Optimizing the performance of a sidebranch array duct muffler Shiu Keung Tang 1 The University of Hull Department of Engineering, The University of Hull, Hull, HU6 7RX, United Kingdom Ho Man Yu 2 The Hong Kong Polytechnic University Department of Building Environment and Energy Engineering, The Hong Kong Polytechnic Univer- sity, Hong Kong, China ABSTRACT Recent studies of the authors show that muffler formed by putting together narrow sidebranches in array form can offer strong and broadband sound attenuation in low Mach number flow ducts. It is also found that such muffler with the increasing branch length in the direction of duct flow has stronger resilience to flow excitation and thus can maintain its sound attenuation performance unless the duct flow is high. However, the arrangement of the branch length has not been optimized. Nu- merical investigation is carried out using a semi-analytical method in the present study to further understand how the branch length variation is affecting the overall sound attenuation of the side- branch array muffler. For simplicity, a branch length variation is assumed to follow a power law of the branch order. A power law index of 1 represents linear variation. It is found that for a muffler with 11 sidebranches, the index for minimum transmission is about 1.6, but that for maximum particle kinetic energy is around 1.38. In term of noise attenuation spectrum, the muffler with index 1 and 3 give very strong attenuation at particular frequencies, while that with index ~1.6 gives a more uni- form attenuation within the operation frequency range. 1. INTRODUCTION

The control of noise from the ventilation and air conditioning systems of modernized commercial buildings has long been a focus of engineers and academics. This noise arrives at the occupied zones of the buildings through the ductwork. The traditional method to reduce this noise is by used of dissipative silencers, which are usually flow constrictions lined with porous materials. However, this type of structures results in significant static pressure drop, which has to be overcome by fans and much energy is consumed. This is undesirable in view of sustainability. Much effort has been made to develop broadband low-static-pressure-loss duct silencing devices in the past few decades. Exam- ples are the coupled resonators of Seo and Kim [1], the drum-like silencer [2] and the narrow side- branch array muffler [3], but this list is by no mean exhaustive. The narrow sidebranch array creates a metasurface on the duct wall to reduce sound transmission and has been proved to be effective and broadband [3]. However, this type of devices, which work based on acoustical coupling and resonance, is susceptible to flow excitation. Their performances 1 S.Tang@hull.ac.uk

2 louisyu125@hotmail.com.hk

will thus deteriorate in the presence of a low Mach number duct flow. Yu and Tang [4] derived a simple method to enhance the resilience of this sidebranch array against flow excitation. They found that such resilience is enhanced if the acoustical pressures inside the sidebranches can be made stronger. Both Tang [3] and Yu and Tang [4] demonstrated the effectiveness of sidebranch array using arrays with linear variation of branch lengths and with linear variation of quarter-wavelength resonance frequencies. However, such branch length variations are not likely to result in the best noise reduction performance. An attempt is made in this study to find out a simple length variation which can result in improved noise reduction and/or provide better resilience to flow excitation. Finite-element method is used. There are efforts in existing literature where optimization schemes, such as the Genetic algorithm, are used (for instance, Červenka and Bednařík [5]). However, their application will result in non- straightforward length variations, which, under the constraint stipulated in Yu and Tang [4], could perform similar to an array consists of the simple length variation studied in the present study. 2. SIDEBRANCH ARRAY AND NUMERICAL METHOD

The sidebranch setting of Tang [3] is used in the present numerical study. The schematics of the array and the nomenclature are given in Figure 1. The width of each sidebranch, w , is fixed at 0.1 a and the branch separation  equals 0.11 a . n = 11 as in Tang [3].

w

Sidebranch Tubes

l 1

l n l i



( i  1)  Main Duct

y

a

x

Origin of Co-ordinate System

Figure 1: Schematics of the sidebranch array muffler and present nomenclature [3]. In this study, the sidebranch lengths are varied according to Equation 1 :

ఈ , ሺ1ሻ

𝑙 ௜ ൌ𝑙 ଵ ൅ሺ𝑙 ଵଵ െ𝑙 ଵ ሻ൬ 𝑖െ1 10 ൰

where  is the index to be varied for optimized array performance. In this study, it is varied from 0.1 to 5. As shown in Yu and Tang [4], the following arrangement will give better resilience to flow excitation, and it is adopted in the present study : 𝑙 ଵ ൑𝑙 ଶ ൑⋯൑𝑙 ௜ ൑⋯൑𝑙 ଵଵ . ሺ2ሻ

In the present study, l 1 = 0.5 a and l 11 = a . The semi-analytical approach of Tang [3] is used to estimate the sound transmission loss, TL . A plane wave of unit magnitude is set to propagate towards the sidebranch array with rigid walls. In this approach, it is assumed that the width of each sidebranch is narrow so that only plane wave can propagate with the sidebranch. This sidebranch mouths together act as a ‘meta-surfaces’, resulting in sound transmission loss. In the numerical approach, the main task is to estimate the acoustical particle velocity, V , at each sidebranch mouth by solving Equation 3 :

ቌ 𝛽 ଵ,ଵ ⋯ 𝛽 ଵ,ଵଵ ⋮ ⋱ ⋮ 𝛽 ଵ,ଵଵ ⋯ 𝛽 ଵଵ,ଵଵ ቍ ൭ 𝑉 ଵ ⋮ 𝑉 ଵଵ ൱ൌ൭ 𝐽 ଵ ⋮ 𝐽 ଵଵ ൱, ሺ3ሻ

where J i is the average pressure resulted from the upstream plane wave at the mouth of the i th side- branch and the  s represent the interactions between the sidebranches. The mathematical workings are tedious and thus not repeated here. Details can be found in Tang [3]. The sound power transmission coefficient,  , is thus given by the square of the magnitude of the transmitted plane wave :

ଶ

ଵଵ

𝜏ൌอ1 ൅ 1 𝑘𝑎 sin ൬𝑘𝑤 2 ൰෍𝑉 ௜ 𝑒 ௝௞௅ ೔

௜ୀଵ อ

, ሺ4ሻ

where k is the excitation wavenumber, L i the distance between the first and the i th sidebranch and 𝑗ൌ√ െ1 . V i s are normalized by the particle velocity induced by the upstream incident wave. The TL is 𝑇𝐿ൌെ10log ଵ଴ ሺ𝜏ሻ. ሺ5ሻ As the results of Yu and Tang [4] show the resilience of the sidebranch array against flow excita- tion is improved when the total acoustical kinetic energy, KE , at the mouths of the sidebranches is higher, this KE is also discussed in this study :

ଵଵ

𝐾𝐸ൌ෍|𝑉 ௜ | ଶ

௜ୀଵ . ሺ6ሻ

A weak acoustic damping is added to the analysis as in the existing literature (for instance, Hart and Lau [6]). In this study, this is implemented by using a complex wavenumber k c : 𝑘 ௖ ൌ𝑘ሺ1 ൅0.02𝑗ሻ, ሺ7ሻ which replaces k in the above calculations. 3. RESULTS AND DISCUSSIONS

It is understood from the results of Tang [3] and Yu and Tang [4] that the narrow sidebranch array under the abovementioned configuration is effective within a certain frequency range. In the study,

the frequency range of focus is taken to be 0.3  ka  3. The total sound transmission coefficient within this range  tol is discussed in the first place, followed by the spectral variation of  . Same is done for the KE .

3.1. Sound Transmission Coefficients

Figure 2 illustrates the variation of  tol with the index  . For the present sidebranch array configura- tion, minimum overall sound transmission is observed at  ~ 1.62. It is also noticed that the linear sidebranch length variation array can give a performance not so bad compared to that at optimal  . It is also observed a convex shaped array (  < 1) gives a noise reduction which varies faster with increasing  that of its concave counterpart (  > 1), except for  near to the optimal value. One can expect that for   1 or at very large  , the performance of the array will be more narrow-banded as the corresponding sound transmission loss will depend of sidebranch(es) with similar lengths.

0.70

0.68

0.66

0.64

 tol

0.62

0.60

0.58

0.56

0 1 2 3 4 5



Figure 2 : Effect of  on the total sound power transmission Figure 3 shows the spectral variations of  for  = 1, 1.62, 2 and 3. One can notice that low frequency performance is better for  < 1. At  = 1, the sound transmission loss is very significant at ka ~ 1.45. As  increases, the low frequency performance is weaken while those at higher fre- quencies become better. At the optimal  (  = 1.62),  within the major operation range is relatively uniform and this operation band is the widest. As  increases, the working bandwidth becomes nar- rower. At  = 3, the overall performance is biased to within a narrow frequency range at around ka ~ 2.1. The noise reduction capacity of the array is further reduced as  is increased further. At  = 5, there is a broadband reduction in  within the major operation frequency range of the array (not shown here), while the frequency of minimum  is very similar to that observed at  = 3.

3.2. Acoustic Kinetic Energy

The kinetic energy of the air at the mouth of a sidebranch is very much related to the acoustic pressure magnitude within the sidebranch [4]. The stronger the former, the stronger the resilence of the side- branch array against flow excitation.

1.0

0.8

0.6



0.4

 = 1  = 1.62  = 2  = 3  = 1.39

0.2

0.0

0.5 1.0 1.5 2.0 2.5 3.0

ka

Figure 3 : Spectral variation of sound transmission coefficient at various 

10

9

8

KE

7

6

5

0 1 2 3 4 5



Figure 4 : Effect of  on KE Figure 4 illustrates the variation of KE with  . The KE peaks at ka ~ 1.39 which is slightly lower than that for minimum  tol . At this frequency, the corresponding  tol is just 0.2% higher than that at ka =1.62. One can also observe from Figure 3 that the sound transmission spectra of  = 1.39 and 1.62 arrays are very similar. Thus, this  = 1.39 array should be a better choice for use in a low Mach number flow duct, that of  = 1.62 is slightly more broadband. The applied air damping affects only the magnitudes of the KE ,  tol and  , but has insignificant effect on the critical  s discussed above (not shown here).

30

25

20

KE

15

 = 1  = 1.62  = 2  = 3  = 1.39

10

5

0

0.5 1.0 1.5 2.0 2.5 3.0

ka

Figure 5 : Spectral variation of KE and the effect of  One can notice from Figure 5 that the low  frequency range (Figure 3) is largely correlated to the frequency range of high KE . Though KE does not contain any phase information, strong air motion at the sidebranch mouths appear to be essential for noise reduction. One can also observe that the difference between the corresponding results of  = 1.39 and 1.62 array is very small. The major difference is that the KE spectrum at  = 1.39 is more uniform within the strong sound transmission loss frequency range. 4. CONCLUSIONS

A semi-analytical approach is adopted in the present study to explore the configuration of a narrow sidebranch array muffler which can result in improved noise reduction than those studied previously by the authors. For design simplicity, the branch length variation is taken to follow a power law, and the index  of this power law is the focus of the present study. Owing to the expected flow excitation of this array muffler, the total kinetic energy of air at the mouths of the sidebranches is also consid- ered. Results suggest that for the muffler configuration adopted in this study, the best index is around 1.3 to 1.7 with  = 1.62 for minimum sound power transmission and 1.39 for strongest resilience to flow excitation. At the lower end of this index range, the sound power transmission loss is more related to the lower end of the active operation frequency range of the array muffler. The opposite is observed at the higher end of this index range. In conclusion,  = 1.39 array should be a better performer for low Mach number flow duct application. 5. REFERENCES

1. Seo, S. H. & Kim, Y. H. Silencer design using array resonators for low frequency band noise reduction. Journal of the Acoustical Society of America, 118(4) , 2332–2338 (2005). 2. Huang, L. & Choy, Y. S. Vibro-acoustics of three dimensional drum silencer. Journal of the Acoustical Society of America, 118(5) , 2313–2320 (2005).

3. Tang, S. K. Narrow sidebranch arrays for low frequency duct noise control. Journal of the Acous- tical Society of America, 132(5) , 3086–3097 (2012). 4. Yu, H. M. & Tang, S. K. Sound transmission across a narrow sidebranch array duct muffler at low Mach number. Journal of the Acoustical Society of America, 148(3) , 1692–1702 (2020). 5. Červenka, M. & Bednařík, M. Optimal reactive silencers with narrow side-branch tubes. Journal of the Acoustical Society of America, 144(4) , 2015–2021 (2018). 6. Nelson, P. A., Curtis, A. R. D., Elliott, S. J. & Bullmore, A. J. The active minimization of har- monic enclosed sound field. Part I : Theory. Journal of Sound and Vibration 117(1) , 1 – 13 (1987).