AERODYNAMICS I

LECTURE 6

AERODYNAMICS OF A WING

FUNDAMENTALS OF THE LIFTING-LINE

THEORY

AERODYNAMICS I

The Biot-Savart Law

The velocity induced by the singular vortex line with the

circulation can be determined by means of the Biot-

Savart formula

3

( )( )

4VL

d

l x ξυ x

x ξ

Special case – induction of the

straight vortex line:

, ( )x x yd d x y l e x ξ e e

( ) x x zd yd yd l x ξ e e e 3/23 2 2( )x y x ξ

AERODYNAMICS I

From the Biot-Savart formula one gets

2

1

2 2 3( , ,0)

4 [( ) ]z

yx y d

x y

υ e

where 22

2

111

2 3 22 2 3 2

1 2

2 2 2 2

1 2

( ) 1 1 1

/ (1 )[( ) ] 1

1

( ) ( )

xx

yy

xx

yy

s x yy sd ds

ds d y y s yx y s

x x

y x y x y

Case 1 – induction of the infinite vortex line (equivalent to the 2D point vortex!)

2 2 2 2( , ,0) lim

4 2( ) ( )z z

x xx y

y yx y x y

υ e e

AERODYNAMICS I

Case 2 – induction of the semi-infinite vortex line segment [0, )

2 2 2 2 2 2( , ,0) lim 1

4 4( )z z

x x xx y

y yx y x y x y

υ e e

If 0x then (0, ,0)4 zy

y

υ e

AERODYNAMICS I

Flow past a finite-span wing – physical properties

AERODYNAMICS I

Lifting-line model of a finite-span wing

Flow past a wing is modeled by the superposition of the uniform free stream and the

velocity induced by a plane vortex sheet “pretending” to be the cortex wave behind the

wing.

The vortex sheet behind the wing is “woven” from continuum of infinitesimally weak

horseshoe vortices. These vortices are “attached” to the lifting line leading to a continuous

distribution of circulation along the wing span.

AERODYNAMICS I

The vortex sheet induces vorticity all around. The idea is to calculate the calculate the

velocity induced by this sheet on its front edge, i.e., along the lifting line. Next, it is

assumed that each infinitely thin slice of the wing generates the (differential)

contribution to the total aerodynamic force as it were a two-dimensional airfoil.

Each slice “senses” its individual direction of “free stream”, which results from the real

free stream vector V and the vertical (normal to the vortex sheet) velocity induces at

the lifting line in the point corresponding to the position of the wing slice.

According to the Biot-Savart formula, the infinitesimal contribution to the velocity induces

along the lifting line at the point 0(0, ,0)y is

0

( )

4 ( )

y dydw

y y

The total velocity induces at this point is obtained by integration

/2

00/2

1 ( )( )

4

b

b

y dyw y

y y

AERODYNAMICS I

Due to (generally) non-uniform distribution of the induced velocity along the wing

span, the effective angle of attack has an individual value of each wing section – see

figure below.

The direction of flow “sensed” by the

wing section at 0y y is rotated

clockwise by the induces angle

0 0( ) atan[ ( ) ]i y w y V

For small angles … /2

00

0/2

( ) 1 ( )( )

4

b

i

b

w y y dyy

V V y y

Clearly, an effective angle 0( )eff eff y .

AERODYNAMICS I

For small angles one can assume that the local lift coefficient changes linearly with the

(local) angle. Hence

0 0 0 0( ) [ ( ) ( )]L eff

c y a y y

Here, a denotes the slope of the lift characteristics for the wing section, 0 is the

angle of attack corresponding to the zero lift. Note that 2a if the thin-airfoil

theory is used. Note also that – in general – the angle 00 0( )y .

Next, we assume that the spanwise density of the lift force developed on the wing can

be computed from the Kutta-Joukovski formula, namely

210 0 0 02

( ) ( ) ( ) ( )LL y V c y c y V y

where 0

( )c y is local chord of the wing section

Hence, the local lift coefficient is 00

0

2 ( )( )

( )L

yc y

V c y

AERODYNAMICS I

Assuming that 2a , the local effective angle of attack is

00 0 0

0

( )( ) ( )

( )eff

yy y

V c y

Finally, the sum of the two local angles: 0( )eff y and 0( )i y is equal to the geometric

angle of attack . If the wing has geometric twist, this angle also depends of the

spanwise location, i.e., 0( )y .

Hence, we have obtained the following integro-differential equation for the

spanwise distribution of the circulation

/20

0 0 00 0/2

( ) ( )1( ) ( )

( ) 4

b

b

y y dyy y

V c y V y y

AERODYNAMICS I

One this equation is solved, then the spanwise distribution of the circulation is known.

The lift force developed on the wing can be calculated as follows

/2 /2

/2 /2

( ) ( )b b

b b

L L y dy V y dy

The (global) lift coefficient is

/2

/2

2( )

b

Lb

LC y dy

q S V S

The local contribution to the drag force is

sini i iD L L

The induced drag force is equal

/2 /2

/2 /2

( ) ( ) ( ) ( )b b

i i i

b b

D L y y dy V y y dy

AERODYNAMICS I

Thus, the coefficient of the induced drag is equal

/2

/2

2( ) ( )

i

b

iiD

b

DC y y dy

q S V S

Important case - elliptical distribution of the circulation

Assume 2

0

2( ) 1 yb

y

meaning that 2

02( ) 1 yb

L y V

We have 02 2 2

4( )

1 4 /

yy

b y b

Hence

/2 /2

00 2 2 2

0/2 /2 0

1 ( )( )

4 ( ) 1 4 /

b b

b b

y dy ydyw y

y y b y y y b

AERODYNAMICS I

Let us apply the following change of coordinates

1 12 2cos , siny b dy b d

Thus

0 00

00

cos( )

2 cos cos 2w d

b b

Conclusion: for the elliptical distribution of the circulation, the downwash

velocity is constant!

The induced angle is 0

2i

i

w

V bV

The lift force

/22 2 2

0 0 00/2

12 4

1 4 sinb

b

bL V y b dy V d V b

Thus, the maximal circulation is 0

4L

V b

AERODYNAMICS I

On the other hand 21

2 LL V SC

Hence 0

2 LV SC

b

and the induced angle is 02

2 1

2 2L L

i

V SC SC

bV b bV b

We define the aspect ratio of the wing 2b

S

Then, the alternative form of the formula for the induced angle for the elliptical

distribution of vorticity is

Li

C

AERODYNAMICS I

The coefficient of the induced drag is calculated as follows

/220 0

/2 0

2 22( ) sin

2 2 2i

bi ii L L

D

b

b C V SCb bC y dy d

V S V S V S V S b

Thus, we have obtained the formula

2

i

LD

CC

Consider the wing with no geometrical or aerodynamic twist.

Then, both and 0 are constant along the wing span. For the elliptical load

distribution the angle i is also constant, hence the effective angle of attack eff and

the lift coefficient 0

( )L effc a are also constant along the wing span.

Since ( ) ( )LL y c q c y then ( )

( )L

L yc y

c q

.

AERODYNAMICS I

Conclusion: the spanwise variation of the wing chord follows the variation of the

aerodynamic load. Hence, the planform of such wing is also elliptical!

AERODYNAMICS I

General lift distribution

Again, we use the transformation 12 cos , [0, ]y b

The elliptic distribution is expressed now as 2

0 0( ) 1 cos sin

Generalization: 1

( ) 2 sinnn

V b A n

We will need the derivative … 1

2 cosnn

d d d ddy d dy dy

V b nA n

The central equation of the lifting-line theory takes the form

0 0 0 01 10 00

2 1 cos( ) ( ) sin

( ) cos cosn nn n

b nA n nA d

c

AERODYNAMICS I

The Glauert integral appears again 0

0 00

sincos

cos cos sin

nnd

Hence, the main equation is transformed to the algebraic form

00 0 0 0

1 10 0

sin2 1( ) ( ) sin

( ) sinn n

n n

nbA n nA

c

In order to find approximate solution, we first truncate the infinite series …

00 0 0 0

1 10 0

sin2 1( ) ( ) sin

( ) sin

N N

L n n

n n

nbA n nA

c

… and make use of the collocation method, i.e., require fulfillment of this equation at

N different point [0, ], 1,..,m m N .

This way, the linear algebraic system is obtained for the unknown coefficients

1 2{ , ,..., }NA A A , which can be solved, e.g., by the Gauss Elimination Method.

AERODYNAMICS I

Once ( ) is known, one can calculate all aerodynamic characteristics of the wing.

We have

/2 2

01/2

2 2( ) sin sin

b N

nLnb

bC y dy A n d

V S S

We use the orthogonality of the Fourier modes 0

2 , 1sin sin

0 , 1

nn d

n

and obtain the formula

2

1 1L

bC A A

S

We see that only the first coefficient of the Fourier series is needed to calculate the

lift force coefficient!

AERODYNAMICS I

The calculation of the induced drag is more complicated …

We have

/2 2

10/2

2 2( ) ( ) ( )sin sin

i

b N

ni iDnb

bC y y dy A n d

V S S

We need expression for the induced angle, namely

/2

0

01 10 0 0/2

sin1 ( ) 1 cos

4 cos cos sin

b N N

n nin nb

ny dy nnA d nA

V y y

Hence, the formula for the iDC can be transformed as follows

2

1 10

2

, 1 0

2 sinsin sin

sin

2sin sin

i

N N

nD kk n

N

nkk n

b kC kA A n d

S

bkA A k n d

S

AERODYNAMICS I

Again, using the orthogonality property of the Fourier modes

120

0 ,sin sin

,

k mk n d

k m

the formula for the induced drag coefficient simplifies to the form

222 2 2 2 2

1 1 21 1 2 2 1

2( ) 1

2i

N N N Nn

n n nDn n n n

AbC nA nA A nA A n

S A

We can write shortly

2 2

(1 )i

L LD

C CC

e

where 2

22 1

Nn

n

An

A

and 1(1 )e (Oswald aerodynamic efficiency parameter).

Note that 0 , hence i iD D

elliptic wing any wingC C (optimality!)

AERODYNAMICS I

The most famous airplane with the elliptical wing:

Trapezoidal wings are easier to construct and to build. Theoretically, they are nearly as

good as elliptic ones if only the taper ratio (i.e., /tip rootc c ) is near the optimal value. In

the wide range of aspect ratios, ( 4 10 ), the smallest values of are achieved

when the taper ratio is close to 0.3. Other factors, like the stall pattern also matters!

AERODYNAMICS I

LEFT: Spanwise lift distribution for trapezoidal wing with different taper ratios.

RIGHT: Stall patterns for different planforms.

AERODYNAMICS I

Reduction of the lift slope

The finite span not only leads to the appearance of the induced drag – it also changes

(reduces) the slope of the “lift vs angle of attack” characteristic.

Denote:

Ldca

d - slope of the lift characteristic for the 2D wing section (equivalent to )

LdCa

d

- slope of the lift characteristic for the 3D wing.

Due to appearance of the

induced angle, the 3D wing

achieves the same value of

the lift coefficient at larger

geometric angle of attack –

see figure.

We have …

( )iLC a const

AERODYNAMICS I

Hence, for the elliptic wing ( )LL

CC a const

Thus 1

LdC aa

d a

For other planforms … 1 ( )(1 )

aa

a

Correction factor typically ranges from 0.05 to 0.25. The value of this factor can be

expressed by the coefficients 1 2{ , ,..., }NA A A