JAVAFOIL User s Guide Marin Hepperle -Deember-017 Conens Conens... 1 JAVAFOIL... Limiaions... JAVAFOIL s Cards... 3 The Geomery Card...3 JAVAFOIL S Geomery Generaors...5 The Modify Card...17 The Flowfield Card...19 The Airraf Card...1 The Panel Mehod... Boundary Layer Analysis... 3 Transiion Crieria...3 Effe of Roughness...5 Sall Correions...6 Compressible Flow... 8 Criial Pressure Coeffiien...8 Compressibiliy Correions...9 Finie Wings in JAVAFOIL... 9 Swep Wings in JAVAFOIL... 34 The Aerodynami Cener... 36 Effe of a Ground Surfae... 37 Effe of a Waer Surfae... 38 Muli-Elemen Airfoils... 40 Auomaing JAVAFOIL wih a Srip... 41 Referenes... 4 Version Hisory... 43 1
JAVAFOIL JAVAFOIL is a relaively simple program, whih uses several radiional mehods for he analysis of airfoils in subsoni flow. The main purpose of JAVAFOIL is o deermine he lif, drag and momen haraerisis of airfoils. The program will firs alulae he disribuion of he veloiy on he surfae of he airfoil. For his purpose i uses a poenial flow analysis module whih is based on a higher order panel mehod (linear varying voriiy disribuion). This loal veloiy and he loal pressure are relaed by he Bernoulli equaion. In order o find he lif and he pihing momen oeffiien he disribuion of he pressure an be inegraed along he surfae. Nex JAVAFOIL will alulae he behavior of he flow layer lose o he airfoil surfae (he boundary layer). The boundary layer analysis module (a so alled inegral mehod) seps along he upper and he lower surfaes of he airfoil, saring a he sagnaion poin. I solves a se of differenial equaions o find he various boundary layer parameers. The boundary layer daa is hen be used o alulae he drag of he airfoil from is properies a he railing edge. Boh analysis seps are repeaed for eah angle of aak, whih yields a omplee polar of he airfoil for one fixed Reynolds number. Addiional ools for he reaion and modifiaion of airfoils have been added o fill he oolbox. These ools are wrapped in a Graphial User Inerfae (GUI) whih was designed o be easy o use and no overly ompliaed. The GUI is organized ino a sak of ards, whih will be desribed laer. All alulaions are performed by a ompuer ode of my own. JAVAFOIL is neiher a rewrie of Eppler PROFIL nor of Drela s XFOIL program. The boundary layer module is based on he same equaions whih are also used in he iniial version of he Eppler program. Addiions inlude new sall and ransiion models. The panel mehod was developed wih he help of he exensive survey of panel mehods found in [14]. Compared wih similar programs, JAVAFOIL an also handle muli-elemen airfoils and also simulae ground effe. Limiaions As already noed, JAVAFOIL is a relaively simple program wih some limiaions. Like wih all engineering ompuer odes, i is up o he user o judge and o deide how far he wans o rus a program. As JAVAFOIL does no model laminar separaion bubbles and flow separaion, is resuls will beome inaurae if suh effes our. The boundary layer mehod does no inlude any feedbak o he poenial flow soluion, whih means ha i is limied o mosly aahed flows. Flow separaion, as i ours a sall, is modeled o some exen by empirial orreions, so ha maximum lif an be esimaed for onvenional airfoils. If you analyze an airfoil beyond sall, he resuls will be quie inaurae. On he oher hand, i is somewha quesionable, wheher any wo-dimensional analysis mehod an be used a all in his regime, as he flow field beyond sall beomes fully hree dimensional wih spanwise flow and srong vories developing. In he ase of muli elemen airfoils, one mus be aware, ha in he real world very omplex flows an develop due o ineraion of railing wakes and he boundary layers of he individual elemens or if he boundary layers separae loally. An aurae analysis would require a more sophisiaed solver for he Navier-Sokes equaions, whih would also imply an inrease in ompuer ime in he order of 1000. Neverheless a simple ool like JAVAFOIL an be helpful o esimae he main effes and o improve a design o avoid suion peaks and flow separaion.
JAVAFOIL s Cards The user inerfae of JAVAFOIL is divided ino a sak of ards. Eah ard onains inerfae elemens for a speifi ask. The onen of some ards is also relevan for aions exeued on oher ards, for example he Mah number speified on he Opions ard affes he analyses on all oher ards. The Geomery Card The Geomery ard is used o sore and prepare he geomery of your airfoil. I onains he urren or working airfoil. The geomery is desribed by a se of oordinae poins, eah having an x and a y value. The working airfoil is used and modified by he aions you perform in JAVAFOIL. The Geomery ard shows a lis of x- and y-oordinae pairs and a plo of he resuling airfoil shape. You an ener or pase arbirary oordinaes ino his field and press he Updae View buon o opy he oordinaes ino he working airfoil. The oordinaes mus be ordered so ha hey desribe he shape in a oninuous sequene. The order mus be railing edge upper surfae nose lower surfae railing edge. JAVAFOIL omes wih a se of shape generaors for a variey of airfoils whih is aessible from his ard. These airfoils represen lassial airfoil seions for whih analyial desripions exis (e.g. NACA seions) or whih an be onsrued from geomerial onsrains (e.g. wedge seions). Despie heir age, many lassial airfoil seions are sill appliable o many problems or form a good saring poin for new developmens. Today, modern airfoil seions are usually developed for speifi purposes and heir shapes are usually no published. More reen developmens lead owards he dire design of hreedimensional wing shapes, eliminaing he lassial seps of wo-dimensional airfoil design and hree-dimensional wing lofing. In mos ases, modern airfoil seions are no desribed anymore by analyial formulas, jus by a se of poins. The row of buons a he boom allows for opying, saving, loading and prining of airfoil oordinae ses. 3
Figure 1: View of JAVAFOIL s Geomery ard. Exporing airfoil geomery JAVAFOIL an wrie airfoil geomery o he following file ypes: *.x muli-elemen airfoil geomery in form of simple x-y oordinae ses arranged in wo olumns. Muli-elemen airfoils mus be separaed by a pair of x and y-values eah being larger han 999. *.xml muli-elemen airfoil geomery in JAVAFOIL s hierarhially sruured xml forma. *.dxf muli-elemen airfoil geomery in AuoCad drawing exhange forma. Many CAD programs an read his file forma, bu he inerpreaion is no always perfe. *.igs or *.iges muli-elemen airfoil geomery in Iniial Graphis Exhange Sandard forma. Many CAD programs an read his file forma. Noe ha JAVAFOIL seles he oupu file forma aording o he file name exension. Imporing airfoil geomery JAVAFOIL an read airfoil geomery from he following file ypes: 4
*.x muli-elemen airfoil geomery in form of simple x-y oordinae ses arranged in wo olumns. The elemens of muli-elemen airfoils mus be separaed by a pair of x- and y-values eah having a value of 999.0 or larger. *.xml muli-elemen airfoil geomery in JAVAFOIL s hierarhially sruured xml forma. *.png, *.gif, *.bmp, *.jpg single elemen airfoil geomery from an image file. For deails, see nex seion. Noe ha JAVAFOIL seles he inpu file forma aording o he file name exension. Imporing sanned images You an also load an airfoil from a bimap image in GIF, PNG, BMP or JPG forma. JAVAFOIL hen ries o find an airfoil shape in his image by omparing he image poins wih he olor found in he upper lef orner of he image. Therefore he image should have no border, and a monohrome bakground. Before sanning he image, a smoohing filer is applied o remove spurious poins from he image. To ahieve aepable resuls an image widh of 1000 or more pixels is reommended. The inerior of he airfoil shape an be empy or arbirarily filled, beause he algorihm searhes from he op and boom edges of he image and sops when i dees he border of he shape. The resuling poins are filered again o improve he smoohness of he shape. Neverheless he resuls will no be perfe, bu his mehod an be onsidered as a las resor o quikly deermine airfoil oordinaes if only a sanned image is available. I is reommended o inspe he resuling veloiy disribuion and o use he inverse design mehod for smoohing he airfoil shape. Figure : Airfoil image (op) and omparison of he original (dashed) and he reonsrued airfoil shape (solid) using JAVAFOIL s bimap impor apabiliy on he Geomery ard. JAVAFOIL S Geomery Generaors Noe on NACA airfoils The onsruion of he ambered NACA airfoil seions requires ha he hikness disribuion is ereed a righ angles o he amber line. Some ompuer programs do no 5
follow his onsruion priniple and add he hikness jus o he y-oordinaes of he amber line. This leads o larger deviaions from he rue airfoil seion when he amber line is inlined, e.g. lose o he leading edge or lose o he railing edge of airfoils wih a high amoun of af amber. The orre onsruion mehod may lead o poins exending slighly ino he negaive x- range, when a large amoun of amber is loaed lose o he leading edge. This is a orre behavior and an expeed resul. Noe also ha mos NACA seions have a hik railing edge by definiion. In order o produe a hin, sharp railing edge, JAVAFOIL has an opion o lose he airfoil shape by bending he upper und lower surfaes o lose he railing edge. NACA 4-digi airfoils The alulaion of hese lassial airfoils is easy beause heir shape and he assoiaed amber lines are defined by raher simple formulas. The maximum hikness is loaed a x / = 30%, whereas he maximum amber is ypially loaed a x / = 40%. See [3] and [4] for more deails. The amber lines are omposed of wo paraboli ars, whih are joined wih equal angens, bu a kink in he urvaure. This kink an be seen in he veloiy disribuions, espeially when he posiion of he maximum amber is differen from he ommon 40% hord saion. f x f Figure 3: Parameers of NACA 4-digi airfoil seions. Parameers: Free: /, f/, x f / Fixed x/ = 0.3 Naming Sheme The firs wo inegers define he amber line, while he las wo inegers define he hikness. 1s digi: maximum ordinae of amber 100 f / nd digi: loaion of maximum amber 10 x f / 3rd and 4h digi: maximum hikness 100 / Example: NACA 41: % amber a 40% hord, 1% hikness The hikness disribuion for he 10% hik seion is given by he polynomial: y = 0.9690 x -0.1600 x -0.35160 x + 0.8530 x -0.10150 x 3 4 ( ) The oeffiiens of his hikness disribuion had been hosen aording o he following four onsrains [4] (for a 10% hik seion): y maximum hikness ours a x/ = 0.3 x ( 0.3 ) 0 =, 6
finie hikness a railing edge of y = 0.004, y finie railing edge angle a x/ = 1.0 x ( 1.0) =- 0.34, nose shape defined by y / = 0.078 a x/= 0.1. Modified NACA 4-digi airfoils The modifiaion adds he posiion of he maximum hikness as well as he nose radius o he parameer se of he 4-digi series (see [3]). r x f x f Figure 4: Parameers of he modified NACA 4-digi airfoil seions. Parameers: Free: /, f/, x/, x f /, r Naming Sheme The name onsiss of a 4 digi prefix whih is idenial o he 4-digi series designaion, followed by a dash and wo addiional digis. 1 s digi: maximum ordinae of amber 100 f / nd digi: posiion of he maximum amber 10 x f / 3 rd and 4 h digi: maximum hikness 100 / dash 5 h digi: indiaes he leading edge radius and is usually one of 0, 3, 6, or 9: o 0: sharp leading edge, o 3: ¼ normal radius, o 6: he normal radius of he 4-digi series, o 9: 3 imes he normal radius. 6 h digi: posiion of he maximum hikness 10 x / Example: NACA 1410-35: 1% amber a 40% hord, 10% hikness, redued leading edge radius, maximum hikness a 50% x/ NACA 5-digi airfoils These seions use he same hikness disribuions as he 4-series, bu have new amber lines leading o lower pihing momens. The amber line is omposed of a ubi urve in he forward par o whih a sraigh line is aahed whih exends o he railing edge. Insead of he amber f/, a design lif oeffiien C is now used o define he maximum heigh of he amber line. In praial appliaions, hese airfoils are ofen used wih a maximum amber a x/ = 0.15, i.e. relaively far forward. design 7
x f Figure 5: Parameers of NACA 5-digi airfoil seions. Parameers: Free: /, x f /, Fixed x/ = 0.3 C design Naming Sheme 1s digi: design 10 / 3 C design nd and 3rd digi: 100 x f /. Noe ha he 3 rd digi is usually a zero, i.e. he posiion of he maximum amber is a muliple of 5%. 4h and 5h digi: maximum hikness 100 / Example: NACA 301: design lif oeffiien 0.3, maximum amber a 15% hord, 1% hikness. Modified NACA 5-digi airfoils The rear par of he amber line of hese seions has been modified o a ubi urve whih provides a reflexed amber line. Therefore he pihing momens are redued furher or may beome even posiive. x f Figure 6: Parameers of he modified NACA 5-digi airfoil seions. Parameers: Free: /, x/, f Fixed x/ = 0.3 C design Naming Sheme 1s digi: design 10 / 3 C design nd and 3rd digi: 100 x f / plus 1. Assuming ha he posiion of he maximum amber is a muliple of 5% he 3 rd digi is always a 1. 4h and 5h digi: maximum hikness 100 / Example: NACA 311: like NACA 301: design lif oeffiien 0.3, maximum amber a 15% hord, 1% hikness, bu wih a reflexed af amber line. 8
NACA 1-series airfoils The developmen of hese airfoils was aiming a high subsoni speed appliaions like propellers (see [5]). Their shape was designed wih he help of he new (ha is, in he 1930s) numerial design mehods. JAVAFOIL an reae airfoils of he NACA-16 ype, whih are he only members of he 1-series published by NACA. The maximum hikness and he maximum amber are loaed a 50% hord, whereas he minimum pressure is reahed a 60% of he hord lengh. Figure 7: Parameers of NACA 1-series airfoil seions. Parameers: Free: /, C design Fixed x f / = 0.5, x/ = 0.5 Naming Sheme 1s digi 1 : series designaion nd digi: posiion of minimum pressure of he hikness disribuion 10 x / a dash 3rd digi: 10 C design 4rd and 5h digi: maximum hikness 100 / Example: 16-1: 1-series, minimum pressure a 60% hord, design lif oeffiien 0., 1% hikness. While hese airfoil shapes are no based on analyial expressions, he published oordinaes have been approximaed o produe an aurae represenaion of hese airfoils. The amber lines used are of he uniform load ype (a=1.0, see nex seion abou NACA 6-series airfoils). NACA 6- and 6A-series airfoils These airfoils were he firs NACA airfoils whih had been sysemaially developed wih he inverse design mehod by Theodorsen. The onformal mapping algorihm was able o deliver a shape for a given pressure disribuion. This means ha no losed form equaions desribing he hikness disribuions exis. Earlier JAVAFOIL versions used a very approximae algorihm whih had been lifed from he "Digial Daom" programs, bu i was disovered ha his produed very inaurae represenaions of he 6-series airfoils. Therefore, sine version.09 (Augus 009) JAVAFOIL uses a more elaborae algorihm, whih is based on he work of Ladson [6]. This new mehod is using quie aurae ables of he sream funion for mos of he 6-series airfoils. JAVAFOIL an generae individual members of he 63, 64, 65, 66 and 67 as well as he 63A, 64A, and 65A families. The "A" modifiaion leads o a less usped railing edge region. The 63, 64, 65, 66 and 67 families an be ombined wih amber lines of he a = 0 o a = 1 ype. 9
The 63A, 64A, and 65A seions use a modified a = 0.8 amber line whih is sraigh af of x/= 0.8. The hikness disribuion of hese airfoils has also been modified o yield sraigh lines from x/ = 0.8 o he railing edge. The a amber line shapes are speified in erms of he design lif oeffiien and he posiion x/ where he onsan loading ends. This is indiaed wih an addiional a = x.y label in he airfoil name. If you speify a ³ 1 in JAVAFOIL s inpu, he amber line has a onsan loading from leading edge o railing edge. The resuling airfoils do no arry he a label. Noe ha offiially no inermediae airfoils (e.g. a NACA64.5-01) exis. Naming Sheme 1s digi 6 : series designaion nd digi: hordwise posiion of minimum pressure of he hikness disribuion 10 x / single digi suffix following a omma, whih is 10 DC. I represens he range DC above and below C design where favorable (aeleraing) pressure gradiens for laminar flow exis (herefore DC is approximaely he semi-widh of laminar buke) a dash 3rd digi: design 10 C design 4rd and 5h digi: maximum hikness 100 / A amber line shape differen from a = 1.0 is indiaed by he addiional designaion a = x.y, where x.y is replaed by he loaion x/ where he onsan par of he loading ends and he linear drop owards he railing edge sars.. TsAGI "B" airfoils The TsAGI (also ZAGI, CAGI) was and is Russia's leading aeronauial researh organizaion. No muh is known abou early airfoil developmen, bu he available lieraure [6], [9] shows ha similar o oher naions Russia has developed airfoil families based on analyial shape desripions. The TsAGI series-b is jus one suh airfoil family. The very simple shape desripion is using jus he maximum hikness. The resuling seions have a reflexed amber line and hene low pihing momen. Figure 8: Parameers of TsAGI B airfoil seions. Parameers: Free: / Fixed x / = 0.3388, maximum (posiive) amber a x f / = 0.3018, minimum (negaive) amber a x f / = 0.904. 10
The maximum amber is linked o he hikness by he expression f / = 0.168 /.!!! I am sill looking for more informaion abou Russian airfoil developmens. NPL-EC, ECH and EQH airfoils These Briish symmerial airfoil seions are omposed of an ellipial forward porion (E) and a ubi (C) or quari (Q) rear end. The ail losure is buil from a hyperboli urve (H series). The loaion of he maximum hikness an be varied beween 30 and 70% of he hord lengh. A limied desripion is onained in [10], [13]. x Figure 9: Parameers of NPL airfoil seions. Parameers: Free: /, x/ Afer some reverse engineering, I have used he following assumpions for hese airfoils: he railing edge hikness is % of he airfoil hikness, in ase of he C and Q series he rear end is aahed wih C0, C1, C oninuiy (posiion, angen, urvaure) o he ellipi fron par, in ase of he Q series he seond derivaive a he railing edge is se o -0., (his gave he bes approximaions for 140 o 160 airfoils), he H modifiaion loses he hik railing edge by a hyperboli urve whih is aahed wih C0, C1 oninuiy (posiion, angen) o he hikness disribuion a x / = 0.965. Camber lines are 3 rd order polynomials whih allow o plae he loaion of he maximum amber approximaely beween 30 and 60% of he hord lengh. Noe: I am sill looking for he offiial desripion of he airfoil geomery of he EQ and EQH aerofoils, espeially how he quari urve was defined and how he hyperboli losure was aahed o he quari urve. I seems o be ha he proedure o generae hese shapes was no published. Bionvex airfoils These are symmerial airfoils, formed by wo ars. They an be represened by he following formula: b y = a ( x- x ) The exponen b an be found from he loaion of he maximum hikness, i.e. he poin where y/ x = 0 æ1ö xmax = ç çè b, ø while he faor a depends on he value of he maximum hikness: 1 b-1 11
b ( ) = a x - x max max max If he maximum hikness is plaed a x/ = 0.5, he airfoil is omposed of wo equal irular ars. These airfoils are normally used for appliaion in supersoni flow. x Figure 10: Parameers of bionvex airfoil seions. Parameers: Free: /, x/ Double Wedge airfoils These are symmerial airfoils omposed of sraigh lines. They are inended for supersoni flow. x Figure 11: Parameers of double wedge airfoil seions. Parameers: Free: /, x/ Plae airfoils The hikness disribuion of hese seions represens a plae wih a rounded nose and a sharp railing edge. The nose shape is formed by a so alled Cassini urve, whih provides a smooh urvaure ransiion o he fla par of he surfae. The railing edge losure is modeled by a ubi parabola. This hikness disribuion is superimposed over a NACA 4-series amber line o produe a ambered plae. x f f Figure 1: Parameers of ambered plae airfoil seions. 1
Parameers: Free: /, f/, x f / Fixed: Leading edge shape. The railing edge losure begins a x/= 0.8 Newman airfoils These seions onsis of a irular nose o whih sraigh apered ail is aahed. I an be manufaured easily, bu has a urvaure jump a he junion beween he nose and he railing wedge, leading o suion peaks and a risk of flow separaion. Figure 13: Parameers of Newman airfoil seions. Parameers: Free: / Joukowsky airfoils These lassial airfoil seions are generaed by applying a onformal mapping proedure. They were he firs praial airfoils developed on a heoreial model. Besides produing he airfoil shape, he mapping proedure was also used o find he flow field around he airfoil as well as he fore and he momen aing on he wing seion. The airfoils have very hin usped railing edges and are herefore diffiul o analyze wih panel mehods and diffiul o manufaure. The onformal mapping is performed using he Joukowsky ransformaion of he omplex poins z irle on a uni irle wih is ener a ( x,y 0 0). l zairfoil = zirle +, where l=- x0 + 1- y0. zirle In order o mah he presribed airfoil hikness and amber, JAVAFOIL performs an ieraive searh for he ener of he irle. As usual, he resuling oordinaes are saled o uni lengh. Figure 14: Parameers of Joukowsky airfoil seions. Parameers: Free: /, f/ Van de Vooren airfoils In onras o he lassial Joukowsky airfoils, hese airfoils have a finie railing edge angle. The ransformaion funion is of he ype f 13
k ( 1- zirle) zairfoil = k-1. ( e- zirle ) They an be used o reae seions wih hik railing edge regions e.g. for fairings. A desripion of his shape an be found in [14]. Noe ha no all hiknesses an be ahieved for all railing edge angles; herefore he final maximum hikness may no be wha was desired. Also only symmerial seions are generaed in JAVAFOIL. TE Figure 15: Parameers of Van de Vooren airfoil seions. Parameers: Free: /, f TE Helmbold-Keune airfoils In he 1940s many aemps were made o exend he hen lassial NACA airfoil seion mehodology o more general airfoil shapes. Helmbold and Keune [15] developed elaborae mehods o haraerize and parameerize airfoil seions. While he mahemaial approah allowed for represenaion of a wide range of shapes, he mehodology was no really suessful in hese years of manual alulaion. Laer in he age of numerial shape opimizaion similar mehods have been developed, e.g. he Parse shape funions. The parameers of he symmerial airfoil mus be arefully hosen o generae a realisi airfoil shape. The ener urvaure mus be large enough o avoid self-rossing of he ouline. r urvaure r nose TE x Figure 16: Parameers of Helmbold-Keune airfoil seions. Parameers: Free: /, x/, railing edge angle, urvaure radius a middle, nose radius. Roßner airfoils Anoher algorihm o generae analyial airfoil shapes based on onformal mapping was published by Roßner [16]. Like all mehods using onformal mapping, his soluion also allowed for he exa analyial deerminaion of he orresponding pressure disribuions. 14
r nose TE x Figure 17: Parameers of Roßner airfoil seions. Parameers: Free: /, x/, railing edge angle, nose radius. Parse airfoils The Parse geomery parameerizaion was developed by H. Sobiezky in he 1990s. I ries o model airfoil shapes by superposiion of seleed polynomial erms. The parameers resemble he Helmbold-Keune approah and are mainly inended o be used for numerial shape opimizaion. JAVAFOIL implemens he so alled Parse-11 formulaion whih uses 11 parameers. The parameers of he airfoil mus be arefully hosen o generae realisi airfoil shapes. The ener urvaure parameers, he nose radius as well as he railing edge wedge angle mus be arefully adjused o avoid self-rossing of he ouline. x 1 y 1 urvaure upper y/ x TE r nose y urvaure lower x Parameers: Free: leading edge radius parameer P, ( P, P ) and ( P, P ) he oordinaes of 1 3 4 5 onrol poins on he upper and he lower surfaes, P and P urvaure onrol 6 7 parameers for upper and lower surfaes, P he railing edge verial posiion, P he 8 9 gradien of he amber line a he railing edge, P he railing edge slenderness, and 10 P he blunness angle a he railing edge. 11 Horen airfoils The Horen brohers are well known for heir developmen of flying wing airplanes. For mos of heir wings, hey used airfoil seions wih a reflexed amber line. These were based on a amber line of low or zero pihing momen (following he hin airfoil heory of Birnbaum) o whih a hikness disribuion was added. A desripion of hese raher simple funions an be found in [17]. f 15
Figure 18: Parameers of Horen airfoil seions. Parameers: Free: /, f/ Fixed x / = 0.93, maximum amber a x/ f = 0.5. DHMTU airfoils The sable fligh of ground effe vehiles depends on wing planform and airfoil shape. This family of fla boom airfoils has been developed for his speifi appliaion a he Deparmen of Hydromehanis of he Marine Tehnial Universiy in Sain Peersburg, Russia. Similar o he NACA 4-digi series, he airfoil shape is omposed of polynomial segmens and a sraigh lower surfae. Their amber line is slighly reflexed and he ouline beween poins and 3 is a sraigh line segmen. A desripion of he shape an be found in [18]. x 1 y 1 r nose -y -y 3 y/ x x x 3 Figure 19: Parameers of DHMTU airfoil seions. Parameers: Free: poin x/,y/ 1 1 on he upper and poins x /, y / and x 3/, y 3/ on he lower surfae, a nose radius parameer k = ( r/ )/( y ) 1 / and he gradien y/ x on upper surfae a he railing edge. Guderley airfoils This shape was derived from heoreial onsideraions of soni flow and is of mosly aademi ineres. I is haraerized by an aeleraing flow wih a linear pressure disribuion in he forward porion followed by a se of expansion waves. The maximum hikness is loaed a x/= 3/5. A desripion of he shape an be found in [6]. Figure 0: Parameers of Guderley airfoil seions. Parameers: Free: /. 16
The Modify Card This ard an be used o perform various modifiaions o he airfoil geomery. I onsiss of an inpu and aion area and a geomery view below. The modifiaion of parameers is performed by enering new values ino a ex field and hen pressing he buon a he lef of he ex field or pressing he Ener key while he fous is sill in he ex field. Thus i is easy o apply erain operaions several imes. Any modifiaion will only be applied o he airfoil elemens whih are urrenly seleed in he Elemen lis box. The geomery view is auomaially saled o fi all airfoil elemens. The urrenly seleed elemens are highlighed in red. Figure 1: View of he Modify ard showing a wo-elemen airfoil wih elemen # seleed. 17
Figure : View of he onrols on he Modify ard. The following modifiaions an be performed: NAME Changes he name of he airfoil NUMBER OF POINTS Changes he number of oordinae poins of he seleed elemen(s). THICKNESS Sales he hikness of he seleed elemen(s) by deomposing he shape ino a hikness disribuion and a amber line. Only he hikness disribuion is saled, so ha he amber line is mainained. Noe ha small hanges o he amber may our due o numerial errors. CAMBER Sales he amber line o a new heigh. This works only if he airfoil is already ambered. Saling he amber line of a symmerial airfoil aomplishes nohing. SCALE BY Sales he airfoil shape by muliplying he oordinaes wih he given saling faor. FLAP DEFLECTION Modifies he oordinaes by defleing a plain flap of he given hord lengh. The axis of roaion is always he middle beween upper and lower surfae. TRAILING EDGE GAP Modifies he shape so ha he presribed railing edge gap is produed. Generally i is reommended o use losed railing edges for analysis, exep if he airfoil is exremely hin owards he railing edge. This funion an also be applied before exporing airfoil shapes suiable for manufauring. ROTATE Roaes he seleed airfoil elemen(s) around he speified Pivo poin. TRANSLATE X Moves he seleed airfoil elemen(s) by he given disane horizonally. TRANSLATE Y Moves he seleed airfoil elemen(s) by he given disane verially. DUPLICATE Creaes a opy of he urrenly seleed elemen(s). Noe ha you have o move he new elemen from is iniial loaion so ha i is no overlapping wih oher elemens. DELETE Delees he seleed elemen(s) 18
FLIP Y Refles he seleed elemens aross a horizonal line passing hrough he pivo poin. SMOOTH Y This ommand uses he smoohing faor speified in he ex field o he righ of he buon. Currenly i suppors wo smoohing varians: If he smoohing faor is posiive, he oordinaes are approximaed by a smoohing spline urve. A reasonable smoohing faor is 0.1. If he smoohing faor is negaive, a filer is applied o he y-oordinaes o redue waviness. This filer applies a weighed average o eah poin and is wo neighbor poins. If for example he smoohing faor is -0.1, he y oordinae of he smoohed poin is 90% of is iniial value and 10% of he linear inerpolaion beween he wo neighboring poins aording o: æ si - s ö i-1 yi ( 1-f) yi + f y i- 1+ ( yi+ 1 yi -1) - ç si+ 1 s è - i-1ø This filer an be applied several imes, bu subsequen appliaion will also smooh ou deails like a poined airfoil nose. You an also modify individual poins by dragging hem up or down wih he lef mouse buon depressed. This modifiaion mehod is resried o movemens in he y-direion. If you need more freedom, you have o modify he numerial oordinae values on he Geomery ard. The COPY (TEXT) ommand buon a he boom of he ard opies he airfoil geomery o he lipboard wih he following daa: a able wih he x-y oordinaes, similar o he opy on he GEOMETRY ard, bu addiionally wih he loal urvaure 1/r, and a seond able wih he oordinaes of he amber line and he hikness disribuion as reonsrued from he airfoil shape. Noe onerning muli-elemen airfoils Modifiaions are applied only o he airfoil elemen(s) seleed in he Elemens lis box. The seleion is also used by oher ards. Only seleed elemens are aken ino aoun when oal fore, momen and drag oeffiiens are deermined. The Flowfield Card This ard is inended o visualize he flow around he airfoil in various ways. Basially he Analyze I! ommand firs performs an analysis of he airfoil for he given angle of aak. The resuls are presened in form of he global oeffiiens in he able. In order o be onsisen wih he display on he Boundary layer ard, hese resuls inlude friion using parameer aken from he boundary layer ard (Reynolds number and ransiion loaion) as well as from he Polar ard (sall model). Then he loal veloiy over a reangular grid of poins is alulaed. This alulaion uses he voriiy disribuion on he surfae and negles friion. Therefore you will no see flow separaion or a visous wake behind he airfoil. I is possible o display eiher he raio of he loal veloiy o he freesream veloiy v/v or he loal pressure oeffiien. 19
Moving he mouse poiner over he olored field shows he orresponding pressure oeffiien or veloiy raio a he boom of he sreen. ANALYZE IT! Performs he analysis of he flow field and if seleed he inegraion of he pah of sreamlines. Noe ha he lassial Runge-Kua sheme used o inegrae sreamlines wih inreased auray an ake quie some ime. Sandard auray uses a fas bu simple forward sepping Newon algorihm whih inrodues larger errors in regions of high urvaure. Progress is indiaed in he saus line. COPY (TEXT) opies he field daa o he lipboard in abular forma suiable for ploing wih he Teplo sofware. Remember ha you an opy or expor he piure using he onex menu of he graph window. INTEGRATE uses he momenum equaion o inegrae he momenum and pressure field along a irular pah wih a radius of 50 around he airfoil. As we negle friion and hene follow d Alember s houghs, he resul should produe zero drag, bu a lif oeffiien lose o wha we obain from he inegraion of he surfae pressure. Remember ha he lif oeffiien given in he able inludes friion and he effe of he sall model, so ha boh resuls an only agree if he Reynolds number is raher high. The resul is displayed in a message box and also opied o he lipboard. Figure 3: View of he Flowfield ard showing an airfoil wih imes sreamlines. 0
Inegraion along irular pah wih R = 50.0 x-y sysem momenum + pressure = oal Fx = -0.0596 + -0.0596 = -0.1193 Fy = 0.338 + 0.338 = 0.6763 aerodynami sysem ( = 10.0º) Cl = 1.3735 Cd = 0.0000 Inegraion over surfae panels (for omparison) Cl = 1.373 Cd = 0.0005 Figure 4: Resul of momenum and pressure inegraion over a irular pah around he airfoil. Compare he lif oeffiien wih he value obained from surfae pressure inegraion (The sall model was se o none on he Polar ard). The oal fore is he resul of he hange of he momenum passing hrough he onrol volume and he pressure aing on is surfae. Noe ha he pressure par is very imporan, even if he inegraion boundary is raher far away from he airfoil. The Airraf Card This ard is similar o he Polar ard bu is inended o be used o analyze he airfoil under ondiions wih are lose o he appliaion on airraf or hydrofoil wings. I is assumed ha he wing has o arry a erain load (he weigh of he airraf) a all fligh speeds. In order o produe he same lif he lif oeffiien of he airfoil seion a low speed mus be higher han a high speed. Thus lif oeffiien and fligh speed (and hene Reynolds number) depend on eah oher. To esablish he relaions using airraf design parameers we sar wih he definiion of he lif oeffiien C L for he omplee wing C g m =. S L r v Solving he definiion of he Reynolds number wih he hord lengh v Re = n for he fligh speed v and insering ino he lif oeffiien gives μs Solving for he Reynolds number C g m =. r ære ö S n ç çè ø L g m Re = n r C S Yields he desired relaion beween lif oeffiien and Reynolds number as i is seen by and airraf wih a reangular wing planform. The design parameers direly relaed o he airraf and is wing are wing loading ms and hord lengh. L 1
The Panel Mehod JAVAFOIL implemens a lassial panel mehod o deermine he linear poenial flow field around single and muli-elemen airfoils. In JAVAFOIL he airfoil surfaes arry a linearly varying voriiy disribuion. This is he same ype of disribuion as used in XFOIL bu simpler han he higher order (paraboli) disribuion used in Eppler s PROFIL ode. The resuling equaion sysem onsiss herefore of a (# of panels +1)² sized marix and wo righ hand sides. These are for 0 and 90 angle of aak and an be solved effiienly a he same ime for he wo orresponding voriiy disribuions. The voriiy disribuion for any arbirary angle of aak is hen derived from hese wo soluions (remember ha poenial heory is linear and allows for superposiion). There is no ineraion wih he boundary layer, as in XFOIL, hough. For a shape disreizaion by N panels, he equaion sysem of his lassial panel mehod onsiss of he marix of influene oeffiiens, he unknown irulaion srengh a eah panel orner poin and he wo righ hand side veors. These represen he no flow hrough he surfae ondiions for 0 and 90 angle of aak. Eah oeffiien C i,j refles he influene of he riangular voriiy disribuion due o he vorex srengh g i a eah orner poin on he ener poin of eah panel j. The las row onains he angenial flow ondiion a he railing edge (he Kua-ondiion ). This is needed o obain a soluion whih is ompaible wih he experiene ha he flow normally separaes smoohly a he sharp railing edge. Noe ha his assumpion will no be orre when large regions of flow separaion our. éc1,1 CN+ 1,1 ù g RHS 1, 0 1,90 1, 0 RHS g é 1,90 ù é ù g, 0 g,90 RHS, 0 RHS,90 ê ú = C C ê ú ê ú 1,N N+ 1,N 1 0 1 gn+ 1,0 g N+ 1,90 RHSN+ 1,0 RHS ê ú êë úû ê N+ 1,90ú ë û ë û Like wih mos panel mehods he soluion ime for he sysem of linear equaions inreases wih he square of he number of unknowns. Therefore i is advisable o limi he number of poins o values beween 50 and 150. This relaively small number already yields suffiien auray of he resuls (in onras o more omplex CFD mehods for solving he Navier- Sokes equaions, where you may need several 100 poins on he airfoil surfae and many more poins o fill he spae around he shape). ime per soluion [ms] 0 18 16 14 1 10 8 6 4 JavaFoil Panel Mehod y = 0.0016x - 0.0084x + 0.0438 0 0 0 40 60 80 100 10 number of poins on airfoil Figure 5: Graph of he soluion ime versus number of poins on he airfoil (Penium 4,.4 GHz).
Boundary Layer Analysis The boundary layer analysis module implemens an inegral boundary layer inegraion sheme following publiaions by Prof. Rihard Eppler. Suh inegral mehods are based on differenial equaions desribing he growh of boundary layer parameers depending on he exernal loal flow veloiy. These equaions are hen inegraed saring a he sagnaion poin. While aurae analyial formulaions are available for laminar boundary layers, some empirial orrelaions are needed o model he urbulen par. Noe: he loal skin friion oeffiien as given on he Boundary Layer ard is wie he value r as used by Eppler o follow he more ommon onvenion C f =0/ ( v ). In JAVAFOIL here is no ineraion beween he boundary layer and he exernal flow, as in XFOIL, hough. Therefore largely separaed flows anno be analyzed a shor flow separaion ( s separaed / < 10% ) a he railing edge does no affe he resuls very muh. Also laminar separaion bubbles are no modeled; when laminar separaion is deeed he ode swihes o urbulen flow. Transiion Crieria Mehods o predi ransiion from laminar o urbulen flow have been developed by many auhors sine he early days of Prandl s boundary layer heory. While i is possible o analyze he sabiliy of a boundary layer numerially, all mehods whih are praial and fas are more or less approximae and rely on empirial relaions (usually derived from experimens). Beause he loal boundary layer parameers a a saion s are he resul of an inegraion proess saring a he sagnaion poin, hey onain a hisory of he flow. Loal Crieria Many mehods predi ransiion by applying a rierion based on loal boundary layer parameers. These rieria are based on relaions, whih an be evaluaed a any saion along he surfae. They do no need an exra inegraion of some insabiliy parameer, bu of ourse are affeed by he hisory of he flow. Mos of hese rieria are relaing Re d o he shape of he boundary layer profile. Eppler 18.4 H3-1.74-0.36 r Transiion is assumed o our when Red ³ e. Eppler enhaned Transiion is assumed o our when Re 18.4 H - 1.74+ 15 ( H -1.573) -0.36 r 3 3 d ³ e. Mihel (1) 0.444 This simple rierion assumes ransiion o our when Red ³ 1.535 Re s. Mihel () 0.46 Transiion is assumed o our when Red ³ 1.174 ( 1 + 400/ Re ) Re. See [4]. s s 3
H1-Res Transiion is assumed o our when.1<h 1<.8 and 3 log ( Re ) ³- 40.4557 + 64.8066 H -6.7538 H + 3.3819 H. See [5]. 10 s 1 1 1 Crieria based on a region of insabiliy, n- Re d envelopes These mehods firs deermine a loal poin of insabiliy and hen begin a his poin o inegrae a measure for he amplifiaion of insabiliy. Drela approximaes he envelopes of he amplifiaion rae n versus Re d by sraigh lines of he form n = f( Re d,h ) 1. Two versions of his approximaion were used in his odes of he XFOIL and MSES/ISES family. The approximaion is expressed by n n = ( Red -Re ) d,ri. Re d Transiion an our when Red > Re d and n > n,ri ri. In JAVAFOIL ransiion is assumed o our when he value nri = 9- r is exeeded. Drela, XFOIL 1.1 and 5.4 n = 0.01 (.4 H1 +.5 anh( 1.5 H1-4.65) - 3.7) ) + 0.5 Red æ 1.415 ö æ 0 ö 3.95 log10 ( Red ),ri = -0.489 anh - 1.9 + + 0.44 çh 1 H -1 è - ø çè ø H -1 These approximaions an be found in [1] and []. 1 1 1 Drela, XFOIL 5.7 Modifiaion in 1991 n 0.0345 = 0.08 ( H1-1)- æ 3.87 ö Re - -.5 d ç çèh1-1 ø e 0.43 æ 14 ö æ 1 ö log10 ( Red ),ri = 0.7 anh - 9.4 +.49 + 0.66 çh -1 çh -1 è ø è ø 1 1 Drela, XFOIL 6.8 only a iny modifiaion (erm 0.66 0.6) n 0.0345 = 0.08 ( H1-1)- æ 3.87 ö Re - -.5 d ç çèh1-1 ø e 0.43 æ 14 ö æ 1 ö log10 ( Red ),ri = 0.7 anh - 9.4 +.49 + 0.6 çh -1 çh -1 è ø è ø 1 1 4
Mehod of Arnal: A se of ables produed by D. Arnal has been approximaed by W. Würz wih polynomials: n = a1 + a H1 + a3 H1 Re d ( d ) log Re = b + b H + b H 10,ri 1 1 3 1 Here he envelope is no a sraigh line as in Drela s mehod. For deails see [1]. In JAVAFOIL ransiion is assumed o our when he value nri = 9- r is exeeded. Mehod of Granville This mehod is no desribed here. I also works by inegraing a sabiliy parameer saring from a poin of insabiliy. Abbreviaions: approximaion of n n roughness faor (0 = smooh) r displaemen hikness d 1 momenum hikness d = q shape faor displaemen hikness / momenum hikness H1 d1 = d Reynolds number based on loal momenum hikness Red = Re Reynolds number based on loal ar lengh Re Effe of Roughness The effe of roughness on ransiion and drag is omplex and anno be simulaed auraely. Even modern resoure hungry dire numerial simulaion mehods have diffiulies o simulae he effe. In JAVAFOIL wo effes of surfae roughness are modeled: laminar flow on a rough surfae will be desabilized leading o premaure ransiion, laminar as well as urbulen flow on rough surfaes produe a higher skin friion drag. The effe on oughness is modeled in he following ransiion models Eppler 18.4 H3-1.74-0.36 r Transiion is assumed o our when Red ³ e. Sandard Eppler 18.4 H ( ) Transiion is assumed o our when 3-1.74+ 15 H3-1.573-0.36 r Red ³ e. enhaned Drela, n Transiion is assumed o our when he value nri = 9- r is exeeded. e approx. Arnal (Würz) Transiion is assumed o our when he value nri = 9- r is exeeded. The global effe on drag is aken ino aoun by a simple saling of he oal drag oeffiien C = C ( 1+ r/10) d The roughness faor r is mean o represen he following surfae ondiions r = 0 perfe smooh surfae as for example on a omposie maerial sailplane wing d 5 s q
r = 1 smooh, bu slighly rough surfae as for example a pained loh surfae r = similar o he NACA sandard roughness r = 3 diry surfae wih spos of dir, bugs and flies Noe ha he NACA sandard roughness is usually applied o he leading edge only. I onsiss of a sparse (5-10% of he area) leading edge oaing up o 8% x/. The grain size is abou 0.45 of he hord lengh. Thus for a wing hord lengh of 1m he grain size would be 0.45mm. Sall Correions Empirial Sall Correion #1 ( CalFoil ) if ( a> 0 ) { // handle separaion on upper surfae // drag inremen ( ) ( ) C = C + sin a x - x + 0.05 os a x - x d, upper d, upper TE sep, upper TE sep, upper // lif muliplier redues lif linearly wih lengh of separaed lengh C = C 1-0. x -x } else if ( a< 0 ) { } ( ( )) TE sep, upper // handle separaion on lower surfae // drag inremen ( ) ( ) C = C + sin a x - x + 0.05 osa x - x d, lower d, lower TE sep, lower TE sep, lower // lif muliplier redues lif linearly wih lengh of separaed lengh C = C 1-0. x -x ( ( )) TE sep, lower // momen muliplier C = C 0.9 x x m, orreed m, panel mehod sep, lower sep, upper // lif muliplier due o suion peak rierion 1 C = C, where D CP, max is he differene beween he minimum pressure æd CP, max ö ç + 1 çè 0 ø oeffiien lose o he nose of he airfoil and he pressure lose o he railing edge. Empirial Sall Correion # ( Eppler ) if ( a> 0 ) { // handle separaion on upper surfae 6
if ( xsep, upper < xte ) { // railing edge angle of upper surfae æ ysep, upper y ö - TE q TE = aran - ç xsep, upper x çè - TE ø } else { q TE = 0 } // drag inremen ( ) ( ) C = C + 0. sin a +q x - x d, upper d, upper TE TE sep, upper ( ) ( ) D C = C a +q p x -x l, max, fudge TE TE sep, upper if ( D C > 0 ) { // lif reduion C = C -DC } else { // lif muliplier C = C 1-sina x -x } ( ( )) TE sep, upper // momen inremen C = C -sin a x -x 0.5 1 + x - 0.5 ( ) ( ( ) ) m m TE sep, upper sep, upper } else if ( a< 0 ) { // handle separaion on lower surfae if ( xsep, lower < xte ) { // railing edge angle of lower surfae æ ysep, lower y ö - TE q TE = aran - ç xsep, lower x çè - TE ø } else { q TE = 0 } // drag inremen 7
( ) ( ) C = C -0. sin a +q x - x d, lower d, lower TE TE sep, lower ( ) ( ) D C = C a +q p x -x l, max, fudge TE TE sep, lower if ( D C < 0 ) { // lif reduion C = C -DC } else { // lif muliplier C = C 1-sina x -x } ( ( )) TE sep, lower } // momen inremen ( ) ( ( ) ) C = C -sin a x -x 0.5 1 + x - 0.5 m m TE sep, lower sep, lower // lif muliplier due o modified suion peak rierion 1 C = C, where D CP, max is he differene beween he minimum pressure æd CP, max ö ç + 1 çè 30 ø oeffiien lose o he nose of he airfoil and he pressure lose o he railing edge. Compressible Flow JAVAFOIL analyzes airfoils in inompressible flow, whih means low Mah numbers as hey are ommon in model airraf of general aviaion airplanes. In praial appliaion his means Mah numbers below M = 0.5. I is possible however o exend he Mah number range somewha by applying ompressibiliy orreions o he inompressible resuls. This is only possible, as long as he flow speed is subsoni all over he surfae of he airfoil and ompressibiliy effes are small. Criial Pressure Coeffiien The haraer of he flow hanges dramaially when soni speed is exeeded anywhere on he surfae. The pressure oeffiien assoiaed wih soni speed is alled riial pressure oeffiien ( C p, ri ). In mos ases pressure reovery from supersoni o subsoni speeds (from Cp < Cp, ri o Cp > Cp, ri) is leading o an abrup reompression wih a shok. The analysis of suh flows requires more omplex mehods han implemened in JAVAFOIL. Suh mehods mus be apable of handling ompressible flows (for example by solving he full, ompressible poenial equaions or by solving he Euler equaions). 8
In order o indiae how lose he loal flow is o supersoni speeds, JAVAFOIL alulaes he riial pressure oeffiien if a Mah number is speified on he Opions ard. The riial limi is drawn as a wavy line in he graph on he Veloiy ard. Addiionally, a ompressibiliy orreion is applied o he inompressible soluion o model firs order ompressibiliy effes. Noe however, ha he heory beomes invalid, when flow reahes or exeeds soni speed. In JAVAFOIL, he riial pressure oeffiien is alulaed from he relaion k æ ö çæ k-1 ö çç æ k- ö ç 1 Cp, ri = 1 M 1 + - k M çèç k+ 1 è ç ø ø ø In erms of he veloiy raio he riial limi is found from æ v ö 1+ M = 1+ çv è ø ( k+ 1) M ri Compressibiliy Correions There are differen ways o orre inompressible flow resuls for ompressibiliy effes. One should keep in mind ha hese are only orreions hey an never produe he orre physial effes when he flow loally reahes or exeeds supersoni speed. Therefore he appliabiliy of all ompressibiliy orreions is limied o ases where he loal flow veloiy (whih an be muh higher han he onse flow veloiy) is well beyond he speed of sound. In praial appliaion one an use suh orreions well up o abou M = 0.5, he error grows very rapidly when he onse Mah number exeeds 0.7. In JAVAFOIL, he inompressible panel analysis is always performed for he given airfoil he shape is never geomerially disored. The ompressibiliy orreion is applied laer o he loal surfae pressure aording o he Kármán-Tsien approximaion C P, CPi, 1 1M 1 1 M M M CP, i The orreed pressure oeffiien is hen used o alulae he lif and momen oeffiiens.. Finie Wings in JAVAFOIL In he 190s i has been found by Prandl and also by Lanheser ha he finie span of wings affes heir aerodynami performane. They found ha he effes ould be expressed as a funion of he aspe raio (a.k.a. slenderness or finesse ) of he wing. Prandl s Lifing Line heory was developed and suessfully applied o design wings up o he 1940s and even oday i is useful for unswep wings of relaively high aspe raio ( L> 5 ). The aspe raio an be deermined from L= b/ = b /S (span b divided by he mean hord lengh or span squared divided by wing area S ). The main resul of his heory is ha he airfoil drag is inreased by an addiional drag fore ( indued drag a.k.a. vorex drag ) whih is aused by he finie wing span and he assoiaed wake downwash behind he wing. I is physially unavoidable when a wing produes lif. The vorex drag oeffiien of a wing an be expressed by C D, indued = k C L /( p L ), where C L is he lif oeffiien of he whole wing 9
and k is a faor o aoun for he shape of he lif fore disribuion along he span (for good high aspe raio wing designs and for very low aspe raios k» 1). Linked o he downwash is also a loss of lif a he same angle of aak and a reduion of he pihing momen. Now, JAVAFOIL is a program for he analysis of wo dimensional airfoils. Neverheless i suppors a very simple model of finie wings o allow for a more realisi omparison of airfoils. When he user supplies a value for he aspe raio on he Opions ard lassial wing heory formulas are used o deermine an approximaion of he 3D effes on lif, drag and pihing momen. These effes an applied o he polars produed by JAVAFOIL and make i possible o ge a firs impression of he relaions beween indued drag and airfoil drag. For example he imporane of he airfoil drag is diminishing for higher lif oeffiiens and lower aspe raios. These hree dimensional orreions an also be applied o he resuls for onsan Reynolds number (Polar ard) as well as more realisially for he resuls assoiaed wih a onsan wing loading (Airraf ard). C L 1.5 NACA 001 m/s = 50kg/m, = m/s = 50kg/m, = 5 m/s = 50kg/m, = 10 m/s = 50kg/m, = 0 1.0 0.5 0.0 0.000 0.010 0.00 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 C D Figure 6: Lif versus drag oeffiien polars for a NACA 001 airfoil and wings of differen aspe raio. The graph above shows he effe of he wing aspe raio on lif over drag oeffiien. Saring wih infinie aspe raio (aspe raio = 0 on he Opions ard) hree wings wih inreasing aspe raio have been analyzed. For eah urve he maximum of he lif over drag (L/D) raio is indiaed by a filled irle. I an be learly seen, ha depending on he aspe raio he addiional indued drag disors he polar so ha he opimum L/D raio is shifed o lower lif oeffiiens. While he wo dimensional airfoil ahieves is maximum of L/D a slighly above C 1.0, he low aspe raio wing of 5 requires o operae he airfoil a C 0.5 beause his is he opimum C L of he whole wing. If we ompare wih anoher airfoil we would beer ompare he airfoils a he lif oeffiiens orresponding o he wing aspe raios. Noe ha he resuls as shown above are aurae for a wing having an ellipial lif disribuion and an ellipial, unwised planform. Due o he spanwise lif disribuion on a generi wing, he airfoils along he span of he wing will operae somewha above and below he oal lif oeffiien of he wing. To analyze suh effes requires a more sophisiaed hree dimensional wing analysis ode (e.g. lifing line, vorex laie or panel mehods). Also 30
no addiional wing effes (like Reynolds number variaion due o aper) are aken ino aoun. Polars for Consan Wing Loading Airfoil daa has radiionally been presened in form of graphs and ables for onsan Reynolds numbers. This form resuls from he ypial way wind unnel experimens and numerial analyses are ondued. In a wind unnel i is relaively easy o mainain a onsan wind speed and Reynolds number. Now he lif oeffiien of a real airplane depends on he speed beause he wing loading is usually fixed during fligh flying a low lif oeffiiens resuls in high speeds (and high Reynolds numbers) and vie versa. Therefore he operaing poins during fligh would slie hrough a se of polars having onsan Reynolds numbers. I is possible o reae polars more losely relaed o he ondiions during fligh. This would require adjusing he wind speed o eah lif oeffiien, whih is umbersome and expensive in a wind unnel, bu feasible in a numerial ool like JAVAFOIL. Here you an use he Airraf ard o alulae polars for a given wing loading. Abbreviaions: mass of airraf m kg graviy onsan g m/s densiy of medium r m/s kinemai visosiy n m /s fligh speed v m/s wing area S m hord lengh m Reynolds number Re - Basi Equaions m g The definiion of he lif oeffiien is CL = r. Solving he definiion of he v S v Reynolds number Re = for he veloiy v yields v n ino he definiion of he lif oeffiien produes Solving for he Reynolds number yields C m g =. n S L r Re Re n =. Insering his resul g m Re = n r C S L. g m Noe ha his equaion an also be wrien Re C =, whih means ha we L n r S an also alulae polars of onsan Re C o mah a given airraf. L Using hese resuls one an derive an airraf oriened airfoil polar for a given wing loading m S and given mean hord lengh. Due o he dependeny beween lif oeffiien and Reynolds number an ieraive alulaion proedure is used: 31