Aerodynamic Analysis of Finite Span Wings - McGill...

Aerodynamic Analysis of Finite ... Span Wings

by

Awot M.Berhe

Department of Mechanical Engineering Mc Gill University

Montréal, Québec, Canada

August 2003

This thesis is submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements of the degree of Master of Engineering.

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Abstract

This thesis presents a new method of solution for the aerodynamics of finite-span

wings, which overcomes the difficulties of the previous methods. The present method

uses velocity singularities in the Trefftz plane (situated downstream at infinity) to derive

the contributions in the solution of the circulation distribution caused by the changes in

the spanwise variation of the wing chord and incidence. The new specifie functions

derived for these contributions contain both natural and forced symmetry and

antisymmetry terms, and thus represent a correct mathematical modeHng of the physical

problem. The correct mathematical representation of these contributions leads to a highly

accurate theoretical solution which is not the case in the previous methods.

The method has been vaHdated in comparison with the results obtained by theoretical

methods such as Rasmussen & Smith, and Carafoli for rectangular and tapered wings of

uniform incidence, and with panel method (Katz & Plotkin) results. Accurate theoretical

solutions have been derived for various wing geometries of aeronautical interest, such as

wings with curved leading and trailing edges, and wings with asymmetric incidence

variations caused by symmetric and antisymmetric deflection of flaps and ailerons(which

are more difficult to model using the panel methods).

Furthermore, the present method of solution has been extended to solve the problem

of swept wings. A procedure has been developed to specifically treat this problem.

Introduction

Résumé

Cette thèse présente une nouvelle méthode de solution, pour des ailes d'envergure

fini, qui surpasse les difficultées des méthodes précédentes. La présente méthode utilise

singularités des vitesses dans le plan de Trefftz (situé en aval à l'infinit) pour calculer les

contributions des changements de l'incidence et de la corde dans l'expression de la

variation en envergure de la circulation. Les nouveles fonctions derivés pour ces

contributions contient les termes de symétrie et d'anti-symétrie naturelles et forcées, et

représent une modélisation mathématique correcte pour ce problème physique. La

représentation mathématique correcte de ces contributions mène à une solution théorique

extrêmement précise, ce qui n'est pas le cas pour les méthodes précédentes.

La méthode a été validée pour les ailes rectangulaires et trapézoïdales à incidence

uniforme avec les solutions obtenues par Carafoli, Rasmussen & Smith, ainsi qu'avec une

méthode des panneaux (Katz & Plotkin). La solution théorique a été facilement obtenue

pour les ailes avec bords d'attaques courbes et avec incidence variable (les bords courbes

sont plus difficiles à modéliser avec la méthode des panneaux).

De plus, la présente méthode de solution à été aussi dévéloppée pour les ailes en

flèche. Une procédure a été développée pour le traitement spécifique de ce problème.

Introduction 11

Acknowledgement

1 wish to extend my sincere gratitude to my research supervlsor, Professor Dan

Mateescu, for his guidance, understanding and support throughout the course of this

pro gram.

1 would also like to thank Ms. Anna Ciand and Ms. Joyce Nault of the Department of

Mechanical Engineering for their help with administrative details throughout my stay at

McGill University.

Thanks to my family, my friends, aIl members of the local WUSC committee, and my

church congregation for their love, friendship and prayers, in the time when 1 needed

them most.

Introduction Hl

Table of contents

List of Figu:res............................................................................... VB

Chapte:r 1 Int:roduction................................................................ 1

Chapte:r 2 P:roblem fo:rmulation and p:revious studies. ................... 6

2.1 Problem formulation.. ................. . ......... ...... ...... . ............. . .. .. ...... . ... 6

2.2 Previous theoretical studies....... .......... ..... ......................... ............... 10

2.2.1 Physical considerations. ...... ... ...... ... .. . .... ... .. . ... ..... .. .. .......... 10

2.2.2 The classicallifting hne the ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 13

2.2.3 Method developed by Carafoli.... ....................................... ... 20

2.2.4 Method developed by Rasmussen & Smith.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22

2.3 Panel Method. .... ............ ... .. ......... ............ ....... ...... ... ... ...... ...... .... 26

2.3.1 Definition of the problem................. ................................... 26

2.3.2 The boundary condition on the wing................................... .... 27

2.3.3 The vortex distribution model........................................... .... 28

2.3.4 Lifting-surface solution by vortex ring elements......................... 30

2.4 Comments on the previous methods 37

Chapte:r 3 P:resent method of solution fol' st:raight wings................ 38

3.1 Method of solution........... ............................................................. 38

3.1.1 The prototype problem.......... ............................... .......... .... 40

3.1.2 The complete problem........................................................ 43

3.1.3 Determination of the unknown coefficients. . .. . . . . . .. . . . . . . . . . . . . . .. . .. .. 47

3.1.4 Aerodynamic characteristics............................ ............. ... .... 49

3.2 Solutions for symmetric wings with continuous incidence variation. ... ... .... .... 50

3.3 Solutions for symmetric wings with incidence and chord changes............. .... 51

------------------------------------------------------------ IV

3.4 Solutions for wings with asymmetric incidence variation........................... 53

3.4.1 Wings with antisymmetric incidence variation........................... 53

3.4.2 Wings with asymmetric incidence variation.. ... .. . .. . ... ...... . . . . . .. . .. 54

Chapter 4 Present method of solution for swept wings.................. 55

4.1 Method of solution ....................................................................... , 56

4.2 Collocation procedure........... ...................................................... .... 64

Chapter 5 Method validation and results.......... ............................ 65

5.1 Solutions for symmetric wings with continuous incidence variation........... .... 65

5.1.1 Method validation: rectangular and tapered wings............. .......... 65

5.1.2 Wings with curved leading and trailing edges and variable incidence 74

5.2 Solutions for symmetric wings with incidence and chord changes .............. 00. 76

5.3 Solutions for wings with asymmetric incidence variation........................... 79

5.3.1 Wings with antisymmetric incidence variation ................. '" .... ... 79

5.3.2 Wings with asymmetric incidence variation. .......................... ... 81

5.4 Solutions for swept wings. ................ .......... ...... ............................. ... 83

Chapter 6 Conclusion.................................................................. 86

References................................................................. ................... 88

Appendix - A................................................................................. 91

A.l The vortex filament, the Biot-Savart law, and Helmholtz's theorems.. .......... 91

A.2 Vortex singularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

A.3 Subroutine VORTEXL.................................................. ................ 95

v

Appendix - B Derivatives and IntegraIs appearing in the present

method.................................................................... 96

B.l Important derivatives............................................................ ......... 96

B.2 Important integrals. . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. .. . . .. . . . .. . .... 97

B.2.1 Integral J k (s ) . .. .. .. .. . . .. ... .. .. . . . .. . .. .. .. .. .. .. .. . .. .. . .. . .. . . .. .. . .. . ... 97

B.2.2 Integral Ik .............................................................. ....... 98

- 98 B.2.3 Integral Ek(s,y) .......... ................................................... .

B.2.4 Integral Ek(s). ... ....... ................... .......... ...... ......... ...... ... 99

B.2.5 Integral Êk(s). ..... ....... ..................... ..................... ......... 100

B.2.6 Integral Rk ........... ...................................................... ... 101

B.2.7 Integral ~..................................................................... 102

Appendix - C................................................................................. 103

C.l The prototype problem ....................................... '" ..... .... ... ... .... .. .... 103

C.2 The complete problem.................................................................... 106

C.3 Determination of coefficients.. ................ ...... ....... ... . ...... .. ... ... . .. .... .... 109

C.3.1 IntegraIs related to the improved solution procedure ......... , ...... .... 110

Y Integral Q;k ..... ...... ......................... .................. ... 110

~l 111 Y Integral Qi,k .......................................... " ............ .

A 2 112 Y Integral Qi k ••••••••••••••••••••••••••••.••••••••••••••••••••••••••••

C.3.2 Solution for symmetric wings with continuous incidence variation... 114

C.3.3 Solution for symmetric wings with incidence and chord changes. .... 115

C.3.4 Solution for wings with antisymmetric incidence variation.. .. ........ 118

Appendix - D..................................... ......... ....... .................. ...... .... 120

D.1 IntegraIs related to the swept wing problem .......................... '" ...... ... .... 120

D.2 Method ofsolution.............................................. ............ ......... ..... 125

----------------------------------------------------------Vl

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

Figure 2.17

Figure 3.1

Figure 3.2

Figure 4.1

Figure 4.2

Figure 5.1

Figure 5.2

List of figures

Typical wing geometry ...................................................... , 6

Various geometrical shapes ofunswept (straight) wings.... ... ...... .... 8

General planform of a swept wing.. .. . . . .... ... ... .... .... . ... . ... ... ..... 9

A wing section: a cambered Jouk:owsky airfoil.. .......... ... ............ 9

Top and front views of a finite span wing.. ... ............................ 10

Schematic representation of wing tip vortices. . . . . .. . . . . . . . . . . . . . . . . . . . ... Il

Effect of downwash on the local flow over a section of fini te span

wlng ................................................... , .... ...... ... ............ 12

Lumped vortex positioned at the aerodynamic center of the airfoiL . .. 14

Finite span wing replaced by a single horseshoe vortex line......... ... 14

Horseshoe vortex hne model and the induced velocity on the wing... 15

Lifting line model developed by Prandtl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 17

Vortex representation for the lifting surface model. . . . . . . . .. . . . . .. . . . . ... 29

Vortex ring model for a thin lifting surface .............................. ,. 30

Nomenclature for the vortex ring element .. '" .. , ...... .... ............ ... 31

Arrangement ofvortex in a rectangular array.......................... .... 32

Array ofwing and wake panel corner points(dots) and of collocation

points............................ ............. ........ .......... .... .......... ... 34

Sequence of scanning the wing panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . ... 34

Two sudden changes in the geometry placed at Ri and R2.. ..... .... .... 41

The complete problem.. ... ............. ....... ............... ... .......... ... 44

A bound vortex of variable intensity r(s) and a semi infinite free

vortex sheet of distributed intensity <5 r along the Ene of

aerodynamic centers ............ :.......... ....... ............. ............ .... 56

A vortex loop element for symmetrical swept wings..... ... ... ...... .... 59

Rectangular and tapered wings. Influence of the aspect ratio A on

the distribution of f(y) ...................... ... .......... .............. ...... 70

Tapered wings. Variation of 'l'and <5 with the taper ratio, q , for

--------------------------------------------------------------vn

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

various aspect ratios, Â ...................................................... 71

Influence of the taper ratio q on: (a) local lift coefficient

distribution, Cl (y) / CL' for Â = 9 (b) variation of the lift

coefficient distribution, C La = 2K Â , for various Â ..................... 72

Variation of f(y) for trapezoidal (q = 0, 0.25, 0.5, 0.75 and 1)

and elliptic wings. Comparison with Carafoli's solutions for unifonn

wing incidence................................................................. 73

Wings with curved edges and variable incidence: parabolic chard

distribution. Variation of rAy) = r(y )/[ 2 baR U J along span............ 75

Wings with incidence changes due ta control surface dejlection

(!laps).Variation of r(y) along span....................................... 77

Symmetric wings with incidence changes due ta control surface

dejlection (!laps). Influence of the location, s, of incidence changes

on variation of fT (y) = r(y )/[ 2 b a T U J ................................ 78

Wings with anti-symmetric incidence variation. Distribution of

fT (y) = r(y) /[ 2 baT U 00] along span for: (i) Linear incidence

variation a(y) =a rY . (ii) Incidence changes due ta anti-symmetric

dejlections of ailerons at three locations, y = ± s ......................... 80

Wings with asymmetric incidence variation due ta anti-symmetric

aileron dejlections. Distribution of rR (y) = r(y )/[ 2 baR U",] along

span.............................................................................. 82

Figure 5.10 Variation of r(y) for rectangular swept wings X = 20· and X = 25· 84

Figure 5.11 Variation of r(y) for rectangular swept wings X = 30· and X = 35° 85

Figure A.t The velocity induced by a vortex filament of arbitrary shape.... .... .... 91

Figure A.2 Velocity induced by a straight vortex filament at point P.. ................. 92

Figure A.3 The velo city induced within and outside the core of a vortex segment 94

Figure A.4 Nomenclature used for the velocity induced by straight vortex

filament ........ , . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . ... 95

-------------------------------Vlll

Chapter 1

Introduction

The first practical theory for predicting the aerodynamic properties of a finite span wing

was developed by Ludwig Prandtl and his collaborators at G6ttingen, Germany, during

the period 1911-1918, spanning World War 1. The classicallifting line theory developed

by Prandtl and Glauert [1, 2] established the foundation of the wing aerodynamics and

provided the tool for the computation of the aerodynamic forces, lift and induced drag.

Numerous other methods of solution have been developed for this problem in the last

decades based on the lifting surface theory, on boundary element (or panel) methods, or

using flow field solvers based on various computational formulations, such as finite

difference, finite volume or finite element. Sorne of these methods (including those based

on the lifting line theory) are presented in books and papers elaborated by Anderson (2),

Bisplinghoff et al [3), Carafoli [4], Karamcheti [5], Katz & Plotkin [6], Katz [7], Kuethe &

Chow [8], Milne-Thomson [9], McCormick [10], Robinson & Laurmann [11], Schlichting

& Truckenbrodt [12], Thwaites [13], Von Mises [14], and others. Although the recently

developed methods provide a good accuracy and can aiso take into account more complex

wing features, the methods based on lifting Une theory are still used for a fast estimation

of the spanwise load distribution and aerodynamic characteristics of unswept wings. The

lifting line theories may provide more physical insight on the influence of the wing

geometrical shape and incidence variation on the overaH aerodynamic forces (lift and

induced drag), and on the spanwise distribution of local lift.

In the classicallifting Hne theory, the wing is modeled by a spanwise-variable bound

vortex and a sheet of free semi-infinÏte vortices, wmch are introduced to satisfy the

Helmholtz vortex theorem. This leads to a mathematical formulation represented by a

singular integro-differential equation, which relates the spanwise distribution of the

circulation with the spanwise variations of the local wing chord and incidence, via the

normal-to-wing velocity component induced by the free vortex sheet. This formulation

leaves one with two types of problems; the direct and the indirect problems. In the

indirect problems, the circulation distribution is specified. If the spanwise variation of

introduction 1

either the local chord or the local incidence is given, it is possible to determine the other

variable which is not given, such that the requirement of having the pre-specified

circulation distribution is satisfied. This is easier than the direct (physical) problem,

where one is required to determine the circulation distribution for specified spanwise

variation of the wing shape and incidence.

For the direct problem, the classicallifting line theory uses an infinite Fourier series

expansion, which leads to an algebraic equation for the a priori unknown coefficients to

be satisfied everywhere on the wing. This infinite series is eventually truncated to a finite

number of terms, and the manner in which these unknown coefficients are determined

distinguishes the numerous lifting line methods developed for this problem.

The most popular schemes, such as that developed by Glauert [1], are based on a

collocation approach, which requires the resulting algebraic equation to be satisfied at an

equal number of points along the wing span, leading to the solution of a linear system of

algebraic equations. Multhopp [15] developed a more accurate approach using Gaussian

quadratures. However, the accuracy of the collocation methods depends substantially on

the selection of the collocation points, especiaUy when the wing chord and incidence

display sudden spanwise changes. Regardless of this disadvantage however, these

methods are widely used in recent books, such as [8, 16] to make calculations of finite

span wmgs.

One group of methods [17-19] have succeeded in avoiding sorne of the difficulties

encountered by the collocation procedures, by using variational formulations to determine

the Fourier coefficients of the circulation, which replace the need for specified discrete

collocation points.

Another interesting class of methods uses Fourier expansions for the wing incidence

and the inverse of the local chord, such as those developed by Carafoli [4, 20],

Kararncheti [5] and Lotz [21]. In particular, the method developed by Carafoli [4] is

based on efficient cosine expansions of the inverse of the chord, containing only three

terms to appropriate1y represent the tapered (trapezoidal) wings, and uses a Fourier

identification procedure to determine the Fourier coefficients of the circulation.

Recently, Rasmussen & Smith [22] developed a very interesting method based on a

rigorous Fourier series analysis to determine the coefficients of the infinite series

Introduction 2

expansion of the circulation. This method, which was developed only for symmetric

wings with symmetric incidence variation, uses Fourier-series expansions for the

spanwise variations of the local chord (not Hs inverse) and incidence in the case of right

left symmetry. The resulting infinite set of equations is then truncated to determine the

first N unknown coefficients by solving an algebraic system of linear equations. The

method was found to converge faster and is more accurate for the same level of truncation

than the collocation methods.

AU the lifting line methods mentioned in the foregoing are directly based on the

bound and free vortex scheme developed by Prandtl and use an infinite Fourier series

expansion of the spanwise variation of the circulation to express the integro-differential

equation in algebraic form. By using various schemes, aH these methods reduce this

algebraic equation (to be satisfied everywhere on the wing span) to an infinite system of

equations, which is then truncated to a finite number N of equations to determine the first

N unknown Fourier coefficients of the circulation. In these methods, the truncation

procedure is based on the assumption that the coefficients decrease rather monotonically

with their rank, which is not always the case for various wing configurations (e.g. for

several tapered wings, the fourth coefficient was larger by an order of magnitude than the

second and third ones, as calculated in [18] considering a Fourier expansion truncated to

four terms).

In general, the Fourier series expanSlOns are not convenient for the wings with

changes in the spanwise variation of the chord and/or incidence (the latter may be

generated by the deflection of control surfaces situated on the wing). In these cases, a

larger number of terms in the truncated Fourier expansions of the circulation is needed,

and the solutions are less accurate and contain spurious spanwise oscillations. The

inability of the Fourier series expansions to mode! the sudden geometrical changes is

shown by Mateescu and Newman [23] for flapped airfoils.

The aim of this thesis is to develop a new method of solution for the wings of finite

span based on velocity singularities, whlch represents a different approach in comparison

to the Prandtl & Glauert's lifting Hne theory and to the methods based on the Fourier

series expansion of the circulation. This method takes more rigorously into account the

sudden changes in the wing geometry and incidence variation (e.g. due to the deflection

Introduction 3

of control surfaces along the wing span), by deriving special functions representing the

contributions of these changes in the solution of the circulation. This method avoids thus

the difficulties introduced by the use of Fourier series expansions.

The methods based on velocity singularities (name first introduced in [24]) consist in

the determination of the specifie contributions in the velocity solution related to the

singular points on the wing or airfoil, which are characterized by sud den changes in the

boundary conditions. These contributions are determined by taking into account the

singular behavior of the fluid velocity at these points and satisfy aH other boundary

conditions, including Kutta condition at the trailing edge of an airfoil (these contributions

are similar to the Green functions associated to these changes).

In subsonic flows, methods based on velocity singularities have been developed for

the first time by Mateescu et al. [25-26, 23] for the analysis of the steady flows past rigid,

flexible or jet-flapped airfoils. More recently, velocity singularity methods have been also

developed for the nonlinear aerodynarnic analysis of airfoils (Mateescu [27]) and for the

Ullsteady flows past oscillating airfoils (Mateescu & Abdo [28]). Methods based on

velocity singularities have also been successfully developed for the analysis of wings and

wing-body systems in supersonic flows (Mateescu [24]). In aH these subsonic and

supersonic flow problems, the velocity singularity methods led to efficient theoretical

solutions in closed form.

In this thesis, the analysis of finite-span wings is carried out by considering velo city

singularities on the trace of the free vortex sheet in the Trefftz cross-flow plane (situated

down stream at infinity), in contrast to the velocity singularities placed directly on the

airfoil [23,26,27] or on the wing [24]. New special functions are derived in this method

for the contributions of the changes in the wing incidence and chord variation in the

solution of the circulation. They contain both natural and forced symmetry and anti

symmetry terms, leading to a correct mathematical modeling of the physical problem. The

forced-symmetry terms are not considered in the previous methods, which consider only

the natural symmetry terms in the Fourier series expansions, although in reality the chord

variation of tapered wings contains a forced-symmetry term, due to its sudden change at

the mot chord.

Introduction 4

For straight wings, the method is successfully va!idated by comparison with the lifting

line solution obtained by Rasmussen & Smith [22] for the rectangular and tapered wings

ofuniform incidence, and with results obtained with a panel method (Katz & Plotkin [6]).

Then this method is used to obtain efficient new solutions for wings with changes in the

wing incidence and chord variations, due to symmetric and anti-symmetric deflections of

flaps and ailerons, and for wings with curved leading and trailing edges and continuously

variable incidence (which are more difficult to be solved by panel or numerical methods).

The main results obtained with the present method have been presented by Mateescu,

Seytre and Berhe in Ref. [29, 30].

A procedure has also been developed to obtain solutions for swept wings. The

solution of the circulation distribution in the Trefftz plane (with a priori unknown

coefficients) is used to derive the velo city induced on the wing. Then a collocation

procedure is applied to solve for the coefficients. The results from this procedure are

cornpared with a panel method (Katz & Plotkin [6]). Although the results for swept wings

are not as accurate as the case of straight wings, the results are encouraging enough to

continue an extensive study to improve their accuracy.

Following is a brief summary ofwhat has been presented in each chapter of the thesis.

In Chapter 2 the general problem definition for finite span wings and previous

theoretical methods as weIl as a computational rnethod are discussed.

In Chapter 3 the velocity singularity method for finite span straight wings is fully

presented. In this chapter the general solution of the circulation in the Trefftz plane is

derived and specifie problems of various straight wing configurations solved.

In Chapter 4 the capabilities of the present method is extended to solve the problem of

symmetrical yawed wings. The general solution in the Trefftz plane (derived in Chapter

3) is used to derive the downwash in the plane ofthe wing.

In Chapter 5 the results obtained using the present method are validated for several

wing configurations in comparison with previous solutions and with numerical results

based on panel method. Results are given both for straight and swept wings.

In Chapter 6 concluding remarks and sorne suggestions for future work are pointed

out.

In the Appendix sections the details of the mathematical derivations are presented.

Introduction 5

Chapter 2

Problem formulation and previous studies

The geometric characteristics and physical considerations for thin wings of fini te span

are introduced in this chapter. The Prandtllifting tine theory, theoretical solutions based

on Fourier series, and as weB as the panel method by Katz & Plotkin [6] are presented.

2.1 Problem formulation

Consider a thin wing of semi span 2b placed in a uniform stream of velocity U co' The

fluid flow past this wing is referred to a Cartesian reference system bx, by and bz ,

where x, y ,and z are nondimensional coordinates, with the bx axis parallel to U co as

shown in Figure 2.1.

bY

Figure 2.1 Typical wing geometry

Problem formulation and previous studies 6

The wing geometry is defined by a specified spanwise variation of the local chord,

b c(y) , where c(y) is the nondimensional local chard, and by a specified variation of the

local wingincidence, a(y)=ag(y)+ac(y), where:

a g (y) is the geometric incidence of the local wing section with respect to the freestream

velo city, and

a c (y) is the added-by-camber incidence representing the effect of the wing section

cam ber that includes the effect of a local flap or aileron deflection of wings.

The following are also important characteristics of the wing geometry:

Wing area: S (includes part inside the fuselage)

Wing aspect ratio:

Taper ratio:

Sweepback angle:

Line of aerodynamic centers:

Â=(2b)2;S

q = 1-cr ; CR where cr is the chord at the tip

and CR at the root (centerline).

X = 0 for unswept or straight wings.

X ;::: 0 for swept wings.

connects the aerodynamic centers of the

wing sections (situated at one fourth of the

local chard from the leading edge of the

wing section).

Various straight wing plan forms are presented in Figure 2.2 and also a general plan form

of a sweptback wing is shown in Figure 2.3. A wing section (a cambered Joukowsky

airfoil) taken at a distance y along the span is also shown in Figure 2.4.


y

x

.... 1"

cry) = CR Y

h h

X if

x

y

X

Figure 2.2 Various geometrical shapes ofunswept (straigbt) wings


y

~!-____ ----------r-------L-______ l1-CT

b =1

x

Figure 2.3 General plan form of a swept wing

Camber line

Local chord c(y)

Figure 2.4 A wing section: a cambered Joukowsky airfoil


2.2 Previous theoretical studies

2.2.1 PhysicaI considerations

The physical mechanism for generating lift on a wing is the existence of a high

pressure on the bottom surface and a low pressure on the top surface. The net imbalance

of the pressure distributions on the two sides of the wing creates the lift. Throughout this

process of lift production however, a flow from the region of high pressure to the region

of low pressure is established around the wing tips. This flow around the wing tips is

shown in Fig. 2.5. Hence, on the top surface of the wing, there is generally a spanwise

component of the flow from the tip toward the root, causing the streamlines over the top

surface to bend toward the root, as sketched on the top view of Figure 2.5.

Top view (plan form)

( ::.eamline over U.I ~ top surface +

, ____ Streamline over the bottom surface

Wing area= S

Wingspan2b

0· Lowpressure ~

F~ont c:: ::::::::> Vlew ------------------------------------------

High pressure

Figure 2.5 Top and front views of a finite span wing

Problem fàrmulation and previous studies 10

Similarly, on the bottom surface of the wing, there is generally a spanwise component of

flow from the root toward the tip, causing the streamlines over the bottom surface to bend

toward the tip. These additional components of the flow in the spanwise direction clearly

suggest that the aerodynamic properties of a finite wing differ from those of its airfoil

section in which the flow is two dimensional.

The tendency for the fluid to flow around the wing tips has another important effect

on the aerodynamics of the wing. This flow establishes a circulatory motion which trails

downstream of the wing, i.e., a trailing vortex is created at each wing tip. These wing tip

vortices are sketched in Fig. 2.6. And these vortices downstream of the wing induce a

small downward component of air velocity in the neighborhood of the wing itself. This

surrounding motion of the flow induces on the wing a small velocity component (called

downwash and denoted by the symbol w) in th~ downward direction.

Figure 2.6 Schematic representation of wing tip vortices

The downwash combines with the free-stream velo city to pro duce a local relative wind

which is canted downward in the vicinity of each airfoil section of the wing, as sketched

in Figure 2.7.

Let us examine Figure 2.7 in detai!. The angle between the chord hne and the

direction of the freestream is the geometrical angle of incidenceag (for uncambered

airfoil). In Figure 2.7, the local relative wind is inclined below the direction of non

disturbed flow by the angle ai' called the induced angle of incidence.

Problem formulation and previous studies Il

Local airfoil section

a

1 __ ,a.. W 1

Local relative 'Wind

a : Twist angle

w ai = - - : lnduced angle

Uoo

a eJJ = a + ai: Effective angle,

Figure 2.7 Effee! of downwash on the local flow over a section of a finite span wing.

This vertical additional component of the velocity has two important effects on the local

section of a finite wing:

1. The angle of incidence actually seen by the local section is the angle between the

chord line and the local relative wind. This angle is given by a eJJ in Figure 2.7 and is

defined as the effective angle of incidence. Hence, the local section of the wing is

seeing a smaller angle than the geometric angle of incidence a eJJ = a + ai .

2. The local lift vector is aligned perpendicular to the local relative wind, and hence is

inclined behind the vertical by angle ai' as shown in Figure 2.7. Consequently, the lift

vector has a component in the direction of U 00' which creates an induced drag denoted

by Di in Figure 2.7,

Hence, we see that the presence of the downwash over a finite wing reduces the angle of

attack that each section effectively sees, and· moreover, it creates a component of drag

(the induced drag), Therefore, for a finite wing, the downwash reduces the angle of

incidence effectively seen by each section, and even for an inviscid flow, where there is

no skin friction, an induced finite drag exists, and d'Alembert paradox is not contradicted

since this drag is induced by the component of the lift vector.

From the above discussion, the behavior of a flow past a finite span wing is not

identical to that of its airfoil sections. The following sections will present the main

theoretical developments which enable the aerodynamic analysis of finite span wings.


2.2.2 Lifting line theory

The classical theory of lifting line was first developed by Prandtl and his

collaborators, for a thin cambered finite span wing at an incidence .

• :. Sindy of the local wing section

The first step was the analysis of the two dimensional flow around a local section of

the wing, that is an airfoil situated at the spanwise position y .

For such a flow, the circulation can be expressed as:

where,

K = (O.85 ... 0.9)2T to account for viscous effects

c (y) is the chord at the spanwise position y.

a ejJ is the effective angle of incidence. .

a g defines the geometrical angle of attack (geometrical twist).

ac defines the effect of the wing camber (or the angle of incidence at zero lift).

ai is the induced angle of attack.

a is the incidence relative to the uniform stream velo city U 00 •

(2-1)

(2-2)

As shown in the classical theory of thin airfoils, a model for this section of the wing

can be represented by a bound vortex sheet of distributed vortex strength r ( x). The total

circulation around the airfoil is hence obtained

r(y)= f r(x) <lx. (2-3)

The simple st mode! for the wing section, situated at the distance y from the Ox axis, is

based on the Lumped-Vortex method, which replaces the vortex sheet by a single

concentrated vortex, r (y) as given by (2-1), situated at the aerodynamic centre of the

airfoil (XCA = c /4), as shown in Figure 2.8.


z

r B

x c

Figure 2.8 Lumped vortex positioned at the aerodynamic center of the airfoil

This idea is the basis for the lifting-line model which replaces a finite wing by a

bound vortex tine along the line of aerodynamic centers as illustrated in Figure 2.9.

Finite wing

y

'-----l'--X ~ Finite wirig

replaced by a horseshoe vortex

Free-trailing vortex

Free-trailing vortex

Horseshoe vortex line

Figure 2.9 Finite span wing replaced by a single horseshoe vortex Une

However, to respect Helmholtz' theorems (stated in Appendix A), tms bound vortex

has to be extended by two free vortices trailing downstream from the wing tips to infinity.

This vortex (the bound plus the two free vortices), due to hs shape, is called a horseshoe

vortex .

• :. Horseshoe vortex Une model

As we discussed above, the simplest vortex line model for straight wings, which

respects Helmholtz' vortex theorem, is formed by a straight bound vortex filament

(perpendicular to the undisturbed free stream velocity) of constant strength r (y) , which


is extended at its extremities by two semi-infinite free trailing vortex lines of same

strength r (y) , paraUel to the fluid velocity.

z

(y)

\ \ \ \

F(OJ',O}

Figure 2.10 Horseshoe vortex Hue mode} and the induced velocity ou the wing

The bound vortex induces no velocity along itself; however, the two trailing vortices

both contribute to the induced velocity along the bound vortex, and both contributions are

in the downward direction.

Looking at Figure 2.10, the velocity induced by the two free trailing vortices at the

point p(x = O,y,z = 0) is:

w,(y}= ~(y) }[C05(0)-C08("/2)J- ?y) )[C08("./2}-005(iT)], (2-4) 4n s + y 4n s - y

r(y) s Wi(y)=-- 2 2 • (2-5)

2n s -y


This vertical induced velocity changes the direction of the fluid at the location P (0, y, 0)

situated on the bound vortex line by an angle,

(2-6)

Using (2-5),

r(y) s a;(y)=--- 2 2 •

21fUoo S - Y (2-7)

The effective angle of incidence, a ejf (y) , becomes hence,

aejf (y) = ag (y)+ ac (y)+ai (y). (2-8)

For a non-cambered thin wing section, ac(Y) = 0, and we get:

aefJ (y) = ag (y )+ai (y). (2-9)

Introducing (2-7) and (2-9) in (2-1), we get:

ï(Y)=KU.c(y)(ag(y)-~~~ s' ~y') (2-10)

The expression of the circulation satisfies Helmholtz' theorems (stated in Appendix A)

only if the product c(y) a ejf (y) is constant. However, this restriction is not satisfied by

actual finite wings. Moreover; the downwash is infinite at the tips, which is especially

disturbing .

• :. Lifting Une theO/y for unswept finite span wings

During the early evolution of finite span wing theory, this problem perplexed Prandtl

and his coHaborators. After several years of effort, a resolution of this problem was

obtained. Instead of representing the wing by a single horseshoe vortex, the idea came to

supenmpose an infinite number of horseshoe vortices. This concept, illustrated in Figure

2.11, allows a spanwise variation of the circulation r(y), and respects the assumption of

constant infinitesimal strength (df'ldy) filaments that do not end or start in the fluid.


r(y)

y

P(O,y,O)

y

x

Figure 2.11 Lifting Une model developed by Prandtl

In Figure 2.11, the infinÏtesimal downward vertical velocity induced at a point P ( 0, y, 0)

by the infinitesimal trailing free vortices is hence expressed as:

andhence,

Wi (y) =_1 1+1> dr dYl .

41l' -b dYl Y - YI

The following assumptions have to be considered for the lifting line model:

(2-11)

(2-12)

o Each wing section is modeled by a concentrated vortex of strength r(y) located at

the aerodynamic centre (approximately 1/4 of the chord from the leading edge).


oThe free vortices, fOlming the free vortex sheet, are assumed to extend to infinity

in the direction of the undisturbed stream, although:

1) They are deflected slightly dowl1wards by their own downwash field.

2) They eventually roll-up downstream mto two concentrated free vortices.

oThe induced vertical velocities, w j (y) are assumed small with respect to U",.

The downward vertical velocity (dowl1wash) induced at P(O,y,O) by the free vortex

sheet, given by the relation (2-12), changes the direction of the fluid velocity at this

location by the induced angle of incidence (using 2-6 and 2-7):

(2-13)

Thus, as the effective angle of incidence is expressed as

a ejf (y) = a g (y) + a c (y) + ai (y). (2-14)

The circulation r(y) around this wing section is given by equation (2-1), which leads to

the Prandtl 's integro-ditfèrential equation:

(2-15)

The only unknown remaining is r (y) ; aIl the other quantities, ag (y), c (Y), U 00'

and a c (Y), are known· for a finite span wing ûf given design at a given geometric angle

of incidence in a free stream of given velocity. This equation has to be solved in order to

determine the distribution of the circulation along the wing span. Various mathematical

studies have been made to solve this problem, and in particular a straightforward scheme

was developed by Glauert [1] .

• :. Glauert's solution for the lifting fine theory

This method [1,2] is based on Fourier series expansions of the circulation distribution

and uses a collocation procedure to determine the unknown coefficients of the truncated

Fourier series.


Assurning a Fourier expansion of the circulation in the form:

'" r(y)=4bU"" L4,sinne,

n=1

where B is related to the spanwise variable y by the following relations,

y = -bcosB,

dy = b sin B dB.

(2-16)

(2-17)

(2-18)

The induced vertical velocity, wj ' defined by (2-12) can easily be calculated taking into

account (2-17) and (2-18), one obtains the following expression:

Wj

= U'" ~ n4, [71" cosnB, dB]. 7T L....J Jo cosB] -cosB

n=]

(2-19)

U sing the classical solution of Glauert' s integrals we get:

00

L sinnB

W j =U", nA,,-. -. smB

(2-20)

n=]

By introducing (2-16) and (2-20) in (2-15), .the Prandtl's integro-differential equation

becornes:

'" '" C; sinB L 4, sinnB + f.1R L n4, sinnB = f.1R (ag +7 )sinB,

n=] n=1

where f.1R = KCR , and CR is the root chord. 4b

(2-21)

The coefficients 4, are the unknown and can be determined with a collocation method

by truncating the Fourier series after a certain order N and evaluating the expression (2-

21) at N different values of B to create a linear system of N equations and N unknowns

4,. The truncation process is based on the assumption that the 4, coefficients decrease

monotonously with their rank, which is not always the case for tapered wings [30). Thus,

the solution obtained for this type of wings is less accurate and contains spurious

spanwise oscillations.

Several other methods have been proposed by numerous scientists for the solution of

Prandtl' s integro-differential equation of circulation given by (2-15).


2.2.3 Method developed by Carafoli

The method developed by Carafoli [4] is based on an efficient co sine expansion of the

inverse of the chord. The circulation distribution is expanded in Fourier series, using the

same polar variable as in (2-17) and (2-18). Then (2-21) is used. But, the spanwise

variation of the inverse of the wing chord, C is expressed in the fonn:

CR sin e = /30 + 2/32 cos 2e + 2/34 cos 4e +. . . + 2/32m cos 2me, (2-22) C

where CR sine is part ofPrandtl's integro-differential equation in the form (2-21). C

The expression (2-22) is very convergent and usually only three terms, those are /30'

/32' and /34' are needed to represent the cornmon wing plan fonns.

Thus, in the general case of trapezoidal wings, for which the spanwise variation of the

chord is linear:

or, using (2-17)

C -=l+qcose, CR

where q is the taper ratio defined by q = 1- cr . C .

R

(2-23)

(2-24)

The expansion (2-22) reduces to orny three terms with the following expressions for the

corresponding wing form coefficients fJ2m , which are called wing-form coefficients.

/30 = 0.5 ( 0.383 + 0.924 ), 1-0.924q 1-0.383q

2/32

= 0.707( 0.383 0.924), 1-0.924q 1-0.383q

2/34=0.S( 0.383 + 0.924 )_ 0.707 , 1-0.924q 1-0.383q 1-0.707q

Problem fàrmulation and previous studies

(2-25)

(2-26)

(2-27)

20

The relation (2-25,26,27) have been obtained by imposing equation (2-22) with m = 2 to

be rigorously satisfied at three convernent locations.

Thus for example, these coefficients, P2m' have the following values:

For a rectangular wing (q = 0):

Po = 0.6535, 2P2 = -0.384, 2/34 = 0.0536,

and trapezoidal wing with q = 0.5 :

Po = 0.9265, 2P2 = -0.304, 2/34 = 0.1855.

And in the case of elliptic loading, these coefficients are obviously:

Po =1, 2P2 =0, 2/34=0,

Furthermore, the spanwise variation ofthe incidence can be expressed in the form:

N

(a+r)sinB = Lan sinnB. n=J

By introducing (2-22) and (2-31) in the equation (2-21) one obtains,

N

J-iR LnA" sinnB+(/3o +2P2 cos2B-:2/34 cos4B+·

N

= J-iR L an sinnB n=J

The equation (2-32) can be recast in the form:

(nJ-iR + Po) A" + P2 (An_2 + A,,+2) +. . . + P2m (A,,-2m + A,,+2m) = J-iRan .

(2-28)

(2-29)

(2-30)

(2-31)

(2-32)

(2-33)

Since the coefficients A,,+2,An+4 , ••• are usually very small starting from a certain order

n = N , equations (2-33) become a recurrence relation which can be used to determine the

A" coefficients of the circulation.

Problem fOrmulation and previous studies 21

2.2.4 Recent method developed by Rasmussen & Smith

Recently, a very interesting analysis for wings of finite span with symmetric

incidence distribution with respect to the root chord has been proposed by Rasmussen &

Smith [22]. The forthcorning rnethod solves the classicallifting line equation (2-15) with

a rigorous Fourier series analysis, using Fourier series expansions for the chord and

incidence distribution. The circulation distribution is then obtained explicitly in terms of

the Fourier coefficients and does not depend on an arbitrary distribution of the collocation

points. An infinite set of algebraic equations is obtained for the circulation coefficients,

which must be truncated arbitrarily for a desirable degree of convergence.

Consider equation (2-21), wruch can be recast for wing of serni-span b :

where,

mo = 2iC is the section lift curve slope.

mOr is the value of mo at the root chord.

CR is the value of C at the root chord.

SnCB) = sin nBjsin B ,

(2-34)

(2-35)

(2-36)

The variation of a along the span is the twist, the variation in a g is the geornetric twist,

and the variation in aOL = -ac is the aerodynarnic twist in the present notation.

);;> The Fourier-Series analysis.

The Fourier sine series representation of the planforrn shape c = c(B) Cafter substitution

in polar coordinates y = b cos e for the right-half of the wing in case of the assumed

symmetry ofwing incidence) can be written as:

(2-37)

where only odd terrns are taken into account for right-left symmetry


i"/2 c( B) . C2n+1 = --sm(2n+l)BdB,

o CR

n = 0,1,2,3 .... (2-38)

The plan fonu area Sis:

S= Î c(y) dy=b f" c(B)sinBdB= TCbcR CI' lb Jo 4 (2-39)

The absolute angle of incidence variation a (B) can be represented by a Fourier cosine

series (with even terms only for right-left symmetry):

00

a(B) = L a2n cos2nB, (2-40)

n~l

ao is the average angle of incidence ao,

2 i"/2 ao=- a(B) dB, TC 0

(2-41 )

4 i"/2 a2n = - a (B)cos 2nB dB, TC 0

n = 1,2,3, .... (2-42)

Considering the odd values of n for the function Sn (B), defined by (2-35), we note

S3 = 1 + 2 cos 2B

S5 = 1 + 2cos2B+2cos4B '

hence,

n

S2n+l = 1+ 2 cos 2mB . (2-43)

m=1

Using (2-43) the induced angle of incidence can be vvritten in tenus of a Fourier cosine

senes,

(2-44)

where,


'" Bo = L(2m+l)A2m+p

m=O (2-45) 00

B2n = 2 L (2m + 1) A2m+p n = 1,2,3 ... m=n

With the above notations and definitions, the classical lifting hne equation (2-34)

becomes:

:;, [~:C'''I sin(2k + I)e][ ~) a" - ~:R B" )COS2se ] ~ ~o: 4"'1 sin(2n+ I)e.

The following normalizations are useful to be used:

a~ = 1, n=0,2,3, ... ,

c; = 1, c· = C2n+1 2n+1 C

1

n = 0,2,3, ... ,

A• = A2n+

1 ° 2 3 2n+1 C ' n = , , , ... , laO

<Xl

B~ = L(2m+l)A;m+1' m=O

00

B;n = 2 L (2m + 1) AL+p n = 1,2,3, ... , m=n

(2-46)

(2-47)

Assuming that mo is the same at every spanwise location (mor = mo) , equation (2-46) can

be recast as:

[ ~/;hl sin (2k + 1)8 ][ ~};, - ;; B;}OS2S+ ~: A;"éin (2n + 1) e .

(2-48)

The product of the sine series and the co sine series on the Left side of equation (2-48) can

be resolved into a single sine series by using the trigonometric identity:

sin(2k + 1)8 cos2sB = ~ [sin (2k+ 1 + 2s)B +sin(2k+ 1-2s)BJ. (2-49)


After the sine terms are collected on the left side of equation (2-48) the respective

coefficients of each term of sin (2n + 1) e on the left and right sides can be equated.

The results are:

(2-50)

n = 0,1,2,3, ...

For the direct problem, the A;n+l contained in the B;n (equation 2-47) can be

separated out in equations (2-50), and like terms collected. When this is done, the result

can be cast in the form:

cc L C(2m+IX2n+1)A;n+l = D2m+1, ln = 0,1,2,3, ... , 12=0

(2n+l)mo ~ • C(2m+l)(2n+l) = b(2m+l)(2n+l) + 7Z'A .L..J C 2k+1 '

hlm-ni

b(2m+I)(2n+l) is the .Kroneker delta function.

(2-51)

m,n = 0,1,2,3, ... , (2-52)

n=0,1,2,3, ... , (2-53)

Equations (2-41,42 & 43) constitute an infinite set of linear equations for A;n+!' when

C;n+l and a;" are given, together with mo /7Z'A. The system is then truncated and the

remaining finite algebraic linear system can be solved. The elements of the square matrix

and the right hand side are fixed (depending on the spanwise chord and incidence

distribution) and do not change with the level of truncation. This is in contrast to

collocation methods, for which the clements and number of elements change according to

the location and the number of collocation points.


2.3 Panel method (Katz & Plotkin)

Numerical approach for three dimensional wings have extensively been developed

over the last several years, especially because of the improvement of the computer

processing speed which allows us to solve complex type of flow even on a personal

computer. Here the panel method developed by Katz & Plotkin [6] is presented.

2.3.1 Definition of the problem

Consider a finÏte wing in a unifonn stream velocity U 00 related to a Cartesian

coordinate system, in which the components of the free-stream velocity U 00 are

respectively (;00' V"" = Ü and W"", in the (x,y,z) system ofreference.

Assuming that the flow field in the vicinity of the wing and the wake is inviscid,

incompressible and irrotational, the resulting velocity field can be obtained by solving the

following classical Laplace equation:

V 2Cl>*=ü ,

where Cl>' = Cl> + Cl> "" is the velocity potential, .

Cl> is the perturbation velocity potential,

Cl> 00 is the velo city potential due to the free stream flow.

The boundary conditions are:

limr~oo V Cl>' = U co ,

(2-54)

(2-55)

(2-56)

The first condition (2-55) assures that the perturbation induced by the fluid will

vanish far from the wing and the second one (2-56) verifies that the normal component of

the velocity is null on the solid boundaries.

Equation (2-55) is automatically fulfilled by the classical singular solution of the vortex

elements, so the problem reduces to find a singular distribution satisfYing equation (2-56).

The analytical solution of this problem for an arbitrary wing shape is complicated by

the difficulty of specifying the boundary conditions of equation (2-56) on a curved

surface, and by the shape of the wake.


If the wing is modeled by singularity elements that introduce vortices, these need to

be extended into the flow in the form of a wake to respect the Helmholtz theorern.

In addition to that, sorne simplifying assumptions have to be made to define the zero

normal flow boundary conditions on a wing of arbitrary shape.

2.3.2 The b01mdary condition on the wing

Consider a wing defined by the following solid surface in the (x, y, z) Cartesian

system of reference:

z = s{x,y),

Defining a functionF{x,y,z) as:

F(x,y,z) = z-s{x,y) = 0,

the normal on the wing surface is:

The velocity potential due to the free stream is:

By linearity of (2-54),

<1>' =<1>+<1>00'

Introducing equations (2-59), (2-60) and (2-61) in (2-56),

The problem is then reduced to the determination of the perturbation potential:

V 2<1> = 0,

with the boundary conditions,

0<1> = os(O + 0<1»+ OS(D<DJ-w onz=r. oz ax '" oz ayay '" '='


(2-57)

(2-58)

(2-59)

(2-60)

(2-61)

(2-62)

(2-63)

(2-64)

27

Using a Taylor series expansion of the perturbation potential <D(x,y,z) in the vicinity

of the wing to transfer the boundary conditions from the wing surface to the x-y plane, we

can write:

a<D a<D a2eD 2 -(x,y,Z=S)=-(x,y,O)+S-2 (x,y,O)+O(S ). az az az

(2-65)

Only the first term remains, and the boundary conditions can be transferred from the wing

surface to the x - y plane.

The following system of equations involving the perturbation potential determine the

solution of the problem.

V 2<D = 0

a<D - (as ) -(x,y,O) = Uao --a az ax

(2-66)

Now, the equations to be solved are set, and we need to use a model for the perturbation

potential, in the following section a vortex distribution will be presented.

2.3.3 The vortex distribution model

To establish the lifting surface equation in terms of vortex line distributions, the

velocity !J.Q due to a vortex line element dl with a strength !J.1 will be computed by

the Biot-Savart law, where, is the distance from the singularity point located at

(xo,yo, zo) to the point of calculation situated at (x, y, z). Thus,

(2-67)

. Denoting by Yy and rx the vortex element pointing respectively in the direction x

and y, as shown in Figure 2.12, the downwash velocity induced by these vortices is,

1 J w(x,y,z)=-4 1C Wing + wake


r y (x - xo ) + r x (y - Yo) dx d o 010 • ,3 (2-68)

28

z

Figure 2.12 Vortex representation for the lifting-surface mode!

For a vortex distribution we have:

( O+)=8<D=+ry(x,y) U x,y, _ _ "'

8x 2 (2-69)

( 0+)= 8<D =+rx(x,y) v x,y, _ . 8y 2

(2-70)

Hence, the velocity potential at any point x can be expressed as:

(2-71)

and,

(2-72)

P~r~ob~l~em~&~rm~u~w~t~w~n~an~d~p~r~~~w~u~s~st~ud~fu~s ____________________________________ 29

The lifting surface equation is then obtained with the boundary conditions on the wing,

expressed in terms of surface vortex distribution:

(2-73)

The following section presents the discretization of the wing as weIl as the solution of

equation (2-73) using vortex ring elements.

Lifting-surface solution by vortex ring elements

These singularity elements solutions are based on the vortex line solution of the

continuity equation satisfying the boundary condition (2-73). In order to solve this lifting

surface problem numerically, the wing is divided into elements containing vortex ring

singularities as shown in Figure 2.13.

z

Bound vortices

'Free' wake vortices

rT,E.= rw

iWingL.E.

y Collocation point

Nonnal vector fi

WingT.E.

x

Figure 2.13 Vortex ring model for a thin lifting surface


.. :. Choice of singularity element

The thin wing plan form is divided into panels (Figure 2.13) and sorne typical panel

elernents are shown in Figure 2.14. The leading segment of the vortex ring is placed on

the panel's quarter chord line and the collocation point is at the centre of the three-quarter

chord line as illustrated in Figure 2.14.

u~

Leading edge----

Figure 2.14 Nomenclature for the vortex ring element

Rectangular patches (arrays) indexed by (i,j) are used to store the vortex ring

elements, as shown in Figure 2.15. Denoting by VORTEXL (Appendix A), the subroutine

used to calculate the velocity induced ai an arbitrary point P(x,y,z) , by a typical vortex

ring at location (i,j). It calculates the induced velocity (u, v, w) at a point P(x,y, z) as a

function of the vortex line strength and its edge coordinates (Xl> YI' zJ and (X2' Y2' Z2 ) ,

such that:

(2-74)


Denoting by:

(UI,V1,W1) thevelocitycreatedbythesegment (x,y,z .)-(x l'Y 'l'Z,, 1) l,) l,) l,) I,j+ 1,)+ 1,)+

(U2 , V2 , w2 ) the velocity created by the segment (Xi,j+l' Yi,)+]' Zi,j+l ) - ( Xi+1,j+l' Yi+l,j+l' Zi+l,j+])

(u3 , v3 ' w3 ) the velocity created by the segment (Xi+l,j+l' Yi+l,j+l' Zi+l,j+1 )-(xi+U ' Yi+l,j' Zi+l,j) .

(u4 , v4 ' w4 ) the velocity created by the segment (xi+l,j'Y,+I,),Zi+l,j )-( Xi,j'Yi,),Zt,j)

Iy p (x, y, z)

JODD r. I · ,- J 7/7:7 Ui M ----Jt----- r+

J .

1 J

j-lUDD i - l i i + 1 i+ 2 ·X

Figure 2.15 Arrangement of vortex in a rectangular array

We can forrnally write for each segment,

(uJ , VI' w1) = VORTEXL(x, y, z, x . . ,y .. ,Z .. ,X . . J'Y' . 1'Z . . 1,r .), l,) l,) l,j 1,)+ 1,)+ 1,)+ l,)

(uz' v2 ' w2 ) = VORTEXL (x,y,z,Xi,j+PYi,j+P Zi,j~pXi+l,j+l'Yi+l,j+P Zi+l,J+pri,j)'

(u3 , v3' w3 ) = VORTEXL( X,Y, Z,Xi+l,j+l'Yi+l,j+l,Zi+l,j+I,Xi+J,j'Yi+l,j' Zi+l,j,ri,J)'

(u4 , V4' w4 ) = VORTEXL(x,y,z,x. 1 .,y. 1 .,Z. 1 .,X .. ,y.,Z . . ,r .. ). l+ ,} l+ ,) 1+ ,} l,) l,) l,) I,J

The velocity induced by the four vortex segments is then:

(u, V, w) = (uj ' vI' w1) + (u2 , v2 ' w2 ) +(u3 , v3 ' w3 ) + (u4 , v4 , w4 ).

For convenience, the following notation will be used,

(u, V, w) = VORING(x,y,z,i,j,rj,J)'

(2-75)

(2-76)

The total induced velocity is then obtained by the applying four times the VORTEXL

subroutine, itself included in the VORING routine taking into account the four segment.


+:+ Discretization and gl'id generation

Elements with which the thin wing planform is modeled are shown in Figure 2.13,

and sorne typical panels are shown in Figure 2.14. For symmetric reasons, only a wing

semi-span is modeled and symmetric or anti-symmetric relations will be used to take into

account of the entire wing. The leading segment of the vortex ring is placed on the

pane!'s quarter chord line and the collocation point is at the centre of the three-quarter

chord lÏne.

One has to notice that the circulation of the leading edge is equal to ri' but for all the

other elements behind it, the circulation is equal to ri - ri-] . In order to satisfy the Kutta

condition at the trailing edge, the vortex of the last panel must be cancelled.

rTE. = 0 (2-77)

For steady flow we have hence,

rTE = r w' (2-78)

For this example (in Figure 2.13) the semi span of the wing is equally divided in

M = 3 panels in the x direction and N = 4 panels in the y direction. An the geometrical

information, such as the coordinates of the ring corner points and the collocation points,

the panel area Sk' as weIl as the normal vector IlK are calculated and stored and this point

through the panel sequential counter K, which has values from K = 1 ~ (M x N) .

• :+ Influence coefficient

A procedure calculating the influence of each point to the other ones is implemented,

using the counter K . Starting from the leading edge with i = 1, j = 1 (K = 1), the counter

scans an the points as illustrated in Figure 2.3.5 and Figure 2.17 up to the last point of the

trailing edge, i = M, j = N (K = Mx N) .


x x

}o--~--+ __ -;o-________ Wake points

/ ~--~~---++----++---------------~

x x

x x

x x

j=l~----~----~------~----------__ i= 1 2 3

Figure 2.16 Arnay ofwing and wake panel corner points (dots) and of collocation

points (symbols)

y

K=MxN

x

Figure 2.17 Sequence of scanning the wing panels


Considering the collocation point 1, the velocity induced by the first vortex ring is

then:

(U;, vi' WJII = VORING(x,y,z,i = l,j=l,r = 1.0),

adding the contribution of its image,

(uü ' Vii' wJll = VORING(x,-y,z,i = l,j=l,r = 1.0).

Hence, the velocity induced by the unit strength rI and its image at collocation point 1 is

(2-79)

In this process, the influence of the first vortex at the first collocation point is represented

by the subscript ( )11' note that both subscript have a value from 1 ~ Mx N . A unit

strength vortex is used to evaluate the following influence coefficient.

(2-80)

An inner scanning loop with the counter L = 1 -? N x M is needed to scan aH the vortex

rings influencing this point. The L counter will scan aU the vortex rings on the wing

surface with the counter K staying at 1. AH the influence coefficients a IL are computed:

(2-81)

Now, a particular attention has to be taken when a particular vortex ring is at the

trailing edge. A "free wake" vortex ring with the same strength has to be added to cancel

the spanwise starting vortex !ine, and hence, the contribution of this segment is added

when the influence of such trailing edge panel is calculated. For example, in Figure 2.16

the first trailing edge panel is encountered for L = 9 (i = 3 ), and in this case, the

following two velocities have to be taken into account:

(u, V, W)J9 = VORING(xpYj,zpi = 3,j = 1,r = 1.0 ),

(u, V, W)J9W = VORING(xl'Ypzpi =3+ l,j = 1,r = 1.0 ).

Resulting in this case,

(2-82)


For the first collocation point the zero normal flow across the wing boundary condition

can be written as:

(2-83)

where,

(2-84)

Using the same procedure for aIl the collocation points lead to the fol1owing system of

linear equations

(2-85)

.......... ~ ....................... ~''''''''' .. ''''''. = .............. .

.............................................. = .......... .

One can write the above set of equation in matrix form:

ail an a1m rI Ua;) -Dl

a 21 a22 a2m r2 U",-D2

= (2-86)

amI a m2 amm r m U oo -Dm

The solution gives the value of the circulation at different locations of the wing, and

using the consideration developed in this part, the spanwise distribution of the circulation,

as well as the aerodynamic coefficients can bé obtained. This method has been used as a

method of validation and as a comparison for the results for various wing configurations

obtained in chapter 5.


2.4 Comments on the previous methods

AH of the theoretical methods discussed in the above sections are based on the bound

and free vortex scheme developed by Prandtl and use infinite Fourier series expansion of

the spanwise variation of the circulation to express the integro-differential equation (2-15)

in algebraic form.

Glauert's method uses a collocation procedure to solve the resulting algebraic

equation. However, the accuracy of the collocation methods depends considerably on the

selection of the collocation points, especially when the chord and incidence variations

display sudden spanwise changes.

The method developed by Rasmussien and Smith is more accurate than the

collocation methods. However, this method was developed only for the case of symmetric

incidence distributions without considering the forced symmetry terms (although the

chord distribution for tapered wings contains forced symmetry terms). Also, the problem

of wings with sudden incidence changes is not treated by this method.

AIl of these theoretical methods use truncation procedure to reduce the infinite system

of equations to a finite number of equations based on the assumption that the coefficients

decrease monotonically with their rank, which is not always the case for various wing

configurations and therefore causing accuracy problems in the solution. None of these

methods are able to provide efficient and accurate solutions for wings with sudden

changes of incidence or chord. In fact, the Fourier expansions can not model accurately

sudden changes in the boundary conditions on the wing, as shown in [23].

The numerical panel method discussed in Section 2.3 1S a widely used method.

However, it is difficult to model wings with curved leading edges using panel methods. In

order to obtain a good level of accuracy one needs to use a large number of panels.

The present approach, which will be presented in the following sections, aims to

provide accurate and efficient solutions for aH cases of finite-span wings with symmetric

and antisymmetric incidence distributions, including sudden incidence changes, as weIl as

the wings with curved leading and trailing edges.


Chapter 3

Present method of solution for straight wings

The aim of this Chapter is to present a solution to unswept wings based on a velocity

singularity method in three dimensions. First a prototype problem is solved, which is

followed by the presentation of the complete problem and its solution for the circulation

distribution. New functions related to the changes in wing incidence and chord variation

are identified or defined in the general solution of the circulation.

This method is applied to various configurations of unswept wings, for instance,

rectangular, tapered or elliptic wings with symmetric or antisymmetric or asymmetric

incidence variations.

3.1 Method of solu.tion

Consider a thin wing of span 2b placed in a uniform stream of velocity U 00' The

Cartesian coordinate system, bx,by,bz , is used as a reference for the flow past this wing,

where x, y, and z are nondimensional coordinates, with y -axis along the wing span and

the x axis parallel to U 00 •

The wing geometry is defined by a specified variation of the local chord, b c(y) ,

where c(y) is the nondimensional local chord, and by a specified variation of the local

wing incidence a(y) == a g (y) + a c (y); a g (y) is the geometric incidence of the local wing

section with respect to the velocity U 00' and a c (y) is the added-by-camber incidence

representing the effect of the wing section camber including the effect of a local flap or

aileron deflection.

The following notations win be used for the incidence and nondimensional chord at

the wing center (root chord) and at tip: tlR(y)==a(O) , CR ==c(o) and aT =a(l),

CT = C(l) , respectively. In the following sections, b = l, is used.

The mean (average) nondimensional chord and incidence of the wing are calculated as

Present method ofsolution (or straight wings 38

1 fI Ca = - c{y)dy, 2 -]

aa = - c(y )a{y )dy. 1 fI 2 -1

(3-1)

The following nondimensional notations will also be used for convenience:

c(y) = c(y)/ca , a(y) = a(y)/aa . (3-2)

A three dimensional wing can be represented by a bound vortex of variable intensity,

r{y), along the hne of aerodynamic centers (situated at a quarter of the local chord

behind the leading edge), and a semi-infinite free vortex sheet of distributed intensity

r{y) = df'{y)/ dy , introduced to satisfy Helmoholtz' vortex theorems), which originates at

the trailing edge and extends to infinity. If we introduce a plane perpendicular to the x

direction and situated downstream at an infinite distance from the wing, the vortex sheet

intersects it. This plane is known as the Trefftz plane [4, 31]. The vortex sheet continues

to infinity behind the Trefftz plane.

Let v{y, z) and w(y, z) be the velocity components along the y - axis and z - axis

induced by the free vortex sheet in the wing cross-flow plane, and v'(y,z) and w*(y,z),

the corresponding velo city components induced by the same vortex sheet in the Treftz

plane situated at x ~ 00. For the case of incompressible potential flows, the velocity

components v , w and v' , w· are harmonic functions, satisfying the Laplace equation.

Thus, a complex conjugate velocity U(X) ll)ay be defined in the Trefftz plane in the

nondimensional complex form:

U(X) = v*(y,Z)-iW'(y,z), aaUoo

where X = Y + i z . (3-3)

It can be shown that, the velocity component ~(y,z), induced by the semi-infinite vortex

sheet on the wing is half of w * (y, z) which is induced by an infinite vortex sheet at the

same point (y, z) in the Trefftz plane, i.e.,

w(y,z) = tw' (y,z) . (3-4)

Any sudden change on the wing chord and/or incidence variations (as it is the case in

many aeronautical applications) affects the velocity distribution on the wing and on the

Present method ofsolution fOr straight wings 39

trace of the free vortex sheet in the Treftz plane. Consider the case of two such changes

located at y = sJ and y = -S2 on the vortex sheet trace in the Trefftz plane, the

corresponding boundary condition expressed in terms of w' (y, z) as:

(3-5)

where w * (y) = w' (y,O)j(aaU J is the non-dimensional normal velo city defined by

w * (y) = Wo (y) + H(sj ,y )Wj (y) + H(S2'Y )W2 (y), (3-6)

H(s,y)~ {~ for s < y <1 (3-7)

for y < s or y> l'

ll(s,y)= {~ for -1 < y < s (3-8)

for y <-1 or y >-s ,

in which Wo (y) , ~ (y) and W2 (y) are usually polynomial functions of y .

Outside the vortex sheet, y{y) and hence the velocity v' (y,O) = 0, which results in the

following boundary condition:

REAL{U(X)}x=y = 0 for y < -1 and y > 1 (3-9)

3.1.1 The prototype problem

The incidence and chord variations along the span of a wing may consist of either

continuous or sudden change or a combination of the two. The study of a prototype

problem makes it possible to determine the contribution of a discrete change of geometry

on the solution of the circulation distribution without having to specifY whether the

discrete geometrical change under consideration is infinitesimaUy small or not. Then

whenever one needs to extend the solution to a continuous variation the prototype

solution is used to model the infinitesimally smaU changes. On the other hand if one

needs to calculate a sudden change again th~ prototype solution can be applied for the

discrete change under consideration.

For the prototype problem, first consider the case when Wo (y) = /30' WJ (y) = 0 Wj and

W2 (y) = oW2 are constant (where oW] and oW2 may or may not be infinitesimally

small). The problem in the Trefftz plane is in this case identical to the two dimensional

incompressible flow problem past a thin plate .having the same boundary conditions (3-5)


and (3-9). The singularities 1 .JIh-, In(s] - X), In(s2 + X) characterize the .JI-X' l+X

complex conjugate velocity U(X) (see [23, 24, 29]) representing the changes at the

locations y = 1, Y = -1, Y = SI' and y = -S2 respectively.

WJ

-------------------

----------------------------l1 -- W,(s,) ~ 5W, 1 WzCS2) - ÔW2 1 ----.----------- -------.------------_.- --, , , , ,

wo= ~o , , , , ,

..:. .i.. A2 ~T S2

0 S1 TR1 A1

1 1

1 1

Figure 3.1 Two sudden changes in the geometry placed at RI and R2

As shown in the previous papers [23, 24], one can determine the singular

contributions to the plate edges and ridges, which satisfy the boundary conditions (3-5)

and (3-9), leading to the following solution for the complex conjugate velo city in the

Treftz plane:

U(X) = -1' Po + Ao + AX ~m G~( X) s:rXT G~( X) ury! SI' +uYV 2 S2'- , .J1- X 2

(3-10)

where

G(s,X) = ~cosh-l (3-11 ) 7T

The functions G(sp X) and G(S2' --X) represent the singular contributions of the ridges

situated at y = Sj and y = 82 respectively, and the second term in the right-hand side of

(3-10) represents the singular contributions of the plate edges (corresponding to the wing

tips). The constants A and Ao are constaI!ts to be determined later from boundary

conditions.


'" ?

y

The circulation dr(y) for an infinÏtesimal control volume around a portion bdy of

the free vortex sheet, and hence the local free vortex intensity r(y), is related to the

spanwise velo city component v' (y, z) on the vortex sheet trace in the Trefftz plane in the

form:

aaU ",v' (y,O) = - r(Y)/2, r(y) = df(y)/bdy. (3-12)

Rewriting (3-12) in the integral form:

(3-13)

where f(y) = r (y )/(2baaU",) is the nondimensional circulation. Integrating (3-13)

results in:

f(y)= A~l-l + Jl'AoC(y )-8~ [(SI - y)G(spy)+~l-s;C(y) ]

-8W2 [( S2 + y)G(S2' -y )-~1- s~C(y) J, (3-14)

respectively, and are defined as

~ cosh-I

Jl'

Jl' o

() 2 -lff+S Cs =-cos --. Jl' 2

The details of the integrations in (3-13) are given in Appendix C.

(3-15)

(3-16)

Now the constant, Ao is determined using the boundary condition at the wing tips where

the circulation is zero, i.e.,

f(l)=f(-l)=O. (3-17)

P~r~~~ffl~t~m~et~ho~d~o~fs~o~m~tio~n~w~r~s~~a~~~h~t~wl~ng~s~ ______________________________ 42

Substituting this in (3-14) results in:

(3-18)

To determine the constant A, consider the condition at infinity where the induced

velocity components become zero, i.e.,

lim {U ( X)} = 0 . X-+'"

Applying this to the complex conjugated velocity (3-10) one obtains

A = /30 +bW]C(s])+bW2C(S2)'

Finally introducing these constants in (3-14) a simpler form is obtained as:

f(y)=~l-i /30 + ~ft;[Rc(s])+(Y-S])G(Sl'Y)J Contributi~ 5Wo Contribution ~elated to 5~

+~w;[ R C(S2)+(-Y- S2)G(S2'-Y)} Contribution relatid to 5W2

(3-19)

(3-20)

(3-21)

In this expression the contributions in the circulation distribution related to t5Wo' t5ft;,

and t5W2 are clearly indicated for the prototype problem.

3.1.2 The complete problem

Consider two sudden changes at the location y = s] and y = -S2 which are

modeled by a sudden change of contribution Wj(Sj) G(SI'X) and W2 (S2) G(S2'-X) in

U (X). These sudden changes may be generated by the deflection of flaps or ailerons

situated at this locations. The variation of the boundary condition (3-5) related to Wo (y )

can be modeled by a continuous distribution of elementary ridges defined by the

contributions [W~(a) da] G(a,X) and [-W;(-a) da] G(a,-X), respectively, for

o < a < 1 , while those related to Wj (y) and W2 (y) can be modeled by two continuous

distribution of elementary ridges defined by the contributions [Wj'(a) da] G(O",X) , for

SI <0"<1 and [-W;(-a) dO"] G(O",-X) for S2 <a <l,respectively.


Figure 3.2 The complete problem

The contributions in the nondirnensional circulation related to each of these velocity

variations in the Trefftz plane (shown in Figure 3.2) are defined using appropriate

substitutions in the solution of the prototype problem (3-21).

Consider first the contribution related to W) (y) in the nondirnensional circulation,

f) ( SI' y). This contribution constitutes two parts:

1. The contribution of the sudden change at y = SI , which IS glVen by

n. The continuous change in s) ~ y ~ 1 related to the velocity variation PV; (y) is

rnodeled by a continuous distribution of elementary ridges on the wing. The effect

of these elementary ridges is displayed in the Trefftz plane as infinitesirnal velocity

changes given by (dpv; / da) da, where. SI ~ a ~ 1. Henee, aceording to (3 -21) the

corresponding contributions related to each of this changes is given by

(dpv;/da)da[ )1- y2 C(a)+(y-a)G(a,y)]. Summing up an such

contributions finally gives: ft (dpv;/da)da[ )1-l C( a )+(y- a)G( a,y) J. Summing the above two parts of the contribution one obtains:


rI (SI ,Y) = ~ (S])[ )1- y2 C(s]) +(Y -SI )G(SI'Y) ]

+ (( d~/da )da[ )1- y2 C( a) +(y- a )G( a,y) J. (3-22)

Integrating by parts (details shown in Appendix C), this contribution is found to be:

rj(Spy)= fwj(a) G(a, y) da. 1

(3-23)

A similar procedure is applied to calculate the contribution r 2 ( S2' y) of a sudden change

(R2) at y = -S2 and of the continuous change in -1 ~ a ~ S2 ' and is given as:

(3-24)

And finally, we have to calculate the contribution, f 0 (y) , related to the velocity variation

Wo (y). This can be seen as the contribution of a ridge situated at y = O. This type of

ridge may appear, for instance, in the case of tapered wings in which case we have a

discontinuity in the chord variation. The contribution thus can be calculated from (3-23)

and (3-24) by setting SI = 0 and S2 = O.

(3-25)

Consider the general polynomial representations for the functions Wo (y), W] (y) and

W2 (y) on the vortex sheet trace in the Trefftz plane

2n fi fi

Wo(Y) = l Pk yk , WI(y) = l 13; yk, W2 (y) = l pi yk ,

where in general

k=O k=û k=O

for y>O

for y<O

(3-26)

(3-27)

P~r~~~m~t~m~d~ho~d~o~f~so~lu~u~'on~fi~w~s~ff~a~œ~ht~w~m~&~s ________________________________ 45

where the superscripts S and A correspond to the symmetric and anti-symmetric

variation of the wing incidence, respectively (this refers to right-left wing symmetry).

The sudden changes at y = SI and y = -S2 in the boundary conditions (3-5) and (3-

26) correspond to the case when the spanwise variation of the wing incidence and/or the

chord presents sud den changes at these locations (the coefficients f3 k are not yet

specified).

By introducing expressions (3-26) in (3-23) -- (3-25), one finally obtains the

following general solution of the nondimensional circulation on the wing:

n n

r(Y)=I [13; Fko(y)+PkA Fi (y)] + I [Bk Fk(spy)+(-lY fi; Fk(S2'-Y)], (3-28)

where,

k;O k;O

Go (y) = G (0, y) = ~ cosh-I

1(

1+lyl

21yl '

J (s) = J~I J. (s) + -.L. sj-l .Jl=7i ] ] J-2 Ir] '

Jo(s)=2 C(s), J 1(s)=1;;.Jl=7i ,

Ij=Jj(O),Ij =[(j-l)/j]Ij_2 , 10=1, IJ=2/7r,

f!.k =[l+(-l)k ]/2.

(3-29)

(3-30)

(3-31)

(3-32)

(3-33)

(3-34)

(3-35)

The solution (3-28) is a general solution vaUd for wings of arbitrary shapes, defined

by a non dimensional distribution of the chord c (y) = c (y) / Ca and incidence

a (y) = a (y) / a a , and includes an cases of the natural and forced symmetry and anti

symmetry of the wing incidence and chord distribution.

In this solution we can identify the physical meaning of the different terms, the new

functions Fko (y) and Fkl (y), corresponcÜng to the superscripts r = 0 and r = l ,


represent respectively the continuous contribution (without sudden change) of the

symmetry and anti-symmetry of the wing geometry, in which the function Go (y) takes

into account the discontinuity at the root chord of the wing y = 0 introduced by the

forced symmetry and anti-symmetry terms. For example, such a discontinuity is found in

the chord distribution of a tapered wing containing a forced symmetry term

c(y) = CR (l + q y) whereas the wing is symmetric. These forced symmetry terms have

not been considered in the previous methods using Fourier series expansions and using

the natural symmetry hypothesis for the sake of simplicity. A A

The new functions Fk(spY) and Fk(S2'-y) in the general solution (3-28) take into

account the changes in the wing incidence and chord at y = SI and y = -S2 (for example

due to the deflection of control surfaces). These specifie contributions related to the wing

discontinuities, derived here for the first time, were not taken into account in the previous

methods using Fourier expansions of the circulation, which were shown to be not

convenient to model aerodynamic discontinuities [23].

The general solution (3-28) is considerably simplified In the case of symmetric

variations of incidence, when /3 kA = 0 and p; = (- 1 Y P k2 = Pk' or in the anti -symmetry

case, when /3/ = 0 and P; = (_ly+1 Pk2 = Pk ,as shown in the following sections.

The velocity singularity methods lead to the solution of the circulation (3-28)

expressed in terms of the coefficients fik' which are not known at this time. Hence in

order to solve the problem completely we need to determine these coefficients.

3.1.3 Determination of the unknown coefficients

Consider the following expression of the local nondimensional circulation around the

wing section, rL (y) = r L (y)/[2 b a a U col, whlch depends on the effective incidence,

a eff (y) = a(y)-w(y,O)/Uoo ,where w(y,O)=tw*(y,O) represents the normal-to-wing

velocity induced by the free vortex sheet on the wing (at x = 0),

(3-36)

Present method ofsolution {or straight wings 47

where w * (y) == w*(y, 0 )/(aa u "J= 2 w(y) is defined by (3-6) and (3-26), and

K = C La /2 is haif of the lift coefficient sI ope ( C La = dC L ! da ). The thin airfoil theory

gives K=rc, while K=(O.85-0.95)rc is recommended by Carafoli [4] to include the

viscous effects on the lift coefficient slope and to be thus in agreement with the

experimental results.

Obviously, the expressions (3-28) and (3-36) of the circulation have to be equal

(3-37)

This equation has to be satisfied at any location along the span, and can be used to

determine the unknown coefficients fJk .

However to get an improved accuracy of the solution a special procedure similar to

the Galerkin technique is developed for this problem to obtain an enhanced accuracy (this

is especially important for complex wing geometries with incidence variations involving

sudden changes, such as those due to the deflections of the ailerons or flaps). In this

procedure, a system of N + 1 linear equations (with the same number of unknown

coefficients fJk ) is obtained by multiplying equation (3-30) with yi, where

i E {o, 1,"', N}, and integrating along the wing span:

(3-38)

This procedure permits a better resolution of the incidence variation and wing

geometry, and displays a better accuracy than the collocation procedure for the same

number of unknown coefficients. To formally justify this improved solution procedure,

one may note that the polynomial terms can be expressed as a sum of Tchebyshev

polynomials, / = -;-! C;k ~-2k (y) , where C; are the binomial coefficients. Thus, the 2 k=O

system (3-38) is equivalent to the system of equations L [r(y)- r L (y)] ~ (y) dy == 0,

where the Tchebyshev polynomials ~ (y) represent orthogonal test functions.

If is aiso worth noting that the integrals appearing in (3-38) are also present in the

calculation of the lift coefficient (those for i == 0), as shown in (3-39), and those for i;::: 0

Present method ofsolution {or straight wings 48

in the calculation of the induced drag coefficient (3-40), since w*(y) has a polynomial

representation defined by (3-6) and (3-26). This enhances the accuracy of the improved

solution approach and simplifies the calculation of the aerodynamic characteristics of the

wing, by using the simpler expression r L (y) of the circulation, instead of r(y) which

usually is more complicated.

3.1.4 Aerodynamic characteristics

The lift and induced drag coefficients, CL and CD i , can thus be calculated using

rL (y) by the equations

CL = Aaa frL (y) dy, (3-39)

CDi=Aa; r tw*(y)rL(y)dy, (3-40)

(3-41)

(3-42)

where c(y) = c(y )/ ca' and Il., = 4 b 2 / s = 2/ Ca denotes the wing aspect ratio, in which

S = 2b 2 Ca is the wing area. By analogy with the elliptic wing one cau define:

C 2

CDi =-L-{1+8), ;rA

1 1 1+r --=-+--2K). 2K ;rA

(3-43)

(3-44)

where 2 K). and 2 K are the lift coefficient sI opes for the wing of aspect ratios Il., and

for a two-dimensional airfoil (or Â ~ 00 ), respectively, and where the lift and induced

drag factors 1: and 8 are

• ;r Â a ja

1:=- • -1, 2K l-a;a

(3-45)

(3-46)

P~r~~~e~nt~m~e~ffi~o~d~o~(s~o~fu~tl~'on~w~r~st~ra~~~h~t~w~m~gs~ __________________________________ 49

3.2 Solutions for symmetric wings with continuous incidence

variation

Consider the general symmetric variation of the wing chord and incidence in the forrn

(3-47) m=O }=O

which can represent any continuous variation of the chord and incidence along the wing

span, including the case of the wings with curved leading edges. For this type of

symmetric wings the solution (3-28) of the circulation is substantially simplified by

considering fJ/ = 0 and fJ/ = fJ k , resulting thus

2n

r (y) = I fJ k F kO (y) . (3-48) k=O

This solution includes both the even and odd terms in k , corresponding to the natural

and forced symmetry terms, respectively. This ensures a correct and more general

mathematical formulation of the physical problem (e.g. the chord variation of tapered

wings, c(y) / CR =1 + q y , contains a forced symmetry term). Moreover, by considering

both even and odd terms, the highest order k = 2n for the sarne nurnber of coefficients

Pk is substantially reduced (by aimost a half). This avoids the presence of spurious

oscillations in the solution of the spanwise variation of the circulation, by contrast with

the methods using Fourier series wruch consider only the natural-syrnmetry terms.

The functionF:(y) contains, as shown in (3-28), the function Go(Y) defined by

equation (3-31), which takes into account the geometrical discontinuities at y = 0 (where

for exarnple the chord variation slope has a sudden change for trapezoidal wings),

associated with the forced syrnmetry terms.

To deterrnine the coefficients Pk , the improved solution procedure (3-38) is used,

leading to the following system of N = 2 n + l linear equations in matrix form, for

i E {O, 1, "', 2n} and k E {O, 1, "', 2n},

{Bi,k }[f3k] = [D;], (3-49)

Present method ofsolution (Or straight wings 50

where,

B =E ~ ~ f!. 1 P 2.é'k+l Ik+i+1 i,k i,k + k+l ~ j k-j J+i + k+l k+i+2' (3-50)

(3-51)

(3-52)

(3-53)

This system of equations can be easily solved to determine the unknown coefficients f3 k .

The aerodynamic coefficients can be determined from equations (3-43,44), where

(3-54)

(3-55)

3.3 Solutions for symmetric wings with incidence and chord changes

In many cases, the wings have sudden changes in the spanwise variations of the

geometrical shape (chord variation) and incidence, the latter being usually the result of

the deflection of control surfaces, such as ailerons or flaps. Consider such a symmetric

wing with sudden changes of incidence and/or geometry occurring at y = s and y = -s

defined by

nm

c(y} = L [cm +H(s,y)c; +H(s,y)c; ] Iylm , (3-56) m~O

ni

a(y)= L [a) +H(s,y)a; +H(s,y)a;] Iyl) , (3-57) j~O

P~r~~~e~nt~m~e~lli~o~d~o~(s~o~fu~ti~on~h~r~st~ra~œ~h~t~w~m~~~ __________________________________ 51

where H(s,y) and H(s,y) are defined in (3-7,8). In this case, the general solution (3-28)

AS' 1 ( )k A 2 A of circulation is simplified by considering f3k = 0, f3k = f3 k and f3k = -1 f3k = Pk ,

resulting thus

Zn fi

f(y) = If3k Fko(y) + Ifik [Fk(s,y)+ Fk(s,-y)], (3-58)

where aH functions are defined in (3-29,30). These new functions Fk (s, y) and Fk (s,-y) ,

which are derived in Section 3.1.2, are the specifie contributions of the changes in the

wing incidence and chord at y = s and y = -s.

The unknown coefficients f3 k are detennined from the system of equations (in matrix

fonn)

K nm 1 m+k+i+1

B~ l -* -s =-c c ----i,k 2 a m=O m m + k + i + 1 '

~1 2{2 k i } -..jl-S- k-j k+1+i-j + . --I f! j+l 1j+i+1 S - l gi,) 1j s ,

k+l+2 k+l )=0 )=0

2f!. i+k

gi,k=~'

where Di and Ei,k are defined in (3-52,53).

(3-59)

(3-60)

(3-61)

(3-62)

(3-63)

(3-64)

P~r~~~e~nt~m~e~ffi~o~d~o~Û~o~fu~tl~'on~0~r~s~~a~œ~h~t~w~m~gs~__________________________________ 52

3.4 Solution fo:r wings with asymmet:ric incidence va:riation

3.4.1 Wings witb anti-symmetric incidence variation

The wing incidence may have an anti-syrnrnetric variation of incidence, either

continuous, such as a (y) = a T y (which rnay correspond to the effect of a rolling

rotation of the wing), or with sudden changes, such as a (y) = H(s, y) al' - H(s, y) aT ,

which corresponds to anti-syrnrnetrically deflected ailerons. Consider such a wing with

the syrnrnetric chord variation (3-56) and an anti-syrnmetric variation of incidence

defined by

2n+l

a(y) = l Sgn (y) [a j + H(s, y) a; + H(s, y) a;] Iyl j , (3-65) )=0

where H(s,y) and H(s,y) are defined in (3-7,8), and Sgn (y) represents the sign of y:

for y> 0

for y < 0 (3-66)

In this case, the solution (3-28) of the circulation is simplified by considering

" 1 ( )k+l A 2 A AS, Pk = -1 Pk = Pk , Pk = Pk and Pk = 0 , resultmg thus

r(y)=tpk F;(y) + ÎPk [Fk(s,y)-Fk(s,-y)], (3-67) k=O k=O

where all functions are defined in (3-29,30).

These new functions Fk{S,Y)-Fk(S,-y), which are derived in Section 3.1.2, are the

specifie contributions of the anti-symmetric changes of the wing incidence and chord at

y = s and y = -s .

The unknown coefficients fJ k can be determined from the following system of

equations

(3-68)

where

(3-69)


r:-71 2{2 k i } " 1 - s - k- j k+l+i- j + . --I f j Ij+i+1 S - I gi,j+1 Ij S ,

k + 1 + 2 k + l j~O j~O (3-70)

where Di , Ei,k , Bi,k , Di" , Ei~k and gi,j are defined in (3-52,53) and (3-60,62,63,64).

3.4.2 Wings with asymmetric incidence distribution

In many flight evolutions, asymmetric incidence distributions may occur as a result of

the combined effects of the wing incidence, symmetrically de:flected :flaps and anti

symmetrically de:flected ailerons. In this case, the wing circulation distribution is given by

the general solution (3-28), or, altematively, it may be obtained by a convenient

summation of the symmetric and anti- symmetric solutions (3-48), (3-58) and (3-67).

Present method o{solution fOr straight wings 54

Chapter 4@

Present method of solution for swept wings

The use of sweep angle to finite span wings for the purpose of delaying and redueing

the effeets of compressibility was first proposed by Adolf Busemann in 1935. A swept

wing delays the formation of the shock waves encountered in transonic flow to a higher

Mach number [32]. This action enables the transonic airplane to eruise faster before

encountering wave drag. Apart from this sweep angles are used in the design of low

speed airerafts to shi ft the overall aerodynamlc center of the airplane without having to

move the entire wing. Sweep improves the laterai stability of an airplane, giving the

same effeet as dihedral.

However sweep does have sorne disadvantages. A significant part of the air veloeity

is now flowing spanwise and not contributing to the lift. This will raise the stan speed,

and the resulting takeoff and landing distance 'over an equivalent straight wing; however,

much of this performance can be recovered with the use of extensive flaps or other high

lift devices. Spanwise flow will also cause the tips to staIl first, which could cause the

10ss of aileron effectiveness before the entire wing is stalled. Again there are some design

solutions for this, however this makes the co st ofusing swept wings more expensive [33].

This chapter deals with the method of solution for swept wings in subsonic flow based

on the method of velocity singularity formulated in Chapter 3. The general solution (3-

21) represents the distribution of circulation in the Trefftz plane. The velocity induced by

the bound and the local free vortices on the line of aerodynamic centers of the wing will

be derived for the case of symmetric wings with continuously varying incidence, by using

the circulation in the Trefftz plane. Then collocation procedure will be used to determine

the unknown coefficients in the solution ofthe circulation.

Present method ofsolution (or yawed wings 55

4.1 Method of Solu.tion

Consider a thin sweptback wing of semi-span b placed in a uniform stream of

velocity U 00' Once again the fluid flow past this wing is referred to a Cartesian reference

system bx, by, bz where x, y and z are nondimensional coordinates, with the bx axis

paraUd to U 00 as shown in Figure 4.1.

o

C

by

"'- ... ....... Thewing

.A ~~(fffl"\~~:----È-___ / ~I 'î! r-r-I"'-I"'-r- .'" - ...

bs

bx

r-r-r-r- !G- ............ r-r-r-r- "'-- ... r-r-r-r- ......... -L

r-r-r-;-- _~ 1 -;-i--+--El

.. ·-----1--- __ ------r ~or ~{

D

b = l

by

Figure 4.1 A bound vortex of variable intensity r(s) and a semi infinite free

vortex sheet of distributed intensity8 r along the Une of aerodynamic cent ers

Present method ofsolution fàr yawed wings 56

The wing geometry is defined by the swept angle x, by a specified variation of the

local chord, b c(y), where c(y) is the nondimensional local chord, and by a specified

variation of the local wing incidence a(y) = a g (y) + a c (y); a g (y) is the geometric

incidence of the local wing section with respect to the velocity U ao' and a c (y) is the

added-by-camber incidence representing the effect of the wing section camber including

the effect of a local flap or aileron deflection.

The following notations will be used for the incidence and nondimensional chord at

the wing center (root chord) and at tip: aR(y)=a(O) , CR =c(O) and aT =a(l),

CT = C(l), respectively. In the following sections, b = 1, is used. The mean (average)

nondimensional chord and incidence of the wing are calculated from (3-1) and (3-2)

respectively. The line of aerodynamic centers connects aU points at one quarter of the

local chord from the leading edge.

Let the variable s represent the position of an arbitrary free vortex of strength r{s)

and y the position of a point on the line of aerodynamic centers of the wing at which the

induced velocity is to be calculated calculated.

A three dimensional swept wing can be represented by a bound vortex of variable

intensity placed at the line of aerodynamic centers (along segment AE in Figure 4.1) and a

semi-infinite free vortex sheet of distributed intensity. Arbitrarily chosen vortex element

is highlighted in Figure 4.1 for the purpose of illustration. Each vortex element is made

of three segments, two semi-infinite free vortices(segments A C and BD) and part of the

bound vortex connecting them. The three segments make an element having uniform

intensity y(s) = dr(s )j ds along the filament CABD , thereby satisfying Helmholtz' vortex

theorems (stated in Appendix A). In Figure 4.1 only half of the wing is shown because

the type of wing under consideration is a wing with symmetrical incidence variation.

In the case of straight wings, the bound vortices of the two sides of the wing are

parallel and both lie along the hne of aerodynamic centers and thus the velocity induced

by the two bound vortices on the Hne of aerodynamic centers is zero. Hence the

relationship between the velocity component w(y,z), induced by the semi-infinÎte vortex

sheet on the wing and w' (y,z) which is induced by an infinite vortex sheet in the Trefftz

Present method ofsolution {Or yawed wings 57

plane is straight -forward and is given by w' (y, z) = 2 w(y, z). Hence this enables one to

proceed to the process of determining the unknown coefficients in the general solution of

the circulation distribution(as shown in Chapter 3). However in the case of swept wings,

due to the presence of the sweep angle x, the bound vortex on one wing induces a

velocity on the Hne of aerodynamic centers of the other side (clearly seen in Figure 4.2).

Thus including the contribution of the bound vortex makes it difficult to obtain easier way

of determining the induced velocity on the wing .

• :. Derivation of the induced velocity on the wing

Following, this section will be devoted to developing a way to determine the velocity

component w(y,O) on the wing.

The solution of circulation distribution obtained by using the method ofvelocity

singularity for finite span of arbitrary shapes in the Trefftz plane for the case of

symmetrical wings, with a continuous variation of incidence (derived in Chapter 3) is

given by:

where,

n

f(y) = L.Bk Fko (y), k=O

o 2 yk+1 .,Jï=Y2 k .

Fk(y)=fk+1--GO(Y)+ L fjlk_jyJ, k + l k + 1 j=O

2 l+lyl Go (y)=G(O,y) = Ji cosh-

1 21yl'

1j=[(j-l)/j]lj_2' 10=1, 1j =2/Ji,

f. k =[l+(-l)k ]/2.

(4-1)

(4-2)

(4-3)

(4-4)

(4-5)

The function F kO (y) contains Go (y) , defined by (4-3), which takes into account the

geometrical discontinuities at y = 0, associated with the forced symmetry terms (for

example the chord variation has a sudden change at y = 0 for trapezoidal and swept

wings).

Present method ofsolution fàr yawed wings 58

Taking the derivative of (4-1) and changing the variable y to s,

(4-6)

l k (j-l j+l J] +-LR/k - j jp (j+1) P . k + 1 j=O l - s 1 - s

(4-7)

A single element of vortex intensity 8 r is shown in Figure 4.2, along with the

geometrical dimensions useful for the calculation of the induced velocity at y .

ycos 2X

y

y

s s

x

Figure 4.2 A vortex loop element for symmetrical swept wings

Present method o(solution for yawedwings 59

Since the wing under consideration is a symmetrical wing only the velocity induced

on the right hand side of the wing may be considered while taking into consideration the

effects of the left hand side of the wing.

Let y be the point at which the nondimensional total velocity, w(y), Îs going to be

calculated, then:

w(y) = 1 b'W1f + 18Wlb + lb'w2f + 18W2b , (4-8)

W(y) = W(y,O) = w(y) , U""aa

(4-9)

where

For an element in the left hand si de of the wing:

b' W 1b = The nondimensional velocity induced by the bound vortex part of the element.

b'w1f = The nondimensional velocity induced by the free vortex part of the element.

For an element in the left hand side of the wing:

b'w2b = The nondimensional velocity induced by the bound vortex part ofthe clement.

b'w2f = The nondimensional velocity induced by the free vortex part ofthe element.

AH parts of the vortex clements are straight segments, therefore the velocities induced

by each ofthese vortices can be calculated from equation (A-12) (see Appendix A for a

detailed derivation). Substituting the appropriate angles for each case, as shown in Figure

4-2, one results for b'w1f :

__ 1 (1 + sin X) or 2:r (s - y)

b'w1f = b'w

_ 1 (1 - sin X) or 2:r (s- y)

for O~s~y-&

for y - & ~ S :::; y + & , (4-10)

for y+&:::;s ~ 1

where & is the radius of the core of a vortex filament and ow is the variation of the

velocity induced with in the core of the vortex filament. The actual function for this

variation need not be expressed explicitly. This condition i8 illustrated in Figure A.3 in

Appendix A.

Present method ofsolution {or yawed wings 60

Thus,

(4-11)

and b' w1j given by (4-10) is continuous with in the intervals SE [0, Y - &], [y - &, Y + s] &

[y + &,1] with the condition y "* 0 or 1. Thus:

The '+' and the '-' signs in the superscript position indicate from which si de that point is

approached, '-' indicates from the left and '+' from the right side.

Now from the condition given by (A-16) we can state that :

(4-13)

and

(4-14)

Thus:

(4-15)

One must be careful, however, not to take (4-15) as one integral, because the derivatives

at the two extremities are represented by two different functions as given by (4-10).

Appropriately introducing the relations in (4-10) in the integrals of (4-15) results in:

- l (1 . )1 b'r 2. i b'r w1j =-- +smx --+-smx --. 2re s - y 2n s=l S - Y

(4-16)

Obviously the contribution of the bound vortex in the RHS (right-hand-side) of the wing

to the velocity induced on itself is zero:

8W1b = O. (4-17)

Using relation (A-12) in Appendix A, the contribution of the free vortices in the LHS

(left-hand-side) is:

b'w2

= __ 1 (1-sinOJ8r, (4-18) j 2n (s + y)

where, from geometry(as shown in Figure 4.2),

P~r~~~e~nt~m~e~M~o~d~of~s~ol~u~tio~n~h~r~y~aw~e~d~w~m~g~s ___________________________________ 61

· e (s - y)sinX sm J = --.=============

~y2 +S2 +2sycos2X

Thus substituting (4-19) in (4-18) we have:

aW2f

= - 1 [1- (s - y)sinX ]ar. 2n(s + y) ~ i + S2 + 2sycos2X

And finally the bound vortex of the LHS of the wing:

s:- ____ 1 [cosez -cos2X] s:-r uW2b - u ,

2n 2y sin X

where, from geometry (shown in Figure 4-2):

cose2

= s + ycos 2X . ~i + S2 + 2sycos2X

Thus substituting (4-22) in (4-21) we have:

a W2b

= - l [ s + y cos 2 X cos 2 x] ar . 4JT ysinx )l +S2 + 2sy cos 2X

(4-19)

(4-20)

(4-21)

(4-22)

(4-23)

Substituting ar from (4-7) in equations (4-16,20 & 23)and integrating equation.( 4-8)

results in:

n

w(y) = IfJkFk(y) , (4-24) k;O

where:

Pk (y) = { (1 +~inZ) y' _

-Si:Z l',,t k,"I{Y)- 1."I{y}f2(k+I)}-~ k,,k" (y)- f2,kH{y)j2(k+l)}]-

-4(~;I)t l' /,-t V 1.,1{y)-V + l)I.J+,{y)}- ~ Vf2J{y)-V + l)f',1+2{y l}] +

P~r~~~en~t~m~et~ho~d~o~[s~ol~u~tio~n~œ~r~yaw~e~d~w~mg~s~_______________________________ 62

2 sin X kA ( ) } + 7r R k+JY R Y ,

Vk for p=O

fp,k (y) = l (- y)' fp,o (y)+ t, (- y y+' f'-',I (y) for p;' 1

fp,o(Y)=

for p = 0 Ik

2(k + 1) S-p,k (y) = k-l

(- YY S-p,o(Y)+ I(- yY-i-1 S-P-l,j(Y) for p ?:.1

i~O

for p=l

l ~(P-l) p-l-r {" () fi 2 2( -1) p-l L..J r Y J p-I,r-l Y or p?:. P Y r~J

(4-25)

(4-26)

T

(4-27)

for p ?:. 2

(4-28)

(4-29)

, (4-30)

P~r=~e=n~tm=e=fflo=d~~~so=m=ti=on~&~r~y~==ed~w=m=~~--------------------------63

where ao, a J , a2 , e and the dilogarithm integral Li2 (x) are defined in Appendix D .

They are constants used to approximate sorne functions in the pro cess of integrating sorne

difficult integrals. The integrals ç p,k (y), fp,k (y), l k and R(Y) are solved in detail in

Appendix D. From (4-24) one can see that for a straight wing , X = 0, and expression

11

reduces to w(y) = t L.Bkyk = tw' (y) , which is the case for straight wings. k;O

4.2 Collocation Procedu.re

Consider the local nondimensional circulation around the wing section, f L (y), which

depends on the effective incidence a eff (y) = a(y) - w(y )/U 00' The expression of r L (y)

is repeated here for convenience,

(4-31)

The velocity distribution on the wing, w(y), is obtained from equation (4-24).

The equality of expressions (4-1) and (4-31) of the circulation at any location

along the span yields an equation that can be used to determine the unknown coefficients

(4-32)

Thus introducing (4-24) and (4-31) in (4-32) results in:

11 •

L.Bk {Fk°(y)+ tKcac(y )Fk (y)} = t Kci:(y )a(y), (4-33) k;O

where the function Fko (y) is defined in (4-2)

This equality is imposed to be satisfied at n specified locations along the wing span, in

order to obtain a system of linear equations for the coefficients .Bk' Once the coefficients

are calculated, they can be substituted to equation (4-1) to calculate the solution of the

circulation for the swept wing.

Present method of solution for yawed wings 64

Chapter 5

Method validation and results obtained

In this Chapter, the method of velocity singularities for finite span wings developed in the

previous Chapters is applied to various wing configurations and the results for specifie

wing geometries are given. The method is successfully va!idated for rectangular and

trapezoidal wings in comparison to theoretical solutions of Rasmussen and Smith [22],

and solutions obtained by panel method developed by Katz & Plotkin [6]. Then the

present method is also applied to wings with curved leading and trailing edges and

variable incidence. Moreover the solutions to wing problems having sudden changes in

their geometries (due to the presence of ailerons and/or flaps) are presented. These

include the case of rectangular and trapezoidal wings with symmetric, anti-symmetric and

asymmetric variation of incidence. Finally, the results obtained for swept rectangular

wings with continuous incidence variation are also given. In aU cases comparisons have

been made with panel method results.

5.1 Solutions for symmetric wings with continuous incidence variation

5.1.1 Method validation: rectangular and !apered wings

Consider a thin trapezoidal wing of uniform incidence defined by the foHowing

geometrical parameters:

Aspect ratio: À

Taper ratio: q =l-cT /cR

Chord variation: c(y) = CR (1- qlyj)

Mean chord: ca =(l-tq)cR =2//1.

Method validation and results 65

Chord coefficients:

Incidence variation:

Incidence coefficients:

Co = CR/Ca =2/(2-q), CI =-qco' and cm =0 for m'22

these coefficients correspond to (3-40).

a(y) = a R = aa

ao = l, and a j = 0 for j '2 l , the coefficients correspond

to (3-40).

Substituting the above geometric characteristics of the wing in (3-45) and (3-46) results

m:

D - K Co CI E -_ C 0 + 1 [

- -] K [C C] 1 - ca i + 1 + i + 2 ' i,k - 2 a k + i + 1 k + i + 2 ' (5-1)

where K = 7r is used in most of the following numerical examples.

The present solutions for the spanwise variation of the circulation, f(y), and the lift

and induced drag factors, rand <5, are compared for validation in Tables 5.1 and 5.2

with the results obtained by Rasmussen & Smith [22] for a rectangular wing ( q = 0 ) of

aspect ratio A = 6 and for a trapezoidal wing of taper ratio q = 0.6 and A = 9. For a

more meaningful comparison, in the same tables are also given the results obtained with a

panel method (Katz & Plotkin [6]) for J = 8 p~els along the chord and J = 10 panels

along the wing semi-span (or 20 panels along wing span), and for tbis reason the values

of 1(Y) are given at the panel centers (the solutions calculated with J = 20 panels

displayed very small changes compared with J = 10 ).


Panel method Rasmussen y Present solution of r(y) }Catz 8G Plotkin 8G Smith

[6] [22] n=l n=2 n=3 n=4 n=5 1=8, )=10 8-term

0.00 0.4396 0.4332 0.4323 0.4321 0.4320 - 0.4312 0.05 0.4377 0.4319 0.4315 0.4316 0.4317 0.4285 0.4317 0.15 0.4299 0.4289 0.4303 0.4304 0.4302 0.4269 0.4303 0.25 0.4214 0.4268 0.4274 0.4267 0.4269 0.4234 0.4270 0.35 0.4143 0.4231 0.4214 0.4215 0.4217 0.4184 0.4214 0.45 0.4084 0.4152 0.4130 0.4139 0.4135 0.4107 0.4135 0.55 0.4018 0.4019 0.4021 0.4019 0.4020 0.3996 0.4021 0.65 0.3909 0.3829 0.3855 0.3845 0.3848 0.3821 0.3847 0.75 0.3687 0.3564 0.3580 0.3583 0.3579 0.3554 0.3579 0.85 0.3217 0.3140 0.3117 0.3124 0.3127 0.3104 0.3127 0.95 0.2106 0.2150 0.2139 0.2131 0.2127 0.2118 0.2127

r 0.1481 0.1589 0.1602 0.1605 0.1606 0.1635 0.1606 5 0.0540 0.0495 0.0486 0.0484 0.0483 0.0474 0.0483

CLa 4.5442 4.5323 4.5309 4.5306 4.5305 4.5273 4.5305

CDi 1.1547 1.1438 1.1420 1.1416 1.1415 1.1390 1.1415

t:r 0.453 % 0.124 % 0.052 % 0.026% 0.015 % -0.635 % -

t:L 0.302 % 0.040 % 0.009 % 0.002 % 0.000 % -0.070 % -

t:D 1.156 % 0.198 % 0.046 % 0.013 % 0.003 % -0.223 % -

Table 5.1 Values of r(y), rand 5 for a rectangular wing (q = 0) of Â = 6

The present solutions based on velocity singularities were found in very good

agreement with those of Rasmussen & Smith [22] for both the rectangular and trapezoidal

wings. The present solutions obtained with' n = 5 displayed a very good accuracy,

although those for n = 3 and n = 4 were also very accurate (even n = 1 provides an

acceptable engineering accuracy). This cau also be seen from the average relative

differences, t: r , between these solutions for f(y) , and the relative differences for the

lift and induced-drag coefficients, t: L and t: D , also shown in these tables.

U~e~ffi~o~d~va~b~'M~tl~'M~a~n~d~re~su~h~s________________________________________ 67

Panel method Rasmussen y Present solution of f(y) lCatz ~ Plotkin ~ Smith

[6] [22] n=l n=2 n=3 n=4 n=5 1=8, J=10 8-tenn

0.00 0.3813 0.3775 0.3765 0.3762 0.3761 - 0.3735 0.05 0.3782 0.3741 0.3737 0.3738 0.3740 0.3724 0.3737 0.15 0.3633 0.3622 0.3638 0.3639 0.3636 0.3625 0.3643 0.25 0.3441 0.3486 0.3492 0.3483 0.3486 0.3470 0.3486 0.35 0.3244 0.3324 0.3300 0.3304 0.3305 0.3291 0.3299 0.45 0.3053 0.3116 0.3089 0.3101 0.3095 0.3089 0.3098 0.55 0.2868 0.2863 0.2871 0.2866 0.2869 0.2855 0.2871 0.65 0.2674 0.2586 0.2625 0.2611 0.2614 0.2600 0.2611 0.75 0.2432 0.2306 0.2322 0.2331 0.2324 0.2320 0.2325 0.85 0.2061 0.1998 0.1958 0.1966 0.1973 0.1957 0.1973 0.95 0.1321 0.1400 0.1391 0.1376 0.1368 0.1376 0.1369

r 0.03231 0.04711 0.05044 0.05127 0.05153 0.05250 0.0514 0 0.01353 0.01670 0.01583 0.01538 0.01521 0.01494 0.0152

CLa 5.1108 5.0971 5.0941 5.0933 5.0931 5.0922 5.0932

CDi 0.9363 0.9342 0.9323 0.9316 0.9314 0.9308 0.9314

er 0.536 % 0.280 % 0.150 % 0.084 % -0.033% -0.300 % -

eL 0.345 % 0.077 % 0.017 % 0.002 % -0.002% -0.020 % -eD 0.526 % 0.303 % 0.097 % 0.023 % -0.003% -0.065 % -

Table 5-2 Values of f(y) , rand 0 for a trapezoidal wing of q = 0.6 and À = 9

The spanwise variation of f(y) in the present solutions was smoother than in

Rasmussen ~ Smith solution, which displayed slight oscillations especially near the

central chord (e.g. the values at y = 0.05 were slightly larger than those at y = 0, instead

of smaller), probably due to the absence of the forced symmetry tenns. The effect of the

forced syrnmetry terms in the circulation distribution, which is not continuous at y = 0 ,

can be se en in Figure 5.1 and more clearly in Figure 5.9 for flat tapered wings

( a A / a R = 0 ). For trapezoidal wings, the present solutions were found slightly closer to

the panel method results than Rasmussen ~ Smith solutions.

M~e~ffi~o~d~w~lz~'M~fl~on~a~n~d~re~su~h~s________________________________________ 68

The variations of the lift and induced drag factors, r and li , versus the wing taper

ratio q calculated with n = 5 are shown in Figure 5.2 for four values of the aspect ratio,

Â, and they are compared with the panel method results (Katz & Plotkin [6]). A good

agreement was found between the present solutions and the panel method results,

especially for Â > 3 (even for triangular wings, q = 1 ). One can notice that the induced

drag of the trapezoidal wings with a taper ratio between q = 0.6 and q:= 0.65 IS very

close to the minimum induced drag of the elliptic wings (for which 6 = 0).

The influence of the aspect ratio, Â, on the spanwise distribution of the circulation is

shown in Figure 5.1 for rectangular (q = 0) and trapezoidal (q = 0.6) wings. The

influence of the wing taper ratio, q, on the spanwise distribution of the local lift

coefficient, Cl (y) / CL = (2/ K;.) f(y)j c(y), is illustrated in Figure 5.3a for Â = 9, and

the variation of the lift coefficient slope, Cl a = 2K L ,in function of q for four values of

Â is shown in Figure 5.3b.

A comparison between the spanWlse distributions of the circulation, f(y) , for

tapered wings of various values of the taper ratio q and elliptic wings is illustrated in

Figure 5.4 for Â = 2 K , where K = 0.9 Ji was used instead to take into account the

viscous effects on the lift coefficient slope; as recommended by Carafoli [4]. Good

agreement was found with the solutions derived for this particular case (using a

completely different method) by Carafoli [4].


0.7

r 0.6 .. -.... -.------+------+-----t---:~--_+_----

0.5 +-----_f_

OA

0.3

0.1 q=O

o -.~.--------~--------~-------4--------~--------~

0.9

0.8 f

0.7

o 0.2 OA

0.6 -+------4·-·--·····--.. ·--··· .... ·--

0.6 0.8 y

- Present solutions

L::..Â=3 ) o Â = 6 panel method <) Â=9 (Katz & o Â = 12 Plotk:in

6)

0.4 -+------4-----~--~~~~----_+~~-~

0.3 ·I~~=e==;=r_=:=~ 0.2 +----+----+---~~~~=---=~~.,.....--~~

0.1 q=O.6

o o 0.2 OA 0.6 0.8

y

Figure 5.1 Rectangular and tapered wings. Influence of the aspect ratio Â on the

distribution of f(y).

M,~e~~o~d~m~û~·@~ti~on~a~n~dr~e~su~~~ _____________________________________ 70

8

0.3 .,..-.. ·-----.-.. --·~~r-·-----------------_,__-~-··-· .. ·-.. -·-... , -- Present solutions o Â= 12 <> Â,=9 o 1=6 6 1=3

panel method (Katz & Plotkin6

)

0.2 +->-..:-----"';:--r----------,---"---

0.1

0-0 0.2 0.4 0.6

q

0.2

-- Present solutions 0 Â= 12

0.15 ... <> 1=9 ) panel method 0 Â=6 (Katz & Plotkin6

)

6 1=3

0.1

o 0.2 0.4 0.6 q

0.8

0.8 1

Figure 5.2 Tapered wings. Variation of rand 8 with the taper ratio, q, for

various aspect ratios, Â.

M,~et~ho~d~v~al~m~at~w~n~a~nd~r~~~u~~~ ________________________________________ 71

1.4· i -1

1.2

Cl (y) 1

CL

0.8

0.6

OA

0.2

0

5.5

4.5

4

3.5

i ! +-----+-_._---- ------ -+----------------_ ... j

~~~*--~----~----~-------E----~--~~!~ . ..... -... ---.------ -----+-------t-----t---"r~!'r\__I

0

... ""'"

..... ~

.. _-_ ... -'-. --.

-

.... -....... - ... -.------'-----...........L.-----t---............,,~

0.2

Present solutions for Â = 9 : o q=O /::). q=0.4 o q = 0.6 panel method

(Katz & Plotkin6) o q=0.8

+ q=l

OA 0.6 y

~ " ~ ~

..... ..... ~ ..... ~ -

0.8 1

1

~I) ~

~

~I Panel method (Katz & Plotkin6

) : qJ

- Present solutions !

1

/::). ..1,=3, 02=9 0 Â=6, o.ii= 12

':'" ......

~ il.J>

,r;> il.J>

, 1

o 0.2 0.4 0.6 0.8 1 q

Ca)

(b)

Figure 5.3 Influence ofthe taper ratio q on: (a) local Uft coefficient distril.:mtion,

CI (y) / CL ,for Â = 9; (b) variation of the lift coefficient slope,


0.6 ~--------'----T--'.~'--------'-'-'r----~====-P-res-.-e-n-t-s-o-lu-t-io-n-s-----.

05

o q=O 6. q=0.25 <) q=O.5 o q=O.75 + q=l

------ EUiptic wing

Carafoli's solutions

0.4 ....... -_ .. __ .. - ... __ ._. __ .. _--+----_ ...... _--------!

0.3

0.2 .

0.2

0.1- .--~ ..

0.1

0

0.9 0.95 1 0

0 0.2 0.4 0.6 y 0.8

Figure 5.4 Variation of r(y) for trapezoidal ( q = 0, 0.25, 0.5, 0.75 and 1) and

elliptic wings. Comparison with Carafoli's solutions for uniform wing

incidence.

M,~effi~o~d~va~li=M~ti~on~a~n~dr~e~su=ili~ _____________________________________ 73

1

5.1.2 Wings with curved leading and traHing edges and variable incidence

The solution (3-41) of the circulation r{y), as well as the general solution (3-21),

includes a1so the case of curved leading and trailing edges with a variable incidence along

the span. The wings with curved edges are more difficult to model with the panel

methods(and in general by numerical methods) since a large number of panels in the

spanwise direction are required to correctly represent the curved edges.

The solution (3-41) for wings with curved leading and trailing edges is illustrated for a

wing with a parabolic chord variation.

Aspect ratio:

Taper ratio:

Chord variation:

Mean chord:

Chord coefficients:

q = l-cr ICR

c(y)= CR (l-q y2)

Ca =(I-q/3)cR =2/J..

Co =l/Q-q/3), CJ =0, C2 =-qco ,and Cm =0 for m~3

Two incidence variations have been considered for this wing:

(i) a parabolic incidence variation a (y) = a R (1 + a y2 ), for which

a o =aRlaa =(3-q)/(3-q+a-3qaI5) and a 2 =aao ;

(ii) a uniform incidence a{y) = aa = a R , with a o = 1 (which corresponds to

a=O).

Substituting the above geometric characteristics ofthe wing in (3-45) and (3-46) results

m:

(5-2)

(5-3)


The solutions obtained for these wings with a parabolic variation of the chord along the

span are illustrated in Figure 5.5 for A = 9, q = 0.6 and for three cases ofparabolic

incidence variation (a = 0, 0.4 and 0.8). A different nondimensional circulation,

r R (y) = r(y) I[ 2 baR U", ], is used in this figure to compare the influence of incidence

variation. The solution for a trapezoidal wing of uniform incidence with the same root

chord CR and taper ratio q = 0.6 , and hence À = 9 (1- q 12) 1(1- q 13) , is also shown for

comparison. No other comparisons are shown for these wings, since panel method

solutions are difficult to obtain for curved leading and trailing edges.

r R

0.2 +-.. ------1--..... - ..... --.......... - ....... ---_'---___ -+-_____ .....

0.1

Present solutions for q = 0.6 :

Parabolic chord distribution -0- 0..=0 -~- o..=O.4} paraboIic inèidence -0- a = 0.8 distributions

-0- Trapezoïdal wing(a=constant) o +--------T--------T--------+--------+--------m

o 0.2 0.4 0.6 0.8 l y

Figure 5.5 Wings with curved edges and variable incidence: parabolic chord

distribution. Varnatimn of rR (y) = r(y )/[2 baR U C1J] along span.


5.2 Solutions for symmetric wings with incidence and chord changes

Consider a rectangular( q = 0 )and trapezoidal (q = 0.6) wings with sudden changes

of incidence at s = 0.5 from a R to a r . The wing geometries defined in the following

manner.

Aspect ratio:

Taper ratio:

Chord variation:

Meanchord:

Chord coefficients:


c (y) = CR (1- q Iyl)

ca =(1-q/2)cR =2/}"

Co = CR/Ca =2/(2-q), CI =-qco, and Cm =0 for m~2

these coefficients correspond to (3-40).

a (y) = {a R = a R / a a for 0 ~ y S; s ar=ar/aa for s~yS;l

For a meaningfuI comparison, the wing solutions compared in Figure 5.6 correspond

to the same average (me an) incidence, aa.' and hence a r = a r / aa is related to

ŒR =aR 1 aa through the relation Œr =aR +(1-aR )(2-q)/[(1-s)(2-q-qs)].

Three cases are considered in each case as shown in Figure 5.6 for (s = 0.5). The

present solutions for Â = 9 (calculated with n = 3 and n = 4 ) have been found in very

good agreement with the panel method results (Katz & Plotkin [6]), for both cases of

rectangular and trapezoidal wings.

The influence of the location of sudden changes in incidence (e.g. at

s = 0.4, 0.5 and 0.6) is shown in Figure 5.7 for rectangular (q = 0) and trapezoidal

( q = 0.6 ) wings of aspect ratio Â = 9 for a R = 0 . In this case, a different nondimensional

circulation, rr (y) = r(y )/[ 2 b a r U co], was used for a more meaningful comparison. The

present solutions were found in good agreement with the panel method results (Katz &

Plotkin [6]), which are also shown in Figure 5.7.


0.6 _ Present solutions Panel method (Katz & Plotkin6

) :

0.5. 0 aR = l ,aT= 1 o aR =0.5, aT=1.5

r DaR=O ,ar=2 0.4 - ---------r------+

0.2

Â=9 q=O

o +---------~------~--------_+--------_r--------m o 0.2 0.4 0.6 0.8 l

y

0.6 _ Present solutions Panel method (Katz & Plotkin6

) :

0.5 - 0 aR = l , aT= 1

r 0 aR = 0.5, ar= 1.773 o aR = 0 ,ar=2.545

0.4 ----.-.... -.-~----__l_--_I__ -:::::::I;:::;:;;;;,;::::::::---t--~--j

0.2 +------+-----+--lé:t-----+---.--t--~--~

0.1 +-------~~~---r---~--~------~------~

Â=9 q=O.6

o +---------~------~---------+--------~--------~ o 0.2 0.4 0.6 0.8 l

y

Figure 5.6 Symmetric wings with incidence changes due to control surface deflection

(/laps). Variation of f(y) along ~pan.


0.3 - l . -- Present so utlOns

Panel method (Katz & Plotkin6) :

rr 0 3=OA, a)aT =0.6 o 3=0.5, a)aT =0.5 o 0.2 -t------'-r-----'"-'--'----r---f"'or.---+--+-----;;--"'---t--?~'<__-__l

a =0 R

.IL =9

q=O

o +--------4--------~--------+--------+--------~ o 0.2 OA 0.6 0.8 1

y

0.3 ---_._------, '-----.....,-----..,-------, -- Present solutions

Panel method (Katz & Plotkin6) :

o 3 = OA > a)ar = 0.497 o 3 = 0.5, a)aT = 0.393

o s=0.6, a)aT =0.297 0.2 +----~~-L---r--A---~~~~~1__---------j

a =0 R

Â =9

q=O.6 o +-------~--------~--------+---------t--------~

o 0.2 0.4 0.6 0.8 1 y

Figure 5.7 Symmetric wings with incidence changes due to control surface

deflection (jlaps). Influ.ence of the location, s, of incidence changes on the variation of

M,~et~ho~d~v~al~~=Œ~io~n~a~nd~r~~~u=~~_______________________________________ 78

5.3 Solutions for wings with asymmetric incidence variation

5.3.1 WhIgS with antisymmetric incidence variation

The solution (3-60) of the circulation is illustrated in Figure 5.8 for the following

cases of rectangular ( q = 0 ) and trapezoidal ( q = 0.6 ) wings with Â = 9 :

(i) Wings with a continuous anti-symmetric linear variation of incidence,

a (y) = a r Y (which may correspond to the effect of a rolling rotation of the

wing).

(ii) Wings with sudden anti-symmetric changes of incidence, from a R = 0 to

± a r ,at y = ± s ,for s = 0.4, 0.5 and 0.6.

The anti-symmetric distribution of the incidence results m an average incidence

aa = 0, hence the nondimensional 'circulation derived m Chapter 3,

r(y) = r(y) /[ 2 b a aU", ], is no more meaningful. Thus a different nondimensional

circulation, rr (y) = r(y )/[ 2 b a r U <Xl]' is used in tbis case for a meaningful comparison.

In an these cases, the present solutions were found in good agreement with the panel

method results (Katz & Plotkin [6]), which ar~ also shown in Figure 5.8. This shows the

effeetiveness of the functions Fk (s, y)- Fk (s,-y), representing the specifie contributions

of the anti-symmetrie changes of the wing incidence at y = s and y = -s .


0.3 ..,------

- Present solutions

r T Panel method (Katz & Plotkin6) :

-1--...,..--__

b. a(y)=ary

0.2 o s = 0.4 } ~ncidence o s=0.5 Jump <> s=0.6 O-»±aT

a =0 R

0.1 Â =9

o 0.2 0.4 0.6 0.8 l y

0.3 ,...---------------------,----------r------.--.------,

- Present solutions

0.2

0.1

o

Figure 5.8

Panel method (Katz & Plotkin6):

b. a(y)=aTy

o s = 0.4 } ~ncidence o s=O.5 Jump <> s=0.6 O~±aT

a =0 R

Â =9

q=0.6

0.2 0.4 0.6 0.8 1 y

Wings with anti-symmetric incidence variation. Distribution of

fT (y) = r(y) I[ 2 baT U 00] along span for: (i) Lineal' incidence

variation a(y) = a T y . (fi) Incidence changes due to anti-symmetric

deflections of ailerons at three locations, y = ±s .


5.3.2 Wings with asymmetric incid.ence variation

In many flight evolutions, asymmetric incidence distributions may oœur as a result of

the combined effects of the wing incidence, symmetrically deflected flaps and anti

symmetrically deflected ailerons. In this case, the wing circulation distribution 1S given by

the general solution (3-21), or, altematively, it may be obtained by a convenient

summation of the symmetric and anti- symmetric solutions (3-41), (3-51) and (3-60).

The solutions for wings with asymmetric incidence distribution, generated by a

symmetric wing incidence and anti-symmetrically deflected ailerons, are iHustrated in

Figure 5.9 for rectangular (q = 0) and trapezoidal (q = 0.6 ) wings of aspect ratio /l., = 9

with the incidence distribution given by

a (y) = a R + H(s,y) a A -H(s,y) a A

where H(s,y) and H(s,y) are defined by equations (3-6).For this case s = 0.5 is taken.

The nondimensional circulation r R (y) = r(y) 1[2 baR U 00 ] 1S used in this figure to

compare the influence of the anti-symmetric incidence changes,

a A 1 a R = 0, 0.25, 0.5 and 1. One can notice that aU circulation distributions for the

trapezoidal wings have a sI ope discontinuity at y = 0, in contrast to the rectangular

wings, which stresses the importance of the forced symmetry and anti-symmetry terms,

taken here into account for the fist time.

The present solutions were found in good agreèment, in a11 cases, with the panel method

results (Katz & Plotkin [6]), which are aIso shown in Figure 5.9.


0.6 ,------,----,-.

0.5 +-------+---t----j----+--+----f---t------;;l---....::=+---j

rR

0.4

-1 -0.8 -0.6 -0.4 -0.2 o 0.2

q=O À,=9

0.4 0.6 0.8 1 y

0.5 ..,..---,-----,---,------,----,--------.. ----------- ----r---i

rR 0.3 +----l----+----:;.~"Y'f',y;.._-I__--+--_+_--+__~--+--"""'--f'<:-\

-0.8 -0.6 -0.4 -0.2 o 0.2 0.4 0.6 0.8 y

Figure 5.9 Wings with asymmetric incide~ce variation due to anti-symmetric

j 1

1

aileron deflections. Distribution of r R (y) = r(y) I[ 2 baR U 00] along span.

M,~e~~o~d~va~l~m~at~w~n~a~nd~r~~~u~fu~________________________________________ 82

5.4 Solutions for swept wings

Consider a symmetric rectangular or trapezoidal wing with continuous variation of

incidence. Then the geometrical parameters for the swept wing are defined as:

Aspect ratio:

Taper ratio:

Sweep angle

Chord variation:

Mean chord:


x

c(y) = CR (1- qlyl)

ca = (I - t q) CR = 2/ Â

a(y)= aR = aa

The solution of the nondimensional circulation f(y), is calculated using the

procedure derived in Chapter 5. As shown in Figure 5.10 & Il the solutions are given

for Â = 3, Â = 6 , and Â = 9 , and for various sweep angles X .

It is evident from the result figures that as the aspect ratio gets smaller and the sweep

angle gets larger, the accuracy generally tends to reduce. The same holds true for the case

of straight wings of small aspect ratio. The contribution of the left-hand si de bound

vortex part of each vortex element goes to infinity each time the point at which the

induced velocity is calculated approaches zero or y ~ O. This could be the cause of

reduction in the accuracy of the solution. In addition to this, in the process of solving

sorne integrals, which are more difficult to solve, sorne functions have been approximated

by other appropriate functions that behave in a similar manner; and thus this could

contribute to the reduction of accuracy. However the results are close to the solutions

obtained by using panel method ( Katz & Plotkin [6]) especiaHy for Â > 4 and not too

large sweep angles. Thus further investigation is required to improve the situation.


0.7 -

%=20" 0.6 -+----------+~-------+----~----+--~-------->------------"

r 0.5 -------+------t--------j----------+---------------;

0.4 --------4-------+--------l------------..I.-.--------------

0.3 +-----+------l--------I--~

0.2 -+------'------"------1~-------'----""' __ ~"---,

- Present Solutions

o A = 6} Panel Method o A = 9 (Katz & Plotkin [6] )

o +------~~-----~-----~~----4----~

0.1

o 0.2 0.4 0.6 0.8 y l

0.7 -

%=25" 0.6

r 0.5 -

1

,-, r1_ ,-, r1 r1 n -= ~

V v V A --«---~D -0 v v

1 ~I d

0.4

0.3

0.2

0.1 i

o 1

o 0.2 0.4 0.6 0.8 y 1

Figure 5.10 Variation of r(y) for rectangular swept wings X = 20° and X = 25°


0.7i-----T 1 i

0.6 t--r

0.5 1

0.4 -t------t-------t-------t------+

0.2 +-----------1--------+------

0.1

o -.~----~---_4----_+----

0.6

0.5

r 0.4

o 0.2 0.4 0.6

-----,----_. __ ._---- ._. __ ... ,-_._--~_._--_ .... _---_ ... __ .. _-------_.-

0 0 0 0 0 u

0.8

1

~I

%=30'

y 1

=35'

~D <> <> <> 1 <> <> 0 ~ <>

0.3

0.2 1 ~

~I 0.1

o o 0.2 0.4 0.6 0.8 y 1

Figure 5.11 Variation of r(y) for rectangular swept wings X = 30° and X = 35'

M=e~~=o=d~va=li=~=tio=n~a=nd~r~~=u=fu~_____________________________________ 85

Chapter 6

Conclusion

A new method of solution has been presented for finÏte span wings of arbitrary shapes

In incompressible flow, which avoids the difficulties of the previous methods. This

method uses velocity singularities placed in the Trefftz plane to derive the contributions

of the spanwise changes in the wing chord and incidence variations along span in the

solution of the spanwise distribution of the circulation. The new functions derived for

these contributions represent a correct mathematical modeling of the physical wing,

containing both natural and forced symmetry and anti-symmetry terrns (due to these

terrns, the circulation distribution of trapezoidal wings has a discontinuous slope at

y = 0, in contrast to rectangular wings, as shown in Figure 5.9). The new forced

symmetry and anti-symmetry terms and the new contributions of the changes in the

incidence and chord variations (which were missing from the previous methods)

determine a high level of accuracy of the solution, which is free of spurious spanwise

oscillations. An improved Galerkin-type method for solving the unknown coefficients

also contributed to this.

This method has been successfully validated by companson with the solutions

obtained by Rasmussen & Smith [22] and Carafoli [4] for the rectangular and tapered

wings ofuniform incidence, and with panel method results (Katz & Plotkin [6]).

The method has been then used to obtain solutions of aeronautical interest for wings

with symmetric and anti-symmetric incidence changes due to symmetric and anti

symmetric deflections of the control surfaces situated on the wing, such as flaps and

ailerons.

Solutions have been also easily obtained for wings with curved leading and trailing

edges and variable incidence (the curved edge~ of wings are more difficult ta be modeled

in panel methods).

The method led to very efficient, robust and accurate theoretical solutions in aH cases

studied, including the asymmetric incidence variations occurring in various flight

evolutions.

Conclusion 86

A procedure has been developed to find the solution for swept wings. This procedure

uses the solution of the circulation distribution in the Trefftz plane to derive the induced

velocity on the wing. The effect of the bound vortex on the wing has been considered in

this process. Because of this, sorne difficult integrals have been encountered and sorne

approximations have been used in the process of integration. Apart from this the

contribution of the bound vortex at y = 0 tends to go to infinity which might have caused

sorne accuracy problems in the formulation. Although the results obtained are very close

to the ones from the panel method the accuracy is not as good as for the case of straight

wings. Therefore more effort has to be invested for the future in the improvement of the

accuracy of the solution. Furthermore a more efficient procedure similar to the one used

for the determination of coefficients (Galerkin method) for straight wings may be used to

avoid the deficiencies of using the collocation procedure.

This method may further be extended to consider finite wings in a compressible

subsonic flow. It may also be developed to include the effects ofviscosity.

Conclusion 87

References

[1] Glauert, H., The Elements of A erofo il and Airscrew Theory, 2nd ed., Cambridge Univ.

Press, Cambridge, England, UK, 1959, Chap. 10,11.

[2] Anderson, J. D., Fundamentals of A erodynamics, 2nd ed., McGraw-Hill, New York,

1991, Chap. 5.

[3] Bisplinghoff, R. L., Ashley, H., and Halfman, R L., Aeroelasticity, Addison-Wesley,

Longman, Reading, MA, 1955, pp. 229-238.

[4] Carafoli, E., Tragflügeltheorie, VebVerlag, Berlin, 1954; Aerodinamica, ed. Tehnica,

Bucuresti, 1951, Chap. 15-18.

[5] Karamcheti, K., Ideal-Fluid Aerodynamics, Wiley, New York, 1966, Chap. 19.

[6] Katz, J., and Plotkin, A., Low-Speed Aerodynamics, McGraw-Hill, New York, 1991,

Chap.8.

[7] Katz, J., "Calculation of the Aerodynamic Forces on Automotive Lifting Surfaces",

Journal ofFluids Engineering, VoU07, No. 12, 1985, pp. 438-443.

[8] Kuethe, A. M., and Chow, C.-Y, Foundations of Aerodynamics, 4th ed., Wiley, New

York, 1986, Chap.6.

[9] Milne-Thomson, L. M., Theoretical Aerodynamics, Dover, New York, 1966,

Chap.ll.

[10] McCormick, B. W., Aerodynamics, Aeronautics, and Flight Mechantes, 2nd ed.,

Wiley, New York, 1995, Chap. 3,4.

[11] Robinson, A., and Laurmann, J. A., Wing theory, Cambridge Univ. Press,

Cambridge, England, UK, 1956, Chap. 3.

[12] Schlichting, H., and Truckenbrodt, E., Aerodynamics of the Airplane, McGraw-

Hill, New York, 1979, Chap. 3.

[13] -Thwaites, B., Incompressible Aerodynamics, Oxford Univ. Press, Oxford,

England, UK, 1960, Chap. 8.

[14] Von Mises, R, Theory ofFlight, Dover, New York, 1959, Chap. 3.

[15] Multhopp, H., "The Calculation of the Lift Distribution of Airfoils," British

Ministry of Aircraft Production, R T. P. 2392, translated from Luftfahrtforschung,

Bd. 15, NI. 14, 1938, pp. 153-169.

Refèrences 88

[16] Bertin, J. A., and Smith, M. L., Aerodynamies for Engineers, 3rd ed., Prentice

Hall, Englewood Cliffs, NJ, 1998, Chap. 7.

[17] Andreson, R. c., and Millsaps, K., "Application of the Galerkin Method to the

Prandtl Lifting-Line Equation," Journal of Airerafi, Vol. 1, No.3, 1964, pp. 126-128.

[18] Bera, R J., "Sorne Remarks on the Solution of the Lifting-Line Equation," Journal

of Airerait, Vol. Il, NoJO, 1974, pp. 647-648.

[19] Berbente, C., "On a Method for the Aerodynamic Calculation of Finite-Span Wings

in Incompressible Flow," Buletinul Institutului Politehnic Bucuresti, Vol. 35, No. 5.

1973, pp. 1-13.

[20] Carafoli, E., Aérodynamique des Ailes d'Avions, Chiron, Paris, 1928.

[21] Lotz, I., "Berechnung der Auftriebsverteilug beliebig geformter Flugel,"

Zeitschrift für Flugtchnik und Motorlufisehiffahrt, Vol. 22, No. 7, 1931, pp. 189-195.

[22] Rasmussen, M. L., and Smith, D. E., "Lifting-Line Theory for Arbitrarily Shaped

Wings," Journal of Airerait, Vol. 36, No. 2, March-April1999, pp.340-348.

[23] Mateescu, D., and Newman, B. G., "Analysis of Flexible-Membrane and Jet-

Flapped Airfoils using Velocity Singularities," Journal of Aircraft, Vol. 28, No. 11,

1991, pp. 789-795.

[24] Mateescu, D., "Wing and Conical Body of Arbitrary Cross-Section in Supersonic

Flow," Journal of Airerafi, Vol. 24, No. 4,1987, pp. 239-247.

[25] Mateescu, D., "A Hybrid Panel Method for Aerofoil Aerodynamics," Boundary

Element XII, Springer-Verlag, Berlin & New York, Vol. 2, pp.3-14, 1990.

[26] Mateescu, D., Nadeau, Y., "A Nonlinear Analytical Solution for Airfoils in

Irrotational Flows," Proe. of the 3rd International Congress of Fluid Meehanies, Vol.

4, Cairo, 1990, pp. 1421-1423.

[27] Mateescu, D., "Steady and Unsteady Flow Solutions Using Velocity Singularities

for Fixed and OsciHating Airfoils and Wings," Computational Methods and

Experimental l\.1easurements X, Comp. Mech. Pub., Southampton & Boston, 200 l,

pp.3-12.

[28] Mateescu, D., Abdo, M., "Theoretical Solutions for Unsteady Flows Past

Oscillating Airfoils Using Velocity Singularities," Journal of Airerajt, Vol. 40, No.l,

2003, pp. 153-163.

References 89

[29] Mateescu, D., Seytre, J. F., and Berhe, A. M., "Theoretical Solutions for Finite

Span Wings of Arbitrary Shapes Using Velocity Singularities", Journal of Aircrajt,

Vol. 40, No.3, 2003, pp. 450-460.

[30] Mateescu, D., Seytre, J. F., and Berhe, A. M., "Aerodynamic Solutions for Finite

Span Wings of Arbitrary Shapes", AIAA Conference, 4yt Aerospace Sciences

Meeting and Exihibit, Reno, Nevada, AIAA 2003-1098, January 2003.

[31] Carafoli, E., Mateescu, D., and Nastase, A., Wing Theory in Supersonic Flow,

Pergamon Press, Oxford, London & New York, 1969.

[32] Talay, T. A., Introduction to the Aerodynamics of Flight, NASA, Springfield, Va,

1975.

[33] Smith, H.C., The Illustrated Guide to Aerodynamics, 2nd ed.,McGraw Hill, Blue

Ridge Summit, PA, 1992, Chap. 6,7.

[34] Mateescu, D., and Abdo, M., "Unsteady Aerodynamic Solutions for Oscillating

Airfoils", AIAA Conference, AIAA 4Ft Aerospace Sciences Meeting and Exihibit,

Reno, Nevada, AlAA 2003-227, January 2003.

[35] Mateescu, D., and Abdo, M., "Nonlinear Theoretical Solutions for Airfoil

Aerodynamics", AIAA 21 st Applied Aerodynamics Conference, Orlando, Florida,

AIAA 2003-4296, June 2003.

[36] Lewin, L., Polylogarithms and Associated Functions, North Holland, New York,

1981, Chap. 1.

[37] Dziubinschi, A., Analysis of Finite Span Wings Based on Velocity Singularities,

McGiH University, Montreal, Canada, 1999.

[38] Seytre, J. F, Theoretical Solutions for Finite-Span Wings of Arbitrary Shapes

Using Velocity Singularities" Mc Gill University, Montreal, Canada, 2002.

[39] Mateescu, D. Subsonic Aerodynam~cs (Course Notes), McGiH University,

Montreal, Canada, 2000.

[40] Mateescu, D. Computational Aerodynamics (Course Notes), McGiH University,


[41] Mateescu, D. High-Speed Aerodynamics (Course Notes), McGill University,


R~e~k~re~n~ce~s______________________________________________________ 90

Appendix -A

A.1 The vortex filament, the Biot-Savart law, and Helmholtz's

theorems

The concept of vortex filament and sorne theorems related to it are very important to

establish a rational aerodynamic theory for a finite wing. A vortex filament is a curve

around which a vortex flow field is induced. A vortex filament could assume the shape of

any curve. If the circulation is taken about any path enclosing the filament, a constant

value, r , is obtained. The strength of the vortex filament is defined as r.

B

A

o

Figure A.t The velo city induced by a vortex filament of arbitrary shape

Consider a vortex filament of strength r located arbitrarily in space as shown in

Figure A.I. Then the velocity induced by the vortex can be calculated by as:

where,

dV=~ Rxd~ 4n- R 3

'

R = T(x-xJ+ J(y- yJ+k(z-zJ,

R=~~-~Y+(y-hY+~-~Y.

(A-l)

(A-2)

(A-3)

(A-4)

A~p~p~m~d~ü~A~ ______________________________________________________ 91

The velocity induced by the vortex filament from point A to B is thus given by:

V =~ JRxŒ, . 4;r R3

(A-5) AB

Equation (A-l) is called the Biot-Savart law and is one of the most fundamental

relations in the theory of inviscid, incompressible flow. For detailed derivation see

Ref.[5]. The vortex filament and the associated Biot-Savart law and Helmholtz's

theorems are simply conceptual aerodynamic tools to be used for synthesizing more

complex flows of an inviscid, incompressible fluid. When a number of vortex filaments

are used in conjunction with a uniforrn freestream , it is possible to synthesize a flow

which has a practical application. The flow over a finite wing is one such example.

Apparently, straight line vortex, shown in FigureA.2, is of importance to our study

and thus the induced velocity is derived for this group of vortex filaments.

ALL----7 v

, ,

Figure A.2 Velocity induced by a straight vortex marnent at point P

The velocity induced by a straight vortex segment AB can be derived from (A-5) by

using appropriate substitutions. From the geometry it is clear that

and

rs = -a cotan (J ,

a drs =-'-2-d(J,

sm (J

R=_a_ sin (J ,

(A-6)

(A-7)

(A-8)

A~p~p~ffl~d=a~A~ ______________________________________________________ 92

and we know that

IR x r ~.I = R r sin a drs .

Substituting (A-7) to (A-9) in equation (A-l):

=> dV =~sinada. 47ra

Integrating (A-11) from BA = ~ to aB = rp we get:

V = ~(cos~ -cosrp). 47ra

(A-9)

(A-IO)

(A-11)

(A-12)

The velocity induced by an infinite vortex filament can simply be obtained by substituting

~ = 0° and rp = 18W , resulting:

V=~. 21ra

(A-13)

Similarly for the case of semi-infinite line vortex substituting ~ = 90' and rp = 18W or

~ = W and rp = 90° (depending on which si de the line vortex extends to infinity),

resulting in either case:

V=~. 47W

(A-14)

The direction of the velocity induced can easily be detected by using the right hand mIe.

The great German mathematician, physist, and physician Hermann Von Helmholtz

(1821-1894) was the first to make use of the concept of vortex filament in the analysis of

inviscid, incompressible flow [2]. In the process, he established several basic principles,

which are now known as Helmholtz's vortex theorems:

1. The strength of a vortex filament is constant along its length.

2. A vortex filament cannot end in a fluid; it must extend to the boundaries of

the fluid (which can be at infinity) or form a closed path.

3. The fluid that forms a vortex tube or filament continues to form that vortex

tube or filament with time.

In the study of finite span wings these theories are the foundational concepts.

A·~pxp~en~d~~~A~ ____________________________________________________ 93

A.2 Vortex singularity

It is clear frorn equations (A-12) to (A-14) that as a ~ 0 the value of the induced

velocity goes to infinity (as shown in Figure A.3 by the dashed lines). The flow around

the vortex filament is necessarily irrotational except in the central region of the vortex

flow where aH the vorticity is concentrated. This region of the flow is called the vortex

core. The size of the core is very smaU, which basically defines the vortex filament itself.

Let & be the radius of the core, where & is very small (there is no clear eut nurnber to

define &). In contrast to the flow surrounding the core, the velocity induced by the

vortex filament within the core is not governed by the Biot-Savart law. Although there

exist different suggestions as to what function to use to represent the pattern of the

velocity induced within the core region, generally the velocity is considered to decrease in

sorne pattern to zero at the center of the core as shown in Figure A.3.

\ ··.\·.c

.~ ................. .

••• ~ •...••.. c.

The vortex core

a

Figure A.3 The velocity induced within and outside the core of a vortex segment

Looking at the variation of the velocity induced around and with in the core, we cau write

the following boundary conditions.

A~p~p~en~d=à~A~________________________________________________________ 94

At the center of the vortex core,

V = 0 at a = 0

and at the boundary of the core the velocity is continuous,

A.3 Subnmtine VORTEXL

2

1

z

p (x,y,z)

x

Figure A.4 Nomenclature used for the velocity induced by straight vortex

filament

(A-15)

(A-16)

The velocity induced by a constant vortex segment from point 1 to point 2, as shown

in the above figure, at an arbitrary point P can be wrÏtten as:

where,

~ = ( x p - Xl) T + (y p - YI) J + (~p - ZI ) k ,

r; =(Xp -x2)T +(Yp -Y2)J +(Zp -z2)f,

10 = r; - ~ = (X2 - XI) T + (Y2 - YI) J + (Z2 - ZJ) f ,

and the velocities u, v and w are the X,Y and z components of q12 •

(A-17)

(A-lg)

(A-19)

(A-20)

A~p~p~en~d~rr~A~____________________________________________________ 95

Appendix ... B Derivatives and integrals

appearing in the present method

B.I Important derivatives

Let X = Y + iz be a complex variable, then:

G(S,X)=~COSh-l (l-X)(l+s) 7T 2(s-X)

(B-1)

~[6(sx)J=_2. ~ ds' 7T (s-X)~

(B-2)

~[6(s,x)J=2. ~ dX 7T (s-X).JI-X2 (B-3)

G(s,y) = Re{6(s,X)} = ~COSh-l (1-Y)(l+s) 7T 2(s-y)

(B-4)

~[G(s,y)]=- 1 FI ds 7T(s-y).Jl-s2

(B-5)

~[G(s,y)]= l ~ dy 7T (s - y) ~l-l

(B-6)

Go{Y)= ~ Re {COSh-1 l;:} (B-7)

~[G ()J-_2. 1 dy 0 Y - 7l y~l- y2

(B-8)

2 ~+y C(y)=-cos-1 -

7T 2 (B-9)

~[C(Y)J=- 1 dy 7T~1- y2

(B-IO)

A~pcpe~nd~ù~B ________________________________________ 96

B.2 Important integrals

B.2.! Integral Jk (s)

Let:

for k = °

Using the result of derivative B-IO,

where C (s) is defined in (B-9).

For k = 1,

il .

2 cr JJ(s)=- ~dcr,

7r sI-cr

For k 2. 2,

integrating by parts:

J, (s) = ~ [-,11-0"' CT'-' [-(H) f _,11_,,20"-2 dO"

(B-11)

(B-l2)

(B-13)

(B-14)

(B-15)

(B-16)

A~p~pe~n~dù~B~ ______________________________________________ 97

and thus,

k -1 2 Sk-I. 2

Jk(S)=-k- Jk-2(S)+ TC k~ k?:.2

Jo(s)=2C(s), JI (S)=l:~ TC

B.2.2 Integral Ik

The integral Ik is defined as:

Substituting s = 0 in (B-ll), we see that:

Ik = Jk (0),

thus from (B-17) , we get:

k-l Ik =TIk-2' k?:. 2,

B.2.3 Integral Ek (s,y)

Let:

(B-17)

(B-18)

(B-19)

(B-20)

(B-21)

where G( o-,y) is defined by (B-4) and its derivative 1S given in this case by (B-5). Thus:

We can expand o-k as:

k

ak+l = l+l +(0-- y) L a k-

j yi. j=o

(B-22)

(B-23)

A~p~pe~n=d~~B~ _______________________________________________ 98

Inserting this in (B-22) and simplifying, we obtain:

- Sk+J yk+l il1 ~l-l Ek(S,y)=--G(s,y)+- ~ da

k + l k + 1 s TC ( () _ y) l _ (}2

~l-l Ik ·2 fI a

k-

j

+ yJ_ da 2(k+l). TC s .Jl-a2

J=O

But the integral J k (s) is defined in (B-ll) and its solution given in (B-17).

_ l+1 yk+1 CT=1 ~l- y2 Ik .

Ek(s,y)=--G(s,y)+-[-G(a,y)] + ( ) yJJk_J(s). k + l k + 1 CT=S 2 k + l

j=O

(B-24)

Let:

(B-25)

where G(s,y) is defined by (B-4) and its derivative is given in this case by (B-5). Thus:

- [yk+l ]1 1 fI yk+1 .Jl- S2 Ek(S)= - G(s,y) - - - dy.

k + 1 0 TC 0 k + 1 (s - y) ~l-l

We can expand yk+l as:

k

/+1 = Sk+l -Cs - y) Lsk-j yj

j=O

Inserting fuis in (B-26) and simplifying, we obtain:

E ( ) -V 1 - s- S d k- j Y d r;--:;l 2 {fI k+l k fI j }

kS=-TC(k+l) o(s-y)R Y-~S oR y .

(B-26)

(B-27)

A~p~p~en=~~x~B __________________________________________________ 99

But the integral Ik is defined in (B-18) and its solution given in (B-20).

r:-?l 2 {[ hl ]1 k } - _ --...jl-S- S 7r k-j

Ek(S)-- ( ) 7r Jï=7 G(s,y) --I> 1) . 7r k + 1 1 - S 0 2 j=O

Therefore,

(B-28)

~

B.2.5 Integral Ek (s)

Let:

(B-29)

where G(s,y) is defined by (B-4) and its derivative is given in this case by (B-5). Thus

integration by parts could be used to solve the integral. However cafe must be taken

when making the derivative of G ( s, - y) with respect to y .

d d d -[G(s,-y)] = [G(s,-y)]._[-y]. dy d(-y) dy

(B-30)

Thus:

[hl II 1 il hl r:-?l 2 A Y Y ---...jl-S-

Ek(S)= - G(s,-y) - - - dy. k + 1 0 7r 0 k + 1 (s + y) ~l-l

(B-31)

We can expand /+1 as:

k

/+1 = (_S)hl +(s+ y) ~)_l)k-j sk-j y}. (B-32) }=o

Inserting tbis in (B-31) and sirnplifying, we obtain:

Ê () --...j 1-s- -s d + (- t- j y d r:-z {I ()hl k 1 j }

k s "(k+l) I(s+y}R y f,: s J, R y .

A~p~p~en~m~x~B __________________________________________________ I00

But the integral Ik is defined in (B-18) and its solution given in (B-20), hence:

r:--2 {[ ()k+l ]1 k } A "\I1-s- -s lf k-j

Ek(S)= ( ) -lf 1ï=7 G (s,y) +-I(-s) IJ . lf k + 1 1- s 2_0 o J-

Therefore,

( )k+l r:--2 k

A -s "\Il-S-I k-j

Ek(S)= ( ) Go(s)+ ( ) (-s) Ij k + 1 2 k + 1 j=O

(B-33)

B.2.6 Integral Rk

Let:

(B-34)

where Go (y) is defined in (B-7) and its derivative given in (B-8). Integrating by parts,

(B-35)

=0 +! 1 y dy fI k

lf ( k + 1) 0 ~l-l . (B-36)

And finally:

R = Ik k 2(k+l) "

(B-37)

A~pxp~en~d~u~B ________________________________________________ IOl

Let:

(B-38)

then,

2 il l 2 il yk+2 P,=- ~-- ~

k TC 0 ~1- y2 TC 0 ~1- y2 ' (B-39)

(B-40)

P, =J..L k k+2

(B-41)

A~PLP~en~d~~~B ____________________________________________________ I02

Appendix - C

C.I The prototype problem

The nondirnensional cornplex conjugate velocity is given by:

. Ao +AX - -U(X)=-lfJo+ ~ -6~G(sl'X)+6W2G(S2'-X),

l-X2 (C-l)

And the nondirnensional circulation, f (y) = r (y) / (2baaU <1J) is given by:

f(y) = - r REAL[U(X)Jx~y dy. (C-2)

Substituting (C-l) in (C-2) and integrating:

f(y)=Re{[ifJoXr + f~+AX aX-6~El(SpX)+6W2Ê2(S2'X)}, (C-3) y Jy l-X2

using (B-9) and (B-l 0)

(C-4)

(C-5)

and where in (C-3)

El (s),x) = s: G(spX)dX, (C-6)

E2 (S2'X)= s: G(s2'-X)dX, (C-7)

Where G(s,x) is defined in (B-l). Integrating (C-6) by parts and using the derivatives

of G(s,X) and C(X) given in (B-3) and (B-lO) we have:

A~p~pe~n=m~x~C _______________________________________________ I03

=i-YG(S!,X)-SlfJ{~ R }dX-~l-S~ fJ- dX y ;rr (sJ - X) ~l- X2

y ;rr-/l- X 2

= i(l-sJ )+(sJ - y )G(s],y)+ ~l- s( C(y).

Therefore,

El (spx) = i(l-sl )+(sJ - y )G{spy )+~l-s( C(y). (C-8)

Similarly integrating (C-7) by parts:

E2 (S2'X) = J: G(s2'-X)dX

=-yG(S2,-X)+S2 fJ{- 1 R }dX -~l-si Jl- dX y ;rr (S2 +X)-/l-X2

y 7f-/I-X2

= -yG(S2'-y )-S2 G(S2' -y)+ ~l-si C(y).

Therefore,

(C-9)

A,.:.t:.pc::.pe.:.::::nd:::.::..ix.:::..C ___________________ I04

Now inserting the integrals (C-4), (C-5), (C-8) and (C-9) in (C-3) we have:

f(y) = A~1-l +JTAoC(y)-oW; [(SI - y)G(spy)+~l- s~ C(y)]

-oW2 [(S2 + y )G( S2'-Y)- ~l- si C(y)]. (C-lO)

However the constants A and Ao must be determined from boundary conditions.

Applying the boundary condition at the wing tips:

f(-1)=f(1)=0.

using this in (C-l 0) we get:

Ao = ~ [o~~l-SI2 -ow2~1-si J. Substituting Ao back to (C-l 0):

(C-ll)

CC-12)

f(y) = A~l-l-o~ (s] - y)G( s],y)-oW2 (S2 + Y )G( S2'-y). (C-l3)

To determine the constant A, we need to go to the complex conjugate velocity (C-l). The

velo city induced at infinity is zero. Thus:

lim [U(X)] = O. x~c()

Evaluating each of the limits in (C-14), we not~ that:

1. [ X ] 1 . lm =--=1 x ~oo -Jl- X 2 i'

[-( )] 2 1 (X-l)(l+s]) .2 _I~+S] . ( ) lim G spX =i-cos- ( ) =l-cOS --- = lC SI ,

X --"00 JT 2 X - SI JT 2

Introducing the results in (C-15) to (C-l 7) in (C-14) we have

Jim [U(X)] = -i Po +iA -Ù)~C(SI )-ioW2C(S2) = o. x--"oo

Solving this we get:

A = /30 + O~C(SI)+OW2C(S2)'

Writing the circulation (C-13) in its final form:

(C-14)

(C-15)

(C-16)

(C-lg)

(C-19)

A~p~pe~n~drr~C ________________________________________________ I05

r(y) = ~l-l /30 + ~W; [.JI=-! C(Sj)+(y-Sj)G(Spy)]

Contributi~ 5Wo Contribution "related to orY;

+8W2 [ ~l- y2 C( S2) +( -y- S2)G(S2'-y)], (C-20) ~ /

Contribution relaÛd ta oW2

where the function G (s, y) is defined in (B-4).

C.2 The complete problem

Consider equation (3-22). For convenience it is rewritten here.

r j (spy)=W; (SI)[ ~l-l C(sl)+(y-sl)G(spy)]

+ (dW;/da)da[ ~l-l C(a)+(y-a)G(a,y)]. CC-21)

We use the derivatives (B-5) and (B-IO) of the functions G(s,y) and C(s)in the process

ofintegration (by parts), thus:

f (51' y) = W; (51) C ( 51 ) ~1-l - W; (SI) (5] - y) G (5] , y)

= W; (5] ) C ( SI ) ~l-l - W; (SI) (SI - y) G ( s], y)

+[ ~l-l ~ (a)c(a)l, - f ~ (a) d~[ ~l- y2 C(a)Jda

-[~ (a)(a-y)G(a,y)l + (W; (a) :a[ (a- Y)G(a,y)Jda

= ~ ( SI) C (SI) ~l-l -~ (SI) (s] - y) G (s], y) - W; (S1 ) C ( 51 ) ~l-l

2 J.I ~1_y2 -W;(l)(l-y)G(l,y)+W;(Sj)(sl-y)G(sPY)+- W;(a) ~da Jr SI 2 l-a

fi 2 fi .JI=-! + W;(a)G(a,y)da-- W;(a) ~da SI 1t: SI 21-a

=-~ (1)(1- y)G(l,y)+ f W; (a)G(a,y)da.

A~p~pe~nd~ù~C~ _________________________________________ 106

But -TV; (1) (1- y) G (1, y) doesn't have a real part. Therefore,

r\ (Sl'Y) = f TV; (a )o( (J,Y )da. 1

Similarly integrating r 2 (S2' y) we arrive at:

r 2 (S2'Y) = 1: W2 (-(J )G( (J, -y)d(J .

And finally r 0 (y) can be calculated from:

of course by using the velocity variation Wo (y).

Now introducing general polynomial functions for the variation of the velocities,

(C-22)

(C-23)

(C-24)

2n il il

W()(y) = L Pk yk , W] (y) = L fi {yk, W2 (y) = l P -; yk , (C-26) k=O k=O k=O

and the general solution is the sum of the three contributions r 0 (y), r] (s], y), and

r2 (S2'Y) as:

r(y) = ro (y)+ f\ (Sj,Y)+ r2 (S2'Y) (C-27)

= J: Wo (a)G( 0-, Y )d(J + J: Wo (-(J )o( o-,-y )do-...

+ f-w;(o-)G(a,y)do-+ (w2 (-a)G(a,-y)do-

= t,p; {f:u'G(o-,Y)da}+ t,p; u: (-a)' G(o-,-Y)da}.

+ t,Pi{ J~ a'G(",Y)da} + t,Pi{ {(_,,)k G(a,-Y)da} (C-28)

But the integrals appearing in (C-28) are solved in Section B.2.3 and are given here for

convenience,

A~p~pe~n=d~~' ~C ________________________________________________ 107

~ fI k yk+l_i+l ~I- y2 k . Ek(S,y)= u G(u,y)du= G(s,y) + ( )LyJJk_J(s),

s k + 1 2 k + 1 j~O

Substituting appropriately in (C-28):

where Go (y) and Ik are defined in (B-7) and (B-18),and the coefficients are:

{P; for y>O

P= p~ for y<O

(C-29)

(C-30)

(C-3I)

The supers cri pts rand 1 are used to denote the coefficients for the left side and the right

side of the wing respectively. It can be noted that in the case of left-right-symmetry of

the wing geometry , P; = ( -1 t p~, and for the case of antisymmetry we have

P; = ( -1 t+1 p~. Thus a convenient notation is employed as follows.

P; = t (p; + ( _1)k p~),

P: =t(p; -(-lt pi).

(C-32)

(C-33)

Where the superscripts S and A are used to denote the case of symmetry and antisymmetry

respectively. Or altematively:

P; =P; +P:,

( -1 t p~ = P; -P: . (C-34)

(C-35)

A~p~pe~n~m~x~C __________________________________________________ I08

The coefficients p~ and pi represent the unknown coefficients for the two sudden

changes placed at y == s] and y = S2 •

Introducing these definitions, the circulation distribution becomes:

FI {( )k+1 k+l Pk} + 2:)-lt pi -y -S2 G(S2'Y) + (-y )L(-YYJk - j (S2) .

k~O k + l 2 k + l j~O (C-36)

Defining functions F; (y) and ~ (s, y) , it can be written down as follows:

~ FI

f(y)== 21,B: F,,0(y)+,B: F"l(y)J+ L[p~~(spy)+(-lt p; ~(S2'-Y)J, (C-37) ~o ~o

e k == [l + (_l)k ] /2. (C-40)

Again where G(s,y), Go (y) and Ikiyare given in (B-4), (B-7) and (B-18) respectively.

C.3 Determination of the coefficients,Bk

Applying the improved solution procedure equation (3-38) is given as:

(C-41)

This can also be recast in the fonn:

A~p~p~en~d~~~C __________________________________________________ I09

f: f (y) i dy = L f L (y) i dy . (C-42)

Considering the left-hand-side integral:

f ['(y) y' dy ~ f y' {tcP;, Ft (y)+ P: 1';' (y)J

+ t[p; ft; (s"y)+( -1)' if: ft; (s,,-y) ] }dY,

+ Î,8i !iFk(SPy)dY + Î(-lt,8; !iFk(S2,-y)dy. (C-43) k~O k~O

Now each of the integrals in (C-43) will be calculated.

C.3.1 Integrais related to the improved solution procedure

Let:

(C-44)

where F; (y) is given by (C-38) and is repeated here for convenience.

2 k+1 )l-l k .

F;(Y)=fk+r+l-y-GO(Y) + ( ) ~ej+ryJlk_j" k+l k+l f:t

Thus,

Qr = 2f k+r+1 rlyk+i+IG (y)dy+_l_~ e. 1 . tyj+i~1_y2dy. I,k k+l JI 0 k+l.L.t J+r k-j J(

o ;"=0 0

A~pxp~en~d~a~C~ ________________________________________________ 110

But the two integrals are Rk and 11c, which are defined in (B-34) and (B-38), and the

solutions are given in(B-37) and (B-4l), respectively. Hence:

k

Qr = e k+r+J lk+i+J + Jr '"" e 1 P i,k k+l k+i+2 2(k+l)i:o- i+ r k-i i+i'

(CA5)

)0> Integral Q;,k

Let:

Q:,k = i i~ (spy)dy, (C-46)

where Pk (spY) is defined in (C-39) and is repeated here for convenience.

(C-47)

Thus,

Q:,k = ( l ) { fI yk+i+IG(Spy )dy- SJk+1 fI iG(sl'Y )dy+~ t Jk - i (SI) [1 yi+i ~l- y2 dY}. k + 1 Jo Jo 2 i~O Jo

But the integrals appearing in the above equatlon are Ek (SI) and 11c, which are defined in

(B-25) and (B-38), and the solutions are given in(B-28) and (B-41), respectively. Hence:

QAI _ SI G (s ) + "1 - SI '"" Sk+i+I-j] l { k+i+2 ~ k+i+l

i,k - (k + 1) k + i + 2 0 1 2 (k + i + 2) i:o- 1 i

= 1 { ( -k -1) Sk+i+2G (s ) (k+l) (i+l){k+i+2) 1 0 1 , ••

R[~ s;+i+l-ilj ~S;+i+l-jlj] Jr~ J ( )p } + 2 i:o- ( k + i + 2) - f:t (i + 1) + 4" f:t k- j SI j+i

A~p~p~~~d~~~C~ ________________________________________________ 111

Therefore,

+ " 1- SI k- j 1 _ __ l k+i+I-} ~[1 kil ] 2(k+i+2) (k+l) ~SI j+i+1 ~(i+l) jS]

(C-48)

Let:

(C-49)

where Fk (S2' -y) is defined in (C-39) and is repeated here for convenience.

( )k+1 k+l . ~ k

A -y -S2 \jl-y ~ j

Fk(S2'-Y)= G(S2'-Y)+ ( )L)-Y) Jk_j (S2)' k+ 1 2 k+ 1 j~O

(C-50)

Thus,

But the integrals appearing in the above equation are Êk (S2) and ~, whieh are defined

in (B-29) and (B-38), and the solutions are given in(B-33) and (B-41), respectively.

Henee:

A=p=p=en=~=x~C ________________________________________________ l12

Q2 = l (-lt+1 -S2 G (s)+ ...;1-s2 ~(-s t+ i

+1- i l

{ [

( )k+i+2 ~ k+i+l ]

I,k (k + 1) k + i + 2 0 2 2 (k + i + 2) it 2 J

1 { ( -k -1) ()i+1 k+i+2 ( ) TC Ik

()J () =( ) (. )( . ) -1 S2 Go S2 +- -1 J k- i S2 ~+i'" k + 1 1 + 1 k + 1 + 2 4 }=o

Therefore,

+ ...; 1- S2 1 -1 i+l l Sk-J _ -1 1 Sk+i+l-i ~ [k i ( )i+i ]

2(k+i+2) (k+1) ~( ) i+i+l 2 ~ (i+l) j 2 • (C-51)

Now consider the right-hand si de integral of (C-42),

J:ii\(Y)dY=tK ca 11ic(Y) [a(y)-w(y)Jdy, (C-52)

where the sectional circulation distribution is given by equation (3-36).

It can be seen from (C-52) that the result depends on the particular geometry of the wing.

Thus in the following sections the system of linear equations (to solve for the unknown

coefficients Pk in the circulation distribution function ) for various wing configurations

will be derived.

A~PLpe~n=Œ~x;C __________________________________________________ 113

C.3.2 Solution for symmetric wings with continuous incidence variation

For this case the general symmetric variation of the wing chord and incidence are

given in the form:

C(y} = ICm \ylm , ni

a(y}= La} Iyl} , (C-53) m=û }=o

And for this case, fJ/ = 0 and /3/ = fJ k , thus the solution of the circulation distribution

(3-28) reduces to:

2n

r(Y)=L/3k FkO(y) k=O

The left-hand-side of the integral equation (C-42) is:

2n

= LfJkQ~k' k=O

where the integral Q\ is defined in (C-44) the solution given in (C-45) for r = 0 .

i~l1d solving the right hand side of the integral equation (C-42) we have:

Lyi rL (y) dy = ±K Ca Li c(y) [ a(y)-±w' (y)Jdy

= t K c. J>'L.o Cm Iyl" [ t, il, IYI' -t ~>V ]} dy

(C-54)

(C-55)

=tK c·{~tCmil, S>j''''dy -t ~tCmfl, J: ;/'""dy }

=1.K C {~~ c,ixr _1.~.ç., cm/3k }. 2 a fot~i+m+r+l 2ft-ft i +m + k +1

(C-56)

A~pxp~en~d~Œ~C~ ____________________________________________________ 114

Using (C-55) and (C-56) in (C-42) we have:

f f(y) i dy = f f L (y) i dy,

Now defining:

nl1J _

E _1 K L cm ik -"2 Ca. , , m=O 1 + m + k + 1

Bk = 2QOk + E k , " l, 1 ~

(C-57)

(C-58)

(C-59)

we can finally write the system of linear equations with unknown coefficients fik as:

{ Bi,k } [fi k ] = [ Di] . (C-60)

C.3.3 Solution for symmetric wings with incidence and dwrd changes

Consider a symmetric wing with sudden changes of incidence and/or geornetry

occurring at y = s and y = -s defined by:

nm

c(y) = L [cm +H(s,y) c; + H(s,y) c; ] Iy\m

r t [Cm + C; ] Iyl' for yE {( -1,-s )v(s,l))

~1~>" Iyl" for yE {(-s, O)v (0, s)}

(C-61)

A~PLP~m=d=~~C~ __________________________________________________ 115

n,

a(y)= L [ar+H(s,y)a; +H(s,y)a;] lylr r~O

t [ar+a; ] Iylr for YE{(-l,-s)v(s,l)} = r~O (C-62) t arlylr for y E {( -s,O)v (O,s)}

r~O

Where H(s,y) and H(s,y) are defined in (3-7) and (3-8), respectively. In this case, the

general solution (3-28) of circulation is simplified by considering fi/ = 0, fi/ = fJ k and

~ 1 ( )k ~ 2 ~ • fik = -1 fik = fik ,resultmg thus

2n n r(y) = Lfik FkO(y) + LPk [Fk(s,y)+ l\(s,-y)], (C-63)

k~O k~O

where aB functions are defined in (3-29,30).

The left-hand-side of the integral equation (C-42) becomes:

f f(y) y' dy = f / [ t/v;(y) + tpk [ Pk (s,y)+ Pk (s,-y)] ]dy

= t,p, f/F~O(y)dy+ tA [f y't, (s,y )dy+ f /p,(s,-y )dY ] ,

1 ~ n 1 f(y) i dy = LfikQ~k + LPk (Q:,k +Q:~) , k=O k=O

(C-64)

where the integrals Q~k' QI,k and Qi\ are defined in (C-44,46 & 49) and their solutions

given in (C-45,48 & 51).

And solving the right hand side of the integral equation (C-42) we have:

11

/ Ï\ (y) dy =t K Ca Li c(y) [ a(y)-t:w* (y)Jdy

= t K Ca {J: / c(y) [ a(y)-tw' (y)Jdy

+ f/c(yJ[ a(y)-tw'(Y)]dy}

A~p~pe~n=d~~' ~C ________________________________________________ 116

Integrating and eliminating simplifying we get:

(C-65)

Now introducing the notations:

(C-66)

(C-67)

(C-68)

The definitions of Ei,/c, and Di are given in equations (C-57,58).

(C-69)

A~p~p~en~d=~~C~ ________________________ ~ ____________________________ 117

Now, we have the results ofboth the left-hand-side and the right-hand side integrals given

by (C-64) and (C-66) respectively. Equating and rearranging them we get:

(C-70)

Defining:

* ..... 1 ..... 2 * Bj,k = 2Q,k + 2Q,k + Ej,k , (C-71)

and using the definition of Bj,k given in (C-59), the system oflinear equations becomes:

(C-72)

C.3.4 Solution for wings with antisymmetric incidence variation

Consider a wing with the symmetric chord variation (C-61) and an anti-symmetric

variation of incidence defined by

n,

a(y)= L Sgn(Y) [ ar+H(s,y)a; +H(s,y)a;] Iylr r=O

nr L Sgn(y) [ ar+a;] Iyl' for YE{(-l,-s)v(s,l)} = nr

(C-73)

L Sgn(y)arlyl' for YE{(-S,O)v(O,s)} r=O

where H(s,y) and H(s,y) are defined in (3-7,8), and Sgn(Y) representsthesignof y:

for y> 0

for y < 0 (C-74)

In this case, the solution (3-28) of the circulation is simplified by considering

AI ( )k+l" 2 A AS, Pk = -1 Pk = Pk Pk = Pk and Pk = 0, resultmg thus

2n fi

f(y) = LPk Fk1(y)+ LPk [ Fk (S,Y)-Fk (s,-y)], (C-75) k=O k=O

where aH functions are defined in (3-29,30).

A~pxp~en=d~~~C~ __________________________________________________ 118

The left-hand-side of the integral equation (C-42) becomes:

f f(y) y' ~ ~ f y' [ t./i,F,'(y)+ tA [ F, (s,y)- F, (s,-y) J]~

~ I;p, f y'Fi(y)dy+ t,ô, [f iF, (s,y )dy-f y' F, (s,- y)dY] .

Finally,

1 2n fi l f(y) i dy = Lf3kQ;,k + L/Jk (ii,k -Q,\) . k=O k=O

(C-76)

Where the integrals Q;,k' Q;,k and Q;'k are defined in (C-44,46 & 49) and their solutions

given in (C-45,48 & 51).

And observe that the right hand si de of the integral equation (C-42) is exactly the same as

the result obtained for the case of symmetrical incidence changes. Thus repeating this

integral:

(C-77)

Equating (C-76) and (C-77) and rearranging results in:

where,

- 1 Bk =2Qk +Ek 1, 1. I,

l Al "'2 '-' * >1< ~ 1"'"

And Q,k' Q,k' Q,k' Ei,k' Di' Bi,k' Bi,k' Ei,k and Di are defined ln (C-45,48,J 1,57,58,59,

66,67 & 68), respectively.

A~pzp~en~d=ù~C~ __________________________________________________ 119

Appendix ~ Solution to the swept wing problem

D.I Integrais related to the swept wing problem

Let

2 il l f p k (y) = - ~ ds for p, k E {O,1,2, .... } . , 7r 0 (s + y y 1- S2

Using

k-l

Sk = {- y y + (s + y) I (- y Y-l-) sj '.

j:O

k-l

fp,k (y) = (- y y fp,o (y)+ I (- y Y-J-l fp-I,) (y).

Forp=k=O,

2 il ds 10,0 (y) = - r:-z 7r O,\/l-s-

2 -I( )1 1 = -- cos S

7r 0

=1.

For p= 1, k=O,

2 fi ds Ao(Y) = - ( ) r:-z

7r 0 S + y ,\/1- s-

(D-l)

(D-2)

(D-3)

(D-4)

(D-5)

A~p~pe~nd~ù~D~ __________________________________________ 120

Now integrating for any p> 1, k=O, by parts,

f () _2! 1 ds p,O y -- ( ) ~ , TC s+yP"l-s-

/, 0 (y) = - 2 [ 1 - fI S ds J . p, TC(p-l) (s+ yy-l ~1-s2 0 (s+ yy-J (~)3

Again integrating by parts,

From equation(D-3), we have:

fp,l (y) = -y fp,o (y) + fp-l,O (y),

and,

f p,2 (y) = Y 2 fp,o (y) - Y fp-I,o (y) + f p-l,1 (y) again substituting f p-l,1 (y) we get

=> f p,2 (y) = Y 2 fp,o (y) - 2 Y fp-l,O (y) + f p-2,O (y )

Inserting these equations in equation (D-6) and simplifying gives:

(D-6)

(D-7)

(D-8)

Jp,,(l' ) ~ (p _ al _ y' ) [" :P~I - (2 p - 3 )y Jp~I,O (y) + (p - 2 )Jp~"o (y)] for p ~ 2,3,4, .. "

(D-9)

And,

CD-lO)

In surnrnary:

A~p~pe~n=dŒ~D~ _____________________________________________ 121

(D-ll)

where,

/p,o{y) =

for p ?:. 2

(D-I2)

).> Integral r; k (y) p,

Let

(D-13)

U sing the identity:

k-l

Sk = {- Y Y + {s + y)2: (- y Y-I-i si , (D-14) i~O

k-l

r; p,k (y) = (- y y r; p,o (y) + 2: (- y Y- j-I r; p-I,i (y) , (D-15) j~O

Forp=k=O,

r;o,O (y) = lao (s )ds , (D-16)

2 il h-1 Nf+ S d 2 [ h-I Nf+ sIl Sa ds ] =- cos - s=- scos - +-TC 0 23 TC 2s 0 2 0..J1-S2

2[lf ds" = TC 2 Jo)1-7 J

1 Il 1 = --cos-I (3) =-7r 0 2

(D-17)

A=p=pe=n=d~~D~ ____________________________________________ 122

For p=l,k=O,

r Go(s) ç] 0 (y) = ( ) ds , s+y (D-18)

Here we use the Approximation:

cosh- I )1 ;ss ~ ln(s )[ao +als +a2s2 J, (D-19)

where the constants ao' al and a2 can be calculated by setting s at three positions in the

above equation. Thus integrating results:

ç] 0 (y) = (ao +als+ a2s2 )[ln(y )ln(l + YJ+ ln( -1)ln(1 + YJ - Li2 (1 + YJ +!C] , y y y 6, (D-20)

where,

L · ( ) - IX In(1- y) d 12 X - - y, o y

(D-21)

is a dilogarithm integral [36]. Standard numerical procedures can be found in Ref. [36] to

solve this integral.

Forp>l,

() r Go(s) ç p,O Y = ( )p ds

o s+ y

1

= -1 Go ( s ) _ 1 rI 1 ds

(p-1) (S+yy-I 0 ïr(p-l) Jos(s+Yr-I~1-s2

o

1 [rI ds - ïr(p-l)yP-1 Jo s.Jl-i

p- p-l-r S d p-l (lJ 1 r-l ]

-~ r y f. (H y)'-l.jj:::; S

noting that

Go(l)=O,-l f p=Go(s) ,and JI ::;l.jj:::;ds~" fp-I,~I(Y) wehave 1C s 1-s2 o(s+y) 1-s2 2

A~p~p~~~du~'~D __________________________________________________ 123

(D-22)

Finally,

ShI Il! k

=-Go(s) + r s ds k+l 0 7r(k+l)Jo~

(D-23)

In summary,

Ik 2(k + 1)

ç p,k (y) =

for p = 0

k-I

(-yYçp,o(Y)+ 'L)-yY-J-Içp_l,j(Y) for p21

(D-24)

j=O

where:

(a, +a,s+a2s2 l[ln(Y)lnC :Y)+ ffi( -l)ln( 1 :Y)-Li2 C :Y )+ :']--( al +ta2)+a2y for p = l

(D-25)

Appendix D ====~ ____ ~ ___________________________________ 124

D.2 The swept wing problem

The elementary velocities induced by a vortex element at a point y along the span are

given by (4-10, 17, 18 & 20). Thus:

- () 1 (1 . ) fi gr 2. i gr wlj Y =-- +smx --+-smx --, 21C 0 S - y 21C s=1 S - y

(D-26)

(D-27)

- () 1 i l l [ ( S - y) sin X ]-W 2j Y =-- 1- gr

21C 0 (s+ y) )i +S2 +2sycos2X ' (D-28)

- () Iii 1 [S+YCOS2X 2 ]~r-W y = --- -cos X U

2b 21C 0 2ysinX ~i +S2 +2sycos2X ' (D-29)

where,

n

f(y) = Lf3k FkO(y), (D-30) k=O

and,

(D-3I)

And functions Go (y), Ik and R. k are defined in (B-7, 18) and (4-5). Making the

derivative of the circulation distribution we get:

(D-32)

The total velocity is given by:

(D-33)

A~p~p~en~d~~~D~ ________________________________________________ 125

Substituting equations (D-26) - (D-29) in (D-33) results in:

- () 1 (1 . ) fI <5 rI. f <5 r w y =-- +smx --+-smx --2Jr 0 S - y Jr s~l S - y

__ 1 rI 1 [1- (s-y)sinx ]<5r 2Jr Jo (s+ y) ~l +S2 +2sycos2X

__ 1_ r 1 [ s+ ycos2X Cos2x]<5r 2Jr Jo 2ysinx ~l +S2 +2sycos2X

___ 1_ il{(l+sin x ) l _ 4sysin2 X - + l 2 ••• 2Jr 0 S - y 2 Y sin X ( s + y)

1 sin Xl} s: ï- 1. i <5 r ---+--+ u +-smx --2 Y sin X y ( s + y) Jr s~1 S - y

U sing the approximation:

4sysin2 X l sy. 2 l 2 ~ -e 2 sm X,

(s+y) (s+y)

_() l il{(l+sin x ) es. w y =-- -- smx 2Jr 0 s - y 2 (s + y)2

+ sinX + l }<5r +!sinX r <5r' y ( s + y) Jr JS~I S - y

1 il {(l . )[ l 1] es. =-- +smx --+-- -- 2 smx 2Jr 0 s - y s + y 2 (s + y)

sin X s } s: ï- 1. l <5 r +-- u +-smx --y ( s + y) Jr s~l S - y

lntroduce the notations,

R(Y)= f br, Js=l s- y

(D-34)

(D-35)

(D-36)

A~p~p~en=Œ~x~D __________________________________________________ 126

2 fI Sk fp,k(Y)=- ( )P ~ ds,

1( 0 S+Y -vl-s-(D-37)

(D-38)

but these integrals are solved in Section D-l and substituting (D-32) in equation (D-35),

results in:

n

w(y) = I.8kFk(Y)' (D-39) hO

where,

{ (1 + sin X) k

Fk(Y) = 2 Y-

- s~ X f .. ,[~ (q"",{y)- 1., .. , {Y)j2(k + 1)}-~{q2,h' (y)- J2, .. , (y)j2(k + 1)}}

-4(~n:l) t f J.-t VI.Jy)-(j + 1)1.,,., {y)HV J2,J{y)-(j + I)J2,J+2 (y)}] +

2sinx k ~ ( ) } + JT R k+1Y R Y , (D-40)

(D-41)

A~p~pe~nd=u~D~ ____________________________________________ 127

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