İSTANBUL TECHNI CAL UNI VERSI TY INSTI TUTE OF SCI ENCE AND TECHNOLOGY
AERODYNA MI C SHAPE OPTI MI ZATI ON OF ROTOR AI RFOI LS VIA A GENETI C ALGORI THM
M. Sc. Thesi s by A. Tayl an KÖKS AL, B. Sc.
De part me nt : Space Engi neeri ng Progra mme: Space Engi neeri ng
İSTANBUL TECHNI CAL UNI VERSI TY INSTI TUTE OF SCI ENCE AND TECHNOLOGY
M. Sc. Thesi s by A. Tayl an KÖKS AL , B. Sc.
(511001204)
Dat e of sub missi on : 5 May 2003 Dat e of defence exa mi nati on: 30 May 2003
Supervi sor ( Chai r man) : Prof. R. Alsan MERİ Ç Me mbers of t he Exa mi ni ng Co mmi ttee Prof. A. Rüste m ARSLAN
Assi st. Prof. Hayri ACAR
MAY 2003
AERODYNA MI C SHAPE OPTI MI ZATI ON OF ROTOR AI RFOI LS VIA A GENETI C ALGORI THM
İSTANBUL TEKNİ K ÜNİ VERSİ TESİ FEN Bİ Lİ MLERİ ENSTİ TÜS Ü
GENETİ K ALGORİ TMA İ LE ROTOR KANAT PROFİ LLERİ Nİ N AERODİ NAMİ K ŞEKİ L
OPTİ Mİ ZAS YONU
YÜKSEK Lİ SANS TEZİ Müh. A. Tayl an KÖKSAL
(511001204)
Tezi n Enstitüye Veril diği Tari h : 5 Mayı s 2003 Tezi n Savunul duğu Tari h : 30 Mayı s 2003
Tez Danı ş manı : Prof. Dr. R. Alsan Meriç Di ğer Jüri Üyel eri : Prof. Dr. A. Rüste m Arsl an
PREFACE
First of all, I woul d li ke t o t hank t o my super visor Pr of.. R. Al san Meri ç for hi s support and encourage ment t hroughout t his st udy. I’d also li ke t o t hank t o my f a mil y, friends and especi all y t o my bel oved one for t heir e moti onal support i n my ti mes of need.
CONTENTS PREFACE iii LI ST OF TABLES vi LI ST OF FI GURES vii LI ST OF SYMBOLS viii ÖZET i x ABSTRACT x 1. I NTRODUCTI ON 1
1. 1. Introducti on and Rel at ed Wor k 1
2. AI RFOI L GEOMETRY AND AERODYNAMI CS 3
2. 1. Airf oil Geo met r y 3
2. 2. Airf oil Aer odyna mi cs 4
2. 2. 1. Lift, Dr ag and Mome nt on Airf oils 4
2. 2. 2. Airf oil Pressure Distri buti ons 6
2. 3. Kutt a-Jouko ws ki Theore m 8
3. POTENTI AL FLOW 9
3. 1. Pot enti al Fl ow 9
3. 2. Si mpl e Sol uti ons of Lapl ace Equati ons 11
3. 2. 1 Freestrea m Pot enti al 12
3. 2. 2. Source/ Si nk Pot ential 13
3. 2. 3. Poi nt Vort ex Pot enti al 14
4. S MI TH&HESS PANEL METHOD 16
4. 1. Panel Met hods 16
4. 2. Kutt a Conditi on 17
4. 3. Smit h &Hess Panel Met hod 18
4. 3. 1 Introducti on 18
4. 3. 2. I mpl e ment ati on 21
5. OPTI MI ZATI ON AND GENETI C ALGORI THMS 27
5. 1. Opti mizait on Over vie w 27
5. 2. Aif oil Opti mizati on 27
5. 3. Geneti c Al gorit hms 28
5. 3. 4. Struct ure of Genetic Al gorit hms 31
5. 3. 4. 1 Encodi ng 32
5. 3. 4. 2 Initi al Popul ati on 33
5. 3. 4. 3 Eval uati on 34
5. 3. 4. 4 Repr oducti on 34
5. 3. 4. 5 Sel ecti on 34
5. 3. 4. 6 Cr ossover and Mut ati on 35
5. 3. 4. 7 Elitis m 37
5. 3. 5. Micr o Geneti c Al gorit hms 37
5. 3. 6. Para met ers of Geneti c Al gorit hms 38
6. B- SPLI NE CURVES 40
6. 1.Introducti on 40
6. 2. For mul ati on of a cubi c B- Spli ne 40
7. PROBLE M FOR MULATI ON AND RES ULTS 43
7. 1. Appr oach 43
7. 2. Airf oil Surface Represent ati on 45
7. 3. Aer odyna mi c Anal ysis Vali dati on 46
7. 4. Geneti c Al gorit hm Or gani zati on 47
7. 5. Results and Discussion 48
7. 6. Concl usi on 52
REFERENCES 54
LI ST OF TABLES Page No Tabl e 7. 1 Tabl e 7. 2 Tabl e 7. 3
: Best airfoil pr operti es fro m several generati ons and co mpari sons wit h NACA 0012 aer odyna mi c coeffi ci ent s ……….. ... : Aer odyna mi c pr operti es of t he best desi gn ( NACA0012)... : Aer odyna mi c pr operti es of t he best desi gn ( VR7) ...
53 54 56
LI ST OF FI GURES Page No Fi gure 1. 1 Fi gure 1. 2 Fi gure 1. 3 Fi gure 1. 4 Fi gure 2. 1 Fi gure 2. 2 Fi gure 2. 3 Fi gure 2. 4 Fi gure 3. 1 Fi gure 3. 2 Fi gure 3. 3 Fi gure 4. 1 Fi gure 4. 2 Fi gure 4. 3 Fi gure 4. 4 Fi gure 5. 1 Fi gure 6. 1 Fi gure 6. 2 Fi gure 7. 1 Fi gure 7. 2 Fi gure 7. 3 Fi gure 7. 4 Fi gure 7. 5 Fi gure 7. 6 Fi gure 7. 7 Fi gure 7. 8 Fi gure 7. 9 Fi gure 7. 10 Fi gure 7. 11 Fi gure 7. 12 Fi gure 7. 13 Fi gure 7. 14 Fi gure 7. 15
: Aer oacousti c best airfoil ……… : Aer odyna mi c best airfoil ………... : Co mpr o mise airfoil ………...
: Co mpr o mi se airfoil pressure pr ofil es ……… : The basi c parts of an airfoil... : Lift and drag on an airfoil ... : Pressure di stri buti ons on an airfoil... : Generati on of lift... : Unif or m fl ow... : Sour ce fl ow... : Vort ex fl ow... : Traili ng edge vel ociti es ... : Airfoil anal ysi s no menclat ure for panel met hods... : Local Panel Coor di nate Syst e m... : Geo met ri c Int er pret ati on of Sour ce and Vort ex Induced
Vel ociti es ...
: The struct ure of a genetic al gorit hm ...
: B- Spli ne and Cont r ol Poi nt s... : Successi ve B- Spli nes Joi ned Toget her... : Li mit s of Contr ol Poi nts Repr esenti ng Airfoil Surface... : Spli ne contr ol poi nt s and represent ati ve surface of
NACA 0012... : Co mpari son of CL Val ues... : Co mpari son of CM Val ues... : Conver gence Hist or y of t he Best Run...
: An airfoil shape fro m generati on- 0 t hat vi ol at es the CM constrai nt... : The best airfoil i n generati on- 10 ... : The best airfoil from generati on- 50... : The best airfoil from generati on- 200... : Fi nal Desi gn - The best airfoil fro m generati on- 1000 …. ..
: Pressure distri buti ons of t he fi nal airfoil ………... : Fi nal Desi gn - The best airfoil fro m generati on- 1000 …. ..
: Pressure di stri buti ons of t he fi nal airfoil ……….. . : Fi nal Desi gn - The best airfoil fro m generati on- 1000 …. ..
: Pressure distri buti ons of t he fi nal airfoil ………...
3 3 3 3 4 6 8 9 14 l 4 16 18 20 22 24 30 42 43 46 47 48 48 49 50 50 51 51 52 52 54 54 55 56
: Pressure distri buti ons of t he fi nal airfoil ………... 3 3 3 3 4 6 8 9 14 l 4 16 18 20 22 24 30 42 43 46 47 48 48 49 50 50 51 51 52 52 54 54 55 56
LI ST OF SYMBOLS
: Densit y of t he fl ui d
V : Freestrea m vel ocit y c : Chor d l engt h
q : Freestrea m dyna mi c pressure S : Charact eristi c body area
L
C : Lift coeffi ci ent D
C : Dr ag coeffi ci ent M
C : Mo ment coeffi ci ent p : Local pressure
p : Freestrea m pressure P
C : Pressure coeffi ci ent F : Force per unit dept h
: Circul ati on
V : Vel ocit y
u : Vel ocit y i n x directi on v : Vel ocit y i n y directi on
: Vel ocit y pot enti al
n : Unit nor mal
: Strea m f uncti on
: Angl e of att ack
q : Source strengt h
: Vort ex strengt h
,
r : Pol ar coor di nat es of x and y
r : dist ance bet ween panel s
pop
GENETİ K ALGORİ TMA İ LE ROTOR KANAT PROFİ LLERİ Nİ N AERODİ NAMİ K ŞEKİ L OPTİ Mİ ZAS YONU
ÖZET
Bu çalış mada, geneti k al gorit ma met odu, bir heli kopt er kanat pr ofili ni n aer odi na mi k özelli kl eri göz önünde t ut ul arak yapıl acak bir di zayn pr obl e mi ni çöz mek i çi n kull anıl mıştır. Bir mi kro geneti k al gorit ma, aerodi na mi k anali z a macıyl a kull anıl an Smit h&Hess panel met oduyl a bir arada çalıştırıl mıştır. Kull anılan mi kro geneti k al gorit ma modeli “bi nary t our na ment sel ecti on”, “unifor m cr ossover” ve elitiz m yönt e mleri yl e çalış makt adır. Kanat pr ofili ni t e msil eden B-spli ne eğrisi ni n 20 kontrol nokt ası geneti k al gorit ma opti mizasyonu i çi n di zayn para metrel eri ni ol uşt ur makt adır. Bu di zayn para metrel eri her üç uçuş dur umu i çi n kal dır ma katsayısı nı maksi mize et mek a macı yl a opti mize edil miştir. Bu opti mizasyon probl e mi nde kı sıtlar mo ment katsayıları içi n kon muşt ur. Geneti k al gorit manı n bu a maçlara eriş mek a macı il e sürekli ol arak sıradışı kanat pr ofilleri bul duğu sapt anmıştır. Mo ment katsayıları üzeri ndeki kı sıtlar bul unan kanat pr ofili nin NACA 0012 pr ofili nden daha i yi aer odi nami k özelli kl er sergile mesi göz önünde t ut ul arak a maç f onksiyonuna yerl eştiril miştir. Ger çekt ende, bul unan opti mu m pr ofilin NACA 0012 pr ofili nden daha fazl a mo ment ol uşt ur madı ğı ve verilen çeşitli hücu m açıları i çi n daha yüksek kal dır ma sağl adı ğı gözl enmi ştir. Sonuç ol arak, geneti k al gorit ma yönt e mi kanat pr ofili di zaynı nda ü mit veri ci sonuçl ar ort aya çı kar mıştır. Fakat heli kopt er kanat pr ofili di zaynı bir çok farklı mühendi sli k di si pli ni ne bağı mlı ol duğu i çi n, bu alanda disi pli nl er-arası bir şekil opti mizasyonuna ihti yaç vardır.
AERODYNA MI C SHAPE OPTI MI ZATI ON OF ROTOR AI RFOI LS VI A A GENETI C ALGORI THM
ABSTRACT
In t hi s wor k, t he genetic al gorit hm met hod has been used t o s ol ve a heli copt er r ot or airfoil desi gn pr obl e m t hat addr esses aer odyna mi c concer ns. A mi cr o geneti c al gorit h m wi t h bi nar y t our na ment sel ecti on, unif or m cr ossover and wi t h an elitis m operat or i s used i n conj uncti on wi t h Smi t h &Hess panel met hod aer odyna mi c code. The geneti c al gorit hm oper at ed on 20 vari abl es whi ch constit ut ed t he contr ol poi nt s of a B- spli ne cur ve r epresenti ng t he ai rfoil surface. These desi gn vari abl es wer e opti mi zed t o ma xi mi ze lift coeffi ci ent at t hree fli ght conditi ons wi t h a constrai nt on mo me nt coeffi ci ent. It was f ound t hat t he geneti c al gorith m coul d consi st entl y desi gn a non-traditi onal airfoil t o achi eve it s obj ecti ves. The constrai nt s are e mbedded i n t he fit ness functi on i n s uch a way t hat t he r esulti ng desi gns exhi bit charact eristi cs mor e f avor abl e t han t he f a mous NACA 0012 airfoil. I ndeed, it was f ound t hat t he r esulti ng opti mal desi gn has bett er lift coeffi ci ent s t han t he NACA 0012 airfoil at vari ous val ues of angl e of att ack whil e not exceedi ng t he mo ment coeffi cient s of NACA 0012 airf oil i n any fl ow conditi on. As a concl usion, geneti c al gorit hm appears t o be quit e pr o mi si ng i n airfoil shape desi gn, but si nce t he heli copt er r ot or airfoil desi gn i s dependent on t he effect s of vari ous di sci pli nes, an i nter di sci pli nar y desi gn optimi zati on shoul d be applied.
1. I NTRODUCTI ON
1. 1 Introducti on and Rel ated Wor k
Thi s wor k di scusses t he appli cati on of a geneti c al gorit hm t o an airfoil desi gn pr obl e m i n t he consi derati on of aer odyna mi c concer ns.
The Geneti c Al gorit hm is a co mput ati onal version of nat ural sel ecti on and repr oducti on obser ved in bi ol ogi cal popul ati ons [1]. Because an anal ogy can be ma de bet ween sur vi val-of-t he-fittest and opti mizati on, different for ms of the geneti c al gorit hms have been appli ed t o engi neeri ng design and opti mizati on. Recent l y, t hese appli cati ons have also i ncl uded rot orcraft desi gn pr obl e ms.
A geneti c al gorit hm f or t he shape desi gn of airfoils is preferred i n t hi s wor k due t o several disadvantages t hat ot her desi gn met hods suffer whil e t he geneti c al gorit hms don‟t.
Di r ect airfoil desi gn pert ur bs an i niti al airfoil shape t o i mpr ove the perf or mance of t he airfoil. Alt hough successf ul, t his appr oach generally pr oduces airfoils devi ati ng onl y slightl y fro m t he i niti al design. Cal cul us- based search met hods especi all y encount er t his li mit ati on t hey fi nd the nearest l ocal opti mum t o t he ori gi nal desi gn. Airfoil feat ures li ke traili ng-edge tabs, dr oop-snoot s, and co mpl ex ca mber woul d be diffi cult t o di scover usi ng a traditi onal met hod t hat pert ur bs a kno wn shape [2].
In contrast, i nverse airfoil desi gn pr oduces an airfoil whose pressur e di stri buti on mat ches a desired di stri buti on. The i nverse appr oach risks defi ni ng a di stri buti on t hat no physical shape can pr oduce [2].
As one of its central feat ures for engi neeri ng desi gn appli cati ons, t he GA begi ns its search fro m a rando ml y generat ed popul ati on of desi gns. Thi s feat ure all ows opti mal desi gns to be conduct ed wit hout the need of a st arti ng desi gn, whi ch is different fro m most opti mizati on met hods. For rot orcraft airfoil desi gn, t hi s offers gr eat pr o mise si nce most previ ous airfoil desi gn st udi es have been conduct ed usi ng
appli cati on recentl y, most of t hese efforts att e mpt ed t o sol ve an i nverse pr obl e m [4, 5]. Of t hose t hat att empt t he direct pr obl e m [6], many pert ur b an i nitial shape, t hus li miti ng t he chances of fi ndi ng an untraditional airfoil shapes [ 2]. Furt her GA appli cati ons have st art ed t o expl ore multi obj ective and multi disci pli nar y pr obl e ms [2, 7].
By defi ni ng a desi gn space t hat uses t he upper surface and l ower surface coor di nat es as desi gn variabl es, t he GA may fi nd airfoil desi gns t hat i ncl ude feat ures t hat can not be acquired usi ng traditi onal opti mizati on met hods. In fact, t he resulti ng airfoil may be si gnifi cantl y different fro m t hose found usi ng t hese met hods. Ho wever t his appr oach has t he drawback of requiri ng an aerodyna mi c anal ysi s t hat can predi ct t he perf or mance of ver y poor shapes, as well as good shapes.
As a concl usi on, a genetic al gorit hm met hodol ogy is devel oped t o generat e a fa mil y of t wo- di mensi onal airfoil desi gns t hat address aer odyna mi c concerns t o sol ve a heli copt er rot or airfoil desi gn pr obl e m. The geneti c al gorit hm code [8] was used i n conj uncti on wit h a panel met hod code. The genetic al gorit hm operat ed on 20 desi gn vari abl es whi ch constit uted t he contr ol poi nt s for a spli ne representi ng t he airfoil surface.
A ver y si mil ar appr oach t o rot orcraft airfoil design pr obl e ms has been t aken pr evi ousl y [2]. In t hi s wor k a parall el geneti c al gorit hm met hodol ogy was devel oped t o generat e a fa mil y of t wo- di mensi onal airfoil desi gns t hat address rot orcraft aer odyna mi c and aer oacousti c concer ns [2]. The multi pl e obj ecti ves of t hi s wor k wer e t o mi ni mize t he drag and overall noi se of t he airfoil. Constrai nt s wer e pl aced on lift coeffi ci ent, mo me nt coeffi ci ent, and boundar y l ayer conver gence. The aer odyna mi c anal ysi s code XFOI L pr ovi ded pressure and shear di stributi ons i n additi on t o lift and drag predi cti ons. The aer oacousti c anal ysi s code, WOP WOP, pr ovi ded t hi ckness and loadi ng noi se predi cti ons. The fi nal set of desi gns gener at ed by t hi s met hodol ogy is sho wn i n Fi gure 1. 1, Fi gure1. 2 and Fi gure 1. 3. The pressur e di stri buti ons of t he fi nal co mpr o mi se airfoil for thr ee fi ght conditi ons are present ed i n Fi gure 1. 4.
Fi gur e 1. 1 Aer oacousti c best airfoil
Fi gur e 1. 2 Aer odyna mi c best airfoil
Fi gur e 1. 3 Co mpr o mi se airfoil
Fi gure 1. 4 Co mpr o mise airfoil pressure profiles : ◊, fl ow conditi on 1; □, fl ow conditi on 2 ; ○, fl ow conditi on 3.
2. AI RFOI L GEOMETRY AND AERODYNAMI CS
2. 1 Ai rf oil Geo met ry
Ai rf oil geo met r y can be charact eri zed by t he coor di nat es of t he upper and l ower surface. It is often su mmari zed by a few para met ers such as: ma xi mu m t hi ckness, maxi mu m ca mber, positi on of max t hi ckness, positi on of max camber, and nose radi us. One can generat e a reasonabl e airfoil secti on gi ven t hese paramet ers.
The basi c parts of an airfoil are shown i n Fi gure 2. 1, wit h t heir defi nitions bel ow.
Fi gur e 2. 1 The basi c parts of an airfoil
The chord li ne is a straight li ne connecti ng t he leadi ng and traili ng edges of t he airfoil.
The chord is t he l engt h of t he chor d li ne fro m l eadi ng edge t o traili ng edge and is t he charact eristi c longit udi nal di mensi on of t he airfoil.
The mean camber li ne is a li ne dra wn half way bet ween t he upper and l ower surfaces. The chor d li ne connect s t he ends of t he mean ca mber li ne.
The shape of t he mean ca mber is i mport ant i n deter mi ni ng t he aer odyna mi c charact eristi cs of an airfoil secti on. Maxi mu m ca mber (di spl ace ment of t he
mean ca mber li ne from the chor d li ne) and t he l ocati on of maxi mu m ca mber hel p t o defi ne t he shape of t he mean ca mber li ne. These quantiti es ar e expr essed as fracti ons or percent ages of t he basi c chor d di mensi on.
Thi ckness and t hi ckness di stri buti on of t he pr ofil e are i mport ant pr operti es of an airfoil secti on. The maxi mu m t hi ckness and its l ocati on hel p defi ne t he airfoil shape and are expressed as a percent age of the chor d.
The l eadi ng edge radi us of t he airfoil is t he radi us of cur vat ure gi ven t he leadi ng edge shape.
2. 2 Ai rf oil Aerodyna mi cs
2. 2. 1 Lift, Drag and Mome nt On Ai rf oils
The aer odyna mi c forces and mo ment s acti ng on an airfoil are due t o t wo basi c sources :
Pr essure distri buti on over t he body surface
Shear stress distri buti on over t he body surface
The net effect of t he pressure and shear stress di stri buti ons i nt egrat ed over t he co mpl et e body surface is a result ant aer odyna mic force R and mo ment M on t he body. [ 9]
An airfoil' s aer odyna mic force may be separat ed i nt o lift and drag co mponent s. Thi s force i nt ersect s wit h its chord li ne at a poi nt desi gnat ed as its cent er of pressure. The lift, drag, and cent er of pressure for a ca mber ed airfoil var y as its angl e of att ack is changed. No aer odyna mi c mo me nt s (t he t endency of an airfoil t o t ur n about its cent er of gravit y) are present at t he cent er of pressure because t he li ne of acti on of t he aerodyna mi c force passes t hr ough t hi s poi nt. If one has t he airfoil mount ed at so me fi xed poi nt al ong t he chord, for exa mpl e, a quart er of a chor d lengt h behi nd t he l eadi ng edge, t he mo ment is not zer o unl ess t he result ant aer odyna mi c force is zero or t he poi nt corresponds t o t he cent er of pressur e. The mo me nt about t he quart er-chor d poi nt is generall y a functi on of t he angl e of att ack.
Fi gur e 2. 2 Lift and drag on an airfoil
In t he fl ow of real fl ui d about a body, t he aer odyna mi c resist ance depends on t he si ze, shape, and attitude of t he body (its directi on wit h respect t o t he airfl ow— angl e of att ack); t he pr operti es of t he fl ui d, e. g., its densit y and pressure; and t he rel ati ve vel ocit y bet ween t he body and t he fl ui d (air). To ill ustrat e, consi der t he lift force, defi ned as t he aerodyna mi c reacti on perpendi cul ar t o t he directi on of t he airfl ow. Lift depends on si ze, shape, attit ude, fl ui d pr operti es, and vel ocit y. For an i deal fl ui d, t he fl ui d pr operti es (except for densit y) do not i nfl uence t he lift force. For a real fl ui d, however, viscosit y, el asti cit y (t he reci pr ocal of co mpr essi bilit y), and t ur bul ent pr operti es are al so i mport ant. In addition t o t he shape and attitude of t he body, t he surface roughness has an effect on t he force. Furt her mor e, t he effect s of attit ude and shape of a body are l umped t oget her i nt o t he fact or call ed K. Then,
K S V
Lift 2 (2. 1)
Si nce we are i nt erest ed in t wo- di mensi onal shapes, we shoul d repl ace t he ar ea S, wit h t he chor d l engt h , c.
c
S (2. 2)
The dyna mi c pressure of airfl ow was previ ousl y defi ned as 2
2 1 V q , so
if a val ue of 1/ 2 is i ncl uded i n equati on (2. 1) and t he val ue of K is doubled t o keep t he equati on t he sa me, 2K may be repl aced by CL. Fi nall y ;
c q C
L
C is kno wn as t he coeffici ent of lift. The equati on st at es si mpl y t hat
aer odyna mi c lift is det ermi ned by a coeffi ci ent of lift ti mes t he free-stream dyna mi c pr essure ,q , ti mes t he charact eristi c body area S.
c q Lift CL (2. 4)
The aer odyna mi c drag is t he aer odyna mi c resist ance parall el t o t he free-strea m directi on (t he directi on of t he airfl ow). Simi l ar equati ons can be obt ai ned f or t he drag coeffi ci ent, na mel y ,
c q Drag CD (2. 5)
The mo ment acti ng on a body is a measure of the body' s t endency t o tur n about its cent er of gravity. Thi s mo ment represents t he result ant aer odynami c f orce ti mes a mo ment dist ance. A si mil ar deri vati on may be appli ed t o the mo me nt equati on as used for t he lift and drag equati ons such t hat,
2 c q Moment CM (2. 6)
wher e C is t he coeffi ci ent of mo ment and “ Mo me nt ” is t he measured mo ment per M
unit lengt h acti ng on the airfoil ( whet her at the quart er-chor d poi nt or at t he aer odyna mi c cent er or any ot her poi nt desired).
2. 2. 2 Ai rf oil Press ure Distri buti ons
The aer odyna mi c perf orma nce of airfoil secti ons can be st udi ed by reference t o t he di stri buti on of pressure over t he airfoil. This distri buti on is usuall y expr essed i n t er ms of t he pressure coeffi ci ent:
q p p CP (2. 7)
P
C is t he difference bet ween l ocal st ati c pressure and freestrea m st ati c
pr essure, nondi mensi onalized by t he freestrea m dyna mi c pressure. To see airf oil pr essure di stri buti on C is pl ott ed versus P x /c whi ch vari es fro m 0 at t he l eadi ng edge t o 1. 0 at t he traili ng edge. C is pl ott ed "upside- do wn" wit h negati ve val ues P hi gher on t he pl ot so t hat t he upper surface of a conventi onal lifting airfoil corresponds t o t he upper cur ve. Vari ous parts of t he pressure di stributi on are depi ct ed i n Fi gure 2. 3.
Fi gur e 2. 3 Pressure di stri buti ons on an airfoil
Whe n t he chor d is t aken as 1 unit, t he secti on lift coeffi ci ent is rel at ed t o t he Cp by:
dx C C CL
pl pu 1 0 (2. 8)In ot her wor ds it is t he area bet ween t he cur ves of upper and l ower surface pr essure coeffi ci ent s.
2. 3 Kutt a- Jouko ws ki Theore m
An i sol at ed t wo- di mensional airfoil i n an i nco mpressi bl e i nvi sci d fl ow feels a force per unit dept h of
V
F (2. 9)
Si nce t he lift and drag forces are defi ned as t he forces i n t he directi ons normal and parall el t o t he free strea m, a direct consequence is t hat
Fi gur e 2. 4 Generati on of lift
V L (2. 10) 0 D
The Kutt a-Jouko ws ki t heore m i s si mpl y an alt ernat e way of expressi ng t he consequences of t he surface pressure di stri buti on si nce lift is caused by t he net i mbal ance of t he surface pressure di stri buti on, and circul ati on is si mpl y a defi ned quantit y det er mi ned fro m t he sa me pressures.
3. POTENTI AL FLOW
3. 1 Pot enti al Fl ow
For i nco mpr essi bl e fl ows, t he conti nuit y equati on, t hat is t he conser vati on of mass is gi ven by;
0
V (3. 1)
An i nvi sci d, i nco mpr essibl e fl ui d is also call ed an i deal fl ui d, or perfect fl ui d. In a vel ocit y fi el d, t he curl of t he vel ocit y defi nes t he vorti cit y. Si nce an i deal i nco mpr essi bl e fl ui d with zer o vorti cit y at an i nstant will never generat e vorti cit y, it is mat he mati call y consi stent t o consi der a fl ui d wit h
0
V (3. 2)
ever ywher e and at all ti mes. Such fl ui d fl ows are call ed irrot ati onal flo ws. Thi s i mpli es t hat t he fl ui d el eme nt s have no angul ar vel ocit y, t heir moti on t hr ough space is a pure transl ati on. [9] The irrot ati onalit y conditi on for a t wo di mensi onal fl ow i s ;
0 y u x v (3. 3)
These fl ows are oft en called pot enti al fl ows also, because t he vel ocit y f or an irrot ati onal fl ow can be writt en as t he gradi ent of a scal ar pot enti al;
0
Thi s equati on st at es t hat for an irrot ati onal fl ow, there exi sts a scal ar function
such t hat t he vel ocit y given is gi ven by t he gradient of whi ch is denot ed as t he vel ocit y pot enti al.
For t wo di mensi onal flo ws, t he vel ocit y co mponent s , u and v can be expr essed as,
y v x u , (3. 5)
Substit uti ng t he vel ocity co mponent s, u and v, for t wo di mensi onal fl ows, t he equati ons for conti nuit y and irrot ati onalit y woul d t hen beco me,
0
2
(3. 6)
In t he above equati on , t he operat or,
2 2 2 2 2 y x (3. 7)
is ter med t he Lapl aci an operat or, aft er Pi erre de Lapl ace (1749- 1827).
Si nce t he nor mal co mponent of t he fl ui d vel ocit y has t o be zer o at a soli d surface at rest, t he appr opri at e boundar y conditi on is
0
n (3. 8)
on a st ati c soli d boundary wit h unit nor mal n .
Consequentl y, t he Lapl ace equati on is t he gover ni ng equati on for t he sol uti on of t he pr obl e ms of t his i nvi sci d, i nco mpr essi ble fl ui d. The assu mpti on made i n sol vi ng Lapl ace‟s equation is t hat t he fl ows satisfy t he equati ons for conti nuit y and irrot ati onalit y.
3. 2 Si mpl e Sol uti ons of Lapl ace Equati ons
Pot enti al Fl ow i s an i deali zed met hod of modeling fl ow. If a fl ow is i nvi sci d and i nco mpr essi bl e t hen a vel ocit y pot enti al functi on can be defi ned such that
y v x u , (3. 9)
Substit uti ng t hese expressi ons for hori zont al (u ), and verti cal ( v ) vel ocit y i nt o t he
gover ni ng conti nuit y equati on for a fl ow pr oduces a Lapl ace equati on t he sol uti on of whi ch is rel ati vel y si mpl e.
0 x v x u beco mes 2 0 2 2 2 y x (3. 10)
In t wo- di mensi onal fl ow, strea m f uncti on ( ) can be defi ned as a measure of t he vol u me fl ow rat e of fl uid bet ween a pair of streaml i nes. Strea mli nes are defi ned by j oi ni ng a conti nuous li ne of poi nt s i n t he fl ow fiel d by foll owi ng t he l ocal vel ocit y vect ors.
St rea mli nes have a constant val ue of strea m f uncti on si nce all t he fl ow must be parall el t o t he strea mli nes. No fl ow cr osses a strea mli ne. For t wo- dime nsi onal, i nvi sci d, i nco mpr essi bl e fl ow, conti nuit y makes t he l ocal pr oduct of di st ance bet ween strea mli nes and vel ocit y a const ant. Thus t he vel ociti es i n a fl ow fi el d can al so be found by differenti ati ng strea m f uncti on wit h respect t o t he fl ow fi el d coor di nat es y and x.
u y and v x (3. 11)
If t he t wo di mensi onal flo w i s irrot ati onal (0) t hen strea m f uncti on can al so be used t o defi ne a gover ni ng Lapl ace equati on.
0 x v y u beco mes 2 0 2 2 2 y x (3. 12)
Vel ocit y pot enti al () and strea m f uncti on ( ) are ort hogonal functi ons defi ni ng i deal, i nvi sci d i nco mpr essi bl e and irrot ational fl ow i n t wo di mensions.
Si mpl e strea m f uncti ons or vel ocit y pot enti al functi ons can be found whi ch are exact sol uti ons for t he above Lapl ace equati ons.
One way t o obt ai n sol utions t o Lapl ace‟s equati on (subj ect t o t he appr opriat e boundar y conditi ons) is to expl oit its li near nat ure and t he pri nci pl e of superpositi on.
Superpositi on
If 1,2,3,4 is each a sol uti on of , 2 0
t hen A1B2C3 Xn is also a sol uti on of 2 0, wher e A, B ... X are const ant s.
Thus, t he sol uti on for a co mpl ex pr obl e m can be expressed as t he sum of sol uti ons of several si mpler pr obl e ms.
Obvi ousl y, t he pri nci pl e of super positi on can be used t o add up an ar bitrar y nu mber of el e ment ary sol uti ons t o Lapl ace‟s equati on. Thi s met hod is rat her po werf ul and will be used i n t he re mai nder of t his lect ure t o construct t he sol uti on of t he fl ow about an ar bitraril y shaped lifti ng airfoil.
Ther e are t hree t ypes of el e ment ar y sol uti ons. Wi t h a l arge nu mber of each of t hese t hree ki nds of soluti ons we will be abl e t o construct t he fl ow about rat her general airfoils at ar bitrary angl es of att ack.
3. 2. 1 Free Strea m Pot enti al
The pot enti al functi on and strea m f uncti on for a free stream of magnit udeV ,
ali gned wit h t he x- axi s is gi ven by
x V
Vy (3. 13)
Taki ng t he gradi ent of thi s pot enti al we see t hat t he resulti ng vel ocit y fiel d is gi ven by
V y x u( , ) (3. 14a) 0 ) , (x y v (3. 14b)
That is, t he vel ocit y is unifor m ever ywher e i n t he do mai n. The pot enti al functi on can be rot at ed at an ar bitrar y angl e so t hat
) sin (cos x y V . (3. 15) Fi gur e 3. 1 Unif or m fl ow
3. 2. 2 Source/ Si nk Pot enti al
A sour ce/ si nk t hat expel s/absor bs an a mount of flui d vol u me/ unit ti me qcan be construct ed fro m t he foll owi ng pot enti al ;
) ln( 2 r q S (3. 16)
and t he strea m f uncti on ;
2 q S (3. 17)
wher e x r.cos() y r.sin() and q is source strengt h.
The resulti ng vel ocit y component s are; 2 2 2 ) , ( y x x q y x u (3. 18a) 2 2 2 ) , ( y x y q y x v (3. 18b)
3. 2. 3. Poi nt Vortex Pot enti al
2 V ln( ) 2 r V (3. 19)
wher e is t he circul ati on strengt h of t he vort ex. is defi ned positi ve if the i nduced circul ar fl ow i s cl ock wi se. is t he angl e meas ured i n pol ar coor di nat es
from so me ar bitrar y ori gi n radi al li ne. Taki ng t he gradi ent of t his function, we see t hat t he vel ocit y fi el d of a vort ex is gi ven by
2 2 2 ) , ( y x y y x u (3. 20a) 2 2 2 ) , ( y x x y x v (3. 20a)
and has strea mli nes t hat are concentri c circl es cent ered about t he l ocati on of t he poi nt vort ex. The circul ati on ar ound any cont our that encl oses t he poi nt vort ex is const ant and equal t o . Furt her mor e, t he fl ow out si de of t he poi nt vort ex i s full y irrot ati onal. All of t he vorti cit y i n t hi s fl ow i s cont ai ned at t he si ngul ar l ocati on of t he poi nt vortex. Noti ce also t hat t he for m of t he pot enti al for a poi nt vort ex is rat her si mil ar t o t hat of a source/ si nk, wit h t he substit uti ons x y, and y -x i n t he appr opri ate for mul a for t he vel ocity fi el d: t he vel ocit y fi elds are
Fi gur e 3. 3 Vort ex Fl ow
A wi de range of fl ows and fl ow ar ound obj ect s can be built up usi ng t hese si mpl e sol uti ons. The more co mpl ex pot enti al fl ows can be construct ed by t he super-positi on of exact sol uti ons or nu meri cal appr oxi mati on t echni ques can be used wit h t his gover ni ng equati on to sol ve a l ar ge range of aer odyna mi c pr obl e ms.
4. S MI TH&HESS PANEL METHOD
4. 1 Panel Met hods
Panel met hods are wi dely used i n t he aer ospace and aut o moti ve i ndustr y and are effecti vel y boundary-el e ment met hods for co mput ati onal fl ui d dyna mi cs pr obl e ms. These met hods e mpl oy t he surface of the body over whi ch fl ui d is fl owi ng t o be used as t he co mput ati onal do mai n rat her t han usi ng t he whol e regi on i n whi ch t he body is e mbedded. Thi s is not onl y co mput ationall y mor e effi ci ent t han a fi nit e difference met hod, for exa mpl e, but also all ows mor e co mpli cat ed body shapes t o be st udi ed t han woul d be tract abl e if t he body wer e embedded i n a regul ar mesh.
Panel met hods have been ext ensi vel y i nvesti gat ed and successf ull y appli ed i n t he aer onauti c i ndustry for mor e t han 30 years, and at present t here are a nu mber of pr oducti on codes avail able t hat perf or m co mput ati ons of fl ows ar ound very co mpl ex geo met ri es. The met hod is based on t he for mul ation of li near pot enti al fl ow ar ound a soli d body i n t er ms of an i nt egral equati on over t he surface for t he vel ocit y or pot enti al. The surface int egral is appr oxi mat ed by panel el e ment s on whi ch a si ngul arit y di stri buti on is assu med t o exi st. The nu meri cal sol uti on of t he pot enti al fl ow i s rendered i nt o a mat ri x li near syst e m, where t he si ngul arit y di stri buti ons over t he panel s are t he unknowns. The attracti veness of t hi s for mul ati on li es in t he fact t hat t he sol uti on can be obt ai ned fro m t he surface pr obl e m, and it is not necessar y t o model t he co mpl et e t hree- di mensi onal fl ow fi el d. Thus, panel met hods reduce t he di mensi on of t he pr obl em, fro m t hree t o t wo dime nsi ons, or fro m t wo di mensi ons t o a one- di mensi onal pr obl e m. Ot her advant ages are t he capabilit y of obt aini ng fl ow sol uti ons about co mpl et el y ar bitrar y confi gurati ons and t he fact t hat t he pr edi cti ons by panel met hods have proven t o agree well wit h experi ment i n a wi de range of fl ow conditi ons [10].
Ge ner al aspect s rel ati ng t o t he i mpl e ment ati on of panel met hods wer e di scussed, cat egori zi ng the m i n four di sti nct pr ocedur es, as foll ows:
1. Geo met r y defi niti on and generati on of t he surface panel el e ment s.
2. Cal cul ati on of t he coeffi ci ent matri x (vel ocity i nduced by t he panel s on each ot her‟s control poi nts).
3. Sol uti on of t he resulti ng syst e m of li near equations for t he si ngul arit y strengt hs. 4. Cal cul ati on of t he fl ow fi el d para met ers of i nterest.
One of t he i mport ant advant ages of panel met hods is t heir abilit y t o analyze t he fl ow around compl ex confi gurati ons or “arbitrary bodi es”, as is co mmonl y expr essed. Ot her co mput ati onal met hods for fl ui d dyna mi cs, e. g., fi nit e el eme nt s and fi nit e vol u me, require a gri d on t he fl ui d do mai n whi ch is not easil y adapt ed t o co mpl ex body geo met ries. Consequentl y, panel met hods have found massi ve appli cati ons i n i ndustry, and practi call y all maj or aer onauti c co mpani es have a pr opri et ar y panel code or have cust o mi zed a public- do mai n panel met hod. [11]
The mai n i nt erest is t he possi bilit y of si mul ating fl ows ar ound co mpl et e aircraft, or si gnifi cant parts of an aircraft.
4. 2 Kutt a Condi ti on
Kutt a conditi on st at es t hat t he pressure above and bel ow t he airf oil trailing edge must be equal, and t hat t he fl ow must s moot hl y l eave t he traili ng edge i n t he sa me directi on at t he upper and l ower edge. Consider t he fi gure bel ow:
Fr o m t hi s sket ch we see t hat pressure will be equal, and t he fl ow will l eave t he traili ng edge s moot hly, if, and onl y if
upper upperV
lowerVlower (4. 1)
4. 3 Smit h &Hess Panel Met hod
4. 3. 1 Introducti on
Ther e are many choi ces as t o how t o for mul at e a panel met hod (si ngul arit y sol uti ons, vari ati on wit hi n a panel, si ngul arit y strengt h and di stri buti on, et c.) The si mpl est and first trul y practi cal met hod was due t o Hess and Smit h, Dougl as Ai r craft, 1966. The Hess- Smit h t echni que co mbi nes source panel s and vortices for a si ngl e-el e ment, lifti ng airfoil i n i nco mpr essi bl e flo w. It is based on a di stri buti on of sour ces and vorti ces on the surface of t he geo met ry. In t heir met hod
V
S
(4. 2)
wher e, is t he t ot al pot ential functi on and its t hree co mponent s are t he pot enti al s
correspondi ng t o t he free strea m, t he source di stributi on, and t he vort ex distri buti on. These l ast t wo di stri buti ons have pot enti all y l ocally var yi ng strengt hs q(s) and (s), wher e s is an arc-l engt h coor di nat e whi ch spans the co mpl et e surface of t he airf oil i n any way desired.
The pot enti als creat ed by t he di stri buti on of sources/ si nks and vorti ces are gi ven by:
q s rds s ln 2 ) ( ) ( (4. 3)
s ds v 2 ) ( ) ( (4. 4)wher e t he vari ous quantities are defi ned i n t he Figur e bel ow, wher e s is the di st ance measured al ong t he surface and (r,)are pol ar coor di nates of t he „fiel d poi nt‟ (x,y),
Fi gur e 4. 2 : Airfoil Anal ysi s No mencl at ure for Panel Met hods
In t hese for mul a, t he int egrati on is t o be carried out al ong t he co mpl et e surface of t he airfoil. Usi ng t he super positi on pri nci pl e, any such di stributi on of sources/si nks and vortices satisfies Lapl ace‟s equati on, but we will need t o fi nd conditi ons for q(s)and (s)such that t he fl ow t angency boundar y conditi on and t he Kutt a conditi on are satisfied.
In t heor y, we coul d:
Us e t he source strengt h di stri buti on t o satisfy fl ow t angency and t he vortex di stri buti on t o satisfy t he Kutt a conditi on.
Us e ar bitrar y co mbi nati ons of bot h sources/si nks and vortices t o satisf y bot h boundar y conditi ons si mult aneousl y.
Hess and Smit h made the foll owi ng vali d si mplifi cati on; t ake t he vort ex strengt h t o be const ant over t he whol e airfoil and use t he Kutt a conditi on t o fi x its val ue, whil e all owi ng t he source strengt h t o var y fro m panel t o panel so t hat, t oget her wi t h t he const ant vort ex di stri buti on, t he fl ow tangency boundar y conditi on is satisfi ed ever ywher e.
In or der t o sol ve t he equati on, consi sti ng chall engi ng i nt egral s t o eval uat e, si mplifi cati on is done by sel ecti ng a nu mber of point s, N, on t he body cont our, call ed nodes. The nodes are t hen connect ed wit h strai ght li nes, whi ch beco me t he panel s, as sho wn i n fi gure 4. 2. Fi gure 4. 2 ill ustrat es t he represent ati on of a s moot h surface by a seri es of li ne seg ment s. The nu mberi ng syst e m st arts at t he l ower surface traili ng edge and pr oceeds forwar d, ar ound t he l eadi ng edge and rear war d t o t he upper surface traili ng edge. N+1 poi nt s are used t o defi ne N panel s.
Fi gur e 4. 2 Defi niti on of Nodes and Panel s
we can di screti ze Equation 4. 2 i n t he foll owi ng way:
ds r s q y x V N j panelj
2 ln 2 ) ( ) sin (cos 1 (4. 5)Si nce Equati on 4. 5 i nvol ves i nt egrati ons over each di scret e panel on t he surface of t he airfoil, t he vari ati on of source and vort ex strengt h wit hi n each of t he panel s must be para met eri zed. Si nce t he vort ex strengt h was consi dered t o be a const ant, t he att enti on shoul d be on t he source strengt h di stri buti on wit hi n each panel. Thi s is t he maj or appr oxi mati on of t he panel met hod. Ho wever, it can be seen ho w t he i mport ance of t his appr oxi mati on shoul d decrease as t he nu mber of panel s,N (of course t his will i ncrease t he cost of t he co mput ati on consi derabl y, so t here are mor e effi ci ent alt er nati ves.)
Hess and Smit h deci ded t o t ake t he si mpl est possi bl e appr oxi mati on, t hat is, t o t ake t he source strengt h t o be const ant on each of t he panel s
N i i panel on i q s q( ) ( ) , 1,..., (4. 6) Ther ef ore, we have N + 1 unkno wns t o sol ve for i n our pr obl e m: t he N panel sour ce strengt hs q and t he const ant vort ex strengt h i . Consequentl y, we will need
N + 1 i ndependent equati ons whi ch can be obt ai ned by for mul ati ng t he fl ow
tangency boundar y conditi on at each of t he N panel s, and by enf orci ng t he Kutt a conditi on. The sol uti on of t he pr obl e m will require t he i nversi on of a matri x of si ze
) 1 ( ) 1 (N N .
In t he Smit h &Hess met hod, t he fl ow t angency boundar y conditi on is i mposed on t he poi nt s l ocat ed at t he mi dpoi nt of each of t he panel s. Alt hough t his appr oach
used f or t he i mpositi on of t he Kutt a conditi on (on t he l ast panel s on upper and l ower surfaces of t he airfoil, assu mi ng t hat t heir midpoi nt s re mai n at equal di st ances fr o m t he traili ng edge as t he nu mber of panel s is i ncreased).
4. 3. 2 I mpl e me nt ati on
Consi der t he it h panel t o be l ocat ed bet ween t he it h nodes, wit h its ori ent ati on t o t he
x-axi s gi ven by i i i i l y y 1 sin (4. 7) i i i i l x x 1 cos (4. 8)
wher e l is t he l engt h of the panel under consi derati on. The nor mal and tangenti al i
vect ors t o t his panel, are t hen gi ven by
j i nˆi siniˆcosiˆ (4. 9) j i tˆi cosiˆsiniˆ (4. 10)
The t angenti al vect or is ori ent ed i n t he directi on fro m node i t o node i+1, whil e t he nor mal vect or, if t he airfoil is traversed cl ock wi se, poi nt s i nt o t he fl ui d.
Letti ng t he it h panel be defi ned as t he one bet ween t he it h and (i +1)t h nodes, and its i ncli nati on t o t he x- axi s be , as shown i n Figur e 4. 3.
Furt her mor e, t he coor di nat es of t he mi dpoi nt of t he panel are gi ven by 2 1 i i i x x x (4. 11) 2 1 i i i y y y (4. 12)
and t he vel ocit y co mponent s at t hese mi dpoi nt s are gi ven by
) , ( i i i u x y u (4. 13) ) , ( i i i v x y v (4. 14)
The fl ow t angency boundar y conditi on can t hen be si mpl y writt en as(u.n)0, or, for each panel
0 cos
sin
ui i vi i f or i =1, ……, N (4. 15)
whil e t he Kutt a conditi on is si mpl y gi ven by
N N N N v u v
u1sin1 1cos1 sin cos (4. 16)
wher e t he negati ve si gns are due t o t he fact t hat t he t angenti al vect ors at t he first and last panel s have nearl y opposit e directi ons.
No w, t he vel ocit y at the mi dpoi nt of each panel can be co mput ed by super positi on of t he cont ri buti ons of all sources and vorti ces l ocat ed at t he midpoi nt of every panel (i ncl udi ng itself). Si nce t he vel ocit y i nduced by t he source or vort ex on a panel is pr oporti onal t o t he source or vort ex strengt h i n t hat panel, q and i can
be pull ed out of t he i nt egral i n Equati on 4. 5 t o yi eld
N j vij N j sij j i V q u u u 1 1 cos (4. 17)
N j vij N j sij j i V q u u v 1 1 sin (4. 18)perf or m t he i nt egral s i n Equati on 4. 5 by noti ci ng t hat t he l ocal vel ocit y co mponent s can be expanded i nt o absol ut e ones accor di ng t o the f oll owi ng transf or mation:
j j v u u *cos *sin (4. 19) j j v u v *sin *cos (4. 20)
No w, t he l ocal vel ocit y co mponent s at t he mi dpoi nt of t he it h panel due t o a unit-strengt h source di stri bution on t hi s jt h panel can be writt en as
dt y t x t x u j l sij
0 2 * 2 * * * ) ( 2 1 (4. 21) dt y t x y v j l sij
0 2 * 2 * * * ) ( 2 1 (4. 22)wher e (x*,y*)are t he coor di nat es of t he mi dpoi nt of panel i i n t he l ocal coor di nat e syst e m of panel j. Carr ying out t he i nt egrals i n Equati on 4. 21 we fi nd t hat
t lj t sij x t y u 2 0 1 2 * * * ln 2 1 (4. 23) j l t t sij t x y v 1 * * 0 * tan 2 1 (4. 24)These results have a si mpl e geo met ri c i nt er pret ation t hat can be di scer ned by l ooki ng at t he Fi gure 4. 4. One can say t hat
j l t t ij ij sij r r u* ln 1 0 2 1 (4. 25) 2 2 0 * l ij sij v v v (4. 26)
ij
r is t he di st ance fro m t he mi dpoi nt of panel i t o t he jt h node, whil e ij is t he
angl e subt ended by t he jth panel at t he mi dpoi nt of panel i. Noti ce t hat u*sii 0, but
t he val ue of v*sii0is not so cl ear. When t he poi nt of i nt erest appr oaches the
mi dpoi nt of t he panel fro m t he out si de of t he airfoil, t his angl e, ii . Ho wever, when t he mi dpoi nt of t he panel is appr oached fro m t he i nsi de of the airfoil,
ii . Si nce we are i nt erest ed i n t he fl ow out si de of t he airfoil onl y, we will al ways t ake ii .
Si mil arl y, for t he vel ocit y fi el d i nduced by t he vort ex on panel j at t he mi dpoi nt of panel i we can si mpl y see t hat
( ) 2 2 1 0 2 * 2 * * * ij l sij dt y t x t x u j
(4. 27) ij ij l sij r r dt y t x y v j 1 ln 2 1 ) ( 2 1 0 2 * 2 * * *
(4. 28)and fi nall y, t he fl ow t angency boundar y conditi on, usi ng Equati on 4. 17 and 4. 18 , and undoi ng t he l ocal coor di nat e transf or mati on of Equati on 4. 19 and 4. 20 can be writt en as i N j iN j ijq A b A
1 1 (4. 29) wher e i sij i sij ij u v A sin cos (4. 30) whi ch yi el ds ij j i ij ij j i ij r r A sin( )ln 1 cos( ) 2 (4. 32)Si mil arl y for t he vort ex strengt h coeffi ci ent
ij
N r
A
cos( )ln 1 sin( )
The ri ght hand si de of t his matri x equati on is gi ven by ) sin( i i V b (4. 34)
The fl ow t angency boundar y conditi on gi ves us N equati ons. We need an additi onal one pr ovi ded by t he Kut ta conditi on i n or der t o obt ai n a syst e m t hat can be sol ved. Accor di ng t o Equati on 4 1 ` , 1 2 , 1
j N N N N j j N q A b A (4. 35)Aft er si mil ar mani pul ati ons we fi nd t hat
) cos( ) cos( 1 1 N N V V b (4. 36)
These vari ous expressi ons set up a matri x pr obl e m of t he ki nd ;
b Ax
wher e t he matri x A is of si ze (N1)(N1). Thi s syst e m can be sket ched as foll ows :
Noti ce t hat t he cost of i nversi on of a full matri x such as t his one isO(N1)3, so t hat, as t he nu mber of panel s i ncreases wit hout bounds, t he cost of sol vi ng t he panel pr obl e m i ncreases rapi dl y. Thi s is usuall y not a pr obl e m f or t wo- dime nsi onal fl ows, but beco mes a seri ous pr obl e m i n t hree- dime nsi onal fl ows where the nu mber of panel s are much hi gher.
Fi nall y, once we have sol ved t he syst e m f or t he unkno wns of t he pr obl e m, it is easy t o construct t he t angenti al vel ocit y at t he mi dpoi nt of each panel accor di ng t o t he foll owi ng for mul a
ij ij j i ij j i N j ij i tir
r
q
V
V
1 1ln
)
cos(
)
sin(
2
)
cos(
N j ij j i ij ij j i r r 1 1 ) cos( ln ) sin( 2 (4. 37)And kno wi ng t he t angenti al vel ocit y co mponent, we can co mput e t he pressur e coeffi ci ent (no appr oxi mati on si nceVni 0) at t he mi dpoi nt of each panel accor di ng t o t he foll owi ng for mul a
2 2 ) , ( V V y x C ti i i p (4. 38)
from whi ch t he force and mo ment coeffi ci ent s can be co mput ed assu mi ng t hat t hi s val ue of Cp is const ant over each panel and by perfor mi ng t he discret e sum.
5. OPTI MI ZATI ON AND GENETI C ALGORI THMS
5. 1 Opti mizati on Overvie w
Opti mizati on t echni ques are used t o fi nd a set of desi gn para met ers,
x x xn
x 1, 2,..., , t hat can i n so me way be defi ned as opti mal. In a si mpl e case t hi s
mi ght be t he mi ni mization or maxi mi zati on of so me syst e m charact eristic t hat is dependent on x. In a more advanced for mul ati on t he obj ecti ve functi on, f(x), t o be mi ni mi zed or maxi mi zed, mi ght be subj ect t o constrai nt s i n t he for m of equalit y constrai nt s, e i x i m G( )0 1,..., ; i nequalit y constrai nt s m m i x
Gi( )0 e 1,..., and/ or para met er bounds, x ,l xu A general pr obl e m descripti on is st at ed as
Mi ni mi ze f(x) xn subj ect t o e i x i m G( )0 1,..., m m i x Gi( )0 e 1,..., u l x x x
wher e x is t he vect or of desi gn para met ers (xn), f(x)is t he obj ecti ve f uncti on t hat ret ur ns a scal ar value, and t he vect or functi on G(x)ret ur ns t he val ues of t he equalit y and i nequalit y constrai nt s eval uat ed at x .
An effi ci ent and accurate sol uti on t o t hi s pr obl em depends not onl y on t he si ze of t he pr obl e m i n ter ms of t he nu mber of constrai nt s and desi gn vari abl es but al so on charact eristi cs of t he obj ecti ve functi on and constrai nt s.
5. 2 Ai rf oil Opti mizati on
In t he desi gn of airfoils typi cal t ar get s i ncl ude prescri bed pressure or vel ocit y di stri buti ons, lift range, maxi mu m lift, mini mal drag, shock-free suction si de i n transoni c fl ow and t ype of st all at subsoni c speeds, under geo met ri cal constrai nt s t hat ma y i ncl ude one or mor e of t he foll owi ng: t hi ckness rati o, maxi mu m ca mber, l eadi ng edge radi us, traili ng edge angl e, or even t he whol e geo met r y itself defi ned by coor di nat es of contr ol poi nts.
5. 3 Geneti c Al gorit hms
5. 3. 1 Overvi e w of Geneti c Al gorit hms
Evol uti onar y al gorit hms are st ochasti c search met hods t hat mi mi c t he met aphor of nat ural biol ogi cal evol uti on. Evol uti onar y al gorit hms operat e on a popul ati on of pot enti al sol uti ons appl yi ng t he princi pl e of sur vi val of t he fitt est t o pr oduce bett er and bett er appr oxi mati ons t o a sol uti on. At each generati on, a ne w set of appr oxi mati ons is created by t he pr ocess of sel ecti ng i ndi vi dual s accor ding t o t heir level of fit ness i n t he probl e m do mai n and breedi ng t he m t oget her usi ng oper at ors borr owed fro m nat ural geneti cs. Thi s pr ocess l eads t o t he evol uti on of popul ati ons of i ndi vi dual s t hat are bett er suit ed t o t heir envir onme nt t han t he i ndi vi duals t hat t hey wer e creat ed from, j ust as i n nat ural adapt ati on [12].
Evol uti onar y al gorit hms model nat ural pr ocesses, such as sel ection, reco mbi nati on, mut ati on, mi grati on, l ocalit y and nei ghbor hood. Evol uti onar y al gorit hms wor k on populati ons of i ndi vi dual s i nstead of si ngl e sol uti ons. In t hi s way t he search is perf or med in a parall el manner. Fi gure 1 shows t he struct ure of a si mpl e geneti c al gorit hm:
Fi gur e 5. 1 The struct ure of a geneti c al gorit hm
At t he begi nni ng of t he co mput ati on a nu mber of i ndi vi dual s (t he popul ati on) are rando ml y i niti ali zed. The obj ecti ve functi on is t hen eval uat ed f or t hese i ndi vi dual s. The first/i niti al generati on is produced.
If t he opti mizati on crit eria are not met t he creation of a ne w gener ati on st arts. Indi vi dual s are sel ect ed accor di ng t o t heir fit ness for t he pr oducti on of offspri ng. Parent s are reco mbi ned t o pr oduce offspri ng. All offspri ng will be mut at ed wi t h a cert ai n pr obabilit y. The fit ness of t he offspri ng is t hen co mput ed. The offspri ng are i nsert ed i nt o t he popul ati on repl aci ng t he parent s, pr oduci ng a ne w generati on. Thi s cycl e is perf or med until t he optimi zati on crit eri a are reached.
Such a si ngl e popul ati on evol uti onar y al gorit hm i s powerf ul and perf or ms well on a br oad cl ass of pr obl e ms. Howe ver, bett er results can be obt ai ned by i ntroduci ng many popul ati ons, call ed subpopul ati ons. Ever y subpopul ati on evol ves for a fe w generati ons isol at ed (li ke the si ngl e popul ati on evol uti onar y al gorit hm) bef ore one or mor e i ndi vi dual s are exchanged bet ween t he subpopul ati ons.
The Multi-popul ati on evol uti onar y al gorit hm model s t he evol uti on of a speci es i n a way mor e si mil ar t o nat ure t han the si ngl e popul ati on evol uti onar y al gorit hm.
Fr o m t he above di scussion, it can be seen t hat evol uti onar y al gorit h ms differ subst anti all y fro m mor e traditi onal search and opti mizati on met hods. The most si gnifi cant differences are:
Evol uti onar y al gorit hms search a popul ati on of poi nts i n parall el, not a si ngl e poi nt.
Evol uti onar y al gorit hms do not require deri vati ve i nf or mati on or ot her auxili ar y kno wl edge; onl y t he obj ecti ve functi on and correspondi ng fit ness level s i nfl uence t he directi ons of search.
Evol uti onar y al gorit hms use pr obabilisti c transiti on rul es, not det er mi ni sti c ones.
Evol uti onar y al gorit hms are generall y mor e strai ghtf or war d t o appl y
Evol uti onar y al gorit hms can pr ovi de a nu mber of pot enti al sol uti ons to a gi ven pr obl e m. The fi nal choi ce is l eft t o t he user. ( Thus, i n cases wher e the parti cul ar pr obl e m does not have one i ndi vi dual sol uti on, for exa mpl e a fa mil y of paret o- opti mal sol uti ons, as i n t he case of multi obj ective opti mizati on and scheduli ng pr obl e ms, t hen t he evol uti onar y al gorit hm i s pot enti all y usef ul for i dentifyi ng t hese alt er native sol uti ons si mult aneousl y [12].
5. 3. 2 A Bri ef Hist ory of Ge neti c Al gorit hms
Ge neti c al gorit hms ori gi nat ed from t he st udi es of cell ul ar aut o mat a, conduct ed by John Hol land and hi s coll eagues at t he Uni versit y of Mi chi gan. Holl and‟s book [15], published i n 1975, is generall y acknowl edged as t he begi nni ng of t he research of genetic al gorit hms. Until t he earl y 1980s, t he research i n geneti c al gorit hms was mai nl y theor eti cal , wit h fe w real appli cati ons. Thi s peri od is mar ked by a mpl e wor k wit h fi xed l engt h bi nar y represent ati on i n t he do mai n of functi on opti mizati on by, a mong ot hers, De Jong and Holstei n. Hol st ei n‟s wor k pr ovi des a caref ul and det ail ed anal ysi s of t he effect t hat different sel ecti on and mati ng strat egi es have on t he perfor mance of a geneti c algorit hm. De Jong' s wor k att e mpt ed t o capt ure t he feat ures of t he adapti ve mechani s ms i n t he fa mil y of geneti c al gorit hms t hat constit ut e a robust search pr ocedure [14].
Fr o m t he earl y 1980s t he co mmunit y of geneti c al gorit h ms has experi enced an abundance of appli cati ons whi ch spread acr oss a l arge range of di sci pli nes. Each and ever y additi onal appli cation gave a ne w perspecti ve t o t he t heor y. Furt her mor e, i n the pr ocess of i mpr ovi ng perf or mance as much as possi bl e vi a t uni ng and speci alizi ng t he geneti c al gorith m operat ors, ne w and i mport ant fi ndi ngs regar di ng t he generalit y, robust ness and appli cabilit y of geneti c al gorit h ms
Foll owi ng t he l ast couple of years of furi ous devel op ment of geneti c al gorit hms i n t he sci ences, engi neeri ng and t he busi ness worl d, t hese algorit h ms i n vari ous gui ses have now been successf ull y appli ed t o opti mizati on pr obl e ms, scheduli ng, dat a fitti ng and cl ust eri ng, trend spotting and pat h fi ndi ng.
5. 3. 3 Bi ol ogi cal Background
Al l li vi ng or gani s ms consist of cells. In each cell t here is t he sa me set of chr o mos o mes. Chr o moso mes are stri ngs of DNA and ser ve as a model for t he whol e or gani s m. A chr o mos o me consi sts of genes, bl ocks of DNA. Each gene encodes a parti cul ar pr ot ei n. Basi call y, it can be sai d t hat each gene encodes a trait, for exa mpl e col or of eyes. Possi bl e setti ngs for a trait (e. g. blue, br own) are call ed allel es. Each gene has its own positi on i n t he chr o mos o me. Thi s positi on is call ed l ocus [13].
Co mpl et e set of geneti c mat eri al (all chr o mos omes) is call ed geno me. Parti cul ar set of genes in geno me is call ed genot ype. The genot ype is wi t h l at er devel op ment aft er birt h base for t he or gani s m's phenot ype, its physi cal and ment al charact eristi cs, such as eye col or, i nt elli gence et c.
Duri ng repr oducti on, reco mbi nati on (or cr ossover) first occurs. Genes from parent s co mbi ne t o for m a whol e ne w chr o mos o me. The ne wl y creat ed offspri ng can t hen be mut at ed. Mut ati on means that t he el e ment s of DNA are a bit changed. Thi s changes are mai nl y caused by errors i n copyi ng genes fro m parent s.
The fit ness of an or gani sm i s measured by success of t he or gani s m i n its life (sur vi val).
5. 3. 4 Struct ure of Geneti c Al gorit hms
Ge neti c al gorit hms are i nspired by Dar wi n' s t heory of evol uti on. Sol uti on t o a pr obl e m sol ved by geneti c al gorit hms uses an evol uti onar y pr ocess.
Al gorit hm begi ns wit h a set of sol uti ons which are represent ed by chr o mos o mes, call ed popul ati on. Sol uti ons fro m one popul ati on are t aken and used t o for m a ne w popul ati on. Thi s is moti vat ed by a hope, t hat t he ne w populati on will be bett er t han t he ol d one. Sol uti ons whi ch are then sel ect ed t o for m new sol uti ons (offspri ng) are sel ect ed accor di ng t o t heir fit ness - t he mor e suit abl e t hey are t he mor e chances t hey have to repr oduce.