Rotating airfoils in steady flow

MATHEMATISCH INSTITUUT

RIJKSUNIVERSITEIT

GRONINGEÑ

REPORT TW-111

ROTATING AIRFOILS

IN

STEADY FLOW

by Felix Hess

da

ARCHL

Téchñische HogeschoI

Onderaf deli nische Hogeschoo, DJCUMENTATIE DATUM: oo C J N £ T A Il

t

ti

air sus

in

y

REPORT TW-1 11 by Felix Hess Completed March

1972

This work as supported by the

Netherlands organization for the advancement of pure research (z.w.o.)

Contents.

§ 1 Introduction. ₃

Chapter I. Motion of the system parallel to its plane.

§ 2

Winglet structures.

₄

§ 3 An example of a winglet structure. 11

§ 4

Linearized aerodynamic theory. ₁₃

§ 5

General character of the inteal equation and its solution. 17

§ 6

The collocation method. ₂₅

§ 7

Tackling the singular integral. ₂₈

§ 8

Derivation of the expressions (7.12), (7.18), (7.19).

₃₄

§ 9

Some elementary "spanwise" integrals.

₃₉

§10 _{The resulting forces and torques.}

Chapter II. Motion of the system under an angle r with its plane.

§11 Linearized aerodynamics.

₄₅

§12 Derivation of the expressions

(11.12)

and (11.13).

49

§13 Application to wingl.et structures. ₅₃

§14 General character of the integral equation and its solution. ₅₆

§15 Tackling the singular integral (first method). ₅₈

§16

Derivation of the expressions

(15.12)

and

(15.16).

62

§17 Tackling the singular integral (second method).

₆₅

§18 Derivation of the expressions

(17.28)

and

_(17.32).

71

Chapter III. Numerical calculations.

§19 The method of nunieri2al integration. 71#

§20

The distribution of pivotal points.

₇₉

§21 A numerical comparison with Van Spiegel's theory _{for circular wings.}

₈₂

§22

_{The winglet structures used in numerical examples. 84}

§23 Influence of the accuracy of the numerical integrations. 88

§24 Influence of the distribution of pivotal points. ₉₀

§25

_{A numerical comparison between programs B, CB and C.. 95}

§26

An experiinent. ₉₉

§27 Comparison between experiment and theory. ₁₀₅

§28 Pictures of some lift distributions. ₁₂₂

§29

References. 141

This report presents a method of tackling the aerod.ynamic problem associated with devices such as boomerangs or helicopter rotors, where a number of airfoils perform a rotational and a translational motion at the same time, while remaining approximately in one plane.

The problem in which we are particularly interested concerns the _{case where}

the shape and the motion of the airfoils are given and. the aerodynamic forces are

to be found. It is clear that, since the interaction between such devices and_the

medium through which they move is very complicated, a simplified model has to be

chosen, which can be taken as a basis for numerical calculations, in order to be

of practical use.

We have chosen the following model:

The medium is an ideal, inviscid., incompressible fluid. The theory is linearized, since in general non-linear theories present nearly insurmountable difficulties.

The airfoils are infinite in number, each having a vanishing chordlength; _they

form a continuum filling a plane region S. We name such infinitesimal airfoils:

winglets.

In this way a continuous distribution of lift forces is produced. _{The motion}

of the wingletsis such that it gives rise to a steady flow. Hencé the structure

of winglets as a whole can be looked upon as a sort of pervious lifting surface,

fixed in a steady flow.

A strip theory applied to the winglets gives the aerodynamic forces _{in terms}

of the motion of the winglets and the induced velocity field. _{To this steady flow}

problem lifting surface theory is applied, yielding the induced velocity in terms of the lift distribution on S. The resulting integral equation can be numerically solved by a collocation method.

In chapter I we treat the case where the motion of the undisturbed fluid

is directed parallel to the plane of winglets, and in chapter _{II we deal with the}

case where this motion is directed under an angle with the plane of winglets. Numerical results are given in chapter III. The accuracy of the numerical

results is influenced by: i2 the number and distribution of pivotal _{points, and}

22 _{the accuracy of the numerical integrations. In our method the second factor}

is completely separated from the first one. The number of steps in the numerical integrations is determined by a preset tolerance requirement.

In the last part of chapter III the theory is compared with the results of experimental measurements.

§2 Winglet structures.

The systems in which we are interested consist of a number of airfoils, not necessarily straight, lying approximately in one plane, named the B-plane. (e.g. boomerang, rotor of a helicopter). Such a system moves through a fluid (e.g. air, water), which is originally at rest with respect to an inertial frame I.F. and which, for our purposes, can be assumed to extend to infinity.

We introduce a right-handed cartesian coore.iate system (x,y,z), moving with the structure of airfoils in such a way that this structure occupies a fixed region

fig. 2.1. Structureof airfoils.

with respect to the (x,y,z)-system. The B-plane is taken to be the (x,y)-plane. The airfoils thusl.ie approximately in the (x,y)-plane, their projections on this plane occupy a region s. (see fig. 2.1). The (x,y,z)-system, and with it the region S, moves with a uniform velocity U in the negative x-direction with respect to I.F. through

the fluid. (Hence the (x,y,z)-system itself is an inertial frame). In addition to

this motion the airfoils may have velocities with respect to the (x,y,z)-system,

which differ from place to place. The airfoils may also change their shapes and their angles of incidence.

/Y

)X

an airfoil-'

its projection on S 3 31 Z 1 2

fig. 2.2. The local, coordinate system.

We also introduce a local right-handed cartesian coordinate system (1,2,3), which is related to the orientation of the airfoils (which, for the present, be a function not only of position but also of time). We do this in two steps

(see fig. 2.2). First we define the (1',23')-system. The 3'-direction is parallel

to the z-direction. The _{1 '-direction is parallel to the (x,y)-plane, pointing from}

the local leading edge towards the local trailing edge of an airfoil. The

2'-direction is parallel to the local spanwise 2'-direction of the airfoilts projection on the (x,y)-plane. The angle t3(x,y) is defined by identifying the 2'-direction with the direction of the vector (cosç, sine, o) as notated in the (x,y,z)-systeni. The spanwise direction of the airfoils deviates from the (x,y)-plane by a small angle y(x,y). Now we take the 2-direction parallel to the airfoilYs local spanwise

direction. Thus the 2-direction deviates from the 2'-direction by the angle _y,

also the 3-direction deviates from the 3'-direction by the same angle. The 1-direction

is parallel to the 1 '-direction. The transformation matrix for the rotation from

the (x,y,z)-system to the (1,2,3)-system is:

Next we introduce the Ttrelative velocity" _{'r(xY) which is the local velocity}

of the fluid relative to the airfoils:

_,

-,

_-,

_-4

Vr=V+V_W

where

x-direction

(2.2)

y = (vv,v) is the local velocity of the fluid with respect to I.F.,

-v = -(u,o,o) is the velocity of the (x,y,z)-system with respect to I.F.,

W= (WW11) is the local velocity of the airfoils with respect to the

(X, y, z)-system.

The components of these velocities are notated in the (x,y,z)-system. The components sine cosç3cósy - cossiny - cos sin3cosy - sin3siny o -sin7 cosy (2.1)

V = (U _{- W)sinJ3} W cosf3 . + y

sin3 -

V CO5

V = [(u - W)cos

- W sin]cosy

(-w _{+ y )siny}

+ cose +

+ vsin?cosy

V

1'3 - [(u - W )cosç3 + W sin]siny

+ (-w + V )cos

z z

The meaning of the dotted Une will be explained later on.

By definition of the 1-direction we have V O always. The function (x,y) may

have jumps of magnitude ir at points where _{Vn =} O, and at these points an individual

airfoil may suddenly have its leading edge turned _{into a trailing edge and reversely.}

Further we introduce the local "effective angle of incidence" _{of the airfoils,}

cL(x,y). By _{a. is meant the angle between the direction the vector (VrIOV}

), as

notated in the (1,2,3)-system, should have in order _{for the local lift force to}

fig.2.3. Angle of incidence.

direction of zero lift

vanish (direction of zero lift) and. the actual direction of this vector. The effective angle of incidence cx. is the sum of the geometrical angle of incidence a0(x,y) and the angle:

-. V

a.1(X,y) =

arctan ...!

_(2.11.)

V

ri

(see fig. 2.3).

-We assume that the interaction between fluid and airfoils is determined by

the

following conditions:

The airfoils cannot locally exert or experience forces parallel to their _local

spanwise direction, i.e. the 2-direction.

The airfoils experience a local force in the (1,3)-plane, normal to the direction

of , i.e. parallel to the vector (-V,O,V). The magnitude of this _force,

per unit of length along an airfoil, is given by

l(x,y)

=

i(V + y2), b(x,y) _CL(x,y) (2.5)

where p. is the density of the fluid, b is the local chordlength of the airfoils

and

CL their local lift coefficient.

e) CL is taken to be proportional to sins:

Remark: for a thin airfoil in two dimensional flow the factor of proportionality theoretically is 2ic,see [i].

These conditions mean that we apply a strip theory to the airfoils. The assumptions AB,C appear to be valid for a reál straight, airfoil of large aspect

ratio placed in a uniform flow, provided that the angle between the spanwise

direction and the flow direction is not too small, and that effects due to viscosity may be neglected. (see .11]). We expect that the assumptions will hold approximately in cases where the radiüs of curvature of an airfoil is large compared to its

chordl.ength. Generally, the conditions will not be satified everhere on S.

Exceptions are for instance the regions close to the wingtips of the airfoils and the -region where the airfoils are attached to an axle or a central disk.

The aerodynamic prOblem presented by the sems as they are described so

far would be rather difficult to solve mathematically. A time-independent case

would be considerably simpler to deal with. This is our motive for approximating the structures of airfoils by the föllowing.mod.el.

.

Devide-the reg-ion S into subregions s, each centered around apoint (x,y).

The number of s should not be too small, and in the.neÌghboiirng s. the physical

situationshould. not betoo different. Hence we assume that thè number of air-foil,s is rather large and that thé local properties of the airfoils, determined

-

by,,y,Ç7

do not differ much at neighbouring points (x,y).. We define the local

"filling factor" d by naming the fraction of s1 which is occupied by airfoils

d(x,y)... We shall, replace d, ,,y,bycontinuous'fuictions of (x,y) by replacing

every airfoil by a large number of similar airfoils with smaller chordlengths, in such a way that the local filling factor d remains unchanged. In the limit we have the region S occupied by a continuum of airfoils with vanishing chordlengths. We

shall, neme such infinitesimal airfoils winglets Such a system of winglets exerts

forces upon the fluid which are distributed continuously over S (to be moreexact,

over a two dimensional region close to s): (x,y) per unit of area. In general this

field of forces _{will be time-dependent. However, we shall. òonfine' our attention}

to those cases, only where _{is independent of time. Hence wé only consider cases}

where d(x,y), a(x,y), p(x,y) _{7(x,y), t(xy) do not, have an explicit time-dependence.}

From a somewhat different -and perhaps more rewarding- point of view we could say: the original airfoils have been "smeared out", so that at anr poit the

original situation has been replaced by a kind of time-average. The wiñglet model thus would correspond to a sort of time exposure of the original. airfoils.. If the original' system was periodic in time, its periòd shoúld be small cpared .o the

characteristic time D/U, where D is the diameter of S Thus even a two-armed

boomerang could be simulated b. a winglet model, provided that it would spin fast enough. A theory based.on the winglet model, of course, would at most yield

and on average velocities of the fluid.

The wingletz move (approximately) within the region S, but as far as the

interaction with the fluid is concerned, only the stationary field of forces -,

f(x,y) is of importance, and a steady flow results. For the magnitude _{of the}

forces per unit of area acting on the fluid we have, taking into account (2.6):

a(x,y) (V2

+V ):

2

ri r

where c depends on the local density and profile shape of the winglets. Comparing (2.7) with (2.5) we find:

f(x,y) = -.1c(x,y)sin

CL(x,y) c(x,y) = d(x,y)

2sin a.(x,y)

For thin airfoils we would obtain theoretically (see remark under C):

c(x,y) it . d(x,y)

We remark that the condition

(2.7)

(2.8)

(2.9)

(2.10)

which is part of condition B, follows from a general argument, which is valid if

viscosity is absent. Locally at the point (x,y,O) the winglets, per unit, of area

of S and per unit of time, do the work (x,y) . (?(x,y) _- on the fluid. On the

other hand the fluid there wins, per unit of area and per unit of time, the energy '(x,y) . '(x,y). Therefore: _'.

(i'-')

=

'. and

(+ V-

).

The components of in the (1,2,3)-system are:

fi = - f3

f2 = O

2 2

f3=-IcsinV

Jv

+v

ri ri r3

After substitution of (2.3) and (2.ti.), _{(2.1 1) shows how the forces ? depend on the}

fluid's velocity v. (2.11) gives a description of the interaction between winglets and. fluid which is a property of the winglet structure. On the other hand in section

a relation between. and will be derived which is aproperty of the fluid. This

relation will be a linear one, since we will linearize the equations of motion

for the fluid. An exact, non-linear theory would. be _{extremely difficult to work}

out. *)

Under what conditions is the application of the linearized theory to a winglet structure justified? The angle of incidence a.(x,y) should. be small, this means:

wz/J( 2 2

u-w) +v

«1

X

y

«1

«1

The conditions can be forniulated more exactly this way: Take a system with U,

a(x,y), f3(x,y), y(x,y), d(x,y), (x,y) given. Now reduce

'

y and w by a small

factor e, and let e tend to zero. Then y, w are of O(e) and so will be

V ,

y , y f . However, f , f will be of O(e2). The essence of the linearized

x y z z x y

theory is that velocities and forces of O(e2) are neglected with respect to those

of o(e).

In (2.2) the velocity components of

o(i)

are denoted by capital letters,

those of 0(c) by small letters. In (2.3) the ternis at the right-hand 'side of the

dotted line are of O(e). We shall now linearize. The sign _{between two expressions}

means that the relative difference between these expressions is of O(e). (2.3) becomes:

y

(u

_-

w)sin +

w cose

ri _Y

V

(u

_-

W)cos

- w sine

r2

Y

V

-,

[(u

_-

Wx)cos

- W sin]y - w

₊

We define the local "effective velocity" V(x,y) by:

.Jv 2 V 2 (U - W )sin

+ w

cose

_v

(2.1li.)

ri r x

_y

e

And the angle

e by:

[(UW)cosWsin]yw+v

V +

Sifl.+

(UW)s

o e

in+W cos3

' e

The linearized version of (2.7) becomes:

T(x,y) - c(c,y) [0(x,y)

v(x,y,o)

V(x,y)

(2.16)

V(x,?T)

fi - 3

e +

= o

f3 f

-4

The linearized components of f in the (x,y,z)-system are

As far as the linearized theory is concerned the fluid is acted upon by the force field

-4

f(x,y) _(0,0,f(x,y))

situated in the region S of the (x,y)-plane. Substitution for ie and V yields: fz - ic{a. +

{(U-W)cos-W sinj3]y-w +v

X

y

z z

(u -

W )sin + W cose) X

_y

2

x [(u -

W)sin

+ WY cosJ

X

(2.20)

This represents one relation betwe,en

_Ç

and v, which is a property of the winglet

structure. The linearized aerodynamics in section i. will lead to an integra1 relation

of the form

v(x,y)

ffK(x,y,,i-)

_f(,r1)ddr1 (2.21)

S

The equations (2.20) and (2.21) together constitute an integral equation for the

function f (x,y).

In some cases be known before hand, but in other cases some

of these quantities may depend on the forces which the winglet experience from

the fluid. (for instance cases with elastic bending). In those cases one or more additional equations may be needed to determine the behaviour of the system.

(Bemeniber that we only consider systems giving rise to a steady flow). A siinple

example of such a system is discussed in section

_3.

f X f y f z + .tc(a - tc(a. o - ic(a + a. e + a. e + a. e 2

+ )v

(a. + _!)sin e e e e 2

+ _)v

(a. + _)cos e e e e 2 + e e (2.18)

In this section a simple example is described of a system where Wz depends on the forces exerted by the fluid on the mechanism.

Up

fluid

fig. 4.1. Winglet structure as viewed

from

z-direction.

The wingletsin this case are straight, have length R and are directed radially

butward from a centre 0. They rotate with a constant angular velocity w around the z-axis. The winglets are hinged at O so that they can move in a direction

perpendicular to the (x,y)-plane. (There is a resemblance to the rotor of a

helicopter). The angle by which the winglets deviate _{from the (x,y)-plane towards}

the positive z-axis as viewed from O is supposed to be small _{(0Cc)). We call it 5.}

The region S is a circle with radius R.

We introduce the polar coordinates (r,cp):

x

r coscp , y= r sinq _(3.1)

We thèn have:

?= (-

wr sincp , wr coscp

, w(r,cp)) (3.2)

At points where wr + U sincp = _{O individual winglets interchange their leading and}

trailing edges. Hence:

f3

=

p . si(wr

¡J sincp) _(3.3)

y =

. sign(wr + U sincp) (3.4)

Ve

=

¡wr + U sinp (3.5)

(For a definition of W,f3,y,V see section 2).

dS

w(r,cp) =

r -

₌

_wr

-dt dcp

Substituting (3.2) through (3.6) into (2.20), we obtain:

which can be ritten as:

cIS

f

-U8coscp-wr+v

dcp z

f LC +

L o ₊ }twr + U

smp]2

which together with the expression for v of the form

(2.21)

constitutes an equation

for f and. 8.

z

The ang1e 8(p) is determined by the condition that the angular acceleration of a wingLet times its moment of intertia with respect to its hinge at O must equal the torques exerted upon the winglet. f(r,p) being the force in z-direction exerted upon the fluid per unit of area, the torque per unit of angle cp exerted on the winglets by the fluid is given by:

M(p)

= - f

_{(3 .8)}

The torque per unit of angle p due to the centriftgal forces is:

2

- 8(1)1

where I denotes the moment of intertia of the winglets per unit of angle _{cp. We}

now obtain as the equation of motion for 8: 2 I -_ = M(cp) dt2 2 R

r.

dp 1w2 o

(3.7)

(3.9)

(3.10) (3.11)

Thus we bave a set of coupled equations (3.7) and (3.11) for the two unknown functions f(r,p) and 8(p).

It may be remarked that for a stationary solution of

_(3.11)

the Fourier

expansion o± M(cp) cannot contain terms with sincp or coscp. This means that the load. distribution f(r,p) can have no first moments with respect to any line through the centre 0. This, of course, could be expected beforehand, since the hinges at O cannot transfer such moments.

§li. Linearized aerodynamic theory.

The behaviour of an ideal inviscid. incompressible fluid is determined by the following two equations (see [2]), the equation of motion:

dv-'

p = F - grad p

and the equation of continuity:

div '= O (Ii..2)

The used s,mbo1s have the following meaning: -9

y: velocity of the fluid, p: pressure in the fluid,

p: density of the fluid (constant), -3

F: external, forces per unit of volume acting on the fluid.

d

-The operator stands for + y

y

We shall, use a right handed cartesian coordinate system (x,y,z).

The-left-hand side of (1..1) is not linear in which makes an exact

treat-ment of the f'uid's behaviour very difficult. Therefore we shall work with a linear

-3 -3

approximation.. We assume F and y to be small of O(e) (for an explanation see

section

2),

and we neglect quantities o± 0(e) with respect to those of O(e). The

linearized theory tends to be exact as e tends to zero.

Taking the divergence of both sides of (li..1), we obtain: -3

div = div - div grad p

(li..3)

As to the left-hand side:

with -9 dv d. . -div = (div V) a,b= x;y,z.

Here the first term vanishes because of (4.2), and the second term is of O(e2) and can be negl.ected. in the linearized theory. Thus we obtain, correct to o(e):

-3

div F = div grad. p J

(4.5)

This Poisson's equation yields for an infinite region (see

[31, p. 323):

va Vb

p(x,y,z,t)

₌

fff -

div F(,,t)ddd

G

-4

We assumed that F vanishes outside and on the boundary of a finite region G. A more convenient expression for the pressure field can be obtained. Since

i -) 1_ -,

div - = - div F +

)-i-,tr itr

.r

"ItT and

-,

¡fi div

ddid =

= O

G

where the last integration is performed over the boundary of G, we have

-, div F

-«J

1tr dr,d1d = fff

ddrd

G

and instead of (ii..6) we obtain:

with

p(x,y,z,t)

₌

jtjj

ddd

G irr

'= (x - , y - , z

-From (ti..8) it can be seen that the pressure field can be regarded as a superposition of pressure dipoles.

From now on we confine our attention to cases where a field of external

-,

forces F moves uniformly through a fluid which was originally at rest. The cartesian coordinate system (x,y,z) is now chosen in such a way that its origin moves with the field of forces and the negative x-axis points in the d,irection of motion. With respect to this inertial frame we have a stationary field of forces in a

uni-form flow with a constant velocity U in the positive x-direction. This presents a

steady flow problem.

If the velocity of the fluid with respeet to thi system iE _with

'= (u,o,o) of o(i) and

_{= (v,v1,v)}

of o(e), the equation of motion (i.i) becomes

-) -, _,

1-4

1

(V.v)v + (v.v)v= - F - - grad p

and the linearized equation of motion is:

-+ _{-, i}

(V.v)v = - F - - grad p

(11.9)

We are particularly interested in fields of forces acting on a finite region S in the (x,y)-plane, with the forces directed parallel to the z-axis. In these cases the field of forces can be represented by:

with

(E) denotes Dirac's delta function). The pressure

field. (!..8)

now becomes:

-) -,

F(x,y,z,t) = f(x,y)5(z)

-4

f

₌

_(O,O,f(x,y))

p(x,y,z

The first of the equations

i..iIi.) yields

v=--X

where we have used. the condition _v - O for x -'

-.

An equation of the type

v(x,y,z) with ! -*0 for x

-,-has the solution

(x,y,z)

=

f

,y,z)d

where again we have used the condition

_-*

O for x -, -

.

In this way we

obtain

frcmi (li..1!i.) and (ij..13):

f (,T)z

3 S

4itr

with 2 2

21

r

= [(x-e)

(y-y)

+ z ]2

Differentiating under the integral

sign,

we obtain from (l..12):

- - .... 3z(x-) - ! r5 f(E,,1Ì)ddr1 .2

__Ljj3z(-iì)

1

r5

22

- -

j._jj

3z

-r

(, )

ddî1

z_ 1S

_r5

Outside S i..io) reduces to:

ddr1

}4.11)

The equations 1..i6) give the velocity field outside S if the field of

external forces on S is known. For our calculations we need, however, in particular

an expression for v on S. From (li-.16) it can be seen that, while v and. v are

odd functions of z y is an even function of z. We define:

z

v(x,y,O)

= hm y (x,y,z) (li-.17)

z-40 Z

From

the third equation

(ii.. 16)

we obtain after perfonning the integration

with respect to :

def

K(x,y,,i)

=

i

-v(x,y,z) _4-3TLV II

K(x,y,z,,i)f(,i)ddi

S with + 2 2 = 2

2 2

L [z ₊ (y-r1) I [(x-e)

+ () +

2-2

z (x-a)

2 2 [z ₊

_{(i) ][(x-)2}

+

z2]3/2

Substitution of z=O into (1.I..19) yields:

-1

2

_{

2

(.2o)

(i)

+ [(x)2

(Y-i)

]2 J

Obviously this expression has a second order singularity for r=y. Because of this

singular behaviour it is not permitted. to

simply

substitute z0 into

.i8)

in

order to obtain an expression for the induced velocity v on S. A careful analysis

yields

(see for instance R-I):

i

v(x,y,O)

=

him y (x,y,z)

=

z-,Q Z

where K is given by (1I..2O) andJ denotes the _Hadamard/ principal value of the

integral with respect to _i defined by

b

RPJ}

ft:ji)

_{di -}

f()]

¿

I

f(n)

j

thj=lim

2 2 _ß-40 (r-î)

a

t

a

(Ii..18) (ii.. 19)

(li..21)

v(x,y,z) ₌

v(x,z)

-v(x,y,z)

=

-1 ,,. z

f(,i)dd

.16) 2 2

23/2

'S'

_[(x-)

+ (y-n)

+ z _] X

3z(y-)

If ! _{2 2 2}

_5/2

d

S

-

[w _+(y'-r1)

+z I

x-E 2

_{- (3rT)j}

fI ! _{2 2 2}

_5/2

dw S

-

[-r

+(r-r)

+z]

In this section we shall investigate the general behaviour of the solution of the integral equation resulting froni (ii..21) and (2.20). It can be written in

the forni:

with

L(x,y) = P(x,y) - Q(x,y)

-f L

_{= -i}

itU i L(x)

=

P(x) - Q(x) -S

where K(x,y,,i) is given by (i..2o) and where we have used the abbreviations:

2

V

P =

c( -I-ct

U V e

Q,=

c-fl-The meaning of the symbols has been explained in section 2. Note that _Q, 0.

(5.1) is an integral equation (with singular kernel) of the second kind, and it may be expected that the behaviour of its solution near the boundary of S will

differ from the solution of the integral equation for an ordinary lifting surface, which is of the first kind.

First we shall, investigate the behaviour of the solution in x-direction. Some essential information can already be obtained by studying the siinpl,e

two-dimensional problem where the region S is a strip in y-direction: -1 x 1,

and where P and Q, do not depend on y. In this case (5.1), after integration with respect to r, reduces to:

+1

k(x,)

=

The solution of this integral equation can be explicitly found (see 15]), but for our purpose it is sufficient to consider the case where P and Q are

constants. We then have:

+ (x,y,,i1) L(F,,,1)ddi (5.1)

K(x,) L()d

(5.2) (5.3) (5.5)

Its general solution is:

_[5]

1 +1 1

L(x)=

E' [

f

(\-arctanQ

1-a)

.(1-).r-

+

c'}]

l-I-jQ2 -1 (5.6)

where C' is an arbitrary constant. Computing the integral in (5.6) yields:

1 L(x) P (i-x ' arctan-Q +

Ji2+x)

.{i 1-xJ

where C is an arbitrary constant. Thus L(x) has a singularity at x-1 ('pleading

-s,

edgett) of 0[(1x) U] with

i i

s1

= -

arctanQ

and a singularity at x.1

("trailing edge") of O[(i-x)

5]

with

1 -' arctan-Q

If _{Q tends to zero with P/Q remaining finite, we obtain the solution}

2P fi-x C

L(x)=----.

(5.8)

(5.9)

(5.10)

which corresponds to an

ordinary two dimensional flat plate placed in a homogeneous

fLow under an

angle P/Q.

For each value of C

_(5.7)

_{represents a different mathematical solution to the}

integral

equation (5.5). _{A definite choice for C has to be made to obtain the} unique solution corresponding to the physical system to which we want to apply

the theory. In the case of ordinary lifting surfaces the _choice

C

=

0

(5.11)

is generally made. It corresponds to the Kutta condition: the trailing

_edge

singularity vanishes. This condition is generally

_found

to be

in

agreement w:ith

experiments.

Further down in this section it is shown that porous lifting

sur-faces obey

an

_{integral equation similar to ours. It seems plausible that the}

Kutta condition would apply to these pervious lifting surfaces as well. We decide to make the choice (5.11) also for our winglet systems, but we recognize

We then have:

L(x) P

(i-x

arctanQ

1+x}

This solution has the leading edge singularity of order (5.10), but near the trailing

edge it tends to zero of OE(l_x)zt] with

zt = - arctan-Q, (5.13)

(5.12)

resultant flow through surface

fig.

5.2.

Porous lifting surface.

integral equation

(x,y)

L(x,y) = a(x,y) - i _L(,i1)ddy (5.15)

s

where K(x,y,,i) is the same as in (5.1). The two-dimensional _{case is treated in [6].}

Thus a winglet structure can be exactly simulated by a porous lifting

sur-face with angle of incidence

V

a(x,y) = _{= o+OEe)}

u

(5.16)

and porosity coefficient

o-(x,y) =

=

(5.17)

a z

We see that the limit Q, - corresponds to a vanishing porosity coefficient, Figure 5.1 gives an impression of the general behaviour of L(x) for different values

of Q.

Returning to the original three-dimensional equation (5.1), we see that the behaviour of L(x,y) near the boundary of S varies from place to place with Q(x,y) hence with the local density of the winglets and their local effective velocity V.

Thare exists a remarkable analogy between winglet structures and porous

lifting surfaces. With a normal non-porous lifting surface placed in a homogeneous flow the induced velocity must be such that the resulting flow is tangential to the lifting surface. With a porous lifting surface, howevér, there may be a leakage

through it. If this leakage is taken proportionalto the pressure difference between the two sides of the lifting surface, the induced velocity satisfies the equation:

v(x,y)

= -a(x,y) + cr(x,y) L(x,y) (5.11i.)

Here a denotes the local angle of incidence (see fig.

5.2)

and o the local porosity

After substibition of -i-1 z

v(x)l

f

L()

-1 (5.19) can be written as 4-1 +1

P P

L(x) L() 2t

JJ

x-F

ddx

-1 -1

which is easily seen to vanish. We stress that these integrals should include possible singularities at the leading and trailing edges.

The energy taken from the fluid at the leading edge thus must equal the work done on the fluid between leading and trailing edge. This per unit of time and per unit of length in y-direction, is given by:

=

-u3

f

L(x)

(5.20)

(5.21)

and the integral equation (5.1) tends to one of the first kind. On the other hand, in the limit Q - O the integral equation tends to the algebraic equation:

L(x,y) = P(x,y) (5.18)

In this case the induced velocity is of no importance. (In fact essentially (5.18) was used in a simple theory for boomerangs [7:1).

Because of the leading edge singularity a concentrated "suction" force of 2

o(e )

acting on the leading edge will exist. We assume this to be a local phenomenon,

depending only on the local singularity in the lift distribution. Therefore the energy associated with "suctioiiY forces taken locally from the fluid at the leading edge can easily be calculated by considering again the simple two-dimensional case. The total work done on the fluid in this ('ase is zero. This follows from a

general arg.mtent: the total work done on the fluid, per unit of time and per unit

of length in T-direction is given, correct to

_O(E2)

by:

+1

s

v(x)

1.LU

f

L(x) dx (5.19)

where the integral does not include singularities at the leading edge itself. L(x) is given by (5.12) and v(x)/U follows from

With a two-dimensional porous lifting surface the suction force is simply

directed in the negative x-direction; its strength per unit of length in y-direction is given by:

Table 5.1

L

=

P

_{+ Q}

This leads to:

+1 ,.., 2 2 = -.IU3

{

(i-X 'Ç

arctanQ

(!

)

arctanQ -1 ç.,..i

..Ji j2

T,)

- i +

which, after integration, yields:

122

-

i

22tQ

(1)

2 1 arctan

-W

22

2o--f=2U.

2 2 (Ii.o +1)

where we have used the relations (5.16) and (5.17). The factor

2 1

arctan

-A_It

2 2

(1,.o. +1)

contains the porosity dependence of the suction force (A1 for the non-porous case). That the strength of the suction force decreases rapidly with increasing porosity cari be seen from this table:

(5.

25)

(5.

211.) (5.25) (5.26) a

= -

arctan

r

₎ o A

o

o O .1 0.65 1.5!i. 0.002

.2

1.11.5

0.69

0.011.8

.3

2.75

0.36

0.256

.11.

6.16

o.i6

0.655

.5

o

T

For a three-dimensional winglet structure we have to use a modification of (5.2!i). Suppose that in the "chordwise" expansion of the lift flincton L(x,y) the local strength of the singularity is represented by a term

x (y)-x arctan {Q(x1(y),

ï)1

L*(x,y)

=

c*(y) (

t

)

(5.27)

XXl

where x1(y) and x(y) are the x-coordinates at the leading and. trailing edge respectively. We thenmust replace the factor

2

P

12

in (5.21i), which originates from (5.12), by c*2(y). Now we obtain for the local value of W:

3 arctan-Q

c*2

W

= iU

_(xt-x1)

1

where aU quantities have their local values.

Denoting by -f, -f1 the cartesian components of the local suction force per unit of length in y-direction, we have:

r (w-u)r

W =-W

Ex

x

sy y

with W from (5.28). We also have, because of condition A in section 2, or (2.18):

f cose + f sine ₌ 0

(5.30)

sx

sy

Thus, once c*(y) is

_known,

f and f51 are determined by

_(5.28)

through (5.30).

It is perhaps somewhat surprising that the lift distribution over a winglet structure should have leading edge singularities at all, the boundary of S being the locus of the wingtips of the individual winglets. It is true that systems

consisting of a finite (though possibly large) number of airfoils have a lift

distribution which vànishes towards the wingtips. Our theoretical winglet

structure, however, is a limit case, and it might be misleading to think of it as

being composed of individual winglets. It resembles a porous lifting surface.

Let us now consider the behaviour of the solution of the integral equation

(5.1) in y-direction. Since we are particularly interested in cases where the

region S is circular, we shall consider the case where S. is a circle with radius 1. obably we can obtain some essential information as to the behaviour of the

lift ftnction near the "wingtipst' y =

ti,

by considering the simple case where

(5.28)

=

L(î),

independent of . The integral in (5.1) then becomes:

L(i1)

2f

(y)

2 ..

(x_)2

()2}

dd

We choose x=O. Then the integral with respect to yields simply 2.J1_q2 and. (5.31)

becomes:

where we have put

a/i2 L(i1) ₌

_r()

_(5.33)

and made use of the assumption that r(1)

= r(-i) =

_{0. (5.1) now becomes (for x=O):}

2

(y-.q)

dr(r1) dr

=

-dr1

yi

r(y) i ' dr(n) di P(0,y)

+ Q,(0,y)

-2J1_y2 _{= !l.it dr} y-y

This integral is of the type which is examined in [5], chapter 17. By using the

results given there, it is easy to show that, in the special case in which P(0,y)

and Q(0,y) are contants,

r(y)

=

2..Ji-y2.L(y)

=

_0(..J1-y2)

_(5.35)

for y close to ±1. We assume that

_(5.35)

_{will generally hold if P(x,y) and Q(x,y)}

are functions corresponding to actual winglet structures.

Finally it may be remarked that the method for solving the _{integral equation}

(5.1) outlined in the following sections obviously can also be applied to porous lifting surfaces satisfying equation (5.1!,.).

(5.31)

(5.32)

(.3".)

+

§6 The collocation method.

From now on we shall, consider systems where the region S is circular. The

structure of winglets performs a translational motion and a rotational motion

around an axis through the centre of the circle parallel to the z-axis. A system

like this is described by the equations (Ii-.21) _{and (2.20). In our problem the}

motion of the winglets is given and the forces on the winglets are to be found.

Equations (2i..21) and

(2.20)

_{together constitute an integral equation for the}

load distribution f(x,y), which _{we have to solve. This integral equation can be}

written in the form (5.1)

L(x,y) = P(x,y) - Q(x,y)

_-

K(x,y,,i1)L()ddi1 (6.1)

where the meaning of the snnbols is given by (5.2). In this equation, the kernel

function, given by ()4.2o), is singular. The _{functions P and are known, they}

depend on the structure of winglets and the _{way it moves.}

In the ordinary lifting surface theory the _{integral quation is of the first}

kind; (6.1) however is an integral equation _{of the second kind. Yet our method}

for finding an approximate solution of (6.1) will not be essentially different from Multhopp's collocation method [ti.].

Suppose that L(x,y) can be expanded in _{a series of the te}

with

where G.() and H.() are known functions and. a.. unknown coefficients. Then the integral equation (6.1) reduces to

a. .G.H. = P -

_131J

This presents an equation for an infinite number of a.., which must be satisfied every where on S.

If we approximate L(x,y) by taking only a finite number, say n, of terms in the

expression (6.2), (6.3) presents an equation for the n unknown a.., which in

general. cannot be satisfied in _{every point of S. The a.. can then be determined}

by a least squares _{method. The expression}

G.(x)H. (y)

a.. JJ'KG.H.dd.7

J

(6.2)

where g(x,y) is a chosen (positive definite) weight function, should. be minimal as a function of the n a... In general this method requires an excessive amount

of numerical work, but a relatively simple way to determine the a1 is possible

if g is chosen to be a sum of n 3-peaks. The weight

function

then vanishes every

where on S except for n distinct points where it is positive. With this special

choice for g(x,y) (6.Li.)

is minimal

_if

s

rpr n

Jjlzza

[GH

J

s

ij

n a. .[G.H.

j

13 J. ₃

at each of the n pits.

This method is called the collocation method and the n points are called

pivotal points. For each pivotal point (6.5) gives one linear equation for the a..;

we thus have in stead of (6.3) a set of n linear algebraic equations for the n unknown coefficients a... These can be solved numerically on a computer. The

2 13

computation of the n integrations (one for each pivotal point for each term in

the expansion of L):

if

KG.H.

ddr1

s

will present the bulk of the work required for this method.

It is obvious that the distribution of the pivotal points over S should be

ttreasonablett (e.g. not all, of them clustered together in a small part of s), but

it seems very difficult to give definite rules for an ttLptimal" distribution.

A convenient choice for the coordinate system in our case is indicated

in fig. 6.1. The lines of constant _{are parallel to the x-axis and. the lines of}

constant 3 are lines of constant "chord

fraction".

_{Some of these lines are drawn}

in the figure, which resembles a picture of a world globe with meridians and

latitude circles drawn upon it. The pivotal points are chosen on the intersections

of the lines. A definite choice for the loatioi of the piVotal points shall be made later in section 20.

çrKGHdd) -

p]

g(x,y)ddi (6.1k)

JJ

13

S

(6.5)

fig. 6.1. Possible distribution of pivotal points.

§7

Tackling the singular integral.

We start from equation ¿)-i-.21), which for a circular region

S with a radius a

takes the form

or:

with:

lxU..v(x,y,O)

=

lI.ic v'(x',y',o

where

=

x/a

_,

y'

=

y/a

'

=

/a

_,

_ti'

₌

_ti/a,

f 2 Vf

V/j ft

= z +a -fsja2-112

-('ti)

2

-

('-ti)

+1 _-f..i1-TI'2

-.

If

('ti')

-1 _V.tit2

are

dimensionless quantities. For convenience we

shall drop

the primes from now

on and write: X,Y,,ti,V,f.

We introduce the following symbols:

qJl i12

X=

X X = sJ1_ti2 Y (7.1) + 2 1

[(x-c)

+

(rti)22

x?-,

+

_(7.2)

2 ('-ti')

}

7.3)

17.

X'

and X

will be used

as

new variables instead of

and X, while X and Y are mere

abbreviations. In (X',ti)-space the circular region S is transformed into a square.

We expandf (,ti) in

a series in ("chordwise") direction with coefficients

depending on the ("spanwise") coordinate

_ti:

M-1

=

E

a (ti)H (X')

_(7.5)

The function -(it-cp) replaces the customary cotgç) term of the Birnbaum series. The cotgp guarantees the correct behaviour of the load function near the leading edge of an ordinary lifting surface, while each of the terms yield the correct behaviour near the trailing edge in that case. Unfortunately, we cannot make such a claim for the series expansion (7.6) inouï case. On the other hand, since the behaviour of the load function generally varies from place

to place near the boundary of s (as explained in section 5), it is impossible

to obtain a series expansion of the type (7.5) which would yield the correct behaviour everwhere near the boundary of S. We found by actual numerical calculations that the customary seriès with the cotg-cp term gave satisfactory results only in cases with large values of Q, (as could be expected, see section

5).

Omitting the cotgp term, however, gave rise to solutions with a tendency

to build up a peak near the"leading edge". This peak, in particular for higher values of Q, could only be formed at the cost of a some times strongly oscillating

solution. The choice of the function (ic-cp) is a compromise which seems to give

better results than the two possibilities mentioned above. As to the "trailing

edge", our series (7.6) yields the correct behaviour only for large values of _Q,.

This is not so serious, since the load fuction vanishes there any way.

Note that, since the function (it-cp) remains finite at the "leading edge"

(x'= -1),

we have no suction force to take care of explicitely. More about this is said in section 10.

H(X') = sinpcp , p 1,2,...,M-1

p

H (X') = 1()

=

o

X' = -coscp

The functions a(i) are expanded as N

a(r)

=

a1 G1(i) with sirJ G1(î) = sine l 1,...,N

=

COS,, This expansion can also be written as:

with

N

a*(i)

_{a(t1)..J1_iì2 E a1.h1(i1)}

11

= sinli

}7.6)

(7.7)

}7.8)

J(7.9)

which is in accord.ance with (5.35). Substitution of (7.7) into (7.5) yields the

series expansion for f(,r):

N M-1

E E a1G1(i1)H(X')

1=1 p0

which corresponds to (6.2).

After substitution of (7.l,) and (7.5) into (7.2) we obtain

-c-1 *( \ -i-1 M-1 a rj X_X? i

v(X,y)

=

E 2 L _[(x-x')

21}Hx?'d

(7.11)

The expansion of the integral with respect to X? in (7.11) near o (or iy)

contains a term

-Y21Y1

_(7.12)

(for a derivation see section 8), which should be taken care of. Therefore we

write

(7.11.) in the form

+1 1'. M-1 a.*(y)

r

rr

₁₊

_11-H

_{(x')dx' +}

2 L

_[(x-x')

+

]2

J P 1 JL \Á., y / = '-' 2 2 +1 2 _-

M-lfdH\

( a \

(y-)

_lflIv-it

I

di1 - E

Ç .)

1 2

j

a*(i1) lny-,1d1 \

dx')

2 X _l-y J ç?

Xi-y1

Here we have replaced the factôr ?tnIYf from (7.12)

_by

lTl 2

2 1nI-ìI

l-y

(7.13)

which behaves essentiaUy the same near Y=0. The expression between rectangular

brackets in (7.13) is twice differentiable with respect to

i1 at ipy.

Introducing the functions:

=

₂ H(X')dX'

(dli)

2

lflIi1I

[(x-x')

+ y J2 x l-y g(x,y) we

can

write (7h13) as (7.10)

M-1

4t

v(X,y) = E

p=O

Next we define the function by:

f (X;y,r)

= f(x2)

_{+ (n-y) [}

_s

]

+

Near =y R(XTÌ) contains a factor

(_)2

and

.:!! R(X,y,,1) S(XY1-I) ₂

(y-n)

with:

T(X)

=

¡[(

Further we have: =

2f

H(X')dx'

(7.19)

f.(X,y,r) -1 2y L JII

=

_l-y

2xH(x)

P

The first expression follows directly from

(7.14)

_{and the second one is derived}

in section

8.

Now, using (7.9), we can write

(7.15)

as:

a* () 2 + g(X,y) (y-n)

_(\\

_lax'

_jr

'\

dx'

_{IX] X'-X}

¡1

for O

a*(1))1nIy_d

(7.15)

is continuous at iy. In section 8 an expression is derived for _{in the Um:t}

'1-i y: 2

_¡dH

2

S(XY,)

i p x'

il

2 X + +

lflí2(1_X1_} :

=

2 (X) + 1

_2 '

y

2 l-y 1- y 'XL 1-y for p1,...,M-1

(7.16)

(7.17)

7.18)

4ic v(X,y)

= Then we have: N M-1

Va

LP

1c1 Q h1(,1) 2 th1 +

()

[

_JI

The integrand. of (7.21) vanishes for

0 and

_it.

As a final expression for the induced velocity v(x,y,O) we now have:

NM-i

= ap1[b1(I) _c1(y)

I

10

L + (y) 5(X,y) +

{

f

+

f

}

S(X,y, -cos)sinlsind

o e

where b1(y), c1(y) and d(y) are given by (9.9), (9.10) and (9.il).

h1 ()

d,' +

Finally we have to substitute the expressions (7.6) for the functions H(X') into our formulas.

We put X = -coscl (7.20) + 5(X,y)

f

h1()

_!y-ld

+

f

h1()

The first three integrals in _{(7.20) can be evaluated analrticalLy. This is done in}

section 9. The last integral has to be evaluated numerically.

In order to avoid being forced to compute its integrand for _{too close to y,}

we choose y itself as an integration point and write the integral as:

f

=

{

f }

(7.21)

with

COSi ,

y

=

-cose

(7.22)

f

H(X')dX' ₂ (cossinp - p

sincos)

p2. .

.M-1 -1

p-1

lt

= p

f

[(+coscp)2 cospcp dcp o

lt

lt

r

-'

2

21

=

+ J

[(X-i-coscp) + y J2 coscp dcp o

=

-

sin -

(lt -

)cos)

The integral in the expression (7.lLt.) for f has to be evaluated numerically,

but it can be simplified by integration by parts:

+1

f

{1

2 1

(X')dX' =

+ Y

lJ

lt -' 2

21

=

i [(x+i) +

Y ]) -

f[(i-cosp)2

dq

,

o

By the method outlined in the sections ₇ through 9 the induced velocity at

each of the N.M pivotal points cam be expressed in

terms

of a linear conbination

of the N.M unknown coefficients a1. This reduces the integral equation to a set of N.M linear algebraic equations fòr the N.M unknowns, which can be solved easily with the help of a computer.

(7.25)

7.26)

=

-

sin2) _p=1

p=2...M-1

§8 Derivation of the ecpressions(7.12, (7.18), (7.19).

We want to investigate the behaviour of the "chordwise" integral in (7.11) in the limit ri -*y. Consider the expression:

Using X X

fH(X')dX'

=

f

H(x')dx'

+ (i-x) H(X)

w-x) ()

+ o(y3) (8.4) -1 -1

and

+1

I{1

X-X,

-1

+ [_xt)2

21}H

(x')'

p it can be written as +1 +1

X-X? -

X-X' X-X'

t'f{

2

2'

-i

Ix-x'I

y

[(x-x')

+ I

I-x'I

The first term of (8.2) equals

X

-f H(X')dX'

x

x

LV X 2 3

X-X=

-

-

y+.

y 0(Y)

..Jiri2

.Ji-y

J-y2

i-y2

we obtain for (8.3): r., X X 2 P

=

J

%(X')dX'

2yX

H (X)

+

f

H(X')dX'

Y

-1

i-y2

P

22

H(X)

yX

i-y2

+i2()X+0

The second terni of (8.2) equals

ra

r

-

sign(X-X')

H(X')dX'

J

"

2 2 r.' 2 (x-x') + + x-x'J [(x-x') +

y

]2 (8.1)

H(X')dX

(8.2)

(8.3)

(8.5)

(8.6)

(8.7)

We expand H(X?) in a Taylor series:

Making use of (8.5) we can write

[(i-x)

+ (x'-))

E (x'-)3

(

x

\fl-j flj flj+1\

L

j)

1-y2)

Y

+0Y

(8.9)

We thus have to consider expressions like:

which,

+1

rp0,l,2,...

Y-40 '- (x'-')2 + y2 IX'-I [(x'-)2

q= 0,1,2,...

The integrand in (8.10) is bounded for p 2, hence the expressions (8.10) vanish

for p 2, q. 1. For p 2, q. = O finite contributions are obtained. These

contributions to (8.7) can be taken together in the form:

him {H (x')-H (x)-(xT-x)(

Y-40_1

p

p

Taking the limit yields

+1

I

{H

(X') - H

(X) - (x'-x) (

)}

2(x'-X)Ix'-xI

dx'

p p

Next the case p

=

1:

yq-2

f{

Ix'-Integration yields

"t

since X = X

+ 0(Y),

can be written as

V

-

Y2 + Y21n 1.i.(1-X2

Y-40

03 H (X') = H (X) + E [(i-x)

+ (x'-))

(d

p p

_r1

_n!tflJ

"im Y-40 (x'-)

sign(X'-')dx'

r-, sign(X'-X)dx'

)}

_{(x'-)2Ix'-I[(x'-)}

2 2J

_+y

(x'-x)

2 í(x'-x)

+ y

J

i2

2

-

Y hziY + o(Y (8.8) (8.10) (8.11) (8.12)

(8.13)

"t

2 2 2

x'-x)[(x'-x) y iy

ln,X?_X{(XT_)2Y2]l}

(8.i)

(8.15)

This vanishes for q 1. The only contribution to (8.7) originates from q = O, it is given by

(Ofl\

)

f

lY

-+ ln

(1x2)}

(8.16)

Of the expressions (8.10) we finally have to consider those for which p ₌ 0:

+1 2 (8.17) him q-2

r

{si(Xt_)

-

[(x'-)2

+ y ]2J Y-40 -1 Integration yields

lun

q-2

Y-, O

which

can

be written as

1yq_2i.[ Y

2

Y-40

This vanishes for q 1. Again the only contribution to (8.7) originates from

q

=

O, it is given by

-x

H(X)

₂

p

Taking together (8.6), (8.12), (8.16) and (8.20), we obtain for the expression (8.1) in the limit Y -,0: X i

2y XH(X)(

f

H(x')dX'

_{y1_}

p - 1-X2

I y2

2 1 L

i-y

V)

dV

)I

IXt-XI(X'-x)

)

H(X) +

(8.18) (8.19)

(8.20)

+

o(nnIYt)

(8.21)

From

(8.21) it can be seen that indeed the expansion of the

integral

in (7.11)

contains

a term (7.12).

_{We now obtain for f(XY,ri), as defined by (7.11i.) in the}

limit r -by:

{çxi)

- H(X) -

(V-x

2 2

(y-i')

_(Y-'l)

=

2

_{+ dX'}

2'

[nI

+

lnil_y2)

+ O((y-)3lnIY-ri1)

X

_l-ri

(8. 22) where [...) stands for the expression (8.21).

. Only the second term in (8.21) contributes to

23r (X,y,r)

₌

₂ X H(X) _O((y-ri)1.n l-y If p 1 this equals: 1 +1

(p

1Pdx'

(ip

-()j

x-x'

X'-X+

j

'y

and (7.19) follows immediately.

Substituting

(8.23)

with (8.22) into (7.16) and taking the limit -,y, we obtain

(dJ{p

S(XYY)

₂ F(1+2y2 1

H

(x){2 X2-4ln[1(1 _x2)(1_y2)]}

i-y L i-y

i_2)

+1

dx'

+

f

-

H(X) -(x'-x)

(x'-x)Ix'-xIl

-1

The integral in (8.21i.) can be simplified as follows. Integration by parts leads to: 1 +1

r,'

(dHp

dx' X

+H(X)

₂ p X (dJI1

+H(X)

_2dJ('

p

+ dx' ₁ sinpcI cospp dq

(

)f

= _sine +

Ç

with X = -cose.

The second term of (8.26) has to be evaluated numerically.

If p = 0 (8.25) equals

The first term of (8.26) can be evaluated analytically, it equals:

H(-1)

H(1)

2(1+x) 2(1-X) (8.23) (8.25)

)

(8.26) + (8. 2h.)

)ln (

(8.27)

The first term of (8.28)

equals

1 1 2

___

lnhi.(i-x

rdH

X ₍

o\

ir + -1 i

dx'

(X) j +

i-x2

riJ

'-j

°

dx' Ix

li-(1x)

With these results (8.2ti.) can be written in the

form (7.18).

(8.

28)

§9 Some elementary "spanwise" integrals.

In this section we shall evaluate the first three integrals with respect to q occurring in equation (7.20), and some other integrals as well.

We have b1(y) = +1

-4 (Try)2

j

Try

dr1 =

/

ILt (ri) = sinl

= -cose

Further we put

y

= -cose

Presently we shall obtain expressions for some integrals which may be used

both in chapter I and in chapter II.

sinl sin

(cose - cos®)

-sinl sin

cose - cos®

h1(ri) lnIy-dri

=f

sinl _{sine in! cose - cos®ld}

(9.2)

(9.3)

(9.1.i.)

h(ri)dri

=f

sin1 sine d (9.6)

f

=

f

d

=f_sini

sine cose d (9.7)

g1

f

_(ri) d

f

sinl sin2 d (9.8)

-1 ₀

The first three of these integrals can be reduced to Glauert's integrals

(see for instance [8],

p. 173),

(9.14.) _{directly and (9.3) and (9.5) after integration}

by parts. We obtain

sinle

bL(Y) = -in

s in®

1f

1+1 l-

ìifl2

if i

(9.6), (9.7)

and (9.8)

yield easily:

e =0

=1t

= O

=:1c

= (1,-2)

1(12)

=

-{ cose - 1n2} if i i if i = if i 2 if i

=

2 if 1 is even if i IS odd J(9.1i) J(9.12)

§10 The resulting forces and torques.

In this section expressions are derived for the resulting forces and torques

fig. 10.1. Forces and torques acting on system.

acting on the structure of winglets. We are particularly interested in the

foUowing six quantities (see fig. 10.1): the forces in x-, y- and z-directions and the torques around the x-, y- and z-axes. The dimensionless equivalents of

these will be evaluated. In order to obtain the actual forces ¡nd torques one

should multiply the dimensionless quantities by 1.iU2a2 and 1U2a3 respectively. (a is the radius of the winglet structure).

The lift:

LFZ _{= ¡f - f (x,y)dxdy} (10.1)

s

The torque due to the lift distribution, with components:

LTX _{= ¡f - f(x,y)ydxdy} (10.2)

and

LTY _{= ¡f} + _f(x,y)xdxdy (10.3)

s

The force in the (x,y)-plane, with components:

DC

₌

f!

-

ç (x, y) dxdy

and

D'IZ

₌

If

_{(+ f(x,y)y - f (x,y)x)dxdy (10.6)}

s

y

In the first three of these integrals we substitute the series expansion for f(x,y) as given by (7.5) through (7.10):

N M-1 +1 +1 LFZ =

a1

f h.1,(y) f

H(X)dXd3r N M-1 +1 +1 LTX =

a1

f h1(y)y f

H(X)dXdy

N M-1 +1 +1

LTY =

-a1

f

h1(y) ..J1_y2

f

_H(X)XdXdY

After evaluation of the integrals with respect to X we obtain:

L2 =

LTX=

LTY=

L

DFY

= If -

f (x, y) dxdy y

where f and f are defined by (2.18).

X Y

And the torque around the z-axis:

-(a i-a 2 ol It'

(- a1

12

L=-It(a

i-a

4

01 11 LDC

=-It(a

12

i-a ₎ 02 12 LTY = 7t

ç,

(a

₊ h1 (y) dy h1, (y) dy

+ al)

f

(y)

1y2dy

The integrals with respect to y are evaluated in section

_9,

and. we have

(l-2) 1(1+2)

11

i odd. (10.5) (10.7) (10.8)

(10.9)

(10.10) (10.12)

+ a1) j

N M-1 -f(x,y)

=

E

E b1G1(Y) H(X)

1c1

p0

NM-i

-f7(x,y)

= E

Z c1G,,(7) H (X) p 1=1 p=0 N M-1

1

1

DFX

=

b1

f

_ktcsr)

f

_H(X)dXdy

N M-1 +1 +1

DFY = c

f

h(y)

f

_H(X)dXdy

N M-1

-i-1 +1

IXIZ = {_b.1, fh1(7)ï. fH(x)dxd ₊

The integrals can be evaluated, just like those for the lift distribution, and ve obtain:

When the integral equation for the load function f(x,y) is solved by the

method outlined in the section.s6 through 9, the induced velocity v can be calculated

either according to the algebraic equation (2.20) or according to the integral

equation (Li..21). The results will differ, since the solution for

_Ç

is not exact.

Only at the pivotal points there is agreement, and only there we can apply (5.18).

In order to evaluate the integrals in (10.4), (10.5) and (10.6) _{we shall expand}

the functions f(x,y) and fy(XY) in a series which is quite analogous to the

expansion (7.10) for f(x,y). The coefficients are determined by the requirement

that the series have the calculated values at the N.M pivotal points. Thus we put:

The equations (10.16) and (10.17) have to be satisfied at each of the N.M pivotal points, where

f and f are

known.

There result two sets of N.M linear algebraic

equations for the unknown coefficients b1 and. c1 which can be solved by straight forward methòds.

The expansions (10.16) and (10.17) can be expected to correspond to a y

calculated froni (2.20) rather that to a V calculated from (1i.21). (1i..21) would

yield a v having a logarithmic singularity at the leading _{edge. The "SUCtIOn}

forces" would then be shifted into the interior of S, so that they would be

incorporated. in the forces _{and f7. By using (10.16) and (10.17), however, strong}

fluctuations in f and f near the leading edge cannot be taken _{care of very well,}

and we may expect that "suction forces" will not be taken into account properly

If we accept these series expansions for f(x,y) and f(x,y) _{we obtain:}

(10.18)

(10.19)

DTZ = -

₀₂

+b

₁₂) N

(c1

c2)

E

i=1

(i-2) 1(12)

1, odd

(1 0.22)

(lo. 23)

Ak'Ii1' II. Motion of the system under an angle r with its plane.

§11 Linearized aerodynamics.

In this chapter we will consider the case where a field of external forces pointing in the z-direction is moving through an ideal fluid with a velocity having a component in z-direction. The field of forces is the same as the one considered in section 1i-. The conditions under which this model can be applied to winglet structures are discussed in section 13.

Again we choose a cartesian coordinate system (x,y,z) in which the field cf forces is at rest, and we have a steady flow problem.

fig. Ti.!. The system considered in this chapter.

Let the undisturbed velocity of the fluid with respect to the (x,y,z)-system be:

V = V.(cosjr,O,sinjr)

i with

v>o

,

o<J

it/2

_J

and the disturbance velocity field:

y = (y ,v ,v )

X

yz

The meaning of the symbols is the same as in section 2. As before, we are particularly interested in v. An equation of the type

A + B

= !(x,y,z)

with cp-*O, -4O for (x,y,z) .-,-c(A,O,B) has the solution:

(x,y,z)

=

f

y(x + AT,y,z + BT)dT

Hence we obtain from (11.2)

v(x,y,z)

=

F(x

+ V cos.T,y,z +

V sin.)d

with

-co

Again we consider the case where (to O(s)):

(x,y,z) = (O,O,f(x,y).6(z))

is given by (k.13). Now we

_can

write (11.3) as:

o

v(x,y,z) =

f

+ V cosllI.T,y) 6(z + V sinilr.r)d'r +

-co 0 2

3(z

+ V sinljr..r) - dT R

-+ V cosjr.,y,z V sin'ir.í)d.r (11.3)

The first

term

of

(71.5)

equals

R

=

[(x - + V

costr..r)

+ (y -q) + (z ₊ V sin*..r)2] J

.1p

----1p

1p

(11.2)

dz

it V cosijr + V

sinir

=

F i

V cosr-+V sinc-=-F

i____________

ivIsinivj

f (x

COS1J( z siriir z,y)

if

z.sinir> O and O ifz.sin'jr<Q

Using the abbreviations:

X

=

- x,

Y

=

-

y, Z

=

-z, T

=

-VT, C = CO5%fr,

s

sinir,

2 2 2

R

=

[(XTc) + y

+ (Z+Ts) ]2

we obtain for the second

term

of (11.5):

1LV

_So

[2(Z+Ts)2

-

(X+Tc)2

-

Y2]R5

dT

The integral with respect to T yields

2-

2 2

Z Z

Zs-Xc

c

K(X,Y,Z)=

3+

₂₂₊

_{2 2}

(p+o)p

(p+á) p

(p+a) p

(p+o)

with

1(11.9)

2 2

21

p={X Y

+Z

P,

o=Xc+Zs

We need an expression for v(x,y,O). 'Substitution of zO

in (11.9) leads to:

1

12

oi i

{2

Xc

K(X,Y,0)

=

(p+o)2

L

-

j

[[x2

Y ]

2

+ xc)

2 [X +Y2 2

]2J

(11.8)

This kernelfunction, however,

has a. singularity

for X=0, =0 i.e. for E=x ,

iy.

Therefore we shall consider v(x,y,z) for z0 and approaòh the plane z=0 from the upstream side, where the kernelfunction has no singularities. If sinr> O we must

approach from the side with z < O and conversely. This means that we have Zs > 0.

We only need to evaluate the integral (11.8) over a region G of sufficiently

small diameter containing the point x, ,=y (or X=o,

Y=o).

For the rest of the

region S we

can

simply substitute z0. It

turns out that the limit

Um

fi

K(X,Y,Z)

Z-,OG

Zs>0

over a smaU region G containing the point (x,y) depends on the shape of G. In section 12 derivations are given for two cases:

If G is a strip in e-direction with vanishing width, (11.11) yields: 2t

¡sinI

ç(x,y)

If G is a strip in ri-direction with vanishing width, (11.11) yields:

2ltIsin4r1 f(x,y)

Hence we obtain as expressions for v(x,y,O):

where (11.12) is used, and

f(x,y) lsin*I

v(x,y,o) = _{2jtV +}

where (11.13) is used. K stands for the kernelí'unction:

2 _(x-E)cos*

K(x,y,,ri)

1

[{()22())2

{

coz (11.16)

It can easily be checked that for

rQ

(11.16) reduces to the kernel of the integral

equation in chapter I: (1..2o).

In the special case If* K(x,y,,ri) vanishes and both (11.1ti) and (11.15) reduce to v(x,y,o) = 2i.tV v(x,y,o) = 2.1VIsinjrI + i ¡f K x,y,,r1) f(,ri)ddr (11.11,.)

¡f

K x,y,,ri) )dridZ (11.15)

§12 Derivation of the expressions (11.12) and (11.13).

In this section we evaluate the expression (11.11):

hin ¡f

K(X,Y,Z) f(,i)dd11 (12.1)

Z-30 G Z s>O

where K(X,Y,Z) is given by (11.9). For the small region G we choose two different shapes succesiv&Ly.

G is a part of a strip in e-direction of width 26, confined by a circle of radius e with e > 6. The point (x,y) is situated in the centre of this small region. Later on we shall take the limit 6

G is a region similar to the one described under A, but the strip is in

ii-direction.

A

2.-x,y)

8 ) B

fig. 12.1. The two shapes chosen for the region G.

We assume a Taylor expansion to be valid for

00

pq

=

i

p!q!

₍

îci

)(,)

p=0 q=0 1-dire ction >-direction (12.2)

It turns out that only the term with p=0, q=0 i.e. the term f(x,y) gives a finite contribution to (12.1) in the limit 8 -,0. We shall evaluate this contribution; it is not difficult to see that the contributions of the other terms in (12.2) indeed vanish.

Introducing the polar coordinates R,cp by:

X

=

R coscp , Y

=

R sinp (12.3)