POWER INJECTED INTO A RECTANGULAR PLATE EXCITED BY A TURBULENT BOUNDARY LAYER

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1 OWER INJECTE INTO A RECTANGLAR LATE ECITE BY A TRBLENT BONARY LAYER AC REFERENCE : 3.0.Rj Totaro Niolas ; Guader Jea-Louis Loratoire Vibratios Aoustique INA de Lo Bâtiet t Eupér 0,aveue Albert Eistei 696 VILLERBANNE Cede Frae e-ail : iolas.totaro@lva.isa-lo.fr, guader@lva.isa-lo.fr ABTRACT Te eitatio of strutures b turbulet boudar laers as bee itesivel studied ad differet odels ave bee estlised to desribe te boudar pressure ross spetru. We dealig wit eerg etods lie tatistial Eerg Aalsis EA, oe is ore iterested b boudar laer power ijeted ta boudar pressure ross spetru. Tis paper deals wit derivatio of a odel for ijeted power i a plate eited b turbulet boudar laer. For sae of sipliit it s based o Coros odel for boudar pressure, but ore sopistiated odel a be used. Te estlised odel of power ijetio leads to siple aaltial epressio futio of plate ad boudar laer arateristis. A opariso of te odel wit eat alulatio is preseted.. INTROCTION Te eitatio of a struture b a turbulet boudar laer as bee itesivel studied. everal odels ave bee estlised to desribe te boudar pressure ross spetru eitig a retagular plate. B usig eperietal results, Willart ad Wooldridge [] ave desribed te boudar pressure of a plate eited b a turbulet flow ad Coros [] as estlised a odel of te boudar pressure ross spetru. oe odels, lie te oe of Case [3] or ol aov [], ave bee opared i ters of radiated power b Graa [5]. We dealig wit eerg etods lie EA, oe is ore iterested i ijeted power i sub-sstes. Tis paper proposes to estlis su a id of odel for ijeted power. Te sipliit of tis odel sows soe iterestig properties of te iput power.. ENERGY TRANITTE TO THE LATE BY THE TRBLENT FLOW Figure presets te sste uder stud. Te plate is of legt a, widt b as tiess. Te aterial as a Youg s odulus E, a desit ad a oisso s ratio ν. Te plate is eited b a turbulet boudar laer, flowig i diretio. iretio of te flow b a Fig. : et of te sste uder stud

2 If radiatio is ot tae ito aout, te iput power is equal to te power dissipated i te plate: ij diss ηe Were η is te dapig loss fator ad E te eerg of te plate at agular freque. Te eerg is related to ea-square veloit of te plate b te followig relatio: E V If oe itrodues te displaeet spetral desit ww, te equatio beoes : E ww 3 To estiate iput power, oe as to ow te displaeet spetral desit of te plate. Of ourse, we dealig wit eperiets, radiatio dapig eists, it is owever ofte egligible opared to plate loss fator. I te ase of alulatio, te radiatio a be igored ad relatio is eat, allowig us to alulate te ijeted power ito te plate b te turbulet boudar laer, fro te plate eerg. I te followig, radiatio is ot tae ito aout. 3. CALCLATION OF THE ILACEENT ECTRAL ENITY WW w Te defiitio of te displaeet spetral desit is give b te followig equatio : ww, W, W, Were W* represets te ople ojugate of W. Oe a epress te displaeet as a odal suatio : W, W, a 5 Were W, is te ode sape ad a te odal aplitude give b : F a F H 6 iη Were is te odal ass, F te geeralised odal fore ad te atural agular freque of ode,. δ pq W, Wpq, d F p W,,, d 7 Te, usig equatios, 5 ad 6, oe a epress te displaeet spetral desit suig odal otributios: ww, H H pq W, W pq *, I pq 8 p q Were I pq represets te boudar pressure ross-spetral desit of te turbulet boudar laer. sig Coros odel, oe a epress I pq b te followig equatio : I pq F Fpq ξ, ξ, W ξ Wpq ξ dξ dξ 9 Were ξ resp. ξ are te oordiates, resp., of eitatio poits ad ξ,ξ, is te boudar pressure ross spetru. Oe a obtai ea spae value of te displaeet spetral desit b itegratig over te plate surfae : a b ww ww, dd aig use of epressio 8, ad osiderig odal ortogoalit oe a get for a sipl suorted plate :

3 WW I H Fiall usig te equatios 3 ad, te iput power a be epressed as : 3 η ij I H. THE CORCO OEL After eperiets doe b WILLARTH ad WOORIGE 96 [] o te boudar pressure atig o a rigid plate b a turbulet boudar laer, Coros as proposed te followig law : i e B A,, 3 wit A ep ad B ep Were is te boudar pressure ross spetru, te logitudial positio diretio of te flow, te traverse positio, te ovetio speed of te flow ad ad are ostats. ifferet epressios of te ovetio speed are give i [8], owever i tis paper, te siplest odel will be used : K Were K is a ostat ad is te flow speed. I te Coros odel, ad are obtaied fro eperietal tests. Aerodai oiidee peoeo ours we plate bedig wave speed is equal to te ovetio speed. It produes aiu eitatio of te plate. Te oiidee agular freque is : 5 Were is ass per uit area ad te bedig stiffess of te plate. 5. CALCLATION OF THE INT OWER Equatio gives te epressio of te ijeted power ito te plate. I te followig, oe uses te siplified epressio 7 proposed b AVIE 97 [7] : I 7 Were et are ode wave-ubers : a ad b Fiall oe a epress te iput power b taig ito aout equatios, 3, ad 7 : η η ij 8

4 6. FREQENCY AVERAGE INJECTE OWER Eerg etods use freque averagig ad do ot eed te eat owledge of atural frequeies, but just te uber of odes i a freque bad. I tis paper, oe proposes te derivatio of te freque averaged ijeted power epressio, based o te sae id of assuptios ade i eerg etods. Te ijeted power give i equatio 8 is averaged o a freque bad etered o. Te otatio for tis operatio will be : d Let us suose te bad of iterest is at ediu or ig freque ad arrow eoug to ave <<. Tus, oe a aroiate equatio 8 b te followig epressio : η ij η Tis aroiatio is justified beause, i te bad of iterest <<, te ters i te itegral do t varr u eept te oe represetig te odal respose for atural frequeies loated i te eited bad. oreover, tis ter produes te doiat otributio of te odal suatio. I order to redue te alulatio to te doiat ters, tat is to sa to te resoat odes i te bad of iterest, oe a liit te su to te ode idies ad verifig : 0 a b Ad, as << a b Tus, te idies of resoat odes satisf relatio : 9 b a I additio, ide of resoat odes eists if : a a 3 Oe a alulate, wit equatio, te odal desit of resoat odes avig a fied ide : d b d a Te uber of odes of ide, resoat i te freque bad [-, ] a be sipl alulated : N. 5 Let us aroiate equatio 9 restritig odal suatio to resoat odes. η η η Equatio 6 aroiates te itegral fro - to b te itegral fro 0 to ifiit. It s aurate we te pass bad of te ode is sall opared to te eited bad >>η. 6

5 aig use of equatio 6, oe a get equatio 7 : ij η η 7 A seod aroiatio a be doe usig te uber of resoat odes of ide give b epressio 5 : a ij b 8 Oe a otie tat tis epressio is idepedat of te badwidt. I order to obtai a ore opat epressio, oe a aroiate te su b a itegral. Let us also itrodue te age of varile to a. Oe obtai te aroiate equatio 9 : _a 0 ij d 9 Cagig agai of varile to, oe obtai : Ψ ij 30 Were d Ψ 0 Futio Ψ / depeds ol o / ad tus a be alulated ueriall as a futio of tis paraeter. ifferet values are plotted figure 3, usig iterpolatio based o splie futio. Fig. 3 : Nuerial alulatio of te itegral i futio of f/f Futio Ψ / ireases before f/f of 5dB per otave ad is ostat we f/f teds towards ifiit. Te aerodai oiidee peoeo ivolves a aiu of te itegral. 7. RELT AN ICION Figure sows te opariso betwee uerial alulatio of te ijeted power wit equatio 8 ad te alulatio of te averaged power wit equatio 30 :

6 A B Fig. : Copariso betwee te uerial alulatio of te iput power ad te alulatio of averaged power. Eaple : a, b0.7,, 00s -, steel, freque bad 00z-7000Hz. B : averaged o /3 otave bads. Results opare ver well espeiall for /3 otave bad averagig. Oe a otie tat te averaged ijeted power is alost ostat for freque lower ta aerodai oiidee freque f ad dereases ove of 6dB per otave. Te aerodai oiidee peoeo does ot ipl a real irease of ijeted power. I additio, oe a see tat power flutuatio wit freque due to resoaes, is related to odal overlap ad ot to boudar laer eitatio. Fig. 5 : Copariso betwee te uerial alulatio of te iput power ad te alulatio of averaged power. Eaple : a.5, b0.5, 5, 50s -, Aluiiu, freque bad 00z-7000Hz B : averaged o /3 otave bads. Tis is obvious oparig wit eaple. At a give freque te plate uder stud as a lower odal overlap ta te oe of eaple. Let us osider odal overlap equal to, it ours we f/f is equal to 0.7 for eaple ad to 6 for eaple. For bot ases, flutuatio wit freque of ijeted power is aroud 0dB. CONCLION Te derived epressio for freque averaged ijeted power ito a plate eited b a turbulet boudar laer opares well wit odal alulatio. Te epressio relies o a itegral tat is ol depedat o te ratio / ad a be tulated. Tas to tis forulatio, oe a easil uderstad te ifluee of te arateristis of te plate ad te flow o te level of iput power. Notl te freque averaged ijeted power is idepedat of dapig. REFERENCE [] W. W. Willart et C. E. Wooldridge 96. J. F.., p87-0. [] G.. Coros 967. Joural of oud ad Vibratio 6, p [3].. Case 980. Joural of oud ad Vibratio 70, p9-67. [] A. V. ol aov, V.. Taeo 99. oviet sis Aoustis 376,p [5] W. R. Graa 997. Joural of oud ad Vibratio 06, p [6] W. R. Graa 996. Joural of oud ad Vibratio 9, p0-0. [7] H. G. avies 97. Joural of oud ad Vibratio 5, p07-6. [8] G. Robert 98. Tèse de dotorat.