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Optimizing UAV Aerodynamics with Computational FluidDynamics
João Nuno Dias Carvalho
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisors: Prof. Luís Rego da Cunha de EçaDr. Nelson Pereira Caetano MarquesEng. Ricardo José Cabral Veríssimo
Examination Committee
Chairperson: Prof. Filipe Szolnoky Ramos Pinto CunhaSupervisor: Dr. Nelson Pereira Caetano Marques
Member of the Committee: Prof. José Alberto Caiado Falcão de Campos
December 2016
”One, remember to look up at the stars and not down at your feet. Two, never give up work. Work gives
you meaning and purpose, and life is empty without it. Three, if you are lucky enough to find love,
remember it is there and don’t throw it away.”
Stephen W. Hawking
iii
Acknowledgments
To my thesis advisors, Prof. Luıs Eca and Dr. Nelson Marques I acknowledge all the support, feedback
and guidance provided throughout the entire duration of the thesis. To Prof. Luıs Eca, I would like to
show my appreciation for having accepted the orientation of this thesis and for all the feedback given to
the work developed. To Dr. Nelson Marques I take the opportunity to show my gratefulness for the hos-
pitality in receiving me at BlueCAPE, for the time dedicated guiding me in the long months of work that
are reflected in this document and for making the computational infrastructure, present at BlueCAPE,
available to me.
It is with great gratitude that I thank Major Diogo Duarte of the Department of Engineering and Pro-
grams of the Portuguese Air Force for the opportunity to develop my master thesis in cooperation with
the Portuguese Air Force and contribute to the development of the Air Force and my country.
To Lieutenant Ricardo Verıssimo of the Portuguese Air Force, I would like to show my utmost respect
and friendship and acknowledge the personal time spent advising and guiding me throughout the entire
thesis.
To the personnel at BlueCAPE, Bruno Santos, Jorge Azevedo, Pedro Freitas, Rui Gabriel and Vera
Rodrigues, I leave a word of appreciation for the technical and personal support provided during the time
I spent at the company. Without your goodwill and patience this thesis would not be delivered on time.
To my Mother, Father and Brother, I would like to tell you that without your unconditional love and
support, even at a distance, I could not have finished my degree and deliver this thesis. It is with great
pride and sense of accomplishment that I deliver this thesis and I hope you feel the same. Above all,
this document is dedicated to you.
I would like to thank my classmates, Afonso Martins, Ines Castelao, Joao Ribeiro and Marta Santos
for the patience sometimes needed to be around me, for the countless hours working and for the many
more laughing. I had the pleasure to learn with you and from you, and above everything I have the
pleasure of calling you my friends. I want you to know that our time as classmates is coming to an end
but our friendship is just starting.
Finally, allow me to leave a word in my native language to all my friends, extended family and other
people that crossed my life and, in any way, made me the person I am today:
A todos, obrigado.
v
Resumo
O objectivo deste projecto e empregar teorias de aerodinamica de baixo numero de Mach, atraves
de analises de CFD, de modo a optimizar a asa de um Veıculo Aereo Nao-Tripulado (VANT). Atraves
da inclusao de um dispositivo de ponta de asa optimizado, esta tese procura atingir um aumento no
desempenho superior a 10%. Devido a recente procura de veıculos autonomos por parte da industria da
defesa, a maximizacao do desempenho destas plataformas e extremamente desejado. A configuracao
do VANT foi providenciada pela Forca Aerea Portuguesa e uma analise com RANS e os modelos Menter
SST k − ω e γ −Reθ foi efectuada a numeros de Mach e Reynolds de 0.2 e 2.2× 106, respectivamente,
de forma a avaliar os coeficientes aerodinamicos e de desempenho. De forma a determinar a influencia
dos parametros de projecto de dispositivos de ponta de asa, foi realizado um estudo de sensibilidade
com o metodo dos paineis 3D e confirmaram-se os resultados analisando as melhores configuracoes
em RANS. Efectuou-se um estudo de malha de modo a assegurar precisao numerica nos aumentos
no desempenho previstos. Os resultados mostram que o metodo dos paineis nao atinge a precisao
desejada na previsao dos aumentos no desempenho mas proporciona uma inspeccao qualitativa da
influencia dos parametros de projecto. Simulacoes em RANS determinam aumentos de 20% e 30% no
L/D e C3/2L /CD, respectivamente, em relacao a asa limpa.
Palavras-chave: Veıculos Aereos Nao-Tripulados, Aerodinamica de Escoamentos Exteri-
ores, Mecanica dos Fluıdos Computacional, Reynolds-Averaged Navier-Stokes, Dispositivos de Ponta
de Asa, Estudo de Sensibilidade
vii
Abstract
The purpose of this project is to employ low subsonic aerodynamic theories, through CFD analysis,
in order to optimize the main wing of a MALE UAV. By inclusion of an optimized wing tip device this
thesis aims to achieve an increase in performance superior to 10%. Due to the recent demand for
unmanned solutions in the defense and aerospace industry, maximization of the performance of these
platforms is extremely desired. The configuration for the MALE UAV was provided by the Portuguese
Air Force and an analysis with the RANS equations, the Menter SST k − ω and the γ − Reθ models,
at a Mach number of 0.2 and a Reynolds number of 2.2 × 106, is performed in order to evaluate the
aerodynamic and performance coefficients. To determine the influence of the design parameters of wing
tip devices, a sensitivity study with a 3D panel method is employed and confirmation of the results is
done by analyzing the best configurations with RANS simulations. A grid refinement study is carried out
to ensure numerical accuracy of the increases determined by these simulations and estimate the exact
numerical values for the engineering quantities. Results show that the panel method fails to achieve the
desired precision in predicting performance increases but allows a qualitative insight on the influence
of the design parameters. RANS simulations determine a 20% increase in L/D and a 30% increase in
C3/2L /CD from the cut-off configuration.
Keywords: Unmanned Aerial Vehicles, External Aerodynamics, Computational Fluid Dynam-
ics, Reynolds-Averaged Navier-Stokes, Wing Tip Devices, Sensitivity Study
ix
Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Introduction 1
2 Theoretical Background 3
2.1 Concepts of Endurance and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Airplane Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Parasite Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Lift-Induced Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Lift-Induced Drag Reduction Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Mathematical Models 9
3.1 Panel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Reynolds-Averaged Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.2 Transition Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.3 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.4 Wall Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Spatial Grid Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Analysis of the Initial Configuration 19
4.1 Geometry Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.1 Geometry Clean-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.2 Geometry Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Volume Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Numerical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
xi
4.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Wing Tip Device Study 35
5.1 Sensitivity Study of Wing Tip Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.1 Wing Tip Device: Winglet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.1.2 Wing Tip Device: Raked Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Confirmation of the Results from the Sensitivity Study . . . . . . . . . . . . . . . . . . . . 43
6 Grid Refinement Study 49
6.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 Grid Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Conclusions 55
7.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Bibliography 59
xii
List of Tables
4.1 Target size of the volumetric control refinements . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Prismatic layer total thickness by component . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Results for the aerodynamic coefficients for the initial configuration . . . . . . . . . . . . . 28
5.1 Comparison between the initial configurations in XFLR5 and STAR-CCM+ . . . . . . . . . 36
5.2 Best combination of the winglet design parameters . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Best combination of the raked tip design parameters . . . . . . . . . . . . . . . . . . . . . 42
5.4 Comparison between tip devices and the cut-off wing in STAR-CCM+ . . . . . . . . . . . 43
5.5 Comparison of the cut-off configuration in XFLR5 and STAR-CCM+ . . . . . . . . . . . . . 44
5.6 Comparison of the increase created by the winglet in XFLR5 and STAR-CCM+ . . . . . . 44
5.7 Comparison of the increase created by the raked tip in XFLR5 and STAR-CCM+ . . . . . 47
6.1 Influence of the mesh size in the aerodynamic coefficients for both configurations . . . . . 51
6.2 Uncertainty of the three meshes in the prediction of the lift and drag coefficients . . . . . 52
6.3 Exact numerical solution of the aerodynamic and performance coefficients . . . . . . . . . 53
6.4 Comparison of the increases in performance with the exact numerical values . . . . . . . 53
xiii
List of Figures
2.1 Typical flight profile for surveillance UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Typical drag breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Trailing vortex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Wing tip devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Pareto fronts for optimized wing tips with span constraints . . . . . . . . . . . . . . . . . . 8
2.6 Blended winglet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Generic control volumes used in the discretization of the panel method . . . . . . . . . . . 10
3.2 Discretization of the surface of the geometry and its wake with the panel method . . . . . 10
3.3 Velocity distribution near a solid wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 Initial configuration provided by the Portuguese Air Force . . . . . . . . . . . . . . . . . . 19
4.2 Intervention in the front landing gear due to the existence of high proximity surfaces . . . 20
4.3 Change in the geometry of the rear section of the fuselage . . . . . . . . . . . . . . . . . 21
4.4 Hole filling in the lower surface of the Wing . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Microscopic gaps between surfaces, present in the aircraft . . . . . . . . . . . . . . . . . . 22
4.6 Geometry after the surface wrapper operation . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.7 Examples of meshing strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.8 Leading and trailing edges volumetric refinement controls . . . . . . . . . . . . . . . . . . 24
4.9 Final volume mesh with 21.7 million cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.10 Streamwise velocity of the flow surrounding the UAV . . . . . . . . . . . . . . . . . . . . . 28
4.11 Streamlines over the wing and winglet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.12 Streamline visualization of the wing tip vortex created by the initial configuration . . . . . . 29
4.13 Visualization of the pressure contours in four sections of the wing . . . . . . . . . . . . . . 29
4.14 Pressure (left) and skin friction coefficient distributions in the upper surface of the wing
(right) computed with the Menter SST k−ω and the γ−Reθ models (blue) and the Spalart-
Allmaras model (red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.15 Skin friction behavior of a transitional flow over a flat plate determined through DNS sim-
ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.16 Visualization of the skin friction coefficient on the surface of the UAV . . . . . . . . . . . . 32
4.17 Visualization of the streamwise vorticity contours in the wake of the aircraft . . . . . . . . 33
xv
5.1 Reference wing-tail pair in XFLR5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Simulation with fuselage performed in XFLR5 . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Resulting streamlines of the reference wing-tail pair analysis . . . . . . . . . . . . . . . . 37
5.4 Influence of each design parameter in the objective function L/D - Winglet . . . . . . . . 39
5.5 Interaction plot for L/D - Winglet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.6 Influence of each design parameter in the objective function L/D - Raked Tip . . . . . . . 41
5.7 Interaction plot for L/D - Raked Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.8 Configurations used to validate the results from the Design of Experiments . . . . . . . . 43
5.9 Streamline visualization of the wing tip vortex created by the three configurations analyzed
in STAR-CCM+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.10 Streamwise vorticity contours of the three configurations analyzed in STAR-CCM+ . . . . 46
6.1 Effect of the mesh size in the wing tip streamwise vorticity of the cut-off configuration . . . 50
6.2 Effect of the mesh size in the wing tip streamwise vorticity of the winglet configuration . . 50
6.3 Grid convergence plots for the cut-off configuration . . . . . . . . . . . . . . . . . . . . . . 51
6.4 Grid convergence plots for the winglet configuration . . . . . . . . . . . . . . . . . . . . . 52
xvi
Nomenclature
Roman Symbols
AR Aspect ratio
b Span, m
cP Power specific fuel consumption, lb lbf−1h−1
or lb hp−1h−1
CD Total drag coefficient
CD0Base-drag coefficient
Cf Skin friction coefficient
Cp Pressure coefficient
CL Lift coefficient
e Oswald coefficient
k Turbulent kinetic energy per unit mass,
m2 s−2
L/D Lift-to-drag ratio
M Mach number
p Pressure, Pa = N m−2
Re Reynolds number
s Distance, m
S Area, m2
t Time, s
~T Thrust, N
u, v, w Velocity components in cartesian coor-
dinates, m s−1
u Velocity – mean value, m s−1
u′ Velocity – perturbation, m s−1
~V Velocity (vector), m s−1
~W Weight, N
Greek Symbols
α Angle of attack,o
γ Intermittency
Γ Circulation, m2 s−1
δ Boundary layer thickness, m
δij Kronecker symbol
ε Turbulent kinetic energy dissipation rate
per mass unit, m2 s−3
ηP Propulsive efficiency
θ Momentum thickness
µ Dynamic viscosity, N m−1 s−1
ν Kinematic eddy-viscosity, m2 s−1
ρ Density, Kg m−3
τ Shearing stress, N m−2
φ Velocity potential, m2 s−1
ω Specific dissipation rate, s−1
Subscripts
∞ Free-stream condition
i, j, k Computational indexes
i, f Initial and final values
U,L Upper and lower boundaries
x, y, z Cartesian coordinates
xx, yy, zz Second order partial derivatives in carte-
sian coordinates
xvii
Glossary
AFA Air Force Academy is a military academy for officer cadets of the Portuguese Air Force,
located at Granja do Marques, Sintra.
CAD Computer-aided design is the use of computer systems to aid in the creation, modification,
analysis, or optimization of a design.
CFD Computational Fluid Dynamics is a branch of fluid mechanics that uses numerical methods
and algorithms to solve problems that involve fluid flows.
DNS Direct Numerical Simulation is a simulation in Computational Fluid Dynamics in which the
Navier–Stokes equations are numerically solved without any approximation other than the
discretization of the domain.
EEZ Exclusive Economic Zone is a sea zone prescribed by the United Nations Convention on the
Law of the Sea over which a state has special rights regarding the exploration and use of
marine resources, including energy production from water and wind.
EU European Union is a politic and economic union of 28 member states that are located primar-
ily in the continent of Europe. It has an area of 4,324,782 Km2 and an estimated population
of over 510 million people.
LES Large Eddy Simulation is a mathematical model used in Computational Fluid Dynamics, pro-
posed by Joseph Smagorinsky, that attempts to solve the Navier–Stokes equations ignoring
the smallest length scales, via low-pass filtering.
MALE Medium-Altitude Long-Endurance UAV is an unmanned aerial vehicle that operates at al-
titudes between 10,000ft to 30,000ft for extended durations of time, typically 24h to 48h
hours.
MTOW Maximum Take-Off Weight is the maximum weight at which the pilot is allowed to attempt to
take-off, due to design limitations.
PITVANT Projecto de Investigacao e Tecnologia em Veıculos Aereos Nao-Tripulados, Unmanned
Aerial Vehicle Investigation and Technology Project, was a the first large project of UAV
development and testing of the Portuguese Air Force.
RANS Reynolds-Averaged Navier-Stokes equations are a mathematical model used in Computa-
tional Fluid Dynamics that attempts to solve the Navier-Stokes by averaging all the unsteadi-
ness related to the existence of turbulence.
xix
R&D Research and Development is a general term for activities in connection with corporate or
governmental innovation.
UAV Unmanned Aerial Vehicle is any aircraft that has no human pilot on board and is commonly
known as Drone.
xx
Chapter 1
Introduction
According to TEAL Group Corporation, the development in Unmanned Aerial Vehicles (UAV) in the
twenty-first century has been the most dynamic and promising area within the entire aerospace industry.
In 2011, an estimated 2 200 Million Euros (2 500 Million Dollars) were spent in research and develop-
ment (R&D) of UAVs and the prospects are that by 2020 over 3,800 UAVs will have been produced,
which means a 44% increase relative to 2011. Projections also suggest that by 2020 Medium-Altitude
Long-Endurance (MALE) unmanned aerial vehicles will be the most popular, representing 35.6% of all
UAV production value. [1]
Portugal is not indifferent to these developments. Being aware of the constantly shifting paradigms in
modern warfare and homeland security, and responsible for the largest exclusive economic zone (EEZ)
within Europe (also the 3rd largest in the European Union (EU) and 10th largest in the world), with 18.7
times more water than land, Portugal is a prime location for MALE UAV application. [2]
The use of such UAVs, to support the conventional manned platforms in Surveillance and Search
and Rescue missions, is an integral part of the Vision for the future of the Portuguese Air Force.
To this end, the Portuguese Air Force Academy (AFA) started the PITVANT (Projeto de Investigacao
e Tecnologia em Veıculos Aereos Nao-Tripulados) project in 2009 – an R&D program in partnership with
both national and international industry and academia – and numerous other studies with the purpose
of collecting experience and know-how in UAV technology. [3].
One such project is the construction of a MALE UAV prototype with a maximum take-off weight
(MTOW) of 150 Kg, in collaboration with Instituto Superior Tecnico (IST). This prototype is expected to
be airborne for long periods of time and cover large areas and it is, therefore, necessary to maximize its
endurance and range.
1
In this context, this project aims to achieve an increase of, at least, 10% in endurance and range
regarding the provided configuration. This main objective is achieved through the use of computational
models in two phases. The first phase aims to construct a computational model of the current config-
uration of the wing in order to evaluate the integral quantities, lift and drag coefficients, as well as the
performance coefficients. The second phase attempts to implement theories of drag reduction, in order
to increase L/D and C3/2L /CD, thus increasing range and endurance.
As described by Maughmer [4], early attempts to evaluate the influence of design parameters of
winglets, at the Pennsylvania State University, were performed using trial-and-error approaches using
flight testing. Ning and Kroo [5], at Stanford University, have used multidisciplinary optimization tech-
niques to determine the effects of several tip designs and, in 2014, Panagiotou et al. [6] performed
parametric studies of the design parameters of winglets for a MALE UAV for the Hellenic Air Force.
As to the organization of this document, chapters 2 and 3 provide the tools for a complete understand-
ing of the aerodynamic theories (chapter 2) and mathematical models (chapter 3) employed throughout
this Thesis. Chapter 4 focuses on the base configuration, studying its performance and presenting the
reference values for the aerodynamic force coefficients. Chapter 5 explores different wingtip devices and
their effect on performance by means of a sensitivity study to geometric design parameters. In chapter
6, a grid refinement study is presented. Finally, the achievements of this Thesis are summarized in
chapter 7, followed by suggestions of future work.
2
Chapter 2
Theoretical Background
2.1 Concepts of Endurance and Range
Endurance and range are two concepts that are quite often misused, hence the need to clarify them
with a formal definition. While endurance can be defined as the maximum time an aircraft is airborne
under a certain flight condition without being refueled, range refers to the maximum distance covered. [7]
These concepts are mathematically expressed by the Breguet Equations, that for propeller-driven
airplanes and in the imperial unit system take the form presented in equations 2.1a, for range, and 2.1b,
for endurance. These quantities are determined in nautical miles and hours, respectively. [8]
R =
∫ Wf
Wi
ds
dWdW = −326
(ηPcP
)(CLCD
)∫ Wf
Wi
dW
W= 326
(ηPcP
)(CLCD
)ln
(Wi
Wf
)(2.1a)
E =
∫ Wf
Wi
dt
dWdW = −550
(ηPcP
)∫ Wf
Wi
dW
DV= 550
(ηPcP
)(C
3/2L
CD
)√2ρS
[1√Wf
− 1√Wi
](2.1b)
In equations 2.1, Wi and Wf are the initial and final aircraft weights, ηP the propulsive efficiency
and cP the specific fuel consumption. As a result of the definition of both equations, range depends on
CL/CD and endurance depends on C3/2L /CD. While equation 2.1a departs from the fuel consumption
per unit of distance, equation 2.1b from the fuel consumption per unit of time. Since the concepts are
different, the maximum range and endurance of a given propeller-driven aircraft are not achieved under
the same conditions. Maximum range is obtained with the conditions that minimize thrust and maximum
endurance achieved with the minimum power. Even though range and endurance are function of different
factors, both depend on the amount of lift and drag produced by the aircraft. By inspection, maximizing
lift and minimizing drag is in order if one desires an increase in any of the two performance parameters.
3
A typical flight profile for surveillance UAVs is depicted in figure 2.1. For each step of this flight,
the lift (~L) required is defined by the weight ( ~W ), drag ( ~D) and thrust (~T ) forces acting on the aircraft.
In this way, increasing lift would only benefit the climb segment, since it would allow for a faster climb
and, therefore, less fuel consumption. In cruise flight, increasing lift means that the equality ~L = ~W
would no longer be kept and the aircraft would gain altitude. To maintain altitude, velocity would have to
be reduced and such reduction may not be desired. The best way to improve the overall aerodynamic
performance is to decrease drag.
Figure 2.1: Typical flight profile for surveillance UAVs.
2.2 Airplane Drag
The total drag of an airplane is frequently divided in two components, parasite drag and lift-induced drag.
The parasite drag is all the drag that does not appear from the creation of lift and the corresponding vor-
ticity shed into the wake. It contains the drag created by the boundary layers of the aircraft components,
the pressure drag created by the existence of thickness and the drag created by the interference of the
geometries. The lift-induced drag, as the name suggests, is the drag that is directly dependent on the
production of lift and the vorticity shed into the wake. [9]
Figure 2.2: Typical drag breakdown.
4
2.2.1 Parasite Drag
As can be seen in figure 2.2, parasite drag can be divided in friction/form, interference and wave drag.
For low subsonic applications, wave drag is inexistent since it is the result of shock waves. Friction drag
is a consequence of the effects of viscosity [10] and, although this type of drag accounts for a very
large part of the total aircraft drag, its reduction is the most difficult to accomplish. [11] Interference
drag appears due to intersections of boundary layers generated by different components of the aircraft.
Even though it can be present in any kind of geometry intersection such as wing/wing, wing/struct or
struct/fuselage, the most representative is in the intersection between the wing and the fuselage. [9]
2.2.2 Lift-Induced Drag
Lift-induced drag appears due to the energy that is lost as a result of the trailing vortices, is highly
dependent on the wing planform geometry and may account for almost half of the total drag. [11] In
order to obtain the most generic expression for the lift-induced drag, it is necessary to employ the lifting-
line theory, derived by Prandtl. [12]
CDi=
C2L
π ·AR · e(2.2)
The physics of the wing tip vortex in the near-field is extremely complex as it is a three-dimensional
and turbulent phenomena. Due to the pressure difference between the upper and lower surface of the
wing, a strong cross-flow is induced and a wing tip vortex is formed. [13] This phenomena has been
one of the most researched areas in the past century. These studies started in 1907 with Lanchester,
and, in 1918, Prandtl presented the lifting line theory for computing the lift-induced drag. [12] Prandtl
and his disciples (Betz, Trefftz and Munk) calculated the amount of drag induced by lift in the aft-section
of a large box – the Trefftz plane – by computing the integral of the perturbations of the velocity u, v, w
on that plane. [12]
Di =ρ
2
∫∫Trefftz
(u2 + v2 + w2
)dS (2.3)
Equation, 2.3 as simple as it may look, is extremely difficult to implement. One way of overcoming
difficulties was by projecting the wake of the near-field to the Trefftz plane. [12] If one assumes in-
compressible potential flow in the cross-flow plane, it’s possible to arrive to the expression presented in
equation 2.4.
Di =ρ
2
∫∫Trefftz
(v2 + w2
)dS =
ρ
2
∫∫wake
∆φ∂φ
∂ndl = −ρ
2
∫wake
ΓVndl (2.4)
For simplification purposes, Prandtl assumed the wake extended in straight lines parallel to the freestream
when, in fact, the wake is a three-dimensional phenomena, creating a roll-up effect in the wing tip. Re-
sults show that lift-induced drag is very sensitive to the shape of the wake, which proves the inability of
the lifting line theory to compute lift-induced drag accurately. [12]
5
When surface panel methods appeared, a new approach to drag calculation was implemented. [12]
By integrating the pressures along the surface of the geometry in analysis, one is able to compute drag,
and, since these methods are based on potential flow, they only account for lift-induced drag. The
expression used in order to compute a force in these methods is presented bellow. [14]
~F = −∫∫
S
p · ~n dS (2.5)
Even though this method allows a far more accurate prediction of drag and allows better understand-
ing of the overall phenomena, a full comprehension of the wing tip vorticity is yet to be achieved.
Modern CFD codes, use the control volume formulation to discretize the computational domain and
compute drag by integrating pressure and the stress tensor along the aft-plane created by the domain,
yielding far better results in predicting drag. However, the distinction between skin friction, form, interfer-
ence and lift-induced drag is complicated in these methods as the value for drag is being computed as
whole and not separately, as shown in equation 2.6 for the total drag. [15]
D = −∫∫
Trefftz
[p− p∞ + ρu (u− U∞)] dy dz (2.6)
2.3 Lift-Induced Drag Reduction Concepts
Reducing lift-induced drag can be achieved through the reduction in the intensity of the wing tip vortex.
In fact the concept of eliminating all trailing vorticity was first conceived by Lanchester in 1897 when
he proposed the inclusion of wing end-plates. [12] Lanchester’s wing end-plates managed to reduce
wing-tip vorticity but were unable to accomplish the goal of eliminating all trailing vorticity he set out to
achieve. In fact, this goal is unable to accomplish regardless of the wing tip device employed since it is
an intrinsic characteristic of finite wings.
Figure 2.3: Trailing vortex structure. [16]
As seen in figure 2.3, the trailing vortex structure is quite complex and many components of the
aircraft, such as flaps and engines, can influence it. However, the strongest vortex comes from the wing
itself. This vortex is formed near the tip and attracts most of the smaller and weaker vortices. [16]
6
According to Gudmundsson [17], the shape of the wing tip distorts the flow field around the wing,
pushing the wing tip vortex closer or further away from the symmetry plane. This fact explains the reason
why a cut-off wing has better performance than a round tip wing. In the cut-off wing, the flow experiences
a difficulty in the rolling process due to the sharp discontinuity in the tip. This effect is not seen in the
round wing tip since the flow is guided by the surface. The reduction witnessed in lift-induced drag with
the inclusion of winglets and raked tips may be interpreted as an increase in aspect ratio created by the
translation of the vortex outward. The approach followed by Gudmundsson evaluates the term gain-in-
effectiveness (∆AR) in equation 2.7. Using potential flow theory, several wing tip devices are tested and
∆AR is computed with equation 2.8, derived from equation 2.7.
CD = CD0 +C2L
π · (AR+ ∆AR) · e(2.7)
⇒ ∆AR =
(C2L
π ·∆CDi · e+ C2L
− 1
)·AR (2.8)
Results from these studies suggest that the best wing tip devices are winglets and raked tips with
∆AR up to 1.5.
Figure 2.4: Wing tip devices. [18]
Non-planar geometries, such as those presented in figure 2.4, have been target of many research
studies. Following Lanchester’s theories, Whitcomb [19] developed the first winglet (figure 2.4A) predict-
ing that the lift-to-drag ratio would increase in about 9%.
7
In more recent studies, Ning and Kroo [20] predicted drag reductions when using different types of
wing tip devices. Span extensions, winglets and C-wings were tested and the results show that, for
negative (figure 2.5a) and zero (figure 2.5b) pitching moments, C-wings and winglets present a similar
drag reduction with winglets being slightly more efficient. For positive moments about the aerodynamic
center, C-wings produce better results.
(a) Cmac = −0.1 (b) Cmac = 0 (c) Cmac = 0.1
Figure 2.5: Pareto fronts for optimized wing tips with span constraints. [20]
In the vertical axis of the plots present in figure 2.5 one can visualize the ratio between the drag
created by a wing with a certain wing tip device and the drag created by an elliptically loaded wing.
This ratio is plotted against the ratio of the weights of both wings. Although promising, the inclusion of
C-wings or other non-conventional winglets is highly premature and more research is needed to prove
their viability.
The best solution might be the inclusion of winglets, in particular blended ones. In very recent studies,
it has been proven that an optimized blended winglet in low subsonic flights can increase lift-to-drag ratio
up to 16%. This geometry is shown bellow in figure 2.6. [6]
Figure 2.6: Schematics of a blended winglet. [6]
8
Chapter 3
Mathematical Models
In order to predict the changes in drag due to devices such as those previously discussed, it is neces-
sary to evaluate the dynamics of the air around the aircraft.
The most precise way of solving a fluid flow is through the Direct Numerical Solution (DNS) mathe-
matical model. This model computes the Navier-Stokes without any sort of approximations other than
the discretization of the domain. Other models, that do not solve directly the Navier-Stokes and rely on
equations to model turbulence can be used. Models of such type are the Large Eddy Simulations (LES)
and the Reynolds-Averaged Navier Stokes (RANS). The main difference between these two is the fact
that RANS do not resolve the turbulent velocity fluctuations as they only work with mean values for the
product of fluctuating fluid properties. LES resolve the large eddies and model the mean effects of the
smaller eddies. [21] Other models, deriving from the conservation of mass, can be employed in specific
scenarios. A method of such nature is the panel method. [22]
3.1 Panel Method
The panel method can be extremely useful when performing simple flow analysis on complex geome-
tries. This method is often used when the flow can be approximated by an inviscid and incompressible
potential flow. In these cases, the method is in agreement with experimental results and is extremely
useful in predicting the incremental effects of changes in the aircraft configuration with very little compu-
tational effort. [22] (1−M2
∞)φxx + φyy + φzz = 0 (3.1)
The panel method departs from the Prandtl-Glauert equation for linearized compressible flow (equa-
tion 3.1), converting it into an integral equation for the potential of the flow using Green’s theorem. [14]
φ (P ) = − 1
4π
∫∫S
[n · (∇φU −∇φL)
R− (φU − φL) · n · ∇ 1
R
]dSQ (3.2)
9
If the generic control volumes present in figure 3.1 are used in the discretization of the Prandtl-Glauert
equation, one can arrive to the equation 3.2 for the potential of the flow. [14]
Figure 3.1: Generic control volumes used in the discretization of the panel method. [14]
The panel method is considered a surface model, since it does not discretize the volume around the
geometry. Equation 3.2 is applied to 2D quadrilateral panels, resultant of the discretization process, that
cover both the aircraft and its wake. For this reason the panel method is less resource consuming than
the mathematical models present in subsequent sections. This surface method transforms the geometry
in a panel-like geometry, hence the name of the method. [22]
Figure 3.2: Discretization of the surface of the geometry and its wake with the panel method. [22]
10
3.2 Reynolds-Averaged Navier-Stokes
For most complex engineering applications, using DNS or LES is not an option, due to the computational
times of these two mathematical models.
With the Reynolds-averaged method, all unsteadiness due the existence of turbulence in the Navier-
Stokes equations is averaged. Two ways of averaging are possible, depending on the time related
behavior of the flow. For steady flows, time-averaging, shown in equation 3.3, is done by assuming that
the value of a property is the sum of the mean value across time and a fluctuation around it. [21]
ψ (xi, t) = ψ (xi) + ψ′ (xi, t) (3.3)
However, when the flow has an unsteady behavior, time-averaging is not an option and the ensemble-
averaging approach has to be used. The process, known as Reynolds averaging, when applied to the
Navier-Stokes equations, results in the averaged equations of continuity and momentum, present in
equations 3.4, in tensor notation. [21]∂ (ρui)
∂xi= 0 (3.4a)
∂ (ρui)
∂t+
∂
∂xj
(ρuiuj + ρu′iu
′j
)= − ∂p
∂xi+
∂
∂xj
[µ
(∂ui∂xj
+∂uj∂xi
)](3.4b)
3.2.1 Turbulence Models
In turbulent flows the velocity and pressure distributions are highly unsteady and inspection of the in-
stantaneous fields would show fluctuations in all three spatial directions. Attempts on modeling the
fluctuations of the averaged quantities, solve additional terms, resultant of this method of averaging the
Navier-Stokes equations, and create a closed system of equations can be achieved through turbulence
models. Most of these use the expression proposed by Boussinesq, in 1877 to compute the Reynolds
stress, −ρu′iu′j . [21]
− ρu′iu′j = µt
(∂ui∂xj
+∂uj∂xi
)− 2
3ρδijk (3.5)
Mixing Length Model
The mixing length model is one of the most simple to implement. It has no extra transport equation to
couple with the RANS equations since it models the Reynolds stress terms directly. [21]
µt = ρl2m
∣∣∣∣∂u∂y∣∣∣∣ (3.6)
lm in equation 3.6 is known as the mixing length and is a characteristic of the largest eddies that
interact with the mean flow. [23]
11
This method is known to be applicable for a narrow range of cases. Accurate prediction of the mixing
length is only possible for simple flows with no separation. For highly three-dimensional cases, this
model lacks the necessary accuracy. [21]
Spalart-Allmaras Model
Developed by P. R. Spalart and S. R. Allmaras, in 1992, this turbulence model is a one-equation model
that couples, to the RANS equations a partial differential equation for the kinematic eddy-viscosity pa-
rameter, ν. [24]
∂ (ρν)
∂t+∂ (ρuj ν)
∂xj=
1
σν
∂
∂xj
[(µ+ ρν)
∂ν
∂xj+ Cb2ρ
∂ν
∂xk
∂ν
∂xk
]+ Cb1ρνΩ− Cw1ρ
(ν
ky
)2
fw (3.7)
The eddy-viscosity is then determined by equation 3.8. [25]
µT = ρνfν1 , where fν1 =χ3
χ3 + C3ν1
and χ =ν
ν(3.8)
The model constants that appear on equations 3.7 and 3.8 were determined for external aerodynamic
flows and can be found in reference [24] by Spalart and Allmaras.
Designed for aerospace applications, this turbulence model was specially conceived for wall-bounded
flows, as is the case of airfoils. The model was validated with the RAE 2822 airfoil test case and results
proved to be accurate when compared to experimental values. The model responds well when in the
presence of gradual to steep pressure gradients. Weaknesses in the model have been witnessed in
flows with massive separations, wakes with pressure gradients and free vortices. In general the model
shows error when dealing with free-shear layer flows. [24]
The k -ε Model
The k -ε model, conceived by Hanjalic and Launder [26] as an attempt of achieving a universal model for
turbulence, is considered a classical method and uses two partial differential equations to compute the
turbulent kinetic energy, k, and the rate of dissipation of turbulent kinetic energy per mass unit, ε. [27]
∂ (ρk)
∂t+∂ (ρujk)
∂xj= Pk − ρε+
∂
∂xj
[(µ+
µtσk
)∂k
∂xj
](3.9a)
∂ (ρε)
∂t+∂ (ρujε)
∂xj= Cε1Pk
ε
k− ρCε2
ε2
k+
∂
∂xj
(µtσε
∂ε
∂xj
)(3.9b)
Equations 3.9 compute k and ε and to determine the eddy-viscosity, the model uses equation 3.10.
µt = ρCµk2
ε(3.10)
The k -ε model constants can be found in reference [26] by Hanjalic and Launder.
12
The k -ε model is used for a wide range of engineering cases. With the ability to model turbulence
well in free-shear layer flows, where boundary layers do not govern the flow, this two-equations model
has been tested with jet stream and recirculating flows, rendering good results. The limitations of this
model appear with boundary layers specially when adverse pressure gradients are present. In these
flows, the k -ε model does not achieve accurate results. [27] For this reason, the use of this model in
applications with airfoils is unadvised.
Menter SST k -ω Model
Originally conceived by Wilcox [28], the k -ω model was derived from the k -ε model by transforming the
ε-equation into a ω-equation by substituting ε = kω. The model proposed by Menter is based on the
Wilcox k -ω model. What Menter proposed was to create a hybrid model that behaved like the k -ε model
in the fully turbulent region far from the wall and Wilcox k -ω model near the wall. [29]
∂ (ρk)
∂t+∂ (ρujk)
∂xj= Pk − β∗ρωk +
∂
∂xj
[(µ+ σkµt)
∂k
∂xj
](3.11a)
∂ (ρω)
∂t+∂ (ρujω)
∂xj= γPω − βρω2 + 2ρ (1− F1)σω2
1
ω
∂k
∂xj
∂ω
∂xj+
∂
∂xj
[(µ+ σωµt)
∂ω
∂xj
](3.11b)
Since Menter’s model is a hybrid model between the k -ε and k -ω models, a blending function, F1,
has to be used in order to define the regions where each of the two models is used. Such function is
indeed present in equation 3.11b. [29]
The constants for the model can be found in reference [29] by Menter.
This model was tested with several aeronautical cases with documented experimental and numerical
results. In free shear-layer test cases, the model achieved results in accordance with the k -ε model and
for cases with walll-bounded flows with adverse pressure gradients the Menter SST k -ω model proved
to by highly accurate. [29]
3.2.2 Transition Models
Laminar to turbulent transition is the process in which a flow transitions from a smooth and steady
state – laminar – to an unsteady and fluctuating state – turbulent. Three types of transition can be
found: the natural transition, bypass transition and the separation induced transition. Natural transition
is characterized by appearance of Tollmien-Schlichting waves above a critical Reynolds number. In the
bypass transition the flow transitions from laminar to turbulent due to external perturbations such as a
turbulent wake from a body located upwind. [30] Separation of the laminar boundary layer is known to
induce transition. In these cases, the flow is likely to reattach turbulent, transitioning the boundary layer
from laminar to turbulent. [30]
13
en Model
The en model is extremely used in the aerospace industry for natural transition prediction and is con-
sidered the state of the art when it comes to airfoil analysis. As to the extent of the capabilities of
this method, it is important to state that it does not allow modeling of laminar to turbulent transition or
prediction of laminar separation bubbles, unlike the γ − Reθ. This model cannot be incorporated in a
Computational Fluid Dynamics (CFD) analysis and only provides an estimation of the location of this
transition. [31]
This method uses three steps to predict the laminar-turbulent transition. First, the velocity and tem-
perature profiles along the body in analysis are determined. Then the local growth rates of the unstable
waves are calculated for each of the profiles mentioned before. Finally, in order to compute the n factor,
the local growth rates are integrated along each streamline. The overall idea is to compute disturbance
amplitude ratio, en. When this value exceeds the limiting value, transition starts to occur. [32]
γ −Reθ Model
The γ − Reθ transition model is based on two transport equations, one for the intermittency factor, γ,
and one for the transition onset momentum-thickness Reynolds number, Reθt. [31]
∂ (ργ)
∂t+∂ (ρujγ)
∂xj= Pγ1 − Eγ1 + Pγ2 − Eγ2 +
∂
∂xj
[(µ+
µtσγ
)∂γ
∂xj
](3.12a)
∂(ρReθt
)∂t
+∂(ρujReθt
)∂xj
= Pθt +∂
∂xj
[σθt (µ+ µt)
∂Reθt∂xj
](3.12b)
Equation 3.12a is responsible for triggering the transition process and the production of turbulent ki-
netic energy inside the boundary layer, while equation 3.12b is in charge of avoiding additional non-local
operations that appear when experimental correlations are used. [32]
The γ − Reθ transition model has been tested in several aeronautical cases and the results agree
well with the available experimental results. [31] This model is CFD-compatible is ready to interact with
the Menter SST k -ω turbulence model as described by Langtry and Menter [31].
The functions and constants present in equations 3.12 can be found in reference [31] by Langtry and
Menter.
14
3.2.3 Model Selection
Based on the limitations of each turbulence and transition model, described in previous sections, a
choice of models has to be done. For external CFD analysis with airfoils, the turbulence models that
provide acceptable predictions of the flow are the Spalart-Allmaras and the Menter SST k -ω.
Since the flow is likely to have laminar and turbulent regions, transition has to be modeled. The en
model only presents the location of transition and does not model it. Therefore, the only option is by
performing calculations with the γ −Reθ transition model. This model can only be implemented with the
Menter SST k -ω model.
For applications of the nature of this Thesis, one is left with two options for the choice of mathemat-
ical models. On one hand, the Spalart-Allmaras model assumes a fully turbulent flow and yields good
results. This model only contemplates one extra equation added to the RANS equations. On the other
hand, the Menter SST k -ω coupled with the γ −Reθ transition model allows a far better prediction of the
flow, including transition, but at a computational cost of four extra equations.
For this Thesis, the Menter SST k -ω coupled with the γ − Reθ transition model was chosen, since
this phenomena is important in the estimation of the integral coefficients for the Reynolds number in
question.
3.2.4 Wall Boundary Condition
The mathematical models selected in the previous section need boundary conditions to be able to be
resolved. The application of these conditions on solid walls is not so straight forward as in boundaries
of other natures. One possibility to solve this boundary condition is by solving the turbulence model all
the way to the wall and apply the no-slip condition for the velocity in the wall. [21] A number of turbu-
lence models, valid all the way to the wall, have been tested by Patel et al. [33] and all failed to capture
the essential and well-documented features of the near-wall flow. The problem with this method is that
the viscous sublayer of the boundary layer is, sometimes, so thin that placing cells within is extremely
difficult. To avoid the excessive cell requirements inside the viscous sublayer, wall functions can be em-
ployed. [21]
The theory behind the wall functions is the law-of-the-wall, derived by von Karman [34] for the two-
dimensional case of a flow over a flat-plate. The wall functions are based on the fact that the form of the
solution between the solid wall and the outer edge of the logarithmic layer may be assumed invariant
given that appropriate scaling is employed. [35] It is a known fact that the law-of-the-wall is not so
universal as commonly stated. For separated or three-dimensional flows, the law-of-the-wall provides, in
the best scenario, the magnitude of the velocity. The direction, however, has to be determined by other
means. [36]
15
Figure 3.3: Velocity distribution near a solid wall. [23]
The near-wall flow is divided in two regions according to their behavior. As depicted in figure 3.3,
the viscous sub-layer, the closest region to the wall, is very thin and is characterized by the linear rela-
tionship between velocity and distance to the wall. In this region viscous effects are predominant. The
following region is called the log-law layer and turbulent effects have as much importance as viscous
effects. [35] Some authors consider a third region between the two previously stated, the buffer layer.
[30]
The first wall functions, deployed in CFD codes, required the first point above the wall to be placed
in the logarithmic layer. Most mesh generation codes do not follow this rule and, in these cases, the
log-law layer wall functions are generally very inaccurate. Adaptive wall functions, also called low y+
wall functions, were developed in order to override this problem. By not restricting the location of the first
point to the logarithmic layer, these functions allow better accuracy in the current CFD codes. [35]
3.3 Spatial Grid Convergence
Reliability and accuracy are two concepts frequently confused. In the context of CFD applications, re-
liability is associated with the robustness of the computational method employed and, therefore, the
evaluation of iterative errors. Accuracy is composed of two types, the numerical and the modeling accu-
racy. The first is associated with the error generated by the discretization of the control volume. The last
is a measure of the error created by the modeling of the physics by a certain mathematical model.
A spatial grid convergence study is used to evaluate the numerical accuracy of the solution obtained
with a specific mesh in a CFD analysis. Therefore, the objective of a study of this nature is to determine
the error associated with the discretization of the control volume. The method proposed by Roache
allows an estimation of the exact numerical value and determination of the uncertainty of the computed
solutions. [37]
16
In order to perform such a study, one needs to compute the solution of a certain engineering quantity
of interest with three or more grids. To generate this group of meshes, one should start by defining a
mesh with high definition. Then, coarsening should be performed until the remaining meshes are deter-
mined. [37]
In order to evaluate convergence, it is important to compute the exact numerical value of the engi-
neering quantities of interest determined with the meshes. This value, determined using the Richardson
extrapolation, is an estimation of the value the CFD code would produce if the mesh was so thin that it
would be a continuum. [37]
Any solution of a simulation can be expressed as the series present in equation 3.13, where f is the
engineering quantity being evaluated, h the grid spacing and g1, g2 and g3 are functions independent of
the mesh. [37]
f = fh=0 + g1h+ g2h2 + g3h
3 + ... (3.13)
fh=0 is the estimation of the exact numerical value or the value when the grid spacing is zero. With
some mathematical manipulation, one can arrive to the expression that enables the calculation of this
value. Equation 3.14 and 3.15 assumes that increasing indexes mean coarser grids, e. g. f1 is the
solution of the most refined mesh and f3 the solution of the coarser grid. [37]
fh=0 ' f1 +f1 − f2rp − 1
(3.14)
In equation 3.14, r is the grid refinement ratio, and p is the observed order of convergence obtained
with equation 3.15. [37]
p = ln
(f3 − f2f2 − f1
)/ ln r (3.15)
At last, the uncertainty or numerical error due to domain discretization for each mesh can be com-
puted by equation 3.16, where Fs is a safety factor of 1.25.
ε ' Fs ·fh=0 − f1fh=0
(3.16)
17
Chapter 4
Analysis of the Initial Configuration
4.1 Geometry Modeling
The UAV provided by the Portuguese Air Force is a propeller driven aircraft and has a twin-boom pusher
configuration, which is common for MALE UAVs. It is possible to see the provided geometry in figure
4.1, shown below.
Figure 4.1: Initial configuration provided by the Portuguese Air Force.
The given aircraft couldn’t be directly used in a CFD software, since softwares of such nature require
more detail than Computer Aided Design (CAD) softwares. In fact, a CFD analysis implies a highly
detailed geometry with contacting edges in extremely small scales and no intersecting faces. Therefore,
before importing in STAR-CCM+, it was necessary to model the geometry in a CAD software and perform
further operations in STAR-CCM+.
19
The first step in the geometry preparation process is to remove components outside of the volume of
interest. For external aerodynamics, internal engines and electronics rest outside of the control volume
and are of no interest. On the other hand, some external components have undesired details, that
must be solved. With a small contribution to the overall performance, they tend to generate complicated
meshing regions, due to high surface proximities, forcing the meshing code, present in STAR-CCM+, to
create a large number of cells to capture the exact geometry.
4.1.1 Geometry Clean-up
In order to simplify the problem and reduce the computational effort and time spent on a CFD analysis,
intervention in the CAD of the aircraft had to be performed. The most evident changes in the geometry,
for this reason, were done in the front landing gear and in the rear section of the fuselage.
Figure 4.2 shows the operation performed in the front landing gear. The initial geometry contained
not only countless holes for the insertion of bolts but also a internal region with a large amount of detail.
A region of such nature creates problems when meshing operations are performed. In fact, due to high
proximity surfaces in these regions, the meshing code, present in STAR-CCM+, decreases the size of
the cells and places a large amount, increasing the overall size of the mesh.
(a) Initial geometry (b) Final geometry
Figure 4.2: Intervention in the front landing gear due to the existence of high proximity surfaces.
Even though the CFD software can perform an analysis with the original front landing gear, the
number of cells, created by the details present in it, increase the size of the mesh and the simulation
time. In order to simplify and reduce the computational effort, the landing gear was filled, removing the
internal region it created. Interventions like this have a small impact in the results and a large influence
in the overall spent time.
20
In order to further simplify the problem in hands, a change had to be done in the rear section of the
fuselage. It is possible to see in figure 4.1, that the UAV has two air intakes that provide cooling air for
the engine and, at the same time, air for the core of the propeller. To be able to use the geometry as
it is, one would need to know the mass flow of air entering these intakes. Since this value is unknown,
the only way to determine it is by simulating the internal volume with all the components, thus creating a
very large and complex simulation.
Therefore, a decision was made to remove the air intakes. This change in geometry changes the
aerodynamic coefficients of the aircraft, but not in a compromising way. Implementing this change
makes the aim of this Thesis tangible, since the fuselage does not influence in a considerable way
the optimization process of the wing tip. By choosing the path of the real configuration, the time and
resources spent would be far greater and reaching the desired result could not be feasible within the
Thesis timespan. A detail of the change in geometry is depicted bellow.
(a) Initial geometry (b) Final geometry
Figure 4.3: Change in the geometry of the rear section of the fuselage.
4.1.2 Geometry Preparation
With the geometry simplified, it was necessary to prepare it for the meshing process. In order to be able
to mesh and simulate, it is necessary to create a closed region with no self-intersecting faces. In order
to achieve this goal, it was necessary to inspect the geometry and repair as much problems as possible
in the CAD software.
(a) Initial geometry (b) Final geometry
Figure 4.4: Hole filling in the lower surface of the wing.
21
Throughout the entire geometry, it was required to close holes in order to simulate covers that were
missing. An example of this operation is presented in the previous page, in figure 4.4, where it is possi-
ble to see some internal openings in the wings that had to be closed.
With the major holes closed, the assembly of components was converted into one single body. In
fact, when dealing with low quality geometries, importing them as an assembly in STAR-CCM+ may
generate problems with the meshing operations, such as self-intersecting surfaces or non water-tight
geometries. In other words, although surfaces may look joined in a large scale, when zoomed in, they
may be either intersecting one and other or there might exist a microscopic gap between them.
Figure 4.5: Microscopic gaps between surfaces, present in the aircraft.
After combining the entire aircraft into a single body some superficial gaps were still found. This is
a common problem when performing CFD analysis, since these softwares require higher geometry def-
initions than CAD softwares. These gaps were removed in STAR-CCM+, by using the surface wrapper
operation and the advanced surface repair tool.
The surface wrapper is an automated operation of STAR-CCM+ that, as the name suggests, wraps
the entire geometry, closing all gaps and creating a water-tight geometry that can be used to develop a
finite volume mesh.
After importing the geometry in STAR-CCM+ and before starting the operation of wrapping, it was
necessary to divide the surface of the body into several surfaces. In fact this was done in order to control
the refinement settings for each component of the aircraft.
The surface wrapper was done by choosing two bodies, the UAV and a large box that represents the
computational domain. Due to the existing symmetry, the domain could be reduced in half, intersecting
the symmetry plane of the aircraft.
22
The process of surface wrapping was done by selecting both the aircraft and the domain, choosing
a seed point in middle of the desired volume and by choosing the gap closure option. In this way,
it was guaranteed that the obtained surface was closed. Specific controls were implemented in the
components where the geometry is more complicated, ensuring that the details coming from the CAD
were captured.
Figure 4.6: Geometry after the surface wrapper operation.
After the wrapping operation, advanced manual surface repair had to be done in punctual areas
where the surface wrapper code had difficulties in achieving the correct geometry. The surface obtained
by the wrapping process became the new reference geometry for the volume mesh.
4.2 Volume Mesh Generation
The RANS equations coupled with the turbulence and transition models need to be solved in a mesh,
which is essentially a discrete representation of the domain. [21] Determining the mesh that yields the
best results is a hard task. In fact, this process is essentially dependent on the geometry in analysis,
available computational resources and/or the problem physics.
(a) Structured Mesh (b) Block-Structured Mesh (c) Unstructured Mesh
Figure 4.7: Examples of meshing strategies. [23]
23
In figure 4.7, one can view three different mesh generation methods. Most modern CFD codes solve
all meshes as unstructured, even if they are structured or block-structured, keeping the connectivity ma-
trix and solving the equations using this matrix to determine the neighbors of each cell. On the other
hand, structured meshes can only be achieved for simple geometries which are rarely found in real life
applications. [21]
Another important aspect of the mesh generation is the generic shape of each control volume: tetra-
hedral, trimmed or polyhedral. Tetrahedral and polyhedral meshes are very efficient to employ and adapt
easily to any complex geometry [38], while trimmed meshes are more efficient at placing cells in refine-
ment zones [39], thus making them more suitable when refinements in particular regions of complex
geometries are of key importance. Prism layers are added in order to better predict the behavior of
boundary layers near solid walls, by assuming that the flow behaves as described in subsection 3.2.4.
The process of mesh generation is highly coupled to the process of geometry preparation and with
the simulation itself. In fact, it only became obvious that some details present in the geometry had to
be removed, once the mesh was indeed created. On the other hand, after the simulation, it was verified
that there was the need to return to the meshing process and refine specific areas.
Figure 4.8: Leading and trailing edges volumetric refinement controls.
To begin with, the best practice guidelines, provided by Verıssimo [40], were followed and the trimmed
meshing code coupled with prismatic layers was employed. The base size defined was equal to one
chord (main wing) and, in order, to capture certain details of the geometry and fluid phenomena, it was
necessary to use volumetric control refinements throughout the domain. Examples of these controls,
used to capture the curvature of the leading edge and the velocity gradients in this region, are depicted
in figure 4.8 in pink. Inside the volumetric control refinements, the target size of the cells was adjusted
and defined as a percentage of the base size. These percentages are presented in table 4.1.
24
Volumetric Refinement Target Size (% of base)
General Wakes 3.1250%GPS & Fuel Bransle 1.5625%Initial Wake - Fuselage 1.5625%Initial Wake - Landing Gears 1.5625%Protruding Components 1.5625%Tail LE, TE & Tip 3.1250%Winglet LE, TE & Tip 3.1250%Wing LE, TE & Tip 3.1250%Wing/Fuselage Fairing 6.2500%Wing Tip Vortex Wake 3.1250%
Table 4.1: Target size of the volumetric control refinements.
Regarding the prismatic layers, the stretching function employed was hyperbolic tangent, the stretch-
ing mode was wall thickness and the number of prism layers used was sixteen. The thickness of the
first layer was globally defined with equation 4.1. [40] The value computed by such expression, using a
y+target of 1.0, was 6.174× 10−6m.
y1 =y+target · µρ · Uτ
, where Uτ =
√τwρ
, τw =1
2Cf and Cf = 0.058Re−0.2 (4.1)
The total thickness of the prismatic layer was defined for each component of the aircraft with Prandtl’s
expression for the maximum boundary layer thickness for flat plates in turbulent regime (equation 4.2)
and enforced by surface controls. The several total thicknesses of the prismatic layer are presented in
table 4.2.
δ ≈ 0.37Re−0.2L , where L is the component’s reference length. (4.2)
Component Total Thickness (m)
Booms 3.145× 10−2
Front Landing Gear Axis 3.200× 10−4
Front Landing Gear Fork 5.000× 10−4
Front Landing Gear Struct 3.145× 10−2
Fuel Bransle 1.590× 10−3
Fuselage 1.000× 10−1
GPS Cover 1.078× 10−3
Landing Gear Wheels 3.980× 10−3
Pitot Tube 3.720× 10−3
Tail 7.090× 10−3
Wing & Winglet 1.078× 10−2
Table 4.2: Prismatic layer total thickness by component.
25
Finally it is important to state that the domain and the symmetry plane had specific controls of their
own. Both these surfaces had a surface cell target size of 800% of the base size. The symmetry plane
had a imposed minimum surface cell size of 1% of the wing’s chord. These controls, alongside with a
very slow cell growth rate, allow the mesh to have small cells and high definition near the aircraft and
big cells and less definition near the outer faces where the interest in the flow is lower.
Figure 4.9: Final volume mesh with 21.7 million cells.
The final mesh had 21.7 million cells and, according to Verıssimo [40], a trimmed mesh with about 20
million cells, ensures that the solutions for the integral coefficients show deviations to the exact numerical
solution of less than 3% of this value.
4.3 Numerical Models
With a Mach number of M = 0.2 and a Reynolds number of Rec = 2.2 × 106, at the altitude of 1500m
the flow is considered incompressible and has a Reynolds number above√Rex = 1000, the number
at which the flow is likely to be turbulent. [41] In fact, for this Reynolds number, the flow is not entirely
turbulent and transition from laminar to turbulent has an important role in predicting the flow.
In order to resolve the air flow around the UAV, the Reynolds-Averaged Navier-Stokes equations for
steady-state flow with constant density coupled with the Menter SST k−ω and the γ−Reθ models were
employed. As stated in chapter 3, the transition predictions of this model are consistent with aeronautical
experimental cases, justifying its implementation along with the Menter SST k − ω turbulence model.
The γ − Reθ transition model requires the definition of a free-stream edge and the value employed was
equal to four times the maximum boundary layer thickness. To resolve the boundary layers, low y+ wall
treatment was chosen, following the best practice guidelines of Verıssimo [40]. A maximum wall y+ of
0.25 was implemented.
26
The solver chosen for resolving the RANS equations was the segregated flow solver which is known
for being more stable and less resource consuming than the coupled flow solver. The best practice
guidelines state that, for incompressible flows, the segregated flow solver should be employed.
To prevent the solution from diverging, the cell quality remediation option, available in STAR-CCM+,
was employed. Due to insufficient cell quality, the numerical code from STAR-CCM+ has difficulties in
resolving gradients in punctual cells within the domain. This tool identifies these cells by determining if
the skewness angle exceeds a threshold. Once this process is over, the tool marks the cells and the
neighbors and modifies the gradient calculation in order to achieve a more robust solution. This gradient
modification is only applied locally and has a minimal effect in the solution’s accuracy. The cell quality
remediation runs every time the mesh changes and, since the employed mesh is not a moving mesh, it
only ran in the first iteration of the simulation.
Finally, monitors for the residuals, the integral coefficients (lift and drag coefficients) and the lift-to-
drag ratio were created.
4.4 Boundary Conditions
As stated before in section 4.1, a box was created to simulate the computational domain. Following the
best practice guidelines defined by Verıssimo [40], this box or domain was created with a distance from
each side to the aircraft of 60 chords, in order to make sure the boundaries are not introducing numerical
error due to proximity with the aircraft.
The domain was divided into 4 boundaries types and the respective conditions applied to each type
of boundary. The surface of the UAV, aligned with the flow (α = 0), was considered a no-slip wall.
The upwind and lateral faces of the domain were considered just one boundary and the condition
applied was of velocity inlet. In this case, the velocity magnitude and direction of cruise flight were
prescribed. The velocity was of 68.33 m/s and was aligned with the x-direction. Turbulence was also
specified in this boundary through its intensity and viscosity ratio. Values for these parameters of 0.1%
and 1, respectively, were imposed.
The downwind face was considered a pressure outlet and the symmetry plane was defined with the
boundary condition of the same name.
A recommendation from Verıssimo [40] suggests that the ambient source term of turbulence should
be turned on and the turbulence intensity and viscosity ratio prescribed. The values used for the ambient
source term were the same as the inlet, 0.1% for turbulence intensity and 1 for TVR.
27
4.5 Results
After 420 iterations, with a total simulation time of 430 min (approx. 7 hours and 10 min), the residu-
als had lowered their values down to 10−4/10−5, with the exception of the turbulent kinetic energy that
stabilized at 10−2. The run was stopped when the lift and drag coefficients and the lift-to-drag ratio had
converged in the fourth significant digit. The results of the values of interest are presented bellow in
table 4.3.
Mesh Size (×106) Solver Time (min) CL CD L/D C3/2L /CD
≈ 21.7 ≈ 430 0.3078 0.02310 13.32 7.392
Table 4.3: Results for the aerodynamic coefficients for the initial configuration.
A visualization of the flow around the UAV can be seen in figure 4.10 and a representation of stream-
lines over the wing and winglet in figure 4.11.
Figure 4.10: Streamwise velocity of the flow surrounding the UAV.
Figure 4.11: Streamlines over the wing and winglet.
28
Figure 4.12: Streamline visualization of the wing tip vortex created by the initial configuration.
Besides the values for the aerodynamic coefficients, it is important to analyze the flow and the char-
acteristics displayed by it. Starting from the previous figures, it is possible to see, in figure 4.10, a
volumetric representation of the longitudinal velocity of the flow surrounding the UAV, from which one
can observe the wakes being shed by several components of the aircraft. As expected the fuselage and
the landing gears produce the biggest wakes and, therefore, the biggest percentage of drag. From the
streamline visualization (figure 4.11 and 4.12) one can see the wing tip vortex.
Figure 4.13: Visualization of the pressure contours in four sections of the wing.
In the pressure contours presented in figure 4.13 the stagnation points can be seen inside the blue
contours in the leading edge, while lower pressures are exhibited in the upper surface, just as expected.
The pressure distributions and skin friction plots, over the surface of the wing for the same sections as
the contours, are depicted in the next page.
29
(a) 44.7% of the half span (b) 44.7% of the half span
(c) 59.9% of the half span (d) 59.9% of the half span
(e) 74.4% of the half span (f) 74.4% of the half span
(g) 89.3% of the half span (h) 89.3% of the half span
Figure 4.14: Pressure (left) and skin friction coefficient distributions in the upper surface of the wing(right) computed with the Menter SST k − ω and the γ − Reθ models (blue) and the Spalart-Allmarasmodel (red).
30
In figure 4.14 it is possible to see that the suction peak decreases as the spanwise coordinate in-
creases, decreasing the area inside the plot and ultimately decreasing the amount of lift being generated
in each section. The pressure distributions are in accordance with the expected lift spanwise distribution.
The optimum wing lift spanwise distribution is elliptical, with the maximum value in the root.
Results for the skin friction coefficient in a specific instant of time for flat plates in DNS simulations
show that, in the case of a transitional flow, this coefficient decreases in the laminar region and as it
approaches transition there is an increase in the value. These results can be seen in figure 4.15. [42]
In this way, the skin friction coefficient distributions on the suction side of the wing, depicted in the right
side of figure 4.14, are in conformity with the generic behavior of transitional flows.
Figure 4.15: Skin friction behavior of a transitional flow over a flat plate determined through DNS simu-lations. [42]
Both the pressure distributions and the skin friction coefficients, present in figure 4.14, show unex-
pected oscillations. Two explanations can arise to give clarity on this fact. These variations can be
created either by the lack of smoothness in the surface of the wing or by instability generated by the
γ −Reθ transition model.
In order to determine which of the two explanations is responsible for the variations in the pressure
and skin friction coefficient distributions, a simulation using the Spalart-Allmaras turbulence model was
created. As expected, by running the case with this turbulence model, the integral coefficients showed
deviations from the values obtained with the Menter SST k−ω and the γ−Reθ models. Values of 11.02
and 6.022 for L/D and C3/2L /CD were obtained with the Spalart-Allmaras turbulence model. In fact,
this was to expect since the Spalart-Allmaras turbulence model does not model transition and assumes
turbulent flow from the beginning. This behavior is exhibited by the skin friction coefficient in the suction
side of the wing, depicted in red in the plots on the right side of figure 4.14.
31
With both mathematical models generating the same oscillations in the pressure and skin friction
distributions, one can state that these are not a result of the γ −Reθ model but a lack in smoothness of
the geometry. In fact, the oscillations are a result of discontinuities in the geometry that would have to
be corrected in such a way that would allow introduction of points in the definition of the airfoil and allow
smoothness during meshing.
Figure 4.16 complements figure 4.14 by showing the skin friction coefficient on the surface of the
aircraft, where it is indeed possible to see the locations of laminar to turbulent transition, in the upper
and lower surfaces, predicted by the γ −Reθ transition model.
(a) Upper surface (b) Lower surface
Figure 4.16: Visualization of the skin friction coefficient on the surface of the UAV.
As expected, in the upper surface, in the location of the transition, 60% of the chord, there is a clear
line where the value of the skin friction coefficient increases drastically. The lack of smoothness in the
geometry of the wing can be seen in figure 4.16 and it is possible to see the early transition predicted
by the model at 74.4% of the half span. It’s important to note that, for symmetric airfoils at zero angle
of attack, such as the tail, the γ − Reθ transition model predicts transition in the same location for the
upper and lower surface. The same is not verified for the main wing, which is to expect since the profile
is cambered.
In figure 4.17, one can see the streamwise vorticity contours in the wake of the aircraft. A keen
observer can determine whether the current mesh needs more detail in the wakes or not. The current
mesh is able to capture the wakes from the fuselage, rear landing gear and wing tip device until they
almost diffuse completely. It’s important to keep in mind that the purpose of this mesh is to achieve
numerical convergence of the aerodynamic coefficients to a value that approaches reality with small
deviations from the exact numerical values. In this sense, the current mesh achieves the purpose and
renders acceptable values.
32
Figu
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17:
Vis
ualiz
atio
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stre
amw
ise
vort
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cont
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inth
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ake
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33
Chapter 5
Wing Tip Device Study
5.1 Sensitivity Study of Wing Tip Devices
Following the results presented by Gudmundsson [17], there was the need to explore the design of
winglets and raked tips. Since these two configurations present the best results for drag reduction
understanding which design parameters influence the objective function more and in which direction
was of extreme importance. For this task, the open source panel method software XFLR5 was employed
and, since it doesn’t allow the user to import geometry from CAD softwares, the internal CAD tool had
to be used to design the initial configuration.
Figure 5.1: Reference wing-tail pair in XFLR5.
All the simulations in XFLR5 were performed solely with the wing and tail, and all the increments in
the performance coefficients were determined with the reference wing-tail pair shown in figure 5.1. In
fact, an attempt on simulating with the fuselage was done and the results were far from reality. In this
simulation, XFLR5 created a vortex in the wing root, that should be resultant of the intersection with the
fuselage. Since the fuselage is creating a very small amount of lift and no drag, the section occupied
by this body could be, in fact, considered as empty. As a result, the wing/fuselage intersection vortex
is being over-dimensioned, rendering unrealistic values for drag. This effect is depicted in figure 5.2,
present in the next page.
35
Figure 5.2: Simulation with fuselage performed in XFLR5.
A mesh independence study of lift and drag coefficients as a function of the number of panels was
executed and showed that for 30 panels in the longitudinal direction (chord direction) the coefficients
were within 1% of deviation from the results obtained for the maximum number of panels tested (100
panels). On the other hand, since the wing has several sections of different planform configurations, the
convergence study in the span direction was not done with the number of panels but instead with the
size of each panel. For a size of 43 mm (around 112 panels for the cut-off configuration) the results
showed deviations from the smallest tested size around 2%. Overall, the results obtained show devia-
tions smaller than 2% of the most precise value determined.
To be able to compare results from the panel method with the solution from STAR-CCM+, the parasite
drag was calculated recurring to empirical formulas presented by Corke [41]. These formulas estimate
the base drag of the wings, assuming they behave in the same way as flat plates with minimum amount
of flow separations. In this case, the base drag can be related to the skin friction coefficient multiplied
by form factors to account for thickness effects. As for the fuselage, the drag estimated by the formulas
is being calculated with the skin friction coefficient. The parasite drag calculations present in Corke [41]
are empirical formulas based in the skin friction drag and can be considered a low level drag estimation.
The results for cruise flight (α = 0o) can be seen in table 5.1.
Simulation Solver Time (min) CL CD L/D C3/2L /CD
XFLR5 - Wing & Body & Tail ≈ 3 0.3235 0.03514 9.21 5.237XFLR5 - Wing & Tail ≈ 3 0.4064 0.03426 11.86 7.561RANS - Initial Configuration ≈ 430 0.3078 0.02310 13.32 7.392
Table 5.1: Comparison between the initial configurations in XFLR5 and STAR-CCM+.
36
Figure 5.3: Resulting streamlines of the reference wing-tail pair analysis.
The results obtained for the lifting surfaces only using the panel method show a deviation from the
RANS solution. This is to expect since the panel method code, present in XFLR5, shows deficient wake
roll-up. The wake created by wings is a three-dimensional phenomena and, in order to achieve a better
prediction of drag, it is required that the wake generated by the panel method rolls in the wing tip. In the
present code, this effect is barely seen, as it is shown in figure 5.3. This lack of roll-up yields results for
the drag coefficient that are far from reality. In the same figure, one can see the effect the wing tip has
in the wing wake. This effect can be seen by the curvature developed in the wake near the wing tip.
Another aspect that influences the results are the expressions from the Corke [41]. From experience,
these formulas tend to overestimate the parasite drag, resulting in smaller values for L/D. Nonetheless,
the usage of such formulas returns values closer to reality than the values obtained directly from XFLR5.
On the other hand, the configuration analyzed in STAR-CCM+ has a wingtip device that is not included
in the reference configuration used in XFLR5.
The XFLR5, for exploitation of design spaces through parameter variations is an excellent tool. With
a total solver time more than one hundred and forty times smaller than a RANS simulation, it can sweep
a series of parameter variations and provide information regarding the change in the aerodynamic coef-
ficients. This speed of simulation is the reason why the software is used in this sensitivity study.
Since these studies require a large amount of simulations, it is imperative that the user automates the
process in a way that minimizes user input errors in simulations and allows the programmer to perform
other tasks while the study runs. The introduction of automation options is not very common in most
softwares, thus a need to create a macro arises.
37
As stated by Gudmundsson [17], the design parameters for winglets and raked tips are the same.
These parameters are six and five of them (span, tip chord, sweep angle, dihedral angle and twist angle)
could be analyzied in this sensitivity study. The last parameter, the blending radius couldn’t be included
in this analysis since it influences interference drag and XFLR5 does not evaluate drag of this nature.
The list of parameter variations was compiled in a single text file, with each line containing five values
separated by spaces, in the order presented in the last paragraph. As stated before, the purpose of a
macro is to automate user related tasks. In this case, the tasks involve changing the parameters, choos-
ing the simulation code, running the simulation and exporting the aerodynamic data into output files.
The macro ran for 1024 cycles, corresponding to the total number of combinations of 4 variations of the
5 parameters. After 51.2 hours, 1024 text files with the lift and drag coefficients of each configuration
and respective simulation were created.
In order to process the resulting data from the sensitivity study, a code in MATLAB was developed
with the purpose of reading the parameters.txt file and the 1024 results files and collecting all the in-
formation in a matrix with 1024 lines and 11 columns. The first five columns are used to store the
parameters of each simulation, the sixth column the lift coefficient, from the seventh to the eight column
the drag corrections and in the ninth column the total drag coefficient. The last two columns are used
to store the performance ratios, L/D and C3/2L /CD, already calculated in MATLAB with the results from
columns six and nine.
At last, with the matrix completely defined, the code plotted the results using plots specially conceived
for sensitivity studies, present in MATLAB’s Statistics and Machine Learning Toolbox. The two plots
performed by the code were an interaction plot and a main effects plot. These plots can be seen in
subsections 5.1.1 and 5.1.2 for each type of wing tip device.
5.1.1 Wing Tip Device: Winglet
For the sensitivity study of the winglet, the five design parameters were changed according to the fol-
lowing list:
Span (m) 0.100, 0.200, 0.300, 0.400;
Tip chord (m) 0.100, 0.150, 0.200, 0.250;
Sweep angle (o) 0.0, 20.0, 40.0, 60.0;
Dihedral angle (o) 30.0, 40.0, 50.0, 60.0;
Twist angle (o) 0.0, 1.0, 2.0, 3.0;
Figures 5.4 and 5.5 depict five plots of the variation of L/D as a function of the variation of each
design parameter and the result in L/D of the interactions between all the parameters, respectively.
38
Figure 5.4: Influence of each design parameter in the objective function L/D.
As stated before, the graphics presented in this section come from the MATLAB’s Statistics and Ma-
chine Learning toolbox. The first graphic, the main effects plot, depicts the average L/D ratio for each
parameter variation. In other words, when given the list of input parameters and output solutions, MAT-
LAB searches the entries with the same parameter variation and calculates the average value for the
solution. It then plots the averaged value in each point of the graphic. For example, the value of L/D
plotted for an angle of sweep of 20 is the average of all the values of L/D computed by XFLR5 when
the angle of sweep was prescribed to 20.
Figure 5.4 is extremely relevant to determine which design parameters have the most influence in
the L/D. Therefore, it is possible to conclude that the design parameters with the most influence in the
L/D ratio are the span, followed by the sweep and dihedral angles. It is interesting to note that not all
winglet designs present an increase in performance parameters, since the value of L/D can achieve
values lower than the cut-off configuration, hence the importance of this work.
Figure 5.5 is a common result of any sensitivity study. Known as an interaction plot, it depicts
the objective function, in this case the L/D ratio, as a function of the interactions between the design
parameters. The interaction plot follows the same approach as the main effects plot and averages
the multiple possibilities for the objective function that fit in each point. Therefore, it is important to
understand that in each column of this matrix of graphics, a specific design parameter is considered a
variable and in each line is considered a constant, e. g. if one is analyzing the graphic in position (2,3),
he or she is looking at the value of L/D when the tip chord is considered constant and the sweep angle
variable. For each point in this specific plot (2,3) there are several values for L/D since the rest of the
parameters are changing. The value plotted is the average of all the possibilities.
39
Figure 5.5: Interaction plot for L/D.
As expected, the increase in span renders a substantial increase in the performance parameters,
since the results for C3/2L /CD show the same variations as L/D, with the only difference being in its
value. The main reason for this result is the fact that by employing a wing tip device, one is increasing
the effective aspect ratio and consequently increasing lift and decreasing lift-induced drag.
With no surprises, with the increase in sweep angle there is a decrease in the performance coeffi-
cients. It is a known fact that sweep is extremely beneficial when the flow has high subsonic characteris-
tics. For low subsonic applications, including sweep only decreases the effective aspect ratio, increasing
the lift-induced drag and decreasing L/D and C3/2L /CD. In these cases, including sweep might be done
for stability and control purposes or for aesthetic reasons.
As for the dihedral angle, smaller values are preferable, since they mean the winglet is more hori-
zontal, making, thus, lift more vertical. By reducing the dihedral, the cosine of the angle between the
lift force and the vertical plane approaches one, rendering a bigger magnitude for lift and, therefore, a
bigger value for the performance coefficients.
At last, it is relevant to present the combination of the design parameters that appeared in the study
with the best performance coefficients.
40
Span Tip Chord Sweep Dihedral Twist L/D ∆L/D C3/2L /CD ∆C
3/2L /CD
0.400 m 0.100 m 0.0 30.0 3.0 12.47 5.14% 8.22 8.73%
Table 5.2: Best combination of the winglet design parameters.
5.1.2 Wing Tip Device: Raked Tip
The parameter variations, for the sensitivity study of the raked tip, are presented bellow:
Span (m) 0.100, 0.200, 0.300, 0.400;
Tip chord (m) 0.050, 0.075, 0.100, 0.125;
Sweep angle (o) 0.0, 20.0, 40.0, 60.0;
Dihedral angle (o) 0.0, 2.5, 5.0, 7.5;
Twist angle (o) 0.0, 1.0, 2.0, 3.0;
The same graphics, that were presented for the winglet configuration, are now presented for the
raked tip, in figures 5.6 and 5.7.
Figure 5.6: Influence of each design parameter in the objective function L/D.
Just as in the case of the winglet, the parameter with more influence in L/D is the span followed
by the sweep angle. Since raked tips have small tip chords and dihedral angles, the variation in these
parameters were smaller than for the winglet, rendering smaller influences in the value of L/D. Just as
in the case of the winglet, as the twist angle increases, the incidence angle of the tip airfoil is bigger,
creating an increase in L/D through the increase in lift. Of all parameters, the dihedral angle is the
least important, nonetheless the sensitivity study shows that the performance coefficients increase as
the dihedral approaches 2.5o and decreases from that point forward.
41
By the interaction plot depicted bellow, one can see the lack of effect the dihedral has on the per-
formance coefficients and the lack of interaction it has with the rest of the design parameters. On the
other hand, it is possible to state that the interaction between the sweep angle and the rest of the design
parameters is high, meaning that the decision on which sweep angle to employ is a key process in the
definition of the geometry of the raked tip.
Figure 5.7: Interaction plot for L/D.
Again, it is important to note that the variations of C3/2L /CD are the same as the variations of L/D
with the difference being in the value itself. The combination of parameters, that maximize the perfor-
mance coefficients, is present bellow.
Span Tip Chord Sweep Dihedral Twist L/D ∆L/D C3/2L /CD ∆C
3/2L /CD
0.400 m 0.075 m 20.0 2.5 3.0 12.56 5.90% 8.30 9.81%
Table 5.3: Best combination of the raked tip design parameters.
As stated by Gudmundsson in is work [17], raked tips display slightly better performance coefficients
than winglets. These results were obtained from inviscid calculations for the same initial configuration,
just as the sensitivity study performed in this Thesis. In fact, using Gudmundsson’s results, one can
validate the tendency to prefer raked tips over winglets exhibited by this sensitivity study.
42
5.2 Confirmation of the Results from the Sensitivity Study
Even though the values obtained for the performance coefficients with the sensitivity study were cor-
rected with expressions from Corke [41], the variation in parasite drag due to the inclusion of a wing tip
device was not estimated.
With the fast growth in the performance of computers, the ability to almost simulate reality with RANS
is no longer unthinkable and it is indeed indispensable. With the increases in the performance coeffi-
cients, discovered with the sensitivity study, being promising, it was important to confirm the tendency
showed by these results with more powerful tools such as the Reynolds-Averaged Navier-Stokes equa-
tions.
In order to validate the results from the sensitivity study, the winglet present in the initial configuration
was either completely removed or replaced by the best configurations for winglet and raked tip from the
sensitivity study.
In this section, the configurations were analyzed with the same procedure as the initial configuration
(chapter 4) and in chapter 6, the best of the two wing tip devices and the cut-off configuration are target of
a grid refinement study in order to guarantee that the increases discovered in section 5.1, and confirmed
in this section, are accurate.
(a) Cut-off wing (b) Best winglet (c) Best raked tip
Figure 5.8: Configurations used to validate the results from the Design of Experiments.
The process described in chapter 4 was implemented for the three scenarios and the obtained values
are shown in table 5.4.
Simulation Mesh Size (×106) CL CD L/D ∆L/D C3/2L /CD ∆C
3/2L /CD
Cut-off Wing ≈ 16.6 0.2886 0.02364 12.21 6.558Winglet ≈ 23.3 0.3341 0.02281 14.66 20.07% 8.466 29.09%Raked Tip ≈ 22.4 0.3362 0.02364 14.22 14.14% 8.246 25.74%
Table 5.4: Comparison between tip devices and the cut-off wing in STAR-CCM+.
43
By inspection of the table in the previous page, it is possible to state that the tendency to increase the
performance coefficients with the inclusion of a wing tip device, shown by XFLR5 and the panel method,
is validated when viscid simulations are performed in STAR-CCM+.
Simulation CL Deviation CD Deviation L/D Deviation C3/2L /CD Deviation
XFLR5 0.4064 40.82% 0.03426 44.92% 9.21 -24.57% 5.237 -20.14%STAR-CCM+ 0.2886 0.02364 12.21 6.558
Table 5.5: Comparison of the cut-off configuration in XFLR5 and STAR-CCM+.
Table 5.5 presents the comparison between the results obtained for the cut-off configuration with
the panel method and the RANS equations. The deviations presented, were calculated considering the
values from STAR-CCM+ as the reference values. As stated in previous sections, XFLR5 cannot deter-
mine the drag of non-lifting components, hence the need to implement the correlations from Corke [41].
In previous sections, comparisons between XFLR5 and STAR-CCM+ have already been presented. In-
deed, the difference is that in these sections, the comparison was done between the cut-off wing and the
initial configuration. When comparing equal configurations, in this case cut-off configurations, one can
reinforce the statements written before. Due to the non-existence of a body, XFLR5 overestimates the lift
coefficients in 40%, and, regarding the empirical correlations for parasite drag, these overestimate the
drag coefficient in 45%. Since the overestimation of drag is bigger than lift, the performance coefficients
present negative deviations from the values obtained from STAR-CCM+.
Simulation ∆L/D Deviation ∆C3/2L /CD Deviation
XFLR5 5.14% -74.39% 8.73% -69.99%STAR-CCM+ 20.07% 29.09%
Table 5.6: Comparison of the increase created by the winglet in XFLR5 and STAR-CCM+.
When comparing the increases in the performance coefficients by the winglet, the values from the
panel method simulations are far from the increases obtained with RANS. Nonetheless, the tendency
to improve performance with the inclusion of a winglet is unquestionable. The increase in lift, although
bigger with RANS, is found in both types of simulations. The decrease in drag, however, is not found in
the panel method when, in fact, exists.
The increase in L/D and C3/2L /CD, expressed by the panel method is based only in the increase
of the lift coefficient. When performing Reynolds-Averaged Navier-Stokes simulations there is, in fact,
a decrease in drag, creating the 74% and 70% negative deviation in L/D and C3/2L /CD, respectively,
found in table 5.6.
44
Figure 5.9: Streamline visualization of the wing tip vortex created by the three configurations analyzedin STAR-CCM+. By order of appearance: cut-off wing, winglet and raked tip.
45
x = 0.8m
x = 1.3m
x = 1.8m
x = 2.3m
Figure 5.10: Streamwise vorticity contours of the three configurations analyzed in STAR-CCM+. Fromleft to right, the configurations are cut-off wing, best winglet and best raked tip.
46
It’s important to state that the results from STAR-CCM+ are in conformity with the results obtained by
Panagiotou et al. [6], where the inclusion of winglets increased the L/D in 16% for a null angle of attack
and 19% for a four degrees angle of attack. In the mentioned paper, the numerical models are equal
to the ones used in this Thesis with the exception of the turbulence and transition models. Instead of
using the Menter SST k − ω and the γ − Reθ models, Panagiotou et al. employed the Spalart-Allmaras
turbulence model. The fact, that the results of this Thesis are in conformity with a paper with a different
turbulence model, enforces that the tendency to increase performance with winglets is not derived from
a numerical model but is instead due to physical phenomena.
Simulation ∆L/D Deviation ∆C3/2L /CD Deviation
XFLR5 5.90% -58.51% 9.81% -61.89%STAR-CCM+ 14.22% 25.74%
Table 5.7: Comparison of the increase created by the raked tip in XFLR5 and STAR-CCM+.
Just as in the case of the winglet, the results for the case of the raked tip show a deviation of the
panel method from the RANS equations. This situation was expected. In fact, the panel method only
evaluates a portion of pressure drag, the lift-induced drag. On the other hand, the Reynolds-Averaged
Navier-Stokes equations are able to compute both the pressure and viscous drag. The deviations of the
XFLR5 towards the values obtained with STAR-CCM+ are considerable due to the fact that the method
cannot predict a large amount of drag that is also being influenced by the inclusion of a wing tip device.
From the results obtained in XFLR5, the raked tip showed larger increases in the performance co-
efficients than the winglet. However, the results from the RANS equations (table 5.4) show a different
outcome. These results can be confirmed by an inspection of the wing tip vortex, showed in figure 5.9,
and the streamwise vorticity contours present in the previous page. The fact that the raked tip is not
decreasing the intensity of the wing tip vortex can be seen in the vorticity plots. While the winglet is
diffusing and decreasing the intensity of the vortex, the raked tip is just diffusing the vorticity, hence the
lack of decrease in the drag coefficient showed by this tip device.
Diffusion of the wing tip vorticity enables less time between take-offs and is often a factor to consider
when implementing a wing tip device. Since this work focuses solely in cruise flight performance, the
wing tip device must decrease the magnitude of the vorticity. The winglet achieves this goal but does
not diffuse as much as the raked tip. Nonetheless, this configuration meets the aims of this Thesis.
47
Chapter 6
Grid Refinement Study
As a regular practice when performing CFD simulations, a grid refinement study is developed in order to
evaluate the uncertainty associated with it. A spatial grid convergence study is done when it is desired
to determine the discretization error in a CFD simulation. In the present Thesis the study is of signifi-
cant importance since the outcome will allow confirmation of the increases in cruise flight performance
predicted when a wing tip device is employed.
6.1 Methodology
As explained in chapter 3, the grid refinement study is done by testing three or more volume meshes and
then computing the numerical error for the engineering quantities involved in the simulation. In this case,
one wishes to determine the value for the aerodynamic coefficients when the mesh tends to a continuum.
The current study involves three unstructured finite volume meshes: a coarse, an intermediate and
a thin mesh. The mesh employed in section 5.2 was considered the coarse mesh for the study and
successive refinements were performed to arrive to the remaining meshes. To obtain the intermediate
mesh, the final number of cells was aimed at twice the initial value. This was achieved by successively
decreasing the cell base size until the number of cells was near the target value. As for the thin mesh,
the objective was to achieve 3 times the number of cells of the initial mesh. The cell base sizes that
accomplished these objectives were, respectively, 0.350 m and 0.271 m. The refined simulations were
created in the same case as the previous mesh. By doing the refinement of the intermediate mesh and
running on the converged case of the coarse mesh and the thin mesh on the intermediate mesh, one is
assisting the cases to converge, resulting in smaller simulation times.
Finally a code in MATLAB was created to process the data from the simulations and provide infor-
mation regarding the convergence of the solution.
49
6.2 Results
Since the effective output from STAR-CCM+, in terms of engineering quantities for the current sim-
ulations, are the lift and drag coefficients, the convergence study was only performed for these two
quantities. In fact, the performance parameters can be considered a result from post-processing the lift
and drag coefficients. In this way, the quantities of interest that depend directly from the simulation are
the two coefficients mentioned above.
Before analyzing the variations of the quantities of interest it is important to visualize the variations
in the flow due to consecutive refinements. An important aspect of the flow for the current Thesis is
the wing tip streamwise vorticity. Predicting a correct vortex and capturing as many details as possible
in these flow structures renders a correct prediction of the lift-induced drag. Figures 6.1 and 6.2 show
the streamwise vorticity contours in the wing tip of the cut-off and winglet configuration, for the three
meshes, in section x = 2.3 m after the wing tip trailing edge.
(a) 16.6 million points (b) 33.3 million points (c) 55.6 million points
Figure 6.1: Effect of the mesh size in the wing tip streamwise vorticity of the cut-off configuration.
Figures 6.1 and 6.2 show that streamwise vorticity, specially wing tip vorticity, has more definition
every time the mesh is refined. In fact, for the cut-off configuration the thin mesh shows that the vortex
is not as diffuse as initially thought. As consequence of this decrease in diffusion, the intensity of the
core vorticity is higher. Such fact can be seen in both configurations.
(a) 23.3 million points (b) 42.6 million points (c) 67.2 million points
Figure 6.2: Effect of the mesh size in the wing tip streamwise vorticity of the winglet configuration.
50
Having seen how the results affect the wing tip vorticity, it is important to see the variations in the
aerodynamic coefficients for both configurations.
Cut-off Configuration Winglet Configuration
Mesh Size (×106) CL CD Mesh Size (×106) CL CD
≈ 16.6 0.2886 0.02364 ≈ 23.3 0.3341 0.02281≈ 33.8 0.2882 0.02356 ≈ 42.6 0.3347 0.02277≈ 55.6 0.2881 0.02359 ≈ 67.2 0.3349 0.02276
Table 6.1: Influence of the mesh size in the aerodynamic coefficients for both configurations.
6.3 Grid Convergence
The objective of this study is indeed to estimate the exact numerical solution for the engineering quan-
tities and compute the uncertainty associated with the increases predicted in section 5.2. In figure 6.3
the variations presented in table 6.1 for the cut-off configuration are plotted in blue. Present also, in
black dashed lines, are the estimations of the exact numerical values for the lift and drag coefficients,
obtained through the Richardson Extrapolation. These results are plotted against the grid refinement
number, ri =√hf/hi, where hf is the number of cells of the thin mesh and hi the number of cells of the
intended mesh.
Figure 6.3: Grid convergence plots for the cut-off configuration.
For the cut-off configuration the exact numerical solution for the lift and drag coefficients are, respec-
tively, 0.288065 and 0.023606. These values correspond to an estimation of the engineering quantities,
obtained if the mesh would tend to a continuum. In this case the spatial discretization error would be
zero since there would be no discretization of the volume.
51
The error bars present in figure 6.3 are computed as a function of the deviation from each solution
to the exact numerical value multiplied by a safety factor of Fs = 1.25. It is possible to see that, unlike
drag, where results have difficulty in converging, for lift, results converge throughout the three meshes.
For the intermediate mesh there is an increase in the uncertainty associated with the prediction of drag,
inverted when the thin mesh was applied.
Plots with the variations in the lift and drag coefficients and the Richardson Extrapolation for the
winglet configuration are presented in figure 6.4. For this case the exact numerical values for the aero-
dynamic coefficients are 0.334994 for lift and 0.022757 for drag.
Figure 6.4: Grid convergence plots for the winglet configuration.
Unlike the cut-off configuration, the engineering quantities in the winglet configuration converged
throughout the three configurations. As a mean of evaluating the numerical accuracy of the solutions
computed in section 5.2, the uncertainty for the three meshes of both configurations was calculated and
is depicted in table 6.2.
Thin Mesh Intermediate Mesh Coarse Mesh
CL CD CL CD CL CD
Cut-off Configuration 0.0153% 0.0842% 0.0587% 0.2431% 0.0232% 0.1805%Winglet Configuration 0.0352% 0.0194% 0.1098% 0.0743% 0.3337% 0.2941%
Table 6.2: Uncertainty of the three meshes in the prediction of the lift and drag coefficients.
With information regarding the exact numerical solutions of both configurations, it is the intent of this
study to provide information regarding the deviation of the computed performance increases to the exact
numerical solution, estimated by the Richardson Extrapolation. Table 6.3 shows the exact numerical
solutions of all the coefficients for both configurations.
52
Configuration CL CD L/D ∆L/D C3/2L /CD ∆C
3/2L /CD
Cut-off Wing 0.288065 0.023606 12.20 6.550Winglet 0.334994 0.022757 14.72 20.66% 8.520 30.08%
Table 6.3: Exact numerical solution of the aerodynamic and performance coefficients.
When comparing the results from the coarse mesh and the exact numerical solution, one can state
that the coarse mesh predicts accurately the performance increases. This mesh, employed throughout
the Thesis shows negative deviations of approximately 3% of the increases computed with the exact
numerical solution.
Solution ∆L/D Deviation ∆C3/2L /CD Deviation
Coarse Mesh 20.07% -2.86% 29.09% -3.29%Exact Numerical 20.66% 30.08%
Table 6.4: Comparison of the increases in performance with the exact numerical values.
The confirmation of the predicted increases concludes the work performed for this Thesis.
53
Chapter 7
Conclusions
7.1 Achievements
Having concluded the implementation and obtained results, it is presented in this chapter the main
outputs achieved with the work performed. In chapter 4, Computational Fluid Dynamics models were
implemented to external aerodynamics in order to determine the relevant engineering quantities. In this
way, a successful prediction of the integral coefficients for the initial configuration was achieved with the
Reynolds-Averaged Navier-Stokes equations, the Menter SST k − ω turbulence model and the γ − Reθtransition model.
Following the literature review performed in chapter 2, a sensitivity study of the design parameters,
for two wing tip devices, the winglet and the raked tip, was implemented and results showed that for
both configurations, the wing tip device’s span is the parameter with the most influence in the perfor-
mance coefficients L/D and C3/2L /CD. Such result was expected since, by inspection of equation 2.2,
increasing span reduces lift-induced drag, hence increasing the performance coefficients. Following the
span, the most influential parameter is the sweep angle. It is a known fact that for high subsonic flows,
adding sweep angle proves beneficial since it allows the wing to experience a lower effective Mach num-
ber while traveling at a higher Mach number. This fact allows for a retardation of the shock waves that
appear in transonic regions. For applications at low subsonic flight, adding sweep angle only decreases
the effective aspect ratio of the wing, thus increasing lift-induced drag.
When comparing the inviscid results from the sensitivity study, obtained with panel method simula-
tions, the two wing tip devices prove the tendency presented in chapter 2. In inviscid flow, the raked tip
show higher increases in the performance coefficients than winglets.
An attempt at confirming these increases with the RANS equations was performed and results proved
a different tendency. The same numerical models and mesh definitions of chapter 4 were applied and
results show that, in fact, the winglet has higher increases in L/D and C3/2L /CD than the raked tip.
55
When comparing the results obtained for the winglet, with the RANS equations, the Menter SST
k − ω turbulence model and the γ −Reθ transition model, with the results obtained by Panagiotou et al.
[6], one can prove that the decrease in drag and consequent increase in performance through the in-
troduction of winglets is not dependent on the turbulence model employed in CFD simulations. In fact,
Panagiotou et al. [6] implemented the Spallart-Allmaras turbulence model and achieved increases of
16% in the lift-to-drag ratio for a null angle of attack. Increases, produced by the inclusion of a winglet,
of 20.07% and 29.09% for L/D and C3/2L /CD, respectively, were determined in chapter 5, having the
cut-off configuration as the reference value.
To determine the uncertainty of the results, a grid refinement study was performed and the exact
numerical values were estimated. With small variations, the results determined in chapter 5 can be
considered accurate and the best practices guidelines from Verıssimo [40] recommended for future
works. The estimations for the exact numerical values show increases of 20.66% in L/D and 30.08% in
C3/2L /CD regarding the cut-off configuration.
At last, it is important to state that the main objective was achieved. The proposed wing tip device,
the winglet, increases L/D in 10.51% and 15.26% in C3/2L /CD, with regard to the initial configuration
provided by the Portuguese Air Force.
7.2 Future Work
While performing the work presented in this Thesis, several questions of aerodynamic and structural
relevance arose. Some were addressed and are depicted in several chapters while others must remain
for future work. In fact, this work focused in optimizing the design of wing tip devices solely from the
aerodynamic point of view.
Before exploring the work necessary in other engineering areas, it is important to discuss what re-
mains to be done and what is indeed relevant in the aerodynamic department. In chapter 4 problems
with the geometry were encountered and made difficult better predictions of the integral coefficients and
of the dynamics of the flow. Recommendations for improvements of the definition of the geometry are
unavoidable and of extreme importance.
As stated before, during the sensitivity study, one design parameter was left out. The blending
radius is a design parameter highly connected with interference drag. Since this drag is of viscous
nature, performing a sensitivity study of this parameter in the panel method would result in an unrealistic
outcome. A sensitivity study with RANS equations would be interesting to see in future works. The effect
of this blending radius in the decrease of drag is still to be completely determined and there is space to
develop work in this direction.
56
A sensitivity study of the blending radius could also benefit the optimization of the wing root. The
drag created by the wing/fuselage junction is mainly interference drag and could be reduced by blend-
ing both surfaces and optimizing the radius. Other techniques, such as the adjoint formulation, could
be employed and work developed in order to achieve higher drag reductions not possible by a simple
blending radius.
On the other hand, the results produced in this Thesis concern only cruise flight at null angle of at-
tack. Evaluation of the behavior of the winglet in other flight scenarios would be important to understand
how the performance in these flight segments is affected by the inclusion of this wing tip device. The ef-
fect of the introduction of wing tip devices in take-off, climb and landing scenarios is of great importance
and is not reflected in this work.
The present Thesis set out to maximize the performance coefficients in order to maximize range and
endurance. Since these concepts depend also on the structure of the aircraft, an optimization of the
structure of the aircraft would guarantee that the increases determined in aerodynamic performance do
not vanish due to an increase in structural weight.
At last, the impact of the introduction of the winglet in the structure and in the aeroelastic behavior of
the aircraft should be analyzed. The influence in stability and control of the winglet is also an important
aspect due to the unmanned character of this vehicle.
57
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