19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 DETERMINATION OF THE SOUND RADIATION OF TURBULENT FLAMES USING AN INTEGRAL METHOD

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1 9 th NTERNATONAL ONGRE ON AOUT MADRD, -7 EPTEMBER 7 DETERMNATON OF THE OUND RADATON OF TURBULENT FLAME UNG AN NTEGRAL METHOD PA: 43..Rz Pscoya, Rafael; Ochma, Mat Uvesty of Aled ceces; Lembe t., 3353 Bel, Gemay; scoya@tfh-bel.de, ochma@tfh-bel.de ABTRAT A effcet way to deteme the fa feld adato of tblet flames s by s hybd aoaches that cole fll o lea flow eqatos solves FD codes wth lea oaato acostc methods. Oe ossble acostc method s the Lhthll s acostc aaloy whch eesses the sod esse tems of a volme teal ove the sod soce dstbto. Ths method has a hh comtatoal cost sce the volme occed by the sod soces has to be dscetzed. Pose of ths stdy s to develo a method that edces the tme of comtato by ewt the volme teal tems of sface teals aloe. ths wok, the basc deas of the method ae eseted ad the accacy of the ocede s tested s a smle cofato that has a aalytcal solto. NTRODUTON evos woks [],[], a hybd method col a comessble Lae Eddy mlato LE wth the Eqvalet oce Method EM ad the Boday Elemet Method BEM has bee sed to deteme the sod adato of oe tblet flames. The velocty dstbto ove a cyldcal sface cotol sface sod the flame was obtaed by the LE ad tasfeed to the EM ad BEM as a Nema boday codto. Fom the sectm of these steady data, the sod owe ad the adato attes of the flame wee comted a feqecy ae eted fom 4 Hz to 5 Hz. odtos fo the valdty of the method ae: all soces shold be eclosed by the cotol sface ad otsde the cyldcal sface, the medm shold be homoeeos. The fst codto s ease to flfl tha the secod oe, atclaly by et flames whee the eo of o fom mea velocty cold eted dowsteam tes of tmes the ozzle damete. o case, the sze of the LE comtatoal doma was eteded as lo as ossble ty to dmsh the effect of the o fomty of the medm ad kee the calclato tme easoable lmts LE comtatos wee wth oe ocesso. omaso of the mecal eslts wth measemets showed a oveestmato of the sectal sod owe at mddle ad hh feqeces. Aalyss of the testy secta dffeet ots aod the flame sests that the effect of the homoeety of the medm may ot be elble. Ths wok eteds to move o hybd method by cosde sod oaato homoeeos medm s the acostc aaloy bt avod the dect evalato of the thee-dmesoal volme teal. By oe flames, the fomato abot the eqvalet soce tems shold be take dect fom the FD calclato, whle by eclosed flames, these soces may some cases have to be modelled, sce some codes, o data s avalable otsde the combsto chambe. DERPTON OF THE METHOD We cosde that all acostc soces of the flame ae located sde the cotol sface whose omal vecto s ot to the otsde ad comletely ecloses the flame see F. a. At the sface, the velocty feld s ovded by the FD calclato. Otsde, thee s a homoeeos eo the sace of volme delmted by the sface wth omal vecto also ot to the otsde, whose desty ad sod velocty vay locally, ρ, c.

2 To deteme the sod adato, the sace s dvded two eos ad whee the follow dffeetal eqatos have to be solved: NL k D Reo k Reo Eq. whee kω/c s the waveleth ad D NL eesets tems cota all o homoeetes. ce the sod seed s ot costat, k deeds o the osto. The boday codtos at the teface betwee eos ad demad cotty of esse ad atcle velocty, o Eq., o ρ ρ Fe : Descto of the flame model Follow the oal acostc aaloy, the dffeetal eqato eo ca be wtte as: k Qω Eq. 3 NL wth Q ω k k D ad k a costat abtay wave mbe. We ote, that the soce tem at the ht had sde cotas also the esse less kk,.e. the esse shold be kow. Fo the follow devatos, Q ω s cosdeed to be ve. O ew model has ow a homoeeos medm sod the cotol sface ad a addtoal soce dstbto Q ω see F. b. Us the sal boday elemet ocede, the dffeetal eqatos ad 3 ae tasfomed to the teal fom: d d Qω dv Reo d Reo wth kr kr e e,, R y 4πR 4πR whee defes a feld ot ad y a ot at the sface, ad the costats: Eq. 4 9 th NTERNATONAL ONGRE ON AOUT A7MADRD

3 otsde o.5, otsde o.5 Eq. 4, we ca ecoze the volme teal that ceases the comtatoal cost of the actal eesso fo the sod esse. Accod to the theoy of dffeetal eqatos, the eeal solto ca be wtte as the sm of a homoeeos ad a atcla solto of Eq 3: h, whee h s the solto of the homoeeos eqato ad flfl the boday codtos, ad solves the homoeeos eqato bt does ot flfl the boday codtos. Fo the atcla solto, a elato smla to 4 ales: dv Q d d ω Eq. 5 We ca move the sface teals of Eq. 5 to the left sde ad the volme teal ca be wtte tems of sface teals. dv Q ω set 5 4, the ew eesso fo the sod esse eo s ve by: d d d d Eq. 6 Eq. 6 demostates that f a atcla solto of the homoeeos dffeetal eqato s kow, the esse cold be wtte tems of sface teals aloe. Fo most hybd aoaches, the soce tem Q ω s kow fom the FD calclatos ad ot. Hece, the atcla solto has to be detemed. A sal way to aomate a fcto s ead t a sees of bass fctos α Eq. 7 Whe we elace 7 6, we obta the: d d d d α Eq. 8 whee the coeffcets α ae stll kow. Now we ca se the fact that the soce tem Q ω s ve. ce s a solto of the homoeeos eqato, we ca deteme the vales of α fom the follow elato:, k f f Q α ω Eq. 9 9 th NTERNATONAL ONGRE ON AOUT A7MADRD 3

4 ad by dscetz the sfaces ad, a lea system of eqatos ca be dedced. The same elatos 8 ad 9 ae deved de the cocet of the Dal Recocty BEM [3] bt evese ode, stat by f the fctos f The sccess of the method deeds obvosly o the set of fctos sed ad how ood the soce tem Q s eodced. Fom Eq. 9, t s clea that the set of bass fctos ca ot be the soltos of the homoeeos Helmholtz eqato. NUMERAL EXAMPLE The accacy of the method has bee tested aly t to comte the sod adato of a shecal flame [3]. The flame model cossts of a shecal volme of hot as wth ads a, desty ρ, sod seed c ad a sod soce dstbto Q ω Q, whch s costat fo all feqeces. The flame s soded by a wth costats ρ ad c see F. a The aalytcal solto has the fom: Q A k / k Te k / a a Eq. ad the costats A ad T ae detemed fom the boday codtos Eq. a b Fe : a hecal flame; b Dscetzato ots The cotol sface Eq. 6 s ve by a whle thee s o cotol sface. The ead fctos chose to defe whee the same sed [5]: cos k, y Eq. 4 k k wth coesod fctos: f Eq. The flame sface was eeseted by a shee wth 64 elemets. Fo the detemato of the coeffcets α, besdes the elemets at the shecal sface, L ots the teo of the shee whee take F. b,.e, was aomated wth a total of 84 fctos. 9 th NTERNATONAL ONGRE ON AOUT A7MADRD 4

5 Fe 3: omaso of the sod esse at dffeet ostos F. 3 shows a comaso of the aalytcal ad theoetcal vales of the sod esse. The aeemet betwee aalytcal ad mecal eslts s ecellet. Oly at the teface betwee Reo ad, the eo of the mecal calclatos s otceable, bt ths eo does ot affect the sod owe see F. 4. Fe 4: od owe level the feqecy doma 9 th NTERNATONAL ONGRE ON AOUT A7MADRD 5

6 UMMARY A method to calclate the sod adato of flames cosde the oaato homoeeos medm based o a teal fomlato has bee eseted. The velocty feld at a cotol sface sod the flame, has to be evosly detemed wth a FD code, fo eamle ad the homoeetes of the medm have to be eeseted as soce tems. The ood aeemet betwee mecal ad aalytcal eslts obtaed fom a smle case ecoaes the alcato to moe comle cofatos. AKNOWLEDGEMENT Ths wok was soted by the Gema Reseach Fodato DFG wth the Reseach Ut ombsto Nose. Refeeces [] H. Bck, R. Pscoya, M. Ochma, P. Költzsch: Pedcto of the od Radated fom Oe Flames by ol a Lae Eddy mlato ad a Kchhoff-Method. Poceeds Fom Acstcm Bdaest 5 [] R. Pscoya, H. Bck, M. Ochma, P. Költzsch: Alcato of eqvalet soces to the detemato of the sod adato fom flames. Poceeds 3th teatoal oess o od ad Vbato - Vea 6 [3] P. W. Patde,. A. Bebba: omte mlemetato of the BEM dal ecocty method fo the solto of eeal feld eqatos. ommcatos Aled Nmecal Methods 6 99, 83-9 [4] D. G. hto: Mode methods aalytcal acostcs, hate 3. e-vela, Lodo, Bel, 99 [5] E. Peey-Deba: Aalyss of coveece ad accacy of the DRBEM fo asymmetc Helmholtztye eqato. Eee Aalyss wth Boday Elemets 3 999, th NTERNATONAL ONGRE ON AOUT A7MADRD 6