Two-dimensional, incompressible, inviscid and irrotational flow

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1 AERO 58A THIN AEROFOIL THEORY Leture Notes Autor : Hadi Wiarto Two-dimesioal, iompressible, ivisid ad irrotatioal flow Tis ote is prepared as leture material for te ourse AERO 58A Fudametals of Aerodyamis for te topi of Ti Aerofoil Teory. It begis wit a disussio o te goverig equatios for -dimesioal, iompressible ad ivisid flow, wi is te Laplae equatio. Disrete sigularities tat are te elemetary solutios of te Laplae equatio are te disussed. It is te followed by a disussio o te liearity property of te Laplae equatio, wi leads to a disussio o te oept of otiuous sigularities su as te vortex pael wit ostat ad liear vortex stregt per uit legt distributio. Tis is te basi oept beid te pael metod, wi is explaied i some detail, partiularly for a seod order vortex pael metod. Eve toug te pael metod is desribed i some detail, tis is ot a artile o pael metod as su. Te oept is itrodued so tat studets will ave some familiarity wit te oept of otiuously distributed sigularity, vortex i partiular, wi is a fudametal oept i Ti Aerofoil Teory (TAT). Te basi oept i TAT is tat a aerofoil is replaed by a sigle vortex pael o wi tere is a otiuously distributed vortex sigularities, te stregt per uit legt of wi is ukow ad eeds to be evaluated. A good uderstadig of tis basi oept is ot possible witout some rudimetary kowledge of a vortex pael. Te oept of boudary oditios, tat must be satisfied by te sougt for solutio, is also disussed. Te matematis ivolved is disussed i some detail, but from a egieerig poit of view were matematial kowledge is used as a tool to elp i solvig egieerig problems. Basi matematial formulas are assumed as give witout sowig teir derivatios. Te maipulatio ad appliatio of tose matematial formulas are, owever, sow i suffiiet details so tat studets a gai uderstadig o ow te fial egieerig equatios are obtaied ad ot merely give as formulas to be memorized blidly. Fially te egieerig appliatio of te derived formulas to alulate te aerodyami properties of aerofoils is disussed i relatively great details. Te appliatio disussed iludes aerofoils, wi a be represeted by a flat plate or a ambered (urved) plate, as well as a otrol surfae, wi is represeted by a bet flat plate. Please report ay typograpial or oter errors to te autor at te followig address adi.wiarto@rmit.edu.au Budoora, Melboure, 5 Mar 4 Last modified 5 May 4. Te goverig equatios Te simplest model of airflow is represeted by -D, iompressible, ivisid ad irrotatioal flow. Te goverig equatios for tis type of flow osist of partial differetial equatios, ea represetig te oservatio of mass or te Cotiuity Equatio, ad te irrotatioality oditio.

2 Cotiuity Equatio: u v + = x y Irrotatioality oditio: u v = y x () () Two salar futios (magitude oly, witout diretio) a be defied so tat te two ompoets of veloity (vetors), u ad v, i te above equatios a be replaed by te equivalet but simpler salar futios, amely te stream futio ad te veloity potetial futio. Stream Futio,ψ, is defied to satisfy te otiuity equatio as follows ψ ψ u = ad v = (3) y x u v ψ ψ Terefore + = + = x y x y y x It a be see tat te stream futio automatially satisfies te otiuity equatio. Furtermore, te stream futio must also satisfy te irrotatioality oditios ad tus u v = = + = y x y y x x y x ψ ψ ψ ψ (4) Te potetial futio is defied to automatially satisfy te irrotatioality oditio as follows ϕ ϕ u = ad v = (5) x y u v ϕ ϕ tus = = y x y x x y Sie te potetial futio must also satisfy te otiuity equatio, terefore u v + = + = + = x y x x y y x y ϕ ϕ ϕ ϕ (6) It a be see tat te two first order partial differetial equatios i ukows, amely equatios () ad (), a be replaed by a sigle seod order ellipti partial differetial equatio, amely te Laplae equatio eiter i terms of stream futio (equatio (4)) or i terms of te potetial futio (equatio (6)). Furtermore, it a be sow tat te stream futio ad te potetial futio are armoi ojugate of ea oter, ad tus we a defie a omplex potetial futio as follows Φ ( z) = ϕ( x, y) + i. ψ ( x, y) (7)

3 were z is te omplex variable ad i is te imagiary umber z = x + i. y (8) i = (9) Equatios (4) ad (6) a be ombied as follows Φ ( z) = ϕ( x, y) + i. ψ ( x, y) = ϕ( x, y) + i. ψ ( x, y) = (). Solutio of te Laplae equatio From te teory of omplex variable, it is kow tat te solutio of te Laplae Equatio i terms of a omplex futio is ay omplex aalyti futio. We will ot disuss tis furter exept to ote tat te teory elps us i obtaiig a very large umber of elemetary solutios of te Laplae equatio. Amog te very large umber of elemetary solutios, also kow as sigularities, tere are four (4) tat are espeially useful i te study of aerodyamis, amely te soure, sik, doublet ad vortex sigularities. Soure ad sik are atually te same type of sigularity exept tat tey ave opposite sigs for teir stregt. Te problem of a uiform airflow, wi is disturbed by te presee of a aerofoil loated witi te flow field, is modelled by assumig tat te disturbae a be represeted matematially by te sigularities, solutios of te Laplae equatio. Te Laplae equatio is said to be liear, meaig tat a liear ombiatio of some simpler or elemetary solutios is also a solutio. Let us ow ave a quik look at te elemetary solutios Soure / sik : sigularity loated at (, ) x y wit a stregt of σ σ y y = ± x x Stream futio idued at (x,y): ψ ( x, y).ta () ( ) = ± σ + () Potetial futio idued at (x,y): ϕ ( x, y).l ( x x ) ( y y ) Doublet : sigularity loated at (, ) x y wit a stregt of µ Stream futio idued at (x,y): ψ ( x y) Potetial futio idued at (x,y): ϕ ( x y) µ, =. µ, =. y y ( x x ) + ( y y ) x x ( x x ) + ( y y ) (3) (4) 3

4 Vortex : sigularity loated at (, ) x y wit a stregt of Γ ( ) Γ = + (5) Stream futio idued at (x,y): ψ ( x, y ).l ( x x ) ( y y ) y y = Γ x x Potetial futio idued at (x,y): ϕ ( x, y).ta (6) Te veloity ompoets idued at (x,y) by te presee of te sigularity at x, y a be evaluated as follows. x-ompoet of veloity: y-ompoet of veloity: ϕ ψ u = = x y ϕ ψ v = = y x (7) (8) Soure / sik : sigularity loated at (, ) x y wit a stregt of σ x-ompoet: u ( x y) y-ompoet: v( x y) σ, = ±. σ, = ±. x x ( x x ) + ( y y ) y y ( x x ) + ( y y ) (9) () Doublet : sigularity loated at (, ) x y wit a stregt of µ x-ompoet: u ( x y) y-ompoet: v( x y) µ, =. µ, =. ( x x ) ( y y ) (( x x ) + ( y y ) ) ( x x )( y y ) (( x x ) + ( y y ) ) () () Vortex : sigularity loated at (, ) x y wit a stregt of Γ x-ompoet: u ( x y) y-ompoet: v( x y) Γ, =. Γ, =. ( y y ) ( x x ) + ( y y ) ( x x ) ( x x ) + ( y y ) (3) (4) 4

5 Te matematial expressios for te stream ad potetial futios of a uiform flow, wi is ilied at a agleα to te orizotal axis x, are give as follows Stream futio value idued at (x,y): ψ ( x, y) V.( y.os α x.siα ) = (5) Potetial futio value idued at (x,y) ϕ ( x, y) V.( x.os α y.siα ) = + (6) Te x- ompoet of veloity is: u ( x y) Te y- ompoet of veloity is: v( x y) ϕ ψ, = = = V.osα x y ϕ ψ, = = = V.siα y x (7) (8) Te quatity V is te speed of te udisturbed air or te free-stream air. 3. Liearity property of te Laplae equatio It was previously metioed tat te Laplae Equatio is a liear seod order partial differetial equatio. Te liearity property of te equatio implies tat ay liear ombiatio of some elemetary solutios is also a solutio. It was metioed also tat te problem of air flow over a aerofoil (or i Ameria termiology: airfoil) is modelled as a uiform air flow wi is disturbed by te presee of te aerofoil. We sall ow ave a look weter te aerofoil a be represeted by a sigle soure, or a sigle doublet or a sigle vortex. For simpliity it will be assumed tat te value of te agleα is zero. Liear ombiatio of uiform flow ad a soure: ψ σ y y = + x x ( x, y) V. y.ta (9) Liear ombiatio of uiform flow ad a doublet: µ ψ ( x, y) = V. y +. y y ( x x ) + ( y y ) (3) Liear ombiatio of uiform flow ad a vortex: ψ ( ) Γ = + + (3) ( x, y) V. y.l ( x x ) ( y y ) Te flow patter of a flow field is defied by te streamlies of te flow. For a steady flow, i.e. oe tat does ot age wit time, if we release a partile at a poit i te flow field ad te follow te motio of te partile as it is swept dowstream, te pat of te partile motio is kow as a streamlie. Aoter partile released at aoter poit would desribe aoter streamlie. By drawig a umber of 5

6 streamlies, we a get a piture of te sape of te flow field. For a usteady flow or turbulet flow, obviously te sape of te streamlies will age otiuously wit time. However, for a steady flow te patter of te flow field is ostat. Aoter property of te streamlie is te fat tat at ay give poit o te streamlie, te diretio of te veloity vetor is taget to te streamlie at te give poit. Terefore o flow a ross a streamlie. Furtermore, te stream futio value at ay poit o te streamlie is a ostat. I oter words, we a defie a streamlie as beig te lous of poits witi te flow field were te values of te stream futio at all poits o te streamlie are te same. Tus a streamlie a also be alled a ostat stream futio urve Now let us osider wat sort of a flow field we get from te ombiatio of a uiform flow, wi is disturbed by a doublet. To simplify te disussio let us assume tat te doublet is loated at te origi of te system of axes, or at x = ad y =. Furtermore, i order to get a meaigful result it is assumed tat te sig of te doublet stregt is egative. Wit tose assumptios equatio (3) a ow be writte as follows µ y R ψ ( x, y) = V. y. = V. y. x + y x + y (3) were R µ = is a positive ostat (33) V Let us ow ave a look at a partiular streamlie wit a value of stream futio of zero. If ψ = te equatio (3) is simplified to beome R y. x + y = (34) Tere are solutios to te above equatio, amely y = (35) ad also R = x + y Te above equatio a be simplified furter as follows x + y = R (36) Equatio (36) is te equatio of a irle wose etre is loated at te origi ad its radius is R. Sie fluid a ot ross a streamlie, terefore a streamlie may be replaed by a impermeable wall. Terefore equatio (3) atually represets te flow field of a uiform flow wi is flowig over a irular ylider wit radius R. Te streamlie wit a stream futio value of ψ =, wi iludes te x-axis or equatio (35) ad te irle give by equatio (36), is kow as te dividig streamlie. Tis streamlie divides te flow field ito parts, amely oe tat flows 6

7 over te upper part of te irle ad aoter wi flows over te lower part of te same irle or irular ylider. All te oter streamlies, wi desribe te flow over te upper part of te irle, a be obtaied by solvig te followig equatio R ψ y. = x + y V (37) were V ψ ψ is a ostat ad >. V Similarly, all te oter streamlies tat desribe te flow over te lower part of te ψ irle are solutios of equatio (37) were <. V If we evaluate te oordiates (x, y) of a large umber of poits witi te flow field for a partiular value of ψ /V, ad all te eigbourig poits are oeted to ea oter by sort straigt lies, te we will get te sape of te streamlie for tat partiular value of ψ /V. Te flow patter tat we wis to aalyse is te give visually as a olletio of streamlies for various values of ψ /V. A doublet represets te disturbae of a irular ylider immersed i a uiform flow, wereas a soure or a vortex represets aoter type of disturbae.. However oe of tose would represet te disturbae of a aerofoil. Te flow patter of a uiform wid beig disturbed by a soure or a vortex will ot be disussed ere, but tey a be readily foud i most textbooks o aerodyamis. It is importat to ote ere tat wilst te flow aroud a irular ylider is ot partiularly importat i aerodyamis, owever we a employ te teory of omplex variables to trasform te flow field aroud a irular ylider ito tat of te flow aroud a aerofoil. I te simplest ase, te trasform futio or mappig futio is assumed kow ad te sape of te aerofoil is obtaied as a result. Peraps te most well kow mappig futio is te Joukowski s trasformatio, wi really is a speial form of te more geeral Karma-Trefftz mappig futio. If we wat to obtai te flow field aroud ay arbitrary sape aerofoil, te we will eed to evaluate te mappig futio. Tis is a far more diffiult problem ta for Joukowski or Karma-Trefftz mappig, ad will ot be disussed ere. It is suffiiet to ote tat oe of te possible metods to use is te Lauret Series trasformatio. Te use of omplex variable trasformatio is kow as te oformal mappig metod (see appedix 4). 4. Disrete ad Cotiuous Sigularities Let us ow retur to our origial statemet, wi is tat a more omplex flow a be obtaied by addig togeter a umber of simpler solutios ad see if tis a be used to obtai a more diret solutio to te problem of evaluatig te flow field of a uiform wid wi is disturbed by a arbitrary sape aerofoil. Tis more diret approa is kow as te pael metod. Here we will disuss te basis of te -D pael metod oly, but te same approa may be applied to 3-D problems. Tis is i 7

8 otrast to te oformal mappig metod, wi is oly appliable for -D problems. Te stream futio value at ay poit (x, y) i a uiform flow, wi is disturbed by, x, y respetively, is give by te vorties, ea beig loated at ( x y ) ad followig expressio Γ ( ) (( ) ( ) ) ( x, y ) V. y Γ ψ =.l.l x x y y x x y y (38) If tere are N, rater ta, vorties te we ave ψ ( ) N Γ (39) ( x, y) = V. y +.l ( x x ) + ( y y ) = Let us ow osider te situatio were tere are poits, P ad Q, wit oordiates of (XP, YP) ad (XQ, YQ). Te straigt lie from P to Q is divided up ito a large umber of equal legt iterval of s = PQ / N, were PQ is te distae from P to Q. Let us ow imagie tat at te mid poit of ea small iterval tere is a vortex of stregt G, wi is te same for all itervals. Te stream futio value at ay poit (x, y) for tis ase is give by ψ ( ) N G (4) ( x, y) = V. y +.l ( x x ) + ( y y ) = It sould be oted tat eve toug N a be made to be very large, approaig ifiity, we sall impose te oditio tat te total stregt of te vorties is always te same regardless of te atual value of N, ad tis total vortex stregt is Γ were Γ = N. G Let us deote te mid poit as beig te poit S, su tat te oordiates of te poit S is( XS, YS ) were X = (XQ XP) / N = XPQ / N Y = (YQ YP) / N = YPQ / N s = PQ / N = XPQ + YPQ / N XPQ XS = XP +.. s = XP + XPS PQ YPQ YS = YP +.. s = YP + YPS PQ 8

9 G is te stregt of te disrete vortex loated at te midpoit of a elemetal legt of s. If N is ose to be suffiietly large, te s is suffiietly small to be replaed by a otiuous differetial, i.e. s ds. Furtermore we sall assume tat te stregt of te vortex, G, is distributed evely alog ds ad ee Γ PQ G = = γ. = γ. s γ. ds N N Here γ is te stregt per uit legt of te otiuous vortex seet ds. Sie G is te same for all elemetal legts, ds, tereforeγ is also te same for all ds alog te vortex seet or pael PQ. Equatio (4) a ow be writte as follows ψ ( ) N γ. s (4) ( XT, YT ) = V. YT +.l ( XT XS ) + ( YT YS ) = were (XT, YT) are te oordiates of te poit T, at wi te value of te stream futio is to be alulated. Now it is to be oted tat XT XS = XT XP XPS = XPT XPS Terefore, YT YS = YT YP YPS = YPT YPS. (.. ) S T = XT XS + YT YS = PT XPT XPS + YPT YPS + PS were it is defied tat PT = XPT + YPT PS = XPS + YPS Equatio (4) a ow be writte as follows N N γ. s γ.l ST ψ ( XT, YT ) = V. YT +.l S T = V. YT +. s = = Takig te limit of N approaig ifiity, te above summatio a be replaed by te followig itegral PQ γ ψ ( XT, YT ) = V. YT + l r( s). ds (4) 9

10 r = PT. XPT. XPS + YPT. YPS + s were r(s) is te distae from poit S, aywere alog te lie PQ, to te poit T, ad s is te distae from poit P to poit S. It a also be observed tat Terefore s s XPS =. XPQ ad YPS. YPQ PQ = PQ =.. + = + (43) r PT PR s s TR PR s PR = XPT. XPQ + YPT. YPQ / PQ (44) TR = PT PR (45) Sie γ is a ostat, terefore equatio (4) a be simplified to beome PQ γ ψ ( XT, YT ) = V. YT + l r( s). ds (46) It a be sow (see appedix ) tat Q l r s. ds = PQ PR.l QT + PR.l PT PQ + TR. APTQ (47) P were PQ. TR APTQ = ta PT PQ. PR (48) It sould be oted tat APTQ is te agle subteded by te lies PT ad TQ, or te visible agle of PQ see from T. Equatio (4) is te value of stream futio at poit T, wi is immersed i te flow field of a uiform flow wit a free stream veloity of V, ad is iflueed by te presee of a otiuously distributed vortex. Te vortiity is distributed o a lie PQ, wi is atually te itersetio of retagle wit te x-y plae or te plae of te paper. Te retagle is perpediular to te x-y plae ad is ifiitely log i te z-diretio. It is referred to as a pael ad beause vortiity is distributed o it, terefore, it is alled a vortex pael. Te vortex stregt per uit legt distributio alog PQ is γ ( s), wi is a futio of te variable legt represetig te distae from P to a poit S loated aywere betwee P ad Q. Geerally speakig te futioal form of γ ( s) is ukow ad represets te problem tat as to be solved. Te above disussio sows tat te Laplae equatio a ave a disrete solutio as well as a otiuous solutio. I terms of te stream futio equatio (5) represets

11 te solutio of te stream futio value at ay arbitrary poit (x, y) idued by a x, y. Te solutio of te same disrete vortex wit stregt of Γ ad loated at problem i terms of te potetial futio is give by equatio (6), wereas for te veloity ompoets u ad v te solutios are equatios (3) ad (4) respetively. If te disrete vortex is replaed by a distributed vortex alog pael PQ, wit vortiity γ s, te solutios are give as follows stregt per uit legt of were Q ψ ( x, y) = γ ( s).l r ( s). ds (49) P Q ϕ ( x, y) = γ ( s). θ ( s). ds (5) P θ ( s) Q si u ( x, y) =. γ ( s). ds (5) r s P θ ( s) Q os v( x, y) = γ ( s).. ds (5) r s P ( ) s = XS XP + YS YP (53) ( ) r s = x XS + y YS (54) θ y YS = x XS ( s) ta (55) It sould be remembered tat (XS, YS) are te oordiates of te poit S, wi is loated aywere alog te pael PQ. Te oordiates of poits P ad Q are (XP, YP) ad (XQ, YQ) respetively. Similar expressios a also be derived for te ase were te sigularity is soure ad sik or doublet, rater ta vortex. Te fial results are give below Soure ad Sik: Q ψ ( x, y) = ± σ ( s). θ ( s). ds (56) P

12 Q ϕ ( x, y) = ± σ ( s).l r ( s). ds (57) P θ ( s) Q os u ( x, y) = ± σ ( s).. ds (58) r s P θ ( s) Q si v( x, y) = ± σ ( s).. ds (59) r s P Doublet : θ ( s) Q si ψ ( x, y) = µ ( s).. ds (6) r s P θ ( s) Q os ϕ ( x, y) = µ ( s).. ds (6) r s P θ ( s) ( s) Q os u ( x, y) = µ ( s).. ds (6) r P θ ( s) ( s) Q si v( x, y) = µ ( s).. ds (63) r It sould be oted tat σ ( s) ad ( s) ad doublet distributio respetively. P µ are te stregt per uit legt of te soure It a be see learly tat te itegrals ivolved i te above equatios are very omplex ad a t be solved uless te stregt per uit legt distributio of te ose sigularity (soure, doublet or vortex) is give. Furtermore, eve for te simplest ase were te stregt per uit legt distributio is just a ostat it is already quite diffiult to evaluate te expliit expressio for te itegral. Aoter diffiulty is te geometry of te sigularity pael. If te pael is ot straigt but urved istead, te it is impossible to obtai a aalytial solutio for te itegral. O te oter ad we kow tat wigs or aerofoils are ot flat paels, but are igly urved. I te pael metod tis problem is overome by replaig te otiuously smoot urve of te aerofoil wit a approximate urve osistig of a large umber of paels or straigt lies oetig adjaet poits o te surfae of te aerofoil. Te overall effet of te wole aerofoil o te value of stream futio, or ay of te oter futios tat is ose, at a poit is te obtaied as te sum of te effets of all paels represetig te aerofoil. Sie ea pael is a straigt lie terefore it is possible to derive te expressio for te required itegral, provided tat te stregt per uit legt distributio of te sigularity is kept simple. If te distributio is assumed ostat alog te pael, te te metod is kow as te first order pael metod, sie te distributio oly requires te kowledge of oe ukow ostat. A mu better approximatig distributio is give by a liear futio of s, wi

13 ivolves two ukow ostats, ee it is referred to as a seod order pael metod. Below we will sow te derivatio of te expressio for te itegral o te rigt ad side of equatio (49) for a seod order pael metod. Te vortex stregt per uit legt liear distributio is give by te followig were γ P ad Q γ Q γ P γ ( s) = γ P +. s (64) PQ γ are te values of ( s) γ at poits P ad Q respetively, ad represet te two ukow ostats of te seod order pael metod. Te stream futio value at (x, y) for tis ase is tus give by Q γ Q γ P ψ ( x, y) = γ P s.l r ( s). ds + PQ (65) P Te above equatio a be rewritte as follows ψ x, y = CI. γp + CI. γq γp = CI CI. γp + CI. γq (66) Q CI = l r ( s). ds (67) P Q PQ. CI = s.l r s. ds (68) P Te expressio for te itegral i equatio (67) as already bee worked out i appedix wit te folowig result CI = PQ PR.l QT + PR.l PT PQ + TR. APTQ (69) Details for te evaluatio of te seod itegral are give i appedix () te result of wi is PQ ( PR TR ) PR TR CI = l QT + l PT ( PQ + PR) PQ PQ (7) PR. TR. APTQ + PQ Our disussio so far oly deals wit te solutio of te Laplae equatio i geeral, wereas te real problem tat we wat to solve is ow to aalyse te aerodyami properties of a -dimesioal wig or a aerofoil. Te matematial model for our simplified problem, i.e. ofied to ivisid, iompressible flows oly, is ideed te Laplae equatio, but we ave ot disussed about te boudary oditios tat must be satisfied. To obtai a uique solutio we eed to speify wat boudary oditios te solutio must satisfy. 3

14 5. Te boudary oditios Te real questio we wat to fid te aswer for ere is tat if te sape of a wig or a aerofoil is give, ad te wig is immersed i a give airflow, wat is te lift ad drag atig o te aerofoil as a result of its iteratio wit te flowig air. It sould be remembered tat i a Galilea trasformatio, it makes o differee if te body is statioary ad te air is movig (model airraft i a wid tuel) or weter te body is movig i a stagat atmospere (airraft movig i te atmospere). Drag ad lift are te two ortogoal ompoets of te resultat fore atig o te aerofoil, wi is te summatio of all te pressure atig o te surfae of te aerofoil. From Beroulli equatio we kow tat te total pressure i a isetalpi flow is ostat, ee te stati pressure dereases if te dyami pressure ireases ad vie versa. Te dyami pressure is of ourse alf multiplied by air desity multiplied by veloity squared. Tus pressure is diretly related to te fluid veloity squared. Imagie a airflow movig uiformly from te left to te rigt. Everywere witi te flow field te veloity is te same, i.e. te free stream veloity, tus te stati pressure is also te same wit a value of free stream stati pressure. Now imagie tat suddely a aerofoil is iserted ito te airflow. Obviously te presee of te aerofoil would disturb te uiform airflow, ad te veloity field would age from te previously uiform value, at least i te viiity of te aerofoil surfae. O te oter ad we ave also see tat te sigularities, wi are te elemetary solutios of te Laplae equatio, also as te effet of disturbig a uiform flow of fluid, at least i te viiity of te loatio of te sigularity. From tis observatio we a make te olusio tat peraps te airflow aroud a aerofoil a be simulated by plaig sigularities o te aerofoil s surfae, or o a surfae tat a be assumed to be represetative of te aerofoil surfae. It is a observed fat tat air a t peetrate ito te iside of te aerofoil, te surfae of wi is made out of solid. Sie te surfae of te aerofoil represets part of te boudary of te airflow, terefore te requiremet tat air a t peetrate ito te aerofoil s surfae is alled a boudary oditio. Te aerofoil s surfae is alled te ier boudary sie it represets te boudary of te ier part of te flow. Te oter boudary is te outer boudary, wi ideally is ifiitely far away from te aerofoil s surfae, but from a pratial poit of view may be defied as beig suffiietly far away from te surfae. At te outer boudary te flow is udisturbed by te presee of te aerofoil, terefore i our simulated flow te Laplaia sigularities plaed o te surfae also must ot disturb te free stream uiform flow far away from te sigularities. Tis requiremet is always automatially satisfied by all of te sigularities, amely soure/sik, doublet ad vortex. Terefore, it is oly te ier boudary oditio tat must be satisfied i our simulated flow, i.e. tat te flow must ot peetrate te aerofoil s surfae. Tis requiremet a be expressed i eiter oe of at least two differet ways. Firstly, it may be stated tat te surfae of te aerofoil is part of te dividig streamlie tat splits te airflow ito a upper alf ad a lower alf of te flow field. Sie a streamlie is a urve o wi te value of te stream futio is ostat, tis type of boudary oditio is alled te Dirilet Coditio were te value of 4

15 te futio (te stream futio) is defied (as beig of a ostat value) at te boudary of te flow field. Te oter way of speifyig te boudary oditio is tat te ormal ompoet of te flow veloity at te boudary must be zero, sie oterwise it would imply tat air is allowed to peetrate ito te aerofoil s surfae. Sie flow veloity is te derivative of te futio (i.e. te stream futio) tis is te same as speifyig te values of te futio s derivative at te boudary ad is kow as te Neuma Coditio. Geerally speakig, i some problems te futio values are speified at some parts of te boudary, wile at te rest of te boudary te derivative values are speified. Tis tird type of boudary oditio is kow as te Mixed Boudary Coditio, wi is also kow as te Robi Coditio. Now we a begi to desribe ow our problem is to be simulated usig a seod order vortex pael metod. Figure. Defiitio of parts of a aerofoil (from Georgia Istitute of Teology web site ttp:// Te udisturbed flow is represeted as a uiform flow te veloity vetor of wi is ilied at a agle ofα relative to te orizotal or x-axis. Te aerofoil is fixed i spae ad is so loated su tat its ord lie is alog te x-axis, wit te ose beig at te origi ad te tail beig to te rigt of te ose. Te aerofoil s surfae is represeted by a large but fiite umber of poits o it. Te smootly otiuous urved surfae is approximated by a pieewise liear segmeted otiuous surfae osistig of small segmets of straigt lies or paels. Obviously te approximatio gets better as te umber of poits o te aerofoil s surfae is ireased. However, tis as te impliatio of ireasig amout of omputatioal work to be doe. Sie i reality all wigs ave tails wit fiite tikess, it will be assumed ere tat te aerofoil as a blut trailig edge, wit distit lower ad upper tail poits. Te lower trailig edge (tail) poit is idetified as te poit P ad te upper tail poit is te poit P N, were N is te umber of poits represetig te aerofoil s surfae. Te umber of paels is obviously (N-). Idexatio of all surfae poits is doe i a lokwise diretio, tus te poit ext to ad to te left of te first poit is idetified t as te poit P et. Te pael is te lie oetig poit P to P +, wi is also t idetified as te poit Q. Tus te pael is also referred to as te pael PQ. 5

16 I te followig disussio we will desribe a solutio based o te metod tat utilizes te Dirilet boudary oditios. Tus te solutio obtaied will be i terms of te stream futio. Te stream futio value at a poit T wit oordiates (XT,YT) due to T beig immersed i a uiform flow, wi is ilied at a agle of α to x-axis, is give as follows uw ( XT, YT ) V.( XT.si YT.os ) ψ = α + α (7) Te stream futio value at T idued by te vortex pael PQ is give by equatio (65), were te vortex stregt per uit legt distributio is assumed give by a liear distributio, i.e. we sall use a seod order metod. Tus alog te pael PQ we ave te followig distributio γ P + γ P γ ( s) = γ P +. s (7) PQ For tis ase equatio (65) a be rewritte as follows P + γ P + γ P γ..l. PQ P ψ PQ = P + s r s ds (73) (..( + )) ψ PQ = CI γ P + CI γ P γ P (( ).. + ) ψ PQ = CI CI γ P + CI γ P (74) Te expressios for CI ad CI are give by equatios (69) ad (7) as follows CI = PQ PR.l PT + PR.l PT PQ + TR. APTQ (75) + CI PQ PR TR PT PR TR PT PQ PR PR. TR. APTQ + PQ = l + + l + PQ PQ (76) Te stream futio value at T idued by all te vortex paels makig up te omplete aerofoil sape is give by N N ( XT, YT ) = PQ = (( CI CI ). P + CI. P + ) (77) ψ ψ γ γ vs = = 6

17 If it is ow defied tat C = CI CI T, C = CI CI + CI for =,3,..., N T, (78) C = CI T, N N te equatio (77) a be rewritte more ompatly as follows ψ N XT, YT C. γ P = (79) vs T, = Te value of te stream futio at poit T is te sum of te stream futio due to beig immersed i te uiform wid plus te stream futio idued by all te vortex paels makig up te surfae of te aerofoil ( XT, YT ) ( XT, YT ) ψ = ψ + ψ T uw vs = ( + ) + ψ V. XT.si α YT.os α C. γ P T T, = N (8) Te above equatio a be rewritte as follows N CT,. γ P. ψ T =. V. ( XT.si α YT.osα ) (8) = If te oordiates of all poits o te aerofoil s surfae ( XP, YP ) of poit T, i.e. ( XT, YT ), are give te all te ifluee oeffiiets, CI ee C T, for all values of from to N a be evaluated. Now let us defie a dummy value of γ N + as follows ad te oordiates CI ad γ P N ψ T ad also a dummy oeffiiet + = (8) C = + (83) T, N te equatio (8) a be writte more ompatly as follows N + CT,. γ P =. V. ( XT.si α YT.osα ) (84) = 7

18 All te values of C T, ad te rigt ad side of te above equatio are kow ad te oly ukows are γ P for = to = N+ provided tat te otrol poit T is ose to be o te aerofoil s surfae su tat ψ T is a ostat. Sie tere are N+ ukows terefore we eed N+ equatios to be solved simultaeously to alulate te values of te ukows, γ P. However tere are oly N poits o te surfae of te aerofoil tat a be seleted to be te otrol poits were equatio (84) is applied, tus we eed oe more equatio to omplete te system of equatios to be solved simultaeously. Te extra equatio is obtaied from te pysial observatio tat te airflow leavig te upper surfae of te aerofoil must ave exatly te same veloity as te airflow leavig te lower surfae. Tis meas tat te veloity at te two tail poits (upper ad lower tail poits) must be te same. It a be sow tat te airflow veloity at a aerofoil s surfae poit is exatly te same as te value of te vortex stregt per uit legt, γ P, at te poit. Tis trailig edge flow oditio is kow as te Kutta oditio ad a be represeted by te followig equatio γ P + γ P N = (85) For ea of te otrol poit, wi is ose to be te aerofoil s surfae poit P, we a write dow a equatio based o te geeral expressio of equatio (84). Te ( N + ) t must be derived differetly, amely it is based o satisfyig te Kutta oditio. Now we a defie te followig for te ( N + ) t equatio C T, = C = for =,3,..., N C T, T, N = (86) C T, N + = Wit te above defiitios we ow ave a system of equatios osistig of (N+) equatios ivolvig (N+) ukows as follows N + CT,. γ P = D (87) = were N + D =. V. XT.si α YT.osα for =,,..., N D = (88) 8

19 Te system of equatios (87) a be solved simultaeously to alulate te ukow γ P ad sie te absolute value of γ P is te same as te airflow veloity at te poit P, terefore te distributio of flow veloity alog te aerofoil s surfae is kow. Furtermore, it a be sow tat for vortex pael metod as desribed ere, te value of pressure oeffiiet Cp a be obtaied from te Beroulli equatio ad te fial result is γ Cp = V (89) Te pressure oeffiiet at poit o te surfae of te aerofoil a tus be alulated ad plotted as desired. Te pressure oeffiiets a also be itegrated to give te resultat fore atig o te aerofoil. Tis resultat fore a be resolved ito ompoets, oe beig te lift fore i te diretio ormal to te free stream diretio ad te oter is te drag fore atig alog te free stream diretio. Due to te ivisid flow assumptio, it is expeted tat te drag fore or drag oeffiiet must ave a value of zero. I pratie te omputed drag oeffiiet will be foud to ave a o-zero value due to omputatioal error su as roud off error et. Te momet atig o te aerofoil a also be obtaied from te kow pressure distributio. A simple example of te appliatio of te pael metod is give i appedix 3. Eve toug te pael metod is quite good for omputig te aerodyami properties of a aerofoil, it is basially a umerial metod ad does t give aalytial isigt ito te aerodyami beaviour of a aerofoil. To get su a isigt we eed a aalytial tool, eve if it is very mu simplified. Tis tool is kow as te Ti Aerofoil Teory, wi is te ext topi to be disussed. 6. Ti Aerofoil Teory Te amber lie of a aerofoil is te urve midway betwee te lower ad upper surfaes of te aerofoil. For a symmetri aerofoil te amber lie is simply a straigt lie. If a lie is draw perpediular to te amber lie, te it must iterset bot te upper ad te lower surfaes of te aerofoil. Te distae betwee te two itersetio poits is alled te tikess of te aerofoil. Obviously tis tikess would vary alog te ord of te aerofoil. Te tikess is ormally expressed as a peretage of te ord legt. Te tikess of a aerofoil is defied as te maximum tikess as desribed previously. A ti aerofoil is defied as ay aerofoil wose (maximum) tikess is a very small peretage of te ord legt, su tat it is reasoable to model te aerofoil as a urved or flat plate of zero tikess. Wile tis restritio is quite severe, it eables us to get a aalytial solutio to te problem. Here we sarifie auray to get aalytial isigt. 9

20 6. Flat plate as a aerofoil Let us ow osider a very ti symmetri aerofoil, wi is quite reasoable to be approximated as a flat plate of zero tikess, wit a ord legt. Te aerofoil is loated alog te x-axis wit its ose beig at te origi of te system of oordiates. Te aerofoil is immersed i a uiform wid wose veloity vetor is at a agle α relative to te orizotal or x-axis. Te disturbae to te uiform wid due to te presee of te aerofoil is modelled as te disturbae aused by a vortex seet loated alog te amber lie of te aerofoil. Te vortex stregt per uit legt γ x is ukow ad must be evaluated. distributio of te seet * * Te stream futio at ay poit (, ) x y witi te flow field is give as follows * * * * ψ x, y = V x.si α + y.os α + γ x.l r x. dx (9) * * r x = x x + y y (9) Te veloity ompoets at ay poit are give by te followig θ ( x) * * si u ( x, y ) = V.os α + γ ( x).. dx (9) r x θ ( x) * * os v( x, y ) = V.si α γ ( x).. dx (93) r x * * y x x siθ x = ad osθ x = (94) r x r x If te problem is speified as a Dirilet boudary oditio problem, te we must γ x su tat equatio (9) is satisfied at all fid te aalytial expressio for otrol or boudary poits. Te boudary oditio to be satisfied is tat all te poits o te amber lie, or part of x-axis from to, must ave te same stream futio value beause te amber lie is a streamlie. It is quite easily see tat tis problem is impossible to solve aalytially. Now let us reast te problem as a Neuma boudary oditio problem. Te boudary oditio to be satisfied is tat te ompoet of veloity ormal to te * * v x, y, must be zero at all poits alog te amber lie. Te equatio amber lie, to be solved is tus equatio (93), wi a be rewritte as follows θ ( x) os V.si α γ ( x).. dx = (95) r x

21 Sie te otrol poit is o te amber lie tus * θ ( x) = ad os r x = x x. Equatio (95) is te simplified to * x * y is always zero. Terefore γ x. dx = V.siα (96) x It a be see tat tis problem looks simpler ta te Dirilet formulatio of te same problem. Eve so te problem is ot quite so simple. We kow tat ay otiuous futio a always be approximated by a Fourier series, eve if te exat expressio for te futio is ukow. Terefore, it is γ x by a Fourier series wit ukow reasoable to replae te ukow futio oeffiiets. However, before we a do tat it sould be realized tat te Fourier series is expressed i terms of agles rater ta variable su as x. Terefore, it is eessary tat we perform a oordiate trasformatio from x to θ. It is required tat we x = te θ = ad we x = we wat θ =. A suitable trasformatio futio a be give as follows x = dx = ( osθ ) si θ. dθ (97) Equatio (95) a ow be rewritte as follows γ θ.siθ osθ osθ dθ = V.siα * (98) We will ot go ito te matematial details of ow to solve te above equatio. It is suffiiet to simply apply te kow matematial results to elp fid te solutio. For example it is kow tat * os θ. dθ si θ = for =,,,3... * * osθ osθ siθ si θ.si θ. dθ = θ for = * osθ osθ * os,,,3... (99) We sall ow assume a solutio ad te substitute te solutio ito equatio (96) ad verify tat te trial solutio ideed satisfies te equatio or oterwise. Te trial solutio is + osθ γ ( θ ) = V.si α. () siθ Te left ad side of equatio (96) a ow be expaded as follows

22 V ( + ) γ θ.siθ.si α. osθ siθ. dθ osθ osθ siθ osθ osθ dθ = * * ( ) γ θ.si θ α osθ. dθ osθ osθ osθ osθ V.si + dθ = * * () Te first equatio i (99) for = ad = gives te followig results dθ * osθ osθ os θ. dθ * osθ osθ = = () Terefore equatio (98) a be simplified as follows V.si α + osθ V d * osθ osθ.siα. θ =.( + ) = V.siα (3) Tus it as bee prove tat te trial solutio () ideed satisfies te goverig equatio (96). I te previous setio it was stated tat te flow solutio must also satisfy te Kutta oditio at te trailig edge or at θ = (sie x = ). It sould be oted tat te Kutta oditio for a aerofoil wit a sarp trailig edge is tat te trailig edge must be a rear stagatio poit, were γ = V =. Substitutig te value of θ = ito () we get te followig result γ ( ) = V.si α. (idetermiate value). Te above value is idetermiate ad we sould apply te L Hospital s rule. Te rule states tat if te ratio of futios, say f(z)/g(z), beomes idetermiate as z approaes, te te value a be alulated by replaig te futios wit teir f / g is give derivatives. Tus i te limit of z, te value of ' ' by ( ) / ' ' required value sould be alulated as f ( ) / g ( ). f g. I our example bot f ad g approa as θ. Terefore te Now it is oted tat d ( + osθ ) dθ = siθ ad d siθ = osθ dθ

23 Terefore si γ ( ) = V.si α. = os It a be see tat te solutio also satisfies te Kutta oditio. We a also obtai te solutio as a futio of te Cartesia variable, x, as follows. From equatio (97) we a get te followig Terefore ( x ) + osθ = / siθ = os θ = + x x x x γ ( x) = V.si α. = V.si α. x x ( x) (4) At te trailig edge x =, terefore γ ( ) =, tus satisfyig te Kutta oditio. * γ Te values of γ = a be alulated as a futio of x/ ad te results are V.siα tabulated below x/ * γ Te lift, L, atig o te flat plate a be alulated usig te followig Kutta- Joukowski lift equatio L = ρv Γ (5) Te irulatio aroud te aerofoil is equal to te total stregt of te distributed γ x. vortex, wi a be obtaied by itegratig or summig up all te values of Γ = γ ( x). dx = γ ( θ ).si θ. dθ Γ = V siα + osθ dθ = V siα (6) Te lift oeffiiet is defied as follows 3

24 C l L = = siα (7) ρv. For small values of α (i radia) te followig approximatio is quite aurate siα α (8) Terefore te lift oeffiiet a be rewritte as follows Cl =. α (9) Te fore df atig o a aerofoil elemetal legt dx is give by te Kutta- Joukowski lift equatio as follows df = ρv γ ( x). dx () Te pitig momet about te leadig edge due to te fore df is M = x. df = ρv γ x. x. dx LE + osθ M LE = ρv. Vα. ( os θ ). siθ dθ siθ M LE V = ρ α = αρv () 4 Te pitig momet about ay oter poit a be obtaied quite easily by rememberig te defiitio of momet, wi is simply fore multiplied by te arm legt. Te pressure distributio alog te aerofoil surfae is su tat it a be replaed by a resultat momet ad a resultat fore (lift fore). Tere is a partiular loatio or value of x, were te resultat momet of te pressure distributio is zero. Tis zero resultat momet poit is kow as te etre of pressure ad its loatio a be alulated as follows. Let us assume tat te etre of pressure is loated at x p. At tis poit te resultat momet is zero ad te resultat fore is give by equatio (5). Now imagie tat a equal but opposite fore is loated at te same poit. To ael tis fore we must add aoter fore, wi is equal to te resultat fore but loated elsewere, say at te leadig edge. Addig two equal but opposite fores does t age te resultat fore atig o te aerofoil, but tose fores wi is alled a ouple is equivalet to a momet. Te magitude of tis momet of te ouple is simply x p multiplied by te lift fore. However, to get te momet atig at te etre of pressure to remai zero we must add a momet wi is equal but i opposite diretio of te momet due to te ouple. Now we ave te lift fore at te etre of pressure, a ouple of fores te magitude of wi is te same as te lift fore ad a momet, wi as a magitude of x p.l i te diretio tat te lift fore would rotate te aerofoil about te leadig 4

25 edge as te axis of rotatio. Tis of ourse is i te outer lokwise diretio. A momet is defied as beig positive if it teds to rotate te aerofoil ose up. Terefore te momet atig at te leadig edge is egative. A lift fore atig upwards is defied to be positive ad tus te value of x p a be alulated as follows x p. αρ V. M LE 4 = = = () L ρ. α 4 V Te oeffiiet of momet about te leadig edge is defied as C M αρv = = = α (3) LE 4 m, LE ρv ρv It sould be oted tat xp a also be obtaied from te followig equatio x p Cm, LE α = = = (4) C α 4 l 6. Cambered ti aerofoil I te previous setio it as bee sow ow a simple aalysis based o a very rude model wit severe restritios a still be very useful i givig isigt ito te aerodyami properties of aerofoils. Wilst te flat plate model for a aerofoil is ot apable of givig detailed kowledge of veloity or pressure distributio aroud te aerofoil surfae wit ay auray, it is apable of preditig reasoably aurately te value of te slope of te lift versus agle of attak urve, amely. Tis value is quite lose to te value obtaied from wid tuel measuremet or mu more sopistiated umerial modellig, wi gives sligtly lower value of te slope but differig by a fator of ot more ta peret or so. Te metod is also apable of preditig te loatio of te etre of pressure, wi for tis simplified flow model is te same as te aerodyami etre. For te ase of aerofoils i atual subsoi flows, te aerodyami etre is loated ear te quarter ord poit as predited by te flat plate model. However, te etre of pressure moves as a futio of te agle of attak, α. I tis setio we sall relax te ostraits sligtly by allowig te aerofoil to ave urvature or amber. Te tikess of te aerofoil must still be very small, ad te maximum amber must also be very small su tat te aerofoil is still very mu like a flat plate. However, beause te aerofoil is allowed to ave amber te boudary oditio tat must be satisfied is somewat differet from te flat plate model. At a poit o te amber lie, wi represets te aerofoil s surfae, te ormal to te amber lie is ot exatly i te y-diretio (as i te ase of te flat plate model) but alog a lie tat is sligtly ilied to te y-axis. Let tis agle be η ad is te same as te iliatio of te taget (or slope) to te amber lie at te otrol poit. Tis implies tat dy dy taη = or η = ta (5) dx dx 5

26 It is furter assumed tat te agle is suffiietly small su tat te taget of te dy agle is equal to te agle itself i uit of radia, i.e. η taη =. dx Te equatio to be satisfied is ow sligtly differet from equatio (95) as follows Sie ( α η) γ x. dx V.si ( α η) x = (6) * x is quite small, terefore te above equatio a be simplified as follows γ x. dx V. ( α η) x = (7) * x From te web site ttp:// From te web site ttp:// Te solutio of equatio (6) as ompoets. Te first oe is te same solutio as for te flat plate situatio or equatio (98). Te seod ompoet is to aout for te aerofoil s amber. Te trial solutio is tus give as follows Substitutig (8) ito (7) we get + osθ γ ( θ ) = V A + A si θ siθ (8) = + osθ si θ.siθ A A. * dθ α η * ( θ ) + = osθ osθ = osθ osθ (9) From equatio (99) it is kow tat + θ. dθ = () * os os os θ θ 6

27 Te seod part of equatio (99) gives te followig results si θ.siθ *. dθ =.os θ for =,,.. () * osθ osθ Substitutig () ad () ito (9) we get te followig os * * () = A A θ = α η θ Terefore our trial solutio (8) satisfies te goverig boudary oditio (7) provided te Fourier series oeffiiets satisfy te requiremet of equatio (). We kow from basi alulus tat * * * * * ( m) θ θ mθ θ mθ os + = os os si si (3) * * * * * ( m) θ θ mθ θ mθ os = os os + si si (4) Terefore os m os m os os m * * * * + θ + θ = θ θ (5) Te results above a be used to get te followig results * * * os θ os mθ dθ = ( * * ) * os + m θ + os m θ dθ = if m = if m + m = (6) = or Itegratig equatio () from to, we a get te followig = ( ) A A os θ *. dθ * α η θ *. dθ * = Sie * * os θ dθ = for =,, 3,, terefore ( α η ( θ * )). θ * α η ( θ * ). θ * (7) A = d = d Multiplyig equatio () by * os mθ ad itegratig from to we get = ( ) A os mθ dθ A os θ os mθ dθ α η θ os mθ. dθ * * * * * * * * = 7

28 Te itegrals o te left ad side of te equatio above are all zero exept if i wi ase it as te value of. Te first term o te rigt ad side of te equatio is always zero ee te equatio a be simplified as follows = m, ( * ).os *. * m = η θ θ θ for m =,, 3,.. (8) A m d Sie i tis last result m is just a dummy idex, tus we a age it to or k or ay oter symbol if we so wis. Te total irulatio aroud te aerofoil is give by equatio (6), wi for ambered aerofoil a be writte as follows Γ = V. A ( + os θ ). dθ + A si θ.si θ. dθ = From basi matematis we kow tat (9) ( ) si θ.siθ = os θ os + θ (3) Terefore ( ) si θ.si θ = os θ os + θ (3) si θ.si θ. dθ = for = (3) = if (33) si θ.si θ. dθ = for = (34) Equatio (9) a ow be simplified as follows = if (35) V.. A A. Γ = + = V.. A + A (36) Te lift atig o te aerofoil is give by te Kutta-Joukowski lift equatio (), ee te lift oeffiiet a be alulated as follows V A A ρv ρ.. + Cl = = A + A. (37) Te lift urve slope a be alulated kowig tat A is give by equatio (7) C α l Cl, α = = (38) 8

29 Te pitig momet about te leadig edge a also be evaluated as follows were γ ( x) γ ( θ ) M = x. df = ρv γ x. x. dx LE = is give by equatio (96) ad x ad dx are give by equatio (), ee te above equatio a be simplified furter as follows M LE = ρv A ( os θ ) + A si θ.siθ ( osθ ) d θ = = M LE = ρv A ( os θ ) + A ( si θ.siθ si θ.si θ ). dθ Substitutig equatios (49), (5), (5) ad (5) ito te above, we fially get M LE V A A A = ρ + = ρv A + A A 4 [ ] Te leadig edge momet oeffiiet is te M C = = A + A A (39) [ ] LE m, LE ρv Te loatio of te etre of pressure is give by xp C A + A A = = C A A m, LE l 4 + (4) Te momet oeffiiet at te quarter ord poit is [ ] C = C + C = A + A A + A + A = A A (4) m, / 4 m, LE 4 l 4 Te equatios for te amber lies for NACA aerofoils a be obtaied from te book by I.H.Abbott ad A.E.vo Doeoff : Teory of Wig Setios or it a be obtaied from te followig web address ttp:// As a example of te appliatio of TAT for a ambered aerofoil we will osider a x x simple aerofoil wose amber is give by y( x) 4.. = were te maximum amber is a small positive umber ad is te ord dy x = 4 = 4 os = 4 osθ dx ee ( ( θ )) (4) 9

30 Te Fourier oeffiiets for tis aerofoil a be alulated as follows 4 A = α os θ dθ α = A = os.os. θ θ dθ. = = 4 8 A = os θ.os θ. dθ. = = Note tat equatio (4) as bee used to evaluate te itegrals for A ad A. Te value of te lift oeffiiet is Cl = ( A + A ) = α + (43) Te leadig edge ad quarter ord momet oeffiiets, ad te etre of pressure loatio are 4 Cm, LE = α + (44) C m, / 4 = (45) xp α + 4 / / = = + 4 α + / 4 α + / (46) Let us ow study a similar aerofoil, exept tat te maximum amber is ow loated at.5, rater ta at te mid poit.5. Te equatio for te amber of te aerofoil is ow give by te followig 8x( x) dy 8 For x : y ( x) =. ad ( 4 x ) 4 = (47) dx dy 8 8 = ( 4( osθ )) = ( osθ ) (48) dx Terefore ( x + )( x) For x : y ( x) = 8. ad dy 8 ( 4 x ) 9 4 Terefore ( ( θ )) = (49) dx 9 dy 8 8 = ( 4 os ) = ( osθ ) (5) dx 9 9 It sould be oted tat x/ =/4 is equivalet to osθ =.5 or θ = / 3 Te Fourier oeffiiets a ow be alulated as follows 3

31 /3 8 A = α ( osθ ). dθ ( osθ ). dθ + 9 / 3 8 / 3 A = α ( siθ θ ) ( siθ θ ) + 9 /3 8 A = α.6849 x3.865 α.663 = 9 /3 6 A = ( os θ os θ ). dθ ( os θ os θ ). dθ + 9 / 3 / 3 6 A = θ + si θ siθ + θ + si θ siθ 9 / 3 6 A = si si si si A =.64 + = /3 6 A = ( os θ.osθ os θ ). dθ ( os θ.osθ os θ ). dθ + 9 / 3 6 A = si 3θ + siθ si θ + si 3θ + siθ si θ 3 93 / 3 /3 6 A = si si si si A =.433 x.433 =.96 9 Te lift oeffiiet a ow be alulated as follows Cl = ( A + A ) = α ( ) = α (5) Te pitig momet oeffiiets ad te etre of pressure loatio are Cm, LE = α.663 ( ) α = + (5) 3

32 Cm, / 4 =.6495 (53) xp Cm, LE α / / = = = +.99 C 4 α / 4 α / l (54) Now let us osider aoter situatio were te maximum amber loatio is pused bak eve furter. For te ases were te maximum amber is loated at x/ = ¾, te results are as follows 3 For x y ( x) ( 3 ) 8 x x 4 : =. ad 9 ( 3 4x) dy 8 = (56) dx 9 dy 8 8 = ( 3 4 os ) = ( osθ + ) (57) dx 9 9 Terefore ( ( θ )) 3 For x y ( x) 4 ( x )( x) 8 : =. ad ( 3 4 ) dy 8 x = (58) dx dy 8 8 = ( 4 os ) = ( osθ + ) (59) dx Terefore ( ( θ )) It sould be oted tat x/ =3/4 is equivalet to osθ =.5 or θ = / 3 Te Fourier oeffiiets a ow be alulated as follows / 3 8 A = α ( osθ ). dθ 9. ( osθ ). dθ /3 8 / 3 A = α ( siθ + θ ) + 9. ( siθ + θ ) 9 / 3 8 A = α [ x.6849] = α / 3 6 A = ( os θ os θ ). dθ 9. ( os θ os θ ). dθ / 3 / 3 6 A = θ + si θ + siθ + 9. θ + si θ + siθ 9 / 3 6 A = ( x.93) =

33 / 3 6 A = ( os θ.osθ os θ ). dθ 9. ( os θ.osθ os θ ). dθ / 3 / 3 6 A = si 3θ + siθ + si θ + 9. si 3θ + siθ + si θ / A = si si 9. si si A = si 9x si = 3 3 Te lift oeffiiet a ow be alulated as follows Cl = ( A + A ) = α + ( ) = α (6) Te leadig edge pitig momet ad etre of pressure loatio are Cm, LE = α.78 ( ) α = + (6) C, / 4 = / =.9. / (6) m xp Cm, LE α / / = =. = Cl 4 α / 4 α / (63) Te various results a be tabulated to see te effets of siftig te loatio of te maximum amber poit as follows. Max. amber loatio Cl / α +.679( / ) α +.( / ) α ( / ) ( / ) Cm. LE α +.969( / ) α +4.( / ) α ( / ) x p /.99 / 4 α / C / m, / ( / ) ( / ) α +. ( / ) ( / ) / 4.38 α /.9( / ) From te table it a be see tat as te loatio of te maximum amber is moved bakward towards te trailig edge, for a give value of maximum amber, /, ad agle of attak,α, te te lift oeffiiet ireases ad te leadig edge pitig momet also ireases or beomes more egative. Te quarter ord momet also beomes more egative, but te loatio of te etre of pressure moves furter 33

34 bakward by a amout, wi depeds o bot / ad α as well as o te loatio of te maximum amber. If te maximum amber loatio is fixed, te for a give agle of attak te lift oeffiiet ad te momet oeffiiets all irease wit ireasig value of te maximum amber. However te loatio of te etre of pressure beaves i a more omplex maer ad depeds o te magitude of te agle of attak as well. If te maximum amber, /, ad its loatio are kept ostat, te te lift oeffiiet ad te leadig edge momet oeffiiet bot irease liearly wit ireasig agle of attak. Te quarter ord momet remais ostat ad is idepedet of te agle of attak. However, te loatio of te etre of pressure agai beaves i a omplex maer depedig o te relative values of te maximum amber, its loatio ad te agle of attak. If te agle of attak is zero, te above table a be simplified as follows Max. amber loatio C l.679 C m. LE x p As a be see from te above table, a aerofoil seems to ave a positive value of lift oeffiiet eve if te agle of attak is zero. Tis implies tat at a partiular egative value of agle of attak, te lift atig o a aerofoil is zero. Te agle of attak for a ambered aerofoil we te lift produed is zero is alled te zero lift agle of attak ad is deoted by te symbol of α. Te lift oeffiiet for a ambered aerofoil is usually give as C = α α (64) l From te above table it a be see tat te magitude of zero lift agle of attak for a give amber, /, ireases mootoially as te loatio of tat maximum amber is moved furter ad furter bakward towards te trailig edge. If te loatio of te maximum amber is fixed, bot te lift ad momet oeffiiets irease liearly wit ireasig amber, /. Furtermore, te ostat of proportioality ireases rapidly wit ireasig value of te loatio of te maximum amber (or te furter rearwards te maximum amber poit is loated o te aerofoil). Te magitude of α is obviously determied by te urvature of te aerofoil, wi is depedat o bot te magitude ad loatio of te maximum amber /. Max. amber loatio α (for / =.) α (for / =.) α (for / =.3)