Analytic Time Domain Formulation for Acoustic Pressure Gradient Prediction in a Moving Medium

Transcription

1 Analy Tm Doman ormulaon for Aous Prssur Gradn Prdon n a ovn dum Chuan-Xn B Zhao-Huan Wan and Xao-Zhn Zhan Hf Unvrsy of Thnoloy Hf Anhu 9 Pol s ubl of Chna Ths ar rsns an analy m doman formulaon for aous rssur radn rdon n a movn mdum whh has snfan alaon onal n valuan h aous sarn boundary ondon. Basd on h onvv fows Wllams Hawkns quaon a sm-analy m doman aous rssur radn formulaon wh a form nvolvn h obsrvr m dffrnaon ousd h nrals s frs dvlod and hn h dsrd analy m doman aous rssur radn formulaon s drvd. Baus h drvd formulaons ar rformd drly n h m doman hy ar arularly alabl o h movn obsrvr as. mulaon rsuls for a saonary monool sour a saonary dol sour as wll as a roan monool sour n a movn mdum dmonsra h ffvnss and auray of h roosd formulaons for boh saonary and movn sours wh movn obsrvrs. A = amlud of vloy onal m s - Nomnlaur = sd of sound n undsurbd mdum ms - f = daa surfa funon G = m doman Grn's funon n a sady unform subson flow m - H = Havsd funon L = sour srnh of h loadn sour Pa Profssor Insu of ound and Vbraon sarh; b@hfu.du.n. Ph.D. Candda Insu of ound and Vbraon sarh; zhaohuan@mal.hfu.du.n. Assoa rofssor Insu of ound and Vbraon sarh; zhnzhan@hfu.du.n.

2 L = srnh of h loadn sour omonns Pa L = L / Pas - L = L Pa L = L Pas - L = L Pa L L = Pa = sour ah numbr vor = omonns of sour ah numbr vor = / = = = = movn mdum ah numbr vor = omonns of movn mdum ah numbr vor = = = L = L Pa n = omonns of un vor normal o h daa surfa = sound rssur Pa = rssur of loal flud Pa = rssur of undsurbd mdum Pa Q = sour srnh of h hknss sour km - s -

3 Q = Q / km - s - = aous rad m = / = / r = omral vor bwn sour and rvr y rs = roan radus of sour m = daa surfa T = sour rod s T j = Lhhll srss nsor km - s - = obsrvr m s U = vloy vor of movn mdum ms - U n = loal normal vloy of movn mdum ms - U = vloy vor omonns of movn mdum ms - u = flud vloy vor ms - u = omonns of flud vloy vor ms - u n = loal normal vloy of flud ms - v = omonns of daa surfa vloy vor ms - v n = loal normal vloy of daa surfa ms - = obsrvr oson vor m = Carsan oordna for h obsrvr y = sour oson vor m y y y = Carsan oordna for h sour = Dra dla funon m j = Kronkr dla

4 = vloy onal funon of sour m s - = loal flud dnsy km - = dnsy rurbaon of flud km - = undsurbd mdum dnsy km - j = vsous srss nsor km - s - = sour m s = sour ulsaon anular frquny rads - r = sour roan anular sd rads - ubsrs = alulaon a rardd m L = loadn sour O = obsrvr on = sour on T = hknss sour T I. Inroduon HE aous sarn ff n many nnrn alaons suh as h sarn by a fusla boundary layr [ 6] h roor nos sard by h nrbody [7 9] and h nos sard by a nrfual volu [] should no b nld baus subsanally nfluns h ovrall nos boh n manud and drvy []. Basd on h soluons of h fows Wllams Hawkns W H quaon [] or h Krhhoff formulaon [] numral mhods suh as h boundary lmn mhod [4 4 5] and h quvaln sour mhod [6 8] hav bn dvlod o rd h aous sarn fld n rn yars. Whn solvn aous sarn roblms h ky as s obann h aous vloy on h sarn surfa o srv as h boundary ondon. nly Ghorbanasl al. [9] susd h analy formulaons V and VA for alulan h aous vloy drly n h m doman whl h ounrar n h frquny doman was roosd by ao al. []. Gvn ha h dr drvaon of h aous vloy nvolvd havy albra manulaons h aous rssur radn an also b usd as h boundary ondon baus s rlad o h aous vloy 4

5 hrouh h aous vloy onal []. Howvr h dr numral valuaon of h aous rssur radn for a rals sarn surfa s omuaonally nsv and hrfor many rsarhs hav bn don o oban h analy rssur radn formulaon. arassa and Brnnr [] drvd a sm-analy formulaon o alula h aous rssur radn. L al. [] frs rsnd fully analyal formulaon for h aous rssur radn and mlmnd no numral ods. In ha ar h sm-analyal formulaon was rvsd and namd formulaon G and h fully analy formulaon was namd formulaon GA. I should b nod ha h mdum s assumd saonary n h abov suds. Howvr onvon ffs as n a wnd unnl rmn may b moran n aroaous alulaons. To ralz h mor oml aous sarn rdon for wnd unnl rmns h onvv W H quaon [4 5] whh lly aks no aoun h rsn of h movn mdum should b usd and h aous sarn boundary ondon n h movn mdum should b alulad as wll. nly Ghorbanasl al. [6] drvd an analy aous rssur radn formulaon n h frquny doman whh aouns for h ff of a onsan unform flow wh arbrary dron. Consdrn ha h m doman formulaon an b usful n som ass of aous sarn rdon for aml h movn obsrvr as an analy m doman aous rssur radn formulaon whh lly aks no aoun h rsn of h movn mdum s dvlod n h rsn ar. Insrd by h arlr work of L al. [] w wll us smlar nams for our analy formulaon n a movn mdum n h urrn ar: GA- n whh sands for a movn mdum. Ths formulaon an b sn as h nson of formulaon GA o a movn mdum as. A h sam m a sm-analy aous rssur radn formulaon G- s also vn as ar of h rsn sudy. Ths ar s oranzd as follows. Th onvv W H quaon and s m doman soluon ar frs brfly rvwd n. II A and hn h drvaon of h formulaon wh a modfd sour rm for aous rssur radn s dsrbd n. II B. ubsqunly hr numral s ass ar usd o amn h rforman of h roosd formulaons n. III. nally onlusons ar drawn n. IV. II. Thory A. Th Convv W H Equaon and Is Tm Doman oluon 5

6 Consdr a unform flow ha movs a a onsan vloy U and h dron of h vloy s arbrary. oranzn h onnuy and momnum quaons ha nlud h onsan onvv vloy rm h aous rssur a h obsrvr a m ould b dsrbd by h onvv W H quaon [ D D D H f [ Q f ] [ L f ] [ T H f ] ] j D j wh D D U Q v U [ u v U n n n n n ] L [ j j] n j u[ un vn U n ] 4 T j uu j [ ] j j 5 whr f s h Dra dla funon and H f s h Havsd funon; f dnos h daa surfa; U U n n wh h loal un our normal n f / n h dron = ; j s h Kronkr dla; and ar h dnsy rssur and sound sd n h undsurbd mdum rsvly; s h loal flud rssur s h loal flud dnsy s h aous rssur; h loal flud vloy omonn s dnod by u ; h loal normal omonns o h daa surfa of flud and h body vlos ar u n and v n rsvly; T j s h Lhhll srss nsor; j s h vsous srss nsor. Th frs wo rms on h rh hand sd of Eq. ar h monool and dol sour rms whh ar also known as h hknss and loadn sours rsvly. Th hrd rm s h quadruol sour rm whh s yally small omard o h ohr wo rms whn h flud and movn body s vlos ar boh small and hus s rasonably omd n h subson alulaons. By nln h quadruol sour rm h nral soluon of h onvv W H quaon has bn drvd by Ghorbanasl and Laor n f. [7] as T L 6 wh h nral formulaons ovr h daa surfa 6

7 7 T d Q d Q d Q d Q d Q 4 7 and L L d L L d L d L d L L d L 4 8 Th mahmaal bakround as wll as h drvaon rodur an b sn n f. [7]. In Eqs. 7 and 8 h aous rad and ar r r 9 r whr / y r Th quans n braks should b valuad a h rardd m / and h symbol s h flow ah numbr vor wh / U whr s h manud of ha s h body ah numbr

8 vor wh v /. Th ohr nomnlaurs ar dfnd as follows: / / L L L L L L L L L L. Th dos on h svral quans dno drvavs wh rs o h sour m and h dos on h man varabls do no mly h dffrnaon of any of h assoad vors mld by h subsrs for aml L / and L. rom a horal on of vw h analy aous rssur radn formulaon an b drvd by drly alulan h radns of Eqs. 7 and 8 howvr hs ross rqurs havy mahmaal oraons. An alrnav way o drv h analy aous rssur radn formulaon wll b rsnd n h n sub-son and hs drvaon s asly manulad by adon a modfaon of sour rm a h bnnn of h drvaon. B. Drvaon Produr of ormulaons G- and GA- arn from h onvv W H Eq. and omn h quadruol sour rm w an oban a smlfd onvv W H quaon as [ D D D H f [ Q f ] [ L f ] ] D Emloyn Eq. and adon a modfaon of h sour rm rnly susd by Ghorbanasl al. [6] L QU 4 Equaon an b furhr smlfd o h follown form [ D D H f [ Q f ] [ f ] ] 5 Th Grn's funon usd n a sady unform subson flow wh h ah numbr vor s / G ; y whr 8

9 9 / 7 Usn h abov Grn s funon ylds h soluon of h onvv W H Eq. 5 8 whr dd Q 4 9 dd 4 To oban aous rssur radn formulaons h radn oraon s rformd o Eqs. 8 yldn dd Q 4 dd 4 n h nral varabls and n Eqs. and ar ndndn of h obsrvr oordnas = h radn oraors an b movd nsd h nrals. Usn h follown quaon ha has bn drvd n f. [7] 4 Equaons and an b furhr rwrn as dd Q dd Q 4-5 j j j j dd dd 4 6

10 In ordr o alula h nral ovr d h dny of nralzd funon [7 8] should b usd h h d h / 7 Wh h hl of Eq. 7 Eq. 5 an b wrn as 4 E 8 whr d Q d Q E 9 In ordr o avod h numral valuaon of h obsrvr m dffrnaon ousd h nral n Eq. 9 hy ar onvrd o h sour m dffrnaon hrouh h follown dny [7] E hn boms d Q d Q d Q d Q d Q d Q d Q E

11 ollown h sam ss usd o oban Eqs. 8 and and mloyn an ra rlaon ha s usd o smlfy h quaons Equaon 6 would b furhr wrn as d E / 4 whr d d d d d d E 4 I should b nod ha Eqs. 8 and ar ohr alld formulaon G- whh an b sn as an nson of formulaons G o movn mdum ass. Comarn formulaon G- wh h aous rssur formulaon Eqs. 6 7 and 8 s found ha no mor daa ar ndd o alula h aous rssur radn han hos usd o rd h aous rssur n a movn mdum and hrfor h wo aous varabls ould b alulad a h sam m. Th obsrvr m drvavs ousd h nrals n Eqs. 8 and an b valuad numrally wh varous dffrn alorhm suh as h forward bakward and nral dffrns [9]. Comard wh h dr numral valuaon of h aous rssur radn by usn h aous rssur daa of svral obsrvrs s also an advana ha h formulaon G- dos sav onsdrabl omun rsours baus h nral daa of only on obsrvr ar ndd.

12 Th man drawbak of formulaon G- s ha s nonvnn o dal wh h ass whr h obsrvr s no saonary. If h obsrvr s saonary h numral obsrvr m drvavs of h nrals n formulaon G- ar asy o dal wh baus h m hsory of h nrals n formulaon G- an b oband ohr wh h aous rssur daa a ah obsrvr m s. Howvr f h obsrvr s movn svral ra valuaons of h nrals ar ndd o alula h numral obsrvr m drvavs a ah obsrvr m s. In ordr o lmna h numral obsrvr m drvavs of h aous rssur radn alulaon an analy formulaon alld as GA- s ddud n h follown. Th rodur for lmnan h obsrvr m drvavs s o aly Eq. o formulaon G- and hn valua h sour m drvavs of h rlvan varabls. Insrd by h work of L [] som nw funons and ky sour m drvavs ar vn n h follown o mak h formulaon GA- mor ons: n m n m U 5 n m n m n n n m U n m V 6 W 7 W 8 Z 9 Z ] [ 4 B 4 B 4

13 ] [ A 4 ] [ ] [ ] [ B A 44 D 45 ] [ D 46 H 47 H ] [ 48 K 49 ] [ B K 5 I should b nod ha h sond aral drvav wh rs o h sour m s dnod by wo dos ovr h quany and dos ovr h subsrs man dffrnaon of h assoad vors mld by h subsrs for aml and. Takn h obsrvr m drvavs nsd h nrals of G- and usn h abov dfnons on obans

14 4 Q U W Q U Z Q U QB U d d Q U d d d 5 and 4 / U d B U d A U DU d HU d K U d d U d 5 Th las s s o furhr rwr Eqs. 5 and 5 as 4 I I I I4 I5 5 4 I6 I7 I8 I9 I I I 54 whr I o I orrsond o ah of h nrals n Eqs. 5 and 5. Th forms of I ar vn as follows: I Q U Q U Q V U d 55 4

15 I Q U 4 W Q U 4 W Q V U W Q U 4 W d 56 I Q U 4 Z Q U 4 Z Q V U Z Q U 4 Z d 57 I Q U Q U Q 4 V U d 58 I QB U QB U QB V U d 5 59 I U U 6 V U d I A 7 U AV U d I BU B 8 U BV U d I 9 DU 4 DU 4 DV U DU 4 d d 6 I HU 4 HU 4 HU 4 HV U d 64 I K U K V U d 65 I / U d 66 I should b nod ha and ar dfnd as

16 69 Equaons 5 and 54 ohr wh h dfnons of I ar rfrrd o as formulaon GA-. Th formulaon GA- an b sn as an nson of formulaon GA o a movn mdum as as lly aks no aoun h ffs of onsan unform flow. Comard wh formulaon G- h obsrvr m drvavs of h nrals n formulaon GA- ar no lonr ndd and hus s an advana ha only h mdndn nu daa of h flow fld or a mos numral dffrnaon of hm ar rqurd. orovr h formulaon GA- s mor suabl o oban h aous rssur radn n ass whr h obsrvr s no saonary. Howvr should b nod ha a dsadvana of h susd formulaon GA- s s mahmaal omly n s of h fa ha som nw funons and ky sour m drvavs ar dfnd o mak h rsson of GA- ons. III. Numral mulaons In hs son numral smulaons of hr s ass n a movn mdum ar rsnd o valda h m doman aous rssur radn formulaons dvlod n hs ar. Th frs wo s ass ar h saonary monool and dol sours load n a movn mdum wh movn obsrvrs whl h hrd as onsss of a roan monool wh a movn obsrvr for valdan h orrsondn movn sour and movn obsrvr as. In h frs wo s ass h saonary shral surfas ar usd as h daa surfas. Th aous rssur radn m hsory a h obsrvr s valuad and omard aans h analy soluon. In h hrd s as a movn shral surfa nlosn h monool sour s usd as h daa surfa and h rdd aous rssur radn m hsory a h obsrvr s also omard aans h analy soluon. orovr h ffny of formulaons G- and GA- s omard n all hr s ass. In ordr o avod any rror rlad o flow fld smulaon ods all nu flow fld daa on h daa surfa ar oband from h analy soluons of h flow fld nrad by h sours. In hs ar h wo ordr nral dffrn alorhm s rformd o oban h rsuls from h formulaon G-. A. Ts Cas : onool our n a ovn dum 6

17 Th frs s as s o onsdr a snl-frquny monool sour load a h orn of a Carsan oordna sysm n a unform flow wh an arbrary ornaon. Th vloy onal for h monool onans h unform flow wh an arbrary dron s dfnd as [7]: whr h aous rad from h radn of h vloy onal A [ / ] 7 4 and hav bn dfnd n. II.A. Th aous arl vloy an b oband u 7 Th ndud aous rssur and dnsy n a unformly movn flow wh an arbrary dron ar vn by h unsady Brnoull quaon and U 7 / 7 Th aous rssur radn n h -dron s vn by U j 74 j Th analy soluons of aous rssur radn a an obsrvr on an b oband hrouh Eq. 74. In hs s as h vloy onal amlud of h monool s A m /s. Th anular frquny s rad/s. Th ambn sd of sound s hosn as 4 m/s. Th fr sram flow dnsy s assumd o b.4 k/m. Two dffrn man flow ah numbrs = and =.6..5 ar onsdrd. Th radus of h shral daa surfa s m and hr ar 59 ranular lmns unformly dsrbud on for fn nouh saal rsoluon. Thr ar m ons usd r sour rod T / o nsur nouh moral rsoluon. 7

18 ur shows h rdd aous rssur radn m hsory a h movn obsrvr wh dffrn man flow ah numbrs. Th obsrvr s movn alon h -as a a onsan vloy v o = m/s and s nal oson s m. I an b sn ha h rdd rsuls oband from h formulaons GA- and G- boh auraly mah h analy soluons and hus h auray of h roosd m doman formulaons for h rdon of h aous rssur radn s onfrmd. Whn usn a omur wh an 7 CPU and 6 GB mmory h omuaon m for formulaon G- s.55 s and ha for formulaon GA- s.4 s. Th rason why h omuaon m of formulaon G- s lonr han ha of formulaon GA- s ha h formulaon G- nds o alula svral ra nrals a ah m s whn h obsrvr s movn and h dffrnaon alulaon s also m onsumn.. Color onln Comarson of h rdd saonary monool aous rssur radn m hsory wh ha of h analy soluon for dffrn man flow ah numbrs a h movn obsrvr: a = ; d f =

19 9 B. Ts Cas : Dol our n a ovn dum Th sond s as s a dol sour load a h orn of a Carsan oordna sysm n a unform flow wh an arbrary ornaon and h dol as s alnd wh h -as. Th vloy onal for suh dol an b oband by ] [ 4 A 75 Th rodur for obann h analy aous rssur radn s smlar o ha n h monool sour as. Th follown rlaons may b usd n h alulaon: whr h saal drvavs on h rh hand sds of Eqs hav bn dfnd n. II.A. In hs s as wo dffrn man flow ah numbrs = and =.8..4 ar onsdrd. Th shral daa surfa ohr wh s msh daa usd hr ar h sam as hos n h frs s as and h

20 aramrs A as wll as h m samln ons n on sour rod ar also s o h sam valus as hos n h frs s as. ur shows h rdd aous rssur radn m hsory a h movn obsrvr whh s movn alon h -as a a onsan vloy v o = m/s and s nal oson s 5 5 m. Th lln armn furhr valdas h rlably and auray of h roosd formulaons G- and GA- for aous rssur radn rdon n h m doman. Baus of h aformnond rasons h omuaon m for formulaon G-.55 s s lonr han ha for formulaon GA-.54 s.. Color onln Comarson of h rdd saonary dol aous rssur radn m hsory wh ha of h analy soluon for dffrn man flow ah numbrs a h movn obsrvr: a ; d f = = C. Ts Cas : oan onool our n a ovn dum

21 In ordr o show h fasbly and alably of h roosd formulaons for a movn sour n a movn mdum a roan monool as s onsdrd as shown n... Color onln Th shma of a roan monool n a movn mdum. Th monool roas ounr-lokws around h -as wh an anular sd of r rad/s a radus rs m n h lan and h nal oson s s m whn =. Th orrsondn aramrs A as wll as h m samln ons n on sour rod ar s h sam valus as hos n h saonary monool as. In hs as a shral daa surfa ha s h sam as h on usd n h rvous wo s ass nlosn h monool movn alon wh h sour s adod o rd h aous rssur radn m hsory a h obsrvr. Th obsrvr s movn alon h -as a a onsan vloy v o = m/s and s nal oson s o = 5 5 m. In hs s as hr ar wo flow ah numbrs onsdrd orrsondn o a mdum a rs and a movn mdum of = ur 4 ds h aous rssur radn m hsory rdd by h formulaons G- and GA-. Th lln armn bwn h rdd rsuls and h analy soluons rovs h aably of h drvd formulaons o rd auraly h aous rssur radn n a movn mdum. Hr h omuaon m for formulaon G- s 9.4 s and ha for formulaon GA- s.947 s.

22 . 4 Color onln Comarson of h rdd roan monool aous rssur radn m hsory wh ha of h analy soluon for dffrn man flow ah numbrs a h movn obsrvr: a ; d f = = IV. Conlusons In hs ar basd on h onvv W H quaon boh a sm-analy m doman formulaon G- and an analy m doman formulaon GA- for h rdon of aous rssur radn n a movn mdum wr drvd. Alhouh a movn mdum for aml n a wnd unnl an b quvalnly solvd n a saonary mdum as wll by usn a movn obsrvr whh was jusfd n f. [] h formulaons G- and GA- whh lly ak no aoun h rsn of h unform flow ar mor asly nrrd o amn h onvv ffs.

23 Th valdy and alably of h drvd formulaons wr vrfd hrouh hr omuaonal s ass onssn of a saonary monool sour a saonary dol sour as wll as a roan monool sour. Dffrn flow onfuraons wr onsdrd o oban h rdd aous rssur radn daa and h armn bwn h rdd rsuls and analy soluons was lln. anwhl h omuaonal ffns of h formulaons G- and GA- o dal wh h movn obsrvr ass wr omard and has bn found ha h formulaon GA- lads o a mor ffn alulaon han h formulaon G- baus lmnas h obsrvr m dffrnaon of h nrals. Th drvd formulaons G- and GA- lly ak no aoun h rsn of h movn mdum and hus an b usd o rd h aous rssur radn on h sarn surfa whh an srv as h boundary ondon n h aroaous sarn alulaon. In fuur work h auhors wll onsdr aroaous sarn hnomna n h m doman usn hs formulaons. Aknowldmns Ths work was suord by h Naonal Naural n oundaon of Chna Gran Nos and Th auhors wan o aknowld Profssor Earl H. Dowll of Duk Unvrsy for hs hl n rvsn h manusr o mrov s radably. Th auhors also wan o aknowld Profssor onkyu L of Unvrsy of Calforna Davs h auhor of h ornal formulaons of G and GA for rmn us o and an nson - o formulaons G and GA. frns [] Hanson D. B. and alozz B. Proaaon of Prollr Ton Nos Throuh a usla Boundary Layr Journal of Arraf Vol. No do:.54/.458 [] Aalla N. and Gl. A ay-aouss Aroah o usla arn of oor Nos h Aroaouss Confrn AIAA Par do:.54/ [] Aalla N. and Gl. ay-aouss Aroah o usla arn of oor Nos Journal of h Amran Hlor oy Vol. 8 No do:.45/jah.8.56 [4] Hu. An ffn soluon of m doman boundary nral quaons for aous sarn and s alraon by Grahs Prossn Uns 9h AIAA/CEA Aroaouss Confrn AIAA Par -8. do:.54/6.-8 [5] L. Erwn J. P. and Brnnr K.. A mhod o rd aous sarn of roorraf nos Journal of h

24 Amran Hlor oy Vol. 54 No do:.45/jah [6] L. Brnnr K.. and orrs P. J. Tm-Doman Aroah for Aous arn of oorraf Nos Journal of h Amran Hlor oy Vol. 57 No do:.45/jah.57.4 [7] Gl. A. L. Eff of Cnrbody arn on Prollr Nos AIAA Journal Vol. 9 No do:.54/.6 [8] Knan. J. and lf.h. On oor Ton arn Journal of ound and Vbraon Vol. No do:.6/j.jsv... [9] Knan.J. and urshkumar P. On oor Cnrbody arn Journal of ound and Vbraon Vol. No do:.6/j.jsv..8. [] ao Y. and Q D. Comuaon of oan Blad Nos ard by a Cnrfual Volu Prodns of h Insuon of hanal Ennrs Par A: Journal of Powr and Enry Vol. No. A do:.4/957659jpe794 [] Crhon D. G. and Lnon. G. On h arn of Arodynam Nos Journal of lud hans Vol. 46 No do:.7/775 [] fows Wllams J. E. and Hawkns D. L. ound Gnraon by Turbuln and urfas n Arbrary oon Phlosohal Transaons of h oyal oy A Vol. 64 No do:.98/rsa [] arassa. and yrs. K. Enson of Krhhoff s ormula o adaon from ovn urfas Journal of ound and Vbraon Vol. No do:.6/-46x [4] Ern A. A. hankr B. and hlssn E. Analyss of Transn Wav arn from d Bods Usn a Buron-llr Aroah Journal of h Aousal oy of Amra Vol. 6 No do:./.4876 [5] Tsa C. Iannllo. Brnardn G. and Gnnar. ound ard by a Hlor usla n Dsn lh Condon h AIAA/CEA Aroaouss Confrn AIAA Par do:.54/ [6] Dunn. H. and Tn A.. Aroaous arn va h Equvaln our hod h AIAA/CEA Aroaouss Confrn AIAA Par do:.54/ [7] L. Brnnr K.. and orrs P. J. Aous arn n h Tm Doman Usn an Equvaln our hod AIAA Journal Vol. 48 No do:.54/.45 [8] L. Brnnr K.. and orrs P. J. Assssmn of Tm-Doman Equvaln our hod for Aous arn AIAA Journal Vol. 49 No do:.54/.j576 [9] Ghorbanasl G. Carly. and Laor C. Aous Vloy ormulaon for ours n Arbrary oon AIAA Journal Vol. 5 No

25 do:.54/.j5958 [] ao Y. Zhan Q. Xu C. and Q D. Two Tys of rquny-doman Aous-Vloy ormulaons for oan Thknss and Loadn ours AIAA Journal Vol. 5 No do:.54/.j5 [] L. and Brnnr K.. Commn on Aous Vloy ormulaon for ours n Arbrary oon AIAA Journal Vol. 54 No do:.54/.j54845 [] arassa. and Brnnr K.. Th Drvaon of h Gradn of h Aous Prssur on a ovn urfa for Alaon o h as arn Cod C NAA T [] L. Brnnr K.. arassa. and orrs P. J. Analy ormulaon and Numral Imlmnaon of an Aous Prssur Gradn Prdon Journal of ound and Vbraon Vol. 9 Nos do:.6/j.jsv [4] Wlls V. L. and Han A. Y. Aouss of a ovn our n a ovn dum wh Alaon o Prollr Nos Journal of ound and Vbraon Vol. 84 No do:.6/jsv [5] Xu C. ao Y. and Q D. rquny-doman Aous Prssur ormulaon for oan our n Unform ubson Inflow wh Arbrary Dron Journal of ound and Vbraon Vol. No do:.6/j.jsv.4..4 [6] Ghorbanasl G. Huan Z. ozos-ousouls L. and Laor C. Analyal Aous Prssur Gradn Prdon for ovn dum Problms Prodns of h oyal oy A Vol. 47 No do:.98/rsa.5.4 [7] Ghorbanasl G. and Laor C. A ovn dum ormulaon for Prdon of Prollr Nos a Indn Journal of ound and Vbraon Vol. No do:.6/j.jsv..8.8 [8] arassa. Inroduon o Gnralzd unons wh Alaons n Arodynams and Aroaouss NAA TP- 48 Arl 996. [9] Prss W. H. Tukolsky. A. Vrln W. T. and lannry B. P. Numral s: h Ar of nf Comun rd d. Cambrd Unvrsy Prss Nw York 7 Cha. 5. 5