Vector aeroacoustics for a uniform mean flow: acoustic velocity and

Transcription

1 Veor aeroaouss for a unform mean flow: aous veloy and voral veloy Yjun ao 1,, Zhwe Hu 1, Chen Xu 1, 1 Unversy of Souhampon, SO17 1BJ Souhampon, Uned Kngdom X'an Jaoong Unversy, 7149 X'an, People's epubl of Chna and Ghader Ghorbanasl 3 3 Vrje Unverse Brussel (VUB), Plenlaan, Brussels 15, Belgum Aous and voral dsurbanes n unform mean flow bounded wh sold surfaes are nvesgaed n hs paper. A onveve veor wave equaon and a onveon equaon whh desrbe, respevely, he aous veloy and voral veloy n unform mean flow are dedued by ombnng he Helmholz-Hodge deomposon mehod wh he mehod of ao e al. (AIAA Journal, 16, 54(6): ). Analyal aous veloy negral formulaons for he monopole and dpole soures n unform mean flow are dedued from he developed veor wave equaon, and are also verfed hrough numeral es ases. oreover, hs paper larfes ha aerodynam sound s radaed from he monopole soure as well as he rroaonal omponens of he dpole and quadrupole soures. The solenodal pars of he dpole and quadrupole soures are aousally non-radang bu hey ndue voral dsurbanes n unform mean flow. = speed of sound n flud, m s -1 Nomenlaure f = daa surfae funon f = soure pulsang frequeny, Hz Correspondng auhor. y.mao@soon.a.uk; maoyjun@mal.xju.edu.n. Senor ember AIAA. Leurer. z.hu@soon.a.uk. ember AIAA. esearh Fellow..xu@soon.a.uk; xuhen1983@mal.xju.edu.n. Professor. ghader.ghorbanasl@vub.a.be.

2 G = frequeny-doman Green s funon H = Heavsde funon k = wavenumber,, m -1 L = me-doman srengh of loadng soure, Pa L = L, Pa L = Lr, ˆ Pa r L L r = r ˆ, Pa s -1 n ˆ = he h omponen of he un veor normal o he daa surfae p = aous pressure n frequeny doman, Pa Q = me-doman srengh of hkness soure, kg m - s -1 r = geomeral dsane beween he soure and he reever, x y, m r = omponens of he veor n he radaon dreon, x y,m r ˆ = omponens of he un veor n he radaon dreon, r r = observer me, s u = frequeny-doman flud veloy, m s -1 u = omponens of he flud veloy, m s -1 u n = loal normal omponen of he flud veloy, m s -1 v n = loal normal omponen of he daa surfae veloy, m s -1 x = observer poson veor, m x = omponens of he observer poson veor, m y = soure poson veor, m y = omponens of soure poson veor, m = loal flud densy, kg m -3 = flud densy of unperurbed medum, kg m -3 = densy perurbaon, kg m -3

3 () = Dra dela funon j = omponens of vsous sress ensor, Pa = soure me, s = angular frequeny, rad s -1 Subsrps = flud varable for he unperurbed medum a = aous omponen L = loadng soure r = voral omponen T = hkness soure x = observer quany y = soure quany

4 I. Inroduon THE aous analogy proposed by Lghhll [1] naes a sysema researh on sound generaed aerodynamally. Sne hen, varous developmens of he aous analogy have been arred ou, for nsanes, o aoun for he effes of boundary ondons [, 3], nal ondons [4] and mean flow [5-8] on sound generaon and propagaon, and o undersand he mehansm of aerodynam nose by employng dfferen varables for he wave operaor and expressons of soure erms [7, 9, 1]. The above-menoned nvesgaons used a salar, suh as densy or enhalpy, o desrbe he sound radaon from soures. The perurbaon of aous salars a any observer an be drely obaned from an negral ompuaon usng an approprae Green s funon [11-14]. These salars esablsh a lnear response of aous quanes, suh as aous pressure and densy perurbaon, o aeroaous soures, bu hey are unable o provde nformaon of dealed propagaon pah of he aous energy from soures o observers, whh s mporan for desgnng aous reamens o absorb or nsulae aous energy oupu from soures. Compared wh aous salars, aous veors, suh as aous nensy veor [15, 16] and aous energy sreamlne [17, 18], have an advanage n dsplayng he propagaon pah of he aous energy. eenly, some nvesgaons on aeroaous veors have been arred ou. Ghorbanasl e al. [19] developed me-doman aous veloy formulaons for monopole and dpole soures n arbrary moon, and ao e al. [] dedued frequeny-doman aous veloy formulaons for roang monopole and dpole soures wh a onsan angular speed. Afer ha, aous nensy felds around soures [1] and saerng boundares [, 3] were vsualzed o llusrae he propagaon of aous energy. All he above-menoned nvesgaons on he veor aeroaouss mehod assume ha aous medum s quesen. Ths researh ams o exend he veor aeroaouss mehod o movng medum, whh s meanngful n analyzng aeroaous problems n hgh-speed mean flows. Two ypes of poenal engneerng applaons are lsed below. Vsualzaon of he aous nensy feld around soures, suh as arplanes, hgh-speed rans and roors, s benefal o reveal he propagaon pah of he aous energy. The aous mpedane boundary ondon, whh drely orrelaes he aous pressure wh he aous veloy, an be used o pred aous energy saered by mpedane surfaes mmersed n movng fluds and o show prmary aous-absorbng posons.

5 As a frs sep, we onsder he ase of unform mean flow n hs paper, and earler nvesgaons n hs area an be found n [4-8] among ohers. In order o analyze he effe of unform mean flow on sound generaon and propagaon, Wells e al. [9] frs esablshed a onveve Ffows Wllams-Hawkngs (FW-H) equaon whh onans modfed aeroaous soures and a onveve wave operaor o desrbe sound generaon and propagaon, respevely. By sarng from he onveve FW-H equaon, me-doman [3, 31] and frequenydoman [3] aous pressure negral formulaons for he monopole and dpole soures n unform mean flow have been dedued. The above-menoned formulaons explly aoun for he effe of he unform mean flow on boh he sound soures and sound propagaon. On he oher hand, he Lorenz or Prandl-Glauer ransformaon [33, 34] s an alernave mehod o analyze sound propagaon n unform mean flow. Invesgaons have ndaed ha a movng bakground flow wh hgh ah numbers has a sgnfan effe on sound generaon and propagaon [8, 3-3, 35], mplyng ha he lass exponen law of radaed aous power [1-3] s only vald approxmaely for low-ah-number flows and should be orreed a hgh ah numbers. In a quesen aous medum, aous power an be ompued one he dsrbuon of he aous pressure s known on a losed far-feld surfae wh an enough dsane beween soures and observers. However, he aous veloy s an essenal physal parameer o ompue he aous power oupu from soures n unform mean flow [36]. To he bes knowledge of he auhors, analyal formulaons of he aous pressure graden for soures n unform mean flow have only been derved reenly [37], bu no analyal aous veloy formulaon s avalable n publshed leraures for sound radaed from aerodynam soures n unform mean flow. From he vewpon of general aouss, n eher he quesen aous medum or he unform mean flow, he aous veloy veor an be easly alulaed from he graden operaon of he veloy poenal. Indeed, he onveve wave equaon of he veloy poenal s wdely employed o analyze he effe of movng medum on sound propagaon [38-41], and he aous pressure and aous veloy an be easly alulaed from he veloy poenal. However, s dfful o derve a onveve wave equaon of aeroaouss, n whh he veloy poenal s used as he aous varable on he lef hand sde (LHS) meanwhle aerodynam soures are desrbed on he rgh hand sde (HS). In hs paper, a onveve veor wave equaon and he orrespondng aous veloy negral formulaons are developed. eenly, based on he assumpon of quesen aous medum, ao e al. [4] derved a veor wave equaon of aeroaouss by reasng he generalzed onnuy and momenum equaons, whh provdes a

6 mehod o derve he aous veloy formulaons for soures n he quesen aous medum. The paper exends he veor wave equaon of aeroaouss [4] o aoun for he effe of unform mean flow. Sne he veloy fluuaon smulaneously onans aous and voral omponens n unform mean flow whou hea onduon [43], he Helmholz-Hodge deomposon (HHD) s used o dvde he aous and voral veloy for dervng he onveve veor wave equaon. Afer ha, he analyal aous veloy negral formulaons for he monopole and dpole soures n unform mean flow are dedued from he onveve veor wave equaon. The remanng par of hs paper s organzed as follows. Seon II derves a onveve veor wave equaon and a onveon equaon o desrbe, respevely, he aous and voral dsurbanes n unform mean flow bounded wh sold surfaes. Afer ha, analyal me-doman and frequeny-doman aous veloy formulaons for he monopole and dpole soures n unform mean flow are dedued. In Seon III, numeral es ases for saonary and roang soures are arred ou o verfy he proposed aous veloy negral formulaons, as well as o llusrae ha he lnearzed Euler equaon fals o desrbe he relaonshp beween he aous pressure and he aous veloy for sound radaed from he dpole soures n unform mean flow. Conlusons are drawn n Seon IV. II. Conveve veor wave equaon and negral formulaons A. Helmholz-Hodge deomposon The Helmholz deomposon and HHD are fundamenal heorems n flud dynams. Boh deomposons sae ha a suffenly smooh veor feld an be deomposed no he sum of a longudnal (dvergng, url-free, rroaonal) veor feld and a ransverse (dvergene-free, urlng, roaonal, solenodal) veor feld, ha s κ κ κ (1) a r where κ s a veor, and subsrps a and r represen he rroaonal and solenodal omponens, respevely. The above deomposon guaranees he followng denes: κ ; κ ; κ κ ; κ κ a r r a () For a smooh unbounded flud whh approahes a unform seady sae a nfny, he above deomposon s named he Helmholz deomposon and hs deomposon always exss, s orhogonal, and unque. On he oher

7 hand, f he flud s bounded by sold surfaes, n order o ensure he deomposon s sll unque, he followng boundary ondons should be sasfed on he sold surfaes nκ ; nκ (3) a r where n s he un veor normal o he sold surfaes. The deomposon over he bounded flud s usually named HHD. Deals on hese wo deomposons an be found n [44-46]. By performng he above deomposon over he lnearzed Euler equaon, we an dedue he followng expresson: u u U u U u (4) a r a r p where u a and u r are aous veloy and voral veloy, respevely; U s he veloy of he unform mean flow; p s he pressure perurbaon and s he undsurbed densy. Addonally, for a seond-order ensor, suh as he Lghhll sress ensor T, we have he followng denes T ; T (5) r a B. Conveve veor wave equaon In order o onsder he effe of unform mean flow, we make he followng deomposon of he loal flow veloy u = U u, where u s he perurbed omponen of he loal flow veloy. By usng he above deomposon, generalzed onnuy and momenum equaons whh aoun for he unform mean flow are expressed as follows [9-31]: D H f H f u Q x j D j f H f u u j p j j D H f u D x j L f (6) (7) wh D U D x (8)

8 Q vn U n un vn U n ˆ j j j n n n (9) L p p n u u v U (1) where supersrp denoes soure erms n unform mean flow; veloy n he h dreon and U omponen of flud veloy; U s he omponen of he unform mean flow U nˆ ; p s he sa pressure; j s he vsous sress ensor; u s he h n u n and v n are he normal omponens of he flow and daa surfae veloes, respevely; (.) and H (.) are he Dra dela and Heavsde funons, respevely; j s he Kroneker dela funon; f x, s eher an mpermeable sold surfae or a permeable daa surfae n he flow regon. Sne he sa pressure p, densy and sound speed are onsan, he generalzed momenum Eq. (7) an be expressed as D H f D u H f H f f T L (11) wh Tj u u j p p j j (1) Performng url operaon over Eq. (11) gves he followng equaon D H f u f H f D L r T (13) r The densy perurbaon s muh smaller han he undsurbed densy n he regon ousde he soure f, hus we an have he followng deny by usng Eq. (): H f u H f u + H f u H f u r (14) By subsung Eq. (14) no Eq. (13), we an oban D H f u r f + H f D L r T (15) r In order o ensure ha Eq. (15) s always vald, we mus have he followng deny

9 D H f ur f + H f a or onsan D L r T k (16) r where ka s an rroaonal quany. However, all hree erms on he LHS of Eq. (16) are solenodal, hus her summaon anno be an rroaonal quany. oreover, we sudy he voral and aous dsurbanes a non-zero frequenes, and a onsan only affes he omponen a zero frequeny, hus Eq. (16) aually an be expressed by f u L f H f D H T D r r r (17) Eq. (17) ndaes ha he voral veloy s orgnaed from he solenodal omponens of he dpole and quadrupole soures, and s onveed n he unform mean flow. Eq. (17) also mples ha he monopole soure has no onrbuon o he voral veloy, hus he monopole soure radaes only aous waves n unform mean flow. Now, he veor u s employed as he varable of he wave operaor o dedue he onveve veor wave equaon by followng he mehod gven n [4], bu we emphasze ha han he aous veloy. A spaal dervave u s he oal perurbaed veloy raher x of Eq. (6) s performed o oban he followng equaon f H f u j Q f D H D x x x x j (18) Performng he maeral dervave D D of Eq. (7) and mulplyng by a onsan 1 yeld H f u u j p j j D L f 1 D H f u 1 D 1 D D x D j (19) Subrang Eq. (18) from Eq. (19) gves he followng equaon H f u j Q f D L f 1 D H f u 1 D x x x D j 1 D H f u u p D x j j j j j () Eq. () an also be equvalenly expressed as D L f D H f T H f u Q f (1) 1 D H f u 1 1 D D D

10 The LHS of Eq. (1) s no a wave operaor beause he seond erm s no he Laplae operaor. Applyng he followng deny H f H f H f u u u () would urn he LHS of Eq. (1) no a wave operaor wh an addonal soure erm HS. An equvalen expresson of he generalzed momenum equaon (7) s as follows: H f u on he u u j p j j H f u H f u H f U L f j x j x j (3) Therefore, we an oban he followng expresson from Eq. (3) by employng he deny of s a salar. where u L H f f d T U H f d H f u d (4) Subsung Eqs. () and (4) no Eq. (1) gves 1 D H f D u H f u Q f 1 D L f D d 1 D H f T H f d D T U L H f u d f (5) By usng he followng wo denes L L L f d = f d f d (6) H f d = H f d H f T T T d (7) Eq. (5) an be expressed as

11 D H f u H f u Q f f d D L 1 H f T d 1 D L f f d D L 1 D H f T H f d D T U H f u d (8) As shown n Appendx A, we an prove he followng wo denes: 1 D 1 D H f L f D T L f d L f U 1 UU L f U H f d H f D T T 1 d U U H f T d (9) (3) Thus, we an rewre Eq. (8) as follows: D H f u H f u Q f f d H f d D L T 1 U 1 L f U U L f U 1 H f H f T U U T d U H f u d d (31) By employng he assumpon of a small perurbaon ousde he soure regon, he onveve veor wave equaon (31) an be wren as and gnorng he seond-order small quany u D H f u H f u Q f f d H f d D L T 1 U 1 L f U U L f U 1 H f H f T U U T d U H f u d d (3)

12 The veloy fluuaon n he unform mean flow usually onans boh he aous and voral omponens. Therefore, we perform he HHD over he veloy veor as well as he dpole and quadrupole soures. By performng dvergene operaon over Eq. (3) and by followng he mehod o derve Eq. (17), we an oban he followng onveve wave equaon for he aous veloy D H f ua H f ua Q f ( f ) d H f d a a D L T 1 U L ( f ) 1 a a U UL ( f ) U 1 H f H f T U U T d d a a (33) The las erm on he HS of Eq. (3) s a solenodal veor, hus has no onrbuon o he aous veloy and dsappears n Eq. (33). Equaon (33) s he veor wave equaon desrbng aous veloy generaon from aerodynam soures and propagaon n he unform mean flow, where he soures Q, L and T are gven earler n Eqs. (9), (1) and (1), respevely. Espeally, when he ah number of unform mean flow s zero, he las four erms on he HS of Eq. (33) dsappear, and Eq. (33) redues o he veor wave equaon gven n [4]. C. Aous veloy negral formulaons for monopole and dpole soures Three-dmensonal free spae Green s funon n me doman for he onveve wave equaon (33) s as follows [47]: g (34) 4 wh r (35) 1 r where 1 1 r r (36), ˆ r r, r î r r, U ; r xy s he geomer dsane beween he soure and he reever; s he dsane of sound wave ravellng from he soure o he observer, whh s dfferen from he geomer dsane r due o he onveve effe of unform mean flow. Noe ha hs Green s funon s only suable for subson unform mean flow beause he faor mus be a real number [8].

13 By performng Fourer ransformaon, we an oban he orrespondng frequeny-doman Green s funon as follows: k e Gx, y, g x, y, e d (37) 4 I should be emphaszed ha Eq. (37) s vald for saonary soures as well as soures n a perod moon, suh as he soure wh a onsan roang speed [3], n whh ase boh and are funons of he soure me. The frs-order and seond-order spaal dervaves of he frequeny-doman Green s funon are as follows: k,, e ˆ k ˆ e k xy e k ˆ ˆ G k x k k,, e ˆ e ˆ ˆ ˆ ˆ ˆ ˆ k xy k ˆ e 3 ˆ ˆ k j j j j j j j j j G xx j 3 (38) (39) Smlar o he FW-H equaon, a permeable daa surfae ould be used o redue or even avod he ompuaonal me onsumed by he quadrupole volume soure, herefore only he aous negral formulaons for he monopole and dpole surfae soures are derved n hs paper. By sarng from Eq. (33) and employng he onveve Green s funon (34), we an oban he aous veloy formulaon for he monopole soure n unform mean flow as follows: 4 u a, T Q ( f ), d d x f H f x y (4) where subsrp represens he omponen n he h dreon. By employng he feaure of he Dra dela funon o ransfer he volume negral no he surfae negral and o elmnae he emporal negral, Eq. (4) an be expressed as follows: Q H f 4 u x, ds (41) at, x f 1 re Equaon (41) an be used o numerally ompue he aous veloy, bu s ompuaonally neffen and s prone o numeral errors beause of he spaal dervave alulaon over he observer. By followng he mehod of Farassa [48], he spaal dervave n Eq. (41) an be ransferred no he emporal dervave, hus we an dedue he followng negral equaon whou spaal dervave

14 ˆ 1 Q ˆ Q 4 a, T H f u x, dsd dsd (4) f f Furhermore, we an oban he followng formulaon by usng he feaure of he Dra dela funon o elmnae he emporal negral ˆ 1 Q ˆ Q H f 4 u x, ds ds 1 1 (43) a, T f re f re where subsrp re represens quanes nsde he square brakes should be evaluaed a he rearded me =. Equaon (43) s an exenson of he formulaon V1 of Ghorbanasl [19] for he monopole soure, whh avods spaal dervaon bu sll onans he dervave wh respe o he observer me. The numeral dfferenaon an be employed o ompue he me-doman aous veloy, and a fas Fourer ransform an also be performed o drely ompue he frequeny-doman aous veloy. oreover, by followng he dervaon of formulaon 1A of Farassa [48], an negral formulaon elmnang boh he emporal and spaal dervaves wh respe o he observer an be dedued. However, he dealed dervaon s long and we do no presen he dealed expresson n hs paper. Noe ha he above me-doman formulaon s only vald for soures n subson moon owng o he Doppler faor 1 n he denomnaor. An mproved formulaon suable for soures n superson moon an also be dedued by followng he mehod of Farassa [48] o avod he sngulary aused by he Doppler erm. If he monopole soure s n roaon wh a onsan angular speed or n oher perod moons, performng he Fourer ransform on Eq. (4) and employng he frequeny-doman Green funon gven n Eq. (37), gve he followng frequeny-doman aous veloy negral formulaon ˆ kq ˆ Q k H f 4 u at, x, e e dsd (44) f where varables wh a lde ~ denoe he frequeny-doman omplex quanes. Espeally, when he soure s saonary, we an oban he followng smplfed me-doman formulaon from Eq. (43) 1 Q ˆ Q ˆ H f u x, ds ds (45) a, T f re f re 4

15 The orrespondng frequeny-doman formulaon an also be dedued as follows: kq ˆ Q ˆ H f x S (46) k 4,, e d e d f f k u a T S where he frs and seond erms on he HS of Eq. (46) are, respevely, named he far-feld and near-feld erms owng o her dfferen deayng rae on he radaon dsane. A numeral verfaon of he developed aous veloy formulaons for he monopole soure n unform mean flow wll be arred ou n Seon VI. By sarng from Eq. (33) and employng he onveve Green s funon (34), we an oban he aous veloy formulaon for he dpole soure n unform mean flow as follows: L H f 4 u a, L x, f d dd xx j j j f x j k f a, j y a, L L a, f d dd x jx y k f f dyd (47) The seond and hrd erms on he HS of Eq. (47) exs only n unform mean flow. By performng he frsorder and seond-order spaal dervaves over Eq. (47), we an dedue he followng equaon: L ˆ L H f u, g f dyd a, a, a, L f x ˆ L,, La L a a, La, g f f a, a, f a, f ˆ dyd L L 3 ˆ L 3 1 LaLa g f y g f,, d dd The feaure of he Dra dela funon s used o ransfer volume negrals no surfae negrals and o hen elmnae he emporal negral, one an oban he followng expresson dyd (48)

16 L ˆ L H f u ds a, a, a, L 1 f re 4 x, f f L ˆ L L ˆ L 1, a, a a, a, re ds L a, La, 1 ˆ 3L, 3 1 a L, L, dsd 3 1 f a a re re ds (49) Equaon (49) s jus he exenson of formulaon V1 of Ghorbansal [19] for he dpole soure. Formulaon V1A for he dpole soure n unform mean flow an also be dedued by akng he emporal dervave of he frs erm on he HS of Eq. (49). However, dealed dervaon s very edous, hus we do no presen n hs paper. For he roang dpole soure n unform mean flow, performng he Fourer ransform on Eq. (48) and employng he defnon of he frequeny-doman Green funon,.e., Eq. (37), gve he followng frequenydoman aous veloy negral formulaon L ˆ L H f 4 u, k e e dsd a, a, k a, L f x k f f f L ˆ L L ˆ L e e dd S a, a, a, a, k 3L ˆ 3 1 L L L L a, a, k a, a, a, k e e 3 dd S e e dd S (5) Espeally, when he ah number of unform mean flow s equal o zero, Eq. (5) redues o he formulaon of FV1A gven n []. When he soure s saonary, Eq. (49) an be smplfed o

17 ˆ La, L a, H f 4 u a, L x, ds f f f L ˆ L L ˆ L, a, a a, a, ds L L 3L ˆ 3 1 L L a, a, a, a, a, dsd 3 f re re re re ds (51) The orrespondng frequeny-doman negral formulaon s as follows: L ˆ L H f 4 u, k e ds a, a, k a, L f x f L ˆ L L ˆ, a, a a, a, k e L ds L L a, a, k e f ˆ 3L 3 a, 1 L a, La, k d 3 k e S f ds (5) III. Numeral verfaon and dsussons A. ehod of numeral verfaon In hs seon, numeral es ases are performed o verfy he developed aous veloy formulaons onsderng he effe of unform mean flow. To he bes knowledge of he auhors, no benhmarkng ase has been publshed for he aous veloy relaed o soures n unform mean flow. As shown n Eq. (17), he monopole soure has no onrbuon o he voral dsurbane, hus he followng lnearzed Euler equaon expressed n frequeny doman s used o verfy he aous veloy formulaons for he monopole soure n unform mean flow p u U u (53)

18 However, n order o fler he voral veloy omponen smulaed from he dpole soure, we should use he followng frequeny-doman expresson obaned from Eq. (53) o verfy he aous veloy formulaon for he dpole soure n unform mean flow ( u) a p U ua (54) The erms on he HS of he Eqs. (53) and (54) an be alulaed wh he aous pressure formulaons proposed n [3-3]. A frs-order dsrezaon sheme s used o numerally ompue he spaal dervaves relaed o he aous pressure and he aous veloy n Eqs. (53) and (54), hus we have he followng wo expressons: x e x 1 x e x 3 U j u a, l j ua, p l p u a, x l l j j1 x e x u l u l a, a, a, LHS a, HS x e e x e x e x U u l l u l u l u l j a, j a, a, j a, x e x e + x bhs blhs p l p l p l (55) (56) 6 where l represens he dsane beween wo saonary observers, and l 1 m s used n hs paper. In all he numeral es ases presened n hs Seon, we make he followng assumpons. The densy and sound speed of he amben flow are 1. kg/m 3 and 34 m/s. The ah number of he unform mean flow s seleed as (.3,.4,.5), ensurng ha he unform mean flow s subson. The pulsang frequeny of he soure s f 1 Hz. For sound radaed from a saonary soure n unform mean flow, he soure s loaed a he oordnae orgn. 36 observers are evenly loaed on a rle n a plane a z=1m wh a radus of 1m. The ompuaonal resul oupus he drevy paern o verfy he developed aous veloy formulaons.

19 For sound radaed from a roang soure n unform mean flow. The soure roaes around he z-axs n he plane of z wh he radus of roaon of.8 m, and he frequeny of soure roaon s 5 Hz. Only one observer s loaed a (1,1,1) m, and he aous veloy omponens n hree dreons are ompued o verfy he developed aous veloy formulaons. B. onopole pon soure n unform mean flow The soure srengh of he monopole pon soure s f Q ds.1 kg/s. When he soure s saonary, Eqs. (46) and he aous pressure formulaon n [3] are used o ompue he erms on he LHS and HS of Eq. (55), respevely. Fg. 1 llusraes he omponens n hree dreons, where e, Im and Abs represen he real par, magnary par and he magnude of he omplex quany, respevely. The ompuaonal resuls obaned from he wo mehods are onssen wh eah oher, valdang he developed aous veloy formulaon for he saonary monopole soure n unform mean flow. For he roang monopole soure, Eq. (44) and he aous pressure formulaon [3] are employed o ompue he erms on he LHS and HS of Eq. (55), respevely. Fg. dsplays he spera obaned from he abovemenoned wo mehods, and a good agreemen beween hese wo resuls verfes he frequeny-doman aous veloy formulaons for he roang monopole soure n unform mean flow. oreover, n he es ase of he roang monopole soure, we ompue he aous veloy omponens wh me-doman and frequeny-doman formulaons, respevely. In solvng he me-doman formulaon Eq. (43), he number of samples n one revoluon for he roang soure s 36; he soure-me domnan algorhm [49-51] s used o solve he rearded-me equaon. The nsananeous aous veloy sgnals reeved by he saonary observer are lnearly nerpolaed and hen ransferred no he frequeny doman by performng a fas Fourer ransform. Fg. 3 ompares he omponens of he aous veloy alulaed wh he me-doman and frequeny-doman aous veloy formulaons, n whh TDN and FDN represen he me-doman numeral mehod and frequeny-doman numeral mehod, respevely. The resuls obaned from dfferen mehods aheve a good agreemen, furher verfyng ha he proposed me-doman and frequeny-doman aous veloy formulaons an be used o auraely pred he aous veloy omponens of he sound radaed from he roang monopole soure n unform mean flow.

20 (a) (b) Fg. 1. Verfaon of aous veloy formulaon for he saonary monopole soure va Eq. (55): (a) x- omponen (b) y-omponen () z-omponen. ()

21 (a) (b) Fg.. Verfaon of aous veloy formulaon for he roang monopole soure va Eq. (55): (a) magnude of a x ; (b) magnude of a y ; () magnude of a z ()

22 (a) (b) Fg. 3. Aous veloy omponens ompued from he me-doman formulaon (Eq. (43)) and frequenydoman formulaon (Eq. (44)): (a) x-omponen (b) y-omponen () z-omponen. ()

23 C. Dpole pon soure n unform mean flow For he saonary dpole pon soure, he rroaonal omponens of soure srengh expressed n Caresan oordnae sysem are gven as L a, ds 1 N ( 1,,3 f ). Equaons (5) and he aous pressure formulaon [3] are used o ompue he erms on he LHS and HS of Eq. (56), respevely. For he roang dpole pon soure, he rroaonal omponens of he roang dpole soure srengh expressed n ylnder oordnae sysem are L a, ds 1N (,, f Z ). Equaons (5) and he aous pressure formulaon [3] are used o ompue he erms on he LHS and HS of Eq. (56), respevely. (a) Fg. 4. Verfaon of aous veloy formulaon for he saonary dpole soure va Eq. (6.4): (a) real par; (b) magnary par (b)

24 Sne voral dsurbanes an be smulaed by he dpole soure n unform mean flow, Eq. (56) s used o verfy he developed aous veloy formulaons for he dpole soure. As shown n Fg. 4 and Fg. 5, a good agreemen beween he erms on he LHS and HS for boh he saonary and roang ases verfes he aous veloy formulaons for he dpole soure n unform mean flow. oreover, Fg. 6 and Fg. 7 ompare he ompuaonal resuls of he erms on he LHS and HS of Eq. (55), we an fnd ha obvous devaons always exs for boh he saonary and roang dpole soures, onfrmng ha he lnearzed Euler equaon anno be used o alulae he aous veloy for he dpole soure n unform mean flow. (a) Fg. 5. Verfaon of aous veloy formulaon for he roang dpole soure va Eq. (6.4): (a) real par; (b) magnary par (b)

25 (a) (b) Fg. 6. Comparson of he erms on he wo sdes of Eq. (55) for he saonary dpole soure: (a) x-omponen (b) y-omponen () z-omponen. ()

26 (a) (b) Fg. 7. Comparson of he erms on he wo sdes of Eq. (55) for he roang dpole soure: (a) x-omponen (b) y-omponen () z-omponen. ()

27 IV. Conlusons The HHD an be used o deompose he flow veloy no rroaonal and solenodal omponens. By sarng from he generalzed ompressble Naver-Sokes equaons and by performng he HHD over he veloy veor as well as he dpole and quadrupole soures, a onveve veor wave equaon and a onveon equaon are dedued n hs paper o desrbe, respevely, aous and voral dsurbanes n unform mean flow bounded wh sold surfaes. The developed equaons ndae ha he monopole soure as well as he rroaonal pars of he dpole and quadrupole soures have onrbuon o sound radaon, and he solenodal pars of he dpole and quadrupole soures smulae voral dsurbanes n unform mean flow. Tme-doman and frequeny-doman aous veloy formulaons for he monopole and dpole soures n unform mean flow are dedued from he developed onveve veor wave equaon. Numeral es ases are performed o verfy he developed aous veloy formulaons for he monopole and dpole soures. By employng he developed aous veloy formulaons, we analyze he effe of he ah number of unform mean flow on he aous nensy feld and aous power oupu n he nex paper [5]. oreover, he aous and voral dsurbanes saered by sold boundares n unform mean flow ould be suded, and dealed nvesgaons on hs op are planned n fuure researh. Appendx A. Smplfaon of soure expressons n he veor wave equaon In unform mean flow, we have he followng deny by usng he defnon of he Green s funon g 1 D g g κ κ κ x y (A.1) D x x j j where κ s an arbrary veor. By performng he volume and emporal negrals over Eq. (A.1), we an oban 1 D g d d g d d d d D κ y x x κ y κ x y y (A.) f j j f f When x does no onde wh y and does no onde wh, he erm on he HS of Eq. (A.) s equal o zero. Eq. (A.) an be expressed as 1 D 1 D g d d g d d g d d D κ y U D κ y x x κ y (A.3) f f j j f

28 Performng emporal negral over Eq. (A.3) gves 1 D 1 D d d d d d g g gd dd κ y f j κ y U κ y j f f D x x D 1 1 gd d gd dd U κ y U U κ y f f (A.4) If he veor κ s subsued by L f and H f T, respevely, we an oban Eqs. (9) and (3). Appendx B. Alernave dervaon of he aous veloy formulaon for he monopole soure n unform mean flow The aous pressure formulaon for a monopole soure n unform mean flow s as follows: p D Q ( f ) y T x, d d (B.1) D 4 f Beause he monopole soure radaes only he aous wave raher han smulaes he voral dsurbane n unform mean flow, hus he erms relaed o he voral veloy n he lnearzed Euler equaon (4) an be elmnaed. Subsung Eq. (B.1) no Eq. (4), we an oban he followng expresson DuaT, x, D Q ( f ) p T x, d d D D y (B.) 4 f Furhermore, we an dedue he equaon as follows: at, Q ( f ), d f 4 u x y d (B.3) Equaon (B.3) s jus he same as Eq. (4). We emphasze ha hs dervaon s only suable for he monopole soure n unform mean flow. Aknowledgmens The researh s suppored by he Naonal Naural Sene Foundaon of Chna (No , No ) and Newon Fund (No. IE141516).

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