Finite Wing Theory. Wing Span, b the length of the wing in the z-direction. Wing Chord, c equivalent to the airfoil chord length

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1 4 Fnte Wng Thery T date we have cnsdered arfl thery, r sad anther way, the thery f nfnte wngs. Real wngs are, f curse, fnte wth a defned length n the z-drectn. Basc Wng Nmenclature Wng Span, the length f the wng n the z-drectn Wng hrd, c equvalent t the arfl chrd length Wng Tp - the end f the wng n the span-wse drectn Wng Rt the center f the wng n the span-wse drectn Wng Area, S, D, M tw dmensnal lft, drag and mment, D, M - three dmensnal lft drag and mment ceffcents

2 5 The flw ver a fnte wng s decdedly three dmensnal, wth cnsderale flw pssle n the span-wse drectn. Ths cmes aut ecause f the pressure dfference etween the tp and ttm f the wng. As n tw-dmensnal flud mechancs, the flw wants t mve n the drectn f a decreasng pressure gradent,.e., t wll usually travel frm hgh t lw pressure cndtns. In effect the flw splls frm the ttm t the tp as shwn n the fgure elw. The span-wse rtatn manfests tself as a wng tp vrtex that cntnues dwnstream.

3 6 Interestngly the sense (rentatn, rtatn) f these wng tp vrtces s cnsstent wth takng the tw-dmensnal arfl crculatn, magnng that t exsts ff t nfnty n th drectns and endng t ack at the wng tps. The dea s als cnsstent wth Kelvn s therem regardng the start-up vrtex. mnng the tw deas ne sees that a clsed x-lke vrtex s frmed. Hwever, n much f the thery presented next the start-up vrtex s gnred and we cnsder a hrseshe vrtex. Flw u Bund vrtex Wng u Tp vrtex X u Tp vrtex Start-up vrtex As shwn n the fgure, the vrtces nduce flw dwnward nsde the x and upward utsde the x. Ths flw s called the nduced velcty r dwnwash, w. The strength f these vrtces s drectly related t the amunt f lft generated n the wng. Arcraft nflght spacng s determned n part ecause f these wngtp vrtces. An example s the Arus arcraft that crashed at JFK a few days after 9/. The spacng was t small and the Arus s tal was uffeted y the wake vrtces ff a JA 747 that was ahead f t n the flght path. u

4 7 Angle f Attack The dea that vrtex mtn nduces dwnward flw changes the way we have t lk at angle f attack as cmpared t the arfl thery. Gemetrc angle f attack, " - the angle etween the arfl chrd lne and the freestream velcty vectr. Induced angle f attack, " the angle frmed etween the lcal relatve wnd and the undstured freestream velcty vectr. w tan (5.) V Effectve angle f attack, " eff the angle frmed etween the arfl chrd and the lcal relatve wnd. (5.) eff It s mprtant t nte that ths als changes hw we lk at lft,, and drag, D. Ths s ecause the actual lft s rented

5 8 perpendcular t the lcal relatve wnd (snce that s the wnd that t sees) nt the freestream velcty drectn. Because f that, when we g ack t ur rgnal lft and drag drectns (perpendcular and parallel t the freestream) we nw see a reductn n the lft as cmpared t what we expect frm the arfl thery and an actual drag called the nduced drag, even thugh the flw s stll nvscd. What a Drag Induced drag, D drag due t lft frce redrectn caused y the nduced flw r dwnwash. Skn frctn drag, D f drag caused y skn frctn. Pressure drag, D p drag due t flw separatn, whch causes pressure dfferences etween frnt and ack f the wng. Prfle drag ceffcent, d sum f the skn frctn and pressure drag. an e fund frm arfl tests. Nte the ntatn. d D f + Dp (5.3) q S Induced drag ceffcent, D - nndmensnal nduced drag D D (5.4) q S Ttal drag ceffcent, D + (5.5) D d D Arfl data Fnte wng thery

6 9 Bt-Savart aw Durng ur dscussn f panel methds we develped the dea f a vrtex sheet, essentally a lne alng whch vrtcty ccurs that has a rtatn sense aut an axs perpendcular t the lne. vrtex sheet A smlar ut dstnctly dfferent dea s that f vrtex flament, whch s agan a lne f vrtcty, ut ths tme wth rtatn aut the lne tself.

7 3 Vrtex flament The Bt-Savart aw defnes the velcty nduced y an nfntesmal length, dl, f the vrtex flament as dv dl r (5.6) 3 4 r where dl nfntesmal length alng the vrtex flament r radus vectr frm dl t sme pnt n space, P. dv nfntesmal nduced velcty Nte that ths velcty s perpendcular t th dl and r. If the vrtex flament has nfnte length the ttal nduced velcty s fund y ntegratng ver ts entre length V dl r 3 4 r (5.7) nsder a straght vrtex flament n the y-drectn and a pnt, P, n the x-y plane. Equatn (5.7) can e put nt gemetrc functns y cnsderng the fgure elw

8 3 snθ V dl (5.8) 4 r where s the angle frmed y r and the flament. The gemetry gves whch gves h h h, l, dl dθ (5.9) snθ tanθ sn θ r V snθdθ 4 h (5.) we get l ±f we have r B. Ths leads t the smple result

9 V (5.) h same as the tw dmensnal thery. If we have a sem-nfnte flament we get snθ V dl snθdθ (5.) r 4h V 4 4h Helmhltz Therem. Strength f a vrtex flament remans the same alng the flament.. A vrtex flament cannt end n a flud,.e., t must ether extend t the undares r frm a clsed path. 3 Addtnal Nmenclature Gemetrc twst a twst f the wng aut the span-wse axs that results n a change n the gemetrc angle f attack wth span-wse pstn. Washut gemetrc twst such that < Washn - gemetrc twst such that > Aerdynamc twst a wng wth dfferent arfl sectns alng the span, s that the zer lft angle f attack changes wth span-wse pstn. tp tp rt rt

10 33 ft dstrutn the lcal value f the lft frce. Ths can change wth span-wse pstn. Fr example, snce the pressure equalzes at the tp there s n lft there. area. ft per unt span, - akn t pressure,.e., frce per unt ρ V ( y) (5.3) ( y) dy (5.4) If the lft changes alng the span t mples that there are multple (perhaps and nfnte numer f) vrtex flaments.

11 34 Prandtl s ftng ne Thery Prandtl s lftng lne thery s centered aut a fundamental ntegr-dfferental equatn. d dy ( y ) + dy ( y ) ( y ) + (5.5) V c( y ) 4V y y whch s used t fnd the crculatn dstrutn aut the wng. Equatn (5.5) s useful f ne knws the desred gemetrc angle f attack, the aerdynamc twst (.e., " ), and the wng planfrm (.e., lcal chrd length). Tw appraches are presented t make use f ths equatn. Equatn (5.5) s develped frm the dea f vrtex flaments. Prandtl s lftng lne thery stems frm the dea f replacng a wng wth a und vrtex. Helmhltz therem then requres there t exst tralng vrtces at the wng tps

12 35 The Bt-Savart law allws us t determne the dwnwash alng the wng and results n: w( y) 4 ( + y) 4 ( y) (5.6) r w( y) 4 ( ) y (5.7) Nte that the dwnwash s a negatve numer as yu wuld expect frm the crdnate system. Hwever, the sngle vrtex flament case s nt suffcent t descre the physcal cndtns n the wng, ecause f

13 36 The trule s that the dwnwash at the wng tps s nfnte nstead f zer. Frtunately, ths can e fxed f ne cnsders a dstrutn f vrtex flaments as shwn elw It s mprtant t nte that the strength f the ndvdual vrtex flaments s equal t the jump n crculatn at the pnt where the tralng vrtex meets the und vrtex. Ths can e carred t the lgcal extreme y cnsderng a cntnuus sheet f vrtex flaments and ther asscated cntnuus change n crculatn. In that case the dwnwash nduced at pnt y y the vrtces at pnt y s gven y d dy dy dw( y ) (5.8) 4 ( y y)

14 37 S that f ne ntegrates frm wngtp t wngtp d dy dy w ( y ) (5.9) 4 ( y y) Relatnshp etween ' dstrutn and dwnwash at y Usng Equatn (5.), ut recgnzng that w s a negatve numer w( y ) ( y ) tan (5.) V fr small angles Eq. (5.) gves d dy ( ) dy y (5.) 4V ( y y) Recall the tw dmensnal lft ceffcent fr an arfl [ ] [ ] l a eff eff (5.) where ut eff eff ( y ) ecause f dwnwash ( y ) ecause f aerdynamc twst f ρ V c( y ) l ρv ( y ) (5.)

15 38 then l ( y ) (5.3) V c( y ) mnng Eqs. (5.) & (5.3) gves ( y ) (5.4) eff + Vc( y ) whch s clearly a functn f y. Recall ntes Eq. (5.) (5.) eff and cmne Eq. (5.) wth (5.4) & (5.) t get d dy ( y ) + dy ( y ) ( y ) + (5.5) V c( y ) 4V y y Fundamental Equatn f Prandtl s ftng ne Thery Once y ) s knwn (,, D, fllw drectly. D ( y ) ρ V ( y ) (5.3) ( y) dy (5.4)

16 39 ( y) dy V S (5.5) D dy (5.6) D ( y) ( y) dy V S (5.7) Tw appraches can e taken frm ths pnt. Drect A wng planfrm s gven wth a dstrutn f aerdynamc twst, Eq. (5.5) s slved and lft and drag nfrmatn extracted.. Inverse A lft dstrutn s prpsed and the crrespndng planfrm dstrutn develped.

17 4 Inverse Apprach Ellptc ft Dstrutn Prandtl s lftng lne thery can e used n an nverse apprach y assumng the frm f the lft dstrutn and usng Eq. (5.5) t determne the wng planfrm. The mst famus example f ths s the ellptc lft dstrutn whch s fund drectly frm an ellptc crculatn dstrutn. ( ) y y (5.8) where we can use ( y) ρ V ( y) t shw ( ) y y ρ V (5.9) d dy d Recall that Eq. (5.9) requres, s dy 4 y 4 y (5.3)

18 4 s that the dwnwash ecmes y w( y ) 4 y we can agan nvke gemetry and use dy ( y y) (5.3) y csθ, dy snθdθ (5.3) where θ : as y : S Eq. (5.3) ecmes r w θ ) csθ dθ csθ csθ ( (5.33) csθ w( θ ) dθ (5.34) csθ csθ whch s a standard ntegral frm w( θ ) sn nθ snθ wth n (5.35) w( θ ) (5.36) Dwnwash s cnstant fr an Ellptcal ft Dstrutn

19 4 Hwever, ths als mples: V V w (5.37) Induced a..a. s cnstant fr an Ellptcal ft Dstrutn In the end we want t determne the lft and drag ceffcent fr the ellptc lft dstrutn and als the shape f ths wng. T d ths we g ack t Eqs. (5.4) and (5.9) ρ ρ θ θ ρ θ θ ρ θ θ θ ρ θ θ θ ρ ρ 4 sn 4 sn sn sn sn cs ) ( V V V d V d V d V dy y V dy y (5.38) (5.39)

20 43 We can next use the defntn f and Eq. (5.39) t gve r ρ V ρ S V 4 (5.4) V S (5.4) Then gng ack t Eq. (5.37) we fnd S V (5.4) Nmenclature Aspect Rat, AR (5.43) S S that (5.44) AR We can then get nduced drag frm Eq. (5.7) D ( y) ( y) dy V S (5.7)

21 44 θ θ θ θ θ θ θ sn 4 sn sn cs S V d S V d S V dy y S V D gemetry S V D (5.45) Whch can e rewrtten y susttutng Eqs. (5.4) and (5.44) nt (5.45) S V S V AR D AR D (5.46) D - a typcal drag result AR D - use hgh AR wng (lng and thn) But what s the gemetry?!!!

22 45 The gemetry can e fund y gng ack t the lft ceffcent and Eqs. (5.3), (5.) and (5.) l l ( y ) (5.3) V c( y ) [ ] eff (5.) (5.) eff Then usng the dea that the nduced a..a. s cnstant and f there s n aerdynamc twst we see frm Eqs. (5.) and (5.) that l cnst. (5.47) Ellptc ft Dstrutn mnng Eqs. (5.47) and (5.3) gves ( y ) cnst. c( y ) cnst. ( y ) V c( y ) (5.48) Wth the end result: An ellptc lft dstrutn s fund frm an ellptc wng planfrm.

23 46

24 47 Applcatn f Prandtl s ftng ne Thery fr an Ellptc Wng The prevus sectn demnstrated that an ellptc wng planfrm develps an ellptc lft dstrutn ut dd nt answer the questn f hw ne can fnd the aerdynamc prpertes f an ellptc wng. nsder an ellptc wng ) ( y c y c r (5.49) It s clear frm Eq. (5.3) that l can e wrtten n terms f ut Eqs. (5.) & (5.) als shw that l depends n, whch n turn depends n, whch then depends n an ntegrated value f l. What s needed s a way t clse the lp. T d ths cnsder agan Eqs. (5.3), (5.) and (5.) [ ] [ ] r eff r l V c V c V frm whch we see [ ] [ ] + + c V V c V r r (5.5)

25 48 Therefre, gven the rt chrd, span and arfl shape fund. Upn rearrangng Eq. (5.4) can e VS and frm Eq. (5.46) (5.4) D (5.46) AR Example Prlem: nsder an ellptc wng wth m span and.5m rt chrd. If the wng s made up f NAA 64- wng sectns and s flyng 5m/s at a gemetrc angle f attack f 8 degrees, cmpute. and D. and D 3. The acceleratn f ths wng f t has a mass f Mg at sea level. NAA 64-

26 49 The chart shws that. 8, s that

27 5 ( 5m s) [ 8 (.8 )] m rad 48.3 sec +.5m m cr ( m)(.5m) S 9.63m 4 AR ( m) 5. 9 S 9.63m m rad ( m) 48.3 sec V S 5m s 9.63m ( )( ).77 D AR (.77) ( 5.9).37 ft and Drag calculatns ρ V S 3.5kg m 5m s D ρ V S D 3 D.5kg m 5m s ( )( ) ( 9.63m )(.77) 3.kN ( )( ) ( 9.63m )(.37).kN S t can lft.38mg

28 5 Acceleratn calculatns + 3.kN F net 3.kN 9.8kN 3. 3kN r a. 36g W9.8kN Exercse: Develp + and D ver the range f a..a frm ft effcent - Ellptc Wng - m r.5m V5m/s _ Angle f Attack Induced Drag effcent - Ellptc Wng - m r.5m V5m/s _D Angle f Attack

29 5 Ellptcal Wng essns: Desgn nsderatns Equatns (5.5), (5.4) & (5.46) gve us a clear path fr the aerdynamc analyss f an ellptcal wng f gven dmensns and arfls. On the ther hand, the analyst s j s t determne prpertes f a gven wng, a desgner s j s t decde the gemetry tself s that t reaches a specfc desgn jectve. Ths s a dstnctly dfferent skll. The ave equatns can gve the desgner sme nsght f they are manpulated prperly. Of curse, ne shuld e careful aut drawng cnclusns fr a general frm ths analyss snce t apples strctly t ellptc planfrms, ut t turns ut that ther wngs ehave smlarly fr many parameters, althugh ther analyss s mre cmplcated. S Start y agan cnsderng Eq. (5.5) V [ ] (5.5) + c r V [ ] c rv [ ] c r + c + c r r 4 cr 8SV [ ] 4S 4S c + r 4SV[ ] + S S that f S s a cnstant, ges dwn as ges up. Hwever, ths really desn t tell us anythng aut lft and drag n the wng.

30 53 T get that nfrmatn we need t cnsder Eq. (5.4) (5.4) V S 4SV V S [ ] [ ] ( + S) [ ] + S + AR Therefre f keep the wng area the same ges up wth. Nte that ths result drps ack t the tw-dmensnal arfl result as. _..8 _.6.4. _ AR

31 54 Hw then des ths result affect the nduced drag? Eq. (5.46) says: 4 [ ] D AR + AR AR D whch says that l AR [ ] + [ ] 4 AR 4 4 AR AR D AR as and hence. decreases as ncreases. Ths s ecause Hence the cmprmse cmes t lfe etween what can e ult and what s aerdynamcally effcent. D _D _D Aspect Rat _D

32 55 General ft Dstrutn Our prevus analyss fr the ellptcal wng utlzed an ellptc crculatn dstrutn shwn n Eq. (5.8) ( ) y y (5.8) Ths frm was smplfed y the gemetrc transfrmatn y csθ, dy snθdθ (5.3) When cmned, Eq. (5.8) ecmes ( y) cs θ snθ The utlty f ths transfrmatn s apparent, hwever, ts mpact s much gger ecause t suggests the use f a sne seres t represent any crculatn dstrutn. The asc dea eng that the crculatn can e wrtten as N ( y ) V A sn n nθ (5.5) n Why shuld ths wrk? Frst ff, let s assume that ths seres s a reasnale apprxmatn f the actual crculatn functn and further assume that yu can fnd values fr the N cnstants, A n. Then recall what was dne fr the ellptcal wng planfrm; we started thngs ut nt knwng what s, ut y rearrangng equatns we were ale t determne t n terms f the velcty, rt chrd, span and arfl shape. In addtn, all f ths nfrmatn was taned at a sngle lcatn, the wng rt.

33 56 If we use equatn (5.5) fr a general arfl, there are nw N unknwn ceffcents needed t determne the crculatn. These ceffcents wll e determned later y usng cndtns and gemetry at N lcatns n the wng. Hwever, efre we d that t s mprtant t recgnze why Eq. (5.5) mght e reasnale. Furer Sne Seres A very useful engneerng tl s the Furer Seres, whch s essentally a summatn functn cmpsed f snes, csnes r th dependng n the functn t e represented. Mathematcans have prven that any functn can e represented y nfnte seres f ths frm and practcal experence shws that nly a lmted numer f terms are needed t get a reasnale apprxmatn. The functns themselves lk as shwn elw fr the frst 5 terms Furer Sne sne

34 A severe example f ts applcatn fr a square and a sawtth wave are shwn elw 57

35 58 Dr. Orkws has used ths apprxmatn apprach t represent the wakes ehnd a turne statr and fund that nly 4 terms were needed t get aut 95% f the ttal energy. The apprxmatn functns are generally gd representatns f the actual functns f the actual functn s smth. Dscntnuus functns lke square r saw tths prduce rngng r G s phenmenn unless a very large numer f terms are ncluded. Frtunately, wng crculatn dstrutns are usually qute smth and requre relatvely few terms. Keep n mnd that fnte wng thery requres d th ( y) and, s rngng can e a prlem as the dervatve dy can e adly dstrted even thugh the functn s well represented. Applcatn t Prandtl s ftng ne Thery As stated ave, Eq. (5.5) must e dfferentated t e used n the Prandtl ftng ne Thery. We get d dy d dθ dθ dy V N n na n dθ csnθ dy (5.5) Susttute the ave nt Eq. (5.5) r ( θ ) ( θ ) c N na nθ dθ n cs n ( θ ) + dθ N An sn nθ + ( θ ) n dy c N csθ csθ na nθ dθ n cs n ( θ ) + dθ N An sn nθ + ( θ ) n dy csθ csθ (5.53)

36 59 where the ntegral s the famlar standard frm used earler, s that N N dθ sn nθ ( θ ) An sn nθ + ( θ ) + na (5.54) n c dy n snθ ( θ ) n Whch can e used t fnd the N, s f Eq. (5.54) s evaluated at N, θ values. A system f equatns s thus develped and easly slved. We shuld recgnze that the rgnal ellptc wng stll lves n ths equatn f we cmpare Eq. (5.54) t Eq. (5.5) and nte that N. As efre, everythng fllws nce ( θ ) s knwn. cmes frm Eq. (5.5) A n ( y) dy V S (5.5) N V VS n N An sn nθ snθ dθ (5.55) An sn nθ snθdθ S n and, f curse, an ntegral tale wll reveal that n sn nθ snθdθ (5.56) n S n matter hw many terms yu have n the seres, t s nly the frst ne that matters fr,.e., A A AR (5.57) S Keep n mnd thugh that A s part f a system f equatns and as such depends upn KA. A N

37 6 Next, fr the nduced drag we need as ndcated y Eq. (5.7). Agan, Eq. (5.) says d dy (5.) 4V ( y y) dy whch y cmparsn t Eq. (5.53) s N r frm Eq. (5.54) nan csnθ n dθ csθ csθ sn nθ N na (5.58) n n snθ We then g ack t Eq. (5.7) D ( y) ( y) dy V S (5.7) D N N V A nθ n sn na V S n n n sn nθ snθ snθ dθ Nte nt

38 6 Upn further evaluatn D N N AR A nθ n sn ma n m (5.59) m sn mθ dθ Whch s easly evaluated snce m k sn m θ sn kθdθ (5.56) m k Hence, the nly tme the ntegrals yeld smethng s when nm. S N D AR na n (5.6) n r equvalently D AR A ARA whch we can wrte as D D AR ARA + + N n N n [ + δ ] [ + δ ] na n A n A n (5.6) Eq. (5.6) shuld e cmpared wth Eq. (5.46) fr the ellptc wng. Yet anther way t wrte Eq. (5.6) s D (5.63) ear

39 6 Nmenclature Span effcency factr, e where learly e fr an ellptc wng. e (5.64) + δ Ellptc vs. Rectangular mprmse: The Tapered Wng Nmenclature ct Taper Rat cr The dea s t match clsely the ellptc wng planfrm,.e., chrd lengths f smlar sze, and therey match the ellptc lft dstrutn.

40 learly, * has a mnmum fr a taper rat f aut.3. 63

41 64 Numercal ftng ne Thery The lftng lne thery assumes a lnear lft curve slpe, as such, t des nt predct the nnlnear r stall regme. Andersn descres a numercal lftng lne prcess Assume a ( y) alculate alculate eff Use taulated data fr t cmpute ( y) Iterate untl cnvergence The eauty f the technque s that t wrks n the nnlnear/stall regme, hwever, t requres sgnfcant tale lk-ups, whch are slw. Vrtex attce Methd A dsadvantage f the lftng lne thery s that all f the actn asscated wth the und vrtex ccurs at the quarter chrd pnt, such that nly the lft and drag ceffcents are cmputed ut nt the mment ceffcent. Unfrtunately, the mment ceffcent s essental t the perfrmance calculatns. An answer s fund n the vrtex lattce methd whch nt nly prvdes the pressure dstrutn ut als anchrs the results t the actual gemetry rather than mplctly thrugh the. Ths s essental nt nly fr calculatng mments ut fr many practcal wng planfrms lke delta wngs.

42 65 Essental deas: Panel the wng wth dscrete spanwse, γ, and streamwse,δ, dstrutn f vrtces. Set a cntrl pnt smewhere n ths panel t apply the flw tangency cndtn. Bt-Savart t determne the nduced velcty frm all pnts. Slutn f a system f equatns determnes the dscrete vrtcty dstrutns va the dwnwash equatn: w( x, y) 4 ( x ξ ) γ ( ξ, η) + ( y η) 4 S ( x ξ ) + ( y η) ( y η) δ W ( ξ, η) dξd 3 W [( x ξ ) + ( y η) ] Nte that the wake s als ncluded. [ ] δ ( ξ, η) dξdη η 3 (5.65) Anther way t lk at ths s as a system f hrseshe vrtces, each ne appled ver a sngle panel wth a und vrtex at the

43 66 quarter pnt f the panel. The cntrl pnt s agan defned t apply the surface tangency cndtn. The wng s then tessellated wth a vrtex lattce system lke that shwn n the fgure. Once agan the velcty cntrutns are determned frm each hrseshe vrtex and usng the flw tangency cndtn and we are left wth a system f equatns fr the unknwn crculatns. Nte that ths apprach s appled n the plane f the arfl, nt alng the arfl s that we are really stll makng a thn arfl assumptn.

44 67 3D Panel Methds The next step n the herarchy f technques s a drect extensn f the D surce and vrtex panel methds, a 3D panel methd. The text descres these methds refly and develps the 3D surce and dulet. Basc Idea: Dstrute surces, dulets r vrtces n the surface f a dy. Apply the flw tangency cndtn. Slve fr the unknwn surce, dulet and vrtex strengths. Ths apprach s wdely used n the ndustry fr prelmnary desgn cnsderatns and allws us t apply the surface tangency cndtns t all pnts n the wng. A large cde s wrtten fr ths purpse and generally takes a gd deal f effrt t defne the gemetry and apply the methd.

45 68

46 69 Separated Flw Vrtex ft Whle we are dscussng the nvscd flw aut a wng we need t cnsder what happens f the flw s massvely separated. In mst nstances ths s a ad thng as t wll dsrupt the rderly flw aut a wng and lead t pressure drag. Hwever, the case f a delta wng ths s an entrely dfferent ssue as separated flw actually results n enhanced lft, vrtex lft.

47 7 As llustrated n the fgure, severe flw separatn and reattachment ccurs fr the delta wng. The vrtces accunt fr lft n the wng ecause they cause the lcal pressure t drp cnsderaly. The methds we have dscussed s far are nt gd canddates fr predctng these flws; ne has t resrt t full cmputatnal flud dynamc analyss,.e., the feld slutn f the Euler r Naver- Stkes equatns, whch s eynd the scpe f ths class.

48 7 Mments An advantage f the vrtex lattce methd, the 3D panel methds and FD, as cmpared t the lftng lne thery, s that the pressure dstrutn s cmputed and hence mments can e fund. T recall the detals f ths prcess we need t g ack t sme asc mechanc. nsder an arfl as a eam wth dstruted lads defned frm the pressure. E TE We recall that the pressure s a frce per unt area and that a frce s defned nce the pressure s ntegrated ver an area. In the case f an arfl we nly ntegrated n x, snce they drectn (as defned fr a wng) has unt length. We can then use these deas t cme up wth mments aut a pnt n the wng. M E TE E ( ) xdx() (5.66) PTp PBt Hwever, the mment culd as e represented n terms f the resultant frce, R. M E Rx P (5.67) where x s the pnt at whch the resultant f the ntegrated P frces acts t generate the same mment aut the E.

49 7 et s return nw t an arfl: Nte that. The chrd lne s a straght lne that cnnects the E t the TE. (Nt the mean camer lne). Axal frce s assumed t act alng the chrd lne, therefre n mment s develped ecause f t. 3. Nrmal frce acts nrmal t the chrd lne at the center f pressure. 4. Mment s defned pstve clckwse, therefre, a pstve lft frce results n a negatve M. E x P M E (5.68) N 5. The mment ceffcent aut the center f pressure s zer. 6. Mments can e defned aut the E and the quarter chrd pnt. If then N and c M E + M c xp 4 4 (5.69)

50 73 Ths dea can e easly extended frm the arfl t the wng y turnng Eq. (5.66) nt a dule ntegral. M E TE ( P P ) Tp Bt E xdxdy (5.7)

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