A computational approach for ﬂow acoustic coupling in

A computational approach for flow–acoustic coupling in closedside branches

Paul M. Radavich and Ahmet Selameta)

The Ohio State University, Department of Mechanical Engineering and Center for Automotive Research,206 West 18th Avenue, Columbus, Ohio 43210

James M. NovakFord Motor Company, Powertrain Operations, Dearborn, Michigan 48121

~Received 23 October 1999; revised 22 December 2000; accepted 27 December 2000!

The quarter-wave resonator, which produces a narrow band of high acoustic attenuation at regularlyspaced frequency intervals, is a common type of silencer used in ducts. The presence of mean flowin the main duct, however, is likely to promote an interaction between these acoustic resonances andthe flow. The coupling for some discrete flow conditions leads to the production of both large waveamplitudes in the side branch and high noise levels in the main duct, thereby transforming thequarter-wave silencer into a noise generator. The present approach employs computational fluiddynamics~CFD! to model this complex interaction between the flow and acoustic resonances at lowMach number by solving the unsteady, turbulent, and compressible Navier–Stokes equations.Comparisons between the present computations and the experiments of Ziada@PVP-Vol. 258,ASME, 35–59~1993!# for a system with two coaxial side branches show that the method is capableof reproducing the physics of the flow–acoustic coupling and predicting the flow conditions whenthe coupling occurs. The theory of Howe@IMA J. Appl. Math. 32, 187–209~1984!# is thenemployed to determine the location and timing of the acoustic power production during a cycle.© 2001 Acoustical Society of America.@DOI: 10.1121/1.1350618#

PACS numbers: 43.28.Ra, 43.28.Py@LCS#

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LIST OF SYMBOLS

c speed of soundd side branch length in flow directionf frequencyh height of main ductL side branch length perpendicular to flowM U/c, Mach numberPrms root mean squared acoustic pressureSt f d/U, Strouhal number

I. INTRODUCTION

Quarter-wave resonators are used as acoustic silencenumerous applications. This well-known silencer producelarge acoustic attenuation at frequencies where the lengtthe side branch is an odd multiple of a quarter waveleng( f 5c/4L, 3c/4L, 5c/4L,..., with c being the speed of sounandL the length of the side branch!. In the presence of meaflow, however, a shear layer is created between the mofluid in the main duct and the stationary fluid in the sibranch. Under certain flow conditions, instability in the shelayer creates oscillations, which can then go on to exacoustic resonances in the side branch. The acoustic rnances then amplify the oscillations in the shear layer,the whole process continues to amplify until large-amplituvortices are formed. This coupling can lead to the productof both large wave amplitudes in the side branch and h

a!Electronic mail: [email protected]

1343 J. Acoust. Soc. Am. 109 (4), April 2001 0001-4966/2001/109

T 1/f , acoustic periodu total velocity vector minus acoustic velocityu8 acoustic velocity vectorU average mean flow velocity in main duct

Greek symbols

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noise levels in the main duct. Thus, this interaction cauthe quarter-wave resonator to become a noise source rathan an acoustic silencer.

Numerous works have been conducted on the noisehigh-pressure amplitudes generated by flow over rectangcavities. These configurations can be categorized intogroups: deep cavities and shallow cavities, as illustratedFig. 1. The deep cavities have a length to diameter raL/d, greater than 1, and include the side branches ofcurrent investigation. In these cavities, flow-induced noiseproduced primarily when oscillations in the shear layer cate waves which travel along the length of the cavity,L, andreflect back to interact again with the shear layer. These lside branches are commonly investigated both in confiflows, whereh is a finite dimension, and in half-plane flowwhereh is infinite. The shallow cavities, whereL/d is lessthan or equal to 1, are similar in nature to deep cavitiesshare many common properties. For these short cavihowever, the stronger acoustic interactions take place o

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the longer dimension of the cavityd. The shallow cavitieshave typically been investigated in half-plane flows. Detaireviews of flow noise in cavities can be found, for exampin Naudascher~1967!, Rockwell and Naudascher~1978!, andKomerathet al. ~1987!.

Some of the early experimental work on deep cavitwas performed by East~1966!, who examined low Machnumber flows over a two-dimensional~rectangular cross section! unconfined cavity. East showed that these deep cavexcite primarily discrete frequencies that occur near the fdamental acoustic resonance frequency of the side braand concluded that the tones are produced when oscillatare amplified by coupling between the shear layer flucttions and the cavity acoustic modes. He also noted thatpeak excitation occurred in two discrete ranges of Strounumber St5 f d/U50.3– 0.4 andSt50.6– 0.9, suggestingtwo different modes of shear layer excitation. Visualizatistudies by Ericksonet al. ~1986!, Erickson and Durgin~1987!, and Ziada~1993! showed that the excitation neaSt50.4 is characterized by a single vortex in the branmouth, while the excitation nearSt50.8 is characterized bytwo simultaneous vortices in the branch mouth. Later expmental work by Jungowskiet al. ~1987, 1989! for a circularside branch mounted to a circular duct showed excitationsimilar ranges of Strouhal number.

Shallow cavities withL/d,1, as shown in Fig. 1~b!,also exhibit flow–acoustic coupling similar in nature to thof the deep cavities. Several experimental works docum

FIG. 1. Geometry for~a! deep cavities or side branches (L.d) and ~b!shallow cavities (L,d).

1344 J. Acoust. Soc. Am., Vol. 109, No. 4, April 2001

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ing the properties of these shallow cavities include Krishmurty ~1955!, Ball ~1959!, Dunham~1962!, Heller and Bliss~1975!, Franke and Carr~1975!, Shaw~1979!, and Sarno andFranke~1990!. These studies have shown that, with the lolength of the cavity in the flow direction, as many as fidifferent shear layer modes can be excited. The Strounumber of the first vortex mode excitation occurs near 0which is similar to that of deep cavities. Helmholtz resontors, which consist of a cavity connected to the main floduct through a smaller orifice opening, also have distiacoustic resonances and have shown excitation propesimilar to the side branches~De Metz and Farabee, 1977Anderson, 1977; Hershet al., 1978; Panton, 1988!.

Due to the complexity of the problem with unsteadnonlinear, viscous, compressible, and turbulent flow, themulation of analytical methods becomes difficult andvolves numerous assumptions. Covert~1970! used linear sta-bility theory to couple mean flow with oscillations incavity. For shallow cavities, Tam and Block~1978! addedthe shear layer momentum thickness and acoustic wpropagation and reflection in the cavity to their linear modThey demonstrated that the shear layer momentum thickand the length to depth ratio of the cavity were importantdetermining how the Strouhal number at excitation varwith Mach number. With estimated values for the mometum thickness, their results correlated well with experimein determining the Strouhal number for excitation. Theacoustic model did not allow for wave reflections from topen end of the cavity, however; thus, this method worequire modification to allow for cavity normal-mode resnances. Howe~1981, 1997! used linear theory to modelcavity connected to the mean flow through a small apertuHis predictions of excitation velocity also correlated wewith experimental values. More complicated modelsBruggemanet al. ~1991! and Durgin and Graf~1992! intro-duce concentrated vortices into the flow and are able todict acoustic pressure amplitudes for cavity flows. These vtex models perform well in providing qualitative predictionof changes in the acoustic amplitudes, although the prediamplitudes exceed experimental values by a factor of 4Bruggemanet al. and 5 in Durgin and Graf. In another vortex model by Krieselset al. ~1995!, the interaction betweenacoustic waves and distributed vortex ‘‘blobs’’ was studiat the junction between a duct and a deep side branch. Cparisons between Schlieren visualization and computatiopredictions of the vortex growth and propagation using tmethod were excellent. The model requires that acouboundary oscillations be specified from measured valuhowever, and thus does not reproduce self-sustaining olations. Several authors including Nelsonet al. ~1983!,Bruggemanet al. ~1991!, Durgin and Graf~1992!, and Krie-selset al. ~1995! have combined their vortex models with ththeory of Howe~1984! to examine when and where in thacoustic cycle sound power is produced.

Recently, attempts have been made to use conventiCFD methods to solve for the acoustic field generatedflow over a shallow cavity. The majority of these works cosiders high Mach number flows greater than 0.95, exceptHardin and Pope~1995!, who examine a low Mach numbe

1344Radavich et al.: Flow-acoustic coupling

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of 0.1. Hardin and Pope used an uncoupled approacwhich the acoustic field is solved from incompressible alaminar flow-field results. This method does not allow, hoever, for coupling or feedback between the acoustic and flfields. The works of Hankey and Shang~1979!, Rizzetta~1988!, Baysalet al. ~1988, 1990, 1994!, Kim and Chokani~1990!, Chokani and Kim~1991!, and Tamet al. ~1996! treathigh Mach number flows by solving the compressibNavier–Stokes equations with a simple algebraic turbuleclosure model. These works show that the computatiomethod is adequate for predicting the properties of the mflow field. Rizzetta~1988! also demonstrates the ability tpredict the generated acoustic noise. The first three rnance frequencies of excitation are predicted in the cavalthough the pressure amplitudes are overpredicted acthe frequency spectrum. A most recent work by Lilleyet al.~1997! uses a large eddy simulation to close the turbuleequations for flow over a shallow cavity. No results conceing the acoustic pressure field are provided in this wohowever.

The present approach solves the compressible NavStokes equations with a turbulence closure model, similasome of the previous computational investigations of flover shallow cavities. Rather than focusing on modelingmean flow field, however, the objective of the present wis to accurately reproduce the interaction between the actic waves in the cavity and the vortices that form in the shlayer between the flow duct and the cavity. While othersearchers have studied similar shallow cavities at high Mnumber using this method, the present work considers dside branches which are capable of producing nonlinwave amplitudes from low Mach number flow. As indicatearlier, other authors have used various types of simplimodels to study the same interactions in deep side brancMany of the assumptions and limitations of these modelsbe overcome using the present approach, however. By sing the full Navier–Stokes equations, the nonlinearitiesthe acoustic waves can be properly modeled, thereby circventing the constraint of linear wave amplitude treatmenAlso, rather than approximating the vortex strength, locatiand path as it travels across the mouth of the side brancis done in many of the vortex methods, the present approallows the vortex to develop naturally and interact with bothe acoustic waves and the flow field as it travels overside branch. Many of the analytical and vortex models ause the assumption of incompressible flow for low Manumbers, which requires that the acoustic waves be induced externally from experimental measurements. Suchproach does not allow for a true interaction between the vtices and the acoustic field, as the vortices are unable tothe fixed acoustic amplitude. Thus, this simplification makprediction of variations of the acoustic pressure amplitudedifferent flow velocities impossible. By solving the compressible flow equations with the current method, the voces and the acoustic waves are allowed to interact, with eaffecting the other until a natural equilibrium is reacheThis enables the present work to determine both whenflow–acoustic coupling will occur and how the acouspressure varies as the mean flow velocity and various d


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dimensions are changed. The present computationalproach also has the potential to include irregular-shapedometry which would be difficult to include in many of thforegoing analytical models. Several authors, including Dham~1962!, Franke and Carr~1975!, Jungowskiet al. ~1987,1989!, and Bruggemanet al. ~1991! have shown that smalchanges in the geometry of the interface between the flfield and the cavity, such as the addition of a small rampradius, can help to reduce the acoustic pressure amplitudeliminate the coupling altogether. Finally, the presentproach allows for reflections from the inlet and outlet bounaries of the main duct. Bruggemanet al. ~1986! demon-strated experimentally that when the side branchconnected to another duct, as shown in Fig. 1~a!, the reflec-tions from the inlet and outlet ducts can have a major impon the acoustic pressure amplitudes produced duringflow–acoustic coupling. They emphasized that the acouproperties of the entire system, not just the side branch itsneed to be accounted for, and were able to reduce acoamplitudes by 20 dB by altering the termination length. Theffect has been largely overlooked elsewhere in the liteture, however.

The objective of the present study is to assess the febility of using computational fluid dynamics to solve flowacoustic coupling problems in deep cavities at low Manumbers,0.1. A two-dimensional investigation has beeperformed comparing the numerical predictions to expemental results obtained by Ziada~1993!. His experimentalconfiguration consists of a main duct with two side branchpositioned at the same duct location opposed from eother. Results are provided for the acoustic pressure amtude versus Strouhal number, as well as smoke visualizaof the interaction between the main duct and a side branThe detailed computations also allow the theory of Ho~1984! to be applied to the flow field to approximate thacoustic power produced by the flow–acoustic interactiFollowing this Introduction, a brief description of the computational approach is given in Sec. II. A comparison of tcomputations with the experimental results of Ziada is psented in Sec. III, along with a detailed investigation inhow acoustic power is produced by the coupling. Finaresults and concluding remarks are given in Sec. IV.

II. METHODOLOGY

The effect of acoustic propagation and bulk fluid flocan be obtained simultaneously if the unsteady viscous cpressible flow equations are solved with sufficient accuraIn the present problem, the acoustic waves and the mflow are strongly coupled, with oscillations in the mean floexciting acoustic waves which in turn feed energy back ithe mean flow oscillations. This requires an accurate restion of both acoustic waves and fluid flow in the time dmain. To ensure accurate resolution of the acoustic wathe effects of time-step size and grid spacing were invegated for a single side-branch configuration similar in dimesion to the double side-branch configuration used forpresent study. With the current resolution, acoustic amtudes are underpredicted within about 2–3 dB at the quarwave resonance frequency of the side branch due to num


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cal dissipation. Better accuracy can be obtainedincreasing the temporal and spatial resolution further; hoever, the higher accuracy is not justified due to the excesincrease in computational time. The present computatioapproach solves the unsteady flow equations using the Palgorithm~Issa, 1986; Issaet al., 1986!, which is an implicit,noniterative method for unsteady compressible flow eqtions. The standardk-epsilon turbulence model of Laundeand Spalding~1974! is used to close the flow equations. Thturbulence model requires the solution of only two additiodifferential equations, and has been documented extensiYet, it has not been used for flow–acoustic problems extsively, as limited research is available in this area. While ialmost certain that a better closure exists or can be develospecifically for flow–acoustic-type problems, such develoment is beyond the scope and main objective of the prework.

As discussed in the Introduction, the success thus famodeling flow–acoustic interactions for flow over cavitihas been limited, and the majority of the work is confinedshallow cavities. In order to evaluate the feasibility of tnumerical approach, an investigation on deep side branis undertaken by modeling the configuration used by Zia~1993!, who provides information on acoustic pressure aplitudes versus the Strouhal number and clear smoke visization of the vortex as it travels across the branch mouThe geometry, which is shown in Fig. 2, consists of a squmain duct with two opposed side branches that are rectalar in cross section. The rectangular cross section allowssimple two-dimensional modeling. The main duct heighth50.092 m, and the dimension of the side branch in the fldirection is d50.052 m. Both side branches are of equlength L51.0 m. The visualization study was performedatmospheric pressure and low Mach number (M,0.1) witha loudspeaker placed at the end of one of the side brancas shown in Fig. 2, to control the acoustic excitation levethe system. The effect of the Strouhal number on acoupressure amplitudes was also investigated~without the loud-speaker! by varying the flow velocity. In order to reduce thviscous dissipation of acoustic waves, experiments notquiring visualization were performed at an elevated press

FIG. 2. Geometry for the experiments of Ziada~1993!.


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of 0.35 MPa~approximately 3.5 atm!. Ziada and Buhlmann~1992! showed that increasing the mean pressure from 0.to 0.35 MPa increased the nondimensionalized acoustic psure 2Prms/r0U2 by 60%.

The computational domain is two dimensional with2.0-m inlet duct and a 1.0-m outlet duct leading to a mesh17 250 cells. The inlet boundary condition reflects acouswaves as a solid boundary would, while the outlet imposefixed pressure. The inlet length of 2.0 m was chosen totwice the length of the side branch in order to reduce acotic coupling between the two. A law of the wall modelused to simplify the turbulence model at the solid wboundaries. Investigation of the specified turbulence intsity imposed at the inlet revealed that the intensity abovedoes not affect the acoustic pressure amplitudes. An iturbulence intensity of 10% was specified for the presruns. The inlet velocity was imposed using a 1/7 power-lvelocity profile. All computational runs were initialized wita mean pressure of 0.35 MPa to match the pressure useZiada, and a temperature of 293 K. A mean velocity was aspecified in the main duct initially, with zero velocity in thside branches. Ziada reported that the two side branchescillated out of phase~the acoustic pressure at the top branwas maximum when the acoustic pressure at the botbranch was minimum!; therefore, a small pressure imbalanwas introduced to the side branches at start-up to promotout-of-phase response. The system was then run unquasi-steady state was reached in which the only changtime was the acoustic oscillations. Computational time osingle SGI Origin 2000 processor for the above runs isproximately 2.431025 CPU second/cell/time step or aboutto 3 days per run. Details of the computations can be fouin Radavich~2000!.

Once the computations reach a quasi-steady state,theory of Howe~1984! is used to relate the acoustic powgeneration

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to the integral over a control volume, CV, of the time aveage of a triple product between the acoustic velocityu8, thevorticity v, and the flow velocityu. Here, the acoustic velocity is the unsteady irrotational component of the velocand the flow velocityu excludes the acoustic velocity. Ashown by Jenvey~1989!, Eq. ~1! is a constant entropy, inviscid, low Mach number approximation to the sound powFor the present high Reynolds number, low Mach numflow, it provides a reasonable approximation to the soupower, and is convenient for studying the interaction btween the flow and acoustic fields. Equation~1! is solvedcomputationally by storing the velocity and density for eacell in the domain for an acoustic cycle. The vorticitycalculated from the total unsteady velocity, and the solendal velocity is solved from the vorticity. The irrotational velocity is simply the difference between the total velocity athe solenoidal velocity. Fast Fourier transforms~FFTs! of theirrotational velocity and density are then taken cell by cellseparate the acoustic irrotational velocityu8 and the meandensityr0 . As the mean component of the irrotational v


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III. RESULTS AND DISCUSSION

An example of the transient start-up of a run fSt50.4 is given in Fig. 3, which shows the acoustic pressat the closed ends of the top and bottom side branches vethe time step. The acoustic amplitude at the first time steFig. 3~a! is determined by the pressure discontinuity spefied in the initial conditions. This discontinuity promotes aout-of-phase oscillation in which vortices are shed altnately between the top and the bottom branches. Later in3~b!, the system has reached a quasi-steady state in whicacoustic pressure oscillations are repeating. If there werecoupling between the mean flow and the acoustic wavescombined effect of viscosity and the numerical dissipatwould cause the initial oscillations of Fig. 3~a! to graduallydissipate to zero amplitude, since there is no acoustic soin the system to sustain these oscillations. Instead, theplitudes grow until a final amplitude is reached in Fig. 3~b!that is greater than the initial oscillation. For this to occenergy must be transferred from the mean flow to the acotic oscillations, demonstrating a true coupling betweenacoustic field and the flow field.

FIG. 3. Acoustic pressure amplitude during transient start-up atSt50.4; ~a!start of run;~b! quasi-steady state.


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To determine how this coupling varies with flow condtions, a series of runs was performed by holding the geoetry of the side branches fixed and varying the inlet flovelocity to obtain Strouhal numbers ranging from 0.2 to 0Figure 4 demonstrates how the acoustic pressure ampliat the resonance frequency of the side branches variesthe Strouhal number. Since both branches produce simamplitudes, only results from the bottom branch are shohere. The computations show excitation in the region frSt50.25– 0.5. The nondimensionalized pressure amplitpeaks near a value of 8 atSt50.35 and drops off as thevelocity goes lower or higher. Outside of the region froSt50.25– 0.5 the acoustic oscillations are so small thatnondimensionalized pressure approaches zero amplitThe experimental results from Ziada~1993! are also shownin Fig. 4. According to Ziada, the loop in the experimendata is a nonlinear hysteresis effect which shows that asflow velocity is gradually increased, there is a region whethe feedback loop is ‘‘locked in’’ and the coupling continuabove its normal value. Decreasing the flow through tregion avoids the lock-in region and produces differentsults. It was not attempted to reproduce the loop in this stdue to the additional computational time needed to slowramp up the velocity. In the experiments by Ziada, mufflewere placed upstream and downstream of the side braand because of this the reflection properties of the expmental inlet and outlet are unknown. In the computations,inlet and outlet lengths were chosen so that they wouldcouple with the side branch at its quarter-wave frequencythe mufflers used in the experiments are assumed to rethe interaction between the main duct and the side brancdecreasing reflections at the boundaries, then the experimtal pressure amplitudes should be similar to the computional results, which is observed to be true in Fig. 4. Athough a direct comparison cannot be made betweenexperiments and the computations due to these boundaryferences, it is important to emphasize that the computatiomethod predicts the flow–acoustic coupling in the sa

FIG. 4. Nondimensionalized acoustic pressure amplitude versus Stronumber.


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Strouhal range of 0.25 to 0.5. In terms of sound-presslevel, the difference between the computations and theperiments is only about 6 dB atSt50.3. As discussed in SecII, however, increasing the resolution of the computatiomay increase its acoustic predictions by as much as 3which would drop the difference to only 3 dB.

The paper by Ziada also contains detailed smoke visization of the formation and propagation of vortices at tinterface between the main duct and the side branSketches of these smoke figures are shown in Figs. 5 afor St50.4. This experimental work provides a serieseight figures that shows the formation and propagation ofvortex during a single acoustic cycle; however, neitherphase of the first figure nor the phasing between consecufigures is specified. It is assumed here that the acoustic cwas divided into equal increments with one figure at eaincrement. In order to compare with experiments, the prescomputational method released a large number of partiinto the flow and tracked their positions to simulate smoplots. The computational work divided the acoustic cyinto regular intervals and these figures were then matchethe experimental smoke pictures. Due to the uncertaintythe phase of the experimental figures, some phase differemay exist between the experiments and computationsFigs. 5 and 6. In the experiments performed by Ziada


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loudspeaker was used to increase the oscillation amplitufor improved visualization~recall Fig. 2!, and the experi-ments were conducted at atmospheric pressure to simthe smoke injection process. The computations achiesimilar high amplitudes by imposing an oscillating velociat the top side-branch end. To equalize the comparisonsratio of acoustic velocity to mean flow velocityu8/U wasmatched, with both having a ratio of 0.5. The acoustic velity is approximated in both the experiments and the comtations by normalizing the pressure at the bottom clobranch end byr0c. The results are shown in Figs. 5 andfor one complete cycle where the acoustic period,T, hasbeen divided into eight increments. Overall, the comparisbetween the computational predictions and the experimis excellent, suggesting that the actual physics of the floacoustic coupling is captured in the computations. In thequence of figures, the acoustic pulse pulls downward intoside branch atT/8, enhancing the roll-up of the vortex off othe upstream edge. The vortex continues to grow and is cvected along by the mean flow at 2T/8 and 3T/8. The com-pressed fluid in the side branch then pushes back, forcingvortex upwards from 3T/8 to 5T/8. The cycle then repeatitself as the acoustic pulse reverses at 6T/8 and 7T/8, caus-ing the roll-up of another vortex. The simultaneous existenof two vortices generated one acoustic period apart at 7T/8

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FIG. 7. Acoustic source power time averaged over the acoustic cycleSt50.4, u8/U50.5 ~black lines denote zero amplitude!.


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0.45 for this case withSt50.4 and u8/U50.5, whereasZiada~1993! approximates the value to be closer to 0.6. Tdifference is due primarily to the difficulty in locating thvortex centers from the smoke plots; however, as Figs. 56 reveal, the vortex convection velocity is closely matchbetween the computations and experiments.

The approximate acoustic power produced overacoustic cycle by the interaction between the vortex, thelocity field, and the acoustic velocity is shown in Fig.Since the computations are two-dimensional, the powedisplayed per unit depth in the third dimension. This plwhich focuses on the junction between the main duct andlower side branch, was produced by integrating the triproduct@Eq. ~1!# over each computational cell and time aeraging the results. The computations predict that a largesource of acoustic power is produced in the main ductoutside of the side branch, while a concentrated acousticis located near the upstream corner. The details of wherthe acoustic cycle this acoustic power is produced canexamined by solving for the instantaneous approximsound power

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FIG. 8. Instantaneous sound power forSt50.4,u8/U50.5, timing of acoustic period corresponds to Figs. 5 and 6;~a! 2T/8; ~b! 4T/8; ~c! 6T/8; ~d! T ~blacklines denote zero amplitude!.


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taken at different points in the acoustic cycle before tiaveraging. The instantaneous power at four different timare shown in Fig. 8. These four times correspond to 2T/8,4T/8, 6T/8, andT in the smoke plots of Figs. 5 and 6. Foeach time instant in Fig. 8, further details of the flow field agiven in Figs. 9–12, which display the instantaneous velovectors, acoustic velocity vectors, vorticity, and the croproduct2u3u8. The triple product in Eq.~2! can be rear-ranged as

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indicating that the product of the vorticity with the croproduct 2u3u8 is the main component of the acoustpower.

Focusing on the first time corresponding to 2T/8 in Fig.5, Fig. 8~a! shows a small source with peak amplitudeabout 0.05 W/m located near the upstream corner ofjunction between the main duct and lower side branchweaker sink is also located further down in the side branThe instantaneous and acoustic velocity vectors at this tare shown in Figs. 9~a! and ~b!. The velocity in Fig. 9~a!shows a large recirculation near the upstream corner wthe smoke plot rolls up at 2T/8 in Fig. 5. The acoustic velocity in Fig. 9~b! shows that the acoustic pulse is weakpushing up out of the side branch at this time. With toutward acoustic velocity being primarily perpendicularthe flow in the main duct, the cross product of the velocand acoustic velocity in Fig. 9~c! reveals a negative contribution outside of the side branch. The presence of the vo


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tends to augment the negative cross product near thestream corner, while it creates a weaker positive contributfurther down in the side branch. The negative vorticityFig. 9~d! is located near the transition from positive to negtive in the cross product, thus via Eq.~3! producing a sourcenear the upstream corner and a sink below this as showFig. 8~a!. The peak negative contribution of the cross prodis greater in amplitude than the positive contribution, athus the acoustic source at this time is stronger than the s

At the next time step of 4T/8 in Fig. 5, the instantaneoupower in Fig. 8~b! shows a large strong source with a peamplitude near 2.3 W/m located in the main duct nearly ctered on the junction. Comparison with Fig. 7 reveals tthis portion of the acoustic cycle is a major contributor to ttime-averaged acoustic source. Breaking down the trproduct in Fig. 10 shows that the acoustic pulse is pushstrongly out of the side branch at this time. This strong uward pulse perpendicular to the flow in the main duct pduces a large negative cross-product term in Fig. 10~c!. This,in the presence of the vorticity in the junction, yields a strocontribution to the acoustic power. Comparison with Figreveals that the magnitude of the vorticity near the centecirculation in Fig. 10~d! is approximately the same as thvorticity in Fig. 9~d!. The amplitude of the cross product haincreased greatly from Fig. 9~c! to Fig. 10~c! however, whichcauses the acoustic power in Fig. 8~b! to be much greaterthan the power in Fig. 8~a!. Comparison between the locatioof the acoustic sources for these two times with their corsponding smoke plots in Fig. 5 illustrates that the locat


FIG. 10. Components of the tripleproduct @Eq. ~1!# for Fig. 8~b!,St50.4, u8/U50.5; ~a! u; ~b! u8; ~c!2uÃu8; ~d! v @black lines in~c! and~d! denote zero amplitude, white in~d!is due to amplitudes off of color bar#.

FIG. 11. Components of the tripleproduct @Eq. ~1!# for Fig. 8~c!,St50.4, u8/U50.5; ~a! u; ~b! u8; ~c!2uÃu8; ~d! v @black lines in~c! and~d! denote zero amplitude, white in~d!is due to amplitudes off of color bar#.

1351 1351J. Acoust. Soc. Am., Vol. 109, No. 4, April 2001 Radavich et al.: Flow-acoustic coupling

FIG. 12. Components of the tripleproduct @Eq. ~1!# for Fig. 8~d!,St50.4, u8/U50.5; ~a! u; ~b! u8; ~c!2uÃu8; ~d! v @black lines in~c! and~d! denote zero amplitude, white in~d!is due to amplitudes off of color bar#.

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and size of the sources closely resemble the location andof the smoke vortices.

Next at 6T/8 in Fig. 6, the acoustic source has turneda large relatively weak sink in Fig. 8~c!, with peak amplitudenear20.024 W/m. The vorticity in Fig. 11~d! still shows alarge negative circulation above the downstream cornethe side branch, and the strength of the vortex has decreonly slightly. The acoustic velocity has now reversed diretion in Fig. 11~b!, and is now weakly pulling into the sidbranch. This downward pull creates a mostly positive croproduct contribution outside of the side branch, as showFig. 11~c!. The combination of the positive cross produwith the negative vorticity now produces an acoustic sover the downstream corner. However, the weak acouvelocity does not give the cross product a strong contrition, and the sink is relatively weak because of this.

Finally, the last frame investigated corresponds to timTin Fig. 6. Figure 8~d! shows that the acoustic power is mosnegative during this portion of the acoustic cycle, withlocal concentrated peak sink of approximately24.2 W/mlocated at the upstream corner. The velocity in Fig. 12~a!reveals that the vortex has already begun to roll up nearupstream corner at this time. During this portion of tacoustic cycle, the acoustic pulse in Fig. 12~b! is drawingstrongly down into the side branch. This produces a posicross-product component outside of the side branch in12~c!. Similar to the observations in Fig. 9~c!, but with theacoustic velocity in the opposite direction, the presencethe vortex near the upstream corner tends to augmentcross product near the upstream corner and create a neg


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component further down in the side branch. The presencthe negative vorticity in Fig. 12~d! combined with the strongpositive cross product near the upstream corner creatstrong acoustic source near the upstream corner. Becausvorticity is highly localized near the upstream corner, hoever, the strong sink in Fig. 8~d! is confined to the vicinity ofthe upstream corner.

Comparing the figures at different times in the acouscycle, the acoustic source is produced as the acoustic veity pulse pushes upward out of the side branch, whileacoustic sink is produced as the acoustic velocity pulls binto the side branch. Comparing these two conditions in F10 and 12 reveals that the peak amplitude of the negavalues of the cross product from Eq.~3! in Fig. 10~c! isnearly half the peak amplitude of the positive values ofcross product in Fig. 12~c!. The vorticity in Fig. 10~d! occursover a broader area than the stronger, more condensedticity in Fig. 12~d!, which results in the source of Fig. 8~b!being larger in size than the sink in Fig. 8~d!. Examining thetime-averaged acoustic source in Fig. 7 once again shthat the source and sink produced during the upwarddownward acoustic pulses carry over into the time averapower. Once the entire acoustic cycle has been accoufor, a large source remains over the junction betweenmain duct and the side branch with a localized sink nearupstream corner.

IV. CONCLUDING REMARKS

This work has demonstrated the ability of conventionCFD methods to solve coupled flow–acoustic problems fo


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configuration with two coaxial side branches attached tmain duct. A true coupling is computationally possible btween the flow field and the acoustic field, allowing acousoscillations to grow larger than their initial values at certaflow conditions. Computations with several different flovelocities demonstrated the ability to properly identify whthe flow–acoustic coupling should and should not occComparisons with the experimental smoke visualizatwork of Ziada~1993! show the ability to duplicate the motion of the vortex as it propagates across the side-braopening and interacts with the acoustic waves in the sbranch. Using the theory of Howe~1984!, it was shown thata large net acoustic source is produced in the main ductoutside of the side branch due to the interaction betweenvortices and the flow and acoustic fields.

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A computational approach for ﬂow acoustic coupling in