Lifting Line Theory

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Chapter 14

Lifting line theory

The simplest quantitative model used for predicting the aerodynamic force on a wing of finite

span is the lifting line theory. It is covered in most texts on aerodynamics; e.g. Jones (1942,ch. 5); Glauert (1926, ch. 1011); Prandtl and Tietjens (1957, pp. 185210); Abbott and vonDoenhoff (1959, pp. 927); Milne-Thomson (1973, ch. 1011); Thwaites (1987, pp. 305311);Kuethe and Chow (1998, pp. 169200); Anderson (2001, pp. 351387); Bertin (2002, pp. 230256); and Moran (2003, pp. 135138).

14.1 Basic assumptions of lifting line theory

Lifting line theory in its simplest form assumes that:

the thickness and chord are much shorter than the span;

the wing is unswept; and

the flight is steady and perpendicular to the span.

With the above assumptions, the wing is modelled (on the length scale of the span) as theline segment

x = y = 0,b

2 z

b

2. (14.1)

This line segment is called the lifting line.

14.2 The lifting line, horseshoe vortices, and the wake

If the wing is to be producing lift, the KuttaJoukowsky theorem (3.66a) leads us to expect that

there will be circulation around it in circuits lying in planes parallel to the plane of symmetry(i.e. planes of constant z ) and encircling the wing.

14.2.1 Deductions from vortex theorems

Stokess theorem ( 13.2.1) then requires that there are vortex filaments running along the wing.In the lifting line theory, these filaments are assumed to run along the lifting line (14.1).

However, vortex filaments cannot end in the fluid ( 13.2.2) so this description is physicallyincomplete. Since, by Helmholtzs theorems, vortex filaments move with the fluid, the simplest

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158 AERODYNAMICS I COURSE NOTES, 2006

consistent model is that wherever each filament would end on the lifting line, it instead trailsbehind the wing (back to a starting vortex at the take-off strip, or, essentially, to infinity).

14.2.2 Deductions from the wing pressure distributionThis picture of vortex filaments trailing behind the wing is consistent with the difference in thespanwise flow components over and under the wing induced by the pressure differences associatedwith the generation of lift ( 12.1.2). When the upper and lower streams reunite at the trailingedge, they have the same speed (by Bernoullis equation, as the pressure is single-valued) andalso the same u and v components (by the KuttaJoukowsky condition); however, their spanwisecomponents are different: inboard for the upper stream and outboard for the lower. Thusthere forms a surface in the air behind the trailing edge across which the tangential (specificallyspanwise) component of velocity is discontinuous. This is a vortex sheet, composed of vortexfilaments, being the trailing legs of the vortex filaments inferred above from the vortex theorems.

14.2.3 The lifting line model of air flow

The model of the wing therefore consists of a collection of horseshoe vortices, each of whichconsists of a segment on the lifting line called a bound vortex (since it is constrained to movewith the wing rather than allowed to drift in the flow) and two semi-infinite vortex filamentsbehind the wing called the trailing vortices. Together, the bound vortices of all the horseshoevortices constitute the lifting line and represent the wing, and the collection of trailing vorticesrepresent the wake.

The flow around the wing is then taken as the sum of the contributions from the free stream

q = q(i cos +j sin ) (14.2)

and the horseshoe vortices constituting the lifting line and the wake.

14.3 Horseshoe vortex

The horseshoe vortices are significant in so far as they affect the flow near the wing. The boundvortex part of the horseshoe doesnt induce any velocity on the lifting line (on which it lies), butthe trailing vortices do.

Consider as an example, a horseshoe vortex filament of strength running from i + b2

k tob2

k to b2

k to i b2

k , as illustrated in figure 14.1. The unit vectors of the three legs are i ,k , and +i , respectively. The strength, , must be common to the three legs, by the vortexlaws ( 13.2.2).

The velocity induced on the line of the bound vortex (x = y = 0 ) can be calculated fromtwo applications of the formula for points in the perpendicular plane of the end of a semi-infiniterectilinear vortex filament (13.28)

q =

4

(r r)

| (r r)|2.

Here, we can take = i and r = b2

k for the two trailing vortices so that, for some point onthe lifting line r = zk ,

(r r) = i (z b

2)k = (z

b

2)j . (14.3)


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Lifting line theory 159

x

y

z

(0, 0, +b2

)

(0, 0, b2

)

Figure 14.1: A rectangular horseshoe vortex, lying in the zx-plane with vertices at b2

k .

Therefore,

q(zk) = 4

z

b2

|z b2|2 z +

b2

|z + b2|2

j (14.4a)

=

4

(z + b

2)2(z b

2) (z b

2)2(z + b

2)

(z + b2

)2(z b2

)2

j (14.4b)

=

4

(z + b

2) (z b

2)

(z + b2

)(z b2

)

j (14.4c)

=j

b

1 2zb

2 ; (14.4d)cf. Glauert (1926, p. 134), Kuethe and Chow (1998, p. 174), or Anderson (2001, p. 361).

Notice that, having assumed that the trailing vortices lie in the zx-plane (the y = 0 plane),the velocity induced at the bound vortex by the trailing vortices is purely perpendicular to thisplane; i.e. vertical. For |z| < b

2(i.e. on the bound vortex), v j q < 0, so that this velocity is

called downwash and denoted vw . The downwash is plotted in figure 14.2.

14.4 Continuous trailing vortex sheet

A difficulty with using a single horseshoe vortex to model a wing is that the downwash given by(14.4d) is infinite at z = b

2. This difficulty can be obviated by using a continuous trailing sheet

of vortex filaments. If the trailing vortex filament from zk to i + zk has strength (z)z, itcontributes

q(zk) =

(z

)zi

(z

z

)k

4(z z)2 (14.5)

=(z)j

4(z z)z (14.6)

to the downwash at r = zk. The total downwash vw is

vw(z) j q(zk) = 1

4

b/2b/2

(z) dz

z z. (14.7)


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DOWNWASH,vw

/(/b)

SPANWISE POSITION, z/b

Figure 14.2: The downwash induced along the lifting line by a horseshoe vortex.

14.4.1 Circulation distribution and the wake

Since vortex filaments cannot end in the fluid, and since it is the strength of the bound vortexfilaments that makes up the circulation around a section of the wing, the strength of the trailingvortex filament arriving at the lifting line at z = z must be related to the circulation there by

(z) + (z) z = (z + z) (z) + ddz

z (14.8a)

(z) d

dz; (14.8b)

as in figure 14.3. Thus, in terms of the circulation distribution, the downwash is

vw(z) =1

4

b/2b/2

d

dzdz

z z; (14.9)

cf. Glauert (1926, p. 135), Prandtl and Tietjens (1957, p. 199), Kuethe and Chow (1998, p. 174),or Anderson (2001, p. 364).

14.4.2 The form of the wake

In lifting line theory, the wake is assumed to be confined to a plane strip behind the wing. Infact, the situation is not so simple. We have seen ( 14.3) that the horseshoe vortex filamentsconstituting the wake generate a downwash inside the horseshoe. Outside the horseshoe, theygenerate an upwash. Since vortex filaments move with the fluid, the effect of this is that theouter filaments of the sheet drift upward and the edges of the sheet curl inward. These curlededges form what are called the two wing-tip vortices, and are clearly visible when there is dustin the air; e.g. in crop-spraying (Kuethe and Chow 1998, p. 172, figure 6.5). They are also made


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(z)

(z

+ z

)

(z) z

Figure 14.3: Relation between the strength of the trailing vortex filaments and the distributionof circulation along the lifting line.

visible very commonly because, being regions of concentrated vorticity and therefore high speedand, via Bernoullis equation, low pressure, they cause atmospheric moisture to condense intoliquid droplets, as in figure 7.3 of Bertin (2002, pp. 231233).

In spite of this very visible phenomenon, the model of a flat wake is still appropriate forcomputing the downwash on the wing. This is because the curling occurs while the air is passingdownstream behind the wing, so that immediately behind the wing the wake is flatter thanimplied by the photographs. From the BiotSavart law ( 13.5.1), the importance of a piece of avortex filament decreases like the inverse square of the distance; therefore, it is the nearer partof the wake that is more important.

14.5 The effect of downwash

We have established that the system of trailing vortex filaments in the wake cause a downwashat the wing. What effect does this have on the aerodynamics?

Basically, instead of experiencing the free-stream velocity (14.2), the section at r = zkexperiences that plus the downwash; i.e.

q(z) = (q cos )i + {q sin + vw(z)}j . (14.10)

14.5.1 Effect on the angle of incidence: induced incidence

One effect is that instead of experiencing the geometric angle of incidence

arctanq sin

q cos = , (14.11)


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the section experiences

(z) = arctanq sin + vw(z)

q cos , (14.12)

which is called the effective incidence (Anderson 2001, p. 354). For small angles (tan ) andsmall downwash (vw q) this reduces to

(z) +vw(z)

q. (14.13)

The angle

i(z) (z) =

vw(z)

q(14.14)

is called the induced incidence (Anderson 2001, p. 365).Equation (14.14) may be combined with (14.9) to give

i(z) =

1

4qb/2b/2

d

dzdz

z z ; (14.15)

cf. Anderson (2001, p. 365, equation 5.18).

14.5.2 Effect on the aerodynamic force: induced drag

In lifting line theory, we assume that the aerodynamics of each two-dimensional section is basi-cally the same as in two dimensions, except that the section experiences the free stream modifiedby the addition of the downwash.

Since the free stream is rotated by the induced incidence i , the lift and drag components ofthe aerodynamic force per unit span are altered in accordance with (1.3)(1.4) to

C = C cos i Cd sin i (14.16a)

Cd = C sin i + Cd cos i , (14.16b)

where the unprimed coefficients refer to two-dimensional conditions and the primed to three-dimensional. For perfect fluid flow Cd = 0 , and even when viscous effects are accounted for,Cd Cl for a functioning wing section below stall incidence (Jones 1942, p. 85). With the smallangle approximation for i ,

C = C (14.17a)

Cd = Ci . (14.17b)

Essentially, since the effective free stream has been rotated downwards and the aerodynamicforce acts, in accordance with the KuttaJoukowsky theorem, at right-angles to this, some of the

two-dimensional lift force has been rotated backwards and acts as a drag; this drag is called theinduced drag.

14.6 The lifting line equation

In (14.15), we can substitute for the local circulation using the KuttaJoukowsky theorem (3.66):

=(z)

q=

1

2qc(z)C(z) (14.18)


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Figure 14.4: A construction of the eccentric angle (14.20).

to get Prandtls lifting line equation (Abbott and von Doenhoff 1959, p. 20, equation 1.10)

i(z) = 1

8

b/2b/2

d(cC)

dzdz

z z. (14.19)

14.7 Glauerts solution of the lifting line equationThe lifting line equation (14.19) is more easily solved by changing the variable integration fromthe spanwise coordinate z to the eccentric angle via

arccos2z

b(14.20a)

z b

2cos (14.20b)

illustrated in figures 14.414.5 The lifting line equation (14.19) in terms of the eccentric angle is

i =1

0

dcC4b

d

d

cos cos . (14.21)

Comparing this with Glauerts integral (5.43)

0

cos n d

cos cos =sin n

sin , (14.22)

we see that if we can expand the spanwise lift loading as (cf. Abbott and von Doenhoff 1959,p. 22)

cC

4b=

n=1

An sin n (14.23)


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0

1

2

3

-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5

ECCENTRICANGLE,


Figure 14.5: The eccentric angle (14.20) used to described spanwise lift loadings.

so that

d

d

cC

4b=

n=1

nAn cos n , (14.24)

then the lifting line equation (14.21) gives the induced incidence as (cf. Abbott and von Doenhoff1959, p. 22)

i =1

0

n=1

nAn cos n

d

cos cos (14.25a)

=n=1

nAn

1

0

cos nd

cos cos

(14.25b)

=

n=1nAn sin n

sin . (14.25c)

Note that the lift loading (14.23) goes to zero at the wing tips z = b2

(i.e. = 0 and) , which makes sense since the pressure difference has to vanish there. Any function over b2

< z < + b2

could be represented by a trigonometric series like (14.23) but with cosines as wellas sines, but the cosines wouldnt vanish at the tips and so the coefficients of the cosine termswould have to be zero for a function vanishing at the tips.

Note also that if the lift loading is symmetric, An must be zero for all even n .


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14.7.1 Wing properties in terms of Glauerts expansion

Lift

Assuming we were able to determine Glauerts expansion coefficients An in (14.23), we couldcompute the wings lift coefficient from

CL

b/2b/2

dz

1

2q2

bc(14.26a)

=1

bc

b/2b/2

cC dz (14.26b)

= 4b2c

0

cC4b

d (14.26c)

= 2An=0

An

0

sin n sin d . (14.26d)

Using the trigonometric integrals (for integer m and n )

0

sin m sin n d =

2, m = n ;

0 , m = n ,(14.27)

the lift coefficient is simply

CL = AA1 . (14.28)

Rolling moment

Similarly, the lift distribution (14.23) could be used to compute the rolling moment (the com-ponent of the moment about the x-axis); of course, this vanishes when the lift distribution issymmetric:

c(z)C(z) = c(z)C(z) . (14.29)


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Induced drag coefficient

Using (14.17b) for the sectional induced drag coefficient, the wings induced drag coefficient is

CD = D12

q2

bc(14.30a)

=

b/2b/2

d dz

1

2q2

bc(14.30b)

=

b/2b/2

1

2q2

cCd dz

1

2q2

bc(14.30c)

=

b/2b/2 cCd dz

bc(14.30d)

=1

bc

b/2

b/2

cCli dz (14.30e)

=4

c

b/2b/2

cCl

4bi dz (14.30f)

= 2A

0

cCl

4bi sin d (14.30g)

= 2A

0

n

An sin n

m

mAmsin m

sin

sin d (14.30h)

= 2Am

n

mAmAn

0

sin m sin n d (14.30i)

= 2An

nA2n

2

(14.30j)

= An

nA2n . (14.30k)

Thus only the first term of Glauerts expansion (14.23) contributes to the lift, but all terms inducedrag; moreover, each term induces a positive amount of drag since squares are nonnegative.

14.8 The elliptic lift loading

Solving the lifting line equation for the lift distribution is quite difficult, but an easier problemis the investigation of a given lift distribution. Since only the first term in Glauerts expan-sion (14.23) contributes to the lift, lets examine first the lift distribution just consisting of this

term:cC

4b= A1 sin =

CL

Asin =

CL

A

1

2z

b

2, (14.31)

or in dimensional terms

=4L

b

1

2z

b

2, (14.32)

as plotted in figure 14.6. It is called the elliptic lift loading.


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LIFT,

l(z)/l(0)


Figure 14.6: Elliptic lift loading (14.32).

The corresponding induced incidence is given by the lifting line equation (14.25) as

i =CL

A, (14.33)

which is uniform across the span (Prandtl 1921, p. 191; Glauert 1926, p. 143; Abbott and vonDoenhoff 1959, p. 8; Milne-Thomson 1973, p. 202; Kuethe and Chow 1998, p. 179; Anderson

2001, p. 369; Bertin 2002, p. 243). The elliptic lift loading is clearly the only lift loading withthis property.

14.9 Properties of the elliptic lift loading

By (14.30k), an elliptically loaded wings induced drag coefficient is

CD = AA21 = A

CL

A

2=

C2LA

, (14.34)

(Prandtl 1921, p. 192; Glauert 1926, p. 143; Abbott and von Doenhoff 1959, p. 8; Milne-Thomson1973, p. 203; Kuethe and Chow 1998, p. 179; Anderson 2001, p. 369; Bertin 2002, p. 241; Moran2003, p. 142). Drag polars of relation (14.34) are plotted in figure 14.7.

14.9.1 Same lift coefficient, different aspect ratio

If we take two elliptically loaded wings with different aspect ratios (A1 and A2 ) and vary theirgeometric incidences (1 and 2 ) so that they have the same lift coefficients CL, the inducedincidences i,1 and i,2 must be related by

i,1 i,2 =CL

1

A1

1

A2

. (14.35)


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1.2

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0.0 0.1 0.2

LIFTCOEFFICIENT,

CL

INDUCED DRAG COEFFICIENT, CD

A=1

A=7

Figure 14.7: Theoretical draglift polars for elliptic lift loading and A = 1, 2, 3, . . . , 7 . N.B.:drag includes only induced drag, not profile drag (skin friction and form drag); cf. Prandtl (1921,figure 46, p. 193).

This follows by forming (14.33) for each wing and subtracting the two equations. Similarly, onforming (14.34) and subtracting, the induced drag coefficients must be related by

CD,1 CD,2 =

C2L 1

A1

1

A2

. (14.36)

These two equations, given by Prandtl (1921, p. 194), Glauert (1926, p. 144), Abbott and vonDoenhoff (1959, p. 8), Milne-Thomson (1973, p. 204), Kuethe and Chow (1998, pp. 182183),and Bertin (2002, pp. 244, 242), are extremely useful: they allow the prediction of the propertiesof a wing with one aspect ratio from measurements or computations of the properties at anotheraspect ratio.

14.9.2 Elliptic lift loading minimizes induced drag

The induced drag coefficient for the general lift loading (14.30k) can be expressed as

CD

= A

n=1

nA2

n= AA2

1

n=1

nAnA12

= CD,ell

1 +

n=2

nAnA12

, (14.37)where

CD,ell = AA21 (14.38)

is the induced drag coefficient of a wing with the same lift coefficient but with elliptic lift loading.Since the sum contains only nonnegative terms and vanishes for elliptic loading (for which A1 isthe only nonzero sine coefficient), a most important result follows:

The induced drag coefficient is a minimum for elliptic loading.


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14.10 Liftincidence relation

Say we know the liftincidence relation for infinite aspect ratio:

CL, = f() . (14.39)

Then for a finite aspect ratio, A, with elliptic loading the induced incidence is

i =CL

A, (14.40)

and at geometric incidence the lift coefficient is that for the infinite aspect ratio at geometricincidence

= i . (14.41)

Since, for elliptic loading, the downwash and induced incidence are uniform along the span,if the wing is untwisted, the sectional lift coefficient C will be too and the total lift coefficientCL will change proportionately.

14.10.1 Linear liftincidence relation

If the infinite aspect ratio (two-dimensional) liftincidence relation is linear,

CL = m( i 0) , (14.42)

but if the induced incidence is given by (14.33) then

CL =m

1 + mA

( 0) ; (14.43)

thus, if the two-dimensional liftincidence slope is m , the finite aspect ratio slope is

dCL

d =

m

1 + mA . (14.44)

For 0 A , this is less than m. Note also in (14.43) that the zero-lift incidence 0 isindependent of aspect ratio. The result (14.43) for a thin aerofoil (m = 2) is illustrated infigure 14.8.

14.11 Realizing the elliptic lift loading

If the effective wing section lift coefficient is the same function of the effective incidence acrossthe span (i.e. no twist), C is uniform too:

C = CL . (14.45)

For this to be compatible with the elliptical lift loading, since

c(z) (z)

1

2q2

C, (14.46)

the chord c(z) must vary like

(z) =2

q2

cCL

1

2z

b

2; (14.47)


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0.2

0.4

0.6

0.8

1.0

1.2

1.4

-10o

0o

10o

20o

LIFTCOEFFICIENT,CL

GEOMETRIC ANGLE OF INCIDENCE,

A=1

A=7

Figure 14.8: Theoretical liftincidence relations for thin wings with zero-lift incidence 0 = 5

and aspect ratio A = 1, 2, 3, . . . , 7 ; cf. Prandtl (1921, figure 47, p. 193).

i.e. elliptically

c(z)

c=

4

1

2z

b

2; (14.48)

cf. Kuethe and Chow (1998, p. 178), Anderson (2001, p. 370), or Moran (2003, p. 143).

14.11.1 Corrections to the elliptic loading approximation

Practical wings are rarely constructed with an elliptic variation of chord length, since this is moreexpensive to manufacture than rectangular or trapezoidal planforms (Anderson 2001, p. 374).Nevertheless, the simple results for elliptic loading are appealing and useful at least as a firstapproximation. They are frequently used in a generalized form, with correction factors. Forexample, in place of the elliptic loading induced drag coefficient relation (14.34), Abbott andvon Doenhoff (1959, p. 16) recommend a corrected formula which for untwisted unswept wingsreduces to

CD =C2LAu

(14.49)

where u is an induced-drag factor which depends on the taper ratio and aspect ratio. Thevalues of u may be read off the charts of Abbott and von Doenhoff (1959, figure 10, p. 17).

An alternative induced-drag correction factor is given by (Anderson 2001, p. 376).It should be noted that these correction factors dont change the induced drag coefficient by

more than about ten per cent over the practical range of taper ratio and aspect ratio (Anderson2001, p. 376), so that even the uncorrected formulae are sometimes useful for obtaining roughinitial estimates.

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Lifting Line Theory