Cranfield University
Cranfield University
Guillaume MARTIN
Guillaume MARTIN
COMPARISON OF AERODYNAMIC PERFORMANCE OF
COMPARISON OF AERODYNAMIC PERFORMANCE OF
RAKED WING TIPS AND LARGE WINGLETS
RAKED WING TIPS AND LARGE WINGLETS
School of engineering
School of engineering
MSc thesis
MSc thesis
Cranfield University
Cranfield University
School of engineering
School of engineering
MSc thesis
MSc thesis
Academic Year 2005-2006
Academic Year 2005-2006
G. MARTIN
G. MARTIN
COMPARISON OF AERODYNAMIC PERFORMANCE OF
COMPARISON OF AERODYNAMIC PERFORMANCE OF
RAKED WING TIPS AND LARGE WINGLETS
RAKED WING TIPS AND LARGE WINGLETS
Supervisor: Dr S.T. Shaw
Supervisor: Dr S.T. Shaw
September 2006
September 2006
This thesis is submitted in
This thesis is submitted in partial fulfilment of the requirements
partial fulfilment of the requirements
for the degree of Master of Science
for the degree of Master of Science
© Cranfield University 2006. All rights reserved. No part of this
© Cranfield University 2006. All rights reserved. No part of this publication may bepublication may be reproduced without the written permission of the copyright owner.
Abstract
Abstract
A Computational Fluid Dynamics (CFD) study of the aerodynamic performance of a 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 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
designed to operate at a freestream Mach number M≈≈0.8. The winglet design was0.8. The winglet design was
achieved according to the guidance from Whitcomb (reference 5) and the raked tip 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.
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 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 validation hierarchy enabled to attest the ability of this model to compute problems involving a compressible flow over a
involving a compressible flow over a three dimensional lifting device.three dimensional lifting device.
The comparison of the performance was achieved by studying the wings at three 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 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 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 junction between the wing and the winglet. A high suction peak was also observed at
the leading edge of the raked tip. the leading edge of the raked tip.
The comparison of the integrated data has shown a very high increase in performance 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 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 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. 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 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
junction between the wing and the raked tip. This might be related to the sudden changemight be related to the sudden change in sweep angles at this station.
in sweep angles at this station.
Results obtained enabled to have a better understanding of the way both wing tips are Results obtained enabled to have a better understanding of the way both wing tips are
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.
Contents
Contents
Abstract
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Acknowledgements
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Contents
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Figures
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Notations
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Introduction
Introduction ...
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Chapter 1: Literature
Chapter 1: Literature review
review ...
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1.1.
1.1. Aircraft Aircraft performancesperformances ... 22 1.2.
1.2. Wing tip Wing tip vorticesvortices ... 33 1.3. Induced drag
1.3. Induced drag ... 44 1.4. Wing tip devices
1.4. Wing tip devices... 66 1.4.1. Winglets
1.4.1. Winglets... 66 1.4.1.1. Winglet effects...
1.4.1.1. Winglet effects... 66 1.4.1.2. Winglet de
1.4.1.2. Winglet designsign ... 88 1.4.2. Raked wing
1.4.2. Raked wing tips...tips... 99 1.5. Computational fluid
1.5. Computational fluid dynamics simulation...dynamics simulation... 1111 1.5.1. Grid generation...
1.5.1. Grid generation... 1111 1.5.2. Euler
1.5.2. Euler equationsequations ... 1212 1.6. Aim
1.6. Aim of this of this researchresearch ... 1212
Chapter 2
Chapter 2: Wing
: Wing design
design ...
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2.1. Supercritical aerofoil design...
2.1.2. Performances of
2.1.2. Performances of the aerofoil designed...the aerofoil designed... 1616 2.2. Swept
2.2. Swept wing designwing design ... 2020 2.2.1. Design of the
2.2.1. Design of the outer wing section...outer wing section... 2121 2.2.2. Design of the root section (y=0,
2.2.2. Design of the root section (y=0, ηηηηηηηη=0)=0) ... 2222 2.2.3. Design of the intermediate section (y=4.5,
2.2.3. Design of the intermediate section (y=4.5, ηηηηη=0.1875)...ηηη=0.1875)... 2424 2.2.4. Spanwise loading and
2.2.4. Spanwise loading and twist...twist... 2525 2.2.5. C
2.2.5. Conclusions about onclusions about the mthe methodethod ... 2727 2.3. Design of the
2.3. Design of the wing tipswing tips ... 2828 2.3.1. Design of
2.3.1. Design of the raked the raked tiptip ... 2828 2.3.2. Design
2.3.2. Design of the of the wingletwinglet ... 2828
Chapter 3:
Chapter 3: Validation
Validation ...
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3.1. The
3.1. The choice of Euler choice of Euler equationsequations ... 3030 3.2.
3.2. Validation hierarchyValidation hierarchy ... 3131 3.3.
3.3. Richardson Richardson extrapolationextrapolation ... 3232 3.4.
3.4. Compression Compression and and expansion cornerexpansion corner ... 3434 3.5.
3.5. 2D 2D supercritical aerofoilsupercritical aerofoil ... 3636 3.6. Onera M6 wing 3.6. Onera M6 wing... 3838 3.6.1. Presentation... 3.6.1. Presentation... 3838 3.6.2. Computations... 3.6.2. Computations... 3939 3.7. Conclusion about
3.7. Conclusion about validationvalidation ... 4242
Chapter 4: Comparison of aerodynamic performance of a raked wing
Chapter 4: Comparison of aerodynamic performance of a raked wing
tip
tip and
and a
a winglet...
winglet...
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4.1.
4.1. Grid Grid generationgeneration ... 4444 4.2. Near
4.2. Near field flow studyfield flow study ... 4646 4.2.1. Study of the wings...
4.2.1. Study of the wings... 4747 4.2.1.1. Ro
4.2.1.1. Root sectionot section ... 4747 4.2.1.2.
4.2.1.2. Outer Outer sectionssections ... 4949 4.2.2. Pressure distribution over the
4.2.2. Pressure distribution over the winglet...winglet... 5050 4.2.3. P
4.2.3. Pressure distribution ressure distribution over the over the raked tipraked tip ... 5252 4.3. Far
4.3. Far field flow studyfield flow study ... 5353 4.3.1. Vorticity downstream
4.3.1. Vorticity downstream of the wing of the wing cleanclean... 5353 4.3.2. Vorticity downstream of the
4.3.2. Vorticity downstream of the winglet...winglet... 5454 4.3.3. Vorticity
4.3.3. Vorticity downstream of downstream of the raked the raked tiptip ... 5555 4.3.4. Conclusion
4.4.
4.4. Comparison of Comparison of aerodynamic performanceaerodynamic performance ... 5757 4.4.1.
4.4.1. Lift Lift characteristiccharacteristic ... 5757 4.4.2.
4.4.2. Drag Drag characteristiccharacteristic ... 5858 4.4.3. Wing efficiency...
4.4.3. Wing efficiency... 5959 4.4.4. Loading distribution
4.4.4. Loading distribution over the over the wingswings ... 6060
Conclusions and recommendations
Conclusions and recommendations ...
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References
References ...
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Appendices
Appendices ...
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Appendix A:
Appendix A: SectionD programSectionD program ... 6767 Appendix B:
Appendix B: VGK ...VGK... 6969 Appendix C:
Appendix C: Convert programConvert program ... 7171 Appendix D:
Appendix D: Sweptdes...Sweptdes... 7373 Appendix E:
Appendix E: Downwash programDownwash program... 7575 Appendix F:
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; αααα = incidence; εεεε = downwash angle = w/V; αααα∞ = (αααα-εεεε) = equivalent
two-dimensional incidence; L∞ = two-dimensional lift; L = wing lift; Dv =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 αααα=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
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 ααα=1.5...37α 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, ααα=3.06 and Re=11.72e6...41α 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. ηηηη refers to the dimensionless spanwise ordinate: ηηηη=y/b (ηηη=0.949 corresponds to the junction between the wing andη the winglet) ...51 Figure 29: Pressure distribution over the wing mounted with the raked tip at various stations in the tip region. ηηηη refers to the dimensionless spanwise ordinate: ηηηη=y/b (ηηη=0.893 corresponds to the junction between the wing andη the raked tip) ...52 Figure 30: Vorticity magnitude downstream from the wing for a freestream Mach number, M=0.75 and an incidence ααα=1°. The first measurement planeα 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 αααα=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, αααα=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
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 ααα=0 ...76α 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. ηηηη refers to the dimensionless spanwise ordinate: ηηηη=y/b (ηηη=0.893 corresponds to the junction between the wing andη 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*ΠΠΠΠ*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
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, µµµµ 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, θθθ refers to the angle of the wave withθ 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 ααα=1.5°.'nbr of cells ' refers to theα 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
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
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
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.
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.
Chapter 1: Literature review
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).
Drag breakdown Vortex drag 40% Others 20% Profile drag 40%
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.
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; αααα= incidence; εεεε= downwash angle = w/V; αααα∞= (αααα-εεεε)
= equivalent two-dimensional incidence; L∞ = two-dimensional lift; L = wing lift; Dv =trailing
vortex drag)
This drag component can be evaluated using the formula proposed by Prandtl (reference 13, 19): AR C K C Di L π 2 = (1)
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.
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.
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.
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.
(a) Secondary flow around wing-tip
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
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 the 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°
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 to 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.
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)
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
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.
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
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 be acceptable when all the residuals were below 10-3. 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)
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 0 0,5 1 1,5 0 0,2 0,4 0,6 0,8 1 x/c -CP alpha= 0 alpha= 1.5 alpha= -1.5 alpha= 2
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
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°.
0 0,2 0,4 0,6 0,8 1 1,2 -3 -2 -1 0 1 2 3 4 Alpha CL M=0.725 M=0.75 M=0.7
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 CL=0.65 and the flow remains attached all along its surface.
0 10 20 30 40 50 60 70 80 -3 -2 -1 0 1 2 3 4 Alpha L/D M=0.725 M=0.75 M=0.7
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.
Chapter 2: Wing design -1,5 -1 -0,5 0 0,5 1 1,5 0 0,2 0,4 0,6 0,8 1 x/c -Cp M=0.725 M=0.75 M=0.7
Figure 10: Influence of Mach number on the pressure distribution over the aerofoil at αααα=1.3°
On figure 8, we can see the influence of the Mach number on the lift coefficient. The slope of the curve CLagainst 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.
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).
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
Possible leading edge extension
Chapter 2: Wing design
The average sweep angle is simply computed using the following formula:
)
+))
= °=
Λ
Λ
Λ
mean arctan 0.5*tan LE 0.5*tan TE 24.65 798 . 0 3 ≈⇒
M
D2.2.1. 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.
-0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 0 0,2 0,4 0,6 0,8 1 1,2 x/c -CP
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
Chapter 2: Wing design
2.2.2. Design of the root section (y=0,
η
η
η
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 0 0,2 0,4 0,6 0,8 0 0,2 0,4 0,6 0,8 1 x/c -Cpt
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.
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 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 x/c z/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,
(
)
(
/)
0.22 077 . 0 / max max = = c x c z(
)
(
/)
0.30 057 . 0 / max max = = c x c zChapter 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
η
η
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
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
Chapter 2: Wing design Chapter 2: Wing design
drag. However, some structural issues have to be taken into account and the loading drag. However, some structural issues have to be taken into account and the loading over a wing is
over a wing is chosen to have more like a chosen to have more like a triangular shape. In Figure 18 is triangular shape. In Figure 18 is displayed thedisplayed the loading over the wing designed in
loading over the wing designed in comparison with an elliptic comparison with an elliptic loading.loading.
0 0 0 0,,22 0 0,,44 0 0,,66 0 0,,88 0 0 00,,2 2 00,,4 4 00,,6 6 00,,8 8 1 1 11,,22 η η η η η η η η Loading Loading L
Looaaddiinng g oovveer r ththe e wwiningg EElllilipptitic c llooaaddiningg
Figure 18: Comparison of the loading over the wing with an elliptic loading 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 The span wise loading is then used as an input file for the Downwash program. It provides with values for the induced incidence (
provides with values for the induced incidence ( ααii) which enables to compute the) which enables to compute the geometric incidence (
geometric incidence (ααgg) for each design station:) for each design station: α
αgg==ααii++ ααee
The effective incidence (
The effective incidence (ααee) is given in the Sweptdes output files for each design) is given in the Sweptdes output files for each design section. The table below summarizes these data.
section. The table below summarizes these data.
η
η ααii ααee ααgg
Root
Root section section 0 0 1,1183 1,1183 0,66326 0,66326 1,781561,78156
Intermediate
Intermediate section section 0,1875 0,1875 0,78 0,78 0,80388 0,80388 1,583881,58388 Outer
Outer section section 0,7 0,7 0,3 0,3 -0,01871 -0,01871 0,281290,28129
Table 2: Induced, effective and geometric incidence at the various stations designed Table 2: Induced, effective and geometric incidence at the various stations designed
Chapter 2: Wing design Chapter 2: Wing design
2.2.5.
2.2.5. Conclusions
Conclusions about
about the
the method
method
A 3D swept wing is designed using the computer programs Convert, Sweptdes and 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 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 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.
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. 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 Indeed, creating a precise database is necessary in order to generate an accurate grid and to run computations on t
and to run computations on the wing. However, the 15 Weber points he wing. However, the 15 Weber points given as an outputgiven as an output of the Sweptdes and Convert programs were not enough and I had to define some new 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 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 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 streamwise section from the 2D aerofoil coordinates using an Excel file and the formula: formula:
( (
))
( (
))
( (
))
( (
))
nn local local n n mean mean d d d d cc z z cc z z Λ Λ Λ Λ ⋅⋅
= =
++ cos cos cos cos 11 2 2 3 3This way, I was able to double the number of
This way, I was able to double the number of points defining the sections.points defining the sections.
Another problem encountered is related to the size of the grid necessary to compute the 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 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 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 wing on which the study will be based has a half span of 15 meters, corresponding to the wing designed cut at the crank.
Chapter 2: Wing design Chapter 2: Wing design
2.3.
2.3. Design of
Design of the wing
the wing tips
tips
2.3.1.
2.3.1. Design of
Design of the rak
the raked tip
ed tip
Few parameters are required for the raked tip design as it is a planar device. The Few parameters are required for the raked tip design as it is a planar device. The leading edge sweep was taken to be
leading edge sweep was taken to be ΛΛLELE=55° as it seemed to provide good=55° as it seemed to provide good performances according to studies from Kravchenko (reference 4). The trailing edge 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 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 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
swept than the wing, I decided to design it with a trailing edge sweep ΛΛTETE=23.8°. In the=23.8°. In the same way, the length of the tip device was decided according to an article saying that 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 the semi span resulting from the addition of this wing tip was 1.7 meters longer than the cleaned wing.
cleaned wing.
Concerning the design of the section, the sweep definitely had to be taken into account. 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 Indeed, it was found in the literature that the raked tip does not lose efficiency at high speed because of the subsonic flow over t
speed because of the subsonic flow over the device. Thus, the Convert program appliedhe device. Thus, the Convert program applied to the 2D section computed during the supercritical aerofoil design enabled to compute to the 2D section computed during the supercritical aerofoil design enabled to compute the tip section. No twist was implemented on the
the tip section. No twist was implemented on the raked tip.raked tip.
2.3.2.
2.3.2. Design of
Design of the
the winglet
winglet
The design of the winglet
The design of the winglet was mainly based on the guidance from was mainly based on the guidance from Whitcomb (referenceWhitcomb (reference 6). Only an upper winglet will be designed as it corresponds to the most commonly 6). Only an upper winglet will be designed as it corresponds to the most commonly used configuration.
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.
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.
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.
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.
Wing designed with a wing tip
Study of the Onera M6 wing
Study of the 2D supercritical
aerofoil section designed
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:
( )
r f f f f p ln ln 1 2 2 3
− −= ( f refers to the solution computed, r refers to the grid refinement
ratio taken to be constant, subscripts 1 to 3 refer to the different grids, 1 being the fine grid and 3 the coarse one)
Chapter 3: Validation
Using the value obtained for the order of convergence, the continuum value, f 0, can be computed using the relation below:
1 2 1 1 0 − − + ≈ p r f f f f
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:
) 1 ( 12 − = pS r F GCI ε
where Fs is a safety factor (Fs=1.25 for comparisons over three or more grids) and ε is
the relative error
− = 1 1 2 f f f εFinally 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:
1 12 23 ≈ GCI r GCI p
Richardson extrapolation can either be applied to some solutions at a grid point or to a solution functional. The f value can be seen as an estimation of the value that could0 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.
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 M2 angleµµµµ1 angle µµµµ2222
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 order p 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% asymptotic range of convergence 0.999 0.972 1.027 1.021 Theoretical value 0.3212 0.723 0.444 3.25 23,623,623,623,6 17.9
Table 3: Data computed for the expansion corner with an incident Mach number M1=2.5 and an
Chapter 3: Validation 1/ (nbr cells) P2/P1 T2/T1 ρ2/ρ1 M2 angle θθθθ 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 order p 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% asymptotic range of convergence 1.000 1.000 1.001 1.001 Theoretical value 2.468 1.324 1.867 1.87 37373737
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, θθθθ
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.
