optimum source strength parameters
for a given cut-off Strouhal number. This process is repeated for a range of cut-off
Strouhal numbers to find the set of optimized parameters which yields the lowest er-
41
ror. This exhaustive type of approach for determining the optimum cut-off Strouhal
number is necessary because of the non-linear nature of the filter functions and the
averaged weighted error, which cause difficulties due to both solution non-uniqueness
and the presence of local minima.
The errors between the model predictions and the experimental data are evalu-
ated for a range of angles from 90◦ to 150◦ from the intake axis, in 5◦ increments.
This process results in approximately 400 error values. At each angular location
these error values are weighted based on the experimental data spectra using the
weighting function
Ew (θi, f) = 10[0.1(SPLexp(θi,f)−[SPLexp(θi,f)]max)] (5.8)
This weighting function has a value of 1 at the peak of the experimental data, and
approaches 0 as the differences between a given experimental Sound Pressure Level
value and peak Sound Pressure Level in the experimental data spectrum approach
infinity. This weighting, which is similar to the one implicit in the calculation of the
Overall Sound Pressure Level, will weigh the errors in the predicted Sound Pressure
Level that are closer to the peak in the experimental data more heavily. A Perceived
Noise Level (PNL) type of weighting could also be used to weight the sound pressure
level errors. However, since the differences between these two weightings are expected
to be small, in this study the OASPL type will be used for simplicity.
5.4 Two-Source Model Results
In the following section the performance of two different the Two-Source models
are evaluated. The variable parameters in these models are optimized so that the pre-
dictions best match the experimental data. This optimization process is performed
at three different operating set points for each of the three forced mixer designs. The
resulting optimized parameters are then correlated back to the geometric differences
in the forced mixer designs.
42
5.4.1 Model 1 Results
The first Two-Source model that is evaluated is the Model 1. Using this model,
the upstream portion of the jet plume is modeled as a single stream fully mixed jet.
A spectral filter, which eliminates the low frequency region, is applied to the fully
mixed jet noise spectrum. The downstream portion of the jet plume is also modeled
as a single stream fully mixed jet. However, a spectral filter that eliminates the high
frequency part of the single stream noise spectrum is applied to this noise source.
The same cut-off Strouhal number is used in both spectral filters. In addition, each
of the two noise sources has a variable source strength term which shifts the entire
spectrum up or down.
Low Penetration Mixer
The results of the first step in the parameter optimization process for the low
penetration mixer at Set Point 1 are shown in Figures 5.2, 5.3, and 5.4. In Figure
5.2 the maximum error, average error, and average weighted errors are plotted as
a function of the cut-off Strouhal number. The circles on these plots show the
location of the minimum error for each type of error. It is seen here that each type
of error is minimized for different values of the cut-off Strouhal number. As a result
the optimum parameters for this configuration, as well as those for all subsequent
configurations, will be dependent on which metric is used to determine the optimum
criterion. For this study, the average weighted error is used as the metric to determine
the optimum criterion. This metric is chosen because it provides the best measure of
how well the prediction agrees with experimental data from an acoustics standpoint.
One of the difficulties of the parameter optimization problem is the existence of
non-unique solutions. This problem occurs in the parameter optimization process
when the errors in Figure 5.2 are relatively constant for a large range of cut-off
Strouhal numbers. When this condition occurs, there are multiple solutions to the
optimization process that yield roughly the same error. As a result, it is then not
43
obvious which set of parameters should be later used to correlate to the differences
in the mixer design. To overcome some of the non-linear behavior problems that
result from the optimization of the Two-Source model, an exhaustive type of search
is used to determine the optimum cut-off Strouhal number.
100
101
0
2
4
6
8
10
12
Stc [ ]
Err
or [d
B]
MaximumAverageWeighted
Figure 5.2. Model 1 Parameter Optimization Error Results for theLow Penetration Mixer at Set Point 1
In Figure 5.3 the non-dimensional errors resulting the parameter optimization
for the low penetration mixer at Set Point 1 are shown. These errors are the same
as those plotted in Figure 5.2, except they have been normalized by their respective
maximum values so that the behavior of the errors can be more clearly seen. The
problem of local minima is seen in Figure 5.3 for the averaged weighted error. The oc-
currence of local minima in the averaged weighted error is one reason that warranted
the need for the exhaustive type of search to determine the optimum cut-off Strouhal
number. Figure 5.4 shows the corresponding optimized source strengths for both the
upstream mixed jet and the downstream mixed jet sources. In addition, the vertical
dotted line signifies the optimum cut-off Strouhal number for this test case, which
44
100
101
0.4
0.5
0.6
0.7
0.8
0.9
1
Stc [ ]
Nor
mal
ized
Err
or
MaximumAverageWeighted
Figure 5.3. Model 1 Parameter Optimization Non-Dimensional ErrorResults for the Low Penetration Mixer at Set Point 1
100
101
0
5
10
15
20
25
Stc [ ]
∆ dB
[dB
]
Upstream JetDownstream Jet
Figure 5.4. Model 1 Parameter Optimization Results for the LowPenetration Mixer at Set Point 1
45
was determined from Figure 5.3. The optimum source strengths for this test case
are then taken to be those that resulted from the optimum cut-off Strouhal number.
An interesting point to note is that as the cut-off Strouhal number approaches the
lower bound, the noise prediction essentially consists of only the upstream single jet
source. Likewise, as the cut-off Strouhal number approaches the upper bound, the
noise prediction essentially consists of only the downstream single jet source.
The Two-Source model prediction using the optimized parameters for the low
penetration mixer at Set Point 1 is shown in Figure 5.5. It is seen from this figure
the optimized prediction agrees well with the experimental data. However, there are
some deviations present at angles close to the jet axis where the the predictions are
slightly under-predicted near the spectral peak. It is expected that these predictions
agree well since they were essentially curve-fit to match the experimental data. The
fact the optimized predictions are in agreement suggests that the Two-Source model
contains the necessary physics to model the noise from the forced mixer.
This optimization process was repeated for the low penetration mixer at the two
additional Set Points. The normalized error and optimized parameter results for Set
point 2 are shown in Figures 5.6 and 5.7. Similarly, the corresponding results for Set
point 3 are shown in Figures 5.8 and 5.9.
Once this process was completed for all three Set Points, the three corresponding
averaged weighted error curves for the low penetration mixer are combined onto
one graph, as shown in Figure 5.10. The averaged weighted errors from each Set
Point are then averaged again to yield an error that is representative of all three Set
Points. The final optimum cut-off Strouhal number for this mixer is then chosen
based on this metric. The corresponding optimized source strengths for all three Set
Points are shown in Figure 5.11. Once the final optimum cut-off Strouhal number is
determined, the final optimum source strength parameters are found by averaging the
source strength values for the three Set Points at the final optimum cut-off Strouhal
number.
46
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
2 dBExperimentalMM Optimizied
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
2 dBExperimentalMM Optimizied
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
2 dBExperimentalMM Optimizied
Figure 5.5. Model 1 Optimized Predictions for the Low PenetrationMixer at Set Point 1
47
10−1
100
101
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stc [ ]
Nor
mal
ized
Err
or
MaximumAverageWeighted
Figure 5.6. Model 1 Parameter Optimization Non-Dimensional ErrorResults for the Low Penetration Mixer at Set Point 2
10−1
100
101
0
5
10
15
20
25
Stc [ ]
∆ dB
[dB
]
Upstream JetDownstream Jet
Figure 5.7. Model 1 Parameter Optimization Results for the LowPenetration Mixer at Set Point 2
48
10−1
100
101
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stc [ ]
Nor
mal
ized
Err
or
MaximumAverageWeighted
Figure 5.8. Model 1 Parameter Optimization Non-Dimensional ErrorResults for the Low Penetration Mixer at Set Point 3
10−1
100
101
0
5
10
15
20
25
30
Stc [ ]
∆ dB
[dB
]
Upstream JetDownstream Jet
Figure 5.9. Model 1 Parameter Optimization Results for the LowPenetration Mixer at Set Point 3
49
It is seen in Figure 5.11 that for this particular case there is a fairly large difference
in the upstream jet source strengths between the three Set Points at the optimum
Strouhal number. Ideally, the source strength parameters should collapse on one
another for all three Set Points. This result is expected since changes in aerodynamic
conditions are accounted for in the single jet predictions. The fact that there is a
discrepancy in the upstream jet source for the low penetration mixer suggests that
the Two-Source model does not contain all of the components necessary to model the
physics of the jet with the low penetration mixer. Consequently, an additional source
might be needed to model this case. Alternatively, this discrepancy could result if
the single jet characteristics are not representative of the actual flow field properties
in the jet plume. However, the only way to evaluate these differences would be to
analyze the experimental aerodynamic data in the full jet plume. Fortunately, the
optimized source strength results for the other two forced mixers do collapse fairly
well with respect to the three operating Set Points.
The final optimum parameters will later be used in the parameter correlation
process, described in Section 5.4.3. The optimized parameters for each Set Point
and the final set of optimized parameters for the low penetration mixer are given in
Table 5.1.
Table 5.1 Model 1 Optimized Parameters for the Low Penetration Mixer
Case ∆dBum ∆dBdm Stc
Set Pt 1 3.998 1.668 3.440
Set Pt 2 4.794 1.946 3.792
Set Pt 3 6.357 1.681 5.386
Final 5.050 1.765 4.331
50
10−1
100
101
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Cut−off Strouhal Number []
Wei
ghte
d E
rror
[dB
]
High PowerMid PowerLow PowerAverage
Figure 5.10. Model 1 Parameter Optimization Average WeightedError Results for the Low Penetration Mixer at Set Points 1, 2 and3
10−1
100
101
0
5
10
15
20
25
30
Cut−Off Strouhal Number
Opt
imiz
ed S
ourc
e S
tren
gths
∆ d
B [d
B]
Upstream JetDownStream Jet
Figure 5.11. Model 1 Parameter Optimization Results for the LowPenetration Mixer at Set Points 1, 2 and 3
51
Intermediate Penetration Mixer
The optimization described in the previous section is repeated for the interme-
diate penetration mixer. The resulting averaged weighted error curves are shown in
Figure 5.12. Once again, the final optimum cut-off Strouhal number corresponds to
the location of the minimum of the averaged weighted error curve. The correspond-
ing optimized source strengths for the intermediate penetration mixer at all three
Set Points are shown in Figure 5.13. For this configuration, only a small variability
of the source strengths with respect to the operating condition is seen, which implies
that the flow physics are well represented by the two-source model. Once again,
after the final optimum cut-off Strouhal number is determined for this mixer, the
final optimum source strength parameters are found by averaging the source strength
values for the three Set Points at the final optimum cut-off Strouhal number. These
final parameters will later be used in the parameter correlation process. The opti-
mized parameters for each set point and the final set of optimized parameters for
the intermediate penetration mixer are given in Table 5.2.
Table 5.2 Model 1 Optimized Parameters for the Intermediate Penetration Mixer
Case ∆dBum ∆dBdm Stc
Set Pt 1 7.251 0.284 3.372
Set Pt 2 7.643 0.379 3.706
Set Pt 3 7.911 0.515 5.245
Final 7.601 0.393 4.245
52
10−1
100
101
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cut−off Strouhal Number []
Wei
ghte
d E
rror
[dB
]
High PowerMid PowerLow PowerAverage
Figure 5.12. Model 1 Parameter Optimization Average WeightedError Results for the Intermediate Penetration Mixer at Set Points1, 2 and 3
10−1
100
101
−5
0
5
10
15
20
25
30
35
Cut−Off Strouhal Number
Opt
imiz
ed S
ourc
e S
tren
gths
∆ d
B [d
B]
Upstream JetDownStream Jet
Figure 5.13. Model 1 Parameter Optimization Results for the Inter-mediate Penetration Mixer at Set Points 1, 2 and 3
53
High Penetration Mixer
The previously described optimization process is again repeated for the high pen-
etration mixer. The resulting averaged weighted error curves are shown in Figure
5.14. Once again, the final optimum cut-off Strouhal number corresponds to the
location of the minimum of the averaged weighted error curve. The corresponding
optimized source strengths for the high penetration mixer at all three Set Points are
shown in Figure 5.15. It is once again seen that there us only a small variability in
the source strengths with respect to the operating point for this configuration. After
the final optimum cut-off Strouhal number is determined for this mixer, the final op-
timum source strength parameters are found by averaging the source strength values
for the three Set Points at the final optimum cut-off Strouhal number. These final
parameters will later be used in the parameter correlation process. The optimized
parameters for each Set Point and the final set of optimized parameters for the high
penetration mixer are given in Table 5.3.
Table 5.3 Model 1 Optimized Parameters for the High Penetration Mixer
Case ∆dBum ∆dBdm Stc
Set Pt 1 7.801 -0.283 3.443
Set Pt 2 8.101 0.123 3.795
Set Pt 3 8.105 0.200 4.283
Final 8.005 0.013 3.443
54
10−1
100
101
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cut−off Strouhal Number []
Wei
ghte
d E
rror
[dB
]
High PowerMid PowerLow PowerAverage
Figure 5.14. Model 1 Parameter Optimization Average WeightedError Results for the High Penetration Mixer at Set Points 1, 2 and3
10−1
100
101
−5
0
5
10
15
20
25
30
35
Cut−Off Strouhal Number
Opt
imiz
ed S
ourc
e S
tren
gths
∆ d
B [d
B]
Upstream JetDownStream Jet
Figure 5.15. Model 1 Parameter Optimization Results for the HighPenetration Mixer at Set Points 1, 2 and 3
55
5.4.2 Model 2 Results
The second Two-Source model, Model 2, represents the upstream portion of the
jet plume using as a single stream secondary jet. A spectral filter, which eliminates
the low frequency region, is applied to the secondary jet noise spectrum. The down-
stream portion of the jet plume is modeled as a single stream fully mixed jet. A
spectral filter that eliminates the high frequency part of the single stream noise spec-
trum is applied to this noise source. The same cut-off Strouhal number is used in
both spectral filters. In addition, each of the two noise sources has a variable source
strength term which shifts the entire spectrum up or down.
Low Penetration Mixer
The same optimization process that was used with Model 1 is also used here with
the Model 2. The resulting averaged weighted error curves for the low penetration
mixer are shown in Figure 5.16. The final optimum cut-off Strouhal number corre-
sponds to the location of the minimum of the averaged weighted error curve. The
corresponding optimized source strengths for the low penetration mixer at all three
Set Points is shown in Figure 5.17. It is seen from Figure 5.17 that the variability
in the optimized source strength terms with respect to the Set Points using Model 2
are similar to those obtained with Model 1. After the final optimum cut-off Strouhal
number is determined for this mixer, the final optimum source strength parameters
are found by averaging the source strength values for the three Set Points at the
final optimum cut-off Strouhal number. These final parameters will later be used in
the parameter correlation process. The optimized parameters for each Set Point and
the final set of optimized parameters for the low penetration mixer based on Model
2 are given in Table 5.4.
56
10−1
100
101
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Cut−off Strouhal Number []
Wei
ghte
d E
rror
[dB
]
High PowerMid PowerLow PowerAverage
Figure 5.16. Model 2 Parameter Optimization Average WeightedError Results for the Low Penetration Mixer at Set Points 1, 2 and3
10−1
100
101
0
5
10
15
20
25
30
35
Cut−Off Strouhal Number
Opt
imiz
ed S
ourc
e S
tren
gths
∆ d
B [d
B]
Upstream JetDownStream Jet
Figure 5.17. Model 2 Parameter Optimization Results for the LowPenetration Mixer at Set Points 1, 2 and 3
57
Table 5.4 Model 2 Optimized Parameters for the Low Penetration Mixer
Case ∆dBus ∆dBdm Stc
Set Pt 1 7.788 1.671 4.331
Set Pt 2 8.560 1.983 3.792
Set Pt 3 10.474 1.707 5.386
Final 8.941 1.787 5.452
58
Intermediate Penetration Mixer
The averaged weighted error curves for the intermediate penetration mixer that
resulted from the parameter optimization process are shown in Figure 5.18. In addi-
tion, the corresponding optimized source strengths for the intermediate penetration
mixer at all three Set Points is shown in Figure 5.19. It is seen from Figure 5.19
that much like the results from Model 1, the optimum source strength curves at all
three Set Points for the intermediate penetration mixer collapse on one another. The
optimized parameters for each Set Point and the final set of optimized parameters
for the intermediate penetration mixer are given in Table 5.5.
10−1
100
101
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Cut−off Strouhal Number []
Wei
ghte
d E
rror
[dB
]
High PowerMid PowerLow PowerAverage
Figure 5.18. Model 2 Parameter Optimization Average WeightedError Results for the Intermediate Penetration Mixer at Set Points1, 2 and 3
59
10−1
100
101
−5
0
5
10
15
20
25
30
35
40
Cut−Off Strouhal Number
Opt
imiz
ed S
ourc
e S
tren
gths
∆ d
B [d
B]
Upstream JetDownStream Jet
Figure 5.19. Model 2 Parameter Optimization Results for the Inter-mediate Penetration Mixer at Set Points 1, 2 and 3
Table 5.5 Model 2 Optimized Parameters for the Intermediate Penetration Mixer
Case ∆dBus ∆dBdm Stc
Set Pt 1 10.508 0.291 3.372
Set Pt 2 10.969 0.360 3.706
Set Pt 3 11.089 0.525 5.245
Final 10.855 0.392 4.245
60
High Penetration Mixer
The resulting averaged weighted error curves for the intermediate penetration
mixer are shown in Figure 5.20. In addition, the corresponding optimized source
strengths for this mixer at all three Set Points is shown in Figure 5.21. Similar to
the results from Model 1, with Model 2 the optimized source strengths terms at all
three Set Points collapse for the high penetration mixer. The optimized parameters
for each Set Point and the final set of optimized parameters for the high penetration
mixer are given in Table 5.6.
10−1
100
101
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cut−off Strouhal Number []
Wei
ghte
d E
rror
[dB
]
High PowerMid PowerLow PowerAverage
Figure 5.20. Model 2 Parameter Optimization Average WeightedError Results for the High Penetration Mixer at Set Points 1, 2 and3
61
10−1
100
101
−5
0
5
10
15
20
25
30
35
40
Cut−Off Strouhal Number
Opt
imiz
ed S
ourc
e S
tren
gths
∆ d
B [d
B]
Upstream JetDownStream Jet
Figure 5.21. Model 2 Parameter Optimization Results for the HighPenetration Mixer at Set Points 1, 2 and 3
Table 5.6 Model 2 Optimized Parameters for the High Penetration Mixer
Case ∆dBus ∆dBdm Stc
Set Pt 1 11.256 -0.370 3.443
Set Pt 2 11.399 0.115 3.795
Set Pt 3 11.491 0.197 4.283
Final 11.382 -0.020 3.443
62
5.4.3 Parameter Correlations
Once a fixed set of optimized parameters are determined for each forced mixer
geometry, these parameters can then be correlated to the geometric differences in
the force mixer designs. For this particular family of forced mixers, the primary
geometric difference is the lobe height, or amount of penetration (H). This parameter
is non-dimensionalized by the final nozzle diameter (Dnozzle), and is then correlated
to the optimized parameters. This parameter correlation process is applied to the
results from each of the Two-Source models.
Model 1 Correlations
The optimized parameters for each mixer geometry using Model 1 are shown
in Table 5.7. These optimized parameters are plotted versus the non-dimensional
mixer penetration values in Figures 5.22 and 5.23. It is seen from Figure 5.22 that the
source strength for the single stream fully mixed jet that represents the downstream
portion of the actual jet plume exhibits a linear behavior. In addition, the source
strength for the upstream jet source exhibits approximately a linear behavior. Based
on this result, a linear curve-fit can be applied to each of the source strength curves.
The resulting coefficients for this linear curve-fit are given in Table 5.8.
The variation in the optimum cut-off Strouhal number, Stc, in Figure 5.23 is
fairly small when compared the full range of Strouhal numbers in the experimental
data. In fact the difference between a Strouhal number of 3.4 and 4.4 approximately
corresponds to a span of one 1/3 Octave band. As a result, the variation in the
Strouhal number from 3.4 to 4.4 will have a little effect on the overall noise prediction.
Consequently, a constant cut-off Strouhal number can be used. This final cut-off
Strouhal number is found by simply averaging the values for the three mixer designs.
For this Two-Source model, the cut-off Strouhal number will have a value of 4.01.
63
Table 5.7 Final Optimized Parameters for Model 1
Mixer ID HDnozzle
∆dBum ∆dBdm Stc
12CL 0.1994 5.050 1.765 4.331
12UM 0.2602 7.601 0.393 4.245
12UH 0.2801 8.005 0.013 3.443
0.18 0.2 0.22 0.24 0.26 0.28 0.3−1
0
1
2
3
4
5
6
7
8
9
10
Sou
rce
Str
engt
h ∆d
B [d
B]
Upstream JetDownstream Jet
Figure 5.22. Model 1 Optimized Parameter Correlation of the Source Strengths
Table 5.8 Coefficients from the Linear Curve-fit of the Results from Model 1
Source Strength Slope (A) Intercept (b)
∆dBum 37.887 -2.457
∆dBdm -21.917 6.128
64
0.18 0.2 0.22 0.24 0.26 0.28 0.310
−1
100
101
Lobe Penetration / Nozzle Diameter
Cut
−O
ff S
trou
hal N
umbe
r
Figure 5.23. Model 1 Optimized Parameter Correlation of the Cut-offStrouhal Number
65
Model 2 Correlations
The optimized parameters for each mixer geometry using Model 2 are shown in
Table 5.9. These optimized parameters are plotted versus the non-dimensional mixer
penetration values in Figures 5.24 and 5.25. It is seen from Figure 5.24 that the both
source strengths exhibit a linear behavior. Based on this result a linear curve-fit can
be applied to each of the source strength curves. The resulting coefficients for this
linear curve-fit are given in Table 5.10.
Once again it is seen from Figure 5.25 that the cut-off Strouhal number varies over
a relatively small range for all three mixer designs. For this case, the cut-off Strouhal
number ranges from 3.5 to 5.5, which approximately corresponds to two 1/3 Octave
bands. Once again, the effects of this variation on the noise prediction will be fairly
small. As a result, constant cut-off Strouhal number is also assumed for this Two-
Source model. This final cut-off Strouhal number, which is the average of the values
from the three mixers, has a value of 4.38 for this model. As an alternative approach,
since the behavior of the cut-off Strouhal number for this model is approximately
linear, a linear curve-fit could also be used to find the cut-off Strouhal number.
Table 5.9 Final Optimized Parameters for Model 2
Mixer ID HDnozzle
∆dBus ∆dBdm Stc
12CL 0.1994 8.941 1.787 5.452
12UM 0.2602 10.855 0.392 4.245
12UH 0.2801 11.382 -0.019 3.443
66
Table 5.10 Coefficients from the Linear Curve-fit of the Results from Model 2
Source Strength Slope (A) Intercept (b)
∆dBus 30.550 2.859
∆dBdm -22.522 6.274
0.18 0.2 0.22 0.24 0.26 0.28 0.3
0
2
4
6
8
10
12
Sou
rce
Str
engt
h ∆d
B [d
B]
Upstream JetDownstream Jet
Figure 5.24. Model 2 Optimized Parameter Correlation of the Source Strengths
67
0.18 0.2 0.22 0.24 0.26 0.28 0.310
−1
100
101
Lobe Penetration / Nozzle Diameter
Cut
−O
ff S
trou
hal N
umbe
r
Figure 5.25. Model 2 Optimized Parameter Correlation of the Cut-offStrouhal Number
68
5.4.4 Two-Source Model Performance
In the following section the final optimized parameter correlations are evaluated
for the nine data points from which they were developed. It is important to note
that through the optimization process, a number of intermediate steps required av-
eraging of various optimized results. As a result of this process, the final prediction
method will not agree with the experimental data as well as some of the results from
the intermediate steps. In this section errors from predictions using the final fixed
parameters will be compared to errors from the optimized Two-Source solutions for
each data point. In addition, these errors will also be compared to the errors that
would result from using both a coaxial jet and a single jet prediction.
Model 1 Performance
The average weighted errors for four different prediction methods are given in
Table 5.11. The Two-Source Optimized prediction corresponds to the Two-Source
(Model 1) prediction which was optimized to best match the particular data point.
The Fixed Parameters prediction corresponds to a Two-Source (Model 1) predic-
tion made using the parameters which result from the parameter correlation linear
curve-fits described in Section 5.4.3. The Coaxial Jet prediction corresponds to a
prediction made using the Four-Source method with the jet properties based ’Equiv-
alent Coaxial’ jet. The Single Jet prediction corresponds to a prediction made using
a single stream fully mixed jet with the final nozzle exit diameter. In addition to
the averaged weighted errors, the average errors and maximum errors using these
predictions are given in Tables 5.12 and 5.13 respectively. The most important of
these errors are the average weighted errors since they best describe how well the pre-
dictions match the experimental data from an acoustics standpoint. It is seen from
Table 5.11 that the Two-Source model predictions best match the experimental data
for all three forced mixers at all three Set Points. In addition, it is noted that only
a small amount of error is introduced to the Two-Source model predictions as a re-
69
sult of the parameter optimization process. Through this process a number of steps
required either averaging a set of quantities or curve fitting quantities. As a result,
the Two-Source model predictions made using the final correlated parameters will
not match the experimental data as well as they could if the model was optimized
for just one specific data point. Fortunately, it is seen that errors introduced by
the optimization process are small compared to the difference in the errors of other
current jet noise prediction methods. In general, it is noted that the Two-Source
model predictions produce the best match to the experimental data, followed by the
Four-Source prediction, and finally the single jet prediction.
The Sound Pressure Level spectra predictions for the three forced mixers at all
three data points using Model 1 are given in Figures 5.26 through 5.34. It is seen
from these Figures that for all nine data points the predictions at angles close to
90◦ are in strong agreement with the experimental data. However, similar to the
confluent mixer predictions, the spectrum peak is often slightly under-predicted at
angles close to the jet axis. In addition, for all three forced mixers some deficiencies
in the predictions are seen at the high power Set Point at angles close to the jet
axis. In these cases it appears as if there is an additional high frequency noise source
that is not modeled by the Two-Source formulation. It has been suggested that this
excess noise may be due to sources other than those related to jet mixing. As a
result, at this time no efforts have been made to account for this additional source
in the current forced mixer noise models.
70
Table 5.11 Average Weighted Errors in dB for Model 1
Mixer Set Two-Source Fixed Coaxial Single
ID Point Optimized Parameters Jet Jet
12CL 1 0.31 0.35 0.41 0.64
12CL 2 0.38 0.38 0.56 0.86
12CL 3 0.48 0.53 0.82 0.96
12UM 1 0.26 0.29 0.93 0.68
12UM 2 0.35 0.37 0.98 0.82
12UM 3 0.46 0.52 1.04 0.98
12UH 1 0.27 0.28 1.24 0.96
12UH 2 0.36 0.41 1.30 1.09
12UH 3 0.45 0.55 1.29 1.12
Table 5.12 Average Errors in dB for Model 1
Mixer Set Two-Source Fixed Coaxial Single
ID Point Optimized Parameters Jet Jet
12CL 1 1.35 1.42 2.56 2.53
12CL 2 1.32 1.35 2.54 2.60
12CL 3 1.98 2.02 3.16 3.40
12UM 1 1.09 1.37 3.61 2.85
12UM 2 1.41 1.38 3.65 3.02
12UM 3 1.75 2.00 4.33 4.41
12UH 1 1.08 1.12 4.87 4.06
12UH 2 1.24 1.28 4.97 4.18
12UH 3 1.52 1.81 4.81 4.66
71
Table 5.13 Maximum Errors in dB for Model 1
Mixer Set Two-Source Fixed Coaxial Single
ID Point Optimized Parameters Jet Jet
12CL 1 9.30 7.86 13.18 12.02
12CL 2 6.20 5.90 10.74 9.87
12CL 3 7.09 10.12 16.70 14.31
12UM 1 8.35 7.12 14.48 13.56
12UM 2 5.67 6.15 11.32 11.25
12UM 3 7.46 9.51 18.46 16.22
12UH 1 6.14 5.78 14.13 13.00
12UH 2 7.38 6.84 13.04 12.45
12UH 3 5.73 8.24 17.88 15.58
72
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.26. Model 1 Predictions for the Low Penetration Mixer at Set Point 1
73
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.27. Model 1 Predictions for the Low Penetration Mixer at Set Point 2
74
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.28. Model 1 Predictions for the Low Penetration Mixer at Set Point 3
75
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.29. Model 1 Predictions for the Intermediate PenetrationMixer at Set Point 1
76
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.30. Model 1 Predictions for the Intermediate PenetrationMixer at Set Point 2
77
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.31. Model 1 Predictions for the Intermediate PenetrationMixer at Set Point 3
78
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.32. Model 1 Predictions for the High Penetration Mixer at Set Point 1
79
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.33. Model 1 Predictions for the High Penetration Mixer at Set Point 2
80
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 1 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 1 Optimized
Figure 5.34. Model 1 Predictions for the High Penetration Mixer at Set Point 3
81
Model 2 Performance
The average weighted errors for the four different prediction methods are given
in Table 5.14. The Two-Source Optimized prediction corresponds to the Two-Source
(Model 2) prediction which was optimized to best match the particular data point.
The Fixed Parameters prediction corresponds to a Two-Source (Model 2) prediction
made using the parameters which result from the parameter correlation linear curve-
fits described in Section 5.4.3. The Coaxial Jet prediction corresponds to a prediction
made using the Four-Source method. The Single Jet prediction corresponds to a
prediction made using a single stream fully mixed jet. In addition to the averaged
weighted errors, the average errors and maximum errors using these predictions are
given in Table 5.15 and Table 5.16 respectively. Similar to the Model 1 performance,
it is seen from Table 5.14 that the Two-Source model predictions best match the
experimental data for all three forced mixers at all three Set Points. In addition, it
is noted that once again only a small amount of error is introduced to the Two-Source
model predictions as a result of the parameter optimization process. It is seen that
errors introduced by the optimization process of Model 2 are small compared to the
difference in the errors of other current jet noise prediction methods. In general it
is again noted for Model 2 that the Two-Source model predictions produce the best
match to the experimental data, followed by the Four-Source prediction, and finally
the single jet prediction.
The Sound Pressure Level spectra predictions for the three forced mixers at all
three data points using Model 2 are given in Figures 5.35 thru 5.43. It is seen from
these Figures that for all nine data points the predictions at angles close to 90◦ are
in strong agreement with the experimental data. However, similar to the confluent
mixer and Model 1 predictions, the spectrum peak is often slightly under-predicted
at angles close to the jet axis. In addition, similar to the results from Model 1,
for all three forced mixers some deficiencies in the predictions are seen at the high
power Set Point at angles close to the jet axis. In these cases it appears as if there
82
is an additional high frequency noise source that is not modeled by the Two-Source
formulation.
Table 5.14 Average Weighted Errors in dB for Model 2
Mixer Set Two-Source Fixed Coaxial Single
ID Point Optimized Parameters Jet Jet
12CL 1 0.30 0.36 0.41 0.64
12CL 2 0.38 0.38 0.56 0.86
12CL 3 0.48 0.52 0.82 0.96
12UM 1 0.26 0.29 0.93 0.68
12UM 2 0.35 0.38 0.98 0.82
12UM 3 0.46 0.52 1.04 0.98
12UH 1 0.27 0.31 1.24 0.96
12UH 2 0.35 0.45 1.30 1.09
12UH 3 0.45 0.57 1.29 1.12
83
Table 5.15 Average Errors in dB for Model 2
Mixer Set Two-Source Fixed Coaxial Single
ID Point Optimized Parameters Jet Jet
12CL 1 1.29 1.55 2.56 2.53
12CL 2 1.37 1.47 2.54 2.60
12CL 3 2.08 1.97 3.16 3.40
12UM 1 1.16 1.48 3.61 2.85
12UM 2 1.45 1.43 3.65 3.02
12UM 3 1.77 1.89 4.33 4.41
12UH 1 1.03 1.13 4.87 4.06
12UH 2 1.24 1.31 4.97 4.18
12UH 3 1.57 1.75 4.81 4.66
Table 5.16 Maximum Errors in dB for Model 2
Mixer Set Two-Source Fixed Coaxial Single
ID Point Optimized Parameters Jet Jet
12CL 1 8.35 6.81 13.18 12.02
12CL 2 4.78 4.75 10.74 9.87
12CL 3 7.54 10.36 16.70 14.31
12UM 1 5.40 5.44 14.48 13.56
12UM 2 5.93 5.77 11.32 11.25
12UM 3 7.34 10.00 18.46 16.22
12UH 1 4.24 4.08 14.13 13.00
12UH 2 7.36 6.82 13.04 12.45
12UH 3 6.55 9.04 17.88 15.58
84
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.35. Model 2 Predictions for the Low Penetration Mixer at Set Point 1
85
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.36. Model 2 Predictions for the Low Penetration Mixer at Set Point 2
86
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.37. Model 2 Predictions for the Low Penetration Mixer at Set Point 3
87
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.38. Model 2 Predictions for the Intermediate PenetrationMixer at Set Point 1
88
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.39. Model 2 Predictions for the Intermediate PenetrationMixer at Set Point 2
89
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.40. Model 2 Predictions for the Intermediate PenetrationMixer at Set Point 3
90
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.41. Model 2 Predictions for the High Penetration Mixer at Set Point 1
91
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.42. Model 2 Predictions for the High Penetration Mixer at Set Point 2
92
102
103
104
Frequency [Hz]
SP
L [d
B]
90°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
120°
5 dB
ExperimentalModel 2 Optimized
102
103
104
Frequency [Hz]
SP
L [d
B]
150°
5 dB
ExperimentalModel 2 Optimized
Figure 5.43. Model 2 Predictions for the High Penetration Mixer at Set Point 3
93
6. Conclusions
It has been shown that the current Four-Source coaxial jet prediction method ac-
curately predicts the noise from an internally mixed jet with a confluent mixer.
However, neither a standard coaxial jet nor a single jet prediction are capable of ac-
curately predicting the noise from an internally mixed jet with a forced mixer. The
forced mixer noise spectra can, however, be predicted using a Two-Source model.
The three variable parameters in this Two-Source model are determined for a given
mixer geometry through a multi-step optimization process. These parameters have
then been curve-fit to the differences in the mixer geometry. As a result, for the
family of forced mixers studied here, given the mixer penetration and the aerody-
namic conditions of the co-flowing jet, a noise prediction can be made based on a
Two-Source model.
The fact that a fully mixed jet and a secondary jet can be used to model the
noise from a forced mixer suggests that the differences in the structure of a forced
mixer jet plume essentially eliminates the effective jet component of the Four-Source
model. This hypothesis, which was originally proposed by Mike Fisher and Brian
Tester [27] based on the analysis of the forced mixer experimental acoustic data, is
supported by results found in this study.
A notable deviation in the forced mixer noise predictions is seen near the spectrum
peak at angles close the jet axis. However, it is noted that the same deviations are
present in the confluent mixer predictions using the Four-Source method. Since the
basic components that make up the Two-Source model are taken from the Four-
Source method it is logical that any limitation in the predictions of the Four-Source
method for a coaxial jet prediction would be inherited by the Two-Source model. It
is likely that these deviations in the predictions could result from the differences in
the geometric configurations, such as the presence of the center body or the nozzle
94
wall. In addition, these deviations could also be attributed to the quality of the
single jet predictions, which are common to both the Four-Source and Two-Source
models.
It is seen in the predictions at the high power set point that there appears to be an
additional noise source mechanism that is not modeled by the two single jet sources
in the Two-Source model. This additional noise source mechanism may result from
the stream-wise vortices interacting with the nozzle wall. In addition, this noise
source could also be generated by the test rig in the experimental facility. Since the
origin of this noise source is not yet known, no efforts have yet been made to account
for this noise in the current prediction models.
Two Two-Source models were evaluated in this study, a mixed jet - mixed jet
model (Model 1) and a mixed jet - secondary jet model (Model 2). In general,
the results from Model 2 appear to correlate with geometric differences in a more
linear fashion. However, it is possible that the type of Two-Source model which best
represents the actual flow field could be dependent on the forced mixer geometry.
This case could result due to the fact that as the forced mixer penetration increases,
the flow at the final nozzle exit becomes more like a fully mixed jet. Furthermore, as
the forced mixer penetration decreases, the flow at the final nozzle exit will resemble
more that of a secondary jet. The validity of this hypothesis could be determined
through the analysis of the aerodynamic data of the flow field at the final nozzle
exit. In practice, it is possible that a CFD solution may aide in determining which
form of the Two-Source model is most applicable. Based on the performance of
the two Two-Source models in this study, it is difficult to support this hypothesis
due to the limited range of velocity ratios in the current data set. For the current
set of operating points the velocity ratio between the secondary and primary flows
varies little (from 0.62 to 0.68). In addition, at these velocity ratios the secondary
jet and mixed jet have similar jet velocities. As a result, at this time it is difficult to
definitively determine which Two-Source model is best for a given mixer geometry.
95
Future work on this research topic could include relating information from the
experimental aerodynamic data to the source strength terms in the Two-Source
models. In addition, information from the experimental aerodynamic data may also
be used to assist in the identification of the additional noise source at the high power
set point. A similar effort could also made using the results from a CFD (RANS)
analysis to construct a predictive noise tool for evaluating the noise from jets with
forced mixers. In this approach, information about the predicted turbulent flow field
could be used to determine the source strengths in the Two-Source models.
LIST OF REFERENCES
96
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