Wing Lets

Cranfield University

Guillaume MARTIN

COMPARISON OF AERODYNAMIC PERFORMANCE OF

RAKED WING TIPS AND LARGE WINGLETS

School of engineering

MSc thesis

Cranfield University

School of engineering

MSc thesis

Academic Year 2005-2006

G. MARTIN

COMPARISON OF AERODYNAMIC PERFORMANCE OF

RAKED WING TIPS AND LARGE WINGLETS

Supervisor: Dr S.T. Shaw

September 2006

This thesis is submitted in partial fulfilment of the requirements for the degree of Master of Science

© Cranfield University 2006. All rights reserved. No part of this publication may be reproduced without the written permission of the copyright owner.

Abstract

A Computational Fluid Dynamics (CFD) study of the aerodynamic performance of a wing mounted either with a raked tip or a winglet has been carried out. A wing was designed to operate at a freestream Mach number M ≈0.8. The winglet design was achieved according to the guidance from Whitcomb (reference 5) and the raked tip was designed using the shape of the 2D aerofoil section of the wing.

The study was carried out using the Fluent inviscid solver with a structured mesh. A validation hierarchy enabled to attest the ability of this model to compute problems involving a compressible flow over a three dimensional lifting device.

The comparison of the performance was achieved by studying the wings at three different Mach numbers (M=0.8, M=0.75 and M=0.5). A comparison of the pressure distribution over the various wings designed has shown interferences effects at the  junction between the wing and the winglet. A high suction peak was also observed at the leading edge of the raked tip.

The comparison of the integrated data has shown a very high increase in performance due to the addition of the raked tip in all configurations. The efficiency of the winglet seems to be highly dependant on the lift produced by the wing. The winglet performances seem to be underestimated due to a bad computation of the tip vortex.

A far field flow study exhibited the presence of a second vortex that forms at the  junction between the wing and the raked tip. This might be related to the sudden change in sweep angles at this station.

Results obtained enabled to have a better understanding of the way both wing tips are working and to exhibit the influence they have on the performances of a wing.

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Acknowledgements

First of all, I would like to thank my supervisor, Dr S.T. Shaw, for his guidance and support during all the study.

I also would like to thank my parents, Florence and Olivier, who gave me the chance to carry out a master in aerodynamics. I want to thank Camille, Cécile, Virginie, Eskander, Pierre, Emma, Marie-Livia and Anne for their moral support all along this year.

Finally, I thank all of my friends in Cranfield who made of this year an unforgettable one and who enabled to make the nights spent in the computer room facilities more enjoyable: Ahmed, Antoine, Benoit, Brice, Naomi, Patrick, Vincent and Humann.

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Contents Abstract ................... ............................. .................... ................... ................... .................... ................... ................... ................... ................. ........ i Acknowledgements ................... ............................. ................... ................... .................... ................... ................... .................. ........ii ii Contents.......... Contents ................... ................... .................... ................... ................... .................... ................... ................... ................... ...............iii ......iii Figures .................... ............................. ................... .................... ................... ................... .................... ................... ................... ................. ....... vi Tables......... Tables ................... ................... ................... ................... ................... .................... ................... ................... .................... ................... ........... ix Notations......... Notations .................. ................... .................... ................... ................... .................... ................... ................... ................... ................ ....... x Introduction ................... ............................ ................... .................... ................... ................... .................... ................... ................... .......... 1 Chapter 1: Literature review .................... ............................. ................... .................... ................... ................... .......... 2 1.1. Aircraft performances ............................................. .................................................................... .......................................... ................... 2 1.2. Wing tip vortices ........................................... .................................................................. .............................................. .............................. ....... 3 1.3. Induced drag .............................................. ..................................................................... .............................................. .................................. ........... 4 1.4. Wing tip devices...................... devices ............................................. .............................................. .............................................. .............................. ....... 6 1.4.1. Winglets....................... Winglets .............................................. .............................................. .............................................. .................................. ........... 6 1.4.1.1. Winglet effects............................................................ effects................................................................................... .......................... ... 6 1.4.1.2. Winglet design design ............................................ ................................................................... .......................................... ................... 8 1.4.2. Raked wing tips.......................................... tips................................................................. .............................................. .......................... ... 9 1.5. Computational fluid dynamics simulation..................................................... simulation..................................................... 11 1.5.1. Grid generation......................................................... generation................................................................................ ................................ ......... 11 1.5.2. Euler equations ............................................. .................................................................... ............................................ ..................... 12 1.6. Aim of this research ............................................. .................................................................... ............................................ ..................... 12

Chapter 2: 2 : Wing design ................... ............................. ................... ................... .................... ................... ................. ........ 14 2.1. Supercritical aerofoil design................................................ design....................................................................... ............................ ..... 14 2.1.1. Design procedure .............................................. ..................................................................... ........................................ ................. 14

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2.1.2. Performances of the aerofoil designed............................................... designed.................................................... ..... 16 2.2. Swept wing design ............................................. .................................................................... .............................................. ......................... 20 2.2.1. Design of the outer wing section......................................................... section.............................................................. ..... 21 2.2.2. Design of the root section (y=0, η=0) ............................................. ...................................................... ......... 22 2.2.3. Design of the intermediate section (y=4.5, η=0.1875)............................ 24 2.2.4. Spanwise loading and twist.......................................... twist................................................................. ............................ ..... 25 2.2.5. Conclusions Conclusions about the method method ........................................... ................................................................ ..................... 27 2.3. Design of the wing tips ............................................. .................................................................... ........................................ ................. 28 2.3.1. Design of the raked tip ............................................. .................................................................... ................................ ......... 28 2.3.2. Design of the winglet ............................................ ................................................................... .................................... ............. 28

Chapter 3: Validation ................... ............................. ................... ................... ................... ................... .................... ............ 30 3.1. The choice of Euler equations ............................................. .................................................................... ............................ ..... 30 3.2. Validation hierarchy ............................................ .................................................................... ............................................ .................... 31 3.3. Richardson extrapolation ............................................ ................................................................... .................................... ............. 32 3.4. Compression and expansion corner .............................................. ............................................................... ................. 34 3.5. 2D supercritical aerofoil .............................................. ..................................................................... .................................... ............. 36 3.6. Onera M6 wing....................... wing .............................................. .............................................. .............................................. ............................ ..... 38 3.6.1. Presentation............................................................... Presentation...................................................................................... ................................ ......... 38 3.6.2. Computations............................................................ Computations................................................................................... ................................ ......... 39 3.7. Conclusion about validation ............................................ ................................................................... ................................ ......... 42

Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet.................. winglet........................... ................... .................... ................... ................... ................... ................... ............. ... 44 4.1. Grid generation ............................................. .................................................................... .............................................. ............................ ..... 44 4.2. Near field flow study ............................................ ................................................................... ............................................ ..................... 46 4.2.1. Study of the wings......................................................... wings................................................................................ ............................ ..... 47 4.2.1.1. Root Root section ............................................ ................................................................... ............................................ ..................... 47 4.2.1.2. Outer sections ............................................. .................................................................... ........................................ ................. 49 4.2.2. Pressure distribution over the winglet....................................... winglet.................................................... ............. 50 4.2.3. Pressure Pressure distribution over the raked tip ........................................... ................................................ ..... 52 4.3. Far field flow study .............................................. ..................................................................... ............................................ ..................... 53 4.3.1. 4.3.2. 4.3.3. 4.3.4.

Vorticity downstream of the wing clean ............................................ ................................................. ..... 53 Vorticity downstream of the winglet........................... winglet.................................................. ............................ ..... 54 Vorticity downstream of the raked tip .............................................. ................................................... ..... 55 Conclusion about the far field flow study ............................................. ............................................... 56

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4.4. Comparison of aerodynamic performance ........................................... .................................................... ......... 57 4.4.1. Lift characteristic ............................................. .................................................................... ........................................ ................. 57 4.4.2. Drag characteristic ........................................... .................................................................. ........................................ ................. 58 4.4.3. Wing efficiency.......................................................... efficiency................................................................................. ................................ ......... 59 4.4.4. Loading distribution over the wings .............................................. ....................................................... ......... 60

Conclusions and recommendations ................... ............................. .................... ................... ................. ........ 63 References ................... ............................ ................... ................... ................... .................... ................... ................... .................... ............ 65 Appendices ................... ............................. ................... ................... .................... ................... ................... .................... ................... ......... 67 Appendix A:

SectionD program ............................................ ................................................................... ............................ ..... 67

Appendix B:

VGK ........................................... .................................................................. .............................................. ............................ ..... 69

Appendix C:

Convert program ............................................. .................................................................... ............................ ..... 71

Appendix D:

Sweptdes............................................. Sweptdes..................... ............................................... ........................................... .................... 73

Appendix E:

Downwash program...................... program ............................................. .............................................. ......................... 75

Appendix F:

Figures........................................................... Figures.................................... .............................................. ................................ ......... 76

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Figures

Figure 1: Drag breakdown for a typical transport aircraft. The numbers presented are just estimation because this is highly dependant on the configuration of the aircraft (cruise, high lift configuration…) .................................3 Figure 2: Formation of trailing vortices on a finite wing ............................................4 Figure 3: Drag component of lift resulting from downwash (w= downwash; V = forward speed of wing; VR = resultant oncoming flow at wing; α = incidence; ε = downwash angle = w/V; α = (α-ε) = equivalent twodimensional incidence; L = two-dimensional lift; L = wing lift; D v =trailing vortex drag) .....................................................................................................................5 ∞

∞

Figure 4: Winglets to reduce lift-induced drag ............................................................7 Figure 5: Geometric quantities used to define a winglet.............................................8 Figure 6: Block strategy to produce the grid for a wing/body/winglet configuration on the left hand side and for a configuration with multiple winglets on the right hand side (rear view) ................................................................11 Figure 7: Influence of incidence on pressure distribution over the aerofoil at M=0.725 .........................................................................................................................16 Figure 8: Variation of lift coefficient against incidence for various Mach numbers .........................................................................................................................17 Figure 9: Variation of L/D against alpha for various Mach numbers .....................18 Figure 10: Influence of Mach number on the pressure distribution over the aerofoil at α=1.3°...........................................................................................................19 Figure 11: Planform of Cranfield wing 1: distances are in meter and the sections designed are in red..........................................................................................20 Figure 12: Pressure distribution obtained with Sweptdes on the outer wing section (y=16.8), effect of the power factor on this distribution ...............................21 Figure 13: Comparison of the distribution of pressure due to thickness at the root and at the outer wing section ...............................................................................22 Figure 14: Comparison of the thickness at the root and at the outer wing section.............................................................................................................................23 Figure 15: Maximum thickness position over the wing designed and maximum thickness sweep angles................................................................................23 Figure 16: Root section and pressure characteristic .................................................24 Figure 17: Intermediate section characteristics .........................................................25

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Figure 18: Comparison of the loading over the wing with an elliptic loading ........26 Figure 19: Geometric characteristics of the winglet..................................................29 Figure 20: Validation hierarchy ..................................................................................31 Figure 21: Comparison of the pressure distribution computed with VGK and with Fluent on the aerofoil designed with an incidence α=1.5..................................37 Figure 22: ONERA M6 wing geometry with the positions of the pressure taps.....39 Figure 23: Structured grid on the ONERA M6 wing with the boundary conditions .......................................................................................................................40 Figure 24: Pressure distribution computed with Fluent over the upper surface of the ONERA M6 wing for M=0.8395, α=3.06 and Re=11.72e6................41 Figure 25: Structured mesh generated over the clean wing .....................................46 Figure 26: Pressure distribution over the root section (y=0m) at M=0.8 with zero incidence. The graph displays this pressure distribution for the clear wing, the wing with the winglet and the wing with the raked tip.............................48 Figure 27: Pressure distribution over the section y=14.8m (the tip of the wing being at y=15m) at M=0.8 with zero incidence. The graph displays this pressure distribution for the wing clean, the wing with the winglet and the wing with the raked tip.................................................................................................50 Figure 28: Pressure distribution over the wing mounted with the winglet at various stations in the tip region. η refers to the dimensionless spanwise ordinate: η=y/b (η=0.949 corresponds to the junction between the wing and the winglet) ....................................................................................................................51 Figure 29: Pressure distribution over the wing mounted with the raked tip at various stations in the tip region. η refers to the dimensionless spanwise ordinate: η=y/b (η=0.893 corresponds to the junction between the wing and the raked tip) .................................................................................................................52 Figure 30: Vorticity magnitude downstream from the wing for a freestream Mach number, M=0.75 and an incidence α=1°. The first measurement plane was situated at x=10.73m and the second one is at x=15.13m...................................54 Figure 31: Vorticity magnitude downstream from the wing mounted with a winglet for a freestream Mach number, M=0.75 and an incidence α=1°. The first measurement plane was situated at x=11.36m and the second one is at x=15.76m ........................................................................................................................55 Figure 32: Vorticity magnitude downstream from the wing mounted with a raked tip for a freestream Mach number, M=0.75 and an incidence, α=1°. The first measurement plane was situated at x=11.47m and the second one is at x=15.86m....................................................................................................................56 Figure 33: Loading distribution over the wings and comparison with an ideal elliptic loading ...............................................................................................................61

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Figure 34: Pressure drag distribution over the wings ...............................................62 Figure F 1: Pressure distribution computed with VGK along the aerofoil designed at M=0.725 and α=0 ......................................................................................76 Figure F 2: Comparison of experimental and CFD data on pressure distribution over different spanwise sections .............................................................77 Figure F 3 : Comparison of experimental and CFD data on pressure distribution over different spanwise sections .............................................................78 Figure F 4: Structured mesh generated over the wing with the raked tip ..............79 Figure F 5: Structured mesh generated over the wing with the winglet..................80 Figure F 6: Pressure distribution over the section y=14m (the tip of the wing being at y=15m) at M=0.8 with zero incidence. The graph displays this pressure distribution for the wing clean, the wing with the winglet and the wing with the raked tip.................................................................................................81 Figure F 7: Pressure distribution over the section y=14.8m (the tip of the wing being at y=15m) at M=0.5 with zero incidence. The graph displays this pressure distribution for the wing clean, the wing with the winglet and the wing with the raked tip.................................................................................................81 Figure F 8: Pressure distribution over the wing mounted with the raked tip at various stations in the tip region. η refers to the dimensionless spanwise ordinate: η=y/b (η=0.893 corresponds to the junction between the wing and the raked tip) .................................................................................................................82 Figure F 9: Comparison of the lift characteristics of the various wings at different free stream Mach number............................................................................83 Figure F 10: Comparison of the drag characteristics of the various wings at different free stream Mach number. A trendline was added to have an idea of  the drag produced by the wing at each station and to compare it with the drag produced by the wing mounted with a tip device. ............................................84 Figure F 11: Plots presenting the variation of CD*Π*AR against CL². The slopes of the curves plotted corresponds to the parameter K appearing in the expression of the drag ...................................................................................................85 Figure F 12: Comparison of the lift to drag ratio of the different wings tested......86

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Tables

Table 1: Main characteristics of the designed aerofoil..............................................15 Table 2: Induced, effective and geometric incidence at the various stations designed..........................................................................................................................26 Table 3: Data computed for the expansion corner with an incident Mach number M1=2.5 and an angle of 15°.'nbr of cells ' refers to the number of  connectors on the surface of the corner, µ refers to the angle of the wave with the wall surface, the subscript 1 refers to the flow upstream of the corner and subscript 2 refers to the downstream flow..................................................................34 Table 4 : Data computed for the compression corner with an incident Mach number M1=2.5 and an angle of 15°.'nbr of cells ' refers to the number of  connectors on the surface of the corner, θ refers to the angle of the wave with the wall surface, the subscript 1 refers to the flow upstream of the corner and subscript 2 refers to the downstream flow..................................................................35 Table 5: Data computed for the supercritical aerofoil with an incident Mach number M=0.725 and an angle of incidence α=1.5°.'nbr of cells ' refers to the number of connectors on the surface of the aerofoil, and the error band is the grid convergence index .................................................................................................38 Table 6: Comparison of the lift curve slopes of the various wings at different freestream Mach number.............................................................................................58

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Notations

a: speed of sound A: wing planform area AR: wing aspect ratio b: wing semi span c: wing chord CD: drag coefficient CDW: wave drag coefficient CD0: zero lift drag coefficient CL: lift coefficient Cm: pitching moment coefficient CP: pressure coefficient Cp*: critical pressure coefficient D: drag f: a solution computed FS: safety factor g: gravity GCI: grid convergence index K: lift dependant drag factor L: lift M: freestream Mach number m0: mass at take off  mb: final mass n: power factor p: order of convergence P: pressure r: refinement ratio of the grid

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Re: Reynolds number sfc: specific fuel consumption t: thickness of the aerofoil V: velocity x: chordwise ordinate y: streamwise ordinate, y=0 at the root of the wing z: third coordinate to obtain a right handed coordinate system α: wing incidence ε: relative error η: spanwise dimensionless ordinate θ: angle of the compression wave with the flow direction Λ: sweep angle µ: angle of the expansion wave with the flow direction

Subscripts

e: effective g: geometric i: induced LE: leading edge t: thickness TE: trailing edge 1: flow properties upstream of the corner or fine grid 2: flow properties upstream of the corner or medium grid 3: coarse grid : freestream

∞

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Introduction

The design of the wing is an essential part in the design of an aeroplane. Indeed, it enables to lift the aircraft with its passengers and the cargo. On a transport aircraft, the main objective is to maximise the profit related to the use of an aeroplane mounted with this wing. This can be done by many different ways. Increasing the lift will result in an enhancement of the payload, decreasing the drag will reduce the fuel consumption, reducing the weight of the wing by using new materials will enable to raise the payload…

Recently, an aircraft manufacturer decided to modify the design of its wings, introducing raked wing tips instead of winglets on a large number of its aeroplanes. This research aims at understanding the benefits obtained by making such modifications with using computational fluid dynamics (CFD).

The study was carried out in several steps which are described in this report. Firstly, a literature review is done so as to acquire sufficient knowledge in the field and to be able to tackle the problem. Then, the study goes on with a design of a wing and wing tips. A validation hierarchy is achieved in order to ensure the model is a good representation of  the reality. Finally, a comparison of performances is realized by doing some flow studies before looking at some integrated data.

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Chapter 1: Literature review

Chapter 1: Literature review

1.1. Aircraft performances

Diminishing the fuel consumed by an aircraft has always been one of the main challenges of the transport aircraft manufacturers. The increase in fuel costs does not change this tendency. Environmental issues also press researcher to look for reducing carbon dioxide emissions and European aeronautic research aims at reducing the fuel consumption of 50% per passenger and per kilometre before 2020 (reference 3). As being partly responsible of the fuel burnt during flight, the drag produced by the wings requires special care.

Total drag produced by an aircraft results from the sum of various contributions. According to several studies (references 2, 3), lift-induced drag and skin friction drag are responsible for more than 80% of the total drag produced by an aircraft (figure 1). The relative importance of induced drag is widely related to the configuration of the aircraft (cruise or climb configuration). Based on this analysis, research is undertaken to increase the aeroplanes aerodynamic efficiency by reducing these drag components.

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Chapter 1: Literature review

Drag breakdown

Profile drag

Vortex drag

40%

40%

Others

20%

Figure 1: Drag breakdown for a typical transport aircraft. The numbers presented are just estimation because this is highly dependant on the configuration of the aircraft (cruise, high lift configuration…)

1.2. Wing tip vortices The flow field in a 3D study is very different from the flow features that can be observed in two dimensions. Some important flow features appear in the tip region, increasing the complexity of the problem.

The lift force acting on a wing is the resultant of the difference of pressure between the upper and lower surfaces of the wing. The flow on the upper surface is accelerated and is faster than the flow on the lower surface. Bernoulli equation enables to conclude that there is a pressure difference between both faces of the lifting device which produces an upward force called lift. In the case of a finite wing, this results in the development of a secondary flow called crossflow. On the lower surface the flow tends to be sucked outboard near the tip region and rolls up due to the lower pressure on the upper surface. As a result, counter rotating vortices form around the wing tips (figure 2). The near field properties of the flow in these regions are highly dependent on the wing tip shape and the incidence (reference 17).

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Chapter 1: Literature review

Figure 2: Formation of trailing vortices on a finite wing

The presence of the tip vortex induces modifications on the pressure distribution over the wing in the tip region. A reduction in the peak suction pressure on the wing upper surface and a distortion in the pressure distribution can be observed while getting closer to the tip (reference 18).

Some other problems are related to the presence of these strong vortices as they create an unsteady environment for a following aircraft (reference 14, 16). Indeed, in the wake of an aircraft, large modifications in the aerodynamic environment can be experienced with very small displacements resulting in hazardous rolling moment. That is one of the main issues limiting the capacity of the airports.

1.3. Induced drag

Lift-induced drag contributes for approximately 40% of the drag produced by a large transport aircraft (references 2, 3, 12). This drag component is widely related to the strength of the vortex in the tip region and its interaction with the flow around the wing.

Indeed, the vortex filaments in the wake of the wing induce a downward velocity component on the flow along the wing which is usually called downwash. This

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Chapter 1: Literature review

downward component of the velocity at the trailing edge reduces the effective incidence of the wing. As the lift produced by a wing is perpendicular to the local flow direction, it is tilted away from its vertical position and creates a force, called lift induced drag, which opposes the wing motion (figure 3).

Figure 3: Drag component of lift resulting from downwash (w= downwash; V = forward speed of  wing; VR = resultant oncoming flow at wing; α = incidence; ε = downwash angle = w/V; α = ( α-ε) ∞

= equivalent two-dimensional incidence; L

∞

= two-dimensional lift; L = wing lift; D v =trailing

vortex drag)

This drag component can be evaluated using the formula proposed by Prandtl (reference 13, 19): C  Di = K 

C  L

2

 AR

(1)

π  

This formulation was based on the accepted result that, for a rigid wing with an unswept quarter chord line, minimum induced drag occurs for an elliptic loading and in this case, the K factor in equation (1) equals 1.

Based on the relation above, to reduce lift-induced drag, the most straight forward method consists in increasing the wing aspect ratio. However, wing span is subject to some structural constraints and is limited by the dimensions of the airport faciliti es.

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Chapter 1: Literature review

1.4. Wing tip devices

Wing tip design has been rapidly recognised as having a significant impact on induced drag. The use of fences at the tip of the wing to prevent the flow from rolling up was rapidly recognised as a way to reduce drag without requiring essential modifications on the wing design. Some other tip devices have been designed since then. An aerodynamic comparison of several wing tip devices has been performed by Kravchenko (reference 4, 23). He compared effects of winglets, complex planar wingtips and multi-element sails. These tip devices result in a reduction of fuel consumption but may also improve flight characteristics. They are easy to i mplement as they do not require serious modification in the structure of t he wing.

1.4.1. Winglets

1.4.1.1. Winglet effects

Winglet is one of the most commonly used methods to reduce induced drag acting on a wing. It is a non planar tip device with an aerofoil section which interacts with the tip vortex to reduce its influence.

The first effect of the winglet is to reduce the strength of the tip vortex. Indeed, as a fence, it prevents the flow from the wing lower surface to roll up. Besides, it uses the distorted flow in the tip region to produce a force acting in the flight direction.

The crossflow and the tip vortex distort the flow at the tip inducing on the winglet an effective incidence. When the winglet is correctly designed, it produces a lift force

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Chapter 1: Literature review

resulting from this incident flow with a component in the direction of flight (figure 4). If this force component is greater than the drag penalty due to the addition of the winglet itself, it results in a diminution of overall drag. The design of a winglet is very complex. It requires the same aerodynamic characteristics as a wing and its chordwise position on the tip of the wing require special care to optimize its efficiency and to prevent detrimental flow interactions with the wing. The design requirements will be discussed later.

(a) Secondary flow around wing-tip

(b) Forces acting on a winglet

Figure 4: Winglets to reduce lift-induced drag

According to A.J. Bocci (reference 7), winglets show greater efficiency when there is high loading near the tips of the wing and it is more efficient than a wing tip extension producing the same bending moment at the root. It enables to increase the aircraft efficiency. However, the winglet efficiency depends on the lift produced by the wing and strong aerodynamic interference can be found at the concave junction between the wing and the winglet.

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Chapter 1: Literature review

1.4.1.2. Winglet design

A design approach of winglets has been detailed by Whitcomb (reference 5). These recommendations have been confirmed by Heyson studies (reference 6). The different geometric angles used in the design of a winglet are presented in figure 5. The toe angle can be related to the winglet incidence and has to be defined by taking into account the flow distortions in the region. Thus, a winglet is usually slightly toed out. The Cant angle is useful to reduce the interferences at the junction.

Figure 5: Geometric quantities used to define a winglet

As being a reference in many studies about winglets design guidance from Whitcomb is summed up below (reference 5).

The upper winglet has to be nearly vertical and placed rearward of the tip to prevent the adverse pressure gradient from the wing to add with the one produced by the winglet. This design characteristic is necessary to prevent the formation of a strong shock and an early flow separation at the tip.

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Chapter 1: Literature review

Winglet has to be designed to produce a normal force equivalent to the lift coefficient produced by the wing. The sweep angle should be nearly the same as that of the wing and the airfoil section should be designed to avoid flow separation and the formation of  a strong shock wave. The upper winglet must be toed out. Twist is not necessary as changes in inflow velocity with height produce nearly the same effect. A small amount of outward cant is also required.

The lower winglet is shorter than the upper one and is placed forward. Its trailing edge must square with the leading edge of the upper winglet. It must have significant outward cant to increase its favourable effect on the flow over the upper winglet. The lower winglet is not necessary but might be desirable to improve the upper winglet efficiency in high lift configuration (reference 8).

1.4.2. Raked wing tips

A raked wing tip is a planar tip device consisting of a wing tip extension with more sweep than the wing itself. This tip device has been recently introduced by Boeing on a 737 designed for maritime patrol. It was also implemented on the wing of the Boeing 767-400 to increase its efficiency with not much additional design changes. The recent introduction of the raked-tip on a commercial aeroplane has motivated this thesis.

The raked tip is added to the wing, resulting in an increase of its span. The reduction in induced drag found with the addition of this tip device will be due to both the increase in aspect ratio and the improved efficiency of the wing (reference 12). In order to separate these two effects and to valuate the aerodynamic efficiency, the study of the parameter K (formula 1) is more convenient than a study of th e drag.

According to reference 4, the diminution of induced drag produced by a raked tip is due to a redistribution of the loading across the wing span. Studies carried out on a 60°

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Chapter 1: Literature review

raked tip (reference 12) have shown an increase of the loading at the root in comparison with a wing alone and an improvement in the loading distribution near the tip region.

Non planar tip devices can impair the wing efficiency at low lift coefficient due to an increase in the wetted area and to a tip vortex that is not strong enough. The efficiency of a wing mounted with a raked tip does not seem to be affected in the same way (reference 23) and the improved performances are observed even at low lift conditions. Besides, the greater sweep angle of the raked tip enables it t o operate at transonic speed with keeping good performances. Indeed the flow over the tip region remains subsonic for a wide range of speeds and incidences due to the high sweep angles involved.

As regards of the stability of the aircraft, the position of the aerodynamic centre is shifted back which results in an improvement of the static stability. To end with, the aerodynamic efficiency of the raked tip is also related to the redistribution of the pressure distribution over the tip. Indeed, a high suction peak can be observed at the leading in the tip region which is related to the high sweep angles.

Experiments carried out by Gold (reference 9) have shown the formation of a second vortex inboard of the main tip vortex and with the same rotational direction. It is suggested this vortex might come from the transition between the wing and the tip which have different sweep angle.

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Chapter 1: Literature review

1.5. Computational fluid dynamics simulation

1.5.1. Grid generation

Grid generation is an essential step in the computational fluid dynamics procedure. It corresponds to the space discretization of the problem. Several issues related to the grid might affect data computed. Overlap has to be avoided to be able to run computations and some cells can be skewed in some regions of the volume studied. Clustering of the cells has to be controlled in order to avoid useless computations and to ensure good predictions in the area studied.

The most complex case that will be studied for the purpose of this thesis is the wing with a winglet. A high density of cells is required in the tip regions, at the junction between the wing and the winglet. In order to achieve this, blocks have to be created according to the specifications presented on figure 6 (reference 10).

Figure 6: Block strategy to produce the grid for a wing/body/winglet configuration on the left hand side and for a configuration with multiple winglets on the right hand side (rear view)

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Chapter 1: Literature review

However, using this strategy may result in slope discontinuities of the mesh between blocks 1 and 2 in the case of a single winglet. This might require to be smoothed out before running some computations.

1.5.2. Euler equations

Euler equations enable to calculate the velocity of the flow at each point in a problem where viscous effects are neglected. Viscous effects are only important inside the boundary layer and reasonable predictions can be obtained by solving Euler equations even if viscosity is present.

According to reference 11, the accuracy of the data obtained doing such an approximation depends on the wing studied and the test conditions. Generally, good correlation between the data computed and experiment were found for the pressure distribution.

However, the shock in supercritical conditions was found to be stronger and backwards from its actual position. The same inaccuracies were revealed by the data collected with the Euler method of reference 10. These inaccuracies are due to the absence of the boundary layer which reduces the effective camber and the wing effective incidence.

1.6. Aim of this research

As it was said previously, reducing drag produced by the wings is an important challenge for aircraft manufacturers. Substantial drag reduction can be obtained by improving the design of the wing tips. This certainly led Boeing’s representatives to

- 12 -

Chapter 1: Literature review

choose to implement raked tips extensions on some of its aircrafts rather than winglets which are most commonly used.

The aim of this research will be to use computational fluid dynamics in order to compare the efficiency of both tip devices and to determine their pros and cons. As data presented in the literature review on raked tips was obtained from wind tunnel tests, calculations done with CFD will certainly enable to have more information about the flow field around such a device.

- 13 -

Chapter 2: Wing design

Chapter 2: Wing design

The objective of the design procedure is to create a swept wing showing good performance in supercritical flow conditions. The wing was designed in several steps described below:

•

design of a 2D supercritical aerofoil with the programs VGK and SectionD

•

conversion of the 2D aerofoil section to a 3D aerofoil section using the Convert program with the parameters of a typical wing planform

•

design of several spanwise sections along the wing with the Sweptdes program

•

computation of the twist with the Downwash program

A brief description of all these programs can be found in appendix. As the wing tips studied are likely to be used on transport aircrafts, I decided to take the flow conditions in cruise of a typical commercial airplane as a reference. Thus the wing should be designed for a Mach number M ≈0.8 at an altitude of 35000 feet.

2.1. Supercritical aerofoil design

2.1.1. Design procedure

The design of an aerofoil is a first step in the design of a full 3D wing. A supercritical aerofoil is a section of a lifting device designed to operate at subsonic speed with supersonic flow regions at its surface. The main flow features around such an aerofoil

- 14 -

Chapter 2: Wing design

are the presence of an extended near sonic rooftop on the upper surface, a high rear loading and a supercritical isentropic recompression.

Two programs are used in the design of the aerofoil: -

SectionD enables to compute its shape

-

VGK enables to assess quickly its performance

Thus, the design procedure consists of several iterations where we compute an aerofoil section with SectionD before checking its performances with VGK. Several reference values enable to work out the main flow characteristics around the lifting device from the data given by VGK. A shape factor above 2.2 is associated with a separation of the flow and a shock wave with an upstream Mach number higher than 1.3 is considered to be strong.

A design was considered to be acceptable when the aerofoil could sustain a long rooftop with producing enough lift and avoiding a flow separation at the trailing edge. Indeed, the longer the rooftop is, the higher the adverse pressure gradient will be and the more chances we have to separate the flow. Besides, to optimize the size of the rooftop, the flow should separate at the trailing edge for angles of attack between 0.5 and 1.5. Finally, the wave drag should not exceed four digits when the aerofoil operates in this range of incidences.

The convergence of the computations is essential in such a study. It was considered to -3

be acceptable when all the residuals were below 10 . This has been checked for every data published in this report to ensure their accuracy.

The design procedure enabled me to work out the aerofoil with the characteristics presented in table1. M

Re

RAE number

t/c

Rooftop extent

0.725

10,000,000

101

0.12

62% c

Table 1: Main characteristics of the designed aerofoil, RAE number refers to the thickness distribution (Appendix A)

- 15 -

Chapter 2: Wing design

Figure F1 presents the pressure distribution computed with VGK on the aerofoil designed. We can clearly identify the different features of a supercritical aerofoil: a near sonic rooftop with high rear loading and an isentropic recompression.

2.1.2. Performances of the aerofoil designed

Several computations have been done on the aerofoil designed with changing the angle of incidence to get some general aerodynamic characteristics. Some off-design calculations have also been carried out to check the sensitivity of the aerofoil as regards of the velocity. The data computed during this study are presented in figures 7 t o 10.

1,5 1 0,5 alpha= 0 -CP

alpha= 1.5

0 0

0,2

0,4

0,6

0,8

1

alpha= -1.5 alpha= 2

-0,5 -1 -1,5 x/c

Figure 7: Influence of incidence on pressure distribution over the aerofoil at M=0.725

Many features are noticed from the data obtained at M=0.725. The pressure distribution over the aerofoil for several incidences is presented on figure 7. As we increase incidence, the pressure on the upper surface goes down, below the critical pressure. For incidences above 1°, a weak shock wave appears on the upper surface, ahead of the end

- 16 -

Chapter 2: Wing design

of the rooftop. A little bump in the pressure distribution beyond the shock attests of its weakness. As angle of attack increases, this wave gets stronger and is pushed backward. Separation at the trailing edge occurs at α=1.5°. The adverse pressure gradient beyond the rooftop is too high to keep the flow attached. The wave drag gets too high (CDW>0.001) for an incidence above 1.6°. Separation after the shock wave occurs at 61% of the chord for α=2.8°. The flow reattaches beyond this point and separates again at the trailing edge.

The plot of the lift coefficient against the incidence (figure 8) squares with the theory as a straight line is obtained. Computations could not be carried out until the stall angle because the shock wave was getting too strong and calculations did not converge. According to the calculations, for a freestream Mach number M=0.725, the lift curve slope is approximately 0.19 and zero lift is obtained for α= -2.2°. 1,2

1

0,8

M=0.725

CL

0,6

M=0.75 M=0.7

0,4

0,2

0 -3

-2

-1

0

1

2

3

4

Alpha

Figure 8: Variation of lift coefficient against incidence for various Mach numbers

- 17 -

Chapter 2: Wing design

The plot of L/D against incidence (figure 9) enables to work out the angle of maximum efficiency: α=1.3° for M=0.725. With this incidence, the aerofoil produces C L=0.65 and the flow remains attached all along its surface.

80

70

60

50 M=0.725

L/D

40

M=0.75 M=0.7

30

20

10

0 -3

-2

-1

0

1

2

3

4

Alpha

Figure 9: Variation of L/D against alpha for various Mach numbers

Some off design computations have also been carried out. Figure 10 shows the influence of Mach number on the pressure distribution over the aerofoil at 1.3° of  incidence. We can see that minimum pressure over the upper surface is not changed significantly. However, the shock wave moves backward and gets stronger as Mach number increases.

- 18 -

Chapter 2: Wing design

1,5 1 0,5 -Cp

M=0.725

0 0

0,2

0,4

0,6

-0,5

0,8

1

M=0.75 M=0.7

-1 -1,5 x/c Figure 10: Influence of Mach number on the pressure distribution over the aerofoil at α=1.3°

On figure 8, we can see the influence of the Mach number on the lift coefficient. The slope of the curve C L against α slightly increases as Mach number increases. As Mach number appears in the denominator of the expression of lift coefficient, we can conclude that less incidence is required to produce the same lift when velocity is increased which is physically acceptable.

Mach number also affects the angle of maximum efficiency. Indeed, we can see in figure 9 that as Mach number increases, the incidence to achieve a maximum L/D decreases and the value of this maximum decreases.

- 19 -

Chapter 2: Wing design

2.2. Swept wing design

The design of the 3D swept wing is based on the 2D aerofoil designed previously and presented in the paragraph above. The first step of the design is to define a wing planform on which computations should be run. According to the design Mach number of the aerofoil section, the wing planform referred to as Cranfield wing 1 is completely adapted to the problem (figure 11).

Possible leading edge extension

Figure 11: Planform of Cranfield wing 1: distances are in meter and the sections designed are in red

Three streamwise sections are then considered for design: -

one outer wing section: y=16.8

-

one root section: y=0

-

one intermediate wing section: y=4.5

The wing design Mach number can be easily computed from the design Mach number of the 2D aerofoil:

 M 3

 D

=

 M 2 cos( Λ

 D mean

)

- 20 -

Chapter 2: Wing design

The average sweep angle is simply computed using the following formula:

Λ

mean

= arctan 0.5 * tan

Λ ) + 0.5 * tan Λ )) = 24.65°  LE 

TE 

⇒ M 3 D ≈ 0.798

2.2.1. Design of the outer wing section

The outer wing section does not require many modifications as regards of the design. The 3D streamwise aerofoil is computed from the coordinates of the 2D aerofoil designed previously. Additional input data concerning the wing planform requires to be implemented (leading and trailing edge sweep outboard of the crank) and the power factor which is initially chosen to be n=1.5. The streamwise section computed is then implemented in Sweptdes for a forward run. The pressure distribution obtained (figure 12), shows a near sonic rooftop along the section but more loading near the leading edge is required to increase its performances. Thus, I decided to use a power factor n=2 which gave the expected pressure distribution. 0,8 0,6 0,4

-CP

0,2 0 0

0,2

0,4

0,6

0,8

1

1,2

-0,2 -0,4 -0,6 x/c Pressure distribution for n=1.5

Pressure distribution for n=2

Cp critic

Figure 12: Pressure distribution obtained with Sweptdes on the outer wing section (y=16.8), effect of the power factor on this distribution

- 21 -

Chapter 2: Wing design

2.2.2. Design of the root section (y=0, η=0)

The root section is designed using the Sweptdes program. The design of the section is done in two steps, since thickness and camber need to be computed separately. The main objective while shaping this section is to increase the sweep of the maximum thickness towards the root in order to maximize the wing thickness at this station. This will also counteract loss of isobar sweep due to root and compressibility effects. Indeed, high root thickness is required for structural reasons and to increase the storage volume.

To modify the aerofoil thickness, target Cpt’s at the 15 standard Weber stations have to be implemented in Sweptdes. To move forward the maximum thickness position and to increase the thickness of the aerofoil, decreasing the pressure until the critical pressure and moving it forward are necessary (figure 13).

0,8 0,6 0,4 0,2 -Cpt 0 -0,2 0 -0,4 -0,6 -0,8

0,2

0,4

0,6

0,8

1

x/c Root

Outer wing

Cp critic

Figure 13: Comparison of the distribution of pressure due to thickness at the root and at the outer wing section

However, without implementing any modification on the wing planform, I was unable to sweep enough the maximum thickness. A leading edge extension was required. The final root thickness compared with that at the outer wing section is presented figure 14. We can clearly see the increase in the thickness resulting from the advancement of the

- 22 -

Chapter 2: Wing design

maximum thickness position. The maximum thickness position along the span of the wing is presented figure 15 so as to attest the increase in sweep at the root.

0,09

( z / c )max = 0.077 ( x / c )max = 0.22

0,08 0,07 0,06

z/c

0,05 0,04

( z / c )max = 0.057 ( x / c )max = 0.30

0,03 0,02 0,01 0 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

x/c Ro ot thickness

Out er wing t hickness

Figure 14: Comparison of the thickness at the root and at the outer wing section

Figure 15: Maximum thickness position over the wing designed and maximum thickness sweep angles

Camber design is done by implementing a pressure distribution over the wing upper surface. As for the thickness design, the pressure coefficient was defined at the 15 Weber stations. The objective is mainly to increase the loading on the section designed by implementing a long near sonic rooftop. The section obtained is presented figure 16,

- 23 -

Chapter 2: Wing design

along with the pressure distribution. At this point, it is necessary to check that the aerofoil section does not show any “hook” which is not the case here.

Figure 16: Root section and pressure characteristic

2.2.3. Design of the intermediate section (y=4.5, η=0.1875)

The design of an intermediate section is necessary to ensure the continuity of the aerodynamic properties along the span. The position of this section was determined by doing several forward run of Sweptdes at different spanwise coordinates. The position was set when the root effects start to appear which is checked by looking at the parameter K2A given in the Sweptdes output data. This parameter reveals the influence of the root effects at the coordinate studied (K2A=1 at the root and K2A=0 at the tip). For the intermediate section we must have: K2A ≈0.1 which corresponds to y=4.5m (inboard of the crank).

- 24 -

Chapter 2: Wing design

The design is then carried out with the same methodology as for the root section but with target pressure distribution closer from those on the outer section. The maximum thickness position must coincide with the maximum thickness line drawn before. We end up with the pressure distributions presented figure 17.

Figure 17: Intermediate section characteristics

2.2.4. Spanwise loading and twist

The loading over the wing is computed using the local lift coefficient displayed in the Sweptdes output files at the various stations designed. From an aerodynamicist point of  view, the loading over a wing should be nearly elliptic in order to minimize the induced

- 25 -

Chapter 2: Wing design

drag. However, some structural issues have to be taken into account and the loading over a wing is chosen to have more like a triangular shape. In Figure 18 is displayed the loading over the wing designed in comparison with an elliptic loading.

0,8 0,6 Loading 0,4

0,2 0 0

0,2

0,4

0,6

0,8

1

1,2

η

Loading over the the wing ing

Ellip liptic tic loading ing

Figure 18: Comparison of the loading over the wing with an elliptic loading

The span wise loading is then used as an input file for the Downwash program. It provides with values for the induced incidence ( αi) which enables to compute the geometric incidence (α ( αg) for each design station: αg=αi + αe αe) is given in the Sweptdes output files for each design The effective incidence ((α section. The table below summarizes these data.

η

αi

αe

αg

Root section

0

1,1183

0,66326

1,78156

Intermediate section

0,1875

0,78

0,80388

1,58388

Outer section

0,7

0,3

-0,01871

0,28129

Table 2: Induced, effective and geometric incidence at the various stations designed

- 26 -

Chapter 2: Wing design

2.2.5. Conclusions about the method

A 3D swept wing is designed using the computer programs Convert, Sweptdes and Downwash. The starting point of this design is a 2D aerofoil section designed with SectionD and VGK. The design procedure carried out enables to obtain the shape of  various streamwise sections of a wing designed to fly at M=0.798.

The main problem encountered during this design is related to the use of these data. Indeed, creating a precise database is necessary in order to generate an accurate grid and to run computations on the t he wing. However, the 15 Weber points given as an output of the Sweptdes and Convert programs were not enough and I had to define some new point’s coordinates. As I was mainly interested in the section outboard of the crank  which was designed using only the Convert program, I computed directly the wing streamwise section from the 2D aerofoil coordinates using an Excel file and the formula:

n +1

  z    z  (cos(Λ mean ))   =  ⋅ n  c  3d   c  2 d  (cos(Λ local ))

This way, I was able to double the number of points defining the sections.

Another problem encountered is related to the size of the grid necessary to compute the problem. Indeed, the wing designed with this method has a half span of 24 meters and only data collected on the outer wing are of any interest for this study. As a result the wing on which the study will be based has a half span of 15 meters, corresponding to the wing designed cut at the crank.

- 27 -

Chapter 2: Wing design

2.3. Design of the wing tips

2.3.1. Design of the raked raked tip

Few parameters are required for the raked tip design as it is a planar device. The leading edge sweep was taken to be ΛLE=55° as it seemed to provide good performances according to studies from Kravchenko (reference 4). The trailing edge sweep was decided by having a look at some pictures of the raked tip implemented on some transport aircrafts. As it seems the trailing edge of the raked tip was slightly more swept than the wing, I decided to design it with a trailing edge sweep ΛTE=23.8°. In the same way, the length of the tip device was decided according to an article saying that the semi span resulting from the addition of this wing tip was 1.7 meters longer than the cleaned wing.

Concerning the design of the section, the sweep definitely had to be taken into account. Indeed, it was found in the literature that the raked tip does not lose efficiency at high speed because of the subsonic flow over the t he device. Thus, the Convert program applied to the 2D section computed during the supercritical aerofoil design enabled to compute the tip section. No twist was implemented on the raked tip.

2.3.2. Design of the winglet

The design of the winglet was mainly based on the guidance from Whitcomb (reference 6). Only an upper winglet will be designed as it corresponds to the most commonly used configuration.

- 28 -

Chapter 2: Wing design

In order to produce a normal force equivalent to the wing lift coefficient, I decided to use the same section for the winglet as for the outer wing. As the winglet and wing should have nearly the same sweep angles, they were both designed with the same sweep angles. To reduce interferences between the flow over the winglet and over the wing, the winglet root chord is 60% of the wing tip chord and the Cant angle was decided to be 30°. Some toe out was also implemented to obtain nearly the same pressure distribution on the wing and on the winglet. Several computations with various toe angles enable to find that 3° of toe out was acceptable. All these data are presented figure 19. No twist was added to the winglet as the distorted flow at the wing tip should produce nearly the same effect.

The length of the winglet was decided so as to have nearly the same wetted area as with the raked tip. It was found in the literature (reference 20) that the winglet should have a height of around 10% of the semi span. The winglet designed is 1.7 meters high corresponding to 11% of the semi span.

Figure 19: Geometric characteristics of the winglet

- 29 -

Chapter 3: Validation

Chapter 3: Validation

Validation is the process of evaluating the accuracy of our model in comparison with the real phenomenon studied. To do so, we have to compare the data computed with experimental data or benchmarks and take into account the error due to the experimental procedure.

3.1. The choice of Euler equations

Before running some computations, some approximations have to be made. The point is to ensure that the approximation made still represent accurately what we require and to be aware of the weakness of our model.

The main objective of this study is to compare the performances of two wing tips in cruise configuration. In cruise configuration, we can assume that on a well-designed wing the boundary layer remains attached all over the wing. Moreover, the efficiency of  raked tips and winglets is not related to viscous effects. The formation of a tip vortex is an inviscid phenomenon. As the wetted area of the raked tip and of the winglet designed are nearly the same, we can assume that skin friction drag due to the addition of these tip devices will be nearly the same and carry out some computations using an inviscid model.

Another approximation made is to neglect the effects of the fuselage on the pressure distribution over the outer wing. Thus, computations will be made only on the wing and a symmetry boundary condition will be used at the root section.

- 30 -

Chapter 3: Validation

3.2. Validation hierarchy

Validation enables to check that the model represents accurately the physical phenomena. The complete problem being far too specific and too complicated to be checked with experimental or theoretical data, it is necessary to decompose it in several easier problems. This is the purpose of the hierarchy.

As it was explained before, computations will be carried out using Euler equations. The validation hierarchy resulting from this approximation is presented figure 20.

Wing designed with a wing tip

Study of the Onera M6 wing

Study of the 2D supercritical aerofoil section designed

Compression corner

Expansion corner

Figure 20: Validation hierarchy

The lowest level of the hierarchy is dedicated to unit problems involving a simple geometry and a unique flow feature. As computations on the full model will be run using Euler equations, only compressible effects need to be studied. This explains the study of compression and expansion corners where data computed can easily be compared with theory.

- 31 -

Chapter 3: Validation

At the second stage of the hierarchy, the difficulty of the problem increases. The study of the supercritical aerofoil designed means both compressible effects (compression and expansion) will be considered on a more complex shape. Pressure distribution computed with Fluent can be easily compared with the data computed with VGK.

At the third level of the hierarchy, the ONERA M6 wing enables to study a three dimensional wing in transonic flow. It is a benchmark case on which enough experiments have been carried out to ensure their accuracy. Three dimensional effects are introduced such as crossflow and wing tip vortices.

Finally, the last level corresponds to the complete problem with the wing designed either clean or with one of the wing tips. Some interactions between two flows are introduced, especially when adding the winglet.

3.3. Richardson extrapolation

The extrapolated data are obtained using Richardson extrapolation. It enables to work  out the solution of the discretized equations that would have been obtained with an infinite number of cells using data obtained with three grids having a constant refinement ratio. The first step of this extrapolation is the computation of the order of  convergence:

  f 3 −  f 2      f 2 −  f 1   (  f  refers to the solution computed, r refers to the grid refinement

ln  p =

ln(r )

ratio taken to be constant, subscripts 1 to 3 refer to the different grids, 1 being the fine grid and 3 the coarse one)

- 32 -

Chapter 3: Validation

Using the value obtained for the order of convergence, the continuum value, f 0, can be computed using the relation below:

 f 0 ≈  f 1 +

 f 1 −  f 2 r  p − 1

The grid convergence index can then be estimated. It provides an error band on how far is the asymptotic numerical value from the data computed. Thus, it can be seen as an indicator to know whether further refinement of the grid is required. It is defined as:

GCI 12 =

F S

ε  

 p (r  − 1)

where Fs is a safety factor (F s=1.25 for comparisons over three or more grids) and ε is

 

the relative error  ε   =

 

 f 2 −  f 1    f 1

  

Finally checking the convergence of the computations is required in order to ensure the data used to carry out the extrapolation is within the asymptotic range of convergence. This can be done by checking the relation:

GCI 23  p

r  GCI 12

≈1

Richardson extrapolation can either be applied to some solutions at a grid point or to a solution functional. The  f 0 value can be seen as an estimation of the value that could be computed in the limit where the grid spacing tends toward zero.

This sort of procedure is relatively easy to do in 2D but it becomes computationally expensive in 3D. Indeed, a refinement ratio r=2 would multiply the number of cells in the grid by 8.

- 33 -

Chapter 3: Validation

3.4. Compression and expansion corner

The compression and expansion corners are some basic test cases involving compressible flows. Theory enables to compute rapidly from an angle of deviation and the incident Mach number the outlet Mach number and other physical data relating the inlet to the outlet.

In the case studied, the deviation angle was ±15° and the boundary conditions used were:

•

Pressure far field: M=2.5, T=288K for both inlet and outlet

•

Symmetry for the upper boundary

•

Wall for the corner

Three structured grids were used to carry out the computations, enabling to establish a grid convergence using Richardson extrapolation. The coupled implicit solver was used to complete computations, with an inviscid model. Data computed are summed up in the tables below (tables 3, 4). 1/ (nbr cells) P2/P1

T2/T1

ρ2/ρ1

angle µ1

M2

angle µ2

1.3E-02

0.328

0.748

0.439

3.169

25.0

18,0

6.6E-03

0.328

0.729

0.450

3.234

24.0

17,8

3.3E-03

0.328

0.728

0.451

3.235

23.5

17,8

1.54

4.01

7.19

5.53

extrapolated value

0.328

0.727

0.451

3.235

error

2.03%

0.62%

1.47%

0.45%

convergence

0.999

0.972

1.027

1.021

Theoretical value

0.3212

0.723

0.444

3.25

23,6

17.9

order p

asymptotic range of

Table 3: Data computed for the expansion corner with an incident Mach number M 1=2.5 and an angle of 15°.'nbr of cells ' refers to the number of nodes on the surface of the corner, µ refers to the angle of the wave with the wall surface, the subscript 1 refers to the flow upstream of the corner and subscript 2 refers to the downstream flow

- 34 -

Chapter 3: Validation

1/ (nbr cells) P2/P1

T2/T1

ρ2/ρ1

angle θ

M2

1.8E-04

2.467

1.322

1.866

1.872

37.8

4.4E-05

2.466

1.321

1.867

1.874

37

1.1E-05

2.466

1.321

1.867

1.874

37

1.57

2.56

2.95

4.35

extrapolated value

2.466

1.321

1.867

1.874

error

0.07%

0.22%

0.01%

0.22%

1.000

1.000

1.001

1.001

2.468

1.324

1.867

1.87

order p

asymptotic range of convergence Theoretical value

37

Table 4 : Data computed for the compression corner with an incident Mach number M 1=2.5 and an angle of 15°.'nbr of cells ' refers to the number of connectors on the surface of the corner, θ refers to the angle of the wave with the wall surface, the subscript 1 refers to the flow upstream of  the corner and subscript 2 refers to the downstream flow

The angles measured and displayed in these tables are just presented to show that reasonably good predictions are obtained on this data. However, the measuring device used to obtain the data was not accurate enough to be able to obtain angles with a precision below a quarter of degree which is not enough as regards of the theoretical data. The error displayed in these tables is computed with comparing the computed data with the theoretical one and has nothing to do with the grid convergence index.

We can see on these data that the software is reliable in the computation of  compressible flows and very few cells are required to obtain good predictions of the flow characteristics. Nevertheless, a good grid resolution is required in order to obtain good flow visualisations with a high degree of accuracy in the regions of discontinuities (shock wave and sudden change in angle). This change in angle should not appear that suddenly in the case of the wing. Besides, we can see a lower accuracy is obtained with the computation of the expansion corner than with the compression corner.

- 35 -

Chapter 3: Validation

3.5. 2D supercritical aerofoil

The second stage of the hierarchy involves more complex computations. The aerofoil designed in order to achieve a wing design was tested using VGK. These calculations can now be compared with data computed using Fluent.

Computations were carried out using the coupled implicit solver with an inviscid model at an operating pressure: P=48,474Pa. The grid was achieved under Gridgen using a fully unstructured mesh around the aerofoil. The boundary conditions used were:

•

pressure far field: M=0.725, T=250 for both inlet and outlet (t his was decided to achieve the design conditions: M=0.725, Re=10,000,000)

•

wall for the aerofoil which has a chord measuring 1meter

As with the compression and expansion corners, three different grids were tested with a constant refinement ratio in order to achieve a Richardson extrapolation on lift and drag coefficients. The grid refinement is checked using the number of grid points over the aerofoil section.

Figure 21 shows a comparison of the pressure distribution obtained with VGK and with Fluent. First of all, we can see on this plot that the shock wave is predicted far forward when using a viscous model in comparison with the inviscid case. This is due to the increase of the effective thickness of the aerofoil related to the presence of the boundary layer.

When comparing VGK and Fluent, we can see that predictions from VGK exhibit higher loading. The pressure coefficient over the rooftop predicted by Fluent with the inviscid model is closer from the value given by the viscous calculations of VGK.

- 36 -

Chapter 3: Validation

However, the position of the shock does not change much between Fluent and VGK inviscid calculations. The discontinuity through the shock seems to be better predicted with Fluent but this is only due to a grid resolution. Indeed, the finest grid used by VGK has 160 grid points over the whole aerofoil whereas the fine grid used with Fluent counts 600 nodes over the section.

1,5

1

0,5

-Cp

0 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

-0,5

-1

-1,5

x/c VGK inviscid Fluent coarse grid (150 points) C *

VGK viscous Fluent fine grid (600 points)

Figure 21: Comparison of the pressure distribution computed with VGK and with Fluent on the aerofoil designed with an incidence α=1.5

Table 5 sums up the data related to the grid convergence and Richardson extrapolation. We can see that the convergence was well achieved for the lift coefficient and the pitching moment. Curiously this was not the case for drag. It can also be seen from the data computed that the disagreements between Fluent and VGK on the pressure distribution result in huge differences between the integrated data. These differences are not due to the grid refinement because the coarser grid used with Fluent counts the same number of grid points over the aerofoil as the grid used by VGK.

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Chapter 3: Validation

1/ (nbr cells)

CL

CD

Cm

6,7E-03

7,55E-01

7,66E-03

-3,21E-01

3,3E-03

7,66E-01

6,95E-03

-3,26E-01

1,7E-03

7,70E-01

6,79E-03

-3,27E-01

1,32

2,18

1,74

7,72E-01

6,74E-03

-3,28E-01

1,09%

3,71%

0,77%

1,019

0,886

1,019

8,67E-01

4,68E-03

-1,42E-01

order p extrapolated value error band asymptotic range of convergence VGK value

Table 5: Data computed for the supercritical aerofoil with an incident Mach number M=0.725 and an angle of incidence α=1.5°.'nbr of cells ' refers to the number of connectors on the surface of the aerofoil, and the error band is the grid convergence index

To conclude, VGK and Fluent both seem to deal quite well with this problem. However, even if the pressure distributions computed with both programs seem correct as regards of the physics, they exhibit differences that highly affect the integrated data.

3.6. Onera M6 wing

3.6.1. Presentation

The ONERA M6 wing is a classical validation case for external flows because it exhibits very complex transonic flow features despite a very simple geometry. It is a swept untwisted semi span wing based on the ONERA D symmetric section (figure 22).

Data computed with Fluent will be compared with the experimental one, obtained by Schmitt and Charpin (reference 24). These experiments were carried out at a freestream

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Chapter 3: Validation

Mach number M=0.8395 with an incidence α=3.06°. The estimated error introduced during the experiment is 2%.

Figure 22: ONERA M6 wing geometry with the positions of the pressure taps

3.6.2. Computations

Computations on the ONERA M6 wing were achieved using a structured mesh which is presented figure 23. The far field boundary condition is ten times the root chord far from the wing in order to prevent bad interactions between the boundary condition and

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Chapter 3: Validation

the pressure distribution over the wing. The grid over each face of the wing counts 33*73 connectors redistributed in order to have more cells in the tip region where complex flow features appear (tip vortex).

Symmetry

Pressure far field

Wall

Figure 23: Structured grid on the ONERA M6 wing with the boundary conditions

The coupled implicit solver was used with an inviscid model. The operating pressure was set to 101,325Pa and the temperature in the boundary conditions was decided to ensure that the Reynolds number was similar to the one calculated during Schmitt and Charpin experiments (Re=11.72e6). The boundary conditions imposed to run the computations were:

•

symmetry at the root of the wing

•

pressure far field: M=0.8395, T=223.22K

•

wall: α=3.06°

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Chapter 3: Validation

The pressure distribution computed over the wing is presented figure 24. We can clearly identify on this picture the presence of two shock waves, the first one being near the leading edge and the second one being at around 80% of the chord. In the ti p region, the two shocks merge together resulting in a stronger one. A slight decompression is observed after this strong shock. Such a decompression was also noticed while studying the two dimensional supercritical aerofoil.

Figure 24: Pressure distribution computed with Fluent over the upper surface of the ONERA M6 wing for M=0.8395, α=3.06 and Re=11.72e6

The comparison of the experimental and CFD data is displayed figures F2 and F3. We can see quite good agreement between the two sets of data for η= 0.2 to η=0.65. The two shocks observed figure 24 are confirmed on these plots. The CFD calculation predicts a position of the shock slightly downstream in comparison with the experimental data. This is due to the absence of the boundary layer which increases the effective thickness of the aerofoil during the experiments.

At η=0.8, the two shocks are getting too close and strong shock boundary layer interactions appear. The adverse pressure gradient due to the first shock should result in

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Chapter 3: Validation

an increase of the displacement thickness of the boundary layer. The second shock  being very close should be affected by this enhancement of the wing effective thickness. As the complexity of the interactions with the boundary layer is not taken into account by the inviscid model, we can see major differences between the computed and the experimental data. CFD calculations predict the mergence of the two shocks at this station whereas the experiment has shown two shocks that are still well separated. For the same reasons; huge differences can also be seen at η=0.9, where the two shocks are merging in the experiment. Outboard of these critical positions, the accuracy of the predictions with the inviscid model is recovered which is a good news as regards of the subject of the study.

The good agreement between the computed data with the inviscid solver and the experimental data for supercritical flow confirms what was observed by Hicks (reference 10). This also shows an Euler solver should be adapted to carry out a comparison of the performances of a raked wing tip and a winglet.

3.7. Conclusion about validation

The validation hierarchy enables to get confident in the computations that are carried out. An error minor to 2% with the theory has been computed on the compression and expansion corners. Then, good agreement is found between VGK and the computations with Fluent on the supercritical aerofoil designed. The position of the shock is well predicted even if Fluent computed a lower rooftop than VGK. These differences affect the integrated data over the aerofoil such as lift and drag. However, this should not be a great problem for the job we have to achieve as it is a comparison work and the same error will be introduced in the cases studied.

Finally, very good agreement was found in supercritical flow conditions between the computations with the inviscid solver and the experimental data. Only at the junction

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Chapter 3: Validation

between the two shocks some inaccuracies appear in the computations due to the strong shock and boundary layer interaction. This is due to the importance of the viscous effects in this region where they should not be neglected. Such a problem might also appear in the computations that will be carried out during the following study and we should keep aware of this inaccuracy.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet

The objective of this study is to carry out a comparison of aerodynamic performance of  two wing tips. To achieve this comparison, three sets of computations is carried out involving the wing designed clean and with each of the wing tips. The first set of  computations is done near the design Mach number M=0.8. Then, the wing performance was assessed at a lower supercritical Mach number, M=0.75 before testing its low speed performance at M=0.5. All these computations were done using the flow conditions corresponding to the air at 35000feet, a typical cruise altitude for a transport aircraft. Thus, the air temperature and the operating pressure were respectively 218K and 23907 Pa.

The object of these computations is to test various integrated parameters such as lift and drag to achieve an overall comparison of the two wing tips designed. As this will only enable to conclude about the performances of the raked tip and the winglet I designed, a comparison of the pressure distribution around the wing will also be necessary. Indeed, it will enable to have a better understanding of the way these tips act on the flow to increase the performances of the wing.

4.1. Grid generation

The generation of the grid was done on the three wings with several objectives. The first one was to be able to have an accurate resolution of the pressure distribution over the wing. The second objective was to have a grid that does not exceed one million and a half cells in order to be able to run enough simulations in the short period of time given. The last requirement was to ensure the grids generated on the three wings have

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

the same sort of cells distribution in order to be able to achieve a fair comparison of the devices. This was a critical aspect because no grid convergence could be performed to ensure that the solution found is close enough from the asymptotic solution.

Grids used to carry out the computations are fully structured on most of the flow field. An unstructured domain is created on the tip of the wings and extruded till the end of  the flow field in a hybrid mesh. A C-type mesh is then generated in the chordwise direction and an H-type mesh was generated in the spanwise direction. The far field boundary conditions were positioned at three times the wing span to ensure that they do not affect the flow around the wing. A symmetry boundary condition was used at the root of the wing and a wall boundary was used for the wing itself. For all the grids, the cleaned wing counts 121*201 nodes. Both tip devices count 121*81 grid points.

Figures 25, F4 and F5 show the various grids generated with the boundary conditions used. We can clearly see on these pictures that the grids are very similar. The main problem related to these grids is the very low number of cells present away from the wing, especially in the tip region. This was necessary to keep a reasonable number of  cells but should result in some issues related to the computation of the tip vortex. As this problem appears on all meshes, this should not affect much the comparison.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

Pressure far field

Symmetry

Wall

Figure 25: Structured mesh generated over the clean wing

4.2. Near field flow study

One of the main objectives of this study is to have a better understanding of the way a raked tip can improve the performances of an aeroplane. This way we could manage to understand the different parameters that will influence the choice of either a raked tip or a winglet.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

4.2.1. Study of the wings

Computations are run in steady flow over the clean wing or equipped with one of the tips designed. The pressure distribution at various spanwise stations is collected in order to have a better understanding of the physical phenomenon observed. For the interest of this study, more stations were examined near the tip region than on the inner wing. Only few data is selected to be presented in this report as most of the phenomena observed reappear in all computations. I chose the one computed at M=0.8 to be a reference as they exhibit the most complex flow features.

The pressure distributions presented always refer to the clean wing, called wing in the caption, the wing with the raked tip, called raked tip in the caption, and the wing with the winglet, called winglet in the caption. The critical pressure coefficient (Cp*) over the wing is also displayed to get an idea of the flow Mach number over the section and to estimate the strength of the shocks that might appear. The critical pressure coefficient is not constant over a section because of the difference in sweep angles between the leading and the trailing edge.

4.2.1.1. Root section

A very strange pressure distribution is observed at the root section (figure 26). The first thing to remark about this plot is that this distribution is very different from what was seen during the design procedure. There is no near sonic rooftop, the loading at the leading edge equals almost zero and a shock wave appears near the trailing edge. The acceleration of the flow that should appear near the leading edge is underestimated and the flow does not reach the critical pressure before 55% of the chord.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

M=0.8, α=0, y=0 1 0,8 0,6 0,4 0,2 Wing 0 -Cp

-0,2

0

0,2

0,4

0,6

0,8

1

Raked Tip Winglet Cp*

-0,4 -0,6 -0,8 -1 -1,2 x/c

Figure 26: Pressure distribution over the root section (y=0m) at M=0.8 with zero incidence. The graph displays this pressure distribution for the clear wing, the wing with the winglet and the wing with the raked tip.

At first, I thought this phenomenon was due to the symmetry boundary condition which could be inadequate even if this was not encountered on the ONERA M6 wing. Some computations involving the full span wing have shown that this was not the case. As a result, I think this phenomenon could be attributed to the fact that I cut the wing designed at the crank and the flow over the inner part of the wing might affect the pressure distribution over this section.

The wing loading should not be much affected as the lift that is not produced by the acceleration of the flow near the leading edge should be nearly recovered by the very low pressure appearing before. The main issue related to t his inaccuracy is the presence of a shock wave which might affect dramatically the drag produced. However, as we can see on figure 26, the pressure distribution at this station is the same for the three

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

wings and the increase in drag resulting from this problem should not affect the performance of a wing in comparison with another.

4.2.1.2. Outer sections

The problems encountered at the root section damp out very rapidly when moving outwards. It completely disappears at 25% of the semi span and a rooftop is observed outboard of this station. The pressure distribution over the three wings is nearly the same until 87% of the span (y=13m). There, tip effects start to be observed and the tip devices highly influence the nearfield flow.

Figure F6 shows a comparison of the pressure distribution over the three wings at y=14m, one meter inboard from the wing tip. We can clearly see the influence of the tip vortex on the wing. The aerofoil effective incidence is reduced due to the downwash and the resulting lift produced at this station is reduced. The pressure distribution over the wing with a tip device is less affected by this phenomenon and a higher difference of pressure between lower and upper surfaces can be noticed near the trailing edge.

A peak starts to form near the trailing edge of the wing equipped with a winglet, but this can be better seen at y=14.8m (figure 27). This low pressure peak appears due to interactions of the flow over the wing and over the winglet. The flow is accelerated above both of these lifting devices resulting in very high velocities obtained at the   junction. At this station, the lift produced by the wing is highly affected by the tip vortex leading to a decrease of the loading. The raked tip still shows a near sonic rooftop, the tip of the wing mounted with this aerodynamic device being still 1.9 meters far from this station.

At a freestream Mach number M=0.5 (figure F7), we can see that the interferences on the wing equipped with a winglet are less important. There is no shock wave because of 

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

the low speed. However, the same physical phenomena can be observed over the three wings.

M=0.8, α=0, y=14.8 1 0,8 0,6 0,4 Wing 0,2

Raked Tip

-Cp

Winglet

0 -0,2

0

0,2

0,4

0,6

0,8

1

Cp*

-0,4 -0,6 -0,8

x/c

Figure 27: Pressure distribution over the section y=14.8m (the tip of the wing being at y=15m) at M=0.8 with zero incidence. The graph displays this pressure distribution for the wing clean, the wing with the winglet and the wing with the raked tip.

4.2.2. Pressure distribution over the winglet

As it was observed on figure F6, the flow over the wing mounted with a winglet does not exhibit any bad influence of the tip vortex. However, some interactions between the flow over the wing and the one over the winglet appear in the tip region and were discussed earlier. The suction peak appearing at the junction between the wing and the tip device can lead to the formation of a strong shock wave. At zero incidence and with a free stream Mach number M=0.8, the flow over the wing at the junction reaches Mmax=1.28.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

The adverse pressure gradient due to this shock would certainly result in a boundary layer separation. This obviously could not be computed with t he inviscid solver. Due to the boundary layer separation, the drag produced by the wing should increase. As the wing is swept back, the decrease of the loading at the tip would result in a more nose up pitching moment on the aircraft, increasing the size of the detached flow region. Finally, control surfaces such as ailerons that are situated in the tip region would certainly no longer be efficient.

Figure 28 shows the interference effect disappears very fast when moving outboard over the winglet. The winglet then has a more like supercritical behaviour, showing an isentropic recompression after a near sonic rooftop.

M=0.8, α=0 1,2 1 0,8 0,6 0,4 0,2 -Cp 0 -0,2 0 -0,4 -0,6 -0,8 -1

η=0.937 η=0.949

0,2

0,4

0,6

0,8

1

η=0.975 η=0.987 Cp*

x/c

Figure 28: Pressure distribution over the wing mounted with the winglet at various stations in the tip region. η refers to the dimensionless spanwise ordinate: η=y/b (η=0.949 corresponds to the  junction between the wing and the winglet)

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

4.2.3. Pressure distribution over the raked tip

The pressure distribution over the raked tip changes continuously from the junction between the wing and the tip device to the tip. Indeed, because of the continuous change in sweep angles between the wing and the raked tip, the pressure distribution should become gradually like subsonic.

This can be observed on figure 29 where a suction peak appears near the leading edge. Such a feature is usually seen on subsonic aerofoils. This suction peak corresponds to an acceleration of the flow which does not reach a supersonic speed so that no shock  wave forms. Moreover, this should result in a decrease of the drag over the raked tip because it sucks the wing forward. When increasing the incidence at M=0.8, the speed over the leading edge of the raked tip increases, leading to the formation of a shock  wave in this region (figure F8). M=0.8, α=0 0,8 0,6 0,4 η=0.893

0,2

η=0.923

-Cp

η=0.988

0 0

0,2

0,4

0,6

0,8

1

Cp*

-0,2 -0,4 -0,6

x/c

Figure 29: Pressure distribution over the wing mounted with the raked tip at various stations in the tip region. η refers to the dimensionless spanwise ordinate: η=y/b (η=0.893 corresponds to the  junction between the wing and the raked tip)

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

4.3. Far field flow study

Vorticity is an important parameter of the problem which requires further study i n order to have a better understanding of the flow features observed and to validate the data collected. Indeed, the behaviour of the flow downstream of the wing should widely differ when using the various wing tips. Some experimental studies (reference 8, 16, 17) enable to compare the data computed and to assess the ability of an Euler model to deal with such problems. The software Fieldview is used in this part of the research.

4.3.1. Vorticity downstream of the wing clean

Figure 30 shows the contours of vorticity in two planes situated downstream from the wing. Some vorticity is generated all along the span of the wing resulting from the crossflow. Indeed, the flow over the upper surface of the wing tends to go inboard whereas the flow on the lower surface tends to go outboard. When the flows over both of these surfaces are merging at the trailing edge of the wing, a vortex shear layer is generated.

At the tip of the wing, the main tip vortex can be easily observed even if its shape is distorted by the grid. A more interesting feature which was observed by Zuhal during PIV experiments (ref 16) is the presence of a second vortex which is generated down from the main vortex. These two vortices are merging before reaching the second plane.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

Tip vortex

Presence of a second vortex

Figure 30: Vorticity magnitude downstream from the wing for a freestream Mach number, M=0.75 and an incidence α=1°. The first measurement plane was situated at x=10.73m and the second one is at x=15.13m

4.3.2. Vorticity downstream of the winglet Figure 31 shows the vorticity in two planes situated downstream from the wing mounted with a winglet. We can observe on this figure the influence of the winglet. The crossflow has to overcome the winglet before forming the tip vortex at the tip of the winglet. This crossflow over the winglet results in the formation of a vortex shear layer downstream of the winglet.

The flow field is not computed with enough accuracy to draw conclusions about the strength of the vortex. Indeed, the tip vortex which should be generated at the tip of the winglet can hardly be seen on the pictures due to a lack of grid cells.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

Figure 31: Vorticity magnitude downstream from the wing mounted with a winglet for a freestream Mach number, M=0.75 and an incidence α=1°. The first measurement plane was situated at x=11.36m and the second one is at x=15.76m

4.3.3. Vorticity downstream of the raked tip The computation of the vorticity downstream of the wing mounted with a raked tip (figure 32) has enabled to confirm the formation of a second vortex generated at the  junction between the wing and the tip device. Indeed, such a flow feature was observed during the experiments carried out by Gold (reference 8). During these experiments, the presence of this vortex was attributed to a bad junction between the wing and the wing tip where a gap could appear. This is clearly not the case here.

The formation of a secondary vortex inboard of the tip vortex might be due to a difference in the pressure distribution over the raked tip in comparison with the pressure over the wing. Indeed the difference in the gradient of pressure in the spanwise

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

direction due to the change in sweep angles should affect the crossflow. This perturbation would result in instabilities in the shear layer, leading to the formation of a vortex. In figure 32, only the increase in vorticity could be computed downstream of  the junction and the shape of the vortex remains undetermined due to inaccuracies in the space discretisation.

Tip vortex

Second vortex

Figure 32: Vorticity magnitude downstream from the wing mounted with a raked tip for a freestream Mach number, M=0.75 and an incidence, α=1°. The first measurement plane was situated at x=11.47m and the second one is at x=15.86m

4.3.4. Conclusion about the far field flow study The far field flow study carried out enabled to exhibit a problem related to the grid generation. Indeed, the size of the mesh does not enable to compute accurately the vortex and the contours of the cells distort the actual shape of the vortices. However, some important flow features that appeared during some experiments could be observed

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

on the various figures. This is interesting as regards of a possibility to use an Euler solver with a finer mesh to carry out a more accurate comparison of the two wing tips.

4.4. Comparison of aerodynamic performance The comparison of the performance will be done gradually. We will compare the lift characteristics at the different speeds, then the drag characteristics before going on with the lift over drag ratio which can be seen as the wing efficiency at a given speed. The area used to compute the normalised data (C L, CD…) was the projected area: •

AWing=56.63m²

•

ARaked tip=58.60m²

•

AWinglet=57.51m²

4.4.1. Lift characteristic

Figure F9 displays a comparison of the lift characteristics for the different wings at various freestream Mach numbers. The first thing to observe on these plots is that the clean wing produces less lift than a wing equipped with any of the tip device. This must be due to the reduction of the influence of the tip vortex. Indeed, as it is explained in the first chapter, the tip vortex induces downwash all over the wing. This downward velocity component reduces the wing effective incidence which produces less l ift.

This assumption is confirmed by the study of the lift curve slope done in table 6. Indeed, we can see that the lift curve slope for the clean wing is lower than for the wing with a tip device. When increasing the incidence, lift increases but the strength of the tip vortex increases as well because it is directly related to the difference of pressure

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

between the upper and lower surfaces of the wing. Thus, downwash should be more important and the lift curve slope should be reduced. A higher lift curve slope computed for the wing with a raked tip or a winglet shows that both tip devices reduce the influence of the tip vortex. As regards of the comparison between the raked wing tip and the winglet, no significant difference is observed even if the raked tip seems to produce slightly more lift.

Lift curve slope M

0.5

0.75

0.8

Wing

0.0919

0.1099

0.121

Raked tip

0.0948

0.1141

0.126

Winglet

0.0946

0.1138

0.1256

Table 6: Comparison of the lift curve slopes of the various wings at different freestream Mach number

4.4.2. Drag characteristic

Figure F10 displays the evolution of drag coefficient against lift at different Mach numbers for the three wings. There, some clear differences can be observed.

According to the data computed, the wing mounted with a raked tip exhibits higher drag reductions than the one mounted with a winglet. The drag reduction due to the addition of the winglet seems highly dependant on the lift produced by the wing and it shows very low efficiency in low lift configuration (M=0.5, CL ≈0.28). However, the drag reduction increases when increasing the incidence. The maximum reduction in drag obtained with the winglet was around 3% whereas a drag reduction of 5% was usually obtained with the raked tip. The drag reduction due to the addition of the raked tip shows very low dependence on the wing incidence.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

The expression of the drag coefficient related to the lift coefficient enables to obtain an estimation of the efficiency of the wing. Indeed, we have:

C  D = C  D 0 +

K * C  L2 π  

* AR

Thus, studying the variations of  C  D * π   * AR with CL² should give us straight lines. The slopes of these curves will be the parameter K. Such graphs are plotted figure F11. We can see that the parameter K is most of the time below 1 which should not be possible. This reveals a problem related to the computations carried out. Indeed, as it was said before, the tip vortex is not well computed because the grid gets very coarse when moving away from the wing. As the tip vortex induces downwash over the wing span, the lift induced drag is under estimated by the calculations done. This would explain the strange slopes of the curves plotted figure F11, exhibiting higher efficiency with the wing designed than with a wing featuring an elliptic loading.

4.4.3. Wing efficiency

The study of the ratio L/D at a constant Mach number corresponds to a study of the wing aerodynamic efficiency. Indeed the range of an aircraft can be computed with the Breguet range equation presented below:  L a 1  m0    (R is the range, a  is the speed of sound, g is the gravity, sfc  is  R = M  ln  D g sfc  mb  

the specific fuel consumption, m 0  is the mass at take off and m b  is the final mass)

As a result, to increase range, we have to: •

maximise ML/D

•

minimise the specific fuel consumption

•

maximise m0 /mb

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

From an aerodynamicist point of view, to increase an aircraft range, we have to maximise ML/D.

Figure F12 presents the comparison of the lift over drag ratio at various freestream Mach numbers. These plots confirm what was said previously about the efficiency of  the wing mounted with a winglet. Indeed, in low lift configuration, the winglet does not show good performances, being even less efficient than the clean wing at M=0.5. When increasing lift, the efficiency of the winglet raises. This phenomenon seems completely logical when we know the way a winglet works. Indeed, the winglet uses the distorted flow in the tip region to produce lift with a thrust component. In low lift configuration, the tip vortex is weak and does not distort the flow enough to produce this thrust component. When raising the incidence, the difference of pressure between the upper and the lower surfaces of the wing increases, resulting in a stronger tip vortex. Thus, the effective incidence of the winglet will be modified resulting in better performance.

The effect of the raked tip on the aircraft performances seems to be independent from the wing incidence. This suggests that the increase in performance related to the use of  the raked tip is not related to the strength of the tip vortex. This will require further analysis for confirmation.

4.4.4. Loading distribution over the wings

To have a better understanding of the way the tip devices work, a comparison of the distribution of the loading over the wing is necessary (figure 33). We can clearly see on this plot that the addition of the winglet results in high loading at the junction between the wing and the winglet. When going outboard, over the winglet, the loading collapses very fast in comparison with the loading over the wing. Elsewhere, the wing mounted with a winglet does not show much difference in comparison with the clean wing.

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

In contrast, the raked tip exhibits a very different loading distribution. Indeed the curve does not match at all the ideal elliptic loading. The distribution of the loading is much more like triangular which is interesting from a structural point of view. In fact, there is a complete redistribution of the loading over the wing. Loading at M=0.75, α=0

0,5

0,4

0,3

Wing Raked Tip

Loading

Winglet

0,2

Elliptic loading 0,1

0 0

0,2

0,4

0,6

0,8

1

η

Figure 33: Loading distribution over the wings and comparison with an ideal elliptic loading

The same sort of graph was plotted concerning the drag distribution over the wings (figure 34). It can be seen on this graph that the wing mounted with a raked tip produces thrust all over the tip device. The subsonic peak observed figure 29 is clearly responsible for the increase in efficiency of this wing configuration. As a result, the raked wing tip does not seem to use the tip vortex to raise the performances of the aircraft.

However, the lack of performance of the wing equipped with the winglet can clearly be seen on this graph. Indeed, thrust is produced by the winglet only near the junction with the wing which is clearly not satisfactory. This is due either to a bad design of the winglet or to a bad computation of the tip vortex. Because of the short time available

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Chapter 4: Comparison of aerodynamic performance of a raked wing tip and a winglet 

for this study, I was not able to test these hypotheses. However, the study of the vorticity carried out previously (part 4.3.) clearly exhibited the lack of cells in the tip region.

Drag distribution at M=0.75, α=0 0,12 0,1 0,08 0,06

Wing

Drag 0,04 Loading

Raked Tip Winglet

0,02 0 -0,02

0

0,2

0,4

0,6

-0,04 η

Figure 34: Pressure drag distribution over the wings

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0,8

1

1,2

Conclusions and recommendations

A CFD comparison of the aerodynamic performance of two wing tips, a winglet and a raked tip, has been conducted. The first step of this study was the design of the wing and of the tip devices which was achieved to operate at a tr ansonic velocity (M=0.8).

The comparison was done in several parts. A near field flow study has shown the presence of interferences, which could lead to the formation of a strong shock wave at the junction between the wing and the winglet. A suction peak at the leading edge of  the raked tip was observed, resulting in the production of thrust at the stations concerned.

A far field flow study exhibited the formation of a second vortex at the junction between the raked tip and the wing. This phenomenon must be due to a difference in the pressure characteristic over these two devices, resulting in instabilities in the vortex sheet. On the winglet, the flow rolls up over the tip device leading to the spreading of  the tip vortex.

The comparison of the performances has shown great reduction in drag with the addition of the raked tip. The winglet has exhibited fewer advantages during this study, certainly related to the presence of a coarse mesh in the tip region resulting in a bad estimation of the strength of the vortex. Besides, the increase in performances related to the use of the winglet is highly dependant on the lift produced by the wing. Finally, the distribution of the loading over the wing mounted with the raked tip is more like triangular than elliptic, what is interesting as regards of the structure of the wing.

This study shows the advantages of using a raked tip on a wing. However, this study is clearly not complete because some inaccuracies have been introduced in the computations. First of all, the mesh should be refined in the tip region so as to be able to compute the tip vortex accurately and to carry out a more fair comparison of the two

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tip devices. Then, the inner part of the wing should be added so as to see whether the bad data computed at the root disappear. The far field flow study has shown that the use of turbulence model was not the priority in this study and the Euler solver seems well adapted to deal with this sort of problems.

To end with, some problems were encountered when doing the swept wing design. It seems that not enough Weber stations are computed by the Sweptdes program and one should be aware of that when creating the database for a CFD purpose. This can be easily modified in the input data of the program, changing the default value (16 Weber stations) in 32 stations should be enough.

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References

1. E.L. Houghton, P.W. Carpenter (2003). Aerodynamics for engineering students (fifth edition) 2. Thomas A.S.W. (1985). “Aircraft viscous drag reduction” Lockheed Horizons. October 1985, Issue 19, pp. 22-32 3. P. Vela Orge (1997), Study of the wing-winglet junction flow, MSc thesis supervised by Professor A.J. Bocci (Cranfield University) 4. S.A. Kravchenko (1995), Wing tip lifting surfaces: aerodynamic design and comparative analysis, AIAA 95-3909 5. Richard T. Whitcomb (1976), A design approach and selected wind-tunnel results at high subsonic speeds for wing-tip mounted winglets, NASA Technical Note D-8260 6. Harry H. Heyson, Gregory D. Riebe, Cynthia L. Fulton (1977), Theoretical parametric study of the relative advantages of winglets and wing-tip extensions, NASA Technical Paper 1020 7. A.J. Bocci (1981), Wing W4 fitted with a winglet: winglet design and preliminary test results, Aircraft Research Association, ARA/M-230 8. N. Gold, K.D. Visser (2002), Aerodynamic effects of local dihedral on a raked wingtip, AIAA-2002-0831 9. N. Jong Yu, Hai-Chow Chen, Allen W. Chen and K. Robyn Wittenberg (1988), Grid generation and flow analyses for wing/body/winglet configurations, AIAA-88-2548 10. R.M. Hicks, S.E. Cliff, J.E. Melton, R.G. Langhi, A.M. Goodsell, D.D. Robertson, S.A. Moyers (1989), Euler and Potential computational results for selected aerodynamic configurations, from the book “Applied computational aerodynamics” (volume 125) 11. John C. Vassberg, K.C. Chang, Gary S.Wang and Kin Yu (1990), An Euler method for wing-body-winglet flows, AIAA 90-0436

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12. P.M.H.W. Vijgen, C.P. Van Dam, B.J. Holmes (March 1989), Sheared Wingtip aerodynamics: wind-tunnel and computational investigation, Journal of  aircraft 1989, 0021-8669 vol.26 no.3 (207-213) 13. L.V. Schmidt, R.W. Duren (2002), Wing induced drag, AIAA 2002-4878 14. P. Vlachos, D. Telionis (2003), Wing-tip-to-wing-tip aerodynamic interference, AIAA 2003-0609 15. E.A. Anderson, C.T. Wright, T.A. Lawton (2000), Experimental study of the structure of the wingtip vortex 16. L. Zuhal, M. Gharib (2001), Near Field dynamics of wing tip vortices, AIAA 2001-2710 17. F. De Souza, D. Faghani (2001), Near Field dynamics of wing tip vortex measurements via PIV, AIAA 2001-2451 18. J. Szafruga, B.R. Ramaprian (1994), Pressure measurements over the tip region of a rectangular wing – Part 1 stationary wing, AIAA-94-1948-CP 19. T.M. Chen, J. Katz (2004), Induced drag of high aspect ratio wings, AIAA 2004-38 20. M. Smith (2005), Drag reduction through large winglet technology, paper presented at the symposium: Aerospace 2005 21. K. Kubrynski (2003), Wing-winglet design methodology for low speed applications, AIAA 2003-215 22. C.L. Ruhlin, K.G. Bhatia, K.S. Nagaraja, Effects of winglet on transonic flutter characteristics of a cantilevered twin-engine transport wing model, NASA TP 2627 23. S.A. Kravchenko (1996), The application of the wing tip lifting surfaces for practical aerodynamic, ICAS-96-4.6.4, paper presented at: 20th International Council of the Aeronautical Sciences Congress 24. Schmitt, V. and F. Charpin, Pressure Distributions on the ONERA-M6-Wing at Transonic Mach Numbers,  Experimental Data Base for Computer Program  Assessment. Report of the Fluid Dynamics Panel Working Group 04, AGARD

AR 138, May 1979.

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Appendices

Appendix A:SectionD program

To design the aerofoil we use SectionD. It provides a computation of pressure distribution around the aerofoil using Weber theory. This method enables to work in design mode which is very important as regards of our objectives. We can specify a pressure distribution on the surface of the aerofoil and the program computes the aerofoil geometry and an angle of attack. Calculations of thickness and camber/incidence are run separately. Compressible effects are implemented in this method by using the compressibility factor B = 1 − M 2 (1 − Cpi ) .

SectionD requires the setting of different parameters. We first implement the Mach number and an angle of attack which does not seem to be taken into account. Then we have to choose the Reynolds number and define the position of the transition on each side of the aerofoil. Here we took transition starting at 5% of the chord for both surfaces.

Two parameters have to be guessed for the first step of the aerofoil design: the maximum thickness to chord ratio and the thickness form (RAE number). In order to optimize the aerofoil geometry, these parameters will be modified as the design procedure goes on. Changing the maximum thickness of the aerofoil will affect the minimum pressure coefficient. Increasing the thickness will result in a decrease of the minimum pressure coefficient. The RAE number corresponds to a description of the position of the maximum thickness over the aerofoil. For the choice of RAE number available on SectionD (from 101 to 104), increasing the RAE number will result in moving backward the maximum thickness.

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Thus, increasing the thickness of the aerofoil will increase the loading on it and changing the RAE number will shift the loading forward or backward. Running SectionD after having set all these parameters will give an evaluation of the total pressure distribution on the aerofoil. If the minimum pressure coefficient is close to the critical pressure coefficient, then we can try to set a Cp distribution on the aerofoil and run the computation. The shape of the aerofoil is calculated with the pressure distribution and other aerodynamic parameters. Several iterations of this procedure may be required before getting to an optimized design.

This method is straightforward and enables to improve directly the design of the aerofoil. What is more, procedure is extremely robust and usually converges within few iterations for a realistic choice of target pressures. However, if pressure distribution around the aerofoil is reasonably well predicted over subcritical regions, calculations in supercritical regions are incorrect. Besides, some inaccuracies are presents for thick  aerofoils (t/c>10%), especially at the leading edge.

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Appendix B: VGK

The assessment of the performance of the supercritical aerofoil designed was done using VGK. It uses a full-potential method that provides very accurate predictions except in the region of a boundary layer separation. The program carries out the computations accounting for the effects of the boundary layer and the wake. The inviscid solution is calculated using the equivalent source method of Lighthill to take the viscous layers into account. The viscous solution is computed by defining an “equivalent inviscid flow” which behaves in the same way as the real flow out from the boundary layer. Then, calculations within the boundary layer or the wake are done analytically by applying appropriate boundary conditions. For the aerofoil, this boundary condition is a certain distribution of sources giving the mass deficit of the boundary layer. For the wake it is a jump in normal velocity.

The VGK program carries out the computations using an O-grid. A first solution is calculated using a coarse grid and is taken as a basis for calculations on a fine grid. In each case, 10 iterations are carried out with an inviscid model before calculations of the viscous layers and of the wake. The viscous layer solution is then taken to set the boundary conditions for the next inviscid calculation. The procedure is repeated every 5-10 iterations.

To run calculations with VGK, we have to implement aerofoil geometry and to set different flow parameters: Mach number, angle of incidence, Reynolds number, location of the transition on the upper and lower surfaces. A set of aerodynamic characteristics is computed and presented on a graphical interface: pressure distribution, lift coefficient, pitching moment, total drag, viscous drag, shape factor at the trailing edge and residuals. An example of the data presented on the graphical interface of VGK is given figure F1. More precisions about the data computed and the convergence of the calculations are available in output files. Thus, we can find important data about the

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boundary layer development and the speed of the flow at different stations over the aerofoil. Study of these files is necessary to have a better understanding of the flow features around the aerofoil. Indeed, the strength of the shock can be evaluated by looking at the Mach number before the shock and separation of the flow can be predicted by studying the shape factor.

This program is easy to use and enables to get relevant data very rapidly. However, the strength of the shock seems to be a limiting parameter as VGK cannot compute accurately in the regions of a separation.

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Appendix C: Convert program

The Convert program enables to compute a 3D streamwise section from a 2D aerofoil section. This conversion is necessary to take account of the sweep angles. The input file contains the following information:

Line 1:

Line 2 to 32:

LAM1

leading edge sweep

LAM2

trailing edge sweep

XM

chordwise position for mean sweep

POWER

“n” value

x

chordwise position

zs

camber coordinate

Cpu

upper surface pressure coefficient

Cpl

lower surface pressure coefficient

zu

upper surface ordinate

zl

lower surface ordinate

Information from line 2 to 32 is contained in the output file of the SectionD program. The power factor “n” has to be chosen by the user and is a number from 1 to 2. A default value of 1.5 can be used for the first run.

The conversion of the 2-dimensional aerofoil ordinates (z/c 2D) to the 3-dimensional ordinates (z/c3D) depends on the mean ( Λmean) and local sweep angles ( Λlocal) and a power factor (n) that has to be determined by the user. The formula is valid for both upper and lower surfaces and is given by:

n +1

( cos Λ mean ) ( z c ) 3 D = ( z c ) 2 D ⋅ n ( cos Λ local )

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The local geometric sweep ( Λlocal) between the leading and trailing edges can be determined from:

tan Λ local = (1 − x c ) ⋅ tan Λ LE + x c ⋅ tan Λ TE

 

The mean sweep ( Λmean) is the local geometric sweep at half chord position (x/c=0.5). The value for the power “n” lies between 1.0 ≤n≤2.0, where n=1.5 is a good starting value, but if the pressure distribution on the outer wing calculated from this n-value is not satisfactory (equivalent to 2D distribution) the n-value has to be changed and the calculation repeated until adequate outer wing pressure distribution is given. The output file of the CONVERT program contains the upper (zu) and lower surface aerofoil ordinates (zl) for the section geometry to be used for the outer wing. The ordinate values are given for 15 standard chordwise positions (i.e. standard Weber stations) which can be calculated from the formula:

x c=

1 2

(1 + cos )  jπ   16

with

j = 1...15

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Appendix D: Sweptdes

The SWEPTDES program calculates the pressure distribution over wing spanwise section from the streamwise coordinates of this section. It also can work in design mode, computing the streamwise section coordinates from a pressure distribution. As for the Convert program, the stations computed correspond to the 15 Weber stations. This program can give quite good data for a freestream Mach number below M=0.87.

The input file for the SWEPTDES program is more complex than for the CONVERT program. It is divided in three parts corresponding to t hree different sets of data.

The first part of the input file contains the fixed data enabling to set the wing geometry. Sweep angles, cranks, leading edge extensions can be defined in this part of the file.

The second part of the Sweptdes input file sets data for the thickness calculations. Only one station of the wing can be studied at a time and thickness design has to be undertaken before trying to carry out camber calculations. As it was mentioned previously, the program can work either in design or in forward mode, both of these calculations will require different sets of input data. The thickness data, either the coordinates of the aerofoil thickness or the thickness pressure distribution depending on the mode chosen, must be given at the 15 Weber stations.

The camber data is specified for the same spanwise position as the thickness data. Usually the thickness design is carried out first, without considering the camber. Therefore and option “switching off” the camber calculation can be chosen. When the thickness design is completed the camber design is carried out. The program allows selecting between the option for very thin section geometries or normal sections (which will be used here). An “extrapolation”-option may be chosen to give smooth camber at the leading edge. With this option the leading edge point is relocated on the x-axis by

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shearing about the trailing edge and adjusting the effective incidence ( αe). The effective incidence corresponds to the local geometric sweep subtracting the i nduced incidence.

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Appendix E: Downwash program

The DOWNWASH program is used to calculate the downwash induced by the trailing edge vorticity at eleven spanwise positions across a wing of given planform geometry and spanwise loading. To give the specified loading the geometric incidence of the wing at each spanwise station has to be determined by adding the induced incidence to the effective incidence given in the SWEPTDES output. The incidence varying for the different spanwise locations gives the twist of the wing. The additional incidence results in a rotation of the lift vector producing induced drag C Di. In the input data for the DOWNWASH program values for the streamwise lift coefficient at a sufficient number of spanwise stations is provided to ensure a smooth variation of spanwise loading. Near the tip the loading variation is largest and therefore requires more close spacing of points, for which CL L is defined. The values for CL L are obtained by reading them of a graph that can be plotted from the data calculated in SWEPTDES for the root, intermediate and outer wing sections. The DOWNWASH program is based on a simple integration algorithm but results outboard of 90% spanwise postion (i.e. close to the tip) should be ignored because they show spurious behaviour. It is also worthwhile to check  the output values of induced incidence when the twist is calculated because they may require smoothing.

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Appendix F: Figures

Critical Cp

Viscous calculations

Inviscid calculations

Aerofoil

Figure F 1: Pressure distribution computed with VGK along the aerofoil designed at M=0.725 and α=0

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η=0.2

η=0.44

1,5

1,5

1

1 Experiment

0,5

CFD

-Cp

0,5 Experiment

-Cp

CFD 0

0 0

0,2

0,4

0,6

0,8

1

0

1,2

0,2

0,4

0,6

0,8

1

1,2

-0,5

-0,5

-1

-1

x/c

x/c

η=0.8

η=0.65

1,5

1,5

1

1

0,5

0,5 Experiment

-Cp

CFD

0 0

0,2

0,4

0,6

0,8

1

Experiment

-Cp

CFD

0 0

1,2

0,2

0,4

0,6

-0,5

-0,5

-1

-1

x/c

x/c

Figure F 2: Comparison of experimental and CFD data on pressure distribution over different spanwise sections

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0,8

1

1,2

η=0.9

η=0.95

1,5

1,5

1

1

0,5

Experiment

-Cp

CFD

0 0

0,2

0,4

0,6

0,8

1

0,5 Experiment

-Cp

CFD

0

1,2

0

0,2

0,4

0,6

-0,5 -0,5 -1 -1

x/c

x/c

η=0.99 1,50E+00 1,00E+00

-Cp

5,00E-01

Experiment CFD

0,00E+00 0

0,2

0,4

0,6

0,8

1

1,2

-5,00E-01 -1,00E+00

x/c

Figure F 3 : Comparison of experimental and CFD data on pressure distribution over different spanwise sections

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0,8

1

1,2

Pressure far field

Symmetry

Wall

Figure F 4: Structured mesh generated over the wing with the raked tip

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Pressure far field

Symmetry

Wall

Figure F 5: Structured mesh generated over the wing with the winglet

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M=0.8, α=0, y=14 0,8 0,6 0,4 0,2 Wing 0

-Cp -0,2

Raked Tip 0

0,2

0,4

0,6

0,8

1

Winglet Cp*

-0,4 -0,6 -0,8 -1

x/c

Figure F 6: Pressure distribution over the section y=14m (the tip of the wing being at y=15m) at M=0.8 with zero incidence. The graph displays this pressure distribution for the wing clean, the wing with the winglet and the wing with the raked tip.

M=0.5, α=0, y=14.8 0,6 0,4 0,2 0

-Cp

Wing 0

0,2

0,4

0,6

-0,2

0,8

1

Raked Tip Winglet

-0,4 -0,6 -0,8

x/c

Figure F 7: Pressure distribution over the section y=14.8m (the tip of the wing being at y=15m) at M=0.5 with zero incidence. The graph displays this pressure distribution for the wing clean, the wing with the winglet and the wing with the raked tip

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M=0.8, α=1 1,4 1,2 1 0,8 η=0.893

0,6

η=0.923

-Cp 0,4

η=0.988

0,2

Cp*

0 -0,2 0

0,2

0,4

0,6

0,8

1

-0,4 -0,6 x/c

Figure F 8: Pressure distribution over the wing mounted with the raked tip at various stations in the tip region. η refers to the dimensionless spanwise ordinate: η=y/b (η=0.893 corresponds to the  junction between the wing and the raked tip)

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Lift coefficient for M=0,5 5,00E-01 4,50E-01 4,00E-01 CL

3,50E-01 3,00E-01 2,50E-01 0,0

0,5

1,0

1,5

2,0

2,5

Incidence

Lift coefficient for M=0,75 6,00E-01 5,50E-01 5,00E-01 4,50E-01 CL

4,00E-01 3,50E-01 3,00E-01 2,50E-01 0,0

0,5

1,0

1,5

2,0

2,5

2,0

2,5

Incidence

Lift coefficient for M=0,8 7,50E-01 6,50E-01 5,50E-01 CL

4,50E-01 3,50E-01 2,50E-01 0,0

0,5

1,0

1,5

Incidence Wing 2nd order

Raked tip 2nd order

Winglet 2nd order

Figure F 9: Comparison of the lift characteristics of the various wings at different free stream Mach number

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Drag coefficient for M=0,5 2,50E-02 2,00E-02 1,50E-02 CD

1,00E-02 5,00E-03 0,00E+00 2,50E-01

3,00E-01

3,50E-01

4,00E-01

4,50E-01

5,00E-01

CL

Drag coefficient for M=0,75

CD

4,00E-02 3,50E-02 3,00E-02 2,50E-02 2,00E-02 1,50E-02 1,00E-02 5,00E-03 0,00E+00 2,50E01

3,00E01

3,50E01

4,00E01

4,50E01

5,00E01

5,50E01

6,00E01

CL

Drag coefficient for M=0,8

CD

5,00E-02 4,00E-02 3,00E-02 2,00E-02 1,00E-02 0,00E+00 2,50E-01

3,50E-01

4,50E-01

5,50E-01

6,50E-01

7,50E-01

CL Wing

Raked tip

Winglet

Polynomial (Wing)

Figure F 10: Comparison of the drag characteristics of the various wings at different free stream Mach number. A trendline was added to have an idea of the drag produced by the wing at each station and to compare it with the drag produced by the wing mounted with a tip device.

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M=0.5 3,00E-01

2,50E-01 y = 0,7388x + 0,1108

CD*Π*AR

y = 0,6828x + 0,1101

2,00E-01 y = 0,6895x + 0,095 1,50E-01 5,0E-02

1,0E-01

1,5E-01

2,0E-01

2,5E-01

CL²

M=0.75 5,50E-01

4,50E-01

y = 0,8797x + 0,1641 y = 0,819x + 0,1624

CD*Π*AR 3,50E-01

2,50E-01

1,50E-01 5,0E-02

y = 0,8104x + 0,1427

1,0E-01

1,5E-01

2,0E-01

2,5E-01

3,0E-01

3,5E-01

CL²

M=0.8 6,50E-01 5,50E-01

y = 1,0477x + 0,1809 y = 0,9761x + 0,1795

4,50E-01 CD*Π*AR 3,50E-01

y = 0,9513x + 0,1596 2,50E-01 1,50E-01 5,0E-02 1,0E-01 1,5E-01 2,0E-01 2,5E-01 3,0E-01 3,5E-01 4,0E-01 4,5E-01 CL² Wing

Raked tip

Winglet

Figure F 11: Plots presenting the variation of C D*Π*AR against CL². The slopes of the curves plotted corresponds to the parameter K appearing in the expression of the drag

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