Department of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, Kansas 66506, USA

Transcription

1 Electnic unl «Technicl Acustics» 006, 8 Z. C. Zheng, B. K. Tn, W. Li Deptment f Mechnicl nd Nucle Engineeing, Knss Stte Univesity, Mnhttn, Knss 66506, USA On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins eceived , published Bth cmpct Geen's functins nd sympttic expnsins e widely used t nlyticlly pedict sund geneted by lw Mch numbe M << fluid-dynmic suces, whee the custic cmpctness f the suce egin is stisfied. By mthemticlly investigting the detiled ssumptins invlved in ech f the tw methds nd by using tw clssicl exmples f flw nise pblems, it is shwn tht the pplicbility f cmpct Geen's functin is esticted t eceive lctin,, t the custic f-field with / c whee is the fequency nd c is the speed f sund, nd tht the slutin fm mtched sympttic expnsins cn be pplied less estictively stting t / c ~. Significnt diffeences between the tw slutins e shwn when / c ~. In the custic f-field, the slutins fm the tw methds e nlyticlly pved identicl.. INTODUCTION Afte Lighthill [, ], sund geneted by fluid flw cn be diectly elted t the ne-field fluid dynmic pmetes. Using Lighthill's nlgy, tw ppches cn be develped t nlyticlly pedict lw-mch-numbe-flw sund: cmpct Geen's functins CGF nd mtched sympttic expnsins MAE. Exmples f using these tw methds include but e nt limited t vtex elted f-field sund e.g., [3 7]. Bth f the methds tke dvntge f the lw-mch-numbe cnditin nd the custic cmpctness f the suce t fcilitte the mthemticl nlysis. weve, thee cn be cnfusin egding t the pplicbility nd limits f these tw methds. Pticully, in mny cses the cnsistency f the esults fm these tw methds is nt edily bvius, situtin tht deems ceful investigtin. It is eltively cle tht in using MAE, the sptil dimensins in the custic egin e educed with fct f Mch numbe M t mtch the ne-field expnsins, but it is nt the cse tht the CGF is tuncted t the sme de f Mch numbe. The gl f this ppe is t investigte mthemticlly the detiled ssumptins invlved in ech f the tw methds nd t st ut the cmptibility nd, t the sme time, the diffeences between them. Tw clssicl exmples, flw ve n scillting cylinde nd vtex utside cylinde, e used t demnstte the cmpisn pcedues. Smple esults e clculted f explntin f the nlyticl slutins. Cespnding uth, e-mil: zzheng@ksu.edu

2 f 3. COMPAISON OF ASSUMPTIONS EMPLOYED IN TE TWO METODS In bth f the methds, the flw is lw Mch numbe, M <<, whee M U / c, U is the chcteistic velcity f the ne-field, nd is the speed f sund. Since bth f the methds e used f custic clcultin, the f-field cnditin must be stisfied, which is >> l whee is the distnce f the eceive lctin fm the suce, nd l is the chcteistic length f the suce. The custic cmpctness f the suce is ls equied in bth MAE nd CGF. Tht is, l/ c ~ O M <<, whee is the fequency nd O mens the de f. In the MAE methd, t mtch the ne-field flw with the custic field, the sptil dimensins in the custic field is usully educed in the de f M, i.e., c X i Mx i, whee x is the eceive lctin in the custic field, nd X i is the escled ute sptil dimensins t mtch the inne flw slutin. The time scle in the ne-field is in the sme de s tht in the custic field e.g., [6, 7]. It is then cle tht in the MAE methd, ll the ppximtins e in O M. The eceive distnce in the custic field is O l / M wy fm the suce, ~ l / M. In dditin, becuse the expnsins nly pply t lw Mch-numbe flw, it implies the cmpctness f the suce, Eq.. This cnditin esticts the chcteistic size f the nefield suce egin t guntee tht the cnsideed custic wve length is t lest f O l / M. Ntice this estictin is n the size f the suce, the thn n the eceive lctin in the custic field. The estictin f the custic eceive lctin is impsed in Eq., t be O l / M, which is in the sme de s the custic wve length. In cmbining Eqs. nd, it cn be deduced tht / c ~ O. 3 In the CGF methd, mthemticl equiement f the eceive lctin,, hs t be stisfied e.g., [3 5]. weve, it hs nt been explicitly indicted t wht de this equiement needs t be stisfied in tems f Mch numbe. A ceful exmintin f the mthemticl guments emplyed t deive the CGF shws tht this equiement impses n ext estictin n the eceive lctin: >>. c This mens tht the eceive is lcted t the custic f-field. Tht is, the eceive distnce hs t be much lge thn the cnsideed custic wvelength. Evidently, the cmpct suce cnditin, Eq., nd the lw Mch-numbe ssumptin themselves d nt guntee tht Eq. cn be stisfied t the sme time. In de t stisfy bth f the cmpctness nd the f-field equiement Eq., the fllwing cnditin hs t pply: i On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

3 3 f 3 >>. 5 l M By cmping the bve equtin with Eq. 3, it cn be seen tht, while nly l / ~ O M needs t be stisfied in the MAE methd, me estictive cnditin f l / ~ O M hs t be stisfied whee n is t lest gete thn unity. While bth Eq. nd Eq. 5 equie tht the eceive lctin is f wy fm the suce when M <<, the pplicbility egding t the eceive lctin is me estictive in n CGF l / ~ O M thn tht in MAE l / ~ O M. At the custic f-field, the slutins fm the tw methds shuld be identicl since bth f them descibe the sme custic signl fm the sme suce t the custic f-field. In sme cses, these tw slutins hve the sme expessin. An exmple is the custic pessue pduced by vtex ne hlfplne. In this cse, the esults fm CGF by we in Sectin 6... f [5] nd fm MAE by Cightn in [7] e exctly the sme. weve, in tw clssicl exmples, i.e., n scillting cylinde nd vtex utside cylinde, the tw slutins hve diffeent expessins nd the cnsistency between the tw slutins is nt esily shwn. They e discussed in the fllwing tw sectins t futhe illustte the diffeence in the pplicble limit nd cnsistency in the custic f-field in the tw slutins. 3. SOUND GENEATED BY AN OSCILLATING CICULA CYLINDE The nlyticl slutin f this pblem ws given by Dwling nd Ffwcs Willims [8] which ws develped bsed n cnsevtin f mmentum f inviscid flw. F cicul cylinde with dius f scillting t expessin f the pessue fluctutin is p xt, ρ ε c cs ' / c e whee it 0 '' 0 / c x ε e nd it x 0, s shwn in Fig., the, 6 0 x + x is the distnce fm the igin, nd is the nkel functin f the secnd kind f de ze. Ntice tht this slutin cn be pplied t bth the ne-field nd the f-field, unde the ssumptin tht the flw is inviscid nd lw Mch numbe. K [6] used the MAE methd f this pblem nd deived the custic pessue fluctutin. The detils f the pcedue e in [6], nd the esult is ewitten hee in dimensinl fm: 3 ρε p xt, i cs / c e c it, 7 whee is the nkel functin f the secnd kind f de ne. It is expected tht Eq. 6 educe t Eq. 7. With the eltins in [9], the nkel functin deivtives in Eq. 6 cn be ewitten s n On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

4 f 3 nd ' 0 z z, 8 '' z z. 9 z 0 z Figue. An scillting cylinde α x ε e it X Unde the cnditin f cmpct suce / c <<, the sympttic expessins f nkel functins t smll guments give 0 '' i c / c. 0 Substitutin f the nkel functin expessins f Eqs. 8 nd 0 int Eq. 6 yields Eq. 7. Since the detils f using CGF f this pblem is nt shwn in the litetue, we pesent the deivtin hee fllwing we [5]. The CGF f this tw-dimensinl pblem cn be expessed s Eq in [5] x Y t τ / c G x, y, t τ, 3/ c t t τ / c whee is the eviside step functin, nd Y y is the Kichhff vects f the cicul cylinde. The vect cmpnents, y, epesent the velcity ptentils f incmpessible Y j X On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

5 5 f 3 flw pst the cylinde hving unit speed in the j diectin t lge distnce fm the bdy. F flw ve cicul cylinde, they cn be expessed s Y j y j +, y whee j,. F the fluctuting cylinde in this pblem, the f-field fluctutin velcity ptentil cn be expessed s φ x, t υ y, τ G x, y, t τ ds y dτ, 3 S n whee υn y, τ is the sufce nml velcity tht cn be expessed s υ y, τ n y U τ, n i i nd U τ is the sufce tnsltinl scilltin velcity, which is iε e iτ, 0 in this cse. i Ntice since the cntu integl in Eq. 3 is n the sufce f the cylinde, the Kichhff vect in Eq. cn be expessed s Y csα,sinα, 5 nd ni y Ui τ nu τ U τ csα, 6 whee U τ iεe iτ Eqs. 5 nd 6, t get. Then substitute Eqs. nd int Eq. 3, with expessins in t / c U τ φ x, t dτ x α + α α α 3/ τ 0 cs x sin cs d. 7 c t t / c Theefe, the f-field pessue fluctutin is φ ρ t c U τ p x t ρ dτ t c t /, cs. 8 t τ / c T use Eq. 8, U is tken el nd equl t 3 U τ ε sin τ ε cs τ. 9 A fmul in [5] n p. is used t expess the integtin in Eq. 8. Nticing the expessin in [5] is f < 0 nd hee we ssume > 0, we hve t/ c cs β τ dτ cs β t t τ / c c +, 0 giving the integtin in Eq. 8 t be On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

6 6 f 3 t / c ε U τ t τ / c t / c cs t dτ c 3. ε cs3 / τ dτ t τ / c By substituting the bve integtin int Eq. 8, the pessue becmes / 5/ 3 pxt, ρ ε cscs t/ c c. Equtin is the custic pessue using the CGF methd. Accding t the gument fllwing Eq. 5, we wuld like t see hw it cmpes with Eq. 7 unde the custic ffield cnditin, Eq.. In fct, when / c the nkel functin in Eq. 7 cn be expessed s [9] c / c e i / c 3 /. 3 Substitutin f Eq. 3 int Eq. 7 with tking the el pt leds t exctly Eq.. Figue cntins smple plts f the custic pessue vesus time using the MAE nd CGF methds. The esults e clculted using the MAE methd, Eq. 7, nd the CGF methd, Eq.. Nte tht Eq. 7 is pplicble f ny vlues f / c ~ O, while Eq. is nly vlid in the custic f-field when c pessue is nmlized with the cefficient in Eq., /. In Fig., the custic ε 5/ ρ / c. With this nmliztin, the CGF slutin behves just s sinusidl functin f time, while the MAE slutin shuld symptte t sinusidl functin s in Eq. when / c. The time vitin is in the nge f t fm 0 t. A epesenttive diectin f the eceive lctin t / is selected f cmpisn. Tw diffeent vlues f / c e used: 0. Fig. f / c ~ nd 00 Fig. b f / c >>. The slid lines e the MAE esults, nd the dshed lines e the CGF esults. Figue shws tht when / c is smll vlue, t which the CGF methd des nt pply, the MAE esult is diffeent bth in mgnitude nd in phse fm tht f the CGF. When / c becmes lge in Fig. b, the MAE nd CGF slutins e the sme. The fct tht the esults f Eqs. 7 nd e exctly the sme ls pvides the evidence f the cectness in the pcedue f educing the MAE slutin Eq. 7 t its custic f-field fm Eq.. Figue demnsttes gphiclly the cnsistency nd diffeences between Eqs. 7 nd. On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

7 7 f 3 Figue. F-field pessue t the diectin f / cmpisns using the MAE nd CGF methds f the scillting cylinde cse. The slid lines epesent the nmlized MAE esults fm Eq. 7. The dshed lines epesent the nmlized CGF esults fm Eq.. / c 0. ; b / c 00. VOTEX OUTSIDE A CYLINDE The pblem f vtex utside cylinde is shwn in Fig. 3. Bth the flw-field nd the f field sund using the CGF methd e pesented in [5]. The sund is pduced due t the unstedy mtin f the vtex nd its imge t z, whee is the dius f the cylinde, z is the psitin f the vtex in cmplex vible fmt, nd is the cmplex cnjugte f z. The imge t the igin des nt hve ny cntibutin t the sund becuse it des nt hve mtin. Ntice tht ll the thee vtices hve cnstnt cicultin vlue. A On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins / z

8 8 f 3 vtex with cicultin f Γ < 0 is plced utside the cylinde s tht the vtex tjecty is cicle tvesing in the cunte-clckwise diectin with cnstnt dius t speed Γ υ, nd the ngul speed f the ttin mtin f the vtex is thus psitive, equl t υ Ω. 5 Γ -Γ z 0 Figue 3. A vtex utside cicul cylinde Using υ nd t nn-dimensinlize the system nd denting vtex s the vtex utside the cylinde nd vtex s the imge inside the cylinde esult in 0 υ 0 Γ z 0 x, y csτ,sinτ, x, y csτ,sinτ, 6 whee nd τ e dimensinless ngul speed nd time, espectively, nd e expessed s Ω υ υ τ t., 7 On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

9 9 f 3 The pcedue f deiving the CGF slutin is given in ef. 5 nd thus nt epeted hee. The deivtin f the MAE slutin is pesented. Using K's MAE fmul [6], the dimensinless custic pessue in the fequency dmin cn be expessed s i Γ M Dx Dy i p e d e dτ, 8 j j i τ j τ sin τ cs υ Dτ Dτ whee j, nd the Einstein summtin cnventin is used, Mch numbe defined s υ / c s M / Γ Γ nd Γ, M is the, nd is the dimensinless custic eceive distnce defined with the dimensinl eceive distnce. Fm Eq. 6, it cn be deduced Γ Dx Dy, sinτ,csτ, Dτ Dτ Dx Dy, Dτ Dτ sin τ,cs τ. Substituting the bve expessins int Eq. 8, it cn be btined iγm p υ i τ i τ [sin sin τ e dτcs cs τ e dτ]. Using the δ-functin, Eq. 30 cn be witten s iγm p { isin [ δ + δ ] υ + δ + + δ cs [ ]}. Tking n invese Fuie tnsfm f Eq. 3 t get iγm p 8υ + cs[ Defining A B + { i sin[ e i τ,, Eq. 3 cn be ewitten s iγm p 8υ iγm 8υ + e e i τ i τ ]}. { Acs τ ibsin τ }. e i τ { i sin[ B csτ + iasinτ ] + cs[ Acsτ + ib sinτ ]} ] On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

10 0 f 3 In Eq. 33, ntice x iy x x, 35 whee is the Bessel functin f the fist kind f de ne, nd Y is the Bessel functin f the secnd kind f de ne. The definitin f nd Y in [0] gives. sgn, i Y Y + 36 When >0 nd, Eq. 33 gives > 0., B iy A 37 Substitutin f Eq. 37 int Eq. 3 leds t { } sin cs τ τ υ Y M p + Γ. 38 This is the esult by using the MAE methd f the custic pessue fluctutin in the time dmin. T cmpe the CGF esult t the custic f-field, futhe ssume tht. This cnditin leds t the sympttic expessins s +. 3 cs, 3 sin + + Y 39 Substitute Eq. 39 int Eq. 38 t get + Γ 3 sin τ υ M p. 0 T cmpe with the slutin in dimensinl fm given in [5], Eq. 0 cn be dimensinlized t give. sin 3 sin, + Ω Ω υ ρ υ ρ c t M c t M t p Agin ntice tht the slutin in [5] is f 0 < Ω becuse psitive cicultin vtex ws plced utside the cylinde esulting in clckwise ttin. F the psitive cunte-clckwise ttin, the fmul n pge in [5] gives phse ngle f +/, insted f /. Theefe the CGF esult in [5] f the psitive Ω cse tuns ut t be the sme s Eq., which shws typicl tw-dimensinl diple mplitude pptinl t On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

11 f 3 ρ υ M nd cylindiclly diveging custic wve decying t / te. The diple lbes tte cunte-clckwise t n ngul speed f Ω. Figue cmpes the esults f the custic pessue vitin with time using the MAE nd CGF methds f this cse f vtex utside cylinde, simil t Fig. f the scillting cylinde cse. The esults e clculted using the MAE methd Eq. 38 nd the CGF methd Eq.. Equtin 38 is pplicble f ny vlues f Ω / c ~ O s lng s M <<, while Eq. is nly vlid when Ω / equivlent t. The custic pessue in Fig. is nmlized with the cefficient in Eq., ρ υ M. The time vitin is in the nge f Ωt fm 0 t. A epesenttive diectin f the eceive lctin t / is selected f cmpisn. Tw diffeent vlues f c Ω / e used: 0. Fig. f Ω c ~ nd 00 Fig. b f Ω / c >>. Figue shws tht when c / Ω / c is smll vlue, t which the CGF methd des nt pply, the MAE esult is diffeent bth in mgnitude nd in phse fm tht f CGF. When when Ω / c Ω / c becmes lge in Fig. b, the MAE nd CGF slutins e the sme. Agin, is lge, the MAE esult f Eq. 38 is identicl t its custic f-field fm f Eq., demnstting the cectness f the pcedue when educing the MAE slutin Eq. 38 t its custic f-field fm Eq.. Figue demnsttes gphiclly the cnsistency nd diffeences between Eqs. 38 nd. 5. CONCLUSION While bth f the MAE methd nd the CGF methd e limited t lw Mch numbe flw, it is shwn tht the CGF methd is me estictive in tems f the f-field distnce cnditin. The pplicbility f the MAE methd is t the distnce f O l / M, nd the pplicbility f the CGF methd is t fthe wy lctin f O l / M with n>. This diffeence in the tw methds cn ls be intepeted s the custic f-field equiement f / c f the CGF slutins nd f / c ~ f the MAE slutins. When / c ~, the esults fm the tw methds shw significnt diffeences in mgnitude nd phse. The MAE slutin cn be educed t its custic f-field fm nd is shwn t becme identicl t its cespnding CGF slutin. Althugh thee e the less estictive fmule tht cn be used f flw geneted custics e.g., [] nd [], they e mstly used f numeicl cmputtin. The MAE nd CGF methds cn esult in nlyticl f-field slutins. Thee cn be ecmmended vlues s f wht cnstnt shuld be chsen insted f the de f mgnitude symbl f the f-field distnce in pcticl cmputtins f needed ccucy f the slutin nd wht vlues f n shuld be selected f the sme pupse. In pcticl cmputtins, these vlues depend n the Mch numbe in the specific pplictin. In n exmple f Mch numbe und 0., the f-field distnce f vlid MAE slutin n On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

12 f 3 cn be in the nge f 0.5l / M ppximtely in the nge f l / M nd lge, nd the f-field distnce f vlid CGF is nd lge. Figue. F-field pessue t the diectin f / cmpisns using the MAE nd CGF methds f the scillting cylinde cse. The slid lines epesent the nmlized MAE esults fm Eq. 38. The dshed lines epesent the nmlized CGF esults fm Eq.. / 0. ; b / 00. c c On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins

13 3 f 3 EFEENCES. Lighthill M.. On sund geneted edynmiclly I. Genel they. Pceedings f the yl Sciety f Lndn, 95, A, Lighthill M.. On sund geneted edynmiclly II. Tubulence s suce f sund. Pceedings f the yl Sciety f Lndn, 95, A, we M. S. Tiling edge nise t lw Mch numbes. unl f Sund nd Vibtin, 999, 5, 38.. we M. S. Edge-suce custic Geen's functin f n ifil f bity chd with pplictin t tiling-edge nise. The Qutely unl f Mechnics nd Applied Mthemtics, 00, 5, we M. S. They f Vtex Sund. Cmbidge Univesity Pess, Cmbidge, U. K., K. C. Bdy-vtex intectin, sund genetin, nd destuctive intefeence. AIAA unl, 00, 0, Cightn D. C. ditin fm vtex filment mtin ne hlf plne. unl f Fluid Mechnics, 97, 5, Dwling A. P., Ffwcs Willims. E. Sund nd Suces f Sund. hn Wiley nd Sns, New Yk, Abmwitz M., Stegun I. A. ndbk f Mthemticl Functins. Dve, New Yk, Kkc S., Yene Y. et Cnductin. 3-d editin, Tyl nd Fncis. Wshingtn, DC, Ffwcs Willims. E., wkings D. L. Sund genetin by tubulence nd sufce in bity mtin. Philsphicl Tnsctins f the yl Sciety f Lndn, 969, A 6, Wng M., Min P. Cmputtin f tiling-edge flw nd nise using lge-eddy simultin. AIAA unl, 000, 38, On cmpct Geen's functins nd sympttic expnsins f flw-induced sund pedictins