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Repett No. AM 08-001 ity, College of Eaginecring WALL-CAVITY AEROACOUSTICS AT LOW MACH NUMBER M.S. Howe Boston University, Cateye of Engineering 4110 Cummington Street, Boston MA 02215 Report No. AM-04-(01 3 January, 2004 Final Repert Prepared for Dr. L. Patrick Purtel Office of Naval Rasearch, Cade 333 Grant NIKIOT4-02-2-9540, DISTRIBUTION STATEMENT 4. Approved for Public Release ‘Distribution Untimited ' $6) ogbov0d2 Report No, AR 04-001 Boston University, College of Enginee CONTENTS PREFACE, 1, MECTIANISM OF SOUND GENERATION RV LOW MACH NUMBER FLOW OVER A WALL CAVITY. . a 2. WALL-CAVITY ACOUSTIC GREEN'S FUNCTION AT LOW MACH NUMBER EY 3, CAVITY MODE EXCITATION BY VORTEX SHEDDING FROM A CROSS-BEAM a Report Me. AM 04 001 Buston University, College of Engineering, Preface "This report documents analytical studios performed tn elatfy the mechanisms by which ond is geverntad hy notainally stendy'llow over a rectangular wall cavity. Chapter 1 (Meshanin accepted for publicalion in the Journal uf Sound and Vibrotion. Chapter 2 (Watlcwvity acoustic ereen's funtion al fuse avach number) hass buen published in thi: Jafemnational owe} of Aeronwnunlaes 2, 347 - 965, (2008). Chapter 3 (Caving mode exeitation shedding from a cross bear) has hoon written with the assistance of Ms Alia Winslow and has beon submitted far publication under joint authorship in the Iulernational Joxrnat af ‘of sound genaratinn by low mach mmaber flow over a wall ewvity) has been Aeroaceustion Report No. AM 04-001 Boston University, Collage of Engineering, CHAPTER 1 MECHANISM OF SOUND GENERATION BY LOW MACH NUMBER FLOW OVER A WALL CAVITY Report No. AM 04-001 Boston University, College of Enginewing SUMMARY An analysis is movle uf the mechwniwn of sound provluction by nominally sLeady low Mach umber flow ovr 1 rigid shallow wall cavity. At wry low Mach nuibers the domrigant souroe of sinind is the onateady drag, arul Lie aevagcoustic dipole source accu panying this foree. A monopole nnutee dependent on the compmeasioe of fuid within Lhe’ cavity w smaller by a factur of the order of the How Mreh number Af, The diectieity of eh dipole somnd peaks in diteetions wywircam and downwtivam of the cuvity, and there és a radintior ruil i the direction noemal to the plane of the wall. However, sumerival simulations for 3 as small 43.4 hav: predicted sigaileant redivtion in directions wostaal to the wall, “his anomaly is investigated in Ubis chapter Ivy sncane of en atanistic Green's funcLion tailored to eavity geometry thal accounts for possible aeruacaustic contritmtivns from both the resonance, ‘The Creeu's funetiem ix used to leagedipote ssid fromm the lems! order cavi show thud those sources ate correlitel and thet their slreugths are each proportions to the unsteady drag generate ly vorticity fnterncling with the cavily trailing rage. When M-~ 0.01, the case in mint uaderwater applications, the monopole strengl is alvays negligthle (for a cavity wilh rigid solls). At low Marh aumbers excieding about 0.08 it is stunwn that the cavity stonopale: radiation Is OCM2] <1 relative to the dipole at tow froquencies. Al higher frequencies, near the rasonance frequeacy of the ewvily, the monagWl and dipeit: have similar ondets of ringnituda, and the coubinatiem produces a relatively niforau radiation ditectivity, wil substantial enorgy radiated in ditections normal to the ‘wall, Uuotrative numerical reals are given for a wall cavily subject te ‘shear layer mode! excitation Iyy the Rostiter feedback’ mechanism. Report No, AM 04.001 Boston University, Collage af Engineering 1. INTRODUCTION “Tonal radiation produced hy high Reyachls sumiber nan llow over a rectangular wall ceavily was originally attributed 4a broadband excitation of eavily acoustic rewmnances by uthulence in the shear layer over the cavity reouth fl). Tlowewcr, escillations can also be uaintained by a famirar mean flow, and lazainax flow resinanres ate often observed to be: rors intense [2, ‘Lo cavicy length {ligure 1) generally bear tlle vr no correspwindeuce to cavity modes and are not usnally harmonically relate, Int re mate clavely analogous bo the ‘edge tones exeiied -eneraled by shallow ssvities whose depth dt <= streanansise shen a thin jet impinges on a werlgnsbaped knits elgo, and snaintnincdl via a ‘fendack” mechantor freun Uo wedge to the: jot nozzle A cavity tone of frequency J geacrated by flow of moan stream velocity 27 is rygically ound to lie within certain well defined hands of the Strontod amunbers 7/7 when plotted agsinut flow Mach amabet. Tais iv consistent with the ‘Teedback? scheme proposed by ossiter [1]. involving the periodic formation uf diseeste vortires just dowestrenr: al the leading erige of the cavity, and their schwxquest inceraction with the trailing otge after romvecticn across Ue cavity mewl’s, ‘he inipulbive: siued generated! tiy Ubis interaction propagates upatream wit and omses the boundsry layer to separate just “upstreue of the leseing flye. The travel Utne of a vortex noross the cavity ~ 2/8, where f= 041 - 60%, and the sound radiates back to the lading alge i time: Ley, where rg is the spéed of sind, The remneniag sound theryare arcives fm time i th i the convection velucily in tvinforve posiodie eheddting provided £ aatisties [3] vee (uy ‘This formula erm! be adjusted bu wldain detailed agreement with caperiment [4-6], bY replacing n by #8, when: 2 (~ 0.28) determines a ‘phase lng’ 3/F equal ko tke time delay between (jj the arrival na vorter at thr ailing ede auc the emission of che main acoustic fh ao fos aud iy i8 che rai of specific uoace of the ld. RaW Qe (12) there M a Report Mo, AM 04-001 Boston University, College of Engincoring Drodioronn of the feedback feu (12) for shallow, reclangutor cayiles with 4/2 < | gree well wich obseevalions at M > 0.2 for fxs} and Ui/0! 0.616, The eoatebatons from cavity rencnanees ars important only for doep cavities, and appea lo be unbmportat anles 4/7. resonances cen dominate dhe radiation provided the Slioute! nue he satistce (Lat). An extensive discussion of exporimoutal estlts relating to this wnd thee faffuences of cavity geomndey abd man lene condbtions cm cavity resonances ix give by Ahuja ane Medora [6] for Mach numbers Mf > D2. Rewarch prior to 1980 i reviowed ‘yy Bickel [4] and by Rockwell & Noudaschor [7]. wnd Grave (85 hon wuusarized recent alzompts to wnnulale Bumerieally eaéIk¢ noise raiialion dove nominally wfeady xpotiments conducted by Gharib and Restiko [6 in water with a ianpinging mean flow (at U = Linjs) have shover, Hat Ruwviley Gendback resonances are rela.ed to large flucluations in fhe dag experienred ly’ the cavity. Thy identified twin hydrodynamic modes of rvity flow oeillstions: for ‘shorter’ cavities relative ta Uke upatscasn boundary layer thickness (and, aerording Lo later work [Lf], for lower Via mumbers) Use unsterdy motion over the cavily mouth has thu: characterinries of an unstable, Lin shear layee (‘shear Jajer mod?) chat generates sound by impingetment on rhe eavity tailing aye, essentially in tke mienner propaed by Rossiter [1], Recent smmerical stndies of tno-diniensional cavity ws hig Colonins et at, [20] predict strong madivtiom preferentinlly in the upstream dirootian, frum a ‘source! central on che eavity tailing ‘alge. “‘'hree-dinweasional numerical clmulavions peotocmod lay Fnglsang and Cain [11] of che acoustic fiold withon w shallow caxity (Fd ~ 48} at Af = 0.85 also indated thar har layer instability is Lhe main oxibiug techenism, ad that ft pridutes a source-ike periodic addition and remoral of mass near the trailing odge. An analytical, but empivial represeutation of this edge source had been previously comiidoted by Tarn snd Block {12 Flow over longer cavities (or, sleeruatively, at higher Mach numbers) is charactariaal by a ‘wake mede!, involving large seals vorticity ejection from the cavity. producing, quasi-poriadie separation upstream wf the lending ergs. ‘The ouset of le wake mode accompanied by « lixge increaue in drag ductvatians, This appscntly occurs also a much Iighor Mach mmmbexs. For exaenplo, numerical siulations by “chang [18] for Lye = 3 and M = 1.5, Sxoveal that the violent ejection of vorticity ix slzongly correlated with sign revennls of the eavily diay enetfiient ‘tho eosults of the shesr layer mode thay of Tam and Lilock {12] engyrsted chat at cavity aroun sesonances ums. contsibmte (0 tke very Jow Mach murobass (8 < 0.2) 4 Report No. AM 02.001 Boston University, College af Engineering radisticn, especially for deeper cavities (fd < jy sayl i resonanees with! aocount fir a near ornislizoctionsl character of Ul: radiation pothorn smonopule! nature of auch clwerved at cerlain frequentiog. They Ail not puree Ais theoreticlly, although its likely importance: bad alteady been antyeipuled fy Plumiblee ot of {L4] seul ley East [48]. Salley, experinncnis of Yu [16 have confirmed that ehatlow wail eavities in aix st Maal anmbers M ~ 02 radiate a substantial amcunt of radiation direculy away from the wall, Numeral ‘and experimental stedies al low Mach muubee by Tragaki of uf [17] have: confirmed tis concliwinn for large cevilies with srinll apenings ti the mean flow, but stown star how eniucidonce betwen the cnvity cesanance and the Roseiter frequetey predivtod by (1:2) resulted in wary large aruptiuude radiation, For shallow cavities trailing edge ‘scattering’ of shoar layer pressure fhickantious appeany Ca be the souiisant source. avon when fedbark is not ftnpovtant, ‘This in in accond wilh moacuremeats performed by"Jacabs et al. [28] for bjt > Tand M < 044, for whieh Ihe radiation peakod in the upstream and rowesticam ditections. although vigeilivane radintina la ‘the wallawrmal direction vaus also observed. Ancondling to the theoretical resulls of HTowe 22], the rariativa, rom a shail cavity a4 very low Mach number can be ascrilu! 1c a dipale soarte aligned with the mean flow direction whoce strength is determined by the mnsteady deag, The dipale couree whvemati. i strongly coupled (0 the hytrodyaazaic motions In and near the cavity, bul. js essentially the same ia character for both the ‘shear layer" and "yeke" modee af dhe cavity oscillations, provided M i sulfcienthy srl, The intensity of the dypote radiation pes in thé upstream al on the wal and domnstreen directions, and i wall ia direction ‘This conclusion is apparently incompatible with several af Uke experiments discussed shove and with reoant mumerical simulstions and obgervalioug at low Mock mmmbers. tn Hardin and Pope's [14, 20] low Mark wmaber sdleme, aa incompressible repreventolicn, of the eavity lve is fret simulated umerically, al Ube results 90 uber wed tr ovalnete scowstic ‘sources’ in a modified system of ewmprossible low cquations. AL At =O ther predirtions yield cadiation direchivitles that peal in the upstream: direstion, but alin exhibit ont with the prosenoy of & can substantial level in dizertions. uneamal wo the wall, eons miauopole field. Althongh various details of the approach in 119, 2 and subjected to weslifeation, for cxcmunple ly Hkaterinari [24] and by Shon ane Sura: 2, cho generat characterises of the predicted ralstion are probally eneect iw ant overall sonee, not in detail. Indeed, they accord wich later auunrxieal studies (also based on an ial deteration of an incompressible apyroxionation of the cavity flow} by Graow ef af j have been criticized Report No. AM D001 Gaston University, College of Engineering. [2a] ana by Chrtin Ceauda 124), that similarly prediee large amplivade redintion woorsel fo the wall 1, botwoen, ‘The purpose of the present chaplee is Le resolve those apparent inconsisten« merical and analytical predictions at low Mach uunnbers. Tt will be shown that the low Mach number dipole radiation, pesking in discerions upwirsaun and dowrstcamn of the cavity. is indéed the draainant source aL very low Mack uumbers, tepically auch amnaller than, M =041, howercr. Thus, ic is this source thar determines ewvity radiation in uneerwalet applications (hote M ~ 0.01) yrovided, of caurse, that Qu: cavity walls are slficionthe gid’ to prochnde monopole souren: prculuced by pulsations in the cavity volume, Bur, whem M ~ 0.1 we shall show that the cavity Helmbats anode, slehough very seak and farecaSly vanishingly sual as f£/U -> 0, supplies an additional, ommi-tirectional covtrihntion ther can excued the drag dipole eadistion aver a range of requcicies. Farthwaucue, 3 will br show that the manopole and dip scurce strengths are both slvtecmined at lnwr Much, rmmibuas ty the enviny drag fluctua ‘Dhe low Much nurnber snags will be frame in tens of thu theory of vortex sound fa], and the relovant equations are rwslte in (2. The porsille source types ace identified by introducing an arauslic Green's fuaction that jx valid in the presence: of low Mach rauaber mean strtatt flow pratt he cavity (§0}- AL vou low Maar numbers the neem amplitudes are alvaye sonal! enough for incommpeessils fli to be reper ws aa exeellen first approvimation to the muon in the exvity. Eis flow deteranines the effin: vortex: sound source strengths, ixespective of whother the this is characterize as ‘shear a sake! merle, Predictions of thy theory are therefore ilustrated in §4 for the witmpler ease of shear mode flow by meaus of an idealized modal of shear layer excrtation Report No. AM DE-O01 Boston University, College of Engineering 2, FORMULATION Consider nominally steaily, ow Much aumber, high Reynold mamber inean Alene in the pesitive ay-direction of the rectangulat ooordinaues (04.24.59) over the reeiapgulor wall cavity of Figure 1. The wall and the intusior surfaces of Le cavity ave assamed to be rigid "The fluid has mean slonsity and woul speed respectively equal t» fy ¢y, and fhe wetocity jw the main stream is Z. The eavicy tis epi and breadth. 8, and is aligned with is de of lergth F, parallel vo the mean flow. The couidinate origin is taken at Q in the plane of the wall al the centre of the cavity mouth, with th xy-axis normal to tho wall and directed inte the main etacam, ‘Sound 54 produced fy Hla instability in the nedghbonehowl of the cavity. According to Lighthill's accuscie analogy [2], when the lola] eathalpy B, say, i tabon as the scoustie variable, the radiation enn be exprossed in terms of sources that represent exritaGion by vorticity and entropy @nefuivions, For # aotainally Remogeavous Row af tow: Mach numbers the motion may be rogirded as hwzaenteopie ts # good approximation [25]. Tn that case Ube ja Bape ple) the prawuze, and v densiles velocity, aral Lighthilts total enthatpy becomes Le oo ea) share p ie fluid density, atime analogy mation. becomes D(1Py_1 1 2 (APY Ly. ipm) 2 —baintow a9) 2 & (a2) -3v wos)) Laivtow A, 2) chore eth loal speedo round: To th tational an fae Bet Om’ forms of the morn equation Bvt = TB implies thal = ~dpfy, where ye) the tity potoutit thie dotermines th whole motion in te utstiveal regions a th Mid fio, and arg dstanoss fom ee is thovefore exgual Le & constant in a steady mies sontees perturbations in 2 represent outgoing scund waves c, ix sunaller than about ity wall In the parligalar ewe of los Mar’ munuber flow, whew Mf = sa that AW? £1, the charwclerstics of the anation within aud ose to the ble, the acoustic carmpant constitu Ine essentially tho same ap iF tho shud ie ruses fa very snuill porturbation about this motion, We can Ukem repli p and ¢ were Ley occur explicitly io (2.25 Uy their restive menn values fy and cy. On the Teft band olde we ean also introduce the apprximation (snlid te ist order ia Jf