Experimental verification of the energy dissipation mechanism in acoustic dampers

Journal of Sound and Vibration (1973) 26 (2), 263-267

EXPERIMENTAL VERIFICATION OF THE ENERGY DISSIPATION MECHANISM IN ACOUSTIC DAMPERSt P. K. TANG,D. T. HARRJEANDW. A. SIRIGNANO

Guggenheim Laboratories, Prhlceton University, Prhweton, New Jersey 08540, U.S.A. (Received 30 August 1972) An experimental program is described which verifies the theoretical model that acoustic damping devices undergoing high intensity oscillations dissipate energy via jet kinetic losses. Pressure measurements within the damping devices and flow duct together with detailed surveys of the jet velocities provide the experimental confirmation. The.theory accounts for duct flow effects, both steady and unsteady, as well as the jet dissipation. Discrepancies between theory and experiment can be traced to neglect of higher order terms or ignoring the difficult wall friction term in the case of the quarter-wave tube. 1. INTRODUCTION The subject of this paper is an experimental study of several acoustic damping devices subjected to high intensity oscillation so as to evaluate the valiaity of the jet flow theory of kinetic energy dissipation. Theory and experiments are treated in more detail in reference [1], while the theory alone is summarized in reference [2]. Related aspects of the theory and application are covered in references [3], [4], [5] and [6]. In this investigation the important quantities associated with the damping devices are experimentally measured for comparison with the jet dissipation predicted performance. Some measurements are quite straightforward, whereas others have required rather elaborate experimental apparatus to achieve the required simulation of damping device operational environments. For the quarter-wave tube, measured quantities included gas velocity amplitude at the tube entrance and the pressure amplitude at the tube end. Important quantities for the Helmholtz resonator designs were the gas velocity amplitude inside the orifice and pressure amplitudes inside the cavity. Simulation included both steady and oscillating flow past the damping devices for several frequencies and disturbance amplitudes. Additional velocity survey measurements were taken in the very short orifice Helmholtz resonators to determine the effect of contraction. No experiments were performed on the long damping devices since predicted performance was less than for the other damping devices. In the following sections, the apparatus and the operational procedure for this study will be outlined followed by the experimental results and, finally, the theoretical computations will be presented and discussed. 2. APPARATUS DESCRIPTION A flow schematic of the experimental test arrangement is shown in Figure 1. The principal component of the apparatus is the resonant duct in which a standing wave is t This research was supported by NASA on Grant NGL 31--001-155. 263

264

P.K. TANG, D. T. HAR.RJEAND W. A. SIRIGNANO

generated. Acoustic damping devices are mounted in the test section of the d u c t w h i c h telescopes into stationary upstream and downstream sections. This axial movement allows tests to be performed at pressure and velicity node or antinode locations.

Water bath M~riobleorifice /

: ----Variable gap

Si ren wheel

Test[_Tsection

[

-, Resonant duct Telescopinduct g

Regulator~ Dic _J_~----r ,Itforifice ] [~ .~k.. Plenum tank

Figure 1. General configuration of the apparatus. P = pressure gauge, T = thermocouple.

Air is introduced to the duct from a high pressure source appropriately regulated and monitored. The air is heated to near room temperature levels by passing it through a water bath so that heat transfer effects can be considered insignificant. Mass flow rate control is achieved through proper choice of sonic orifice size, together with upstream pressure control and temperature measurement. A plenum tank at the inlet to the resonant duct provides a pressure node condition: i.e., the entrance is free from pressure oscillations. The mean duct pressure and temperature are measured in this region. The resonant duct is fabricated from seamless tubing with an effective length, L * , of 110.3 inches and an internal diameter o f 1"5 inches 0"46 inches at the telescoping section).t At the duct exit a rotating siren disk opens and closes the rectangular exit orifice. The speed of siren disk rotation and the choice of the number of disk slots or orifices governs the frequency of the exil~ flow oscillations. A one horse power:l:, variable speed motor controls the frequency. The distance separating the siren disk from the exit determines the amplitude of the oscillations. This gap is altered by movement of the exit block (small gaps provide high amplitudes). To maintain the duct velocity constant during an oscillation amplitude adjustment, it is necessary to vary the size of the exit orifice (much like a window shade). Small motors are used for both remote adjustments. The telescoping section, previously described, also is motorized as are the traversing thermocouples and hot-wire probes. Mean duct pressure was maintained at 40 psigw with pressure amplitudes reaching 0"11 of the mean. Duct flow Mach numbers of order 0-1 required an additional duct exit from upstream of the exit block. The entire exit section was enclosed in a rubber foam-lined box with an acoustically treated exhaust duct to the outside to minimize local sound intensity. The resonant character of the duct and the choice of siren frequencies results in high!ntensity sound generation at various organ-pipe modes in the duct. Th e duct exit represents t 1 in. = 2-54cm, ~t 1 h.p...~ 746 W. w1 psi (= 1 lbf/in 2) ~ 6"9 kN/m 2. In the usual way, psig refers to guage pressure: i.e., above atmospheric.

ENERGY DISSIPATION IN ACOUSTIC DAMPERS

265

a pressure antinode. Duct resonant frequency is determined through the following relation (the quarter-wave tube equation) :t f'* =

2 n - 1 coo 4 L*'

(1)

where f * is the resonant frequency, n is a positive integer, and coo is the speed of sound * depends on the air temperature, the resonant it the mean duct condition. Since coo .~requency is therefore not a constant. In many situations the resonant wavelength 2* is t very useful parameter and the expression for it is 4

2,*= 2n--I L~,

(2)

~:hich is a constant for a given harmonic. In these experiments, two frequencies were used ;xclusively: the 13th harmonic (2n - 1 = 13) and the 25th harmonic, which correspond to 'requencies of 400 and 770 Hz, respectively, at a room temperature of 75 ~ F (24 ~ C). Lower section of a Holmholtz resor~for Pressure

Hot wir, probe No. I.

, ,EHot wire probe No. 2

Figure 2. Test section. The test section where the damping devices are mounted is shown in more detail in ?igure 2. The motor-controlled test section can travel approximately 18 inches axially by ,irtue of the telescoping design. The section is located a minimum of 30 diameters downtream of the plenum tank. Two ports are provided for the hot-wire probes and one is ,vailable for a piezoelectric pressure transducer. Hot-wire 1 is designed to measure the tir velocity in the duct and is located 3/4 inch upstream of hot-wire 2, which provides for he velocity measurement inside, or in the vicinity of, the orifice of a damping device under est (see Figure 2). Such an arrangement avoids flow interference between the two probes vhile the distance between them is still quite small compared to the shortest test waveength (17.7 inches). For the measurement of pressure amplitude inside the duct, a pressure ransducer is placed at the same axial location as hot-wire 2 (i.e., at the orifice centerline of he damping device). In the experimental study of the Helmholtz resonator with a very hort orifice the flow contraction becomes an important factor. With this in mind, a precisan driving mechanism for hot-wire 2 was designed and constructed so that the velocity listribution along the center of the orifice could be determined. This system provides an .ccuracy of the location of hot-wire 2 within 0.0005 inch with remote readout. A list of nomenclature is given in the Appendix.

266

P.K.

T A N G , D . T. t l A R R J E A N D W . A. S I R I G N A N O

The construction of a Helmholtz resonator for testing is shown in Figure 3(a). The lower part, which consists of an orifice, was inserted into the test section. To vary theorifice length, this part could be changed as a whole, or orifice disks could be inserted into the cavity to build up the orifice length. Next a cylinder was installed together with a piston to vary the cavity size (employing a lead screw, gears and driving motor). Pressure transducer 2 was installed on the face of the piston so that the pressure oscillation inside the cavity could be recorded for each cavity variation. To prevent direct impingement of the orifice jet flow on the transducer diaphragm, a perforated plate was used to cover the transducer face. Perforations were placed outside of the transducer perimeter and transmitted the pressure through a shallow cavity to the diaphragm.

Fressure ironsd;Jcer

Driving motor

I l

i

Pressure /transducer

y

Piston (b)

,

Driving motor i

rew

Figure 3. (a) Helmholtzresonator assembly with variable cavity length. (b) Quarter-wave tube assembly with variable tube length. Similar construction for the quarter-wave tube is shown in Figure 3(b) with the exception that the diameter of the cylinder is smaller, presenting a tight fit for pressure transducer and wiring. Further details on the instrumentation are covered in reference [1]. 3. OPERATION PROCEDURE Selection of the proper sonic orifice and upstream pressure govern the flow rate for a given test. System temperature stabilization requires a warming time of approximately 30 minutes, The siren gap is minimized initially to maximize the oscillation amplitude. With siren and air on, the speed of the siren is varied to reach the desired resonant frequency of the duct. The waveshape can be varied by moving slightly to either side of the resonant frequency--in these tests near sinusoidal waves were achieved in every case. Through adjustment of the variable area exit, the siren gap and the bypass valve (when required), the desired mean duct Mach number and amplitude of the pressure oscillations are achieved.

ENERGY DISSIPATION I N ACOUSTIC DAMPERS

267

Hot-wire 1 measures the mean duct velocity in addition to measuring local velocity oscillations. Since the velocity measurement is made at the center of the duct, a certain average value over the entire duct section must be used to represent the chamber condition. For simplicity, we use the mass average; the mean to maximum velocity ratio is a function of the local Reynolds number and is determined by calibration [1]. The test section is then allowed to travel along the resonant duct until it reaches a velocity node location. At that point pressure transducer 1 reads a maximum oscillation amplitude and fluctuations in the output from hot-wire 1 become zero. To observe the effect of the oscillating velocity component, one must place the test section somewhat away from this point. Once the duct condition is set, readings from hot-wire 2 and pressure transducer 2 are taken; they represent the velocity oscillation at the orifice (or at the tube entrance) and the pressure oscillation inside the cavity (or at the tube end). These measurements of the velocity and pressure amplitudes are then compared with the theoretical calculations based on the duct conditions. By opening up the gap and closing the exit orifice, one Can reduce the oscillation intensity, while maintaining the mean duct pressure constant. In this manner a new set of data is obtained at a lower duct pressure amplitude. Since the velocity oscillation amplitude of the duct changes with the oscillation intensity, the final reduction of data must include some interpolation or extrapolation so that data can be grouped together and compared as a single constant velocity oscillation amplitude. More details will be given when the experimental results of various dampers are described in the next two sections. 4. QUARTER-WAVE TUBES A 0.325 inch diameter quarter-wave tube was designed at the fundamental resonant frequency of 400 Hz (76~ F) and the assembly is shown in Figure 3(b). A lead screw, one end of which is attached to a small piston containing the pressure transducer 2 provides tube length variability. The innermost position of the piston gives the minimum tube length of 3"830 inches. The tube length required at resonance is equal to

/*=--2"= 4

-L~ .

(3)

2n- 1

At this frequency/* is 8"485 inches long. The quarter-wave tube is known to be sensitive to small variations in frequency [1, 6]. In order to observe the influence on damping of such changes, unfortunately:0ne can not impose other duct resonant frequencies since a 60Hz increment would b e t o o large. Instead, one can change the tube length by any required amount to alter:the tube resonant frequency. What has been accomplished in terms of frequency change can be found by combining the resonant condition of the resonant duct, equation (I),: and the resonant condition of a quarter-wave tube, equation (3); the final relation is l* Lz)* ( f * ) ~r* = 2n----1 ~--, - 1

.

(4)

The frequency ratio in the above equation is the dimensionless frequencyf. Given a value off, we can determine the desired tube length immediately or vice versa. Although from the theory the resonant condition can be determined (i.e., the appropriate tube length of a quarter-wave tube by using equation (3)) some discrepancies are expected because of the mass end correction and the internal friction. Therefore, the first series of

268

P . K . TANG, D. T. HARRJE AND W. A. SIRIGNANO

tests was initiated to check the resonant condition through the measurement of phase between the pressure oscillations in the duct and at the end of the tube. Also measured was the maximum response: namely, the ratio of the square root of the pressure amplitude at the tube end to that in the duct by varying the tube length. The phase, just mentioned, is 90 ~ at resonance according to the theory. Several earlier tests I-7] at different duct pressure levels, without the duct flow effect, have indicated that the length required at

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0.10

ENERGY DISSIPATION IN ACOUSTIC DAMPERS

269

resonance is less than that obtained f r o m equation (3) by an a m o u n t of approximately 0.80 d ~ almost twice as m u c h as that needed for one mass end correction. This difference is considered to be due to the internal friction. As we will see later, the viscous force has some effect on the quarter-wave tube because the tube length is rather long. Once the resonant condition is established, this information is used to d e t e r m i n e the frequency f i n equation (4).

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,s

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,r 8

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0-50

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270

P . K . TANG, D. T. HARRJE AND W. A. SIRIGNANO

The experimental results of a quarter-wave tube are presented in Figures 4 through 7. In those figures the pressure amplitude at the tube end, P2, and the gas velocity amplitude at the tube entrance, ~ , are plotted against the pressure amplitude in the resonant duct, p~. Here one needs to remember that the so-called chamber condition is also the duct condition. Results are given in Figures 4 and 5 with and without mean chamber flow for f = 1"0, at resonance, and for f = 1"15, off-resonance. Two values of C r, the average pressure coefficient [1, 21, namely - 1 . 5 and -2.0, are used in the theoretical calculation when an amount of mean flow (where ct is the mean flow Mach number) is presen t. Investigating the results of no mean chamber flow first, one observes that the experimental data points, denoted by circles in the figures, are below the theoretical prediction. This is attributed to two factors: first, all higher harmonics have been omitted in the theoretical computation which would reduce the total amplitude by approximately 4Yo [7], and this is more severe in the near-resonance region; second, the friction has not been accounted for. With such a long tube, we should expect that the viscous force would reduce the oscillation intensity inside the tube. Future improvement of the theoretical treatment must include the frictional loss. The results with mean chamber flow are also presented in Figures 4 through 7. The value of a is determined from hot-wire 1 and also from the flow condition of the sonic orifice. The difference between these two values is found to be no greater than 5Yo.An experimental point is designated here by an x. The experimental values are found to be less than the theoretical values based on Cp - - 1-5 and -2.0. If the discrepancies come fro m the same soflrces as described in the last paragraph, and if they are assumed to be appri0ximately 9 the same for the two cases with and without imean chamber flow, then a translation of ordinates can be taken to match thel data of no mean flow on the respective curves. Such a step would make the data with mean chamber flow fall onto those curves with Cp = - 1 ; 5 and -2.0. A value of Cp somewhere between - 1 . 5 and - 2 . 0 would be indicated by this model. From the study of Helmholtz resonators, more confidence about the range of Cp will be obtained. It is seen that the general results are not very sensitive to the exact value of Cp. The effects of oscillatory flow only (for which a data point in the figures is denoted byo), and with both mean and oscillatory flow components (denoted by x), are shown in Figures 6 and 7. The mean flow Mach number a is determined by the sonic orifice flow and the duct conditions, while the oscillatory flow component is foufld thi-ough measurements of hot-wire 1. Some iterative procedures were used to keep the oscillatory amplitude a constant in the final comparisons. Thewave in the duet is a longitudinal (cos fl = 1.0, fl being the angle between the two components of the chamber flow) standing mode (~, = hi2, tS,, being the phase between the pressure and velocity oscillations in the duct). The phase tS, was determined by a simultaneous comparison of the traces of hot-wire 1 and transducer 1 on the oscilloscope. The general behavior of these results are similar to those in Figures 5 and 6. Thus, except for the frictional loss and possibly the variation of the mass end correction, when the chamber flow exists, the agreement between the theory and the experiment is considered very satisfactory. 5. HELMHOLTZ RESONATORS The purpose of the experimental studies described in this section is not only.to confirm the theoretical development in a very similar manner as described in the previous section, but also to discover the amount of orifice flow contraction through velocity measurements along the axis of the short orifice. The contraction phenomenon of an orifice flow is wellknown under steady conditions although, in general, it does not distinguish itself from

ENERGY DISSIPATIONIN ACOUSTIC DAMPERS

271

the discharge coefficient which includes both the contraction and frictional effects. It is felt that such a velocity survey can be used to determine the contraction: namely, the ratio of the effective minimum stream tube area A9 to the orifice cross-sectional area A. The assembly o f the Helmholtz resonator is illustrated in Figure 3(a). The lower section consists of an orifice with diameter d* = 0.11 inch and a cavity diameter D* = 0.871 inch (thus the orifice-to-cavity area ratio is 0.016). Two different orifice lengths were used in these tests, / * = 0.011 inch and 0.55 inch, resulting in orifice length-to-diameter ratios of lid = 0.1 and 5, respectively. A cylinder 10 inches long, of the same diameter as the cavity, has been used to increase the cavity volume. Piston movement allows the suitable volume adjustments. The first Helmholtz resonator under investigation had an lid = 0-1. Serious contraction would be expected from such a short orifice. The resonant frequency f~* of this:resonator and those that follow are fixed at 770 Hz, the 25th harmonic o f the resonant duct. If the orifice flow is quasi-steady, it does not really matter at which frequency the contraction is determined. Therefore, the cavity length L* was approximated by using the conventional equation with a single mass end correction included [1]. Once the cavity length was set, the experiment could be performed in the same manner as described for the quarter-wave tube. The exception was that hot-wire 2 was not fixed at one location inside the orifice but was moved in increments of 0.1 orifice diameter to more than ten locations on each side of the orifice. At each location, the velocity amplitude was measured for several duct pressure amplitudes. Later interpolations were made so that only two values of velocity amplitude are shown, corresponding t o the pressure amplitudes of 0"05 and 0.075. Velocity measurements were taken where both the jet and the contraction were present as illustrated in Figure 8. This series of tests for determining the contraction coefficient, Cr was conducted under conditions of no chamber flow. There is some remaining question about its dependence upon chamber flow velocity. I

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.... 0-....0."

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0 .... 0

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x-X-s O1 -I

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Hot 'Mre location x/d

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272

P.K. TANG, D. T. HARRJEAND W. A. SIRIGNANO

The results for various chamber conditions are given in Figures 9 and 10. In these tests the velocity amplitude measurement was made only at the edge of the orifice, but the jet velocity is obtained by taking that value divided by the contraction coefficient. The experimentally measured pressure amplitude at the cavity end p,~ and that predicted from the conventional theory [ l ] (solid lines in Figures 9 and 10) are found to differ somewhat. With this cavity length, at the higher frequency (f* = 770 Hz), wave motion inside the

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ENERGY DISSIPATION IN ACOUSTIC DAMPERS

273

cavity was thought to be responsible for altering the pressure amplitude. Another set of calculations (dashed lines in Figures 9 and 10) based onthe long cavity theory, indicates that agreement with the predicted pressure amplitude is greatly improved. It also explains the difference between experiments and theories found in earlier reports [7].

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002

o

3 m L. 13.

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lid

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lid =

0-1, C p = 1.5, c o s fl = 1 . 0 , 8= = rr/2.

274

P.K. TANG, D. T. HARRJE AND W. A. SIRIGNANO

A n o t h e r series o f tests was performed at the lower frequency, f * = 400 H z , which corresponds to f = 0.52. The results are given in Figures 11 and 12 and they are selfexplanatory. With chamber flow, C~ is found to lie s o m e w h e r e between - 1 ' 5 a n d - 2 ' 0 , a conclusion we have drawn in the last section. As s h o w n in the figures, the agreement between the theoretical prediction and the experimental measurement is again very good. I

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,~I

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Figure 14. Helmholtz resonator, with mean and oscillator'/flow, f = cos ,8 = 1-0, 6, = n/2.

1.0, Cc = 1.0, lid = 5.0, Cp = --1.5,

275

ENERGY DISSIPATION IN ACOUSTIC DAMPERS

An orifice with lid -- 5.0 was installed for the same type of test. It was reported earlier [7] that the theoretical approach without the consideration of contraction could give very good agreement under the no chamber flow condition. Also, it was felt that the deep penetration of the hot-wire probe into the orifice might result in some degree of blockage; therefore, no velocity profile measurements were made. Thus, the velocity measurement in the orifice was taken only at its outer edge--an action justified by the results. The resonant cavity length is found to be 0.212 inch with C~ = 1.0 and f , * = 770Hz. The pressure amplitude at the cavity end 1)4 and the velocity amplitude at the orifice entrance a~ are plotted agains the duct pressure Px under various duct flow conditions and at f = 1.0 and 0.52 (see Figures 13 to 16). 0.20

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Chamber pressure amplitude,;~i Figure 15. H e l m h o l t z resonator, with and without mean f l o w , f = 0.52, Cc = 1.0, 0.20

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I 0"10

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l

002 9

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I

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Chamber pressure amphtude,p t

Figure 16. Helmholtz resonator, with mean and oscillatory flow,/= 0-52, Cc = l-0, lid --- 5"0, Cp = --1"5, cos fl = l'0, ,~ = n/2. 6. CONCLUSIONS The experiments for both quarter-wave tubes and Helmholtz resonators have shown good agreement with the theory that relies upon non-linear kinetic energy losses attributable to jet dissipation. Only where significant frictional losses have been neglected are there

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P. K. TANG, D. T. HARRJE AND W. A. SIRIGNANO

n o t i c e a b l e differences between t h e o r y a n d experiment. F u t u r e theoretical m o d e l i m p r o v e m e n t s w o u l d be required to a c c o u n t for the frictional effects. T h e e x p e r i m e n t a l p r o g r a m h a s also quantitatively e v a l u a t e d the j e t c o n t r a c t i o n coefficient which is i m p o r t a n t f o r the s h o r t orifice H e l m h o l t z r e s o n a t o r designs used for b r o a d e r b a n d d a m p i n g applications. REFERENCES 1. P. K. TANa 1972 Ph.D. Thesis, Prhlceton Uni~'ersity Department of Aerospace and Mechanical Sciences Report No. 1033-T. Acoustic damping devices: theories and experiments. 2. P. K. TANG and W. A. SIPa~NANO 1972 Journal of Sound and Vibration 26, 247-252. Theory of a generalized Helmholtz resonator. 3. W. A. SZRIGNANO 1966 Prhtceton University of Aerospace and Mechanical Sciences Report No. 553-F, 31--40. Nonlinear aspects of combustion instability in liquid rocket motors. 4. T. S. TONON and W. A. SIRIGNANO 1970 American hlstitute of Aeronautics and Astronautics Paper No. 70-128. The nonlinearity of acoustic liners with flow effects. 5. C. L. OBErtG 1970 American blstitute of Aeronautics and Astronautics Paper No. 70--618. Combustion stabilization with acoustic cavities. 6. P. K. TANG and W. A. SIRZGNANO 1971 American btstitute of Aeronautics and Astronautics Paper No. 71-78. Theoretical studies of a quarter-wave tube. 7. P. K. TANG, W. A. SIRZGNANO,D. T. HARRJE and T. S. TONON 1971 Proceedings of the 7th JANNAF Combustion l~Ieethlg CPIA Pt~blication No. 204, Vol. 1,727-742. Quarter-wave tubes t'erstts Helmholtz resonators: theories, experiments and design criteria. APPENDIX NOMENCLATURE a A Acfr 9c Cc C~ De f 1 L LD p u (z fl ~5, 2 a

velocity amplitude of the oscillatory component of the chamber flow effective orifice cross section area effective orifice cross section area speed of sound contraction coefficient pressure coefficient tube or orifice diameter cavity diameter frequency tube or orifice length cavity length resonant duct length pressure gas velocity mean velocity component of the chamber flow angle between the mean and oscillatory components o f the chamber flow velocity phase between the pressure and velocity oscillations in the chamber wavelength orifice-to-cavity area ratio

Superscripts * dimensional quantity " amplitude of that quantity Subscripts 1 at the entrance o f a quarter-wave tube or at the orifice exit of the chamber side I chamber or duct condition 2 quarter-wave tube end or the orifice exit of the cavity side 3 cavity condition on the orifice side 4 cavity end r at resonance oo reference condition