On the length scale and Strouhal numbers for sound 5 transmission across coupled duct cavities at low Mach number

Abstract


I. INTRODUCTION
Ventilation system noise in modernized buildings has long been a problem in building noise control.This type of noise will propagate along the ductwork and enter the occupied zones of the buildings, resulting in poor indoor acoustical quality, which will eventually lead to health problems and lower productivity, if they are not attenuated properly. 1Though there are noise criteria which help limit the noise exposure of building occupants (for instance, Beranek 2 and Hay and Kemp 3 ), the development of effective flow duct silencers has been attracting great attentions of engineers and academics for many façades.
The traditional flow duct silencer is of the dissipative type, which is basically a flow constriction with porous materials installed on the two sides of the constriction. 4However, the low frequency performance of this silencer type is not satisfactory because of the properties of the porous materials.The flow constriction also gives rise to a significant static air pressure drop across the silencer so that a more powerful but noisier fan is often required to deliver the required air flow rate.
Besides, this silencer type is not suitable for application in locations where a high hygiene standard is to be maintained because the deterioration of the porous materials will result in increased air particulate in the indoor air.Neither is this type of silencer suitable for applications where the air is dirty/greasy.The drum-like silencer proposed by Huang and Choy 5 is also not applicable in such condition and the maintenance of the tension in the membrane is not straight-forward.Flow duct silencers containing micro-perforated absorbers, such as that of Allam and Åbom, 6 suffer similar drawback.Active control technique has been implemented in the ventilation systems, 7 but the reliability of the microphone signals in the relatively hostile duct interior remains a big challenge to the professionals.
The development of passive reactive silencers, which do not contain any flexible structure, has also been a hot research topic.Typical examples include the Helmholtz resonators, 8 the expansion chambers/plenum chambers, 9 the Herschel-Quincke tubes 10 and conical tube resonators. 11major drawback of this silencer type is that it can only give satisfactory performance at or close to its resonance frequency.Coupled resonators, such as those of Seo and Kim 12 and Howard et al., 13 can give wider working bandwidths.Recently, Howard and Craig, 14 Tang, 15 Yu and Tang 16 and Červenka and Bednařík 17 proposed the use of sidebranches as silencing devices.However, coupling reactive devices will lead to bulky setup, which is not desirable in the view of the limited ceiling voids available for their installation in practice.Their manufacturing is also not going to be straightforward.
The recent work of Tang and Tang 18 reveals the possibility of creating broadband silencing by coupling two cavities along a rectangular duct.This setup is very simple and is basically that of an expansion chamber with its two cavities offset in the flow direction.It is found that the longer the offset distance, the stronger the sound transmission loss.Also, the cavities need not to be large.
Leung et al. 19 investigated numerically the aeroacoustics of such setup at transonic Mach numbers, and the cavities they adopted were long enough for the shear layers to roll up into individual vortices.
A more recent work of Tang and Tang 20 illustrates the mechanisms of the abovementioned high sound transmission loss and relates explicitly the stopband cut-on frequency to the dimensions of the coupled cavities and the duct in the absence of a duct flow.
Similar to other ducted elements, such as those studied by Yu and Tang, 16 Tang, 21 Davies and Holland 22 and Tonon et al., 23 the performance of the coupled duct cavities is expected to be worsen in the presence of a duct flow.Flow-induced noise is a big problem of duct silencers, regardless of their being dissipative or reactive. 24,25However, the results of Tang 21 illustrate that there is a critical flow speed over which the performance of his resonators will start to deteriorate, but the issue of how such velocity is related to the resonator configurations has not been addressed.Similar phenomenon is also observed in Yu and Tang. 16his study, a series of experiments is derived to study how the sound transmission loss across the coupled cavities is affected by the presence of a low Mach number duct flow.Effort is also made on understanding the critical duct flow velocity over which deterioration of sound transmission loss will be resulted, and how this velocity is related to the peak transmission loss frequency and the cavity dimensions.These results, together with the previous effort of the authors, 20 will help establish a framework for the design of coupled cavities as a flow duct silencing device.
For practical building application reasons, the cavities adopted in this study are kept narrow so that the proposed device is compact and simple.

II. EXPERIMENTAL SETUP
The test rig of the present study was that of Yu and Tang 16 except that their sidebranch array muffler was replaced by two rectangular coupled cavities.It was made of 20 mm thick Perspex panels in order to avoid adverse effect of duct/cavity wall vibration.Figure 1 shows the schematics and internal dimensions of the test section, the nomenclature and the locations of the sensors.The first higher mode was that associated with the duct span s (= 173 mm) and its cut-on frequency was around 990 Hz.The highest frequency of interest in this study was actually below 800 Hz so that one could basically assume that all the evanescent waves at the microphone locations were insignificant and the waves should be essentially planar at the locations of the microphones M1 to M4.These microphones were installed symmetrically about the centerline of the leading cavity.A fan and a loudspeaker with a circular aperture comparable to the duct span were installed at the upstream end of the test section as in Tang. 21The latter was capable of producing sounds with magnitude not less than 100 dB over the whole frequency range of interest.The background noise An absorptive ending, designed according to the recommendations of Neise et al., 26 was attached to the downstream end of the test rig to minimize sound reflection there.The coefficient of sound power reflection by this absorptive ending was less than 0.03 for frequencies higher than 200 Hz. 21The cavities adopted in the present study can be regarded as 'narrow' with w = 100 mm and 70 mm.Given a smaller cavity length w than the duct width a, the type of planar longitudinal wave interactions inside traditional expansion chambers, whose lengths are long compared to the duct widths (for instance, Munjal 9 ), did not take place in this study.This has already been illustrated in Tang and Tang, 20 and thus is not discussed further.
The sound power transmission losses, TLs, in this study were estimated from the signals obtained by the four wall-mounted ¼" microphones (M1 to M4, Brüel & Kjaer Type 4935) using the four-microphone method.This method has been presented in detail in Tang and Li 27 and thus is not repeated here.These microphones were far enough from the coupled cavities to avoid the contamination of the evanescent waves.The separation between the microphones in each pair was set at 20 mm (27 mm in Chung and Blaser 28 ).A trial test was done using a separation of 80 mm with w = 100 mm.The corresponding spectral sound transmission losses do not show significant difference from those measured using a 20 mm microphone separation within 500 Hz to 800 Hz (not shown here), which is the main frequency range of interest in this study, even when U was as high as 16 m/s.M5 and M6 were located at the centres of the cavity ceilings.The data recorder was a Brüel & Kjaer Type 3506D PULSE system.The sampling rate was 4096 samples per second per channel throughout the measurements.
The average longitudinal air velocity across the duct cross-section, U, was varied from 0 and 20 m/s in intervals of 2 m/s.The distance of the leading edge of the leading cavity from the flow entry was about 2 m.The boundary layer at this leading edge should not be laminar even for the case of U = 2 m/s.The air velocity was measured by a TFI Series 100 cobra probe (head width 2.6 mm) upstream of the test section on a vertical plane near to the upstream microphones.This probe was removed during the sound transmission measurements.The air turbulence intensities on the duct centerline on that vertical plane was ~3% of the main longitudinal flow speed U.The mean transverse flow velocities in the main duct were negligible.A single hotwire facing normally to the longitudinal main flow direction was used to measure the longitudinal flow velocity and turbulence intensity profiles across the cavity shear layers. 29loudspeaker was turned on and fed with a white noise signal during the TL measurements with and without flow.The sound pressures at M1 and M2 due to the loudspeaker were kept at least 82 dB (maximum ~ 100 dB) on average between 600 Hz to 800 Hz, which was the major working frequency ranges of the coupled cavities in the present study.This level of artificial sound pressure overrode that due to the flow generating facility in most of the cases.Signal contamination due to flow turbulence was possible when U > 16 m/s at low artificial excitation levels.Those results are not included in the data analysis.

III. RESULTS AND DISCUSSIONS
In this section, the effects of air flow velocity, U, and the offset distance, d, on the TLs across the coupled cavity sections will be examined in details in the first place.The air flow velocity U was varied between 0 to 20 m/s in intervals of 2 m/s, while d was increased from 0 mm to the full cavity length w in intervals of 10 mm.The present interests, apart from obtaining deeper understanding on the sound transmission loss across the coupled cavities, are to find out the threshold flow velocity for reduced cavity acoustical performance and then establish its relationship with cavity dimension and the frequency of peak sound transmission loss.In the foregoing discussions, all lengths and frequency, unless otherwise stated, are normalized by the duct width a and the first transverse odd mode cut-on frequency, f1 (=1143.3Hz), respectively.The results in Fig. 2 show the essential features of a mixing layer. 30The strongest turbulence intensity is observed near the location of the highest mean shear rate.It is observed that the artificial excitation tends to thicken a bit the shear layer into the cavity, reducing the mean shear rate.The  shear layer thickening increases with excitation level (not shown here).This implies that a stronger mean flow speed U is required to maintain a given mean shear rate at increased excitation level.

A. Shear layer profiles
It should be noticed that the turbulence intensity in the shear layer on the cavity-side is increased when an upstream artificial excitation is introduced, and it also increases with increasing excitation level (not shown here).Though this is rather expected, it can be observed that larger and more extensive increase of the turbulence intensity is associated with a lower U.One can even observe a reduction of the maximum turbulence intensity for the case of U = 12 m/s.This tends to suggest that the shear layer aerodynamics is more susceptible to change by the artificial upstream excitation at lower U.One can also conclude that a stronger U will result in a shear layer less vulnerable to acoustic excitation.
Figure 3 shows some turbulence spectra midway within the leading cavity shear layer for d/w = 1 and y/a = 1.Without the artificial excitation, the spectral intensity decays monotonically with frequency [Fig.3(a)] within the frequency range of interest and they resemble those of static pressure fluctuations of a turbulent mixing layer. 31The cavity shear layers are turbulent.Under excitation, there appears some organized motions at frequencies around 0.78f1, around 1.5f1 and between 2f1 to m/s, but only between the frequency range from 2f1 to 3f1.At U = 12 m/s, only a very small and insignificant spectral peak can barely be observed at ~3f1.Apart from these peaks, the rest of these spectra is basically the same as those of their unexcited counterparts.The increase of turbulence level with increasing U tends to mask or dissipate the organized motions within the shear layers.
Though the shear layers are likely to be excited and some organized fluid motions are resulted (more distinguishable at lower U), the effect is small and the frequencies of these motions do not match with the sound transmission loss peak frequencies to be discussed in Section III.B. the trailing cavity shear layers are less affected by the artificial excitation than their leading cavity counterparts.This tends to imply that the leading cavity plays a more important role in the aeroacoustical behaviour than the trailing cavity.The spectral characteristics of the turbulence intensity of the trailing cavity shear layers are similar to those of the leading cavity shear layers.The corresponding results are therefore not presented.
As the offset distance is reduced, the difference between the mean flow speed and turbulence intensity profiles is also reduced (not shown here).However, the corresponding results are largely inline with those shown in Figs. 2 to 4 and thus they are not discussed further.

B. Sound transmission loss
The sound transmission loss, TL, is defined as where I and T are the amplitude of the incident wave (upstream of the coupled cavities) and the transmitted wave respectively.Figure 5 shows the spectral variations of the TLs across the coupled cavity regions with different offset ratios dr (= d/w) and cavity lengths.The two coupled cavities form a regular expansion chamber when dr = 0, and they are 100% offset when dr = 1.The finiteelement simulation results 18,20 of the corresponding 100% offset cases are included for comparison.It should be noted that the TLs for frequency higher than 0.86f1 fluctuate significantly because of the cut-on of the first spanwise higher mode (w/s = 0.86), while the multiple microphone method adopted in this study caters only for planar modes.Though the test section was made essentially two-dimensional, a very small misalignment in the spanwise direction could result in such phenomenon.This is also the situation when a duct flow is introduced because of the threedimensional flow turbulence even if the test section is made perfectly two-dimensional.Results at frequency above 0.85f1 are thus ignored in the present study.It is also not the purpose of this study to examine the condition at higher frequencies as the stopbands of the coupled cavities are well below the first transverse higher mode frequency of the duct. 20Shallow cavities, like those addressed by Oshkai et al., 32 are not within the scope of this study.
In general, the TL magnitude increases with increasing offset distance and there is a slight increase in the frequency of peak sound transmission loss, fp, at the same time.The strong peak is found to be the result of the strong pressure-releasing effect of the first transverse acoustic mode in the middle region of the cavity section. 20One can notice that the magnitude of the peak TL reduces with decreasing cavity length for all offset ratios.This is not surprising as the TL of a regular expansion chamber decreases with chamber length for a fixed area expansion ratio when the chamber length is less than a quarter of the excitation wavelength. 9 There are some discrepancies between the d/w = 1 results obtained by finite-element method and experiment.However, the difference in the TL peak frequency is just about 3 -4 %, and similar discrepancy is not really significant (for instance, Wang et al. 33 and Zhao et al. 34 ).The spectral variation trends of the results obtained by the two approaches are very similar.It will be shown later that the present obtained sound pressure fluctuation patterns within the two cavities also agree with those predicted by Tang and Tang. 20One should note that the actual sound pressure spectra inside the duct and the cavities are not useful as their shapes are dictated by the spectrum of the excitation sound radiated out from the loudspeaker.Therefore, frequency response functions are used hereinafter to relate the sound pressures within the cavities to the incident sound wave from upstream of the cavities.Following the two-microphone method of Chung and Blaser, 28 one can establish, for plane wave motions, before swapping the two wall-mounted microphones M1 and M2, in the frequency domain, ¾¾¾¾ : dr = 0; ----: dr = 0.1; ¾ × ¾ : dr = 0.2; ¾ ×× ¾ : dr = 1.
amplitude responses of the measurement devices are different, but they have been taken into account by calibration already.After swapping M1 and M2, one obtains where a is a general phase variation between the above two sets of measurements (Eqs. 2 and 3).
The target here is to estimate the transfer function / frequency response, HI,S = S/I.One can find from Eqs. ( 2) and (3) that where D = ½x1 -x2½, Hi,j denotes the transfer function pj/pi and ¢ represents the quantity associated with the swapped microphone measurement.The argument kx1 of the exponential function is arbitrary and fs is unknown.However, one can ignore x1 as the present analysis is done with reference to the signal at M1. Eq. ( 4) is useful as the magnitude of HS,I is a main concern in the present study.It should be noted that the incident sound pressure magnitude I, which represents also the artificial excitation level, can be estimated using HM1,I together with the sound pressure recorded at M1.In the foregoing discussions, I is given in decibels.
Two examples of the frequency response functions HM5,I and HM6,I for w/a = 2/3, I ~ 87.8 dB are presented in Fig. 7(a).It is noticed that resonance occurs inside both cavities and the magnitude of the corresponding sound pressure inside the leading cavity is higher than that inside the trailing cavity regardless of the offset ratio.Both sound pressures are stronger than the incident wave.It can also be observed that the resonance frequency of the leading cavity is higher than that of the trailing cavity and both of these frequencies are lower than the corresponding peak TL frequencies fp.For the leading cavity, the sound pressure (M5) is actually in-phase with the incident sound at resonance, while for the trailing cavity, phase lag of p/2 and 3p/4 are recorded for dr = 1 and 0.5 respectively.
It is obvious that such phase differences of the cavity pressures cannot result in complete cancellation of the incident wave.The highest TL is achieved when the cavity pressures are out-of-phase with each other, while none of them is in-phase with the incident wave.This condition is achieved at a frequency higher than the cavity resonance frequencies.
Besides, cavity resonance is stronger at lower dr (except when dr º 0) because of the stronger resonance of the odd transverse dual cavity chamber mode as shown in Tang and Tang. 20 The strength of the artificial excitation does not affect the results in the absence of a duct flow.
The presence of a low Mach number flow along the duct results in flow separations at the sharp edges of the cavities, and these shear flows could cause pressure fluctuations in the coupled cavity region, affecting the overall acoustical impedance and thus the sound propagation and sound transmission loss.These shear flows could also be sound producing (for instance, Davies and Holland, 22 Rossiter 35 and Tonon et al. 23,36 ).It should be noted that the flow Mach number in the present study is well below 0.1.The effect of mean flow Mach number in the calculation of all the required transfer functions (Eq.4) and TL is negligible.As the sound pressures inside the cavities are stronger at decreased dr (Fig. 7), it is believed that the TL drop will decrease with stronger sound pressures and thus decreasing dr at a fixed U.This will be discussed further later.Sharp peaks near to fp can be found in both transfer functions.It appears that a sharp pressure peak at M5 is associated with a TL dip.It will be shown later that this peak is due to the aeroacoustical interference within the cavity region.The peak at ~0.62f1 is also believed to be due to such interference.However, the corresponding TL is low and thus it is not considered further in this study.
It should be noted that the strong sound pressures inside the cavities compared to the incident sound level could lead to the nonlinear roll-up of tiny discrete vortices at the interfaces between the cavities and the main duct flow. 37Linear instability theory 38 could fail in the present circumstance.
It is observed from Fig. 9(a) that the increase in the sound pressure in the leading cavity is higher than that in the trailing cavity at these pressure peaks.The phase difference between M5 and The variations of Ucr/(wfp) with dr are presented in Fig. 13.The corresponding results at dr less than 0.1 are not presented as the TL peaks in those cases cannot be identified reliably, though one can still approximate the corresponding fps using the formula of Tang and Tang 20 or the phase difference between signals at M5 and M6 (c.f.Fig. 6).One can notice that Ucr/(wfp) tends to decrease fairly linearly with increasing offset ratio regardless of the artificial excitation level and the cavity length.The best straight lines obtained again using the method of least square are also given in Fig. 13.More interesting is that the results of the linear fit suggest that a new kind of similitude exists in the aeroacoustics of coupled cavities with a definite relationship between Ucr/(wfp) and dr : where the slope a depends on excitation level and cavity length, while b just varies within a narrow range from 1.83 to 2.17, which is ~ 2 on average.One can also notice that a decreases with decreasing artificial excitation level and a shorter cavity results in smaller a.  electrical power fed to the loudspeaker.It is noticed that Stcr tends to decrease with I.It is rather expected as a stronger shear rate is required to create a pressure fluctuation which is capable of affecting that due to an elevated artificial excitation level.As fp does not change much in the presence of a duct flow for a fixed coupled cavity system, the increase in shear rate, which is achieved through an increase in the flow speed U, results in a lower Strouhal number.Stcr is around 8 at high excitation of ~100 dB regardless of w/a. Figure 14 indicates that there is very likely a definite simple relationship between Stcr and I which could be independent of w/a.The strength and the aerodynamics of the shear layers and the artificial excitation level should play crucial roles in the underlying mechanism.Further investigations are needed for deeper understanding of this  aeroacoustic behaviour and the actual relationship between Stcr and I. Since fp can be estimated using the method given in Tang and Tang, 20 this new Stcr formulation (Eq.7) can in principle be used to determine, for a fixed coupled cavity configuration and excitation level, the flow speed over which significant TL drop is expected.The physics which leads to such a length scale Le is left to further investigations.

IV. CONCLUSIONS
A series of experiments was conducted in the present study in an attempt to understand the mechanism leading to the strong sound transmission loss across two coupled cavities along a rectangular duct.The present coupled cavity system was formed by offsetting the two cavities that made up a conventional expansion chamber.The effects of a low Mach number duct flow on the reduction of the sound transmission loss were also studied in details.For practical application reasons, the dimensions of the cavities adopted were small compared to those of the main duct.
In the absence of the duct flow, broadband increase in the sound transmission loss is observed when the two cavities of a conventional expansion chamber are offset.A peak on the sound transmission loss spectrum is also observed at the same time, and the frequency of this peak is higher than the resonance frequency of each cavity.The magnitude of this peak sound transmission loss increases with increasing degree of cavity offset, but its frequency does not vary much though it does show an increasing trend with increasing offset distance.The performance of the coupled cavities is independent of the artificial excitation level.
The sound transmission loss across the coupled cavities is lowered upon the introduction of the low Mach number duct flow.For any degree of cavity offset, there is a flow speed below which the sound transmission loss remains fairly unchanged.A new length scale is established.This new length scale, together with the critical flow speed and the peak sound transmission loss frequency, gives a Strouhal number, which is independent of the offset ratio, for a fixed cavity length.This dimensionless Strouhal number decreases with increasing excitation level, but does not depend much on the cavity length.All of these collectively suggest that similitude exists in the present low Mach number aeroacoustic problem.The working frequency range and the flow speed limit of coupled cavities as duct silencer are therefore predictable based on such similitude.

Figure 2 (
Figure 2(a) illustrates the vertical profile of the mean flow speed across the leading cavity

Figures 4 (FIG. 4 .
Figures 4(a) and 4(b) illustrate the mean flow speed and turbulence intensity profiles across

Figure 6
Figure 6 illustrates the spectral variation of phase differences between microphones M5 and
However, stronger cavity pressures are associated with a lower TL.Too strong cavity pressures could result in extra sound power radiation downstream, reducing the overall TL.The results with w/a = 7/15, which are shown in Fig. 7(b), are very much inline with those of w/a = 2/3.Therefore, they are not discussed.

Figure 8
Figure 8 illustrates some examples of the TL reductions upon the introduction of a low Mach number
One can then derive a critical Strouhal number, Stcr, based on a new length scale Le equals (b Stcr thus increases with decreasing cavity length or artificial excitation level.A summary of the relationship between Stcr, I, w/a and b are given in TableI.As fp does vary within a narrow range ¢ : w/a = 7/15, dr = 4/7, I = 85.8 dB.for a fixed w/a while dr should also have some effects on the acoustical impedance of the coupled cavities, the excitation level I does vary over a small range in the present study even for a fixed

TABLE I .
Summary of Ucr/(wfp) against dr.