Non-linear Singular Integral Equations for Unsteady Inviscid Flowfields of 2-D Airfoils

Transcription

1 ivesal Joual of Compuaioal alysis 3 (05) -4 No-liea igula Iegal Equaios fo seay Ivisci Flowfiels of -D ifoils Iepape Reseac Ogaizaio 8 Dimaki. es GR Geece elaopoulos@iepape.og bsac iovaive wo-imesioal aeoyamics epeseaio aalysis is iouce fo e ivesigaio of ivisci flowfiels of useay aifoils. Te above poblem of e useay flow of a wo-imesioal NC aifoil is eefoe euce o e soluio of a o-liea muliimesioal sigula iegal equaio we e fom of e souce a voe seg isibuio is epee o e isoy of e above isibuio o e NC aifoil suface. applicaio is fially give o e eemiaio of e velociy a pessue coefficie fiel aou a aicaf by assumig cosa souce isibuio. Key Wo a ases Two-imesioal NC aifoil No-liea Muliimesioal igula Iegal Equaios No-liea eoyamics Cosa ouce Disibuio icaf Velociy & essue Coefficie Fiel.. Ioucio Ove e pas yeas e o-liea sigula iegal equaios ave coceae a iceasig iees because of ei applicaio o e soluio of basic poblems of aeoyamics a flui mecaics especially efeig o useay flows. Te eoy a compuaioal meos by oliea sigula iegal equaios cosis of e laes ig ecology o e soluio of geealize poblems of soli a flui mecaics. Cosequely ee is a big iees o e coiuous impoveme of suc compuaioal meos. Te ew esig aeoyamic poblems ae euce o e soluio of a o-liea sigula iegal equaio wic is use fo e eemiaio of e velociy a pessue coefficie fiel aou a NC aifoil. uc a aeoyamic beavio of e NC aifoils is a vey impoa eleme o e esig of ew geeaio aicafs wi vey ig spees. ece special aeio soul be give o e ew ecology compuaioal meos coceae o e soluio of e befoe meioe aeoyamic a flui yamic poblem..m.o.mi a J.L.ess [] wee e fis scieiss wo ivesigae aeoyamic pael meos fo suyig aifoils wi zeo lif. ccoig o em e aifoil was moele wi isibue poeial souce paels fo olifig flows o voe paels fo flow wi lif. Tis meo was fue eee by R..Djojoiajo a.e.wiall [].E.Robe a G.R.aais [3] J.M.umma [4] D.R.isow [5] D.R.isow a J.D.awk [6] a R.J.Lewis [7] fo suyig ee-imesioal seay a useay flows by combiig souce a voe sigulaiies. Fuemoe e useay pael meos o e moelig of sepaae wakes usig iscee voices wee fue eee by T.apkaya a R.L.coaf [8]. I aiio N.D.am [9] F.D.Deffebaug a F.J.Mascall [0] M.Kiya a M.ie [] a T.apkaya a.k.klie [] ivesigae some oe flow moels. ccoig o em e sepaaig bouay layes wee epesee by a aay of iscee voices emaaig fom a kow sepaaio poi locaio o e aifoil suface. O e coay uig e pas yeas seveal scieiss mae eesive calculaios by usig useay ubule bouay laye meos. mog em we meio: R.E.igleo a J.F.Nas [3] J.F.Nas L.W.Ca a R.E.igleo [4]..Lyio J..Fezige a.j.klie [5] W.J.McCoskey a.i.ucci [6] a J.Kim.J.Klie a J..Joso [7].

2 Recely o-liea sigula iegal equaio meos wee popose by E.G.Laopoulos [8] - [] fo e soluio of flui mecaics a aeoyamic poblems a by E.G.Laopoulos a V..Zisis [3] [4] fo wo-imesioal flui mecaics poblems applie o ubomacies. o by e cue eseac e aeoyamic poblem of e useay flow of a wo-imesioal NC aifoil movig by a velociy is euce o e soluio of a o-liea muliimesioal sigula iegal equaio. Tis olieaiy esuls because e souce a voe seg isibuio ae epee o e isoy of e voiciy a souce isibuio o e NC aifoil suface. Moeove a ubule bouay laye moel is fue popose base o e fomulaio of e useay beavio of e momeum iegal equaio. applicaio is fially give o e eemiaio of e velociy a pessue coefficie fiel aou a aicaf by assumig cosa souce isibuio.. No-liea Flui Dyamics a seay eoyamics ew o-liea useay flui mecaics epeseaio aalysis is iouce a suie fo e aeoyamic poblem of a wo-imesioal NC aifoil. Te popose meo cosiss o e geealizaio of all pas meos by eucig e poblem o e soluio of a o-liea muliimesioal sigula iegal equaio. Te above olieaiy esuls because of e geeal fom give o e souce a voe seg isibuio wile ese ae epee o e isoy of e voiciy a souce isibuio o e NC aifoil suface. I is case e aifoil is movig wi a spee. [8] [] Cosequely cosie a wo-imesioal aifoil movig i a omogeeous a ivisci flui. (Fig.). Fig. wo-imesioal aifoil of suface i a omogeeous a ivisci flui. Te aifoil wi e wake compise s complee lifig sysem i a ioaioal flow oug e ieal flui. ecause of e eisece of suc a ioaioaliy e fo e local flui velociy oe as: 0 (.) lso by eplacig e flui velociy wi e oal velociy poeial oe as: (.)

3 wile (.) ca be fue wie as: (.3) wi e ouwa velociy (Fig. ) a e poeial ue o e pesece of e aifoil. I aiio by usig Gee s eoem [5] follows a basic elaio fo e velociy poeial ( ) wi e ime a ay poi i coiuous acyclic ioaioal flow: g ξ ( ) ξ (.4) W wee is e suface of e aifoil (Fig. ) W e suface of e wake ξ ξ a e souce poi ξ (Fig. ) g e souce seg isibuio seg isibuio a e isace equal o: e suface omal e voe ξ (.5) Te velociy poeial (.4) ca be also wie as followig wic eoes a wo-imesioal o-liea sigula iegal equaio: ξ ξ g ( ) (.6) W Te kiemaical suface agecy coiio o e suface of e aifoil ca be wie as followig: [6] ( ) ( ) 0 (.7) wee eoes e suface omal a e fiel poi (Fig. ). Te above coiio ca be fue wie as followig fo a boy fie cooiae sysem: ( ) ( ) (.8) 3

4 i wic eoes e aifoil aslaio velociy a e aifoil agula oaio. Fom eqs (.7) a (.8) follows: 0 (.9) eyo e above by iseig (.9) io (.6) esuls e followig wo-imesioal o-liea sigula iegal equaio: ξ ξ W g (.0) Te o-liea sigula iegal equaio (.0) ca be fue wie as: ξ ξ W g 3 (.) Cosequely by solvig e o-liea iegal equaio (.) wi e coespoig bouay coiios e e velociy a ay fiel poi will be eemie oug (.7). 3. No-liea essue Disibuio alysis Te pessue isibuio o e aifoil may be obaie by e useay eoulli equaio vali a ay poi i a ioaioal ieal flow: (3.) wee ρ eoes e flui esiy. I aiio by usig e eivaio of e pevious secio e (3.) will be wie as: (3.) lso (3.) euces o e followig fom: 4

5 (3.3) if we eplace e by e suface gaie f : (3.4) ece because of (.9) e (3.3) ca be wie as: (3.5) wic will be use fo e compuaios. 4. Lamia a Tubule ouay Laye Moels eveal bouay laye moels ca be use fo e lamia e ubule pas of e flow a e asiio egio bewee em i oe o eemie e aeoyamic beavio of e aifoils. Tese bouay laye moels ae e fiie iffeece fiie eleme o iegal moels. Te ubule bouay laye moel wic is popose by e pese eseac is base o e fomulaio of e useay beavio of e momeum iegal equaio [5]. Te majo eesio of e above meo by e pese eseac is e iclusio of useay ems i e momeum iegal equaio. Te useay momeum iegal equaio wic is vali fo bo lamia a ubule flow ca be eefoe wie as: (Fig. ) ) ( ) ( F c u u u u (4.) i wic u is e bouay laye ege velociy e ime e isplaceme ickess e momeum ickess e suface isace a c F e ficio faco. 5

6 Fig. Lamia a Tubule ouay Laye Moel fo eoyamics. I aiio cosie e case fo e lamia laye e e pessue gaie paamee μ is give by e elaio: u u R ( ) (4.) u u wee R is e Reyols umbe base o u a. Moeove by cosieig some special elaios bewee e paamees c F / a e a soluio fo e lamia fomulaio may be obaie. Fo e wege flow soluios followig elaios ae vali: [7] c F.9 4.3D R N ( D) D kN (4.3) wee N is e sape paamee D e blockage faco / wi e bouay laye ickess a R e Reyols umbe base o u a. O e coay fo e ubule laye moel followig fomula is vali: u [ u ( )] Λ (4.4) a e fucio Λ is obaie by e fomulas: 6

7 Λ 0.05( Λ Λ K 4.4K e ` w Λ) 0.96 ( ) /5 (4.5) wee τ is e wall sea sess a p/ e seamwise pessue gaie. w lso e sape faco elaiosips ae obaie by followig elaios: u u l( cf cf (sg )( ) 0.4 f ( ) cf u w y y ) f cos ( ) / (4.6) wi u e velociy i e bouay laye a a isace y fom e wall a ρ e flui esiy. Fially e ski ficio law is vali as: cf R 0.68 sg( ) (4.7) iioal eails coceig e eaime e wall sea sess a e skik ficio elaios ca be fou i [5]. 5. Velociy a essue Coefficie Fiel fo Cosa ouce Disibuio (ifoil wi Velociy) Le us cosie e special case of a cosa souce isibuio g. I is case e geeal o-liea poblem pesee i pevious paagaps is muc moe simplifie a is solve as a liea poblem. Te geomeical epeseaio of e poblem is sow i Fig. 3. Fo cosa souce isibuio g e e flui velociy is eemie by e fomula: / g cosi sij (5.) 7

8 wee i j ae e ui vecos o e a y aes especively a eoes e sepaaig wake (Fig. 3). Fig. 3 Cooiae sysem fo e D aifoil of a aicaf. o we y 0 a 0 e e flui velociy will be compue by e followig fomulas: p y p g l i ( ) j y g l i y p 0 p 0 (5.) Moeove we cosie e pessue coefficie C p : C ( ) [ ( ) ] (5.3) p wee ρ eoes e flui esiy a e seam pessue. y usig fue e useay equaio of eoulli e e pessue coefficie will be simplifie oug e elaio: C p ( ) (5.4) wic will be use fo e compuaios. 8

9 6. pplicaio of seay eoyamics o New Geeaio icaf s a applicaio of e pevious meioe wo-imesioal useay aeoyamics eoy we will calculae e velociy fiel pesee aou a aicaf. Te cosucio of ew geeaio uboje egies makes possible e esig of vey fas big jes. eyo e above e iceasig evoluio of aeoelasiciy i aicaf ubomacies coiues o be sill impove accoig o e ees of aicaf powepla a ubie esiges. Cosequely e eoauical Iusies soul acieve a compeiive ecological avaage i seveal saegic aeas of ew a fas evelopig avace ecologies by wic a bigge make sae ca be acieve i e meium a loge ems. uc a iceasig big make sae iclues e esig of ew geeaio lage aicafs wi vey ig spees. I e pese applicaio e leg of e aicaf ue cosieaio is c=50.0m a e aifoil secio NC 00 (Fig. 3). I was suppose ui voe isibuio a ece e velociy fiel o e bouay a aou of e aifoil was compue by (5.). lso e pessue coefficies C p wee calculae oug (5.4) fo seveal aicaf velociies a wi velociy = 5m/sec. Figues a 7 sow e pessue isibuio o e uboje pesee fo aicaf spees 34 Mac especively ( Mac=33 m/sec). lso Figs. 4a o 7a sow e same pessue isibuio o e aifoil i ee imesioal fom. Fig. 4 essue isibuio aou e aicaf of Fig.3 fo cosa souce isibuio a spee Mac. 9

10 Fig. 4a essue isibuio aou e aicaf of Fig.3 fo cosa souce isibuio a spee Mac 3D fom. Fig. 5 essue isibuio aou e aicaf of Fig.3 fo cosa souce isibuio a spee Mac. 0

11 Fig. 5a :essue isibuio aou e aicaf of Fig.3 fo cosa souce isibuio a spee Mac 3D fom. Fig. 6 essue isibuio aou e aicaf of Fig.3 fo cosa souce isibuio a spee 3 Mac.

12 Fig. 6a essue isibuio aou e aicaf of Fig.3 fo cosa souce isibuio a spee 3 Mac 3D fom. Fig. 7 essue isibuio aou e aicaf of Fig.3 fo cosa souce isibuio a spee 4 Mac.

13 Fig. 7a essue isibuio aou e aicaf of Fig.3 fo cosa souce isibuio a spee 4 Mac 3D fom. s i is sow i e above Figues fo e up bouay pois of e NC aifoil e values of e pessue coefficie ae iceasig appoimaely up o /c = 0.5 wile ey eceasig agai up o /c =. O e oe a fo e ow bouay pois e values of C p ae eceasig up o /c = 0.35 a e iceasig up o /c =. 7. Coclusios geeal o-liea moel as bee popose fo e eemiaio of e velociy a pessue coefficie fiel aou a NC aifoil movig by a velociy i wo-imesioal useay flow. uc a poblem was euce o e soluio of a wo-imesioal o-liea sigula iegal equaio wic as o be solve by compuaioal meos. Te olieaiy esule because of e fom of e geeal ype of e souce a voe seg isibuio. lso a bouay laye moel was popose base o e fomulaio of e useay beavio of e momeum iegal equaio. uc a bouay laye moel is vali fo bo lamia a ubule flow a was popose as a geeal meo fo e suy of e aeoyamic beavio of e aifoils. O e oe a by supposig cosa souce isibuio e e velociy a pessue coefficie fiel aou a aicaf movig wi seveal velociies was eemie. Tis meo soul be applie fo e esig of ew geeaio lage aicafs wi vey ig spees. Cosequely e o-liea sigula iegal equaio meos will be i fuue of coiuously iceasig iees as suc meos will be vey impoa fo e soluio of e geealize soli a flui mecaics poblems. pecial aeio soul be eefoe give o e amelioaio of e o-liea sigula iegal equaio meos as may moe soli a flui mecaics poblems wi cosieable complicae foms ae ecely euce o o-liea foms. 3

14 Refeeces. mi.m.o. a ess J.L. Calculaio of e o-lifig poeial flow abou abiay ee imesioal boies Douglas Repo E Djojoiajo R.. a Wiall.E. umeical meo fo e calculaio of o-liea useay lifig poeial flow poblems I J. 7 (969) Robe.E. a aais G.R. Review a evaluaio of a ee-imesioal lifig poeial flow compuaioal meo fo abiay cofiguaios oeig Co umma J.M. oeial flow abou impulsively sae oos J. icaf 3 (976) isow D.R. Developme of pael meos fo subsoic aalysis a esig N CR isow D.R. a awk J.D. ubsoic pael meo fo e efficie aalysis of muliple geomey peubaios N CR Lewis R.I. uface voiciy moelig of sepaae flows fom wo-imesioal bluff boies of abiay sape J. Mec. Egg ci. (98) apkaya T. a coaff R.L. Ivisci moel of wo-imesioal voe seig by a cicula cylie I J. 7 (979) am N.D. eoyamic loaig o a wo-imesioal aifoil uig yamic sall I J. 6 (968) Deffebaug F.D. a Mascall F.J. Time evelopme of e flow abou a impulsively sae cylie I J. 4 (976) Kiya M. a ie M. coibuio o a ivisci voe-seig moel fo a iclie fla plae i uifom flow J. Flui Mec. 8 (977) apkaya T. a Klie.K. Impulsively-sae flow abou fou ypes of bluff boy J. Flui Egg 04 (98) igleo R.E. a Nas J.F. Meo fo calculaig useay ubule bouay layes i wo- a ee-imesioal flows I J. (974) Nas J.F. Ca L.W. a igleo R.E. seay ubule bouay layes i wo-imesioal icompessible flow I J. 3 (975) Lyio.. Fezige J.. a Klie.J. iegal meo fo e compuaio of seay a useay ubule bouay laye flows icluig e asioy sall egie i iffuses afo ivesiy Repo D McCoskey W.J. a ucci.i. Viscous-ivisci ieacio o oscillaig aifoil i subsoic flow I J. 0 (98) Kim J. Klie.J. a Joso J.. Ivesigaio of sepeaio a eaacme of a ubule sea laye: flow ove a backwa-facig sep J. Flui Egg 0 (98) Laopoulos E.G. No-liea sigula iegal compuaioal aalysis fo useay flow poblems Reew. Eegy 6 (995) Laopoulos E.G. No-liea sigula iegal epeseaio fo useay ivisci flowfiels of -D aifoils Mec. Res. Commu. (995) Laopoulos E.G. No-liea sigula iegal epeseaio aalysis fo ivisci flowfiels of useay aifoils I. J. No-Li. Mec. 3 (997) Laopoulos E.G. No-liea muliimesioal sigula iegal equaios i -imesioal flui mecaics aalysis I. J. No-Li. Mec. 35 (000) Laopoulos E.G. igula Iegal Equaios Liea a No-liea Teoy a is pplicaios i ciece a Egieeig pige-velag eli New Yok Laopoulos E.G. a Zisis V.. Eisece a uiqueess fo o-liea sigula iegal equaios use i flui mecaics ppl. Ma. 4(997) Laopoulos E.G. a Zisis V.. No-liea fiie-pa sigula iegal equaios aisig i woimesioal flui mecaics Noli. al. T. Me. ppl. 4 (000) Lamb. yoyamics Dove New Yok Kaamcei K. iciples of Ieal-Flui eoyamics Wiley New Yok