Upload
solomon-rich
View
217
Download
0
Embed Size (px)
Citation preview
1
MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS
Finite Wings: General Lift Distribution Summary
April 18, 2011
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
2
SUMMARY: PRANDTL’S LIFTING LINE THEORY (1/2)
2
20
0
2
20
0
2
20
00
00
4
1
4
1
4
1
b
bi
b
b
b
bL
dyyy
dyd
Vy
dyyy
dyd
yw
dyyy
dyd
VycV
yy
Fundamental Equation of Prandtl’s Lifting Line Theory
Geometric angle of attack, , is equal to sum of effective angle of attack, eff, plus induced angle of attack, i
Equation gives value ofDownwash, w, at y0
Equation for induced angle of attack, i, along finite wing
3
SUMMARY: PRANDTL’S LIFTING LINE THEORY (2/2)
dyyySV
C
dyyyVD
dyySVSq
LC
dyyVL
yVyL
b
biiD
i
b
bi
b
bL
b
b
2
2
,
2
2
2
2
2
2
00
2
2
Lift distribution per unit span given by Kutta-Joukowski theorem
Total lift, L
Lift coefficient, CL
Induced drag, Di
Induced drag coefficient, CD,i
4
PRANDTL’S LIFTING LINE EQUATION
• Fundamental Equation of Prandtl’s Lifting Line Theory
– In Words: Geometric angle of attack is equal to sum of effective angle of attack plus induced angle of attack
– Mathematically: = eff + i
• Only unknown is (y)
– V∞, c, , L=0 are known for a finite wing of given design at a given a
– Solution gives (y0), where –b/2 ≤ y0 ≤ b/2 along span
2
20
000
00 4
1b
bL dy
yy
dyd
Vy
ycV
yy
5
WHAT DO WE GET OUT OF THIS EQUATION?
1. Lift distribution
2. Total Lift and Lift Coefficient
3. Induced Drag
dyyySVSq
DC
dyyyVdyyyLD
LD
dyySVSq
LC
dyyVL
dyyLL
yVyL
b
bi
iiD
i
b
bi
b
bi
iii
b
bL
b
b
b
b
2
2
,
2
2
2
2
2
2
2
2
2
2
00
2
2
6
GENERAL LIFT DISTRIBUTION (§5.3.2)• Circulation distribution
• Transformation
– At =0, y=-b/2
– At =, y=b/2
• Circulation distribution in terms of suggests a Fourier sine series for general circulation distribution
• N terms
– now as many as we want for accuracy
• An’s are unkowns, however must satisfy fundamental equation of Prandtl’s lifting-line theory
2
20
000
00
1
0
2
0
4
1
sin2
sin
cos2
21
b
bL
N
n
dyyy
dyd
Vy
ycV
yy
nAbV
by
b
yy
7
GENERAL LIFT DISTRIBUTION (§5.3.2)
N
nL
N
n
N
n
L
N
n
N
n
b
bL
N
n
nnAnA
c
b
dnnA
nAc
b
dy
dnnAbV
dy
d
d
d
dy
d
dyyy
dyd
Vy
ycV
yy
nAbV
1 0
000
10
00
0 0
100
10
00
1
2
20
000
00
1
sin
sinsin
2
coscos
cos1
sin2
cos2
4
1
sin2
• General circulation distribution
• Lifting line equation
• Finding d/dy
• Transform to
• Last integral has precise form for simplification
8
GENERAL LIFT DISTRIBUTION (§5.3.2)
• Evaluated at a given spanwise location, 0 is specified• Givens:
– b: wingspan– c(0): chord at the given location for evaluation– The zero lift angle of attack, L=0(0), for the airfoil at this specified location
• Note that the airfoil may vary from location to location, and hence the zero lift angle of attack may vary from location to location
• Can put twist into the wing– Geometric twist– Aerodynamic twist
• This is one algebraic equation with N unknowns written at 0
• Must choose N different spanwise locations to write the equation to give N independent equations
N
nL
N
n
nnAnA
c
b
1 0
000
10
00 sin
sinsin
2
9
WING TWIST
10
GENERAL LIFT DISTRIBUTION (§5.3.2)
ARAS
bAC
dn
dn
dnAS
bC
dyySV
C
L
N
nL
b
b
L
1
2
1
0
0
01
2
2
2
1nfor 0sinsin
1nfor 2
sinsin
sinsin2
2
• General expression for lift coefficient of a finite wing
• Substitution of expression for () and transformation to
• Integral may be simplified
• CL depends only on leading coefficient of the Fourier series expansion (however must solve for all An’s to find leading coefficient A1)
11
GENERAL LIFT DISTRIBUTION (§5.3.2)
N
ni
N
ni
N
ni
b
bi
i
N
niD
b
biiD
nnA
nnA
dn
nAy
dyyy
dyd
Vy
dnAS
bC
dyyySV
C
1
1 0
00
1 0 00
2
20
0
0 1
2
,
2
2
,
sin
sin
sin
sin
coscos
cos1
4
1
sinsin2
2
• General expression for induced drag coefficient
• Substitution of () and transformation to
• Expression contains induced angle of attack, i()
• Expression for induced angle of attack
• Can be mathematically simplified
• Since 0 is a dummy variable which ranges from 0 to across the span of wing, it can simply be replaced with
12
GENERAL LIFT DISTRIBUTION (§5.3.2)
Nn
iD
N
niD
N
n
N
niD
N
n
N
niD
N
ni
i
N
niD
A
AnARAC
nAAARC
nAARnAS
bC
km
km
dnnAnAS
bC
nnA
dnAS
bC
2
2
1
21,
2
221,
1
2
1
22
,
0
0
10 1
2
,
1
0 1
2
,
1
2
2
kmfor 2
sinsin
kmfor 0sinsin
sinsin2
sin
sin
sinsin2
• Expression for induced drag coefficient
• Expression for induced angle of attack
• Substitution of i() in CD,i
• Mathematical simplification of integrals
• More simplifications leads to expression for induced drag coefficient
13
GENERAL LIFT DISTRIBUTION (§5.3.2)
eAR
CC
e
A
An
AR
CC
ARAS
bAC
A
AnARAC
LiD
Nn
LiD
L
Nn
iD
2
,
2
2
1
2
,
1
2
1
2
2
1
21,
1
1
1
1
• Repeat of expression for induced drag
coefficient
• Repeat of expression for lift coefficient
• Substituting expression for lift coefficient into expression for induced drag coefficient
• Define a span efficiency factor, e, and note that e ≤ 1
– e=1 for an elliptical lift distribution
14
VARIOUS PLANFORMS FOR STRAIGH WINGS
Elliptic Wing
Rectangular Wing
Tapered Wing
cr ct
15
INDUCED DRAG FACTOR, (e=1/(1+))
16
SPECIAL CASE:Elliptical Wings → Elliptical Lift Distribution
17
ELLIPTICAL LIFT DISTRIBUTION• For a wing with same airfoil shape across span and no twist, an elliptical
lift distribution is characteristic of an elliptical wing planform
AR
CC
AR
C
LiD
Li
2
,
18
SUMMARY: ELLIPTICAL LIFT DISTRIBUTION (1/2)
2
0
2
0
21
21
b
yVyL
b
yy
Points to Note:
1. At origin (y = 0) = 0
2. Circulation and Lift Distribution vary elliptically with distance, y, along span, b
3. At wing tips (-b/2) = (b/2) = 0
– Circulation and Lift → 0 at wing tips
y/b
/
0
19
SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION
Elliptic distribution
Equation for downwash
Coordinate transformation →
See reference for integral
bVV
wb
w
db
w
db
dyb
y
dy
yyby
y
byw
by
y
bdy
d
i
b
b
2
2
coscos
cos
2
sin2
;cos2
41
41
4
0
00
0 0
00
2
20
21
2
22
00
2
220
Downwash is constant over span for an elliptical lift distribution
Induced angle of attack is constant along spanNote: w and i → 0 as b → ∞
20
SUMMARY: ELLIPTICAL LIFT DISTRIBUTION
l
LiD
Li
i
cq
yLc
AR
CC
AR
C
bVL
bV
bw
2
,
0
0
0
4
2
2Downwash is constant over span for an elliptical lift distribution
Induced angle of attack is constant along span for an elliptical lift distribution
Total lift
Alternate expression for induced angle of attack, expressed in terms of lift coefficient
Induced drag coefficient
For an elliptic lift distribution, the chord must vary elliptically along the span
→ the wing planform is elliptical in shape
21
SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION
AR
CC
dyySV
C
AR
CS
bAR
b
SC
bVdy
b
yVL
LiD
b
b
iiD
Li
Li
b
b
2
,
2
2
,
2
2
0
2
2
21
2
2
0
2
4
41
CD,i is directly proportional to square of CL
Also called ‘Drag due to Lift’
We can develop a moreuseful expression for i
Combine L definition for elliptic profile with previous result for i
Define AR because itoccurs frequently
Useful expression for i
Calculate CD,i