1 roceedigs of COEM 005 Copyrigh 005 by CM 8h Ieraioal Cogress of Mechaical Egieerig November 6-, 005, Ouro reo, MG INDICIL RESONSE OF THIN WINGS IN COMRESSILE SUSONIC FLOW Isaac Figueira Mirada Isiuo Tecológico de eroáuica (IT) São José dos Campos - S aulo foso de Oliveira Soviero Isiuo Tecológico de eroáuica (IT) São José dos Campos - S bsrac. umerical mehod o compue he idicial respose of a ig i a compressible subsoic flo is developed. The idicial respose compued is he variaio of he aerodyamic coefficies afer a sudde chage i he agle of aack of he ig. The mehod is a vorex laice mehod, here he useady effecs are compued by he iermie emissio of free vorex i he vorex shee of he ig. The obaied resuls are resriced o hi igs ad small agles of aack, because a liearized mahemaic model is used. The effecs of he Mach umber, aspec raio ad seep agle i he idicial respose are sudied. The resuls have good agreeme ih similar oes foud i previous orks. The mehod ca be also applied o sudy arbirary movemes of he ig, usig he superposiio priciple ad he iegral of Duhamel, due o he lieariy of he sysem. Keyords: useady aerodyamics, guss, idicial respose, vorex laice, compressible poeial flo. Iroducio The aeroelasic pheomea have grea effecs i he desig of moder aircrafs. They ca be divided i o pricipal groups: he saic pheomea ad he dyamic oes. I he aalysis of he saic aeroelasic pheomea, rasie effecs are egligible. Thus, i covers oly he ieracio beee he elasiciy of he srucure ad seady aerodyamics effecs. Hoever, srucural vibraios occurs i he dyamic aeroelasic pheomea, ad, hus, is aalysis require a reasoable dyamic modelig of he srucural, ierial ad useady aerodyamic forces. The pheomea of fluer, dyamic respose o guss ad blade vorex ieracio i helicoper roors are some examples of impora dyamic aeroelasic pheomea ha have grea ifluece i he srucural desig of aircraf igs. s sho, accurae models of he useady aerodyamics are ecessary o a good aalysis of hese problems. experimeal dealig ih useady aerodyamics modelig is very complex ad expesive, beig almos impracicable i he sudy of aerodyamic loads due o arbirary vibraios of lifig surfaces. Thus heoreical aalysis is required. Theoreical compuaio of useady aerodyamic loads due o arbirary vibraios of a lifig surface ca be doe hrough he composiio of he respose o more simple movemes. This kid of dealig is possible oly if he priciple of superposiio is valid, as i liear sysems, for example. If he mahemaical model is liear, accordig o isplighoff, shley ad Halfma (955), i is possible o compue he aerodyamic respose o arbirary movemes, usig he respose for harmoic oscillaios of he ig ad he Fourier iegral. This approach is useful, because he problem of harmoic oscillaios is simpler o solve. oher aleraive, hich also uses he superposiio priciple, is o use he obaied resuls for he idicial respose ad he Duhamel iegral. The idicial respose is he rasie of he aerodyamic loads afer a abrup sep chage i he agle of aack of he lifig surface. The idicial mehodology is more suiable ha he harmoic oe i he modelig of more abrup movemes, such as he respose o verical guss or rapid deflecios of corol surfaces. This occurs because i hese cases he harmoic mehodology has a slo covergece. The prese ork preses a umerical mehod o compue he idicial respose of plae igs submied o a compressible subsoic flo. The scheme is a vorex laice mehod, here he useady effecs are compued by he iermie emissio of free vorex, geeraig he vorex shee of he ig. I he developme of he mahemaical model used, he assumpios of iviscid, irroaioal ad iseropic flo are doe. esides ha, he equaios are liearized. Thus he model is valid oly for small perurbaios (lo agles of aack ad lo hickess of he ig profiles). These assumpios are reasoable for he kid of pracical applicaio he mehod is proposed. The firs heoreical sudies i useady aerodyamics appeared durig he 0s ad 30s decades. Wager (95), Kusser (936) ad Theodorse (935) developed he classical aalyical resuls for he idicial respose, sharp edge gus peeraio ad harmoic respose, respecively, of airfoils i icompressible flo. Sice hose years, may aalyical ad umeric soluios, harmoic or idicial, ere developed. alyical soluios, hoever, are oo complex ad resriced for a fe cases, i geeral. Thus, he search for umeric soluios, more fas ad geeral, becomes aracive. Heasle ad Lomax (949) ad eddoes (984) preseed resuls for he idicial respose of airfoils i supersoic ad subsoic flos, respecively. Joes (940) ad Lomax e al. (95) preseed resuls for he idicial
2 respose of fiie igs i icompressible ad supersoic flos, respecively. There are very fe orks dealig ih fiie igs i compressible subsoic flos, as he prese oe. Vepa (977) preses resuls for his codiio, usig a fiie sae modelig mehod. This ork as used o validae he resuls obaied usig he prese mehodology. Sigh ad aeder (997) ad Siarama ad aeder (004) compue he same hig usig CFD codes. The geeralized vorex laice mehod as developed o harmoic oscillaios of fiie igs i subsoic (Soviero ad orolus, 99), supersoic (Soviero ad Resede, 997) ad rasoic (Soviero ad César, 00) flos. I as also used i he compuaio of he idicial respose of airfoils i subsoic ad supersoic flos (Herades ad Soviero, 003) ad fiie igs i icompressible flo (Mirada ad Soviero, 004). I he prese ork, i is exeded o he compuaio of he idicial respose of fiie igs i compressible subsoic flo.. roblem descripio The problem cosiss i a plae fiie ig (o hickess) submied o a compressible subsoic flo of freesream velociy U. Iiially, his ig is ih zero agle of aack. I a give isa, i suffers a abrup sep chage i he agle of aack, hich becomes a o-zero fiie value (i he prese ork, as said, i is cosidered ha he agle of aack variaio is small eough o maiai he ig C L, lif coefficie, i he liear regio). This problem, amed idicial respose, or respose o a sep chage of agle of aack, cosiss i deermie he ime evoluio of he aerodyamic coefficies of he ig (i he prese ork jus he lifig coefficie is compued), from he perurbaio isa o he developme of he seady flo. Figure, sho belo, illusraes he problem. 3. Mahemaical model Figure. Illusraio of a sep chage i he agle of aack of a plae ig. s specified i he iroducio, he prese ork cosiders a iviscid, irroaioal ad iseropic flo. esides ha, i cosiders small perurbaios of he flo (lo agles of aack ad lo hickess of he ig profiles). Thus, he liear heory ad he superposiio priciple ca be used. Wih hese assumpios, he classical liearized equaio of he compressible ad useady velociy poeial ca be used o compue he velociy field over he ig. M ( M ) φ xx φ yy φ zz φ x φ = 0 () a a The problem ca be modeled umerically usig he sigulariies of he classical aerodyamics (i he prese ork are used he double ad vorex sigulariies). The boudary codiio is he agecy of he flo over he ig o every isa of ime. Tha is, he composiio of he o-disurbed flo ih he iduced flo by he poeial jump geeraed a he ig afer he aleraio of he agle of aack mus produce a velociy field such ha he velociy compoe ormal o he ig is ull i every poi of he ig ad every isa of ime. Thus, callig of W, he ormal compoe of he iduced velociy i a poi of he ig by he poeial jump geeraed (posiive direcio is doard) ad cosiderig si α α (small agle of aack), i follos ha: U = α W () Equaio mus be saisfied i every isa of ime ad every poi over he ig surface. The folloig equaio is valid i he liear compressible heory o compue he C (differece of he pressure coefficie beee he loer ad he upper sides of he ig): C = - ( φ U φx ) (3) U
3 roceedigs of COEM 005 Copyrigh 005 by CM 8h Ieraioal Cogress of Mechaical Egieerig November 6-, 005, Ouro reo, MG The φ erm is he o-circulaory erm ad he φ x erm is he circulaory oe. oher cocep ha is oe of he pillars of he umeric model here used is he Kelvi Theorem, hich guaraees ha ay poeial jump, φ, geeraed over he ig, remais cosa he i is dragged ih he freesream velociy o he vorex shee of he ig. 4. Numerical model The umerical mehod proposed is based o hree ell ko coceps of he heoreical aerodyamics: he impulsive geeraio of circulaio i a perfec fluid, he correspodece beee a superficial disribuio of ormal doubles ad a liear disribuio of vorex ad he emissio of free vorex, creaig he vorex shee of he ig. The igspa ad chord are discreized i recagular paels. Each pael has a corol poi i is cere, here he boudary codiio give i Eq. mus be saisfied i every isa of ime. I he iiial isa (=0), he sep chage i he agle of aack occurs. I his mome, i occurs a impulsive geeraio of poeial jump, ormal o he ig, of iesiy φ 0 (he superscrip 0 idicaes ha he superficial disribuio of ormal doubles as creaed i he isa =0 ad he subscrip idicaes ha i is relaed o he pael ). Figure shos he doubles disribuio, i he iiial isa, for a recagular ad a sep ig. oh igs are discreized ih four paels i he semi-spa ad o paels i he chord. The symmery beee he o semi-spas is ake i accou, o reduce he compuaioal cos. Figure. Recagular ad sep igs ih discreizaio i o paels of chord ad four paels of semi spa. Jus afer he impulsive geeraio of he ormal doubles disribuio, i is subsiued by a liear disribuio of vorex i he boudaries of each pael, i he isa =0, of iesiy Γ 0. s log as he superficial disribuio of ormal doubles is cosa i each pael, he vorex liear disribuio is also cosa i each pael boudaries. s already sho, his is a ell-ko cocep i heoreical aerodyamics. This chage is possible because he iduced flo of boh disribuios is ideical physically. This cocep is used i he prese ork because i is mahemaically easier o deal ih liear disribuios, ha area oes. Figure 3, belo, illusraes his cocep. Figure 3. Equivalece beee ormal doubles area disribuios ad vorex liear disribuios.
4 he same isa (=0), he railig vorex of he mos dosream ro of paels are emied i he vorex shee ad are dragged by he flo. I he ex sep of ime (=), occurs he same process: a e doubles superficial disribuio is geeraed; his disribuio is replaced by a disribuio of vorex i he boudaries of he paels, i he isa =, ad he free vorex are emied io he vorex shee. The process coiues i he subseque isas uil he seady flo is reached. The discreizaio of he ime (d) used is such ha he disace raveled by he free vorex, hich are dragged ih he freesream velociy (U), durig a ime sep is equal o he legh of oe pael i he chord direcio. For a discreizaio of he chord i m paels ad a chord legh c, i implies ha d = c / (U m). This relaio beee ime sep ad chord discreizaio is suiable o sho ell he ime evoluio of he lif over he ig. Figure 4, preses he process for a sep ig, i a op vie. Figure 4. Numerical process: appearig of he vorex ad emissios i he vorex shee i hree differe isas. The iesiy of he poeial jumps i each pael ad isa of ime is compued hrough he applicaio of he boudary codiio (Eq. ), a he corol pois. This applicaio is doe i he half of he ime sep. So, if ij k is he velociy iduced i he corol poi i, i he isa d/, by a uiary poeial jump creaed i pael j, i he isa k, i follos, from he Eq., ha: a a a x φ φ φ = U U U j= k = 0 j= k= 0 j= k= 0 The oly ukos of he Eq. 4 are he compoes of he vecor of he poeial jumps. pplyig Eq. 4 i each isa of ime, i is possible o compue he variaio of he poeial jumps ih ime. The marix ha muliplies he vecor of he poeial jumps is called he ifluece coefficies marix. I eeds o be compued jus oce. The vecor of he righ side of he Eq. 4 eeds o be compued i each sep of ime, because he iduced velociies vary ih he ime. Iside he ifluece coefficies marix, here are he erms a ii, hich are he impulsive velociies i each pael, creaed ih he impulsive geeraio of poeial. These erms appears oly a he pricipal diagoal of he marix, because is value i a give pael depeds oly o he iesiy of he poeial jump over he pael iself. They ca be compued, usig he liear piso heory (isplighoff, shley ad Halfma, 955) ad applyig he boudary codiio i he isa =0. I his isa, all he erms ij k are ull (i a compressible flo, he perurbaio creaed i he flo by he appearig of a poeial jump is o isaaeous). Thus, a isa =0, Eq. 4 reduces o: a φ U (5) ii i = k j k j k j (4)
5 roceedigs of COEM 005 Copyrigh 005 by CM 8h Ieraioal Cogress of Mechaical Egieerig November 6-, 005, Ouro reo, MG The liear piso heory saes ha, i he iiial isa, he pressure differeial over he ig is cosa ad equal o: U 4 = - ρ a C = ; C = (6) M ρ U I he above equaio, a is he free sream soud velociy ad ρ, he free sream desiy. From Eq. 3 ad 6, as φ x is ull, because he poeial disribuio is cosa over he ig i he iiial isa: U U = - ρ a φ = - d (7) M Thus, from Eq. 5 ad 7: a ii = (8) a d s sho i he Eq. 7, he values of he poeial jumps i he isa =0 are give direcly from he piso heory. Thus, he Eq. 4 is applied oly from he isa =. The erms ij k, of he iduced velociies by he vorex disribuios i each isa of ime are compued hrough he expressio for he iduced velociy by a vorex filame i a compressible flo. I he icompressible flo, here he perurbaio creaed by he vorex is fel isaaeously i all he pois of he flo, he expressio for he iduced velociy by a sraigh vorex filame, of iesiy Γ ad exremiies, (X,Y ), ad, (X,Y ), i a poi, (X,Y ), of he flo is give direcly by he iegraio of he io-savar La alog he filame (see Fig. 5). Tha is: W Γ = 4π ( x x ) ( x ( y x ) ( y ( x ( y x ) ( y (9) Figure 5. Illusraio of he io-savar La (lef side) ad vorex pael ih exremiies,, C, D. Figure 6. Ifluece regio, a isa, of a perurbaio (X V, Y V ) appearig i he flo a he isa 0.
6 # "! # This compuaio i a compressible flo is aalogous, bu a lile more complicaed, because he he vorex appears i he flo, he perurbaio iduced by i propagaes ih a fiie velociy. I his case, he he vorex is creaed, i appears a perurbaio ave i each poi of he filame. This perurbaio propagaes i all direcios ih he soud velociy (U / M, here M is he freesream Mach umber) ad is coveced ih he flo velociy. I he plae, he perurbaio ave is a circumferece. ois ou of his circumferece did o feel he perurbaio creaed by he vorex appearig ye. This cocep is sho i Fig. 6. perurbaio appears i he flo a he poi V, (X V,Y V ), a he isa = 0. I he isa, he perurbaio ave has o reached he poi D, (X D,Y D ), ye ad, hus, he iduced velociy by he vorex i he poi D is sill ull a he isa. O he oher side, a he same isa, he perurbaio ave have jus reached he poi C, (X C,Y C ), ad from his ime i ill feel he perurbaio iduced by he vorex. I is possible o compue he value of τ VC =(- 0 ), hich is he ime ecessary o he perurbaio creaed i he poi V o reach he poi C. From Fig. 6: [( x x ) ( y ] U ( xc xv ) U ( yc yv ) a C V C V τ VC = (0) ( a U ) The expressio for he vorex filame i he compressible flo depeds o he ime he perurbaio reaches he poi. I fac, he expressio for he icompressible flo, give i he Eq. 9 is a special case of he compressible equaio, hich occurs he τ=0 (i Eq. 0, he a, hich is he icompressible codiio, τ 0). For he vorex filames of Fig. 6, he folloig equaios should be applied: W W C Γ = 4π ( x x ) Γ = 4π ( y ( y [( x x ) U τ ] ( y [( x x ) U τ ] ( x x U τ ) ( y x [( x x ) U τ ] ( y [( x x ) U τ ] ( x U τ C ( y C ) ( y The equaios for he oher filames, CD ad D, are aalogous o Eqs. ad, respecively. I he above equaios, if τ is egaive for he exremiies of he filame, here are o possibiliies. The firs is he perurbaio has o reached ay poi of he filame. I his case, he value of he iduced velociy is ull. The secod oe is he perurbaio has reached he filame, bu o is exremiies. I his case, he equaios above are applied, jus chagig he exremiies coordiaes for he effecive oes a ha isa. Equaio is valid oly for a fixed vorex. The equaio for a free vorex, hich is coveced ih he freesream velociy, i a compressible flo is he same of he equaio for a fixed vorex filame i a icompressible flo, ha is, Eq. 9. The oly differece is ha he perurbaio las a fiie ime o reach he vorex, i he compressible flo, ad his is ake i accou. The lifig coefficie for ui of agle of aack (i radias), ca be compued iegraig he Eq. 3 over he ig. Thus: k CLα = Ci xi yi CLα () = - φi Γi (3) α αu Ud #$# i= i= i= k= 0 I Eq. 3, he circulaory erm is already rasformed i he iegraio of he circulaio, hrough he Kua- Joukouski Theorem. 5. Resuls ad coclusios () () Figure 7 shos comparisos of he resuls of he prese mehod ih resuls of similar previous orks, for he C Lα of he ig (i radias uis) as fucio of ime, afer a sep chage i agle of aack. The ime, i he graphs, is he o-dimesioal ime (s=u /c, here c is he chord of he ig). Figure 7a shos he resuls for a ig of ifiie spa, obaied ih he prese mehod (discreizaio of 0 paels i chord) ad he aalyical mehod of Lomax e al (95), for Mach umbers of 0.5 ad 0.8. Figure 7b shos he resuls for a recagular ig of aspec raio 6 (R 6), for Mach umbers of 0.3, 0.5 ad 0.7, obaied by Vepa (977), ih his fiie sae modelig, ad he resuls of he prese mehod (discreizaio of 5 paels i chord ad 0 i he spa, ha is, 5x0). oh he comparisos had good agreeme. If a higher discreizaio i chord (ad cosequely i ime) ere used, he agreeme ould be sill beer. Qualiaively, he respose of all he mehods begis, a he iiial isa, ear he value prediced by he piso heory
7 roceedigs of COEM 005 Copyrigh 005 by CM 8h Ieraioal Cogress of Mechaical Egieerig November 6-, 005, Ouro reo, MG (see Eq. 6). fer ha, he C Lα decays rapidly, bu begis o icrease agai, o more sloly, uil reach he seady flo value. Figure 7. Comparisos: a) D resuls of Lomax e. al (95) for Mach 0.5 ad 0.8 ad resuls of he prese mehod; b) resuls of Vepa (977) for a recagular ig of R 6 ad resuls of he prese mehod. Figure 8. Ifluece of he aspec raio for recagular igs a a) Mach 0.3, b) Mach 0.5, c) Mach 0.8 ad d) ifluece of he seep agle i a ig of R 0 for Mach 0.8. Figure 8 shos resuls of he prese mehod for several Mach umbers, aspec raios ad seep agles. The goal is o check he ifluece of hese parameers i he idicial respose. The discreizaio used i his figures as x0, hich is small, pricipally i chord, bu is eough o address he qualiaive ifluece of he parameers of he ig. Figure 8a shos he idicial respose i C Lα for a recagular ig a Mach 0.3 ad several aspec raios. Figures 8b ad 8c sho he same, for Machs 0.5 ad 0.8. Figure 8d shos he respose of a ig of aspec raio 0 (aper raio ), a Mach umber 0.8, ih several seep agles. The resuls sho coclusively ha he seady sae is reached more sloly ih he icrease of he aspec raio ad Mach umber. This is expeced, because he propagaio of he perurbaios becomes sloer ih he icrease of he Mach umber. The effec of he aspec raio ca be explaied by he effec of he ig ip vorex, hich has more effec i smaller aspec raios. he oher side, he seep agle has
8 o sigifica ifluece i he ime o seady sae. The oly ifluece, of course, is he value of he C Lα i he seady sae. The mehod shoed o be able of obaiig good resuls for he idicial respose of igs i compressible flo. Is pricipal advaage is o be applicable o differe siuaios, hile aalyical resuls are resriced. esides ha, he code is very simple ad does o demad much compuaioal resources, differe of CFD codes, hich ca be used for he same purposes (Sigh ad aeder, 997; Siarama ad aeder, 004), bu are much more ime ad moey demadig. For obaiig beer quaiaive resuls, pricipally i he begiig of he respose, i is ecessary a grea chord discreizaio (50 or 00 paels), hich demads some hours of compuaio, bu o more ha CFD demads. s already sho, he idicial resposes for several values of sep chages of agle of aack ca be superposed, hrough he Duhamel Iegral, o sudy he respose o arbirary oscillaios. Thus, he prese mehod ca be very useful i prelimiary projecs of aircrafs. 6. Refereces eddoes, T. S., 984, racical Compuaio of Useady Lif, Verica, Grea riai, V. 8, p isplighoff, R. L., shley, H., ad Halfma, R. L., 955, eroelasiciy, ddiso-wesley, Readig, M. Heasle, M.., ad Lomax, H., 949, To-Dimesioal Useady Lif roblems i Supersoic Fligh, NC Repor 945. Herades, F., ad Soviero,.. O., 003, Modelo Numérico para erfis Fios em Escoameo Compressível Não- ermaee, Maser Degree Thesis i erodyamics, ropulsio ad Eergy, IT, São José dos Campos-S. Joes, R. T., 940, The Useady Lif of a Wig of Fiie spec Raio, NC Repor 68. Küsser, H. G., 936, Zusammefasseder erich über de isaioare ufrieb vo Flügel, Luffahrforsch., d.3, Nr.. Lomax, H., Heasle, M., Fuller, F., ad Sluder, L., 95, To ad Three Dimesioal Useady Lif roblems i High Speed Fligh, NC Repor 077. Mirada, I. F.., ad Soviero,.. O., 004, Modelo Numérico para sas laas em Escoameo Icompressível Não- ermaee, als of 0 o razilia Cogress of Thermal Scieces ad Egieerig - ENCIT 004. Sigh, R., ad aeder, J. D., 997, Direc Calculaio of Three-Dimesioal Idicial Lif Respose Usig Compuaioal Fluid Dyamics, Joural of ircraf, Vol. 34, No. 4. Siarama, J., ad aeder, J. D., 004, Compuaioal-Fluid-Dyamics-ased Ehaced Idicial erodyamic Models, Joural of ircraf, Vol. 4, No. 4. Soviero,.. O. ad orolus, M. V., 99, Geeralized Vorex Laice Mehod for Oscillaig Lifig Surfaces i Subsoic Flo, I Joural, Vol.30, No., pp Soviero,.. O. ad Resede, H.., 997, Geeralized Vorex Laice Mehod for Oscillaig Lifig Surfaces i laar Supersoic Flo, I Joural, Vol.35, No. 7, pp Soviero,.. O. ad Cesar, G.. V., 00, Sep Thi Wigs i Useady Soic Flo, I Joural, Vol.39, No. 9, pp Theodorse, T., 935, Geeral Theory of erdyamic Isabiliy ad he Mechaism of Fluer, NC Repor 496. Vepa, R., 977, Fiie Sae Modelig of eroelasic Sysems, NS CR-779. Wager, H., 95, Uber die Esehug des dyamische ufriebes vo Tragflügel, Z. age. Mah. Mech., d. 5, Hef. 7. Resposibiliy oice The auhors are he oly resposible for he pried maerial icluded i his paper.