Chapter 5: Incompressible Flow over Finite Wings

Transcription

1 Chapter 5: Icompressle Flow over Fte Wgs

2 5. Itroducto fte wgs, dowwash, duced drag 5. ortex Theor prcple: the vortex flamet Bot-Savart law Helmholtz s vortex theorems 5.3 The Classcal ftg-e Theor ellptcal ad geeral lft dstruto the effect of spect Rato Extesos: umercal mplemetato lftg-surfacevortex-lattce REFERECE MTERI: see 5. umercal Example of the Wg Equato

3 rfol : D flow c l, c d Real Wg: 3D flow C, C D fte extet varato of sectos alog the wg spa The flow over fte wgs I what respect s the flow aroud a true wg dfferet from a arfol a fte wg? spawse flow compoet due to leakage flow aroud the tps

4 Tralg vortces ad dowwash Tralg vortces tp vortces Results: tralg vortces tp vortces ad dowwash vertcal flow compoet dowwash Upflow tp vortex

5 Flg geese

6 Tp ortex 3D rfol ftg e theor

7 Dowwash ad the effectve flow drecto. The dowwash modfes the effectve flow drecto ad reduces : effectve agle of attack - geometrc agle of attack - duced agle of attack eff Iduced Drag D. The lft vector s cled ackwards: duced drag D' ' ote: total drag = duced drag + profle drag flght drecto Effectve gle of ttack Iduced eloct

8 DRG FOR SUBSOIC -D IRFOI D THE FIITE WIG For susoc -D arfol: D f : sk frcto drag, C f =D f S D p : prssure darg D f >> D p small agle of attack Profle drag coeffcet C d = D f + D p S d d For susoc fte wg: D : duced drag Total drag coeffcet C D = D f + D p + D S

9 = total lft of the wg Dstruto of lft = sectoal lft, local lft per ut spa log the wg spa varato of: chord c arfol propertes aerodamc twst geometrc geometrc twst duced Hece, also varato of: lft coeffcet sectoal lft crculato c c C q ' cl q c cl c l ote: S eff ' ~ ' d ~ c l c erodamc twst s defed as "the agle etwee the zero-lft agle of a arfol ad the zero-lft agle of the root arfol." I essece, ths meas that the arfol of the wg would actuall chage shape as t moved farther awa from the fuselage.

10 Twsted wg geometr twst & dfferet arfol cross sectos alog the spa aerodamc twst

11 Dstruto of lft ote: ft s zero at the tps pressure equalzato Cetral suject of wg theor: Relato etwee wg shape ad lft dstruto. alss: determe the lft dstruto for gve wg shape. Desg: determe wg shape for desred lft dstruto ftg le theor: the wg s replaced a vortex flamet wth varale crculato at the quarter-chord le + free vortces

12 OUTIE FOR Chapter 5 Helmholtz ortex Theorem Cosder the moto of a vscd flud uder the acto of coservatve od force e.g. gravt, for whch G=gz D Dt

13 HEMHOTZ ORTEX THEOREM for Curved ortex Flamet ortex le vortct veloct q ortex flamet: a ftesmal vortex tue. D Dt ortex tue Referece: ow Speed erodamcs From Wg Theor to Pael method Katz aad Plotk Chapter.9 arale defto: - volume; q - veloct; - vortct

14 3-D ortex Theor: the vortex flamet flow aroud a real wg uform flow + vortces D: Straght vortex le: 3D geeral: curved vortex le r P duced veloct r

15 3-D ortex Theor: Helmholtz s vortex theorems compare the veloct duced the vortex flamet to the magetc feld duced a electrcal curret The crculato stregth remas costat alog the flamet a vortex flamet caot ed the flow, ut: exteds to ft eds at a oudar forms a closed loop cosequece:

16 3-D ortex Theor: The Bot-Savart aw The cotruto d of a flamet secto dl to the duced veloct P: d dl r θ 3 4 r The Bot-Savart aw Drecto: d s perpedcular to dl ad r ote: s the agle: dl r Magtude: s d dl 4 r

17 Propertes of a straght vortex flamet segmet B B dl r s 4 P h r l B θ -θ s ta s h dl h l h r cos cos 4 s 4 B B h d h Fte segmet B, costat

18 Propertes of a straght vortex flamet segmet B θ B B B 4h cos cos B 4h cos cos B h P ote: ad B are the teral agles of BP θ Specal cases: fte vortex flamet : = B = : 4h h same as D vortex sem-fte flamet: =9º; B = : 4h 4h P

19 5.3 The ftg-e Theor The Horseshoe vortex as a smple model of a fte wg the wg tself a oud vortex at the 4-chord le s fxed, hece, expereces lft = the tp vortces free-tralg vortces free to adjust to the local flow drecto, o lft ll vortces have the same crculato stregth ; the free tralg vortces exted to ft dowstream

20 The sgle horseshoe vortex rght tp vortex Sem-fte flamet: h w P W 4h Dowwash duced alog the wg the two tralg wg tp vortces w 4 4 left tp vortex rght tp vortex left tp vortex w 4 Remarks: w < whe > : the duced flow s deed dowwards for postve lft Prolems wth the smple horseshoe-vortex model of a wg: = costat lft dstruto w at the tps ot realstc!

21 Exteso of the horseshoe vortex model towards the lftg-le model Istead of a sgle horseshoe vortex: superposto of ma vortex sstems Each vortex has a dfferet spa ut the oud vortex segmets cocde o the same le ad form the lftg le = the wg The crculato alog the lftg le s o loger costat, ut t vares alog the spa a stepwse fasho Extrapolate to fte umer of horseshoe vortces to ota cotuous

22 Prcple of the lftg le + d d The wg s replaced a oud vortex wth cotuousl varg crculato The tralg vortces create a vortex wake the form of a cotuous vortex sheet local stregth of the tralg vortex at posto s gve the chage : d = dd d the vortex sheet s assumed to rema flat o deformato aldt: good approxmato for straght, sleder wgs at moderate lft

23 Determg the dowwash of the lftg le I Stregth of the tralg vortex at posto alog the wg spa: Take small segmet of the lftg le, d, at posto Over ths segmet the chage crculato of the lftg le s: d = dd d Ths s equal to the stregth of the tralg vortex The cotruto dw to the duced veloct at posto : Total veloct at posto duced the etre vortex wake: w - d d = dd d d dw 4 w 4 d d d

24 Determg the duced agle of attack of the lftg le duced agle of attack: d d d w w 4 ta Total veloct at posto duced the etre vortex wake: d d d w 4

25 The relato etwee crculato ad wg shape Use D arfol theor, ut modfed the effectve flow drecto: From the relato etwee lft ad crculato: comato: ] [ ] [ eff eff l l a a c c ' c c c c l l a c d d d c a 4 d d d w 4 The fudametal equato of Pradtl s lftg-le theor d dc a l = cost

26 Pradtl s lftg-le equato the wg equato a c 4 d d d Some remarks:. Ths equato descres the relato etwee crculato ad wg propertes. It s lear 3. The crculato s proportoal to ft ~ ~ 4. For a wg wthout twst ad = are costat: crculato s proportoal to = for ever value of the lft dstruto has the same form whch depeds o a, c ad, therefore, o the wg shape the total lft s zero whe = = ad the: alog the spawse o drecto 5. For a wg wth twst ad = are ot costat: THIS IS OT SO partcular: total zero lft s geeral ot accompaed : alog the spawse o drecto

27 Wg propertes for gve crculato. ft dstruto:. Total lft: 3. Iduced agle of attack: 4. Iduced drag: ' ' d d d S S q C d d d 4 ' ' d d d D D D d S S q D C

28 The ellptcal lft dstruto Cosder the followg ellptcal lft dstruto: w 4 Compute the dowwash veloct from: d d d = max.crculato - coordate trasformato: cos d s d s d d cos w d d cos cos cos cos w = Dowwash ad duced agle of attack are costat over the spa of the wg!

29 The ellptcal lft dstruto s Calculato of the total lft: 4 s s d d d d d s C S S C 4 4 C S C = S: s called the aspect rato R of the wg slakhed tpcal values: 6-8 for susoc arcraft - for glder arcraft The duced agle of attack Relato etwee ad C :

30 The ellptcal lft dstruto 3 Calculato of the duced drag: D ' d ' d ote that C s costat here C D C C Coclusos: The ducd drag s the drag due to lft Rememer : total drag C D c d C D ~ C : quadratc depedece C D C : large R decreases duced drag D ~

31 The ellptcal lft dstruto - wg shape What wg shape ca geerate a ellptcal lft dstruto? assume: o twst: so ad - are costat assume: lft slope a = dc l d s costat cosequece: wth also costat c l a [ ] costat C c l C Remark: Proof: C cl c d cl S S c d c l requred varato of the chord: ' cl q c ' c ~ ' ~ q c l The wg must have a ellptcal plaform

32 The ellptcal wg shape ellptcal wg plaform: ote straght 4-chord le 4-chord le ellptc lft dstruto, a ellptc wg plaform ad a costat dowwash

33 The Supermare Sptfre

34 erodamc propertes of the ellptc wg We foud that: = costat Comg: solve for C : C C = costat c l c l c l C a a C [ ] where: a dcl d [ a ] a a ote: C = whe = = ad: for a ellptc wg for a geeral wg C dc a d a

35 Effect of spect Rato o the lft-curve C for a ellptc wg: a dc a d a The lft slope s reduced. phscal explaato: the dowwash reduces the effectve agle of attack: dc d dcl d dcl d eff deff. d d a d

36 The ellptcal lft dstruto - summar Costat dowwash alog the spa Iduced drag: ft slope: a C D C dc a d a C effect of creasg the wg aspect rato: - duced drag smaller - lft-slope larger a a Practcal sgfcace of the ellptcal wg: optmum wg shape: mmal duced drag for gve lft referece wg: reasoale approxmato for real wgs C

37 Geeral lft dstruto For the ellptcal wg: wth: ad: s C cos Descre the crculato of a geeral wg wth a Fourer se seres: ote: s The umer of terms should e take suffcetl large = at the tps Questos to e aswered: a costat depedg learl o C, hece, o what are the aerodamc propertes lft, duced drag? what s the relato etwee the coeffcets ad the wg geometr? costats that deped o Ellptcal wg: =; =C

38 Geeral lft dstruto: total lft s s d S d S S q C.. s s s s d S d S Stadard tegrals: = whe = whe = C. Depeds ol o the frst coeffcet Calculato of the lft coeffcet:

39 Geeral lft dstruto: dowwash s Calculato of the duced agle of attack: d d cos 4 d d d d d d cos cos cos cos cos d Stadard tegrals: s s s s

40 Geeral lft dstruto: duced drag s s d S d S S q D C D s s s s d S = whe m = whe = m D C Calculato of the duced-drag coeffcet: s s s s d m S m m

41 Geeral lft dstruto: summar ad coclusos C. Cocluso: the ellptc wg =, e = gves the lowest possle duced drag for gve lft ad aspect rato D C where D C C the "spa effcec factor" where : or e e C C D

42 The relato etwee the ad the wg geometr Solve Pradtl s wg equato: susttute: c a l a c s s s 4 a s s c s umercal soluto method: Take a trucated seres wth ukow coeffcets:,, Take dfferet spawse locatos o the wg where the equato s to e satsfed:,,.. ; ut ot at the tps, so: < < Sstem of equatos wth ukows Solve matx ote: t s ot possle to solve for ol oe coeffcet, as Chapter 4!

43 umercal example of the wg equato Cosder: rectagular wg: c = costat; spa = ; c = ; wthout twst: = costat; = = evaluate the wg equato at the cotrol pots at : 4 a s s The wg s smmetrcal, 4, are zero s s If If s s eve : s s odd : s for s eve umer s s s for s odd umer s,,... s s

44 umercal example of the wg equato evaluate the wg equato at the cotrol pots at : 4 s a s,,... The wg s smmetrcal, 4, are zero take ol, 3, as ukows take ol cotrol pots o half of the wg: < Example for =3: take, 3, 5 as ukows take cotrol pots equdstat : = 6, = 3, 3 = take lft-slope of the arfols a =, ad wg aspect rato =

45 umercal example of the wg equato 3 =, = 6, =, = 3, =3, 3 = a s s 4...,, 6 s5 6 s s3 6 s s 6 s a a a 3 s5 3 s s3 3 s s 3 s a a a s5 s 5 4 s3 s 3 4 s s a a a

46 umercal example of the wg equato 3 4 a =, = 6, s s ,, =, = 3, =3, 3 =

47 umercal example: the rectagular wg =3 The set of equatos ecomes: wth soluto: Evaluato of the propertes of the rectagular wg wth = a = : ote: wth.5: ol 5% more duced drag tha ellptcal wg! C d dc a e =3 =

48 C ~ CD C where

49 Effect of wg plaform ad aspect rato C C a D a alues of deped o plaform ad aspect rato of the wg a Effect of wg plaform o for a tapered wg example tapered wg wth taper rato c t c r =.3 s almost as good as a ellptcal wg!

50 Fal coclusos the effect of wg plaform o the duced drag C D C I order to reduce the duced drag t s more mportat to crease the aspect rato tha trg to approach the ellptc lft dstruto accuratel tapered wg wth taper rato c t c r =.3 s almost as good as a ellptcal wg ad s much easer to maufacture ote that the parameter s a costat.e., depedet of ol for a wg wthout twst! Rememer: total drag = duced drag + profle drag ~ vscost

51 ftg-le theor: Wg theor - a summar The wg s replaced a oud vortex at the 4-chord le of the wg wth varg crculato : the lftg le The tralg vortces form a flat sheet of dstruted vortct: the vortex wake mtatos of the classcal theor: sleder wgs large aspect rato, or: spa>>chord straght wgs o wg sweep moderate aerodamc loadg o deformato of the vortex wake lear relato Extesos: c l ~ eff 5.4 o-lear lftg-le theor: c l eff 5.5 methods where the wg s represeted a vortex-sheet stead of a le: lftg-surface vortex-lattce methods

52 5.4 umercal olear lftg-le method Gve the wg shape ad the agle of attack :. Dvde the wg spawse postos:. ssume a tal crculato dstruto =, e.g. ellptcal 3. Calculate the duced agle of attack: 4. Calculate: 4 eff d d d evaluate the tegral umercall 5. Calculate lft coeffcet: cl cl eff c 6. Update crculato: cl terate utl covergece uder relaxato

53 5.5 ftg-surface theor prcple wg wake streamwse vortct ftg le: wg represeted a vortex flamet ol spawse vortct vald ol for sleder wgs ftg surface: wg represeted a vortex sheet wth dstruted spawse ad chordwse vortct

54 ftg-surface theor - umercal mplemetato 3D vortex-pael methods: the wg s represeted paels wth dstruted vortct three-dmesoal exteso of the vortex-pael method secto 4.9 ortex-attce methods: dstruted vortct s cocetrated to a lattce of horseshoe vortces sgle horseshoe vortex The vortex-lattce sstem o a fte wg