Incompressible Flow Over Finite Wings III

Incompressible Flow Incompressible Flow Over Finite wingsOver Finite wings

Prandtl’s classical lifting-line theory

Prandtl’s classical lifting-line Prandtl’s classical lifting-line theorytheory

2by

2by

y

x

z

V V

iD

L

2b

2b

Finite wing Horseshoe vortex

z y

x

2b

2b

Downwash distribution along the y axis for a single horseshoe vortex

Prandtl’s classical lifting-line Prandtl’s classical lifting-line theorytheory

(5.13)

24

yw

(5.12)

24

24

22

ybb

ybybyw

Trailing from b/2Trailing from - b/2

Superposition of a finite number of Superposition of a finite number of horseshoe vortices along the lifting horseshoe vortices along the lifting lineline

2b

2b

A

B

C

D

E

F

1d

2d

3d

1d

2d

3d

dydx

0 y

V

dw

x

y

z

Prandtl’s classical lifting-line theoryPrandtl’s classical lifting-line theory

dydx

0 y

V

dw

x

y

z

0y

(5.14)

4

by,given is at located vortex trailing

semifinite entire by the induced at Velocity

line. lifting thealong dn circulatioin

change theequalmust at vortex trailing theof

strength theIn turn, is segment over the

n circulatioin change theandΓ is at n circulatio The

0

0

yy

dydyddw

y

ydw

y

dydydddy

yy

.

,


(5.18)

41 (5.17), into (5.15) eq. ngsubstituti

2

20

0

b

bi yy

dydyd

Vy

(5.15)

41

sheet, vortex trailingentire by theat induced velocity Total

2

20

0

0

b

b yy

dydydyw

y

(5.16)

attack of angle Induced

010

Vyw

yi tan

(5.17)

comes be Eq.(5.16)small, is

00

Vyw

y

Vw

i

,

(5.19) 2

at locatedsection airfoil for thet coefficienlift The

00000

0

LeffLeffl yya

yy

,

(5.20) 21

002 yVcycVL l

(5.21)

2 , have we(5.20), eq. from

0

0

ycVy

cl

(5.22)

have we(5.19), into (5.21) eq. ngsubstituti

0L0

0eff

ycV

y

(5.23)

41

have we(5.9), into (5.22) and (5.18) eq. ngsubstituti

2b

2b-0

0L0

00

yy

dydyd

VycVy

y

Fundamental equation of Prandtl’s lifting-line theory


(5.26) 2 L

(5.25)

span over the (5.24) eq. gintegratinby obtained islift totalThe 2.

(5.24)

: theoremJoukowski-Kutta thefrom obtained ison distributilift The 1.

2

2

2

2

2

2

00

dyySVSq

C

dyyVL

dyyLL

yVyL

b

bL

b

b

b

b

(5.30) 2

t,coefficien drag induced

(5.29)

(5.28)

drag induced Total

(5.27)

small is since

spanunit per drag induced The 3.

2

2

2

2

2

2

i

dyyySVSq

DC

dyyyVD

dyyyLD

LD

LD

i

b

bi

iD

i

b

bi

i

b

bi

iii

iii

,

,sin

Elliptical Lift DistributionElliptical Lift Distribution

(5.32) 41

4

(5.31), eq. atingDifferenti Downwash. 3.

ondistributilift elliptical

21

on.distributin circulatio ellipticalan as designated

isit hence, span; thealongy distancely with elliptical n variescirculatio The 2.

5.13 fig.in shown as origin, at then circulatio is 1.

:note

(5.31) 21

bygiven on distributin circulatio aconsider

21222

0

2

0

0

2

0

by

ybdy

d

byVL

byy

dbdyby

dyyyby

yb

ywb

b

sincos

/

/

2

2

nsubtitutio

(5.33)

41

(5.15) into (5.32) eq. ngSubstituti

02

122

2

220

0


(5.36) 2

attack of angle induced (5.17), eq. from

on.distributilift ellipticalan for span over theconstant isdownwash

(5.35) 2

becomes (5.34) eq. hence 1,nfor (4.26) eq.by given form standard is integral this

(5.34) 2

or 2

becomes (5.32) eq. Hence

0i

00

00

20

0

0

02

00

bVVw

bw

db

w

db

w

coscoscos

coscoscos


(5.37) 41

(5.25) into eq.(5.31) ngsubstituti21

2

2 2

2

0 dybyVL

b

b

(5.40) 2

becomes (5.39) eq. hence ,21 however,

(5.39) 4

(5.38) 4

2

(5.37) eq. , cos 2

ation transform theusing again,

0

2

0

00

20

bSCV

SCVL

bVL

bVdbVL

by

L

L

sin

(5.42) AR

becomes (5.41) eq. hence,AR

ratioAspect

(5.41) or

2

12

(5.36) into (5.40) eq. ngsubstituti

2

2i

i

Li

L

L

C

Sb

bSC

bVbSCV

,


High AR (low induced drag)

(5.43) AR

or 2

AR2

obtain we(5.42a), intio (5.42) and (5.40) eq. ngsubstituti

(5.42a) 2

2

22

constant is that noting (5.30), eq. from obtained iscoeffient drag induced

2

2

2 0020

i

LiD

LLiD

b

b

iiiiD

CC

bSCVC

SVbC

SVb

dbSV

dyySV

C

,

,

, sin

Low AR (high induced drag)

b

b

Sb2

AR


(5.45)

(5.44) or

spanunit per lift 2 theory,airfoilThin

tcoefficien

liftsection Local span. thealongconstant also is Hence

span. thealongconstant is twist,caerodynami no and twist geometric no

0

00

l

l

Leffl

ieff

i

cqyLyc

ccqyL

a

ac

,

yelliptic

wingelliptic

constw

V

General lift distributionGeneral lift distribution

N

n nAbV

by

10

00

(5.48) 2

thatcase general for the assume Hence

wing.finitearbitrary an alongon distributi

ncirculatio general for the expression eappropriatan be

wouldseries sineFourier a that hintsequation This

(5.47)

as written is (5.31) eq.by given on distributilift

elliptic the of In term 0by given now

isdirection spanwise in the coordinate thewhere

(5.46) 2

ation transformheconsider t

sin

sin

,.

cos

(5.49)

2 1

N

n dydnnAbV

dyd

dd

dyd

cos


(5.51) 2

becomes (5.50) eq. Hence,

(4.26), eq.by given form standard theis (5.50) eq.in integral The

(5.50) 12

obtain we(5.23), into (5.49) and (5.48) eq. ngsubstituti

(5.49) 2

10

000

10

00

00

100

10

00

1

NnL

N

n

Nn

L

N

n

N

n

nnAnA

cb

dnnA

nAc

b

dydnnAbV

dyd

dd

dyd

sinsinsin

coscoscossin

cos


(5.53) AR

becomes (5.52) eq. Hence

1nfor 0

sin

1nfor 2

is integral the, (5.52) eq.In

(5.52) 22

tcoefficien lifting have we(5.26), into 485eq.(on Substituti

1

2

1

0

01

22

2

ASbAC

dn

dnASbdyy

SVC

L

N

n

b

bL

sin

sinsin

).


(5.54) 2

2

:follow as (5.30) into (5.48) eq. of

onsubstituti thefrom obtained ist coefficien drag induced The

0 1

2

2

2

dnASV

b

dyyySV

C

i

N

n

i

b

biD

sinsin

,

(5.55) 1

41

(5.18) into (5.49) and (5.46) eq. of

n subtitutio thefrom obtained isattack of angle induced The

0 01

2

2 00

dnnA

yy

dydyd

Vyα

N

n

b

bi

coscoscos

(5.58) 2

have we(5.54), into (5.57) eq. ngsubstituti

(5.57) or

(5.56)

becomes (5.55) eq. Hence, (4.26). eq.by

given form standard theis (5.55) eqin integral The

10 1

2

1

0

0

10

dnnAnASbC

nnA

nnA

N

n

N

niD

N

ni

N

ni

sinsin

sinsin

sinsin

,


(5.60) 1AR

AR

AR2

2

becomes (5.58) Eq.

for 0

(5.59) sin

for 2

integral standard thefrom

2

2

1

21

2

221

1

2

1

22

0

Nn

N

n

N

n

N

niD

AAnA

nAA

nAnASbC

km

dkm

km

,

sin

ondistributilift

elliptical for the 1 0,

(5.62) AR

(5.61) eq.then 1

0

(5.61) 1AR

(5.60) eq. intofor

(5.53) eq. ngsubstituti

2

1

2

2

1

2

eeC

C

e

AAn

CC

C

LiD

Nn

LiD

L

,

,

,


Elliptic wing

Rectangular wing

Tappered wing


rctc

r

t

cc

ratio tappered

Effect of Aspect RatioEffect of Aspect Ratio

(5.65) AR

1AR

1

(5.64b) and (5.64a) from

(5.64b) AR

(5.64a) AR

ratio,aspect different w/ wingswoConsider t

(5.63) AR

wing,finite of drag Total

21

2

21

2

2

2

1

2

1

2

eC

CC

eC

cC

eC

cC

eC

cC

LDD

LdD

LdD

LdD

,,

,

,

0

0

: wingfinite

:airfoil

slope,lift The

aa

ddCa

ddca

l

l

(5.68) AR

(5.67) into (5.42) eq. ngsubstituti

(5.67)

follow as related are and

: wingfinite

:airfoil

slope,lift The

0

0

0

0

0

0

constC

aC

constaC

ad

dC

aa

aa

ddCa

ddca

LL

iL

i

L

l

l


0.25. - 0.05

t coefficienFourier theoffunction is

(5.70) 1AR1

platform, general of wingfinite aFor

(5.69) AR1

wing,finite ellipticfor andbetween relation The

0

0

0

0

0

.n

L

A

aa

a

aa

addC

aa


Physical SignificanceA numerical nonlinear lifting-line method

Lifting Surface Theory; vortex Lifting Surface Theory; vortex latice numerical methodlatice numerical method

Lifting surface theory

V V V

Prandtl’s classical lifting-line theory gives reasonable results for straight wing at moderate to high aspect ratio.

For low-aspect ratia straight wing, swept wing and delta wing, classical lifting-line theory is inapproriate

Low aspect ratiostraight wing

Swept wing Delta wing

Schematic of a lifting surfaceSchematic of a lifting surface

V

yx, yx,

yxp ,

x

y

yw

Lifting surface

wake

Velocity induced at point P by an infinitesimal segment of the Velocity induced at point P by an infinitesimal segment of the lifting surface. The velocity is perpendicular to the plane of lifting surface. The velocity is perpendicular to the plane of the paperthe paper

d

d

yxp ,

r

d

d

,y

,x

SRegion

WRegion

(5.73)

sin 4 dl

4dV

is, strength

filament vortex thisof segment aby

at induced velocity lincrementa the

(5.5) eq. law,Savart -Biot theFrom

33 rrdd

rr

dξ

dη

P

V

(5.76)

41

41

is, wake theand surface lifting both theby Pat induced velocity normal The

(5.75) 4

,at velocity induced the

todη strength of vortex chordwise elemental theofon contributi thegConsiderin

(5.74) 4

becomes, (5.73)

eq. Hence-xsin that note Also dV as velocity induced

the to(5.73) eq. ofion constribut thedenote wedirection, positive in the i.e.,

direction, upward in the positive is that conventionsign usual theFollowing

2322

2322

3

3

ddyx

y

ddyxyxyxw

rddydw

P

rddxdw

rdww

z

w

W

w

S

,

,,,

..

(5.76)

41

41

2322

2322

ddyx

y

ddyxyxyxw

W

w

S

,

,,,

zero. is stream fre ofcomponent normal theand of sum thesuch that , and

,for (5.76) eq. solve tois theory surface-lifting of problem central The

x,yw

Incompressible Flow Over Finite Wings III

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