Incompressible Flow Over Finite Wings III
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Transcript of 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
.
,
Prandtl’s classical lifting-line theoryPrandtl’s classical lifting-line theory
(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
Prandtl’s classical lifting-line theoryPrandtl’s classical 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
Elliptical Lift DistributionElliptical Lift Distribution
(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
Elliptical Lift DistributionElliptical Lift Distribution
(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
,
Elliptical Lift DistributionElliptical Lift Distribution
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
Elliptical Lift DistributionElliptical Lift Distribution
(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
General lift distributionGeneral lift distribution
(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
General lift distributionGeneral lift distribution
(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
).
General lift distributionGeneral lift distribution
(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
,
General lift distributionGeneral lift distribution
(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
,
,
,
General lift distributionGeneral lift distribution
Elliptic wing
Rectangular wing
Tappered wing
General lift distributionGeneral lift distribution
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
Effect of Aspect RatioEffect of Aspect Ratio
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
Effect of Aspect RatioEffect of Aspect Ratio
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