design of a morphing wing based on a fluid structural interface analysis
TRANSCRIPT
Design of a morphing wing based on a Fluid Structure Interaction analysis
André Martins Abrantes Leite
Dissertação para obtenção do Grau de Mestre em
Engenharia Aeroespacial
Júri
Presidente: Prof. Afzal Suleman Orientador: Prof. Afzal Suleman
Co-Orientador: Prof. Fernando José Pacharro Lau Vogal: Prof. Pedro Vieira Gamboa
Setembro de 2008
Resumo
Nesta tese é apresentado o projecto de uma asa adaptativa com variação de envergadura e
per�l. Este trabalho é um aperfeiçoamento do estudo �Designing, Building and Wind Tunnel
Testing of a Morphing Wing for Drag Reduction" realizado pelo aluno de Doutoramento José
Vale.
O projecto da asa adaptativa é baseado em análises �uido/estrutura acopladas onde são
criados dois modelos: um modelo de elementos �nitos no software Ansysr Multiphysics para
realizar a análise estrutural e um modelo de volume �nito no software Ansys-CFXr para realizar
a análise computacional de mecânica dos �uidos. Nestas análises os materiais, as secções e as
diferentes con�gurações de nervuras e longarina são analisados de forma a escolher um modelo
com a melhor relação entre peso, resistência estrutural e desempenho. O modelo escolhido é
composto por uma casca de compósito unidireccional com 0.2mm de espessura, reforçada com
compósito bidireccional com 1mm de espessura em torno das nervuras. A estrutura interna é
composta por uma longarina em forma de Z de compósito unidireccional e 8 nervuras também
de compósito unidireccional igualmente espaçadas ao longo da envergadura.
Com este modelo solucionamos os problemas de contornos pouco suaves dos per�s aerod-
inâmicos da asa com pele �exivel projectada em �Designing, Building and Wind Tunnel Testing
of a Morphing Wing for Drag Reduction", substituindo a pele de silicone por compósito. A asa
desenvolvida neste trabalho tem uma massa de 0.868 Kg. Isto traduz-se num aumento de 0.17
Kg quando comparada com a asa do Antex-X2 que serviu igualmente de referência no projecto
acima mencionado.
O estudo aerodinâmico da asa adaptativa revelou uma redução de resistência da asa de
33.74% à velocidade de descolagem e uma redução de 50.78% à velocidade de cruzeiro. Esta
redução torna-se ainda maior com o aumento da velocidade. A pequena deformação resultante
das cargas aerodinâmicas em voo, no modelo escolhido, tem um impacto pequeno no desempenho
aerodinâmico da asa, já que a variação de resistência e sustentação é menor que 5%.
Palavras-chave: Morphing, asa adaptativa, projecto de uma asa adaptativa, análise �u-
ido/estrutura acoplada, análise acoplada, elementos �nitos, volume �nito, mecânica dos �uidos
computacional, análise de desempenho, desempenho aerodinâmico.
i
Abstract
In this thesis it is presented a morphing wing sizing with wingspan variation and airfoil shape
change. The study is a development and improvement of the �Designing, Building and Wind
Tunnel Testing of a Morphing Wing for Drag Reduction" performed by the Ph.D. student Jose
Vale.
The wing sizing is based on coupled �uid/structural analyses where a �nite element model is
created in Ansysr Multiphysics to perform the structural analyses and a �nite volume model is
created in Ansys-CFXr to perform the computational �uid dynamic analysis. In those analyses
materials, sections, ribs and spars con�gurations have been changed in order to chose a model
with the best relation among weight, structural resistance and performance. The chosen model
uses a 0.2 mm thickness shell made of composite rob reinforced with 1mm thickness composite
sheet material around the ribs, a Z shape spar of composite rob and 8 ribs made of composite
rob, equally spaced throughout span direction.
With this model we have solved the wing skin problems exposed in �Designing, Building and
Wind Tunnel Testing of a Morphing Wing for Drag Reduction" replacing the celicone shell by a
composite shell. The new morphing wing has a mass of 0.868 Kg. This traduces to an increase
of 0.17 Kg when comparing with the reference Antex-X2 wing.
A performance study comparing these two wings revealed a performance improvement with
the increase in aircraft speed. A drag reduction of 33.74% has been achieved at take-o� speed
and a 50.78% reduction at cruising speed. The small deformation resulting from the aerodynamic
loads revealed to have a small e�ect in the wing aerodynamic performance as changes in aircraft
lift and drag are smaller than 5%.
Key-words: Morphing, morphing wing, wing sizing, adaptative aerostructures, coupled
�uid/structural analyses, coupled analysis, �nite element model, �nite volume model, computa-
tional �uid dynamic, performance study, aerodynamic performance.
ii
Acknowledgments
I am very thankful to Prof. Afzal Suleman for the opportunity he gave me to work in this
project.
I am also very thankful for the availability and all the guidance provided by Prof. Fernando
Lau.
My friend Jose Vale was also a great support. I think we made a good team and I wish him
the best for his PhD.
Last but not least I would like to thank my parents, specially my mother, and So�a for all
their help.
iii
Contents
List of Figures ix
List of Tables x
1 Introduction 3
1.1 Morphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Historic Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Morphing concepts and their applicabilities . . . . . . . . . . . . . . . . . . . . . 5
2 Problem de�nition 14
3 Coupled �eld analysis 18
3.1 Ansysr coupled �eld analysis procedure . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Ansysr multi�eld analysis using code coupling . . . . . . . . . . . . . . . . . . . 19
3.3 Load transfer between physical �elds . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Analytical Models 25
4.1 Structural formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Fluid �ow formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 Eddy Viscosity Turbulence Models . . . . . . . . . . . . . . . . . . . . . . 30
4.3.2 k − ε turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.3 Shear stress transport turbulence model . . . . . . . . . . . . . . . . . . . 32
4.4 Near wall treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Mesh deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.6 CFX r Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Problem Modulation 40
5.1 Structural �eld modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
iv
5.1.1 Matlab parametric program to create structural model . . . . . . . . . . . 46
5.2 Fluid �eld modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Interface surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Convergence study 52
6.1 CFX r model convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2 Ansysr model convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7 Wing performance analysis 59
7.1 Low speed and high speed airfoil de�nition . . . . . . . . . . . . . . . . . . . . . . 59
7.2 Best wing con�guration de�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.3 Performance improvement analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8 Structural analysis 71
8.1 One way �uid/structural analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.1.1 Shell de�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.1.2 Spar study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.1.3 Ribs de�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.2 Bidirectional coupled structural/�uid analysis . . . . . . . . . . . . . . . . . . . . 86
9 Conclusion 99
Bibliography 101
v
List of Figures
1.1 Wing morphologies for hawks (left) and pigeons (right) . . . . . . . . . . . . . . . 3
1.2 Spider plot comparison of �xed and morphing wing aircraft . . . . . . . . . . . . 4
1.3 AFTI F-111 in �ight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Optimized adaptive wing design obtained in [48] . . . . . . . . . . . . . . . . . . 7
1.5 UAV employing PBP actuated morphing panels . . . . . . . . . . . . . . . . . . . 7
1.6 Goodyears Model GA-468 In�atoplane c. 1950 . . . . . . . . . . . . . . . . . . . 8
1.7 Nastic tensile tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Bump �attening prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.9 Trailing Edge De�ection Morphing Wing Prototype . . . . . . . . . . . . . . . . 9
1.10 Lockheed Martin MAS Wind Tunnel Model During a Morphing Sequence . . . . 10
1.11 Morphing con�gurations of the MFX-1 . . . . . . . . . . . . . . . . . . . . . . . . 11
1.12 Pneumatic Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.13 Experimental model for winglets control study . . . . . . . . . . . . . . . . . . . . 12
1.14 Fully adaptive morphing aircraft design by Virginia Plytchnic Institute and State
University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Di�erent wing span con�gurations . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Di�erent airfoil con�gurations used on the telescopic wing . . . . . . . . . . . . . 17
3.1 Ansysr Multi�eld Solver Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Ansysr and Ansys-CFXr �elds solved simultaneously (on the left) and Ansysr
and Ansys-CFXr �elds solved sequentially (on the right) . . . . . . . . . . . . . 21
3.3 Pro�le preserving interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Globally conservative interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Example of a pro�le preserving interpolation with a coarse mesh in the receiver
side (left) and a coarse mesh in the sender side (right) . . . . . . . . . . . . . . . 22
3.6 Example of a globally conservative interpolation with a coarse mesh in the sender
side (left) and a coarse mesh in the receiver side (right) . . . . . . . . . . . . . . 23
3.7 Improperly mapped nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
vi
4.1 Near wall velocity pro�le of a turbulent boundary layer . . . . . . . . . . . . . . . 34
5.1 Structural �eld, Ansysr Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Di�erent wing con�gurations: a) with the inner wing 25% deployed b) with 50%
deployed and c) 75% deployed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Connections between the two wings . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Interior structure of the wing using di�erent spar sections . . . . . . . . . . . . . 43
5.5 Interior structure of the wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.6 Ansysr model boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.7 Inputs of the Matlab parametric function to generate the airfoil . . . . . . . . . . 46
5.8 Mesh of the CFD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.9 CFXr Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.10 Interface surfaces. Ansysr model on the left side of the �gure and CFXr model
on the right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.11 Non-Matching Area fraction obtained in the solver output �le . . . . . . . . . . . 51
5.12 Force Vectors in the Ansysr model . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.1 CFD convergence study; sparse mesh . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2 CFD convergence study; re�ned mesh . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3 CFD convergence study; re�ned mesh with prism mesh near the wing . . . . . . . 55
6.4 CFD convergence study, graphic of aerodynamic results function of the mesh re-
�nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.5 Structural convergence study; evolution of the maximum deformation, Von Mises
tension, structural energy error and time function of the number of elements . . . 57
6.6 Deformation (left) and Von Mises Stresses (right) obtained with the chosen mesh,
analyzed in this convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.1 Graphics Clα Cdα and ClCd for the high speed airfoils studied . . . . . . . . . . 60
7.2 Airfoil shape of the di�erent airfoils in study . . . . . . . . . . . . . . . . . . . . . 61
7.3 Graphics Clα Cdα and ClCd for the low speed airfoils studied . . . . . . . . . . . 62
7.4 Cl(α), Cd(α), CL(α)Cd(α) and Cl(Cd) for di�erent wing con�gurations . . . . . . . . 63
7.5 Cd(Cl) for di�erent wing con�gurations . . . . . . . . . . . . . . . . . . . . . . . 64
7.6 Cd(Cl), optimal con�guration, for each airfoil . . . . . . . . . . . . . . . . . . . . 65
7.7 Cd(Cl) for wing optimal con�guration . . . . . . . . . . . . . . . . . . . . . . . . 65
7.8 Cd(Cl) for optimal wing con�guration and for Antex wing . . . . . . . . . . . . . 66
7.9 Increase in lift and decrease in drag of the morphing wing comparing with the
antex wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
vii
7.10 Possible increase in weight function of the velocity, accomplished by the morphing
comparing with the Antex wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.11 Decrease in drag function of the velocity, accomplished by the morphing wing
comparing with the antex wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.1 Ribs section used in the structural analysis . . . . . . . . . . . . . . . . . . . . . 72
8.2 Ribs con�guration the shell de�nition analysis (the picture also includes the spar) 73
8.3 Spar section used in the shell de�nition analysis . . . . . . . . . . . . . . . . . . . 74
8.4 1mm epoxy material in all wing shell; wing shell deformation . . . . . . . . . . . 75
8.5 1mm thickness composite rob shell model; wing shell failure study . . . . . . . . 75
8.6 1mm thickness composite rob shell model; wing shell failure study without con-
strained root elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.7 1mm thickness composite rob shell model; wing conections failure study . . . . . 76
8.8 Wing deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.9 Spar, ribs and shell reinforcement in shell to ribs connection region representation 78
8.10 0.2mm composite shell; wing deformation for reinforced . . . . . . . . . . . . . . 79
8.11 Reinforced 0.2mm composite shell; failure criteria . . . . . . . . . . . . . . . . . 80
8.12 Resin reinforced shell model; wing deformation . . . . . . . . . . . . . . . . . . . 81
8.13 Resin reinforced shell model; Von Misesstresses on the resin sections . . . . . . . 81
8.14 Resin reinforced shell model; failure criteria in composite reinforcements . . . . . 81
8.15 Spar sections studied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.16 Wing deformation using the A spar(left) and the B spar (right) . . . . . . . . . . 83
8.17 Detail of wing deformation using the B spar (right) . . . . . . . . . . . . . . . . . 83
8.18 Stresses in spar axial direction on the A spar(left) and B spar (right) . . . . . . . 84
8.19 Failure criteria in the composite when using the A spar (left) and the B spar(right) 84
8.20 Ribs tensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.21 Wing deformation using 6 ribs (left) con�guration and 8 ribs con�guration (right) 85
8.22 Wing failure criteria using 6 ribs (left) con�guration and 8 ribs con�guration (right) 86
8.23 resin model; Lift and Drag convergence . . . . . . . . . . . . . . . . . . . . . . . 87
8.24 resin model deformation comparison with only one stagger loop (left) and with 27
stagger loops necessary to achieve convergence (right) . . . . . . . . . . . . . . . 88
8.25 resin model; 2D Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.26 resin model; Wing surface pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.27 resin model; wall shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.28 Pressure distribution in the symmetry plan and in a middle chord plan (XY plan) 90
8.29 Composite model; Lift and Drag convergence . . . . . . . . . . . . . . . . . . . . 91
viii
8.30 Composite model deformation without iterations (left) and when the result is fully
converged(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.31 Composite model failure criterion without iterations (left) and when the result is
fully converged (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.32 Composite model; 2D Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.33 Composite model; wing surface pressure . . . . . . . . . . . . . . . . . . . . . . . 94
8.34 Composite model; wall shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
8.35 Composite model; pressure distribution . . . . . . . . . . . . . . . . . . . . . . . 95
8.36 Composite model; velocity plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.37 Composite model; Eppler airfoil; wingspan 1.75m; wing deformation . . . . . . . 96
8.38 Composite model; Eppler airfoil; wingspan 1.75m; wing Lift and Drag variation
throughout the coupled �eld iterations . . . . . . . . . . . . . . . . . . . . . . . . 97
8.39 Composite model; Naca airfoil; wingspan 1m (approximately); wing deformation 98
8.40 Composite model; Naca airfoil; wingspan 1m (approximately); wing Lift and Drag
variation throughout the coupled �eld iterations; . . . . . . . . . . . . . . . . . . 98
ix
List of Tables
2.1 RPV Antex-X2 Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.1 Carbon/Epoxy Composite Sheet properties . . . . . . . . . . . . . . . . . . . . . 45
5.2 Carbon/Epoxy Composite Rod properties . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Polypropilene properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4 Epoxy properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.5 Model variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.6 Created �uid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.7 Boundary conditions symbols in CFXr . . . . . . . . . . . . . . . . . . . . . . . 49
6.1 Coarse mesh aerodynamic results . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2 Re�ned mesh aerodynamic results . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.3 Re�ned mesh with prism layer elements; aerodynamic results . . . . . . . . . . . 54
6.4 Properties of the structural model used in the convergence study . . . . . . . . . 56
6.5 Structural convergence study; Maximum deformation analysis . . . . . . . . . . . 56
6.6 Structural convergence study; Von Mises tension study . . . . . . . . . . . . . . . 57
6.7 Structural convergence study; Maximum Energy Error and Computational time
study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.1 Optimal wing con�guration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.1 Properties of the chosen model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
x
Nomenclature
U = Velocity (m/s)
t = Time (s)
ρ = Density (Kg ·m−3)
T = Temperature (K)
P = Pressure (Pa)
σij,i = Cauchy stress tensor (Pa)
Fi = Body forces (N)
ε = Strain
uui = Displacement (m)
Cijkl = Elasticity tensor (Pa)
E = Young modulus (Pa)
ν = Poisson ratio
G = Shear modulus (Pa)
σyield = Yield strength (Pa)
ξstrain = Maximum strain failure criterion
ξstress = Maximum stress failure criterion
ξTsai−Wu= Tsai-Wu failure criterion
εft= Traction failure strain
εfc= Compression failure strain
σft = Traction failure stress (Pa)
σfc = Compression failure stress (Pa)
λ = Thermal conductivity (W ·m−1 ·K−1)
SE = Energy source (Kg ·m−1 · s−3)
htot = Total enthalpy (m2 · s−2)
SM = Momentum source (kg ·m−2 · s−2)
µ = Viscosity coe�cient (kg ·m−1 · s−1))
µt = Eddy Viscosity (Kg ·m−1 · s−1)
Γt = Eddy Di�usivity (Kg ·m−1 · s−1)
Pr = Prandtl number
Prt = Turbulence Prandtl number
δ = Mesh displacement (m)
Γdisp = Mesh sti�ness (m2 · s−1)
rφ = Relaxation factor
L = Lift (N)
xi
D = Drag (N)
T = Thrust (N)
W = Weight (N)
α = Angle of attack (º)
Cl = Lift coe�cient
Cd = Drag coe�cient
xii
Chapter 1
Introduction
1.1 Morphing
Birds are the source of man's inspiration to �y. They are a superb product of nature, with
million years of evolution. When we look at an airplane it shows a resemblance to a giant bird,
but if we look more carefully very little of its morphology is present on modern aircrafts. Birds
possess the ability to constantly adapt and optimize their shape, morphing their wings, body
and tails in a very complex and �uid ways to suit dissimilar �ight conditions, such as high speed
attack and low speed loiter, and to achieve their incredible maneuverability.
Figure 1.1: Wing morphologies for hawks (left) and pigeons (right)
Since the beginning of aviation history, aircraft design has been described by a rigid structure
controlled by a few discrete actuators, elevators, rudder, ailerons and throttle. This restriction
contracts the aircraft �ight envelope and/or the aircraft operates with non optimal conditions in
a large part of the �ight envelope. Expansion of the aircraft �ight envelope would be possible if
aircraft followed bird's morphology by constantly changing its shape in a continuous way during
1
�ight, thus ensuring the optimal �ight conditions throughout the mission. To this achievement
recently research and development have begun on morphing aircrafts.
A morphing aircraft is an aircraft capable of in-�ight gross shape changes, with the purpose of
increasing e�ciency, versatility, and/or mission performance. Morphing technologies may bring
enormous bene�ts to the aircraft industry. On one hand, traditional aircrafts are designed as
a compromise of various performance needs. On the other hand, morphing aircraft can adapt
to speci�c mission requirements. Therefore, they can not only obtain better �ight performance
but also excel at numerous tasks. This is a very important subject in military applications.
Not only it has become more and more cost prohibitive to develop and operate large numbers
of single-mission aircraft, but also the mission needs and the uncertainty of future mission re-
quirements result in the increasing of morphing aircrafts likelihood mission success in future
con�icts when compared with regular aircrafts. In addition, morphing technologies enable new
�ight capabilities, such as perching, urban navigation, and indoor �ight. For all these reasons
morphing technologies are a major thrust in current unmanned aerial vehicle (UAV) research
and micro air vehicles are being developed based on morphing technologies due to their control
di�culties with normal aircrafts surfaces. The morphing bene�ts in aircrafts are well exposed
on the spider plot in Figure 1.2. This diagram is the result of a �rst order study to assess the
potential bene�ts of wing morphing conduced by the Next Gen corporation [74] where �ight
performances are shown for �xed and morphing wing geometries. The �gure 1.2 is meant to be
illustrative and considerations about this diagram are presented in [74].
Figure 1.2: Spider plot comparison of �xed and morphing wing aircraft
1.2 Historic Perspective
Although morphing technologies are a topic of recent research interest in aerospace engineer-
ing, many of the concepts used are not new [68]. In fact, when Wilbur Wright was looking in
2
1899 for a way to control the roll of the Wright B �yer, he twisted a long, narrow box and he
believed that the motion could be applied to the aircraft wing. This concept of twisting the air-
craft wing to produce lateral control is called wing warping. As aircraft speed increased, wings
became sti�er to preclude aeroelastic instabilities and then warping disappeared because the
power required exceeded actuator capabilities: more energy e�cient aileron system started to be
used instead. One of the �rst aircraft to be said to apply morphing concepts was the Advanced
Fighter Technology Integration (AFTI) F-111 [84]. (Figure 1.3)
Figure 1.3: AFTI F-111 in �ight
This airplane had a mission adaptable wing (MAW) with variable sweep and incorporated
a variable camber airfoil capable of variable spanwise camber distribution while maintaining a
smooth and continuous airfoil. The camber was positioned and controlled by �exing the upper
skins through rotary actuators and linkages driven by power drive units [38]. Wind-tunnel and
�ight tests demonstrate a drag improvement specially at o�-design conditions where the MAW's
ability to smoothly recon�gure showed its maximum advantages. [85]. Nowadays, from the
variable-sweep wings on F-14, B-1B, and Tornado, to the variable geometry of the propulsion
system such as that found in the V-22 Osprey and the Harrier [68], to name a few, demonstrations
of large shape changes and their importance to achieve the demanding performance speci�cations
required by aircraft industry are already presented and morphing concepts shall be a constant
in the quest of the sky of tomorrow.
1.3 Morphing concepts and their applicabilities
Many optimization studies have been performed to the application of morphing concepts. In
[80], a light unmanned air vehicle with a takeo� weight of 400N, a constant chord of 0.5m across
the span and a wing area of 1.4m2 was optimized: the optimal airfoils for di�erent stages of
�ight are obtained and although the power reduction achieved was not very high at the major
3
design �ight conditions, the morphing airfoils conduced to a possible extension of the �ight en-
velope and a sustained maneuverability at low speeds. This study concluded that a morphing
mechanism that controls the camber and leading edge thickness of the airfoil will be almost
su�cient to obtain the optimal airfoil at most operating conditions. However, the cruise airfoil
had a relative thickness of 2% and the other airfoils a maximum camber varying up to 8.5% of
the chord, which indicates a structural challenge to construct and implement the actuators. In
[51] another airfoil-shape optimization is presented, this time for rotor blades. In this study the
aerodynamic bene�t associated with deformable compliant structures is exploited to produce a
variable camber leading edge for dynamic stall control. Results were very promising and have
shown that morphing the baseline airfoil can delay or eliminate the formation and shedding of
the dynamic stall vortex and achieve a higher Clmax than maintaining the baseline section high
Mach number advancing blade characteristics. Both these last two studies have been focused
only on the aerodynamic numerical simulation and do not take into account structural impli-
cations and limitations of the morphing optimization. In the studies presented in [69] and in
[48] optimization of morphing wings are presented taking into consideration aeroelastic behav-
ior. The �rst study is focused on minimizing structural weight of morphing wing and the second
is focused on the design of mechanisms and the layout as well as the location and number of
actuators. These studies have shown that the interactions between �ow, structural deformation,
mechanism, and actuator must be considered to �nd the optimal solution. In fact, it is exposed
that while the decoupled procedure allows the designer to employ single-discipline design and
analysis tools improving the single-discipline design optimization, the results presented show that
this sequential approach cannot account for the dependency of the aerodynamic forces in shape
variations. Therefore, the combined aeroelastic design procedure, that considers aerodynamic
and structural design criteria simultaneously (obtained by an aerodynamic shape optimization
and structural optimization simultaneously) has presented better �nal results than the realized
de-coupled result (obtained aerodynamic shape optimization followed by a structural optimiza-
tion). Nevertheless those results were not quite as good as the aerodynamic optimum solution
(obtained by an aerodynamic shape optimization alone).
Conventional roll control in aircraft is typically achieved by aileron de�ection. This de�ec-
tion alters the wing lift distribution and as a consequence the aircraft rotates. Although these
systems are reliable and e�ective they are complex, heavy and add considerably maintenance
and inspections requirements. It has been a long time since a number of approaches have been
made to improve �ight control via adaptative structures. Studies [31]and [3] between airfoils
with convectional ailerons and adaptative airfoils with conformal mobile surfaces have shown
that for the same variation of angle of attack (de�ned by free stream velocity with the chord line
of the airfoil) the conformal mobile surfaces produce higher lift coe�cients as well as higher pitch
4
Figure 1.4: Optimized adaptive wing design obtained in [48]
down moment coe�cient. Conformal control also produces higher maximum roll rates than con-
ventional ones but looses e�ectiveness at lower speeds. Another disadvantage of the conformal
controls is their loss of e�ectiveness for high �ap-to-chord ratios at higher speeds, which is much
more signi�cant than a conventional �ap with the same �ap-to-chord ratio. For low �ap-to-chord
the loss of e�ectiveness is not representative and conformal �ap become a better choice. In a
study of a rotary wing UAV [17], it is shown that the �ight control system weight could be
reduced as much as 40% while simultaneously reducing the drag and power consumptions and
part-count using adapatative aerostructures. In [87] it is shown that postbuckled precompressed
piezoelectric actuators can be applied to roll control of a subscale UAV. Static and dynamic
bench tests have shown a maximum de�ection of more than 3º up to a break frequency of 34
Hz. When comparing conventional aileron actuators with precompressed pizoelectrics actuators,
operating empty weight could be reduced by almost 3.5% for the UAV in study.
Figure 1.5: UAV employing PBP actuated morphing panels
In�atable wings are other morphing mechanisms that have been developed for decades in
manned aircrafts, UAVs and Lighter Than Air (LTA) vehicles. Examples are [79] Goodyears In-
�atoplane from the 50's (�gure 1.6) and the Apteron unmanned aerial vehicle that was developed
in the 1970s by ILC Dover.
Due to the advance in materials and manufacturing, the in�atable wings are nowadays ready
to be applied to aircrafts. In�atable wings are light, can be packed into much smaller volumes
5
Figure 1.6: Goodyears Model GA-468 In�atoplane c. 1950
than their deployed volume without damaging the structural integrity of the wing and their
deployment on the ground or in �ight can be made in less than a second, depending on the wing
size and in�ation system used. In�atable wings can be used to increase aspect ratio. A study of
di�erent morphing concepts, based on in�atable wings, is presented in [37], and nastic structures
are among them. Nastic structures are biomimetic devices whereby materials are activated to
generate large strains while still performing a structural function. It consists of a series of parallel
tubes into which a �uid of varying pressure may be pumped and the cells can transite from a
�at to a circular cross section.
Figure 1.7: Nastic tensile tests
This concept is unsuitable for fast morphing and weight penalty can be problematic, specially
due to required support hardware such as pumps and proportional control valves, because of the
large number of suitably sized nastic cells required to create the necessary loads to morph the
wing.
Bump �attening is another wing morphing concept presented in [37]. In this concept a
piezoelectric actuator is bonded �rst to a rigid substrate, and then to the wing restrain fabric.
When energized, a force is generated to the actuator plane and individual bumps are �atten. By
�attening individual bumps in series, a net increase in run length is generated, resulting in the
de�ection of the wing's trailing edge.
6
Figure 1.8: Bump �attening prototype
In [37] the piezoelectric actuators couldn't provide enough force to overcome the internal
pressure of the bumps, but the approaches appeared viable for scaled cases where the in�ation
and applied loads are more manageable. The estimated low weight of this kind of actuation makes
this concept attractive when piezoelectric actuators technology develops. The last morphing
concept presented in [37] is trailing edge de�ection. This concept consists in the modi�cation of
the baseline in�atable wing con�guration with piezoelectric actuators that �ex the trailing edge
of the wing.
Figure 1.9: Trailing Edge De�ection Morphing Wing Prototype
In contrast with ailerons the actuator is under the wing skin and does not perturb the in�ow.
In this study some experiences were made to investigate the substrate de�ection as function of
material and thickness. The amount of force to �atten the substrate to the initial position was
also calculated. It was observed that there was a trade-o� between force and displacement of the
actuator. For the case in study it was found a maximum control surface de�exion of 3 degrees.
To improve this concept, the rigidization of the in�atable wings is being studied as shown
in [78]. Rigidization of in�atable wings provides several potencial advantages, such as reducing
the vulnerability to punctures, increasing sti�ness and load carrying capability, allowing higher
aspect ratio for high altitude e�ciency and longer missions, and reducing weight by eliminating
the make up pressurization supply. Despite all the studies and all the concepts based on in�atable
wings, this technology is still in its infancy and further developments are required.
Other morphing concepts are based on wing geometry changing, and to achieve it many
concepts are being studied and can be applied. Such is the case of the Lockheed Martin morphing
aircraft [89]. This project is supported by the Defense Advanced Research Projects Agency
(DARPA) Morphing Aircraft Structures (MAS) program. The Lockheed Martin aircraft has a
7
folding wing that allows radical changes in wing span and wet area. This folding wing concept
enables the combination of cruise / loiter e�ciency and dash / penetration capability in a single
vehicle, signi�cantly increasing mission performance over conventional platforms. The Figure
1.10 presents the aircraft at di�erent stages of actuation. Loiter con�guration is obtained with
the wing fully extended and dash con�guration with the wing fully folded.
Figure 1.10: Lockheed Martin MAS Wind Tunnel Model During a Morphing Sequence
Description of half wind tunnel model tests are presented in [76]and [89]. Several wind tunnel
success criteria were identi�ed in this study such as actuation of the morphing structure without
structural failure or binding, demonstration of model integrity and stability for varying con�g-
urations Mach number and altitudes, identi�cation of aeroelastic characteristics, demonstration
of materials resilience to air loads and actuation systems and demonstration of smart actuator
readiness and integrity for �ight typical requirements, to name a few. The results have shown
good correlation with aerodynamic predictions and good agreement with predicted hinge mo-
ments. Nevertheless further research is needed in actuation systems and in materials because
aeroelastic problems due to material behavior were identi�ed.
Another project under a DARPA sponsored program is the �NextGen Aeronautics Morphing
Aircraft Structures� batwing con�guration [74]. The NextGen design is capable of large geometry
changes such as 200% change in aspect ratio, 40% in span and 70% in wing area. Key innovations
of the design include two degree-of freedom system which enables independent control of wing
sweep and wing area, a novel �exible skin design which undergoes over 100% in plane strain while
withstanding air loads of up to 1900 Kg/m2, an actuation system consisting of multiple internal
actuators, centrally computer controlled to implement any commanded morphing con�guration
and a structural e�cient kinematic substructure which enables the morphing geometry changes.
A wind tunnel model of the batwing concept was developed. It incorporated an actuation system
of nine separated hydraulic actuators centrally controlled providing and coordinating control of
the wing geometry. Wing tunnel tests have proven the structural integrity of the overall design by
8
testing at loads up to 2.5g; it also demonstrated controlled morphing at 1-g air-on conditions and
correlated analytical predictions with measured results. In addition to the wind tunnel model,
NextGen developed a jet-powered remotely controlled piloted vehicle (RPV) with success 1.11 -
the Morphing-vehicle Experimental (MFX-1). [66]
Figure 1.11: Morphing con�gurations of the MFX-1
A pneumatic telescopic wing is being developed by the Maryland University [41]. It has a
pressurized telescopic rigid spar, rigid airfoil skins and rib elements. The wing is able to undergo
large-scale spanwise changes while supporting wing loading. A maximum increase in 230% aspect
ratio can be achieved. Wind tunnel analysis were conducted and the data acquired from these
tests compared with theoretical results revealed an increase in parasite drag caused by the seams
of the wing sections. Nevertheless, global aerodynamic performance can be increased due to
change in con�guration throughout di�erent �ight conditions.
Figure 1.12: Pneumatic Wing
Actuation energy was found to be very small even up to stall conditions and beyond. Stability
analysis of an UAV with this variable-span morphing wing (VSMW) was also performed [43] and
the UAV was found to be stable in dutch roll and roll modes, but additional work needs to be
performed to determine if the spiral mode can be controlled by di�erent span inputs. A study
is presented in [25] of a VSMW for a cruised missile. This study reveals an improvement in
aerodynamic characteristics as VSMW reduces drag, which can lead to an increase in range.
Anti-symmetric span control can be used, to replace the conventional missile control by missile
control �n, as this method revealed an increase in roll control authority. Aerodynamic studies of
this concept revealed that the aerodynamic deformation is larger than that of conventional wing
9
for the same �ight conditions and so bending sti�ness has to be increased. Another method for
the control of morphing aircrafts is variable Cant-Angle Winglets, presented in [1] This concept
seems to be a promising alternative to conventional control surfaces as far as basic maneuvers
are concerned. Preliminary �ight tests conducted with a radio controlled model showed a roll
rate comparable to that generated by a pair of conventional ailerons and the concept seem to be
most e�cient at low speed.
Figure 1.13: Experimental model for winglets control study
It was also concluded that a single pair of adjustable winglets cannot be a substitute for all
conventional control surfaces at the same time. At least a second pair of adjustable winglets is
needed to fully control the aircraft. With four independent multi-axis e�ectors, the system is then
overactuated, leading to some redundancy in �ight control systems, which could be exploited to
optimize secondary objectives such as reducing drag and bending moments.
Aircrafts that can bene�t from morphing are micro air vehicles. These aircrafts are suitable
for use in environments with little space and therefore high agility and precision maneuvering
are required. Due to their small size and low velocities ailerons are not suitable for control of
this class of aircrafts and so morphing is used as an aeroservoelastic e�ector for control. A �ight
testing of a micro air vehicle using morphing for aeroservoelastic control is presented in [26] and
as it is exposed the use of morphing concepts is able to command turns and spins with su�cient
authority for precision maneuvering.
In [21] is presented a somehow di�erent approach. Morphing is obtained by the use of
dielectric barrier discharge plasma actuators. In this case the shape of the aircraft used, the
1303 UCAV planform, doesn't change but the �ow�eld is modi�ed by these actuators. Results
from force balances experiences have shown considerable changes in lift and drag characteristics of
the wing for plasma-control actuation. When compared with conventional trailing edge devices,
the plasma actuators demonstrated a signi�cant improvement in the control authority, and as
consequence, the �ight envelope was extended, which demonstrated the feasibility of a plasma
wing concept for hingeless �ight control.
Many of the concepts presented so far can and should be integrated in morphing aircraft. As
an example of integration of di�erent morphing concepts in a single aircraft, a fully adaptable
10
aircraft con�guration is under development in the Virginia Plytchnic Institute and State Univer-
sity [24], where it serves as an experimental testbed for aerodynamic modeling and �ight control.
This aircraft can achieve large scale shape changes through �ve independent planform variations.
The vehicle undergoes a 38% increase in span, 40 degrees of sweep change, 12% change in chord
length and 40 degrees of twist. Wind tunnel tests were performed on this aircraft and results
have shown that the variable planform capability allows low drag to be maintained throughout
a range of lift coe�cients.
Figure 1.14: Fully adaptive morphing aircraft design by Virginia Plytchnic Institute and StateUniversity
11
Chapter 2
Problem de�nition
As it was presented in the introduction, chapter 1, morphing bene�ts in aircrafts are enormous
and many concepts have been and are being developed. Instituto Superior Tecnico (IST) has
been employing a lot of e�ort in this area and many IST sponsored projects are being developed
both in Master and Ph.D. thesis.
The study presented in this thesis is in continuity with Instituto Superior Tecnico, IDMEC-
Instituto de Engenharia Mecanica project POCI/EME/61587/2004 �Designing, Building and
Wind Tunnel Testing of a Morphing Wing for Drag Reduction� performed by the Ph.D. student
Jose Vale [86]. In this project, the improving performance by the developed morphing concepts,
a telescopic wing with span, airfoil shape and chord variation applied to the small radio piloted
vehicle (RPV) Antex-X2 from Portuguese Airforce, are well exposed and wing drag reduction
obtained in this project was up to 69.7% [86]. On the other hand, in the same study it was
concluded that the used materials, such as the �exible natural rubber skin su�ered from some
major problems. The deformation of the rubber skin, due to aerodynamic loads produced non
smooth contours that greatly increased drag. Also the actuation forces to produce the wing
morphing revealed to be excessive due to the rubber skin properties. This thesis has the objective
to improve the morphing concepts presented in [86] and specially resolve the wing skin problems
presented. Thus, the �exible rubber skin will be replaced by other materials. As in [86], the
study of the morphing concepts developed in this thesis will also be applied to the Antex-X2 and
so the geometrical characteristics and some performance data of this RPV are shown in table
2.1.
12
Parameter Value
Airfoil wortman FX-63-137Main Wing Span 2.4 mMain Wing Chord (Root) 0.33 mMain Wing Chord (Tip) 0.33 mMain Wing Aspect Ratio 7.273Main Wing Weight 15 NFuselage Span 0.4 mHorizontal Stabilizer Span 0.72 mHorizontal Stabilizer Chord (Root) 0.18 mHorizontal Stabilizer Chord (Tip) 0.18 mHorizontal Stabilizer Aspect Ratio 4Vertical Stabilizer Span 0.2 mVertical Stabilizer Chord (Root) 0.184 mVertical Stabilizer Chord (Tip) 0.138 mVertical Stabilizer Aspect Ratio 1.242Wheel Track 0.3713 mWheel Base 0.5048 mPropeller Diameter 0.38 mMain Wing Area, Gross 0.792 m2
Flaps Area 0.06336 m2
Ailerons Area 0.06336 m2
Horizontal Stabilizer Area, Gross 0.1296 m2
Elevator Area 0.03786 m2
Vertical Stabilizer Area, Gross 0.0644 m2
Rudder Area 0.007491 m2
Max Takeo� Weight 98.06 NMax Wing Loading 123.86 N/m2
Max Power Loading 0.052612 N/WMax Level Speed (Sea Level) 42.02 m/sCruise Speed (Sea Level, 75% Power) 38.6 m/sStall Speed Clean (Power On) 15.81 m/sStall Speed, 45°Flaps (Power On) 12.39 m/sMax Rate of Climb 8.44 m/s
Table 2.1: RPV Antex-X2 Characteristics.
13
This thesis consists of a morphing wing sizing with wingspan variation and with airfoil shape
change that will replace the Antex-X2 base wing. Chord variation will not be considered as it
was concluded in [86] that the contribution to performance improvements by chord variation is
small compared to the span variation and airfoil changing.
The morphing wing concept model studied is presented in �gure 2.1. The telescopic wing
can continuously change the wing span as it has an inner wing that slides into the outer wing.
The inner wing and the outer wing are de�ned as having 1m span (half wing span) each and the
inner wing can be deployed up to a maximum of 0.75 % as represented in the �gure 2.1.
Figure 2.1: Di�erent wing span con�gurations
The wing can also change the airfoil shape. In this study, for airfoil shape change it is assumed
that it can only change into two di�erent and speci�ed airfoils: a high speed airfoil and a low
speed airfoil. The chosen airfoil for the low speed was Eppler-434 and for the high speed airfoil
Naca-0012 as presented in �gure 2.2. The choice of these airfoils is explained in chapter 7.
14
Figure 2.2: Di�erent airfoil con�gurations used on the telescopic wing
The wing sizing will be based on a coupled �uid/structural analysis that will be the main focus
of this thesis. The coupled �uid/structural analysis presented here is a major improvement to
the �uid/structural coupled analysis used in [86]. The new coupled �uid/structural analysis uses
a �nite element model developed in Ansysr Multiphysics to perform the structural analysis and
a �nite volume model developed in Ansys-CFXr to perform the computational �uid dynamic
(CFD) analysis instead of the nonlinear lifting-line method used in [86]. The iterations between
�elds are performed using the Ansysr standard Mul�-�eld solver MFX, which is exposed later
(chapter 3).
Based on these objectives this thesis has the following structure: Chapter 3 introduces the
�uid/structural coupled analysis in which the wing sizing is based on. Chapter 4 presents
the mathematical formulation required to understand the problem and its application to the
�uid/structural analysis. In chapter 5 we can �nd the description of the problem modulation. It
is described the Finite Element Method (FEM) model used to analyze the structural �eld and
the Finite Volume Method (FVM) model to analyze the �uid �eld. In chapter 6 a convergence
analysis is presented both for the structural and the �uid models. In chapter 7 the results of an
aerodynamic analysis, performed in the �uid models described in chapter 5, are shown. Wing
drag reduction and the result increase in aircraft maximum weight due to morphing are studied
and results are presented. In chapter 8 the results of the �uid/structural analysis are revealed
and the wing sizing based on these results is performed.
15
Chapter 3
Coupled �eld analysis
Coupled �eld analysis or multi�eld analysis is a combination of analyses from di�erent en-
gineering disciplines that interact with each other to solve a global engineering problem, where
the input of one �eld depends on the results of another. There are analysis where there is no
need to interact between the two �elds solutions. For example, this usually occurs in thermal
stress analysis, where the temperature �eld introduces thermal strains in the structural �eld,
but generally the structural strains do not a�ect the temperature distribution. However, in a
�uid-structure analysis, the �uid pressure causes the structure to deform, which in turn causes
the �uid solution to change and therefore interaction between the two physic �elds is required
to achieve a converged solution.
3.1 Ansysr coupled �eld analysis procedure
Ansysr [14] is a powerful tool to perform these coupled �eld analyses and possesses di�erent
methods to this achievement. Next, we present di�erent Ansysr approaches to perform coupled
�eld analysis as well as the reason of our choice to perform the �uid solid interaction exposed in
this thesis. Ansysr coupling between the �elds can be accomplished by either direct coupling,
(matrix coupling) or sequential coupling (load vector coupling).
Direct coupling involves just one analysis that uses a coupled-�eld element type containing
all necessary degrees of freedom and where element matrices or element load vectors contain-
ing all necessary terms are calculated. Direct coupling is advantageous when the coupled-�eld
interaction is highly nonlinear.
Sequential coupling involves two or more sequential or simultaneous analyses, each belonging
to a di�erent �eld. In sequential coupled physics analysis the two �elds are coupled applying
results from one analysis as loads of the other analysis.
For coupling situations which do not exhibit a high degree of nonlinear interaction, the
sequential method is more e�cient and �exible as it performs the two analyses independently
16
of each other. Coupling can be recursive where interactions between the di�erent physics are
performed until the desired level of convergence is achieved.
Ansysr possesses two types of sequential coupled analysis: Physics Files Solver (PFS) and
AnsysrMulti-Field Solver (AMFS).
The Ansysr Multi-Field Solver analysis is a more robust, automated and easy to use tool
for solving sequentially coupled �eld problems than the Physics Files Solver. It is built on the
premise that each physics is created as a �eld with independent solid model and mesh instead of
the single �nite element mesh used in the Physics-Files procedure.
In the Multi-Field Solver analysis surfaces or volumes are identi�ed for coupled load transfer
and the solver automatically transfers the respective loads across these identi�ed surfaces or
volumes which can and usually have dissimilar meshes. In Multi-Field Solver analysis static,
harmonic and transient analyses can be performed.
There are two versions of the Ansysr Multi-Field Solver, the MFS-Single Code and the
MFX-Multiple Code. The Single Code version is used in simulations that involve small models
with all physics �eld contained in a single executable product (e.g. Ansysr Multiphysics or
Ansysr Mechanical).
MFX-Multiple code is used for simulations with physics �elds distributed between more than
one executable product (e.g between Ansysr Multiphysics and Ansys-CFXr ). Much larger
models can be accommodated in the MFX-Multiple Code rather than in the MFS-Single Code
version.
The MFX-Multiple code solvers use iterative coupling where each physics is solved simultane-
ously or sequentially, and each matrix equation is solved separately. The solver iterates between
physics �elds until load transferred across interfaces converge.
As the MFX-Multiple code analysis is primarily intended for �uid-structure interaction(FSI),
in which the Ansysr multiphysics is used for the analysis of the structural �eld and the Ansys-
CFXr for the computational �uid analysis, we preferred this method to accomplish the analyses
exposed in this thesis.
3.2 Ansysr multi�eld analysis using code coupling
In a MFX-Multiple code analysis using Ansysr and Ansys-CFXr there are two codes dis-
tributed by these two programs. The Ansysr code always functions as the master and the
Ansys-CFXr as the slave.
Since the solid models and respective mesh in the Ansysr and Ansys-CFXr models are
independent, the master, Ansysr, reads all Multi-�eld commands, collects the interface meshes
from the Ansys-CFXr code and proceeds to the mapping process. The mapping consists of
an interpolation of the two distinct meshes to transfer loads between the interface surfaces. A
17
more detailed explanation about mapping is presented later in this thesis. The Ansysr code
is also responsible for communicating time and stagger loop controls to Ansys-CFXr . Time
controls are used if a transient analysis is being executed and provides a way for Ansysr and
Ansys-CFXr to track the progress of real time during simulation. Stagger loops consist of
coupling iterations between the two �elds until reaching some convergence criteria. For example,
a transient analysis usually needs a series of coe�cient loops in each time step; in other words,
for a speci�c time there is the need of a series of interactions between the two �elds in order to
achieve the converged solution. During every stagger loop, each �eld solver collects the required
loads from the other �eld solver and then solves its physics �eld. If global convergence of the
load transfer is not achieved another stagger loop is performed. The next diagram summarize
the explanation above.
Figure 3.1: Ansysr Multi�eld Solver Process
It is important to refer that Ansysr and the Ansys-CFXr �elds can be solved simultaneously
or sequentially, as presented in �gure 3.2
Weakly coupled �elds can be solved simultaneously, and the overall simulation time may
decrease because no �eld solver must wait for the results from the other �eld solver. However, if
the �elds are strongly coupled, the simultaneous solution procedure may destabilize the solution
process because less recent results are applied in each �eld solver and so the sequential solution
procedure must be used. Simultaneous solution also requires more computer memory. We have
used the sequential solution procedure.
18
Figure 3.2: Ansysr and Ansys-CFXr �elds solved simultaneously (on the left) and Ansysrand Ansys-CFXr �elds solved sequentially (on the right)
3.3 Load transfer between physical �elds
Load transfer is the process by which one �eld transmits mesh-based quantities to another.
The load transfer occurs from one interface surface to another in MFX-Multicode analysis. For
example, in a Fluid solid interaction analysis the Ansys-CFXr analysis transmits forces produced
by the aerodynamic loads to the Ansysr �eld and the Ansysr �eld analysis transmits the
resulting displacements to the Ansys-CFXr �eld across the interface surface. MFX supports only
mechanical and thermal load transfer between �elds. The MFX solver automatically transfers
coupled loads across dissimilar meshes and two interpolation methods are available for a load
transfer: the pro�le preserving and the globally conservative. In a pro�le preserving interpolation,
each node on the receiver side maps onto an element on the sender side αi. The variable to be
transferred from one �eld to another is then interpolated at αi. Thus, all the nodes on the
receiver side query the sender side. Figure 3.3 is illustrative of pro�le preserving method.
Figure 3.3: Pro�le preserving interpolation
In a globally conservative interpolation, each node on the sender side maps onto an element
on the receiver side. Thus, the transfer variable on the sender side is split into two quantities
that are added to the receiver nodes. Figure 3.4 is illustrative of globally conservative method.
It is important to refer that in a pro�le preserving interpolation, the total quantity transferred
19
Figure 3.4: Globally conservative interpolation
on the interface will not balance in both sender and receiver sides, whereas in a globally conser-
vative interpolation the total quantity transferred will balance locally, but the distribution may
not agree. In general, globally conservative interpolation should be used to transfer quantities
such as heat �ux or forces, whereas displacement and temperatures should be interpolated using
pro�le preserving as pro�les of these quantities should be adequately captured across interfaces.
For a pro�le preserving interpolation a coarse mesh should be used in the sender side and a �ne
mesh in the receiver side. This is well explained in �gure 3.5
Figure 3.5: Example of a pro�le preserving interpolation with a coarse mesh in the receiver side(left) and a coarse mesh in the sender side (right)
For a globally conservative interpolation it is better to have a �ne mesh in the sender side
and a coarse mesh in the receiver side. This is well explained by the �gure 3.6.
20
Figure 3.6: Example of a globally conservative interpolation with a coarse mesh in the senderside (left) and a coarse mesh in the receiver side (right)
In a global conservative interpolation the total quantity in the receiver is equal to the total
quantity in the sender independently of the meshes used.
From the exposed it is evident that for the loads transfer between the Ansys-CFXr and
Ansysr a global conservative interpolation will be used in all the analyses exposed in this thesis
as we want to preserve the total aerodynamic force calculated in CFXr and applied to the
structure model created in Ansysr. On the other hand, in the load transfer from Ansysr
to CFXr a pro�le preserving interpolation is used as it is more �tted to transfer the wing
deformation from the Ansysr model to the CFXr model.
3.4 Mapping
In order to transfer loads across dissimilar mesh interface, the nodes of one mesh must be
mapped to the local coordinates of an element in the other mesh. There is always the need to
perform two mappings for every surface to surface interface. The nodes of one �eld, �eld 1, must
be mapped to the elements of the other �eld, �eld 2 and vice-versa. Ansysr has two mapping
algorithms available, the global and the bucket search algorithms.
In the global method, each node to be mapped loops over all the existing elements of the
other mesh and tries to locate an element that can be mapped to. If there is more than one
element that the node can be mapped to, the element that minimizes the distance is selected.
Sometimes when interfaces edges are not aligned a node does not map any element and then it
is mapped to the closest node. When the number of nodes increases, the global method is less
e�cient than the bucket search. The bucket search method di�ers from the global method as
it restricts the elements over which it loops. In this method all the elements are distributed in
cartesian boxes, designated by buckets. The node to be interpolated is then located in a box and
the global method is used for the node in question; nevertheless the elements are restricted to
that box. The bucket method is computationaly more e�cient than the global method with the
increasing of the nodes and elements to be mapped. In the mapping process Ansysr has the
21
ability to identify improperly mapped nodes, which can be a useful tool to the user in order to
understand if the mapping process is being correctly conducted. Improperly mapped nodes are
nodes that exceed a normal distance tolerance between the node to be mapped and the target
element surface speci�ed by the user, and nodes that are on misaligned surfaces. In this thesis
we have used the bucket method.
Figure 3.7: Improperly mapped nodes
22
Chapter 4
Analytical Models
4.1 Structural formulation
The equations governing a linear elastic boundary value problem [22] are based on the equa-
tion of motion 4.1, strain displacement equations 4.2 and constitutive equations. The constitutive
equations for linear elastic materials can be given by the Hooke's law that relates stresses and
strains in the material 4.3:.
σij,i + Fi = ρ∂ttuui, (4.1)
εi,j =12
(uuj,i + uui,j), (4.2)
σi,j = Cijklεkl. (4.3)
In these equations σij,i is the Cauchy stress tensor, Fi are the body forces, ε is the strain,
uui is the displacement and Cijkl is the elasticity tensor.
For an orthotropic material ( such as a ply of composite carbon-�ber/epoxy matrix ) the
elasticity tensor is given by:
C−1 =
1Ex
−νyx
Ey
−νzxEz
0 0 0−νxy
Ex
1Ey
−νzy
Ez0 0 0
−νxzEx
−νyz
Ey
1Ez
0 0 0
0 0 0 12Gyz
0 0
0 0 0 0 12Gzx
0
0 0 0 0 0 12Gxy
. (4.4)
In 4.4 Ex, Ey, Ez are the Young's modulus in each directions, νyz, νyz, νyz the 3 Poisson's
ratios, andGyz, Gzx, Gxy the 3 shear modulus. The elasticity tensor can be simpli�ed for isotropic
23
materials as Ex = Ey = Ez and νyz = νyz = νyz
As stated before, the structural analysis was performed using the Finite Element Method
(FEM) based on Ansysr program. For information about the �nite element method reference
[22] can be used. For the results post-processing it is important to introduce failure criteria as
presented below. Von Mises yield criterion [22], also known as the maximum distortion energy
criterion, octahedral shear stress theory, or Maxwell-Huber-Hencky-Von Mises theory, suggests
that the yielding of materials begins when the second deviators stress invariant J2 reaches a
critical value k. Von Mises criterion can be also formulated in terms of the Von Mises stress,
σ′, a scalar stress value that can be computed from the stress tensor. In this case, a material
is said to start yielding/failing when its Von Mises stress reaches a critical value known as the
yield strength, σyield. Von Mises stress is used to predict yielding of materials under any loading
condition from results of simple uniaxial tensile tests and satis�es the property that two stress
states with equal distortion energy have equal Von Mises stress. Mathematically it is expressed
by:
σ′ =[
(σ1 − σ2)2 + (σ2 − σ3)2 + (σ1 − σ3)2
2
]1/2
(4.5)
By this criterium the material yielding occurs if:
σ′ ≥ σyelding, (4.6)
where σyelding is the yielding tension of the material.
Von Mises stresses criterion is not adequated when using composite materials. Some of
the most commonly used failure criteria when analysing composite materials are the maximum
strain, maximum stress and Tsai-Wu [15]. In the maximum strain criterion the failure occurs
when ξstrain = 1, where ξstrain is de�ned as:
ξstrain = maximum of
εxt
εfx t
or εx c
εfx c
εy t
εfy t
orεy c
εfy c
εx t
εfx t
or εx c
εfx c
εz t
εfz t
or εz c
εfz c
|εxy |εfxy
|εxz |εfxz|εyz |εfyz
, (4.7)
where εx, y or z represents the strain in the respective x, y or z direction. The symbol t is used
for tension, c for compression and f for failure. So εf(x, y or z) t and εf(x, y or z) c represents the
24
failure strain, in the respective layer x, y or z direction, in traction and the failure strain, in the
respective layer x, y or z direction, in compression.
In the maximum stress criterion the failure occurs when ξstress = 1 where ξstress is de�ned
as:
ξstress = maximum of
σx t
σfx t
or σx c
σfx c
y t
σfy t
orσy c
σfy c
σx t
σfx t
or σx c
σfx c
σz t
σfz t
or σz c
σfz c
|σxy |σf
xy
|σxz |σf
xz|σyz |σf
yz
. (4.8)
The σx, y or z represents the stress in the respective x, y or z direction. As in the strain failure
criterion the symbol t is used for tension, c for compression and f for failure. So σf(x, y or z) t and
σf(x, y or z) c represents the failure stress, in the respective layer x, y or z direction, in traction
and the failure stress, in the respective layer x, y or z direction, in compression.
In the Tsai-Wu failure criterion the failure occurs when ξTsai−Wu = 1 where ξTsai−Wu is
de�ned as:
ξTsai−Wu = A+B, (4.9)
where
A =− (σx)2
σfx tσ
fx c− (σy)2
σfy tσ
fy c− (σz)2
σfz tσ
fz c
+ (σxy)2
(σfxy)2
+ (σyz)2
(σfyz)2
+ (σxz)2
(σfxz)2
+ Cxyσxσy√σf
x tσfx cσ
fy tσ
fy c
+ Cyzσyσz√σf
y tσfy cσ
fz tσ
fz c
+ Cxzσxσz√σf
x tσfx cσ
fz tσ
fz c
, (4.10)
B =
(1
σfx t
+1
σfx c
)σx +
(1σy t
f
+1
σfy c
)σy +
(1
σfz t+
1
σfz c
)σz. (4.11)
4.2 Fluid �ow formulation
The basic equations governing the motion of a viscous heat conducting �uid are called the
Navier-Stokes equations [19]. The Navier-Stokes equations, named after Claude-Louis Navier
and George Gabriel Stokes, arise from the assumption that the stress is the sum of a dissipative
viscous term (proportional to the gradient of velocity) plus a pressure term. The Navier-Stokes
equation 4.13 is supplemented by the mass conservation equation 4.12, also called continuity
equation and the energy equation 4.15. Usually, the term Navier-Stokes equations is used to
refer to all of these 3 equations [10].
25
∂ρ
∂t+∇ · (ρU) , (4.12)
∂ρU
∂t+∇ · (ρU ⊗ U) = −∇p+∇ · τ + SM , (4.13)
where SM is the momentum source and:
τ = µ
(∇U + (∇U)T − 2
3δ∇ · U
), (4.14)
(∂ρhtot)∂t
− ∂p
∂t+∇ · (ρUhtot) = ∇ · (λ∇T ) +∇ · (U · τ) + U · SM + SE , (4.15)
where λ is the thermal conductivity, SE is energy source and htot the total enthalpy given by:
htot = h (T, p) +12U2. (4.16)
The term ∇· (U · τ) represents the work due to viscous stresses and is called the viscous work
term. The term U · SM represents the work due to external momentum sources and is currently
neglected.
The Navier-Stokes equations have no known general analytical solution but can be discretized
and solved numerically. Computational Fluid Dynamics (CFD) is a computer-based tool for
simulating the behavior of systems involving �uid �ow, heat transfer, and other related physical
processes, solving the Navier Stokes equations, in a special form over a region of interest, with
speci�ed conditions on that boundary region. There are a number of di�erent solution methods
which are used in CFD codes. The most common, and the one Ansys-CFXr is based on, is
known as the �nite volume technique.
For information about the �nite volume method the reference [5] can be used.
4.3 Turbulence model
The Navier-Stokes equations describe both laminar and turbulent �ows without the need
of additional information. However, a turbulent �ow span a large range of turbulence length
and time scales (from the energy containing eddys to the small dissipative eddys). This implies
length scales much smaller than the smallest �nite volume mesh, which can be practically used in
a numerical analysis. Nowadays and in the foreseeable future the Direct Numerical Simulation
(DNS) of these �ows is impracticable due to the required computing power. To predict the
turbulence e�ects without the recourse to a prohibitive �ne mesh, a large amount of CFD research
has been concentrated in methods which make use of turbulence models. Most turbulence models
are statistical turbulence model, in spite of the existence in Ansys-CFXr of other non-statistical
turbulence models such as Large Eddy Simulation model and the Detached Eddy Simulation
26
model. Statistical turbulence models are based on the fact that when looking at time scales
much larger than the time scales of turbulence �uctuations, turbulent �ow could be said to
exhibit average characteristics with an additional time-varying �uctuation. Mathematically, the
velocity variable can be represented by the following equation:
U = Uaveragecomponent + uvaryingcomponent, (4.17)
where:
Uaveragecomponent =1∇t
∫ t+∇t
tUdt. (4.18)
In this thesis, U will be used to represent the average component of velocity and u will
represent the average quantity for products of �uctuating quantities. ∇t is the timescale that
is large relatively to turbulence �uctuations, but small relatively to the time scale to which
the equations are solved. This is an important variable because it could and sometimes when
�ow separation occurs it must be changed in the Ansys-CFXr solver parameters to achieve a
converged solution. A statistical turbulence model modi�es the original unsteady Navier-Stokes
equations by the introduction of average and �uctuating quantities. The result is the Reynolds
Averaged Navier-Stokes (RANS) equations:
∂ρ
∂t+∇ · (ρU) , (4.19)
∂ρU
∂t+∇ · (ρU ⊗ U) = −∇p+∇·, (τ − ρu⊗ u) + SM (4.20)
(∂ρhtot)∂t
− ∂p
∂t+∇ · (ρUhtot) = ∇ ·
(λ∇T − ρuh
)+∇ · (U · τ) + U · SM + SE . (4.21)
The mean total enthalpy is given by the following equation:
htot = h+12U2 +
12u2. (4.22)
It is also important to present the Reynolds transport equation for additional variables (non-
reacting scalars (φ)) as it will be used later (to determine turbulence models variables such as
k,ε and ω ):
(∂ρφ)∂t
+∇ · (ρUφ) = ∇ ·(Γ∇φ − ρuφ
)+ Sφ. (4.23)
RANS equations greatly reduces the computational e�ort compared to a Direct Numerical
but also introduces additional unknown terms, di�cult to determine directly, called Reynolds'
stresses, ρu⊗ u. Thus, there is the need to use supplementary relations inspired on the physic
27
process to model the additional unknown terms and close the system. The equations used to
close the system de�ne the type of turbulence model. In this thesis two statistical turbulence
models were used. The K − ε and the K − ω based Shear Stress Turbulence. Both models are
based on the Eddy Viscosity Hypothesis.
4.3.1 Eddy Viscosity Turbulence Models
The Eddy Viscosity Hypothesis [11] assumes that the Reynolds Stresses can be related to
the mean velocity gradients and Eddy (turbulent) Viscosity by the gradient di�usion hypothesis,
analogous to the relationship between stress and strain tensors in laminar Newtonian �ow, as
shown in the following equation:
−ρu⊗ u = µt
(∇U + (∇U)T
)− 2
3δ (ρk + µt∇ · U) , (4.24)
where µt is the Eddy Viscosity or Turbulent Viscosity.
Analogous to the Eddy Viscosity Hypothesis is the Eddy Di�usivity Hypothesis, which states
that the Reynolds �uxes of a scalar (φ) are linearly related to the mean scalar gradient:
−ρuφ = Γt∇φ, (4.25)
where Γt is the Eddy Di�usivity, and can be written as:
Γt =µtPrt
, (4.26)
Prt is the turbulence Prandtl number.
From the equations above we can see that the turbulence �uctuation terms are function of the
turbulence viscosity µt. The two equation turbulence models used in this thesis, k − ε and SST
k − ω provide this variable. Before presenting the two turbulence models used, it is important
to apply these hypotheses to the momentum and energy equations on the RANS equations (the
continuity equation does not change). From these hypotheses the Reynolds average momentum
equation can be presented as:
∂ρU
∂t+∇ · (ρU ⊗ U) = −∇p′ +∇ ·
(µeff
(∇U + (∇U)T
))+B, (4.27)
where
µeff = µ+ µt, (4.28)
and p' is the modi�ed pressure, de�ned by:
p′ = p+23ρK +
23µt∇ · U. (4.29)
28
The Ansys-CFXr solver assumes that p'=p but the contribution of the other terms in the
equation 4.29 can be included by activating an expert parameter on the Ansys-CFXr solver.
In this thesis p' is always assumed to be equal to p.
The Reynolds averaged energy equation becomes:
(∂ρhtot)∂t
− ∂p
∂t+∇ · (ρUhtot) = ∇ ·
(λ∇τ +
µtPrt∇h)
+∇ · (U · τ) + SE . (4.30)
The Reynolds averaged transport equation for additional variables is now given by the equa-
tion:
(∂ρφ)∂t
+∇ · (ρUφ) = ∇ ·(
Γφ +µtσφ
)∇φ+ Sφ. (4.31)
4.3.2 k − ε turbulence model
k − ε turbulence model [12] is considered to be the industry standard model and has proven
to be stable and numerically robust, o�ering a good compromise in terms of accuracy and ro-
bustness. Nevertheless, caution must be taken when using this model in �ows with boundary
layer separation, �ows with sudden changes in the mean strain rate, �ows in rotating �uids and
�ows over curved surfaces as this model may not give an accurate result. It is a two equation
model, it includes two extra transport equations to represent the turbulence properties of the
�ow. This allows a two equation model to account for history e�ects like convection and di�usion
of turbulence energy. In K − ε model, K(L2T−2) is the turbulence kinetic energy and is de�ned
as the variance of the �uctuations in velocity. ε(L2T−3) is the turbulence Eddy dissipation or,
in other words, the rate the velocity �uctuations dissipate. As referred above the k-ε model is
based on the Eddy Viscosity concept and the unknown term, turbulence viscosity is calculated
in this model by the assumption that the turbulence viscosity is linked to the turbulence kinetic
energy and dissipation via the relation:
µt = Cuρk2
ε, (4.32)
Cu is a constant of the model with the value of 0.09. The values of k and ε are obtained by the
di�erential transport equations for the turbulence kinetic energy and turbulence dissipation rate
(the use of equation 4.31):
(∂ρk)∂t
+∇ · (ρUk) = ∇ ·(µ+
µtσk∇k)
+ Pk − ρε, (4.33)
(∂ρε)∂t
+∇ · (ρUε) = ∇ ·(µ+
µtσε∇ε)
+ε
k(Cε1Pk − Cε2ρε) , (4.34)
29
where Cε1 = 1.44, Cε2 = 1.92, σk = 1 and σε = 1.3. The Pk is the turbulence production due to
viscous and buoyancy forces,which are modeled using:
Pk = µt∇U ·(∇U +∇UT
)− 2
3∇ · U (3µt∇ · U + ρk) + Pkb, (4.35)
this term has some considerations in Ansys-CFXr that can be seen in [11].
4.3.3 Shear stress transport turbulence model
The Shear Stress Transport (SST) k-ω turbulence model [13], developed by Menter in 1993 is
a two-equation Eddy-viscosity model which has become very popular. This model has its origin
in one of the main problems in turbulence modeling, the accurate prediction of �ow separation
from a smooth surface. In fact, turbulence models usually fail to predict the onset and the
amount of �ow separation under adverse gradient conditions, specially models based on the ε-
equation which usually predict the onset of separation too late and under-predict the amount
of separation later on. The SST is one of the most prominent two-equation models in this area,
being specially designed to give highly accurate predictions of the onset and the amount of
�ow separation under adverse pressure gradients. This is achieved by the inclusion of transport
e�ects into the formulation of the Eddy-viscosity. The superior performance of this model has
been demonstrated in a large number of validation studies [16]. Basically the SST formulation
combines the best of two worlds. The use of a k-ε, formulation in the inner parts of the boundary
layer makes the model directly usable all the way down to the wall through the viscous sub-layer,
hence the SST k-ω model can be used as a Low-Re turbulence model without any extra damping
functions. The SST formulation also switches to a k-ε behavior in the free-stream and thereby
avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence
properties. Next the SST k-ω model formulation is exposed. A basic k-ω model assumes that
the turbulence viscosity is linked to the turbulence kinetic energy and turbulence frequency via
the relation:
µt = ρk
ω. (4.36)
k and ω are obtained solving the transport equations (applying 4.31) of the two variables. One
problem of a k-ω model is the strong sensitivity to the free-stream conditions. Depending on the
speci�ed value of ω at the inlet, results can be very di�erent. To resolve this problem a blending
between the k-ω model near the surface and the k−ε model in the outer region can be used. This
consists of a transformation of the k - ε model to a k -ω formulation and a subsequent addition
of the corresponding equations. The k -ω model is thereby multiplied by a blending function F1
and the transformed k - ε model by a function 1-F1 .
The SST model uses this formulation and k and ω are given by equations:
30
(∂ρk)∂t
+∇ · (ρUk) = ∇ ·(µ+
µtσk3∇k)
+ Pk − ρβ′kω, (4.37)
(∂ρω)∂t
+∇ · (ρUω) = ∇ ·(µ+
µtσω3∇ω)
+ (1− F1) 2ρ1
σω2ω∇k∇ω + α3
ω
kPk − β3ρω
2, (4.38)
In the SST model F1 is given by:
F1 = tanh(arg4
1
), (4.39)
arg1 = min
(max
( √k
β′ωy,500νy2ω
),
4ρkCDkwσω2y2
), (4.40)
CDkω = max
(2ρ
1σω2ω
∇k∇ω, 1× 10−10
). (4.41)
The values of the coe�cients β′, α, β, σk.σω are presented in [13].
In the SST model to improve the prediction of onset amount of �ow separation from smooth
surfaces, a limiter to the formulation of Eddy-viscosity is also used. This consists in changing
the equation 4.36 to equation 4.42:
µt = ρa1k
max (a1ω, SF2), (4.42)
F2 are blending functions just like F1 and S is an invariant measure of the strain rate.
F2 = tanh(arg2
2
), (4.43)
arg2 = max
(2√k
β′ωy,500νy2ω
). (4.44)
4.4 Near wall treatment
Numerical simulation of a region near a no-slip wall raises some problems because there are
strong gradients in the dependent variables and because viscous e�ects in the transport processes
are large. Commonly there are two approaches to model the �ow in the near-wall region. One is
the wall function method that uses empirical formulas that impose suitable conditions near the
wall without resolving the boundary layer, thus saving computational resources. All turbulence
models in Ansys-CFXr are suitable for a wall function method. Wall functions are based on the
fact that experiments and mathematical analysis have shown that the near-wall region can be
subdivided into two layers, the inner layer and the outer layer. The inner layer can be divided into
3 sub-layers, a linear sub-layer where the only signi�cant shear stresses are laminar/molecular
31
like, a wall or log sub-layer where shear stresses are almost exclusively turbulent and a bu�er
layer where the e�ects of molecular viscosity and turbulence are of equal importance. In �gure 4.1
the velocity pro�le and the subdivisions of the near wall region in semi-logarithmic coordinates
is presented.
Figure 4.1: Near wall velocity pro�le of a turbulent boundary layer
Assuming that the logarithmic pro�le reasonably approximates the velocity distribution near
the wall, it provides a mean to numerically compute the �uid shear stress as a function of the
velocity at a given distance from the wall. This is known as a wall function. Wall functions
have the advantage of saving CPU time and storage, as the near wall region can be modeled
with relatively coarse meshes. Nevertheless, standard wall function formulations are di�cult to
handle, because it has to be ensured that the grid resolution near the wall satis�es the wall
function requirements. If the grid becomes too coarse, the resolution of the boundary layer is
no longer ensured. If the resolution becomes too �ne, the �rst grid spacing can be too small
to bridge the viscous sublayer. In this case, the logarithmic pro�le assumptions are no longer
satis�ed. The user has to ensure that both limits are not overstepped in the grid generation
phase. To overcome this problem Ansys-CFXr has the possibility to use scalable wall functions
[65] which allows a systematic grid re�nement when using wall functions.
The other approach to model the near wall region is the Low-Reynolds-Number method.
This method doesn't use any empirical formulas, integrating the RANS through the viscous
sublayer. This method is generally more accurate but on the other side most low-Re extensions
of turbulence models are quite complex and can reduce the numerical performance or even
32
destabilize the numerical method. In addition, classical low-Re models require very �ne mesh in
the near-wall zone and correspondingly large number of nodes to capture the rapid variation in
variables, which can be very complex in industrial application. Computer-storage and runtime
requirements are higher than those of the wall-function. To reduce the resolution requirements,
an automatic wall treatment was developed by Ansys-CFXr which allows a gradual switch
between wall functions and low-Reynolds number grids, without a loss in accuracy. This hybrid
method is available for all ω-equation based turbulence models (automatic near-wall treatment),
which automatically switch from a low-Re formulation to wall functions based on the grid spacing
provided by the user. These formulations provide the optimal boundary condition for a given
grid. From a best practice standpoint, they are the most desirable, as they allow an accurate
near-wall treatment over a wide range of grid spacings. A minimum of at least 10 grid nodes
inside the boundary layer should be used in a hybrid method.
In this thesis, when using k-ε turbulence model, analyses were performed using scalable wall
functions and when using the SST model an automatic near wall treatment (hybrid method)
was used. As recomended by Ansys-CFXr formulation, a mesh with 10 grids nodes inside the
boundary layer was used, whenever possible, to perform the �uid analysis and it will be seen
later on.
4.5 Mesh deformation
Mesh Deformation is an important component for solving problems with moving boundaries,
as it is the case in this study. The mesh deformation option allows the speci�cation of the nodes
motion in boundaries or subdomain regions. The motion of all remaining nodes is determined
by a mesh motion model. Ansys-CFXr has only the displacement di�usion model available.
In the displacement di�usion model, the displacements applied in the domain boundaries or in
subdomains are di�used to other mesh points by solving the equation:
∇ · (Γdisp∇δ) = 0. (4.45)
In this equation δ is the displacement relative to the previous mesh locations and Γdisp is
the mesh sti�ness, which determines the degree to which regions of nodes move together. This
equation is solved at the start of each iteration between coupled �elds.
In this coupled �uid-structure simulation the motion of the Ansys-CFXr mesh is imposed
by the results of the Ansysr structural �eld analysis. To this accomplishment the Ansysr
Multi�eld mesh motion options were activated in the interface surfaces between the two �elds.
When this option is activated, the nodes are displaced according to the data received from the
Ansysr Multi-�eld solver in regions that match the Ansysr side (interface of the Ansysr
33
�eld), whereas the unspeci�ed condition is imposed in regions that do not match Ansysr side.
This displacement is always relative to the initial mesh.
When applying mesh deformation it is not uncommon for the mesh to become folded. This
was one major problem during this study. Due to large deformations, the mesh deformation
algorithmic can not modify the mesh without folding the mesh and then negative volumes appear,
which imply di�culties in convergence, non convergence results or termination errors. To avoid
these problems several techniques can be used. As stated in [7], to avoid mesh folding large
displacements should be applied gradually because the fewer deformation is applied the easier
it is the mesh algorithm to perform without problems and errors. In Ansys-CFXr/Ansysr
analysis this can be accomplished by the use of under-relaxing factors for displacements applied
per deformation step.
Relaxation is simply using a fraction of the di�erence between the previous global iteration
result and the newly calculated values. Denoting by φ the nodal value of interest, the expression
for relaxation is as follows:
φnewi = (1− rφ)φoldi + rφφcalculatedi , (4.46)
where rφ is the relaxation factor of the variable and can vary from 0 to 1.
As presented in the formula 4.46, if the value of rφ is 1 the previous total iteration will not
be taken into account in the new result. Mesh folding is often avoided with this strategy because
the mesh displacement equations are assembled using the updated meshes from each deformation
step, and as the deformation is reduced by the inclusion of the relaxation factor, mesh deformation
is gradually applied. In general, the desired total mesh deformation should not be larger than
approximately 5 adjacent elements per step. In our study the use of relaxation factors has
the disadvantage of increasing the time to perform the �uid/structure interaction because it
increases the required number of interactions between �elds until reaching convergence. In fact,
when using extremely �ne cells resolution to account for boundary layer e�ects, changing of 5
adjacent elements per step is prohibitive to be performed as it would imply so many iteration
steps that it would take prohibitive time (weeks) to perform one �uid/structure calculation. In
our case, wing deformation is about 10 cm and 5 adjacent cells over the wing would imply a less
than 1mm deformation over each time step.
Folded mesh can also occurs when the displacement equations are incompletely solved. During
each outer iteration or timestep, the mesh displacement equations are solved to the speci�ed
convergence level and the resulting displacements are applied to update the mesh coordinates.
When the displacement function is not converged to a su�ciently low residual the unconverged
displacement solution �eld does not vary smoothly enough to ensure that adjacent mesh nodes
move by similar amounts and mesh folding can occur. To control the convergence of mesh
34
deformation, Mesh Displacement parameters in the Equation Class Settings Tab of the Ansys-
CFXr Solver Control can be speci�ed. From our experience and taking into account that
extremely small cells are used, considering the boundary layer e�ects, a value smaller than 10−3
to the maximum residual should be speci�ed. The Maximum Number of Coe�cient Loops should
be speci�ed high enough to ensure that the number of coe�cient loops is su�cient to achieve the
residual criterion speci�ed before. Another way to control mesh quality and avoid mesh folding
is to use a mesh sti�ness (Γdisp) that suits the problem in question. As stated before, the mesh
sti�ness determines the degree to which regions of nodes move together. If a constant value
of mesh sti�ness is applied, speci�ed displacements are homogeneously di�used throughout the
mesh. When the mesh sti�ness is speci�ed as varying throughout the domain, nodes in regions
of high sti�ness move together (there is little relative motion).
There are three available Mesh Sti�ness options: Increase Near Small Volumes, Increase
Near Boundaries and Value. Increase Near Small Mesh Volumes has the advantage that mesh
quality will bene�t from having larger control volumes absorbing more mesh motion. Using
Increase Sti�ness Near Boundaries (the program assumes boundaries of any type) mesh quality
will bene�t from having the mesh interior (away from boundaries) absorbing more mesh motion.
A more detailed explanation about these two models and how user can change its behavior
can be obtained in [8]. Nevertheless none of these models were used in prejudice to the option
Value. With this option the user can introduce a formula that controls the sti�ness changing
throughout the mesh. The formula used in this thesis is a mix and optimization of the available
mesh sti�ness options presented before. When the mesh deforms, we want to preserve the �ne
mesh distribution near wall boundaries to correctly calculate the boundary layer e�ects, so it is
important to impose large mesh sti�ness near walls. Sti�en the mesh near walls is also important
to prevent mesh folding because the mesh is more dense and the distance between node elements
is smaller and as a consequence their capacity to absorb deformation is also smaller. In opposite
to the options increase sti�ness near boundaries which treats all boundaries indistinctly, we can
use an Ansysr variable Wall Distance to restrict mesh sti�ness to wall boundaries, because it
will not impose high mesh sti�ness in boundaries such as inlets and outlets where mesh is coarse,
can absorb large deformation without folding and where preserving the mesh density distribution
is not important.
Wall Distance(m) is an Ansysr variable which saves for each element the result from the
distance calculation of the element to the near wall. By imposing a value for the mesh sti�ness
of 1Wall Distance(m) , we can secure that mesh near walls will be sti�er. To improve the mesh sti�-
ness distribution we can also introduce the term 1V olume of F inite V olumes(m3)
. Volume of Finite
Volumes is another Ansysr variable that saves for each element (�nite volume) its volume. This
is very important because as the size of the element decreases its capacity to absorb deformation
also decreases. As the mesh deforms, the elements can become so small that they start to fold.
35
Using this variable we are imposing that bigger elements are absorbing higher deformations, and
so it avoids folding. So we have used the following equation for the option Value:
value =1[m6s−1]
WallDistance∗ 1V olumeofFiniteV olumes
. (4.47)
I would like to emphasize equation 4.47 because it allowed the calculation times to be reduced
to less than a quarter of the initial time, when using mesh sti�ness option �Increase Near Small
Volumes� or �Increase Near Boundaries�, as we could increase the relaxation factor without mesh
folding and even perform calculations with higher mesh deformations that could not be performed
with the other mesh sti�ness option.
4.6 CFX r Boundary conditions
Simulation of free surface �ows usually requires de�ning boundary and initial conditions to
set up appropriate pressure and volume fraction �elds. Di�erent boundary conditions can be
speci�ed in Ansys-CFXr.
Inlet
An inlet boundary condition is used where the �ow is predominantly directed into the domain.
For the mass and momentum equations inlet boundary conditions can be set in a number of ways:
de�ning a cartesian or cylindrical velocity, or de�ning the total pressure or the mass �ow. In
this thesis an inlet boundary condition was always de�ned by a cartesian velocity.
Uinlet = uspecifiedi+ vspecifiedj +Wspecifiedk. (4.48)
For all other transport equations, the value is speci�ed directly at the inlet, or speci�ed in terms
of a simple relationship that constrains the dependent variable.
Outlet
An outlet boundary condition is used where the �ow is predominantly directed out of the
domain. The hydrodynamic boundary condition speci�cation for a subsonic outlet involves
some constraint in the boundary static pressure, velocity or mass �ow. For all other transport
equations, the outlet value of the variable is part of the solution. In this thesis the outlet
boundary condition was speci�ed de�ning the relative static pressure:
Pstatic,Outlet = Pspecified. (4.49)
36
Opening
An opening boundary condition allows the �uid to cross the boundary surface in either di-
rection. An opening boundary condition can be speci�ed by di�erent ways as it can be seen
in [9]. In this thesis in all the opening boundary conditions the option Static Pressure for
Entrainment was used. Here the static pressure of the opening is de�ned by the user and �ow
direction is obtained by enforcing the velocity gradient perpendicular to the boundary to be zero.
Symmetry
The symmetry boundary condition imposes constraints which "mirror" the �ow on either
side of the symmetry plane:
Un = 0, (4.50)
and
∂φ
∂n= 0. (4.51)
where φ is a scalar variable.
Wall
Walls are solid (impermeable) boundaries to �uid �ow. Walls allow the permutation of heat
and additional variables into and out of the domain through the setting of �ux and �xed value
conditions at wall boundaries. In all wall boundary conditions used in this thesis the no-slip
condition will be used (Ansys has other options such as Free Slip, Rotating Wall and Counter
Rotating Wall):
Uwall = 0. (4.52)
37
Chapter 5
Problem Modulation
As explained before, this study is based on a Ansysr Multi�eld Code Coupling, Fluid-
Structural interaction analysis. To accomplish this analysis one model for each �eld had to be
created. Thus, one model was created in Ansysr for the structural �eld and one was created
in Ansys ICEM-CFDr for the �uid �eld. To modulate a wing morphing with airfoil shape
variation and span variation a model with continuous and independent airfoil variation and span
variation should be created. This would be extremely complex, even more with the inclusion of
the �uid/structure interaction which is the base of this thesis. Thus, a di�erent simpler approach
was used. We have replaced the continuous span variation model with a continuous airfoil shape
changing for several discrete models that simulate in a speci�ed con�guration the wing in study.
These con�gurations are representative of the complete wing behavior.
5.1 Structural �eld modulation
To create the structural model a program in Matlab was made that writes an Ansysr Para-
metric Design Language (APDL) �le. The Ansysr Multiphysics reads this �le to produce the
model. The idea of using Matlab to create the APDL �le was based on the possibility to use
Matlab standard functions, and specially because it is intended to use Matlab to perform fu-
ture optimization procedures based on Matlab standard optimization functions. The structural
model is based on an inner wing and an outer wing. The outer wing is �xed but the inner wing
is intended to slide, as stated in chapter 2 and as shown in picture 5.1
In �gure 5.1 it is also represented the coordinate system used. The x coordinate is parallel
to the chord and the z coordinate is parallel to the span. The di�erent colors in the wings in
�gure 5.1 represent di�erent materials used in each part. As we will see later the user can specify
di�erent sections in the upper surface and lower surface of the wing to use di�erent materials.
This was specially developed to allow the user to use a rubber section in a part of the airfoil so
that may su�er large chord changes or to specify di�erent materials properties throughout the
38
Figure 5.1: Structural �eld, Ansysr Model
airfoil to ease airfoil shape changing. The user can also specify material changes in the span
direction. The span of the inner wing and outer wing can be speci�ed by the user. In this thesis
we have always considered an inner wing and an outer wing with one meter each.
As stated in the begining of this chapter, the Ansysr modulation of the transient e�ect of the
inner wing sliding into the outer wing has not been represented. Representing a �uid structural
interface and including the motion of the inner wing in the model would imply the CFD mesh,
that is presented later in the CFD model, to deform such an amount that could not be supported
by the CFXr mesh deformation algorithm without completely disarrange the required near to
wall high mesh resolution, and would imply folding mesh problems. So this study will be based on
the analyses of several static con�gurations of the telescopic wing, di�erent inner wing positions
39
relatively to the outer wing. Figure 5.2 is illustrative with di�erent wing con�gurations.
Figure 5.2: Di�erent wing con�gurations: a) with the inner wing 25% deployed b) with 50%deployed and c) 75% deployed.
We have considered a maximum deployment of the inner wing when it is 75% extended
(25% inside the outer wing). To simulate the connection between the inner and outer wing the
connection sections represented in the �gure 5.3 were created.
Figure 5.3: Connections between the two wings
The concept behind the connections modulation is to create di�erent sections with distinct
meshes in the inner wing and the outer wing and then apply constrain equations to the nodes
that are in the connection region. The constrain equations are presented in �gure 5.3 in pink.
The constrains equation applied were zero rotation about all axes and zero displacement in all
the directions except in the Z (span direction) which is unspeci�ed to simulate the possibility of
sliding of the interior wing. Some analyses were made with the displacement in the Z direction
restricted. The two wings connections can be positioned in all six places presented in �gure 5.3.
We can use all six connections, only the four connections presented in J or the two connections
presented in I. The location of the connections J can be changed but only in the chord direction.
The wing shell and the connections between the inner wing and the outer wing were modeled
in Ansysr using the shell element 181. Shell181 is an 4-Node Finite Strain Shell suitable for
40
analyzing thin to moderately-thick shell structures. It is a 4-node element with six degrees of
freedom at each node, translations and rotations in the x, y, and z directions/axes. SHELL181 is
well-suited for linear, large rotation, and/or large strain nonlinear applications. SHELL181 may
be used for layered applications for modeling laminated composite shells or sandwich construc-
tion. The accuracy in modeling composite shells is governed by the �rst order shear deformation
theory also called Mindlin-Reissner shell theory. More information about these Ansysr elements
can be found in [14].
The interior con�guration of the telescopic wing is based on one spar and several ribs. The
number and position of the ribs can be speci�ed by the user. The user can also specify the ribs
and spar sections. In �gure 5.4 di�erent spar sections are represented .
Figure 5.4: Interior structure of the wing using di�erent spar sections
The �gure 5.5 presents an example of the interior con�guration of each wing, as well as some
sections used. As we can see in this con�guration the telescopic wing has one spar, represented in
red. There are two ribs connected to the outer wing and 5 ribs connected to the inner wing. The
connection between the ribs and spar, and between the ribs and the wing shell are modulated
by merging the nodes of each element which implies a coupling in all degrees of freedom. To
modulate the ribs and the spar Ansysr element Beam 188 was used.
Beam 188 is a 3-D Linear Finite Strain Beam suitable for analyzing slender to moderately
stubby/thick beam structures. This element is based on Timoshenko beam theory and is well-
suited for linear, large rotation, and large rotation nonlinear applications. Shear deformation
e�ects are included. Beam 188 can have six or seven degrees of freedom at each node. In this
thesis only six degrees of freedom were used, translation and rotation in the x, y and z. More
information about this Ansysr element can be found in [14].
The wing is �xed at the spar root (all nodes degrees of freedom are restricted) and is con-
stricted in Y and Z directions and in rotation about the X and Y axis, in 2.5% of the shell leading
edge and also in 2.5% of the shell trailing edge as it can be seen in �gure 5.6. Displacements in
X direction and rotation in Z direction are free to allow airfoil morphing. We have considered
restrictions in 2.5% of the shell leading edge and in the shell trailing edge because restricting
only in the shell leading edge and shell trailing edge node will result in a stress concentration
expressing less realistic results. Analyses with all wing root airfoil �xed were also performed.
The wing materials used in the performed structural analysis were the polypropylene, the
41
Figure 5.5: Interior structure of the wing
Figure 5.6: Ansysr model boundary conditions
epoxy, the carbon-epoxy composite rod and the carbon-epoxy composite sheet. The properties
of these materials are presented in tables 5.1, 5.2 , ?? and 5.4.
42
Property Value
Coe�cient of thermal expansion - Longitudinal (x10−6K−1) 2.1Coe�cient of thermal expansion - Transverse (x10-6 K-1) 2.1
Compressive Strength - Longitudinal (MPa) 570Compressive Strength - Transverse (MPa) 570
Density (g cm−3) 1.6Shear modulus - in-plane (GPa) 5Shear strength - in-plane (MPa) 90
Tensile strength - Longitudinal (MPa) 600Tensile strength - Transverse (MPa) 600
Ultimate Compressive Strain - Longitudinal (%) 0.8Ultimate Compressive Strain - Transverse (%) 0.8
Ultimate Shear Strain - in-plane (%) 1.8Ultimate Tensile Strain - Longitudinal (%) 0.85Ultimate Tensile Strain - Transverse (%) 0.85
Volume fraction of �bres (%) 50Young's Modulus - Longitudinal (GPa) 70Young's Modulus - Transverse (GPa) 70
Table 5.1: Carbon/Epoxy Composite Sheet properties
Property Value
Compressive Strength - Longitudinal (MPa) 800-1300Compressive Strength - Transverse (MPa) 50-250
Density (g cm−3) 1.6Flexural modulus - Longitudinal (GPa) 125Flexural strength - Longitudinal (MPa) 1200Tensile strength - Longitudinal (MPa) 1100-1900Tensile strength - Transverse (MPa) 50
Thermal Expansion Coe�cient - Longitudinal (x10−6K−1) -0.3 to -0.7Thermal Expansion Coe�cient - Transverse (x10−6K−1) 28
Ultimate Compressive Strain - Longitudinal (%) 0.8Ultimate Compressive Strain - Transverse (%) 2.5Ultimate Tensile Strain - Longitudinal (%) 1.1Ultimate Tensile Strain - Transverse (%) 0.5
Volume fraction of �bres (%) 55-60Young's Modulus - Longitudinal (GPa) 120-140Young's Modulus - Transverse (GPa) 10
Table 5.2: Carbon/Epoxy Composite Rod properties
Property Value
Elastic Modulus (MPa) 2070Flexural Modulus (MPa) 1725Tensile Strength (MPa) 42
Compressive Strength (MPa) at yield or break 56Flexural Strength (MPa) at yield or break 56
Speci�c Gravity (g/cm3) 0.91
Table 5.3: Polypropilene properties
43
Property Value
Elastic Modulus (MPa) 2089Flexural Modulus (MPa) 2300Tensile Strength (MPa) 74
Flexural Strength (MPa) at yield or break 115Speci�c Gravity (g/cm3) 1
Table 5.4: Epoxy properties
5.1.1 Matlab parametric program to create structural model
As stated before a parametric Matlab function was created to produce a APDL Ansysr log
�le.
The next table presents the Matlab program entries.
Figure 5.7: Inputs of the Matlab parametric function to generate the airfoil
The variables SecA1, SecA2, SecB1, SecB2 divide the upper (A section) and the lower (B
section) surface of the airfoil in three sections to use di�erent properties and materials in the
airfoil based on chord direction. The di�erent sections de�ned by this variable can be seen from
the di�erent colors presented in the �gures 5.1 and 5.2. The value of each variable is the chord
percentage in which the section changes.
The variables actuador1, actuador2 are for posterior use on the position of the airfoil changing
actuators in chords percentage. This variable will not be used in this thesis.
The rib1, rib2 variables are used for the position of the ribs. The rib1 variable is for the
outer wing and rib2 for the inner wing. They correspond to the percentage of the span in which
44
Model Variables Value
Airfoil Naca-0012, Eppler-434, Wortman-FX-63-137Outer wing span 1Inner wing span 1Total wing span 1-1.75Inner wing chord 0.2Outer wing chord 0.22
Wing shell materials Epoxy, Polypropilene, Carbon-Epoxy composite Rob and SheetWing shell sections thickness 0.2-1(mm)
Number of spars 1Spar material Carbon-Epoxy composite RobNumber of ribs 2-10Ribs material Carbon-Epoxy composite Rob
Wing connections type I, J, both I and J)Wing connections materials Epoxy, Carbon-Epoxy composite Rob and Sheet
Table 5.5: Model variables
a rib is �xed.
The ligacaotipo variable de�nes the type of connections to be used. ligacaotipo=0 de�nes
only J connections, ligacaotipo=1 de�nes only I connections and ligacaotipo=2 de�nes the use
of both I and J connections. The ligacaofrente, ligacaotras variables specify the position of the
wing connections for the J type connections. Figure 5.3 is illustrative of these variables function.
The deslocAsa variable de�nes the displacement of the inner wing to the root of the outer
wing. The result is the di�erent wing con�gurations, as presented in the �gure 5.2, function of
this variable.
The cordaper�l1, cordaper�l2 variables are used to specify the respective chord of the outer
wing and inner wing.
The envergaduraAsa1, envergaduraAsa2 variables are the respective span of the outer wing
and inner wing.
Section types, material properties and element types can also be changed and are well identi-
�ed in the code where the user can change it; however to do so it is essential to have a knowledge
of APDL Ansysr code. The output of this function is a text �le to be loaded by the Ansysr
program.
The table 5.1.1 presents the wing properties and con�gurations that can be changed by the
user and the values used in this thesis.
5.2 Fluid �eld modulation
To model the �uid �eld of the FSI analysis, three programs were used. For the wing geometry
modeling Solid Works and Ansysr Workbench were used, and to generate the mesh the Ansysr
ICEM-CFD was used. In the structural model a parametric program was developed as discussed
45
Airfoil chord outer wing (m) chord inner wing (m) Span (m)
NACA 0012 0.22 0.2 1,1.25,1.5,1.75Eppler 434 0.22 0.2 1,1.25,1.5,1.75
Wortman FX-63 0.33 Not Aplicable 1
Table 5.6: Created �uid models
before and any change in the model was extremely fast to accomplish compared to the �uid
model generation. In the �uid model, as it involved three di�erent programs, no parametric or
automated program could be developed to generate di�erent models. In addition, generating a
CFD mesh is a hard process. Therefore only 9 �nal �uid models were created. The nine �uid
models correspond to four di�erent wing con�gurations for the airfoils NACA 0012 and Eppler
434 and one con�guration for the Wortman FX-63 airfoil. The choice of these airfoils is explained
in chapter 7. The models are summarized in table 5.2.
The �gure 5.8 presents a �uid model with the generated mesh, in this case with the airfoil
NACA-0012 and 1.75 wing span. A �ne resolution near the wing was created to account for the
boundary layer e�ects.
Figure 5.8: Mesh of the CFD model
A su�ciently large domain was carefully generated so that the boundaries would be su�-
ciently away from the wing and as consequence the �ow would be su�ciently una�ected so that
the applied boundary conditions would respect the problem physics. For the problem in study a
larger domain is also important to support mesh deformation. Having a larger domain implies
smaller variable gradients near the regions far way from the wing. This allows bigger elements
in these regions which can support more mesh deformation and avoid mesh folding. The domain
is 3.6m, equivalent to 18 chords, long (X direction); 3.2m, equivalent to 16 chords, high (Y
46
Surface symbol Type of boundary condition
black arrows inletyellow arrows outlet
two ways blue arrows openingpink converged arrows symmetry
non symbol wall
Table 5.7: Boundary conditions symbols in CFXr
direction); and 4.5m, equivalent to 2.57 spans, width (Z direction).
After the mesh generation in Ansysr ICEM-CFD the model was imported to Ansys CFXr
where all the problem conditions, such as boundary conditions, turbulence models, mesh dis-
placement options among others, were de�ned for each analysis. The �gure 5.9 obtained from
the CFXr program is representative of the boundary conditions used.
Figure 5.9: CFXr Boundary conditions
The CFXr boundary conditions symbols are explained in the table 5.7.
For all the wing analyses with positive angles of attack the boundary conditions type in each
section of the external volume were the same as presented in �gure 5.9. For obvious reasons, in all
the wing analyses with negative angles of attack the inlet boundary condition of the lower domain
surface was replaced by an opening boundary condition and the opening boundary condition type
47
of the upper domain surface was replaced by an inlet.
5.3 Interface surfaces
To accomplish a FSI analysis interface areas of both structural and �uid models, where loads
transfer occurs, have to be identi�ed so the mesh nodes of the structural model can be mapped
to the mesh nodes of the �uid model and vice versa. After the mapping accomplishement the
Ansysr model can receive the forces resulting from the aerodynamic loads calculated in the
CFXr �eld and the CFXr can receive the node displacements from the Ansysr model. The
areas of Ansysr model de�ned as interfaces were the ones in contact with air�ow: all the outer
wing surface and all the inner wing surface that is not inside the outer wing. There are regions
in the CFXr mesh that have no correspondence in the Ansysr mesh. The interface areas and
the regions without correspondence between the two models are presented in �gure 5.10, marked
as A and B.
Figure 5.10: Interface surfaces. Ansysr model on the left side of the �gure and CFXr modelon the right
Region A of the �gure 5.10 is simply a transition surface from the inner wing to the outer
wing created in CFXr model that has no correspondence to the Ansysr model. This was used
to improve the aerodynamic behavior of the wing. It doesn't have any structural function and
the way the Ansysr model is conceived, generating a transition surface from the inner wing to
the outer wing, would imply complexity without any bene�t. Thus, it was not represented in
the Ansysr model.
Region B is the wing tip (this area can be modeled in Ansysr just altering the APDL code).
In some analyses the wing tip was not modeled in the Ansysr �eld, because it was thought to
use a wing tip very �exible to accommodate airfoil morphing o�ering the minimum resistance
48
through the chord direction and having no structural function. The wing tip functions only as
a shell that doesn't allow the �uid to enter into the interior of the inner wing. As materials
and properties of such a wing tip have not been de�ned yet and assuming the wing tip has no
structural function, it was decided not to model the wing tip in the Ansysr model. When this
occurs there are no quantities transferred between the structural and �uid models through the
wing tip, as CFXr wing tip is not marked as an interface surface. Analyses in the wing tip of
CFXr model demonstrate that normal forces are less than 1N for the studied angles of attack
and therefore we concluded that not including a wing tip in the Ansysr model would not be
signi�cant in the structural analysis. The Non-Matching Area Fraction resulting from the areas
that have no correspondence between the two models is less than 1.5% as shown in �gure 5.11
extracted from the solver output.
Figure 5.11: Non-Matching Area fraction obtained in the solver output �le
In �gure ?? we present an example of the loads transfer from the CFXr to the Ansysr
model. In this �gure it is presented the force vectors that the Ansys model receives from the
CFX model. As we can see the �gure shows a qualitatively expected force distribution.
Figure 5.12: Force Vectors in the Ansysr model
49
Chapter 6
Convergence study
A convergence study was performed both in ANSYS model and in CFX r model.
6.1 CFX r model convergence study
Ideally, for every di�erent wing con�guration study, a convergence assessment should be
performed. However this would be extremely time consuming. Thus, for the CFX r model the
convergence study was performed for the con�guration using the NACA 0012 airfoil and the wing
completely deployed. We have chosen an angle of attack of 8 degrees and a freestream velocity of
30 m/s. We have used two di�erent turbulence models for each mesh studied in order to improve
this convergence study. These turbulence models were already presented in this thesis, the k-ε
and the Shear stress transport. As mesh re�nement reaches convergence and as the conditions
of this convergence study are without �ow separation and aerodynamic loss, it is expected that
the results from both methods in a su�ciently re�ned mesh should not vary a large amount.
A large variety of mesh re�nement was studied and three di�erent re�nements meshes are
presented below. The mesh presented in �gure 6.1 has 417594 elements and has a small re�nement
near the wing. It is also specially re�ned in the wing leading edge to correctly modulate the
airfoil shape (leading edge element is 5 mm width). This mesh has wing elements. Velocity
contour plot of symmetry plane shows that the boundary layer is not correctly simulated by this
mesh re�nement. Comparing the two turbulence models used presented in table 6.1 we can see
that the lift variation is extremely small but the variation in drag is of 7.79%. The Cd value
(0.019) seam to be extremely large comparing with the airfoil drag presented in �gure 7.1
Turbulence Model Lift Drag
SST 142.41 13.97k-ε 141.24 15.14
Variation 0.83% 7.79%
Table 6.1: Coarse mesh aerodynamic results
50
Figure 6.1: CFD convergence study; sparse mesh
An increase in the mesh re�nement is exposed in �gure 6.2. This mesh has 2141695 elements
and has a high re�nement near the wing that slowly decreases through the volume to the bound-
aries. As we can see in the table 6.2 the drag is extremely smaller than the low re�nement mesh
presented before. Nevertheless, once again the drag results of the di�erent turbulence models
used di�er more than 19.18%, which is completely unacceptable and is due to the incorrect sim-
ulation of the boundary layer, as we can conclude from the velocity pro�le represented on the
right of �gure 6.2 where the velocity pro�le varies with wing distance in an incorrect way.
Turbulence Model Lift Drag
SST 148.27 6.94k-ε 144.78 8.59
Variation 2.41% 19.18%
Table 6.2: Re�ned mesh aerodynamic results
51
Figure 6.2: CFD convergence study; re�ned mesh
To solve this problem a near wall prism mesh was created as presented in the �gure 6.3.
Despite an increase in near wing resolution we have decreased the total mesh elements to 1267779,
decreasing the near boundaries resolution mesh because it was found that it has no e�ect in the
results. Comparing the results of the two turbulence models presented in table 6.3 we can see
that the variation on lift is only of 0.4% and on drag is only 2.41%, which is acceptable. The
velocity pro�le with this mesh has an expected shape as presented on the right of �gure 6.3.
Turbulence Model Lift Drag
SST 145.14 7.45k-ε 144.55 7.64
Variation 0.40% 2.41%
Table 6.3: Re�ned mesh with prism layer elements; aerodynamic results
No re�nement on the prism mesh was performed because decreasing the distance from the
closest node to the wall would imply di�culties in mesh deformation. As the results of two
turbulence models were almost equal and analyzing the velocity and pressure contour plots
results seam extremely realistic and respect the recommendations, in terms of the number of
nodes in the boundary layer exposed in Ansysr manual [6], we have used this last mesh to
perform all the CFD analyses exposed in this thesis. The �gure 6.4 presents the aerodynamic
results function of the mesh re�nement.
52
Figure 6.3: CFD convergence study; re�ned mesh with prism mesh near the wing
Figure 6.4: CFD convergence study, graphic of aerodynamic results function of the mesh re�ne-ment
53
6.2 Ansysr model convergence study
For the Ansysr model the convergence study was performed considering only one model
in the most structural demanding con�guration: the wing completely deployed, with maximum
span. To perform the structural convergence analysis we performed a one-way �uid structural
analysis with the wing subject to an upstream velocity of 30 m/s and an 8 degrees angle of
attack. The properties of the structural model are summarized in table 6.4.
Model Variables Value
Airfoil NACA 0012Outer wing span 1 mInner wing span 1 mTotal wing span 1.75 m (0.75 extension of the inner wing)
Wing shell materials carbon-epoxy composite robWing shell sections height 0.004 m
Number of spars 1Spar material carbon-epoxy composite rob
Number of ribs outer wing 2Number of ribs inner wing 5
Ribs material carbon-epoxy composite robWing conections type I
Wing conections materials carbon-epoxy composite rob
Table 6.4: Properties of the structural model used in the convergence study
For this convergence study a variety of structural meshes were used and the evolution of
maximum deformation, maximum Von Mises, structural energy error and computational time as
a function of number of elements are shown in the graphics of �gure 6.5 and in the tables 6.5,
6.6 and 6.7.
Nº of elements Max. Deformation (m) Variation (%)
26850 0.084167 0.15%50850 0.0842 0.10%98750 0.08434 -0.06%147950 0.08431 -0.02%246350 0.08429 0.00%
Table 6.5: Structural convergence study; Maximum deformation analysis
54
Figure 6.5: Structural convergence study; evolution of the maximum deformation, Von Misestension, structural energy error and time function of the number of elements
Nº of elements Max. Tension (Pa) Variation (%)
26850 8.27E+07 36.44%50850 1.15E+08 11.44%98750 1.28E+08 1.36%147950 1.31E+08 -0.48%246350 1.30E+08 0.00%
Table 6.6: Structural convergence study; Von Mises tension study
55
Nº of elements Maximum energy error Computational time (h)
26850 2.306E-03 0:0550850 2.008E-03 0:0798750 1.200E-03 0:10147950 9.660E-04 0:18246350 7.990E-04 00:32
Table 6.7: Structural convergence study; Maximum Energy Error and Computational time study
As the number of elements increases the maximum deformation and the maximum Von Mises
tension converges to 0.0843 m and 130 Mpa respectively. The computational time increases sig-
ni�cantly with the increase of the number of elements and so we have decided to use a structural
mesh with 98750 elements(approximately 100 chord element divisions and 500 span element di-
visions), as the error in maximum deformation compared to the most re�ned mesh was of 0.6%
and 1.36% in the maximum Von Mises stress, o�ering the best compromise between results and
computational time. The �gure 6.6 presents the deformation, stress, and energy error of this
mesh re�nement structural model.
Figure 6.6: Deformation (left) and Von Mises Stresses (right) obtained with the chosen mesh,analyzed in this convergence study
56
Chapter 7
Wing performance analysis
In this chapter it is presented a wing performance study. For this study CFD models described
in chapter 5 are used, and a CFD analysis made in Ansys-CFXr without mesh deformation
(without wing deformation) and no �uid structure iteration is performed. Before presenting this
wing performance analysis a simpler airfoil study is also performed to de�ne the low speed airfoil
and high speed airfoil to be used in the morphing wing (as stated in Chapter 2). From this
performance studies considerations about wing weight sizing and drag reduction are made.
7.1 Low speed and high speed airfoil de�nition
To choose the high speed airfoil and the low speed airfoil a simple study was performed in
Pro�li, Xfoil based, program. Information about Pro�li program can be found in [27]. For the
high speed airfoil it is advantageous to have an airfoil that produces the minimum Cd for low
CLs. For the high speed airfoils we have studied the symmetric Naca 0012, Naca 0015 Eppler
477 and the Eppler EA 6 009. The graphics of Clα, Cdα and ClCd for the respective airfoils are
presented in �gure 7.1.
The chosen airfoil was the NACA-0012 as this airfoil produces the less Cdα for low Clalpha.
For the low speed airfoil it is advantageous to use an airfoil with a high Cl maximum and the
lowest Cd possible. As it is expected, the wing will change in �ight from the high speed airfoil to
the low speed airfoil. Thus, we have also considered the di�erence between the geometric shape
of the two airfoils in terms of thickness and camber. Figure 7.2 presents the di�erences between
the selected low airfoils shapes, comparing with the high speed airfoil NACA-0012. In �gure 7.3
the graphics of ClαCdα and ClCdandClCdalpha
for the low speed studied airfoils are represented.
We have chosen Eppler 434 for the low speed airfoil because it has a high Cl maximum, the
maximum ClCd for the airfoils in study and it is the most similar to the high speed airfoil NACA
0012 in terms of airfoil thickness and camber.
57
Figure 7.1: Graphics Clα Cdα and ClCd for the high speed airfoils studied
58
Figure 7.2: Airfoil shape of the di�erent airfoils in study
59
Figure 7.3: Graphics Clα Cdα and ClCd for the low speed airfoils studied
60
7.2 Best wing con�guration de�nition
As stated in chapter 5 the study of the wing model with continuous independent airfoil
variation and span variation was replaced by discrete models that represent di�erent and repre-
sentative con�gurations of the morphing wing in study. So the discrete con�gurations studied
were the telescopic wing with the inner wing fully retracted (wing span 1m), with the inner
wing 0.25% extended (wing span 1.25m), the inner wing 0.5% extended (1.5m wing span) and
with the inner wing fully deployed, which correspond to an inner wing extension of 0.75% (wing
span 1.75m). For each one of these wing span con�gurations we have used the high speed air-
foil and the low speed airfoil chosen, the Naca 0012 and Eppler 434, and therefore eight wing
con�gurations were studied. These models were the ones exposed in chapter 5.
For the di�erent con�gurations exposed above an aerodynamic analysis was performed in
CFXr. This analysis had the objective to give the aerodynamic wing characteristics as well
as the optimal wing con�guration for each �ight condition. From the aerodynamic analysis the
graphics Cl(α), Cd(α) and CL(α)Cd(α) Cl(Cd) were obtained. For the wing con�guration with the
inner wing fully deployed (wing span 1.75m) these graphics are presented in �gure 7.4 and for
the other con�gurations the same graphics are presented in annex.
Figure 7.4: Cl(α), Cd(α), CL(α)Cd(α) and Cl(Cd) for di�erent wing con�gurations
In order to compare the di�erent wing con�gurations, the coe�cients Cl and Cd were obtained
dividing the Lift(L) and Drag(D) of all those con�gurations by the same area. We have chosen
61
the area of the completely deployed wing:
Cl =L
12ρ
2Scompletely deployed(7.1)
Cd =D
12ρ
2Scompletely deployed(7.2)
where Scompletely deployed is the area of the completely deployed wing con�guration (Scompletely deployed =
0.75× 0.2 + 1× 0.22)
As expected, the variation of the graphics Cl(α) and Cd(α) with velocity (Reynolds number)
is small. This variation increases for the graphics CL(α)Cd(α) and Cl(Cd) as there is contribution from
both Cl(α) and Cd(α). Nevertheless, the variation of Cl(Cd) with the velocity will be neglected
and we have used the curves Cl(Cd) for the velocity of 15 m/s. In annex it is presented the curve
Cl(Cd) for 30 m/s. The curve Cl(Cd) for 15 m/s is more conservative in terms of performance
improvement.
The �gure 7.5 represents the variation of Cd(Cl) for the di�erent wing con�gurations and
for the two airfoils.
Figure 7.5: Cd(Cl) for di�erent wing con�gurations
From the plots presented in �gure 7.5 we can determine the con�guration that for the same
lift produces the minimum drag, as represented in �gure 7.6.
Mixing the curves of the best wing con�guration for each Cd(Cl), for the two airfoils, we
obtain the optimum wing and airfoil con�guration, as a function of Cl.
As we can see from �gure 7.7, the Naca airfoil should be only used in the completely retracted
con�guration, up to a Cl of 0.284. Then the airfoil should morph to Eppler airfoil, before starting
to deploy the inner wing. The wing con�guration with half wing span of 1 m should be used from
a Cl of 0.284 to a Cl of 0.44. The wing con�guration with half wing span of 1.5 should be used
from a Cl of 0.44 to a Cl of 0.7 and then the fully deployed con�guration should be used. For the
62
Figure 7.6: Cd(Cl), optimal con�guration, for each airfoil
Figure 7.7: Cd(Cl) for wing optimal con�guration
di�erent intervals we can use the curves Cl(α) from the respective airfoil and wing con�guration
to obtain the respective range in angles of attack α. This is summarized in the table 7.1.
63
Cl Optimal Optimal wing angles of Cdairfoil con�guration (m) attack (deg.)
≤ 0.284 Naca 0012 1 0-5.92 0.011-0.0170.284-0.441 Epller 434 1.25 0.85-3.62 0.017-0.0240.441-0.702 Eppler 434 1.5 2.28-6.00 0.024-0.0360.702-1.23 Eppler 434 1.75 5.01-12 0.036-0.086
Table 7.1: Optimal wing con�guration
7.3 Performance improvement analysis
Having the optimum curve Cd(Cl) we can �nd the advantages of this morphing wing in terms
of drag reduction and/or increase in maximum take-o� weight.
To accomplish this study we will compare the optimal con�guration of the morphing wing
to the Antex-X2 basic wing, which is exposed in table 2.1. From this table the Antex-X2 has
a wing span of 2.4m and a fuselage span of 0.4m, so we have considered the Antex basic wing
having 1 m span each. We have then performed a CFD analysis in CFXr of the 1m span, 0.33
m chord and with the Wortmann FX 63 137 airfoil, Antex wing.
The graphics Cl(α), Cd(α), CL(α)Cd(α) and Cl(Cd) for the CFD Antex-X2 basic wing are pre-
sented in annex. In �gure 7.8 the curves Cd(Cl) are presented for telescopic wing with optimal
con�guration and for the Antex basic wing. These curves are extremely important to compare
these two wings in terms of performance.
Figure 7.8: Cd(Cl) for optimal wing con�guration and for Antex wing
Next it is described how we found the drag reduction and the increase in aircraft weight with
the telescopic wing comparing with the Antex basic wing.
For level, steady �ight, and with the simpli�cation that the vector trust is parallel to the
64
path direction, the equations of motion, are given by the equations:
LAntexaircraft = WAntexaircraft
TAntexaircraft = DAntexaircraft
(7.3)
where L is the aircraft lift, W is the aircraft weight, T is the aircraft trust and D is the
aircraft drag.
Two approaches can be made. We may increase the weight of the aircraft without performance
penalties or we can maintain the weight of the aircraft and �nd the bene�ts in drag reduction.
Considering that we want to increase the weight of the aircraft without performance penalty
for the new morphing wing we can compare it with the basic Antex wing and determine where:
Lmorphing wing = LbasicAntexwing + δL
Dmorphing wing = DbasicAntexwing
(7.4)
Assuming that the change in the wing doesn't alter the rest of the aircaft aerodynamic we
can say that:
LAntex aircraft with morphing wing = WAntexaircraft + δW
T=DAntex aircraft with morphing wing
(7.5)
The factor δW is the increase in weight that can be added to the Antex without trust penalty.
This factor is equal to the increase in lift generated by the new morphing wing when producing
the same drag as the Antex basic wing.
Thus, considering the Antex maximum take o� weight as 9.985 Kg (98.06N), we can �nd
the respective required Cl of the Antex basic wing and the respective Cd for a speci�c velocity.
Comparing the graphics ClCd of the morphing wing optimal con�guration and the ClCd of the
basic Antex-X2 wing we can �nd a point in the curve of morphing wing where the Cd is equal to
the de�ned for the Antex basic wing. For the given Cd we can �nd the respective Cl and then
calculate the two wings di�erence in Cl (�gure 7.9).
As we have the aircraft weight and the aircraft velocity we can �nd the increase in lift:
δL = δCl × 1/2ρU2Scompletely deployed (7.6)
Doing this procedure for a range of velocities we can �nd the possible increase in aircraft
weight function of aircraft speed without drag penalty. It is important to refer that we have
neglected the variation of Cl and Cd with the velocity (Reynolds number). As the curves of
ClCd were obtained for 15 m/s the error of these graphics will enlarge as the velocity diverges
from 15 m/s. In annex the same graphics for 30 m/s are presented. The �gure 7.10 represents
the increase weight that can be added to the aircraft maximum take o� weight without increase
in drag.
65
Figure 7.9: Increase in lift and decrease in drag of the morphing wing comparing with the antexwing
Figure 7.10: Possible increase in weight function of the velocity, accomplished by the morphingcomparing with the Antex wing
66
The stall speed for the Antex-X2 is 15.81 m/s, as indicated in the table 2.1. For this velocity
the new morphing wing produces more 15N of lift. This is a very important conclusion. In
the structural design of this morphing wing, it is expected a wing increase in weight due to the
morphing mecanisms and the actuations. As the aircraft weight is limited by the maximum take
o� weight, we will limit this wing increase in weight to 15N, so that the morphing wing will
always have a better performance than the Antex-X2 with the basic wing. Nevertheless we can
double the Antex-X2 wing weight as the 15N increase in Lift is equal to the Antex-X2 wing
weight as shown in table 2.1
If we want to maintain the weight of the aircraft and �nd the bene�ts in drag reduction the
approach is similar. If we compare the wings we can determine when
Lmorphing wing = LbasicAntexwing
Dmorphing wing = DbasicAntexwing + δD(7.7)
where δD is the drag reduction. Assuming that the change in wing doesn't alter the rest of
the aircaft aerodynamic we can say that the wing drag reduction is equal to the aircraft drag
reduction and we can �nd the new aircraft drag that is equal to the required aircraft Thrust:
LAntex aircraft with morphing wing = LAntex aircraft with basic = WAntexaircraft
T=DAntex aircraft with morphing wing − δD(7.8)
Once again for the Antex maximum take o� weight, 9.985 Kg (98.06N) and for a speci�c
velocity we can �nd the respective Cl. Comparing the graphics ClCd of the morphing wing
optimal con�guration and the ClCd of the basic Antex-X2 wing we can �nd a point in the
morphing wing curve where the Cl has the same value. In those points we can �nd the di�erence
in Cd for the same Cl of the two wings. Knowing the di�erence in Cd we can �nd the drag for
the chosen velocity.
δD = δCd× 1/2ρU2Scompletely deployed (7.9)
Doing this procedure for a range of velocities we can �nd the reduction in drag when using
the morphing wing instead of the base wing. The �gure 7.11 represents the decrease in drag
obtained by the use of the morphing wing for the Antex maximum take o� weight.
Considering the Antex-X2 cruise speed of 38.6N and for the maximum Takeo� Weight we can
reduce the drag by 5.5 N, which is an enormous reduction. This corresponds to a drag reduction
of 51.2%. At 15 m/s we can reduce the weight 33.74%.
.
67
Figure 7.11: Decrease in drag function of the velocity, accomplished by the morphing wingcomparing with the antex wing
68
Chapter 8
Structural analysis
This chapter reports the wing sizing study. Here the wing structure is de�ned. To accomplish
the wing sizing a series of coupled �uid/structural analyses were performed. First, a preliminary
wing properties selection was made focusing on the wing structural behavior. We studied the
wing structure resistance to the loads. Then, a study of the deformation caused by those aerody-
namic loads and a consequent penalty to the wing aerodynamic behavior was done. Therefore,
for the study of the wing structural behavior only a one way �uid structural analysis was per-
formed. The aerodynamic analysis was performed in CFXr. The resulting loads were applied
to the Ansysr structural �eld and no more stagger loops between the two �elds were performed.
In this analysis several structural parameters were changed: wing materials, wing shell thickness,
number of ribs and spar section, among others. From these analyses we have chosen a number
of wing properties that o�er the best performances in terms of resistance, weight, capability to
perform morphing without penalty, among others. With those chosen models we have run fully
coupled �uid/structural analyses with as many stagger loops as necessary, to achieve conver-
gence and analyzed how wing deformation altered the wing aerodynamic behavior. To all the
�uid/structural analyses presented we have chosen an approximately 3g condition obtained with
an 8 degrees angle of attack and the wing fully deployed. For this con�guration and no wing
deformation the aircraft is subjected to 290.6 N; each wing is subjected to 145.3N (the weight of
the Antex-X2 is 98.06N as exposed in chapter 2). Wing loading is 386.4N/m2 much higher than
the 123.86N/m2 of the original Antex-X2 Wing.
8.1 One way �uid/structural analysis
In this study the shell material, shell section thickness, the number of ribs to be used and
spar section are analyzed. For all the analyses presented below we have considered for the ribs
sections a standard design with a rectangular section of base 0.005 m and 0.01 m thickness as
presented in �gure 8.1. The ribs material chosen was composite rob with �bers in chord direction,
69
since ribs are mainly subjected to axial forces in this direction.
Figure 8.1: Ribs section used in the structural analysis
As it can be seen, in all ribs analyses presented below the ribs are extremely conservative
in their structural properties. We have decided to use conservative ribs because the actuators
to change the airfoil shape are intended to be placed on the ribs and consequently ribs have to
support the actuation forces that have not been accounted on these analyses. Nevertheless, the
weight of each rib is small compared to the total wing weight, as it is presented later.
For the spar we have used composite rob with �bers in span direction as the higher stresses
occur in this direction. We have used a great number of spar sections but the ones presented
below are mostly Z shape spars. Z shape spars are used due to the study performed by Jose
Vale [86] in which this shape was the one that allowed better actuation in changing wing span,
while minimizing asymmetries between the left and right wings, in terms of inertial moments.
The study that led to the use of Z shape sections is intended to be published soon.
The connections between the two wings used in the analyses presented in the next sections
were type I (as de�ned in chapter 5). Although we have analyzed type J connections which are
exposed in the DVD, we have concluded that type I connection has airfoil morphing actuation
advantages: actuators could be introduced on the ribs and actuation forces could be performed
on the ribs axis. This is not possible for type J connections without a more complex mechanism
that connects the ribs to the airfoil type J connections.
70
8.1.1 Shell de�nition
We started this structural analysis with the wing shell de�nition. In this shell de�nition
analysis a total of 8 ribs equal spaced throughout span direction with 0.25% span intervals, as
presented in �gure 8.2, was always used. The spar section used in this subsection is presented in
�gure 8.3.
Figure 8.2: Ribs con�guration the shell de�nition analysis (the picture also includes the spar)
To de�ne the shell we started to analyze an epoxy shell with 1mm thickness section. We
found that this section was not suitable for the wing as the airfoil deformation was unacceptable.
This behavior is independent of ribs and spar properties and is due to the fact that the airfoil
epoxy/propylene material deforms due to the direct aerodynamic load on the airfoil and can't
sustain its shape. Figure 8.4 is representative of the airfoil deformation. The wing mass is 1.514
Kg. We have abandoned the use of epoxy in favor of using composite.
As the use of epoxy was abandoned we have studied a 1 mm thickness Carbon/Epoxy Com-
posite Rod. The use of carbon �bers in the chord direction would render di�cult the airfoil
shape to change and so we have decided to use carbon �bers in the span direction. We have also
used composite rob in the connections between the inner wing and the outer wing. If we analyze
the results applying the structural failure criteria of maximum stress, strain and Tsai Will we
will see that failure occurs in the leading edge of the root wing, where the boundary conditions,
exposed in chapter 5, are applied to the model, as the failure criteria reach more than 4.
Excluding the root elements and analyzing only the shell, �gure 8.6, we can see that shell
failure does not occur. In fact, the security margin is extremely high as the maximum of all
failure criteria applied is below 0.4 and failure occurs when it is 1. The security factor is higher
71
Figure 8.3: Spar section used in the shell de�nition analysis
than 2.5.
From this �gure we can also see that, apart from the root elements where failure actually
occurs, the maximum values of the failure criteria occur near the connections between the ribs
and shell.
Studying the connections between the inner wing and the outer wing from picture 8.7 we
conclude that failure occurs in these connections as failure criteria reach 1.653.
In �gure 8.8 we can also conclude that wing deformation is not acceptable as the inner wing
collides with the outer wing, which can result in morphing mechanism and structural problems.
For this con�guration the shell mass is 1.466 Kg and the total wing mass is 1.974 Kg.
72
Figure 8.4: 1mm epoxy material in all wing shell; wing shell deformation
Figure 8.5: 1mm thickness composite rob shell model; wing shell failure study
73
Figure 8.6: 1mm thickness composite rob shell model; wing shell failure study without con-strained root elements
Figure 8.7: 1mm thickness composite rob shell model; wing conections failure study
74
Figure 8.8: Wing deformation
75
In brief, from this model analysis we have concluded that the maximum stresses in the shell
occur near the connections between the shell and the ribs and especially in the wing root. Stresses
in the connections between shell and ribs may increase as actuation systems to change airfoil
shape are intended to be installed on the ribs. To solve this problem and noticing that the
stress which produces failure occurs in the chord direction, perpendicular to the shell carbon
�ber direction, small carbon-epoxy composite sheet reinforcements, 5 cm width, were used near
the ribs, as shown in �gure 8.9
Figure 8.9: Spar, ribs and shell reinforcement in shell to ribs connection region representation
This allows an advantageous structural behavior with extremely small weight penalty. Be-
tween the reinforcements two options were considered: using only epoxy or using UD carbon-
epoxy composites with the �bers in the span orientation.
As the minimum construction thickness we can produce in our facilities for the composite
rob is 0.2 mm, a 0.2 mm unidirectional composite was studied.
As we can see in �gure 8.10 the maximum deformation is 0.0757 cm. In opposite to the 1mm
thickness composite shell, presented above, the inner wing doesn't touch the outer wing and the
displacement is acceptable, despite the higher maximum displacement. The higher maximum
displacement can be explained by the fact that we have reduced the second moment of area
around the X axis when we reduced the shell section from 1 mm to 0.2 mm in some areas. The
inner wing doesn't touch the outer wing because the reinforcements force the inner and the outer
wing to deform as likely as a rigid body.
In �gure 8.11 we can also see that this con�guration complies with all the failure criteria.
76
Figure 8.10: 0.2mm composite shell; wing deformation for reinforced
The maximum failure criteria value is reached on the reinforcements which are supporting
the majority of the loads the wing is subjected to. As 0.2 is the minimum composite rob
construction thickness no more one way coupled analyses respecting composite in the regions
between reinforcements were required. The analyzed shell has 0.335Kg and the complete wing
has 0.867Kg, which is signi�cantly less than the 1mm composite rob con�guration exposed before.
Another hypothesis to be considered is to use epoxy between the reinforcements (in addition
we used a 0.3mm thickness composite sheet in the connections between the two wings because
the deformation using epoxy and composite sheet reinforcements in the wings type J conections
revealed prohibitive deformation that led to termination errors in the bi-directional analysis
as presented in the DVD). As the epoxy density is 1000 kg/m3 and the composite is 1600
Kg/m3, an epoxy shell thickness of 0,3mm can be used without weight penalty compared with
the unidirectional composite. In �gures 8.12, 8.13 and 8.14 we can see the deformation and the
Von Misestension in the epoxy as well as the failure criteria in the composite. From these �gures
77
Figure 8.11: Reinforced 0.2mm composite shell; failure criteria
we can see that the wing deformation is higher than when using composite; however, the forces
on the reinforcements are smaller as the resin transfers less loads to those same reinforcements.
The higher Von Misestensions are only 14 MPa. So, we conclude that this con�guration also
complies with the maximum stresses allowed in both materials. It is important to refer that
for this con�guration there is a small airfoil shape change that has to be studied with a two-
way coupled �uid/structural method to analyze the e�ects on the airfoil shape change in the
aerodynamic wing behavior.
The mass of this con�guration is 0.838 Kg.
From this shell analysis we concluded that for all the analyses performed these two last shell
con�gurations, with composite shell reinforcements, are the ones that o�er the best relation
between structural resistance and weight. For convenience, in the next analyses we will refer
the model which uses composite rob between the reinforcements as �composite model� and the
model which uses resin between the reinforcements as �resin model�.
78
Figure 8.12: Resin reinforced shell model; wing deformation
Figure 8.13: Resin reinforced shell model; Von Misesstresses on the resin sections
Figure 8.14: Resin reinforced shell model; failure criteria in composite reinforcements
79
8.1.2 Spar study
For the �composite model� presented above we have performed a spar de�nition study. For
the spar de�nition study we changed the spar section used before and analyzed the changes
in structural behavior. We have studied from rectangular to Z shapes spar sections and some
analyses are presented in the DVD. Below, we present only the study of the spar used in the
shell de�nition study and a weaker spar with small inertial moments and less mass. The two
sections studied are exposed in �gure 8.15.
Figure 8.15: Spar sections studied
For convenience, we will refer to the spar presented on the left, which has a higher inertial
moment (both axis), as A Spar and B Spar to the spar presented on the right, which has a
smaller inertial moment. Using the B spar instead of the A spar we can reduce the mass by
0.121Kg as the A spar mass is 0.375Kg and the B spar mass 0.254 Kg. Using the B spar the
wing deformation is twice than using the A spar. It can be seen in �gure 8.16. If we look more
closely, in the connection between the inner wing and the outer wing we can see that when using
the B spar the inner wing touches the outer wing, as presented in �gure 8.17. This can lead to
actuation and structural problems and so the use of the B spar is not recommended.
As we can see in �gure 8.18 the two spars analyzed respect the maximum stress allowed on
the spars which is 1100 MPa (table 5.2). Failure in the shell is not reached as failure criteria
only reach 0.617 (�gure 8.19).
Reducing the wing weight by reducing the spar weight (and consequently reducing the inertial
moment) could be a possible solution as failure does not occur even when using such a thin spar,
as B spar, that allows a large deformation. An optimization algorithm could be used in the
future to reduce the spar weight and �nd the optimum spar, but as wing deformation increases,
actuation and vibration problems also tend to increase. We have decided not to change the
80
Figure 8.16: Wing deformation using the A spar(left) and the B spar (right)
Figure 8.17: Detail of wing deformation using the B spar (right)
conservative A spar because of those same problems.
81
Figure 8.18: Stresses in spar axial direction on the A spar(left) and B spar (right)
Figure 8.19: Failure criteria in the composite when using the A spar (left) and the B spar(right)
8.1.3 Ribs de�nition
In the rib de�nition study we have varied the number of ribs and studied the e�ect in the
wing structural behavior. For this analysis we have also used the �composite model� presented
in the shell de�nition study. We have analyzed two di�erent wing ribs con�gurations with 6
and 8 ribs (0.25% and 0.5% span intervals). It is important to refer that as the composite sheet
reinforcements are used around the ribs, if the number of ribs decreases the reinforcements also
decrease.
The two ribs con�gurations and the axial stresses subjected to aerodynamic loads are pre-
sented in �gure 8.20.
As it can be seen, the tensions are extremely smaller than the limit 1100 MPa. The maximum
82
Figure 8.20: Ribs tensions
spar tension is 140 MPa for the 6 ribs con�guration and 110 MPa for the 8 ribs con�guration.
Wing deformation and failure criteria for both con�gurations are presented in �gures 8.21
and 8.22.
Figure 8.21: Wing deformation using 6 ribs (left) con�guration and 8 ribs con�guration (right)
From these �gures we can see that the e�ect of the ribs on the deformation is large. For the
six ribs con�guration the maximum deformation is 0.813 mm, larger than the 0.757 mm of the
eight ribs con�guration but not as di�erent as when using the B spar exposed in the subsection
�Spar De�nition�. Nevertheless, the deformation is completely unacceptable as the inner wing
collides with the outer wing. This is a consequence of using fewer reinforcements which are wider
apart and so the composite material between the reinforcements is allowed to deform more. For
the same reason, in the six ribs con�guration failure occurs because the composite between the
83
Figure 8.22: Wing failure criteria using 6 ribs (left) con�guration and 8 ribs con�guration (right)
reinforcements is subjected to larger stress. From this analysis we concluded that decreasing the
number of ribs and reinforcements is not possible and so we didn't alter the number of ribs.
For the analyses preformed on the ribs de�nition and spar de�nition we have concluded that
there is no need to change the ribs and spar con�gurations and sections used in the shell de�nition
subsection for the �composite� model. We have assumed the same behavior for the �resin model�.
Many other one way structural/�uid analysis studies were performed. They are presented in the
DVD. We have only presented the ones that led us to the �nal wing de�nition.
8.2 Bidirectional coupled structural/�uid analysis
A bidirectional coupled structural/�uid analysis was performed on the �composite� model
and on the �resin model�. With a coupled structural �uid analysis we can not only study the
full structural behavior of the wing but also the aerodynamic behavior. As shown below, this
analysis revealed to be extremely important as aerodynamic loads cause wing deformations and
consequently those deformations alter the aerodynamic behavior. This interaction (stagger loops)
can originate a completely di�erent result.
First we present the analysis on the �resin model� which revealed to be unsuitable for the
morphing wing. The aerodynamic forces cause the wing shell to deform. This deformation
alters the wing shape which, at �rst, results in a lift increase. This lift increase causes wing
shell to support higher loads and consequently increase the wing shell deformation. Then this
increase deformation starts to have an opposite e�ect. The airfoil shape change starts to traduce
a continuously wing lift reduction. This behavior is presented in �gure 8.23. In this �gure the lift
and drag convergence are shown. Accumulated time steps correspond to the �uid solver iteration
84
to achieve �uid converged solution. The visible large lift and drag value changes, in step form,
occur when there is iteration between �uid and structural �eld. As expected, these changes are
smaller and smaller, as the solution is reaching convergence.
Figure 8.23: resin model; Lift and Drag convergence
The �gure 8.24 shows the di�erent deformations obtained with just a one way coupled analysis
( left), where the loads are transferred from the aerodynamic analysis but no more stagger loops
occur and a fully converged structural/�uid analysis (right) in which convergence occurs after
27 stagger loops (15 hours of CPU processing) (�uid/structural) iterations.
It is interesting to notice that besides the extremely high airfoil shape change in the results
with 27 stagger loops rather than with only one stagger loop, as it is evident in �gure 8.24, the
maximum deformation is smaller in the fully converged result. This occurs because the �nal lift
supported by the structure is smaller. From the �rst stagger loop to the �nal converged solution
the generated lift is reduced 27N, from 145N to 118N (18% decrease). The drag is increased from
7.45 N to 9N (17.2% increase)
The decrease in lift and increase in drag can be explained by the CFXr analysis. In �gure
8.25 we can see the wing surface 2D streamlines and in 8.26 it is shown the pressure on the
wing surface. As we can see, the deformation induces bumps that produce a constantly increase
and decrease in velocity and in pressure (in opposite ways, as expected). The pressure gradients
produce �ow separation and recirculation behind the bumps which traduce into aerodynamic
85
Figure 8.24: resin model deformation comparison with only one stagger loop (left) and with 27stagger loops necessary to achieve convergence (right)
losses well exposed on the 27N reduction in lift and the 1.25N increase in drag. The �gure 8.27
shows the wall shear and from this graphic we can see the areas where separations occur, as wall
shear reaches 0. The �gure 8.28 presents the pressure distribution in the symmetry plan, left
side, where the airfoil isn't deformed and no separation occurs and in a middle chord plan (XY
plan) where we can see the higher pressure in the concave bumps and the lower pressure in the
convex bumps.
86
Figure 8.25: resin model; 2D Streamlines
Figure 8.26: resin model; Wing surface pressure
87
Figure 8.27: resin model; wall shear
Figure 8.28: Pressure distribution in the symmetry plan and in a middle chord plan (XY plan)
88
The bidirectional analysis in the �composite model� is shown below. In �gure 8.29 it is given
the lift and drag evolution throughout the coupled �eld iterations.
Figure 8.29: Composite model; Lift and Drag convergence
From this �gure we see that the lift and drag are a little increased. To be exact, the lift
increases from 145.14N to 147.36N (which corresponds to an increase of 1.54%) and the drag
increases from 7.45N to 7.69N (which corresponds to an increase of 3.23 %). From the �gure 8.30
we can see the deformation when using a fully couple �eld analysis. The wing is barely altered
when compared to one way structural/�uid analysis. The increased deformation is due to the
increase in lift.
Analyzing the �gure 8.31 we can see that all the failure criteria studied are respected by this
wing con�guration and the values are almost unaltered when compared with the results obtained
in the one way �uid/structural analysis.
In �gures 8.33, 8.32, 8.33 and 8.34 are presented the 2D streamlines, the wing surface pressure
distribution, wall shear stresses and pressure distribution. The streamlines show that there aren't
any recirculation on the wing surface and the wall shear plot reveals that there is no separation.
From this analysis we have concluded that this model revealed to be the one that o�ers the
best relation between structural resistance, weight and performance and we will use it for our
morphing wing concept. The table 8.1 makes a summary of the wing model properties.
89
Figure 8.30: Composite model deformation without iterations (left) and when the result is fullyconverged(right)
Model Variables Value
Outer wing span 1 mInner wing span 1 m
Wing shell material carbon-epoxy composite robWing shell sections thickness 0.0002 mShell reinforcements materials carbon-epoxy composite sheetNumber of shell reinforcements 8 (around the ribs)
Shell reinforcements section thickness 0.001m (around the ribs)Number of spars 1Spar material carbon-epoxy composite rob
Number of ribs outer wing 3Number of ribs inner wing 5
Ribs material carbon-epoxy composite robWing connections type J
Wing connections materials carbon-epoxy composite rob
Table 8.1: Properties of the chosen model
90
Figure 8.31: Composite model failure criterion without iterations (left) and when the result isfully converged (right)
Figure 8.32: Composite model; 2D Streamlines
91
Figure 8.33: Composite model; wing surface pressure
Figure 8.34: Composite model; wall shear
92
Figure 8.35: Composite model; pressure distribution
Figure 8.36: Composite model; velocity plot
93
To conclude, we have analyzed the properties of the chosen model in two representative
con�gurations at approximately 1g condition. From this analysis we can verify if the wing
deformation alters the wing aerodynamic behavior and therefore determine if the performance
analyses presented in chapter 7 are changed by the loads the wing is supporting. We have studied
the model with an Eppler airfoil and with a wing span of 1.75m (fully deployed) subjected to a
15 m/s upstream velocity and an 8º angle of attack. The wing deformation and wing Lift and
Drag variation throughout the coupled �eld iterations are presented in �gure 8.37 and 8.38.
Figure 8.37: Composite model; Eppler airfoil; wingspan 1.75m; wing deformation
In �gure 8.37 we can see that the maximum deformation is only 2.49 cm. The aerodynamic
analysis shows that from the one-way coupled �eld analysis to the bidirectional coupled �eld one,
the lift increases from 47.17 N to 48.51 N which corresponds to an increase of 2.85%. The drag
is increased from 2.74 N to 2.80 N which corresponds to an increase of 2.52%.
94
Figure 8.38: Composite model; Eppler airfoil; wingspan 1.75m; wing Lift and Drag variationthroughout the coupled �eld iterations
We have also studied the model with the Naca airfoil, with a wing span of 1m (wing fully
retracted) subjected to an upstream of 30 m/s and 6º angle of attack. Wing Lift and Drag vari-
ation throughout the coupled �eld iterations is exposed in 8.40 . Wing deformation is presented
in �gure 8.39. For the Naca airfoil we have found a lift decrease from 58.74 N to 58.51 N (-0.43%)
and a drag increase from 36.1 to 36.2 (0.28%).
From this study we have concluded that the aerodynamic change due to the wing deformation
this model is subjected to is small and therefore the aerodynamic performance study presented
in chapter 7 is a good base to analyze the realistic wing aerodynamic behavior.
95
Figure 8.39: Composite model; Naca airfoil; wingspan 1m (approximately); wing deformation
Figure 8.40: Composite model; Naca airfoil; wingspan 1m (approximately); wing Lift and Dragvariation throughout the coupled �eld iterations;
96
Chapter 9
Conclusion
Morphing technology is a topic of recent research interest in aerospace engineering, a vast �eld
in the beginnings of its approach in which a lot of e�orts are being applied to get its promising
enormous bene�ts. To take part in this research was a unique opportunity which gave me an
enormous pride and I hope that this �little� contribution could present one more step forward
on this subject. This thesis was focused in a coupled �eld analysis to design a morphing wing.
The morphing design concept consists of a telescopic wing with airfoil shape variation, which
was studied using several discrete models that simulate, in a speci�ed con�guration, the wing in
study. The coupled �eld analysis uses a �nite element model developed in Ansys Multiphysics to
perform the structural analysis and a �nite volume model developed in Ansys-CFX to perform
the computational �uid dynamics (CFD). As far as I am concerned this was the �rst time
in Instituto Superior Técnico that coupled �uid/structural analysis was performed using this
software and so a great part of this thesis, chapters 3, 4 and 5, were focused on how the problem
was modeled, the di�culties we have encountered and the way they have been solved. It was a
tremendous challenge specially due to the considerable large displacements compared with the
small elements volumes in the high resolution mesh near the wing, overcome by using relaxation
factors of 0.6, using mesh displacement maximum residual of 10−4 and a mesh sti�ness given
by the expression 4.47. The knowledge and experience acquired in this study can be very useful
to other IST morphing projects. We can mention, as an example, the development of morphing
wing tips where simulating the �uid structural interface can be extremely helpful to have an idea
of their possibilities.
As stated in chapter 2, with this research we wanted to improve the morphing concepts
applied by Jose Vale in [86] overcoming the wing skin problems that emerged from his work.
This study led us to interesting results concerning this improvement. We managed to change the
wing skin from the �exible natural rubber to a composite skin and combined with the morphing
concepts (some already presented in his thesis which we have analyzed and improved, such as
airfoil shape change and wing span) we could manage to overcome the inability to provide smooth
97
airfoil shapes for the wing section, the major problem of his work. We also managed to always
have a drag reduction independently of aircraft velocity. In Jose Vale thesis he could manage to
reduce the drag between 5% and 39.6% at high speeds with a 6.9% drag penalty at low speed
�ight. In our thesis we always have a drag reduction for all velocities. As presented in chapter
7, we have found the best wing span and airfoil con�guration function of velocity. This was
achieved performing CFD analysis for all the wing con�gurations studied. Considering no �uid
structural interaction (no wing deformation), for 15 m/s we have a drag reduction of 33.74% and
for 30 m/s (approximately the cruising speed) we have a drag reduction of 50.87%. The drag
reduction increases as the velocity increases.
In chapter 8 we have designed our morphing wing based on the �uid structural analysis. In
this chapter, subjecting the morphing wing to an approximately 3g load condition, we have found
that epoxy shell was not suitable for our morphing wing and we have chosen a shell with 0.002
mm thickness of carbon-epoxy composite rob material reinforced with 1mm thickness composite
sheet material around the ribs. We have used 8 ribs of carbon-epoxy composite rob material and
a Z shape spar of the same material. The mass of the most suitable model studied was 1.734 Kg
considering both wings. So the mass penalty of our wing is 0.273Kg as the mass of the Antex-X2
wing is 1.5 Kg.
Analyzing two representative con�gurations subjected to approximately 1g load we have
found that changes in lift and drag produced by the wing deformation are less than 5% and
therefore the performance analysis exposed in chapter 7 isn't much altered by the wing deforma-
tion. To the development of this project further work in actuation systems must be done as the
wing span and chord variation mechanism required by this wing to perform morphing have not
been studied yet.
The morphing wing design with some simpli�cations, like not including airfoil shape variation,
is under construction by Jose Vale in Portuguese Airforce facilities. The construction is in a
preliminary phase and some model properties were changed, as Z shape spars were di�cult to
produce in composite material. Nevertheless, the �rst wind tunnel experiences have given a
positive feedback, and it is expected that the model design throughout this thesis resists the 3g
with good aerodynamic behavior as simulated in the CFD and �nite elements software. Knowing
that Jose Vale is continuing this project in his Ph.D. thesis I wish my contribution could be useful
to his work and I am looking forward to seeing the wing I have designed in one of his morphing
aircrafts being a success.
98
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105
107
Annex I
Performance graphics
Next it is presented the graphics of the aerodynamic parameters obtained in the CFX aerodynamic analyses that were not presented in chapter 7 .
108
109
110
111
112
113
114
Annex II
Performance data
Next it is presented the aerodynamic parameters analyzed in chapter 7 and calculated with the data obtained in the CFX aerodynamic analyses for all the models studied.
115
1 meter halfwingspan, Naca 0012 airfoil naca 0012 1m halfwingspan velocity 15 m/s
naca 0012 1m halfwingspan velocity 30 m/s
angle: 1 angle: 1
Type X Y Z Type X Y Z
Pressure Force 1.45E-01 2.47E+00 6.33E-02 Pressure Force 5.06E-01 9.99E+00 2.51E-01
Viscous Force 3.48E-01 3.11E-03 -5.70E-04 Viscous Force 1.22E+00 9.88E-03 -1.63E-03
Total Force 4.93E-01 2.48E+00 6.27E-02 Total Force 1.73E+00 1.00E+01 2.50E-01
Pressure Torque -1.13E+00 7.38E-02 1.35E-01 Pressure Torque -4.57E+00 2.59E-01 5.48E-01
Viscous Torque -1.51E-03 1.75E-01 5.59E-05 Viscous Torque -4.78E-03 6.15E-01 6.52E-05
Total Torque -1.13E+00 2.48E-01 1.35E-01 Total Torque -4.57E+00 8.74E-01 5.48E-01
inviscid L 2.46969E+00 D 1.88E-01 L/D 1.31E+01 inviscid L 9.97804E+00 D 6.81E-01 L/D 1.47E+01
total L 2.46673E+00 D 5.36E-01 L/D 4.60E+00 total L 9.96656E+00 D 1.91E+00 L/D 5.23E+00
inviscid Cl 5.00692E-02 Cd 3.81203E-03 inviscid Cl 5.05723E-02 Cd 3.45000E-03
total Cl 5.00090E-02 Cd 1.08607E-02 total Cl 5.05141E-02 Cd 9.65741E-03
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force 5.05E-02 4.91E+00 6.87E-02 Pressure Force 1.22E-01 1.99E+01 2.73E-01
Viscous Force 3.47E-01 6.28E-03 -4.48E-04 Viscous Force 1.22E+00 1.95E-02 -1.27E-03
Total Force 3.97E-01 4.92E+00 6.82E-02 Total Force 1.34E+00 1.99E+01 2.72E-01
Pressure Torque -2.25E+00 3.23E-02 2.67E-01 Pressure Torque -9.10E+00 9.06E-02 1.09E+00
Viscous Torque -3.05E-03 1.74E-01 1.04E-04 Viscous Torque -9.46E-03 6.12E-01 1.48E-04
Total Torque -2.25E+00 2.06E-01 2.67E-01 Total Torque -9.11E+00 7.03E-01 1.09E+00
inviscid L 4.90405E+00 D 2.22E-01 L/D 2.21E+01 inviscid L 1.98696E+01 D 8.16E-01 L/D 2.44E+01
total L 4.89825E+00 D 5.68E-01 L/D 8.62E+00 total L 1.98460E+01 D 2.04E+00 L/D 9.75E+00
inviscid Cl 9.94219E-02 Cd 4.49628E-03 inviscid Cl 1.00706E-01 Cd 4.13420E-03
total Cl 9.93043E-02 Cd 1.15241E-02 total Cl 1.00587E-01 Cd 1.03169E-02
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force -3.25E-01 9.75E+00 8.99E-02 Pressure Force -1.41E+00 3.95E+01 3.59E-01
Viscous Force 3.38E-01 1.27E-02 -2.06E-06 Viscous Force 1.20E+00 3.95E-02 4.12E-05
Total Force 1.36E-02 9.77E+00 8.99E-02 Total Force -2.04E-01 3.95E+01 3.59E-01
Pressure Torque -4.47E+00 -1.33E-01 5.28E-01 Pressure Torque -1.81E+01 -5.81E-01 2.15E+00
Viscous Torque -6.15E-03 1.71E-01 2.63E-04 Viscous Torque -1.91E-02 6.05E-01 4.29E-04
Total Torque -4.47E+00 3.78E-02 5.28E-01 Total Torque -1.81E+01 2.35E-02 2.15E+00
inviscid L 9.75250E+00 D 3.56E-01 L/D 2.74E+01 inviscid L 3.94999E+01 D 1.35E+00 L/D 2.92E+01
total L 9.74156E+00 D 6.95E-01 L/D 1.40E+01 total L 3.94559E+01 D 2.55E+00 L/D 1.54E+01
inviscid Cl 1.97717E-01 Cd 7.22339E-03 inviscid Cl 2.00199E-01 Cd 6.85630E-03
total Cl 1.97495E-01 Cd 1.40864E-02 total Cl 1.99977E-01 Cd 1.29468E-02
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force -9.39E-01 1.45E+01 1.26E-01 Pressure Force -3.87E+00 5.88E+01 -3.96E-01
Viscous Force 3.28E-01 1.95E-02 6.16E-04 Viscous Force 1.33E+00 4.66E-02 -4.46E-04
Total Force -6.11E-01 1.46E+01 1.26E-01 Total Force -2.55E+00 5.88E+01 -3.97E-01
Pressure Torque -6.67E+00 -4.04E-01 7.89E-01 Pressure Torque -2.70E+01 -
1.61E+00 3.38E+00
Viscous Torque -9.41E-03 1.67E-01 4.56E-04 Viscous Torque -2.12E-02 6.57E-01 -7.55E-03
Total Torque -6.68E+00 -2.38E-01 7.89E-01 Total Torque -2.70E+01 -9.52E-01 3.37E+00
inviscid L 1.45516E+01 D 5.85E-01 L/D 2.49E+01 inviscid L 5.88330E+01 D 2.29E+00 L/D 2.57E+01
total L 1.45372E+01 D 9.13E-01 L/D 1.59E+01 total L 5.87411E+01 D 3.61E+00 L/D 1.63E+01
inviscid Cl 2.95010E-01 Cd 1.18599E-02 inviscid Cl 2.98187E-01 Cd 1.16008E-02
total Cl 2.94718E-01 Cd 1.85188E-02 total Cl 2.97721E-01 Cd 1.83146E-02
116
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force -1.77E+00 1.92E+01 1.75E-01 Pressure Force -7.33E+00 7.77E+01 7.06E-01
Viscous Force 3.13E-01 2.67E-02 1.38E-03 Viscous Force 1.13E+00 8.28E-02 4.12E-03
Total Force -1.46E+00 1.92E+01 1.77E-01 Total Force -6.20E+00 7.78E+01 7.10E-01
Pressure Torque -8.84E+00 -7.74E-01 1.04E+00 Pressure Torque -3.58E+01 -
3.20E+00 4.26E+00
Viscous Torque -1.28E-02 1.60E-01 7.14E-04 Viscous Torque -3.95E-02 5.76E-01 1.42E-03
Total Torque -8.85E+00 -6.14E-01 1.05E+00 Total Torque -3.58E+01 -
2.63E+00 4.26E+00
inviscid L 1.92299E+01 D 9.14E-01 L/D 2.10E+01 inviscid L 7.79619E+01 D 3.56E+00 L/D 2.19E+01
total L 1.92131E+01 D 1.23E+00 L/D 1.57E+01 total L 7.78864E+01 D 4.69E+00 L/D 1.66E+01
inviscid Cl 3.89857E-01 Cd 1.85316E-02 inviscid Cl 3.95139E-01 Cd 1.80216E-02
total Cl 3.89517E-01 Cd 2.48856E-02 total Cl 3.94756E-01 Cd 2.37667E-02
angle: 10 angle: 10
Type X Y Z Type X Y Z
Pressure Force -2.77E+00 2.35E+01 2.38E-01 Pressure Force -1.15E+01 9.54E+01 9.58E-01
Viscous Force 2.92E-01 3.42E-02 2.24E-03 Viscous Force 1.08E+00 1.06E-01 6.79E-03
Total Force -2.47E+00 2.35E+01 2.40E-01 Total Force -1.04E+01 9.55E+01 9.65E-01
Pressure Torque -1.09E+01 -
1.22E+00 1.29E+00 Pressure Torque -4.42E+01 -
5.06E+00 5.24E+00
Viscous Torque -1.62E-02 1.52E-01 1.05E-03 Viscous Torque -5.01E-02 5.53E-01 2.29E-03
Total Torque -1.09E+01 -
1.07E+00 1.29E+00 Total Torque -4.42E+01 -
4.51E+00 5.25E+00
inviscid L 2.36235E+01 D 1.36E+00 L/D 1.74E+01 inviscid L 9.59699E+01 D 5.26E+00 L/D 1.82E+01
total L 2.36062E+01 D 1.65E+00 L/D 1.43E+01 total L 9.58875E+01 D 6.34E+00 L/D 1.51E+01
inviscid Cl 4.78930E-01 Cd 2.74801E-02 inviscid Cl 4.86410E-01 Cd 2.66797E-02
total Cl 4.78579E-01 Cd 3.34377E-02 total Cl 4.85992E-01 Cd 3.21437E-02
angle: 12 angle: 12
Type X Y Z Type X Y Z
Pressure Force -3.84E+00 2.75E+01 3.10E-01 Pressure Force -1.61E+01 1.11E+02 1.25E+00
Viscous Force 2.63E-01 4.20E-02 2.82E-03 Viscous Force 9.96E-01 1.30E-01 9.16E-03
Total Force -3.58E+00 2.75E+01 3.13E-01 Total Force -1.51E+01 1.12E+02 1.26E+00
Pressure Torque -1.28E+01 -
1.72E+00 1.54E+00 Pressure Torque -5.19E+01 -
7.15E+00 6.18E+00
Viscous Torque -1.97E-02 1.40E-01 1.56E-03 Viscous Torque -6.10E-02 5.20E-01 3.57E-03
Total Torque -1.28E+01 -
1.58E+00 1.54E+00 Total Torque -5.20E+01 -
6.63E+00 6.19E+00
inviscid L 2.76609E+01 D 1.95E+00 L/D 1.42E+01 inviscid L 1.12355E+02 D 7.46E+00 L/D 1.51E+01
total L 2.76473E+01 D 2.22E+00 L/D 1.25E+01 total L 1.12275E+02 D 8.46E+00 L/D 1.33E+01
inviscid Cl 5.60781E-01 Cd 3.95484E-02 inviscid Cl 5.69455E-01 Cd 3.77939E-02
total Cl 5.60505E-01 Cd 4.49408E-02 total Cl 5.69050E-01 Cd 4.28687E-02
117
1.25 meter halfwingspan, Naca 0012 airfoil naca 0012 velocity 15 m/s
naca 0012 velocity 30 m/s
angle: 1 angle: 1
Type X Y Z Type X Y Z
Pressure Force 1.72E-01 3.22E+00 6.30E-02 Pressure Force 5.94E-01 1.30E+01 2.50E-01
Viscous Force 4.28E-01 3.98E-03 -5.64E-04 Viscous Force 1.51E+00 1.26E-02 -1.62E-03
Total Force 5.99E-01 3.22E+00 6.24E-02 Total Force 2.10E+00 1.30E+01 2.49E-01
Pressure Torque -1.85E+00 1.09E-01 1.77E-01 Pressure Torque -7.45E+00 3.79E-01 7.20E-01
Viscous Torque -2.38E-03 2.65E-01 6.81E-05 Viscous Torque -7.50E-03 9.32E-01 7.67E-05
Total Torque -1.85E+00 3.74E-01 1.77E-01 Total Torque -7.46E+00 1.31E+00 7.20E-01
inviscid L 3.21412E+00 D 2.28E-01 L/D 1.41E+01 inviscid L 1.29816E+01 D 8.21E-01 L/D 1.58E+01
total L 3.21065E+00 D 6.56E-01 L/D 4.90E+00 total L 1.29683E+01 D 2.33E+00 L/D 5.57E+00
inviscid Cl 6.51612E-02 Cd 4.61543E-
03 inviscid Cl 6.57956E-02 Cd 4.16186E-
03
total Cl 6.50908E-02 Cd 1.32917E-
02 total Cl 6.57282E-02 Cd 1.18006E-
02
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force 4.45E-02 6.37E+00 6.82E-02 Pressure Force 7.64E-02 2.58E+01 2.72E-01
Viscous Force 4.26E-01 7.98E-03 -4.45E-04 Viscous Force 1.50E+00 2.48E-02 -1.26E-03
Total Force 4.70E-01 6.38E+00 6.78E-02 Total Force 1.58E+00 2.58E+01 2.70E-01
Pressure Torque -3.65E+00 3.84E-02 3.50E-01 Pressure Torque -1.48E+01 9.28E-02 1.43E+00
Viscous Torque -4.77E-03 2.64E-01 1.25E-04 Viscous Torque -1.48E-02 9.28E-01 1.70E-04
Total Torque -3.66E+00 3.02E-01 3.50E-01 Total Torque -1.48E+01 1.02E+00 1.43E+00
inviscid L 6.36826E+00 D 2.67E-01 L/D 2.39E+01 inviscid L 2.58016E+01 D 9.78E-01 L/D 2.64E+01
total L 6.36140E+00 D 6.93E-01 L/D 9.18E+00 total L 2.57742E+01 D 2.48E+00 L/D 1.04E+01
inviscid Cl 1.29107E-01 Cd 5.41146E-
03 inviscid Cl 1.30772E-01 Cd 4.95432E-
03
total Cl 1.28968E-01 Cd 1.40418E-
02 total Cl 1.30633E-01 Cd 1.25554E-
02
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force -4.59E-01 1.26E+01 8.97E-02 Pressure Force -1.97E+00 5.12E+01 3.59E-01
Viscous Force 4.15E-01 1.62E-02 -4.71E-05 Viscous Force 1.48E+00 5.05E-02 -1.22E-04
Total Force -4.32E-02 1.27E+01 8.96E-02 Total Force -4.97E-01 5.12E+01 3.59E-01
Pressure Torque -7.25E+00 -2.40E-01 6.91E-01 Pressure Torque -2.93E+01 -
1.04E+00 2.82E+00
Viscous Torque -9.66E-03 2.58E-01 3.21E-04 Viscous Torque -3.00E-02 9.15E-01 5.06E-04
Total Torque -7.26E+00 1.81E-02 6.91E-01 Total Torque -2.94E+01 -1.26E-01 2.82E+00
inviscid L 1.26382E+01 D 4.24E-01 L/D 2.98E+01 inviscid L 5.11970E+01 D 1.60E+00 L/D 3.20E+01
total L 1.26252E+01 D 8.40E-01 L/D 1.50E+01 total L 5.11439E+01 D 3.08E+00 L/D 1.66E+01
inviscid Cl 2.56220E-01 Cd 8.59533E-
03 inviscid Cl 2.59485E-01 Cd 8.11962E-
03
total Cl 2.55956E-01 Cd 1.70196E-
02 total Cl 2.59216E-01 Cd 1.56000E-
02
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force -1.28E+00 1.88E+01 1.26E-01 Pressure Force -5.33E+00 7.62E+01 5.06E-01
Viscous Force 4.02E-01 2.49E-02 4.96E-04 Viscous Force 1.44E+00 7.76E-02 1.44E-03
Total Force -8.79E-01 1.88E+01 1.26E-01 Total Force -3.89E+00 7.63E+01 5.08E-01
Pressure Torque -1.08E+01 -6.96E-01 1.03E+00 Pressure Torque -4.38E+01 -
2.90E+00 4.20E+00
Viscous Torque -1.48E-02 2.51E-01 5.63E-04 Viscous Torque -4.59E-02 8.96E-01 1.00E-03
Total Torque -1.08E+01 -4.45E-01 1.03E+00 Total Torque -4.38E+01 -
2.01E+00 4.20E+00
inviscid L 1.88349E+01 D 6.92E-01 L/D 2.72E+01 inviscid L 7.63421E+01 D 2.66E+00 L/D 2.87E+01
total L 1.88177E+01 D 1.09E+00 L/D 1.72E+01 total L 7.62692E+01 D 4.10E+00 L/D 1.86E+01
inviscid Cl 3.81848E-01 Cd 1.40225E-
02 inviscid Cl 3.86929E-01 Cd 1.34834E-
02
total Cl 3.81500E-01 Cd 2.21761E-
02 total Cl 3.86560E-01 Cd 2.07816E-
02
118
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force -2.39E+00 2.47E+01 1.76E-01 Pressure Force -9.89E+00 1.00E+02 7.09E-01
Viscous Force 3.82E-01 3.40E-02 1.15E-03 Viscous Force 1.39E+00 1.06E-01 3.35E-03
Total Force -2.01E+00 2.48E+01 1.77E-01 Total Force -8.50E+00 1.01E+02 7.13E-01
Pressure Torque -1.43E+01 -
1.31E+00 1.36E+00 Pressure Torque -5.78E+01 -
5.44E+00 5.54E+00
Viscous Torque -2.01E-02 2.40E-01 8.94E-04 Viscous Torque -6.22E-02 8.67E-01 1.79E-03
Total Torque -1.43E+01 -
1.07E+00 1.36E+00 Total Torque -5.79E+01 -
4.57E+00 5.54E+00
inviscid L 2.48404E+01 D 1.08E+00 L/D 2.30E+01 inviscid L 1.00789E+02 D 4.18E+00 L/D 2.41E+01
total L 2.48209E+01 D 1.46E+00 L/D 1.70E+01 total L 1.00705E+02 D 5.57E+00 L/D 1.81E+01
inviscid Cl 5.03600E-01 Cd 2.19060E-
02 inviscid Cl 5.10835E-01 Cd 2.11888E-
02
total Cl 5.03204E-01 Cd 2.96710E-
02 total Cl 5.10409E-01 Cd 2.82233E-
02
angle: 10 angle: 10
Type X Y Z Type X Y Z
Pressure Force -3.70E+00 3.03E+01 2.37E-01 Pressure Force -1.54E+01 1.23E+02 9.59E-01
Viscous Force 3.54E-01 4.37E-02 1.87E-03 Viscous Force 1.31E+00 1.36E-01 5.54E-03
Total Force -3.35E+00 3.03E+01 2.39E-01 Total Force -1.41E+01 1.23E+02 9.65E-01
Pressure Torque -1.75E+01 -
2.06E+00 1.67E+00 Pressure Torque -7.11E+01 -
8.52E+00 6.80E+00
Viscous Torque -2.57E-02 2.25E-01 1.37E-03 Viscous Torque -7.94E-02 8.25E-01 3.04E-03
Total Torque -1.75E+01 -
1.83E+00 1.68E+00 Total Torque -7.11E+01 -
7.70E+00 6.80E+00
inviscid L 3.04487E+01 D 1.61E+00 L/D 1.89E+01 inviscid L 1.23700E+02 D 6.22E+00 L/D 1.99E+01
total L 3.04306E+01 D 1.97E+00 L/D 1.55E+01 total L 1.23611E+02 D 7.53E+00 L/D 1.64E+01
inviscid Cl 6.17300E-01 Cd 3.26737E-
02 inviscid Cl 6.26955E-01 Cd 3.15084E-
02
total Cl 6.16933E-01 Cd 3.98924E-
02 total Cl 6.26504E-01 Cd 3.81503E-
02
angle: 12 angle: 12
Type X Y Z Type X Y Z
Pressure Force -5.09E+00 3.53E+01 3.08E-01 Pressure Force -2.13E+01 1.43E+02 1.24E+00
Viscous Force 3.13E-01 5.39E-02 2.30E-03 Viscous Force 1.20E+00 1.67E-01 7.45E-03
Total Force -4.78E+00 3.53E+01 3.10E-01 Total Force -2.01E+01 1.43E+02 1.25E+00
Pressure Torque -2.05E+01 -
2.86E+00 2.00E+00 Pressure Torque -8.32E+01 -
1.19E+01 8.01E+00
Viscous Torque -3.14E-02 2.04E-01 2.09E-03 Viscous Torque -9.71E-02 7.66E-01 4.89E-03
Total Torque -2.05E+01 -
2.66E+00 2.00E+00 Total Torque -8.33E+01 -
1.12E+01 8.02E+00
inviscid L 3.55548E+01 D 2.35E+00 L/D 1.51E+01 inviscid L 1.44459E+02 D 8.88E+00 L/D 1.63E+01
total L 3.55426E+01 D 2.67E+00 L/D 1.33E+01 total L 1.44377E+02 D 1.01E+01 L/D 1.43E+01
inviscid Cl 7.20818E-01 Cd 4.76987E-
02 inviscid Cl 7.32173E-01 Cd 4.50323E-
02
total Cl 7.20571E-01 Cd 5.41253E-
02 total Cl 7.31756E-01 Cd 5.11357E-
02
119
1.5 meter halfwingspan, Naca 0012 airfoil naca 0012 velocity 15 m/s
naca 0012 velocity 30 m/s
angle: 1 angle: 1
Type X Y Z Type X Y Z
Pressure Force 1.99E-01 3.95E+00 6.37E-02 Pressure Force 6.89E-01 1.60E+01 2.53E-01
Viscous Force 5.08E-01 4.89E-03 -5.62E-04 Viscous Force 1.79E+00 1.54E-02 -1.60E-03
Total Force 7.07E-01 3.95E+00 6.31E-02 Total Force 2.48E+00 1.60E+01 2.52E-01
Pressure Torque -2.72E+00 1.52E-01 2.19E-01 Pressure Torque -1.10E+01 5.30E-01 8.91E-01
Viscous Torque -3.52E-03 3.75E-01 8.03E-05 Viscous Torque -1.10E-02 1.32E+00 8.62E-05
Total Torque -2.72E+00 5.27E-01 2.19E-01 Total Torque -1.10E+01 1.85E+00 8.91E-01
inviscid L 3.94422E+00 D 2.68E-01 L/D 1.47E+01 inviscid L 1.59356E+01 D 9.67E-01 L/D 1.65E+01
total L 3.94026E+00 D 7.76E-01 L/D 5.08E+00 total L 1.59193E+01 D 2.76E+00 L/D 5.78E+00
inviscid Cl 7.99629E-02 Cd 5.43748E-
03 inviscid Cl 8.07671E-02 Cd 4.90116E-
03
total Cl 7.98826E-02 Cd 1.57299E-
02 total Cl 8.06848E-02 Cd 1.39712E-
02
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force 3.91E-02 7.84E+00 6.91E-02 Pressure Force 3.56E-02 3.18E+01 2.75E-01
Viscous Force 5.04E-01 9.73E-03 -4.40E-04 Viscous Force 1.78E+00 3.03E-02 -1.24E-03
Total Force 5.43E-01 7.85E+00 6.86E-02 Total Force 1.82E+00 3.18E+01 2.74E-01
Pressure Torque -5.40E+00 4.56E-02 4.35E-01 Pressure Torque -2.19E+01 9.52E-02 1.77E+00
Viscous Torque -7.00E-03 3.73E-01 1.51E-04 Viscous Torque -2.17E-02 1.31E+00 1.98E-04
Total Torque -5.40E+00 4.18E-01 4.35E-01 Total Torque -2.19E+01 1.41E+00 1.77E+00
inviscid L 7.83386E+00 D 3.13E-01 L/D 2.51E+01 inviscid L 3.17304E+01 D 1.14E+00 L/D 2.77E+01
total L 7.82595E+00 D 8.17E-01 L/D 9.58E+00 total L 3.16982E+01 D 2.92E+00 L/D 1.08E+01
inviscid Cl 1.58819E-01 Cd 6.33979E-
03 inviscid Cl 1.60821E-01 Cd 5.79678E-
03
total Cl 1.58659E-01 Cd 1.65650E-
02 total Cl 1.60658E-01 Cd 1.48221E-
02
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force -5.93E-01 1.55E+01 9.08E-02 Pressure Force -2.54E+00 6.30E+01 3.63E-01
Viscous Force 4.92E-01 1.98E-02 -1.35E-05 Viscous Force 1.75E+00 6.16E-02 4.55E-06
Total Force -1.01E-01 1.56E+01 9.08E-02 Total Force -7.91E-01 6.30E+01 3.63E-01
Pressure Torque -1.07E+01 -3.76E-01 8.58E-01 Pressure Torque -4.34E+01 -
1.62E+00 3.50E+00
Viscous Torque -1.42E-02 3.65E-01 3.87E-04 Viscous Torque -4.39E-02 1.29E+00 6.12E-04
Total Torque -1.07E+01 -1.15E-02 8.59E-01 Total Torque -4.34E+01 -3.28E-01 3.50E+00
inviscid L 1.55465E+01 D 4.92E-01 L/D 3.16E+01 inviscid L 6.29760E+01 D 1.86E+00 L/D 3.39E+01
total L 1.55322E+01 D 9.85E-01 L/D 1.58E+01 total L 6.29157E+01 D 3.61E+00 L/D 1.74E+01
inviscid Cl 3.15182E-01 Cd 9.98123E-
03 inviscid Cl 3.19185E-01 Cd 9.40486E-
03
total Cl 3.14890E-01 Cd 1.99608E-
02 total Cl 3.18879E-01 Cd 1.82778E-
02
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force -1.63E+00 2.31E+01 1.28E-01 Pressure Force -6.77E+00 9.37E+01 5.13E-01
Viscous Force 4.75E-01 3.04E-02 5.35E-04 Viscous Force 1.71E+00 9.46E-02 1.58E-03
Total Force -1.15E+00 2.31E+01 1.28E-01 Total Force -5.06E+00 9.38E+01 5.15E-01
Pressure Torque -1.60E+01 -
1.07E+00 1.28E+00 Pressure Torque -6.46E+01 -
4.44E+00 5.21E+00
Viscous Torque -2.17E-02 3.54E-01 6.83E-04 Viscous Torque -6.71E-02 1.26E+00 1.24E-03
Total Torque -1.60E+01 -7.13E-01 1.28E+00 Total Torque -6.47E+01 -
3.18E+00 5.21E+00
inviscid L 2.31574E+01 D 7.99E-01 L/D 2.90E+01 inviscid L 9.38721E+01 D 3.06E+00 L/D 3.06E+01
total L 2.31375E+01 D 1.27E+00 L/D 1.82E+01 total L 9.37883E+01 D 4.77E+00 L/D 1.97E+01
inviscid Cl 4.69479E-01 Cd 1.61960E-
02 inviscid Cl 4.75777E-01 Cd 1.55238E-
02
total Cl 4.69077E-01 Cd 2.58428E-
02 total Cl 4.75353E-01 Cd 2.41683E-
02
120
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force -3.01E+00 3.04E+01 1.79E-01 Pressure Force -1.25E+01 1.23E+02 7.20E-01
Viscous Force 4.50E-01 4.15E-02 1.21E-03 Viscous Force 1.64E+00 1.29E-01 3.55E-03
Total Force -2.56E+00 3.04E+01 1.80E-01 Total Force -1.08E+01 1.23E+02 7.23E-01
Pressure Torque -2.10E+01 -
2.00E+00 1.68E+00 Pressure Torque -8.53E+01 -
8.27E+00 6.86E+00
Viscous Torque -2.94E-02 3.38E-01 1.09E-03 Viscous Torque -9.10E-02 1.22E+00 2.23E-03
Total Torque -2.11E+01 -
1.66E+00 1.69E+00 Total Torque -8.53E+01 -
7.05E+00 6.86E+00
inviscid L 3.04952E+01 D 1.25E+00 L/D 2.44E+01 inviscid L 1.23775E+02 D 4.81E+00 L/D 2.57E+01
total L 3.04731E+01 D 1.70E+00 L/D 1.79E+01 total L 1.23676E+02 D 6.45E+00 L/D 1.92E+01
inviscid Cl 6.18242E-01 Cd 2.52920E-
02 inviscid Cl 6.27337E-01 Cd 2.43787E-
02
total Cl 6.17794E-01 Cd 3.44520E-
02 total Cl 6.26835E-01 Cd 3.26816E-
02
angle: 10 angle: 10
Type X Y Z Type X Y Z
Pressure Force -4.64E+00 3.71E+01 2.41E-01 Pressure Force -1.93E+01 1.51E+02 9.73E-01
Viscous Force 4.15E-01 5.32E-02 1.93E-03 Viscous Force 1.53E+00 1.65E-01 5.86E-03
Total Force -4.22E+00 3.71E+01 2.43E-01 Total Force -1.77E+01 1.51E+02 9.79E-01
Pressure Torque -2.58E+01 -
3.10E+00 2.07E+00 Pressure Torque -1.05E+02 -
1.29E+01 8.41E+00
Viscous Torque -3.76E-02 3.15E-01 1.70E-03 Viscous Torque -1.16E-01 1.15E+00 3.84E-03
Total Torque -2.58E+01 -
2.79E+00 2.07E+00 Total Torque -1.05E+02 -
1.17E+01 8.41E+00
inviscid L 3.73192E+01 D 1.87E+00 L/D 2.00E+01 inviscid L 1.51638E+02 D 7.18E+00 L/D 2.11E+01
total L 3.72994E+01 D 2.29E+00 L/D 1.63E+01 total L 1.51539E+02 D 8.71E+00 L/D 1.74E+01
inviscid Cl 7.56589E-01 Cd 3.79119E-
02 inviscid Cl 7.68554E-01 Cd 3.63686E-
02
total Cl 7.56188E-01 Cd 4.63761E-
02 total Cl 7.68054E-01 Cd 4.41650E-
02
angle: 12 angle: 12
Type X Y Z Type X Y Z
Pressure Force -6.33E+00 4.31E+01 3.12E-01 Pressure Force -2.66E+01 1.75E+02 1.26E+00
Viscous Force 3.62E-01 6.56E-02 2.27E-03 Viscous Force 1.39E+00 2.03E-01 7.34E-03
Total Force -5.96E+00 4.31E+01 3.14E-01 Total Force -2.52E+01 1.75E+02 1.27E+00
Pressure Torque -3.01E+01 -
4.28E+00 2.47E+00 Pressure Torque -1.22E+02 -
1.79E+01 9.90E+00
Viscous Torque -4.60E-02 2.80E-01 2.63E-03 Viscous Torque -1.42E-01 1.06E+00 6.29E-03
Total Torque -3.01E+01 -
4.00E+00 2.47E+00 Total Torque -1.22E+02 -
1.69E+01 9.91E+00
inviscid L 4.34275E+01 D 2.76E+00 L/D 1.57E+01 inviscid L 1.76696E+02 D 1.04E+01 L/D 1.70E+01
total L 4.34169E+01 D 3.13E+00 L/D 1.39E+01 total L 1.76603E+02 D 1.18E+01 L/D 1.50E+01
inviscid Cl 8.80425E-01 Cd 5.60189E-
02 inviscid Cl 8.95560E-01 Cd 5.25321E-
02
total Cl 8.80210E-01 Cd 6.34658E-
02 total Cl 8.95089E-01 Cd 5.96240E-
02
121
1.75 meter halfwingspan, Naca 0012 airfoil naca 0012 velocity 15 m/s
naca 0012 velocity 30 m/s
angle: 0 angle: 0
Type X Y Z Type X Y Z
Pressure Force 2.94E-01 8.27E-04 6.24E-02 Pressure Force 1.09E+00 -5.12E-02 2.48E-01
Viscous Force 5.76E-01 6.92E-05 -5.95E-04 Viscous Force 2.07E+00 1.82E-05 -1.70E-03
Total Force 8.70E-01 8.96E-04 6.18E-02 Total Force 3.16E+00 -5.11E-02 2.46E-01
Pressure Torque 3.47E-03 2.52E-01 -4.81E-04 Pressure Torque 4.53E-02 9.36E-01 -4.49E-03
Viscous Torque -6.92E-05 4.95E-01 2.27E-05 Viscous Torque -3.95E-05 1.78E+00 1.33E-05
Total Torque 3.40E-03 7.48E-01 -4.58E-04 Total Torque 4.52E-02 2.72E+00 -4.48E-03
inviscid L 8.26580E-04 D 2.94E-01 L/D 2.82E-03 inviscid L -5.11590E-02 D 1.09E+00 L/D -4.70E-
02
total L 8.95760E-04 D 8.70E-01 L/D 1.03E-03 total L -5.11410E-02 D 3.16E+00 L/D -1.62E-
02
inviscid Cl 1.67576E-05 Cd 5.95046E-
03 inviscid Cl -2.59292E-04 Cd 5.51133E-
03
total Cl 1.81601E-05 Cd 1.76314E-
02 total Cl -2.59201E-04 Cd 1.60079E-
02
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force 4.18E-02 9.25E+00 6.93E-02 Pressure Force 5.43E-02 3.74E+01 2.76E-01
Viscous Force 5.82E-01 1.08E-02 -3.55E-04 Viscous Force 2.06E+00 3.38E-02 -1.01E-03
Total Force 6.24E-01 9.26E+00 6.89E-02 Total Force 2.11E+00 3.74E+01 2.75E-01
Pressure Torque -7.45E+00 5.61E-02 5.15E-01 Pressure Torque -3.01E+01 1.30E-01 2.09E+00
Viscous Torque -9.05E-03 5.01E-01 1.19E-04 Viscous Torque -2.82E-02 1.77E+00 1.24E-04
Total Torque -7.46E+00 5.57E-01 5.15E-01 Total Torque -3.01E+01 1.90E+00 2.09E+00
inviscid L 9.24501E+00 D 3.65E-01 L/D 2.54E+01 inviscid L 3.73513E+01 D 1.36E+00 L/D 2.75E+01
total L 9.23548E+00 D 9.47E-01 L/D 9.75E+00 total L 3.73135E+01 D 3.42E+00 L/D 1.09E+01
inviscid Cl 1.87428E-01 Cd 7.39213E-
03 inviscid Cl 1.89310E-01 Cd 6.88618E-
03
total Cl 1.87235E-01 Cd 1.91994E-
02 total Cl 1.89118E-01 Cd 1.73156E-
02
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force -7.10E-01 1.83E+01 9.14E-02 Pressure Force -3.04E+00 7.43E+01 3.66E-01
Viscous Force 5.71E-01 2.19E-02 6.19E-05 Viscous Force 2.02E+00 6.82E-02 2.66E-04
Total Force -1.40E-01 1.84E+01 9.14E-02 Total Force -1.02E+00 7.43E+01 3.66E-01
Pressure Torque -1.48E+01 -5.32E-01 1.02E+00 Pressure Torque -5.98E+01 -
2.29E+00 4.14E+00
Viscous Torque -1.83E-02 4.92E-01 3.20E-04 Viscous Torque -5.68E-02 1.74E+00 5.43E-04
Total Torque -1.48E+01 -4.04E-02 1.02E+00 Total Torque -5.99E+01 -5.49E-01 4.14E+00
inviscid L 1.83439E+01 D 5.71E-01 L/D 3.21E+01 inviscid L 7.42888E+01 D 2.15E+00 L/D 3.45E+01
total L 1.83260E+01 D 1.14E+00 L/D 1.61E+01 total L 7.42159E+01 D 4.17E+00 L/D 1.78E+01
inviscid Cl 3.71893E-01 Cd 1.15703E-
02 inviscid Cl 3.76522E-01 Cd 1.09085E-
02
total Cl 3.71531E-01 Cd 2.31423E-
02 total Cl 3.76153E-01 Cd 2.11366E-
02
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force -1.94E+00 2.73E+01 1.29E-01 Pressure Force -7.97E+00 1.10E+02 5.15E-01
Viscous Force 5.50E-01 3.35E-02 6.02E-04 Viscous Force 1.95E+00 1.04E-01 1.92E-03
Total Force -1.39E+00 2.73E+01 1.29E-01 Total Force -6.02E+00 1.10E+02 5.17E-01
Pressure Torque -2.20E+01 -
1.50E+00 1.51E+00 Pressure Torque -8.88E+01 -
6.16E+00 6.14E+00
Viscous Torque -2.79E-02 4.76E-01 6.45E-04 Viscous Torque -8.62E-02 1.69E+00 1.34E-03
Total Torque -2.20E+01 -
1.02E+00 1.51E+00 Total Torque -8.89E+01 -
4.47E+00 6.14E+00
inviscid L 2.73198E+01 D 9.17E-01 L/D 2.98E+01 inviscid L 1.10310E+02 D 3.58E+00 L/D 3.08E+01
total L 2.72952E+01 D 1.47E+00 L/D 1.86E+01 total L 1.10205E+02 D 5.53E+00 L/D 1.99E+01
inviscid Cl 5.53867E-01 Cd 1.85891E-
02 inviscid Cl 5.59093E-01 Cd 1.81314E-
02
total Cl 5.53367E-01 Cd 2.97464E-
02 total Cl 5.58561E-01 Cd 2.80372E-
02
122
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force -3.60E+00 3.59E+01 1.80E-01 Pressure Force -1.47E+01 1.45E+02 7.20E-01
Viscous Force 5.10E-01 4.54E-02 1.27E-03 Viscous Force 1.86E+00 1.42E-01 3.92E-03
Total Force -3.09E+00 3.60E+01 1.81E-01 Total Force -1.28E+01 1.45E+02 7.24E-01
Pressure Torque -2.91E+01 -
2.80E+00 2.00E+00 Pressure Torque -1.17E+02 -
1.14E+01 8.07E+00
Viscous Torque -3.76E-02 4.44E-01 1.23E-03 Viscous Torque -1.17E-01 1.62E+00 2.56E-03
Total Torque -2.91E+01 -
2.36E+00 2.00E+00 Total Torque -1.17E+02 -
9.82E+00 8.07E+00
inviscid L 3.60593E+01 D 1.43E+00 L/D 2.51E+01 inviscid L 1.45256E+02 D 5.59E+00 L/D 2.60E+01
total L 3.60329E+01 D 1.95E+00 L/D 1.85E+01 total L 1.45135E+02 D 7.45E+00 L/D 1.95E+01
inviscid Cl 7.31046E-01 Cd 2.90791E-
02 inviscid Cl 7.36210E-01 Cd 2.83175E-
02
total Cl 7.30511E-01 Cd 3.94389E-
02 total Cl 7.35597E-01 Cd 3.77767E-
02
angle: 10 angle: 10
Type X Y Z Type X Y Z
Pressure Force -5.53E+00 4.37E+01 2.42E-01 Pressure Force -2.27E+01 1.76E+02 9.75E-01
Viscous Force 4.67E-01 5.83E-02 1.93E-03 Viscous Force 1.74E+00 1.81E-01 6.19E-03
Total Force -5.06E+00 4.38E+01 2.44E-01 Total Force -2.10E+01 1.77E+02 9.81E-01
Pressure Torque -3.55E+01 -
4.34E+00 2.46E+00 Pressure Torque -1.43E+02 -
1.78E+01 9.86E+00
Viscous Torque -4.80E-02 4.11E-01 1.94E-03 Viscous Torque -1.48E-01 1.52E+00 4.42E-03
Total Torque -3.56E+01 -
3.93E+00 2.46E+00 Total Torque -1.43E+02 -
1.63E+01 9.87E+00
inviscid L 4.40226E+01 D 2.15E+00 L/D 2.05E+01 inviscid L 1.77705E+02 D 8.26E+00 L/D 2.15E+01
total L 4.39986E+01 D 2.62E+00 L/D 1.68E+01 total L 1.77580E+02 D 1.00E+01 L/D 1.77E+01
inviscid Cl 8.92489E-01 Cd 4.35736E-
02 inviscid Cl 9.00672E-01 Cd 4.18832E-
02
total Cl 8.92004E-01 Cd 5.30976E-
02 total Cl 9.00038E-01 Cd 5.07315E-
02
angle: 12 angle: 12
Type X Y Z Type X Y Z
Pressure Force -7.54E+00 5.09E+01 3.16E-01 Pressure Force -3.16E+01 2.06E+02 1.28E+00
Viscous Force 4.09E-01 7.25E-02 1.92E-03 Viscous Force 1.56E+00 2.24E-01 6.98E-03
Total Force -7.14E+00 5.09E+01 3.17E-01 Total Force -3.00E+01 2.06E+02 1.28E+00
Pressure Torque -4.15E+01 -
5.98E+00 2.95E+00 Pressure Torque -1.68E+02 -
2.50E+01 1.17E+01
Viscous Torque -5.93E-02 3.68E-01 3.01E-03 Viscous Torque -1.82E-01 1.38E+00 7.49E-03
Total Torque -4.15E+01 -
5.61E+00 2.95E+00 Total Torque -1.68E+02 -
2.36E+01 1.17E+01
inviscid L 5.13181E+01 D 3.20E+00 L/D 1.61E+01 inviscid L 2.07909E+02 D 1.19E+01 L/D 1.75E+01
total L 5.13036E+01 D 3.61E+00 L/D 1.42E+01 total L 2.07800E+02 D 1.35E+01 L/D 1.54E+01
inviscid Cl 1.04040E+00 Cd 6.47748E-
02 inviscid Cl 1.05376E+00 Cd 6.03273E-
02
total Cl 1.04010E+00 Cd 7.31810E-
02 total Cl 1.05321E+00 Cd 6.82781E-
02
angle: 14 angle: 14
Type X Y Z Type X Y Z
Pressure Force -7.78E+00 5.30E+01 3.78E-01 Pressure Force -3.82E+01 2.29E+02 1.59E+00
Viscous Force 3.31E-01 7.96E-02 1.32E-03 Viscous Force 1.25E+00 2.70E-01 3.23E-03
Total Force -7.45E+00 5.30E+01 3.79E-01 Total Force -3.69E+01 2.30E+02 1.59E+00
Pressure Torque -4.39E+01 -
6.41E+00 3.29E+00 Pressure Torque -1.88E+02 -
3.07E+01 1.40E+01
Viscous Torque -6.60E-02 3.07E-01 4.47E-03 Viscous Torque -2.19E-01 1.15E+00 1.32E-02
Total Torque -4.40E+01 -
6.11E+00 3.29E+00 Total Torque -1.88E+02 -
2.96E+01 1.40E+01
inviscid L 5.32606E+01 D 5.26E+00 L/D 1.01E+01 inviscid L 2.31750E+02 D 1.85E+01 L/D 1.26E+01
total L 5.32582E+01 D 5.60E+00 L/D 9.50E+00 total L 2.31710E+02 D 1.97E+01 L/D 1.17E+01
inviscid Cl 1.07978E+00 Cd 1.06717E-
01 inviscid Cl 1.17459E+00 Cd 9.35239E-
02
total Cl 1.07973E+00 Cd 1.13618E-
01 total Cl 1.17439E+00 Cd 9.99874E-
02
1 meter halfwingspan, Eppler 434 airfoil
123
eppler434 velocity 15 m/s eppler434 velocity 30 m/s
angle: -4 angle: -4
Type X Y Z Type X Y Z
Pressure Force 4.77E-01 -1.70E-01 1.02E-01 Pressure Force 1.80E+00 -6.34E-02 4.07E-01
Viscous Force 2.80E-01 -3.09E-03 -7.06E-04 Viscous Force 9.78E-01 -1.35E-02 -2.04E-03
Total Force 7.57E-01 -1.73E-01 1.02E-01 Total Force 2.78E+00 -7.69E-02 4.05E-01
Pressure Torque 7.20E-02 2.34E-01 4.76E-01 Pressure Torque -1.11E-03 8.85E-01 1.97E+00
Viscous Torque 1.05E-03 1.42E-01 -4.63E-03 Viscous Torque 5.28E-03 4.95E-01 -1.69E-02
Total Torque 7.30E-02 3.76E-01 4.71E-01 Total Torque 4.17E-03 1.38E+00 1.95E+00
inviscid L -1.36089E-01 D 4.88E-01 L/D -2.79E-
01 inviscid L 6.24894E-02 D 1.80E+00 L/D 3.47E-02
total L -1.19648E-01 D 7.67E-01 L/D -1.56E-
01 total L 1.17241E-01 D 2.78E+00 L/D 4.22E-02
inviscid Cl -2.75898E-03 Cd 9.89104E-03 inviscid Cl 3.16719E-04 Cd 9.13842E-03
total Cl -2.42568E-03 Cd 1.55555E-02 total Cl 5.94221E-04 Cd 1.40859E-02
angle: -2 angle: -2
Type X Y Z Type X Y Z
Pressure Force 5.00E-01 4.15E+00 9.13E-02 Pressure Force 1.93E+00 1.76E+01 3.64E-01
Viscous Force 3.13E-01 2.40E-03 -5.03E-04 Viscous Force 1.10E+00 3.66E-03 -1.58E-03
Total Force 8.12E-01 4.15E+00 9.07E-02 Total Force 3.02E+00 1.76E+01 3.62E-01
Pressure Torque -1.92E+00 2.46E-01 6.98E-01 Pressure Torque -8.13E+00 9.47E-01 2.89E+00
Viscous Torque -1.64E-03 1.57E-01 -4.32E-03 Viscous Torque -3.10E-03 5.50E-01 -1.60E-02
Total Torque -1.92E+00 4.03E-01 6.94E-01 Total Torque -8.14E+00 1.50E+00 2.87E+00
inviscid L 4.16661E+00 D 3.55E-01 L/D 1.18E+01 inviscid L 1.76785E+01 D 1.31E+00 L/D 1.35E+01
total L 4.17992E+00 D 6.67E-01 L/D 6.27E+00 total L 1.77207E+01 D 2.40E+00 L/D 7.37E+00
inviscid Cl 8.44715E-02 Cd 7.18761E-03 inviscid Cl 8.96009E-02 Cd 6.63868E-03
total Cl 8.47414E-02 Cd 1.35222E-02 total Cl 8.98149E-02 Cd 1.21870E-02
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force 2.64E-02 1.35E+01 1.12E-01 Pressure Force -2.47E-02 5.59E+01 4.52E-01
Viscous Force 3.53E-01 1.39E-02 3.31E-04 Viscous Force 1.24E+00 3.99E-02 8.18E-04
Total Force 3.80E-01 1.35E+01 1.13E-01 Total Force 1.22E+00 5.59E+01 4.53E-01
Pressure Torque -6.23E+00 4.14E-02 1.20E+00 Pressure Torque -2.57E+01 1.08E-01 4.98E+00
Viscous Torque -7.09E-03 1.77E-01 -3.88E-03 Viscous Torque -2.03E-02 6.21E-01 -1.49E-02
Total Torque -6.23E+00 2.19E-01 1.20E+00 Total Torque -2.57E+01 7.29E-01 4.97E+00
inviscid L 1.35168E+01 D 4.98E-01 L/D 2.71E+01 inviscid L 5.58188E+01 D 1.92E+00 L/D 2.90E+01
total L 1.35185E+01 D 8.52E-01 L/D 1.59E+01 total L 5.58155E+01 D 3.17E+00 L/D 1.76E+01
inviscid Cl 2.74033E-01 Cd 1.01047E-02 inviscid Cl 2.82910E-01 Cd 9.75427E-03
total Cl 2.74066E-01 Cd 1.72731E-02 total Cl 2.82893E-01 Cd 1.60508E-02
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force -5.45E-01 1.83E+01 1.47E-01 Pressure Force -2.39E+00 7.53E+01 5.96E-01
Viscous Force 3.58E-01 2.00E-02 8.77E-04 Viscous Force 1.26E+00 5.96E-02 2.36E-03
Total Force -1.87E-01 1.83E+01 1.48E-01 Total Force -1.12E+00 7.53E+01 5.98E-01
Pressure Torque -8.43E+00 -2.08E-01 1.47E+00 Pressure Torque -3.47E+01 -9.19E-01 6.07E+00
Viscous Torque -9.91E-03 1.81E-01 -3.72E-03 Viscous Torque -2.94E-02 6.35E-01 -1.46E-02
Total Torque -8.44E+00 -2.74E-02 1.46E+00 Total Torque -3.47E+01 -2.84E-01 6.05E+00
inviscid L 1.82675E+01 D 7.31E-01 L/D 2.50E+01 inviscid L 7.52431E+01 D 2.87E+00 L/D 2.62E+01
total L 1.82625E+01 D 1.09E+00 L/D 1.68E+01 total L 7.52147E+01 D 4.14E+00 L/D 1.82E+01
inviscid Cl 3.70346E-01 Cd 1.48136E-02 inviscid Cl 3.81359E-01 Cd 1.45476E-02
total Cl 3.70243E-01 Cd 2.20901E-02 total Cl 3.81215E-01 Cd 2.09627E-02
124
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force -1.35E+00 2.29E+01 1.98E-01 Pressure Force -5.71E+00 9.42E+01 8.04E-01
Viscous Force 3.56E-01 2.63E-02 1.54E-03 Viscous Force 1.26E+00 8.02E-02 4.20E-03
Total Force -9.90E-01 2.29E+01 1.99E-01 Total Force -4.44E+00 9.43E+01 8.08E-01
Pressure Torque -1.06E+01 -5.59E-01 1.72E+00 Pressure Torque -4.35E+01 -
2.37E+00 7.13E+00
Viscous Torque -1.28E-02 1.80E-01 -3.51E-03 Viscous Torque -3.88E-02 6.37E-01 -1.42E-02
Total Torque -1.06E+01 -3.79E-01 1.72E+00 Total Torque -4.35E+01 -
1.73E+00 7.12E+00
inviscid L 2.28815E+01 D 1.05E+00 L/D 2.18E+01 inviscid L 9.42766E+01 D 4.17E+00 L/D 2.26E+01
total L 2.28701E+01 D 1.41E+00 L/D 1.62E+01 total L 9.42241E+01 D 5.43E+00 L/D 1.73E+01
inviscid Cl 4.63886E-01 Cd 2.13099E-02 inviscid Cl 4.77828E-01 Cd 2.11317E-02
total Cl 4.63655E-01 Cd 2.85490E-02 total Cl 4.77561E-01 Cd 2.75464E-02
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force -2.35E+00 2.72E+01 2.62E-01 Pressure Force -9.90E+00 1.12E+02 1.07E+00
Viscous Force 3.48E-01 3.24E-02 2.24E-03 Viscous Force 1.25E+00 1.00E-01 6.30E-03
Total Force -2.00E+00 2.72E+01 2.65E-01 Total Force -8.65E+00 1.12E+02 1.08E+00
Pressure Torque -1.26E+01 -
1.00E+00 1.96E+00 Pressure Torque -5.20E+01 -
4.22E+00 8.12E+00
Viscous Torque -1.56E-02 1.78E-01 -3.15E-03 Viscous Torque -4.81E-02 6.32E-01 -1.33E-02
Total Torque -1.26E+01 -8.26E-01 1.96E+00 Total Torque -5.20E+01 -
3.59E+00 8.11E+00
inviscid L 2.72623E+01 D 1.46E+00 L/D 1.87E+01 inviscid L 1.12426E+02 D 5.81E+00 L/D 1.94E+01
total L 2.72456E+01 D 1.81E+00 L/D 1.51E+01 total L 1.12352E+02 D 7.05E+00 L/D 1.59E+01
inviscid Cl 5.52702E-01 Cd 2.95663E-02 inviscid Cl 5.69816E-01 Cd 2.94248E-02
total Cl 5.52362E-01 Cd 3.66471E-02 total Cl 5.69440E-01 Cd 3.57461E-02
angle: 10 angle: 10
Type X Y Z Type X Y Z
Pressure Force -3.49E+00 3.11E+01 3.39E-01 Pressure Force -1.47E+01 1.28E+02 1.38E+00
Viscous Force 3.31E-01 3.82E-02 2.92E-03 Viscous Force 1.20E+00 1.19E-01 8.36E-03
Total Force -3.16E+00 3.11E+01 3.42E-01 Total Force -1.35E+01 1.28E+02 1.39E+00
Pressure Torque -1.45E+01 -
1.52E+00 2.17E+00 Pressure Torque -5.98E+01 -
6.37E+00 8.98E+00
Viscous Torque -1.82E-02 1.72E-01 -2.54E-03 Viscous Torque -5.65E-02 6.15E-01 -1.16E-02
Total Torque -1.45E+01 -
1.35E+00 2.17E+00 Total Torque -5.98E+01 -
5.76E+00 8.97E+00
inviscid L 3.12124E+01 D 1.96E+00 L/D 1.59E+01 inviscid L 1.28835E+02 D 7.79E+00 L/D 1.65E+01
total L 3.11923E+01 D 2.30E+00 L/D 1.36E+01 total L 1.28746E+02 D 8.98E+00 L/D 1.43E+01
inviscid Cl 6.32782E-01 Cd 3.97924E-02 inviscid Cl 6.52984E-01 Cd 3.94587E-02
total Cl 6.32376E-01 Cd 4.65288E-02 total Cl 6.52531E-01 Cd 4.55290E-02
angle: 12 angle: 12
Type X Y Z Type X Y Z
Pressure Force -4.55E+00 3.38E+01 4.21E-01 Pressure Force -1.95E+01 1.41E+02 1.73E+00
Viscous Force 3.03E-01 4.39E-02 3.45E-03 Viscous Force 1.10E+00 1.38E-01 1.02E-02
Total Force -4.25E+00 3.39E+01 4.25E-01 Total Force -1.84E+01 1.41E+02 1.74E+00
Pressure Torque -1.60E+01 -
2.03E+00 2.31E+00 Pressure Torque -6.62E+01 -
8.61E+00 9.60E+00
Viscous Torque -2.07E-02 1.61E-01 -1.58E-03 Viscous Torque -6.46E-02 5.80E-01 -8.44E-03
Total Torque -1.60E+01 -
1.87E+00 2.31E+00 Total Torque -6.63E+01 -
8.03E+00 9.59E+00
inviscid L 3.40214E+01 D 2.58E+00 L/D 1.32E+01 inviscid L 1.41571E+02 D 1.02E+01 L/D 1.39E+01
total L 3.40015E+01 D 2.88E+00 L/D 1.18E+01 total L 1.41479E+02 D 1.13E+01 L/D 1.26E+01
inviscid Cl 6.89731E-01 Cd 5.22668E-02 inviscid Cl 7.17535E-01 Cd 5.14913E-02
total Cl 6.89327E-01 Cd 5.84609E-02 total Cl 7.17067E-01 Cd 5.71071E-02
125
1.25 meter halfwingspan, Eppler 434 airfoil eppler 434 velocity 15 m/s eppler434 velocity 30 m/s
angle: -4 angle: -4
Type X Y Z Type X Y Z
Pressure Force 5.94E-01 -2.27E-01 1.03E-01 Pressure Force 2.24E+00 -9.93E-02 4.09E-01
Viscous Force 3.41E-01 -4.19E-03 -4.80E-04 Viscous Force 1.19E+00 -1.79E-02 -1.25E-03
Total Force 9.35E-01 -2.31E-01 1.02E-01 Total Force 3.43E+00 -1.17E-01 4.08E-01
Pressure Torque 1.18E-01 3.65E-01 5.79E-01 Pressure Torque 1.36E-04 1.38E+00 2.40E+00
Viscous Torque 2.14E-03 2.12E-01 -5.67E-03 Viscous Torque 9.82E-03 7.39E-01 -2.06E-02
Total Torque 1.20E-01 5.77E-01 5.74E-01 Total Torque 9.96E-03 2.12E+00 2.38E+00
inviscid L -1.85278E-01 D 6.09E-01 L/D -3.04E-
01 inviscid L 5.75093E-02 D 2.25E+00 L/D 2.56E-02
total L -1.65699E-01 D 9.49E-01 L/D -1.75E-
01 total L 1.22668E-01 D 3.43E+00 L/D 3.57E-02
inviscid Cl -3.75622E-03 Cd 1.23384E-02 inviscid Cl 2.91478E-04 Cd 1.13833E-02
total Cl -3.35929E-03 Cd 1.92324E-02 total Cl 6.21726E-04 Cd 1.74073E-02
angle: -2 angle: -2
Type X Y Z Type X Y Z
Pressure Force 6.15E-01 5.34E+00 9.57E-02 Pressure Force 2.37E+00 2.27E+01 3.82E-01
Viscous Force 3.85E-01 2.71E-03 -4.44E-04 Viscous Force 1.34E+00 3.61E-03 -1.39E-03
Total Force 9.99E-01 5.35E+00 9.53E-02 Total Force 3.71E+00 2.27E+01 3.81E-01
Pressure Torque -3.09E+00 3.77E-01 8.69E-01 Pressure Torque -1.31E+01 1.45E+00 3.60E+00
Viscous Torque -2.02E-03 2.38E-01 -5.27E-03 Viscous Torque -3.13E-03 8.26E-01 -1.96E-02
Total Torque -3.09E+00 6.14E-01 8.63E-01 Total Torque -1.31E+01 2.28E+00 3.58E+00
inviscid L 5.36190E+00 D 4.28E-01 L/D 1.25E+01 inviscid L 2.27548E+01 D 1.57E+00 L/D 1.45E+01
total L 5.37802E+00 D 8.12E-01 L/D 6.62E+00 total L 2.28055E+01 D 2.91E+00 L/D 7.83E+00
inviscid Cl 1.08704E-01 Cd 8.67223E-03 inviscid Cl 1.15330E-01 Cd 7.97977E-03
total Cl 1.09031E-01 Cd 1.64642E-02 total Cl 1.15587E-01 Cd 1.47599E-02
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force -3.09E-02 1.74E+01 1.22E-01 Pressure Force -3.02E-01 7.20E+01 4.93E-01
Viscous Force 4.34E-01 1.75E-02 1.51E-04 Viscous Force 1.53E+00 5.03E-02 3.00E-04
Total Force 4.04E-01 1.75E+01 1.22E-01 Total Force 1.23E+00 7.21E+01 4.93E-01
Pressure Torque -1.00E+01 2.43E-02 1.53E+00 Pressure Torque -4.14E+01 -1.85E-03 6.32E+00
Viscous Torque -1.08E-02 2.68E-01 -4.73E-03 Viscous Torque -3.07E-02 9.42E-01 -1.82E-02
Total Torque -1.01E+01 2.92E-01 1.52E+00 Total Torque -4.15E+01 9.40E-01 6.30E+00
inviscid L 1.74335E+01 D 5.78E-01 L/D 3.02E+01 inviscid L 7.20096E+01 D 2.21E+00 L/D 3.25E+01
total L 1.74353E+01 D 1.01E+00 L/D 1.72E+01 total L 7.20062E+01 D 3.74E+00 L/D 1.92E+01
inviscid Cl 3.53436E-01 Cd 1.17159E-02 inviscid Cl 3.64971E-01 Cd 1.12150E-02
total Cl 3.53473E-01 Cd 2.05301E-02 total Cl 3.64953E-01 Cd 1.89712E-02
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force -8.02E-01 2.35E+01 1.59E-01 Pressure Force -3.49E+00 9.70E+01 6.45E-01
Viscous Force 4.39E-01 2.53E-02 5.92E-04 Viscous Force 1.55E+00 7.57E-02 1.48E-03
Total Force -3.63E-01 2.36E+01 1.60E-01 Total Force -1.94E+00 9.71E+01 6.46E-01
Pressure Torque -1.36E+01 -3.99E-01 1.87E+00 Pressure Torque -5.59E+01 -
1.75E+00 7.73E+00
Viscous Torque -1.54E-02 2.72E-01 -4.50E-03 Viscous Torque -4.55E-02 9.60E-01 -1.77E-02
Total Torque -1.36E+01 -1.26E-01 1.86E+00 Total Torque -5.59E+01 -7.88E-01 7.71E+00
inviscid L 2.35426E+01 D 8.42E-01 L/D 2.79E+01 inviscid L 9.70223E+01 D 3.28E+00 L/D 2.95E+01
total L 2.35369E+01 D 1.28E+00 L/D 1.84E+01 total L 9.69899E+01 D 4.84E+00 L/D 2.01E+01
inviscid Cl 4.77289E-01 Cd 1.70771E-02 inviscid Cl 4.91744E-01 Cd 1.66411E-02
total Cl 4.77173E-01 Cd 2.59981E-02 total Cl 4.91579E-01 Cd 2.45154E-02
126
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force -1.88E+00 2.94E+01 2.10E-01 Pressure Force -7.96E+00 1.21E+02 8.56E-01
Viscous Force 4.35E-01 3.35E-02 1.14E-03 Viscous Force 1.55E+00 1.03E-01 2.97E-03
Total Force -1.44E+00 2.95E+01 2.11E-01 Total Force -6.41E+00 1.21E+02 8.59E-01
Pressure Torque -1.70E+01 -9.91E-01 2.19E+00 Pressure Torque -7.00E+01 -
4.21E+00 9.09E+00
Viscous Torque -2.01E-02 2.71E-01 -4.18E-03 Viscous Torque -6.11E-02 9.60E-01 -1.70E-02
Total Torque -1.70E+01 -7.20E-01 2.19E+00 Total Torque -7.00E+01 -
3.25E+00 9.07E+00
inviscid L 2.94560E+01 D 1.21E+00 L/D 2.44E+01 inviscid L 1.21428E+02 D 4.76E+00 L/D 2.55E+01
total L 2.94433E+01 D 1.64E+00 L/D 1.79E+01 total L 1.21366E+02 D 6.31E+00 L/D 1.92E+01
inviscid Cl 5.97175E-01 Cd 2.45048E-02 inviscid Cl 6.15441E-01 Cd 2.41084E-02
total Cl 5.96918E-01 Cd 3.33494E-02 total Cl 6.15125E-01 Cd 3.19601E-02
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force -3.21E+00 3.49E+01 2.75E-01 Pressure Force -1.35E+01 1.44E+02 1.12E+00
Viscous Force 4.22E-01 4.14E-02 1.72E-03 Viscous Force 1.52E+00 1.29E-01 4.70E-03
Total Force -2.79E+00 3.50E+01 2.76E-01 Total Force -1.20E+01 1.44E+02 1.13E+00
Pressure Torque -2.02E+01 -
1.73E+00 2.50E+00 Pressure Torque -8.34E+01 -
7.30E+00 1.03E+01
Viscous Torque -2.47E-02 2.65E-01 -3.65E-03 Viscous Torque -7.62E-02 9.46E-01 -1.57E-02
Total Torque -2.03E+01 -
1.47E+00 2.49E+00 Total Torque -8.34E+01 -
6.36E+00 1.03E+01
inviscid L 3.50280E+01 D 1.68E+00 L/D 2.08E+01 inviscid L 1.44463E+02 D 6.63E+00 L/D 2.18E+01
total L 3.50098E+01 D 2.10E+00 L/D 1.66E+01 total L 1.44381E+02 D 8.15E+00 L/D 1.77E+01
inviscid Cl 7.10138E-01 Cd 3.40677E-02 inviscid Cl 7.32193E-01 Cd 3.35927E-02
total Cl 7.09770E-01 Cd 4.26636E-02 total Cl 7.31777E-01 Cd 4.12883E-02
angle: 10 angle: 10
Type X Y Z Type X Y Z
Pressure Force -4.69E+00 3.97E+01 3.49E-01 Pressure Force -1.98E+01 1.64E+02 1.43E+00
Viscous Force 3.96E-01 4.91E-02 2.29E-03 Viscous Force 1.44E+00 1.53E-01 6.43E-03
Total Force -4.30E+00 3.98E+01 3.52E-01 Total Force -1.84E+01 1.64E+02 1.43E+00
Pressure Torque -2.32E+01 -
2.58E+00 2.76E+00 Pressure Torque -9.54E+01 -
1.09E+01 1.14E+01
Viscous Torque -2.91E-02 2.52E-01 -2.72E-03 Viscous Torque -9.02E-02 9.08E-01 -1.30E-02
Total Torque -2.32E+01 -
2.32E+00 2.76E+00 Total Torque -9.55E+01 -
9.95E+00 1.14E+01
inviscid L 3.99592E+01 D 2.28E+00 L/D 1.75E+01 inviscid L 1.64934E+02 D 8.93E+00 L/D 1.85E+01
total L 3.99387E+01 D 2.68E+00 L/D 1.49E+01 total L 1.64832E+02 D 1.04E+01 L/D 1.59E+01
inviscid Cl 8.10111E-01 Cd 4.62129E-02 inviscid Cl 8.35946E-01 Cd 4.52821E-02
total Cl 8.09695E-01 Cd 5.42917E-02 total Cl 8.35430E-01 Cd 5.25867E-02
angle: 12 angle: 12
Type X Y Z Type X Y Z
Pressure Force -6.00E+00 4.28E+01 4.29E-01 Pressure Force -2.58E+01 1.78E+02 1.76E+00
Viscous Force 3.56E-01 5.63E-02 2.61E-03 Viscous Force 1.30E+00 1.78E-01 7.84E-03
Total Force -5.64E+00 4.29E+01 4.31E-01 Total Force -2.45E+01 1.78E+02 1.77E+00
Pressure Torque -2.53E+01 -
3.36E+00 2.90E+00 Pressure Torque -1.05E+02 -
1.44E+01 1.21E+01
Viscous Torque -3.32E-02 2.31E-01 -1.33E-03 Viscous Torque -1.04E-01 8.38E-01 -8.22E-03
Total Torque -2.53E+01 -
3.13E+00 2.90E+00 Total Torque -1.05E+02 -
1.35E+01 1.21E+01
inviscid L 4.31072E+01 D 3.03E+00 L/D 1.42E+01 inviscid L 1.79763E+02 D 1.18E+01 L/D 1.52E+01
total L 4.30879E+01 D 3.39E+00 L/D 1.27E+01 total L 1.79669E+02 D 1.31E+01 L/D 1.37E+01
inviscid Cl 8.73931E-01 Cd 6.14120E-02 inviscid Cl 9.11102E-01 Cd 5.98616E-02
total Cl 8.73539E-01 Cd 6.87156E-02 total Cl 9.10625E-01 Cd 6.64912E-02
127
1.5 meter halfwingspan, Eppler 434 airfoil
eppler 1.5 m velocity 15 m/s naca eppler velocity 30 m/s
angle: -4 angle: -4
Type X Y Z Type X Y Z
Pressure Force 1.03E-01 7.27E-01 -1.62E-01 Pressure Force 4.10E-01 2.75E+00 4.08E-01
Viscous Force -4.25E-04 4.03E-01 -5.30E-03 Viscous Force -1.09E-03 1.41E+00 -2.21E-02
Total Force 1.02E-01 1.13E+00 -1.67E-01 Total Force 4.09E-01 4.16E+00 3.86E-01
Pressure Torque 6.92E-01 9.93E-02 5.32E-01 Pressure Torque 2.88E+00 -3.41E-01 2.02E+00
Viscous Torque -6.67E-03 3.27E-03 2.99E-01 Viscous Torque -2.43E-02 1.46E-02 1.04E+00
Total Torque 6.85E-01 1.03E-01 8.31E-01 Total Torque 2.85E+00 -3.26E-01 3.06E+00
inviscid L -1.10453E-01 D 7.37E-01 L/D -1.50E-
01 inviscid L 5.98668E-01 D 2.72E+00 L/D 2.20E-01
total L -8.76307E-02 D 1.14E+00 L/D -7.69E-
02 total L 6.74899E-01 D 4.12E+00 L/D 1.64E-01
inviscid Cl -2.23927E-03 Cd 1.49401E-02 inviscid Cl 3.03426E-03 Cd 1.37715E-02
total Cl -1.77658E-03 Cd 2.30973E-02 total Cl 3.42063E-03 Cd 2.09048E-02
angle: -2 angle: -2
Type X Y Z Type X Y Z
Pressure Force 9.67E-02 7.33E-01 6.35E+00 Pressure Force 3.87E-01 2.83E+00 2.73E+01
Viscous Force -4.06E-04 4.52E-01 3.04E-03 Viscous Force -1.28E-03 1.58E+00 3.99E-03
Total Force 9.63E-02 1.19E+00 6.36E+00 Total Force 3.85E-01 4.41E+00 2.73E+01
Pressure Torque 1.02E+00 -
4.41E+00 5.37E-01 Pressure Torque 4.25E+00 -
1.89E+01 2.07E+00
Viscous Torque -6.18E-03 -2.68E-03 3.33E-01 Viscous Torque -2.30E-02 -3.98E-03 1.16E+00
Total Torque 1.01E+00 -
4.41E+00 8.70E-01 Total Torque 4.23E+00 -
1.89E+01 3.24E+00
inviscid L 6.37592E+00 D 5.11E-01 inviscid L 2.73392E+01 D 1.88E+00 L/D 1.46E+01
total L 6.39470E+00 D 9.63E-01 L/D 6.64E+00 total L 2.73984E+01 D 3.46E+00 L/D 7.92E+00
inviscid Cl 1.29262E-01 Cd 1.03612E-02 inviscid Cl 1.38565E-01 Cd 9.51493E-03
total Cl 1.29643E-01 Cd 1.95216E-02 total Cl 1.38865E-01 Cd 1.75244E-02
angle: 1 angle: 1
Type X Y Z Type X Y Z
Pressure Force 1.11E-01 2.60E-01 1.72E+01 Pressure Force 4.49E-01 8.67E-01 7.16E+01
Viscous Force 1.63E-04 5.05E-01 1.66E-02 Viscous Force 4.19E-04 1.77E+00 4.63E-02
Total Force 1.11E-01 7.66E-01 1.72E+01 Total Force 4.49E-01 2.64E+00 7.17E+01
Pressure Torque 1.60E+00 -
1.19E+01 2.26E-01 Pressure Torque 6.66E+00 -
4.95E+01 7.85E-01
Viscous Torque -5.64E-03 -1.22E-02 3.72E-01 Viscous Torque -2.16E-02 -3.38E-02 1.30E+00
Total Torque 1.60E+00 -
1.19E+01 5.98E-01 Total Torque 6.64E+00 -
4.95E+01 2.09E+00
inviscid L 1.72208E+01 D 5.61E-01 L/D 3.07E+01 inviscid L 7.15790E+01 D 2.12E+00 L/D 3.38E+01
total L 1.72290E+01 D 1.07E+00 L/D 1.62E+01 total L 7.15940E+01 D 3.89E+00 L/D 1.84E+01
inviscid Cl 3.49125E-01 Cd 1.13718E-02 inviscid Cl 3.62788E-01 Cd 1.07265E-02
total Cl 3.49291E-01 Cd 2.16243E-02 total Cl 3.62864E-01 Cd 1.97249E-02
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force 1.24E-01 -7.42E-02 2.09E+01 Pressure Force 5.03E-01 -5.24E-01 8.68E+01
Viscous Force 3.48E-04 5.14E-01 2.11E-02 Viscous Force 9.73E-04 1.81E+00 6.08E-02
Total Force 1.25E-01 4.40E-01 2.10E+01 Total Force 5.04E-01 1.29E+00 8.69E+01
Pressure Torque 1.81E+00 -
1.45E+01 4.89E-03 Pressure Torque 7.52E+00 -
6.00E+01 -1.32E-01
Viscous Torque -5.51E-03 -1.54E-02 3.79E-01 Viscous Torque -2.13E-02 -4.39E-02 1.33E+00
Total Torque 1.80E+00 -
1.45E+01 3.84E-01 Total Torque 7.49E+00 -
6.00E+01 1.20E+00
inviscid L 2.09318E+01 D 6.57E-01 L/D 3.19E+01 inviscid L 8.67874E+01 D 2.51E+00 L/D 3.46E+01
total L 2.09349E+01 D 1.17E+00 L/D 1.79E+01 total L 8.67852E+01 D 4.32E+00 L/D 2.01E+01
inviscid Cl 4.24360E-01 Cd 1.33136E-02 inviscid Cl 4.39870E-01 Cd 1.27022E-02
total Cl 4.24422E-01 Cd 2.37455E-02 total Cl 4.39859E-01 Cd 2.18785E-02
128
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force 1.62E-01 -
1.03E+00 2.83E+01 Pressure Force 6.58E-01 -
4.48E+00 1.17E+02
Viscous Force 8.11E-04 5.19E-01 3.05E-02 Viscous Force 2.27E-03 1.84E+00 9.13E-02
Total Force 1.63E-01 -5.06E-01 2.83E+01 Total Force 6.60E-01 -
2.64E+00 1.17E+02
Pressure Torque 2.22E+00 -
1.96E+01 -6.25E-01 Pressure Torque 9.20E+00 -
8.09E+01 -2.74E+00
Viscous Torque -5.20E-03 -2.20E-02 3.84E-01 Viscous Torque -2.07E-02 -6.51E-02 1.35E+00
Total Torque 2.21E+00 -
1.96E+01 -2.41E-01 Total Torque 9.18E+00 -
8.09E+01 -1.39E+00
inviscid L 2.82787E+01 D 9.49E-01 L/D 2.98E+01 inviscid L 1.16967E+02 D 3.69E+00 L/D 3.17E+01
total L 2.82724E+01 D 1.47E+00 L/D 1.92E+01 total L 1.16929E+02 D 5.53E+00 L/D 2.12E+01
inviscid Cl 5.73306E-01 Cd 1.92442E-02 inviscid Cl 5.92833E-01 Cd 1.87085E-02
total Cl 5.73178E-01 Cd 2.97876E-02 total Cl 5.92639E-01 Cd 2.80196E-02
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force 2.13E-01 -
2.34E+00 3.53E+01 Pressure Force 8.72E-01 -
9.99E+00 1.46E+02
Viscous Force 1.37E-03 5.13E-01 4.03E-02 Viscous Force 3.83E-03 1.83E+00 1.23E-01
Total Force 2.15E-01 -
1.83E+00 3.53E+01 Total Force 8.76E-01 -
8.16E+00 1.46E+02
Pressure Torque 2.60E+00 -
2.45E+01 -1.50E+00 Pressure Torque 1.08E+01 -
1.01E+02 -6.40E+00
Viscous Torque -4.77E-03 -2.88E-02 3.81E-01 Viscous Torque -1.97E-02 -8.73E-02 1.35E+00
Total Torque 2.59E+00 -
2.45E+01 -1.12E+00 Total Torque 1.08E+01 -
1.01E+02 -5.05E+00
inviscid L 3.53397E+01 D 1.36E+00 L/D 2.60E+01 inviscid L 1.46254E+02 D 5.33E+00 L/D 2.74E+01
total L 3.53259E+01 D 1.87E+00 L/D 1.89E+01 total L 1.46183E+02 D 7.16E+00 L/D 2.04E+01
inviscid Cl 7.16458E-01 Cd 2.75160E-02 inviscid Cl 7.41268E-01 Cd 2.70159E-02
total Cl 7.16178E-01 Cd 3.79380E-02 total Cl 7.40906E-01 Cd 3.62811E-02
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force 2.78E-01 -
3.95E+00 4.17E+01 Pressure Force 1.14E+00 -
1.68E+01 1.73E+02
Viscous Force 1.96E-03 4.94E-01 4.96E-02 Viscous Force 5.71E-03 1.78E+00 1.53E-01
Total Force 2.80E-01 -
3.46E+00 4.18E+01 Total Force 1.14E+00 -
1.50E+01 1.73E+02
Pressure Torque 2.95E+00 -
2.90E+01 -2.59E+00 Pressure Torque 1.23E+01 -
1.20E+02 -1.10E+01
Viscous Torque -4.03E-03 -3.53E-02 3.71E-01 Viscous Torque -1.77E-02 -1.08E-01 1.33E+00
Total Torque 2.94E+00 -
2.91E+01 -2.22E+00 Total Torque 1.22E+01 -
1.20E+02 -9.63E+00
inviscid L 4.18634E+01 D 1.89E+00 L/D 2.21E+01 inviscid L 1.73393E+02 D 7.43E+00 L/D 2.33E+01
total L 4.18441E+01 D 2.39E+00 L/D 1.75E+01 total L 1.73294E+02 D 9.21E+00 L/D 1.88E+01
inviscid Cl 8.48714E-01 Cd 3.83198E-02 inviscid Cl 8.78819E-01 Cd 3.76680E-02
total Cl 8.48323E-01 Cd 4.83845E-02 total Cl 8.78317E-01 Cd 4.66976E-02
angle: 10 angle: 10
Type X Y Z Type X Y Z
Pressure Force 3.51E-01 -
5.67E+00 4.71E+01 Pressure Force 1.44E+00 -
2.42E+01 1.95E+02
Viscous Force 2.39E-03 4.57E-01 5.87E-02 Viscous Force 7.28E-03 1.67E+00 1.82E-01
Total Force 3.53E-01 -
5.21E+00 4.71E+01 Total Force 1.45E+00 -
2.25E+01 1.95E+02
Pressure Torque 3.23E+00 -
3.30E+01 -3.77E+00 Pressure Torque 1.34E+01 -
1.36E+02 -1.60E+01
Viscous Torque -2.73E-03 -4.15E-02 3.48E-01 Viscous Torque -1.39E-02 -1.28E-01 1.26E+00
Total Torque 3.23E+00 -
3.30E+01 -3.42E+00 Total Torque 1.34E+01 -
1.37E+02 -1.48E+01
inviscid L 4.73272E+01 D 2.59E+00 L/D 1.83E+01 inviscid L 1.96477E+02 D 1.01E+01 L/D 1.95E+01
total L 4.73059E+01 D 3.05E+00 L/D 1.55E+01 total L 1.96365E+02 D 1.17E+01 L/D 1.67E+01
inviscid Cl 9.59485E-01 Cd 5.25170E-02 inviscid Cl 9.95818E-01 Cd 5.10071E-02
total Cl 9.59052E-01 Cd 6.18569E-02 total Cl 9.95250E-01 Cd 5.94811E-02
129
1.75 meter halfwingspan, Eppler 434 airfoil naca eppler velocity 15 m/s
naca eppler velocity 30 m/s
angle: -4 angle: -4
Type X Y Z Type X Y Z
Pressure Force 8.11E-01 -7.47E-01 1.04E-01 Pressure Force 2.97E+00 -
1.53E+00 4.14E-01
Viscous Force 4.71E-01 -4.61E-03 -6.25E-04 Viscous Force 1.71E+00 -2.44E-02 -1.44E-03
Total Force 1.28E+00 -7.51E-01 1.04E-01 Total Force 4.68E+00 -
1.55E+00 4.12E-01
Pressure Torque 5.75E-01 6.99E-01 7.46E-01 Pressure Torque 1.10E+00 2.56E+00 3.12E+00
Viscous Torque 3.03E-03 4.06E-01 -7.12E-03 Viscous Torque 1.83E-02 1.47E+00 -2.69E-02
Total Torque 5.78E-01 1.10E+00 7.39E-01 Total Torque 1.12E+00 4.03E+00 3.10E+00
inviscid L -6.88399E-01 D 8.61E-01 L/D -8.00E-
01 inviscid L -1.31814E+00 D 3.07E+00 L/D -4.29E-
01
total L -6.60174E-01 D 1.33E+00 L/D -4.96E-
01 total L -1.22333E+00 D 4.78E+00 L/D -2.56E-
01
inviscid Cl -1.39562E-02 Cd 1.74548E-
02 inviscid Cl -6.68081E-03 Cd 1.55758E-
02
total Cl -1.33840E-02 Cd 2.69777E-
02 total Cl -6.20027E-03 Cd 2.42206E-
02
angle: -2 angle: -2
Type X Y Z Type X Y Z
Pressure Force 8.50E-01 6.81E+00 9.74E-02 Pressure Force 3.22E+00 2.99E+01 3.89E-01
Viscous Force 5.29E-01 9.83E-03 -6.50E-04 Viscous Force 1.92E+00 1.89E-02 -1.82E-03
Total Force 1.38E+00 6.82E+00 9.68E-02 Total Force 5.15E+00 2.99E+01 3.87E-01
Pressure Torque -5.55E+00 7.28E-01 1.12E+00 Pressure Torque -2.43E+01 2.76E+00 4.73E+00
Viscous Torque -9.07E-03 4.53E-01 -6.16E-03 Viscous Torque -1.79E-02 1.65E+00 -2.47E-02
Total Torque -5.56E+00 1.18E+00 1.11E+00 Total Torque -2.43E+01 4.40E+00 4.70E+00
inviscid L 6.83932E+00 D 6.12E-01 L/D 1.12E+01 inviscid L 3.00123E+01 D 2.18E+00 L/D 1.38E+01
total L 6.86757E+00 D 1.14E+00 L/D 6.02E+00 total L 3.00984E+01 D 4.10E+00 L/D 7.34E+00
inviscid Cl 1.38657E-01 Cd 1.24064E-
02 inviscid Cl 1.52113E-01 Cd 1.10334E-
02
total Cl 1.39229E-01 Cd 2.31141E-
02 total Cl 1.52549E-01 Cd 2.07716E-
02
angle: 0 angle: 0
Type X Y Z Type X Y Z
Pressure Force 6.03E-01 1.52E+01 1.02E-01 Pressure Force 2.17E+00 6.42E+01 4.22E-01
Viscous Force 5.76E-01 2.46E-02 -1.31E-04 Viscous Force 2.09E+00 6.42E-02 -1.82E-03
Total Force 1.18E+00 1.52E+01 1.02E-01 Total Force 4.26E+00 6.43E+01 4.20E-01
Pressure Torque -1.23E+01 5.36E-01 1.56E+00 Pressure Torque -5.20E+01 1.95E+00 6.57E+00
Viscous Torque -2.13E-02 4.93E-01 -5.54E-03 Viscous Torque -5.54E-02 1.78E+00 -2.27E-02
Total Torque -1.23E+01 1.03E+00 1.56E+00 Total Torque -5.20E+01 3.73E+00 6.55E+00
inviscid L 1.51520E+01 D 6.03E-01 L/D 2.51E+01 inviscid L 6.42080E+01 D 2.17E+00 L/D 2.95E+01
total L 1.51770E+01 D 1.18E+00 L/D 1.29E+01 total L 6.42720E+01 D 4.26E+00 L/D 1.51E+01
inviscid Cl 3.07183E-01 Cd 1.22318E-
02 inviscid Cl 3.25429E-01 Cd 1.10166E-
02
total Cl 3.07690E-01 Cd 2.39044E-
02 total Cl 3.25754E-01 Cd 2.15866E-
02
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force -5.35E-02 2.36E+01 1.23E-01 Pressure Force -6.21E-01 9.95E+01 5.18E-01
Viscous Force 6.01E-01 3.95E-02 3.56E-04 Viscous Force 2.18E+00 1.09E-01 -1.45E-03
Total Force 5.47E-01 2.36E+01 1.23E-01 Total Force 1.55E+00 9.96E+01 5.16E-01
Pressure Torque -1.91E+01 2.50E-02 2.02E+00 Pressure Torque -8.05E+01 -2.19E-01 8.53E+00
Viscous Torque -3.37E-02 5.16E-01 -5.02E-03 Viscous Torque -9.24E-02 1.86E+00 -2.15E-02
Total Torque -1.92E+01 5.41E-01 2.02E+00 Total Torque -8.06E+01 1.65E+00 8.51E+00
inviscid L 2.35925E+01 D 7.70E-01 L/D 3.06E+01 inviscid L 9.94901E+01 D 2.85E+00 L/D 3.49E+01
total L 2.36115E+01 D 1.37E+00 L/D 1.72E+01 total L 9.95231E+01 D 5.03E+00 L/D 1.98E+01
inviscid Cl 4.78301E-01 Cd 1.56174E-
02 inviscid Cl 5.04251E-01 Cd 1.44573E-
02
total Cl 4.78686E-01 Cd 2.78220E-
02 total Cl 5.04419E-01 Cd 2.54937E-
02
130
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force -1.14E+00 3.20E+01 1.59E-01 Pressure Force -5.28E+00 1.34E+02 6.73E-01
Viscous Force 6.07E-01 5.47E-02 1.01E-03 Viscous Force 2.20E+00 1.53E-01 -7.03E-04
Total Force -5.28E-01 3.20E+01 1.60E-01 Total Force -3.09E+00 1.35E+02 6.72E-01
Pressure Torque -2.59E+01 -8.19E-01 2.48E+00 Pressure Torque -1.09E+02 -
3.84E+00 1.05E+01
Viscous Torque -4.61E-02 5.24E-01 -4.52E-03 Viscous Torque -1.28E-01 1.89E+00 -2.05E-02
Total Torque -2.60E+01 -2.95E-01 2.47E+00 Total Torque -1.09E+02 -
1.95E+00 1.04E+01
inviscid L 3.19723E+01 D 1.10E+00 L/D 2.91E+01 inviscid L 1.34391E+02 D 4.10E+00 L/D 3.28E+01
total L 3.19848E+01 D 1.71E+00 L/D 1.87E+01 total L 1.34388E+02 D 6.30E+00 L/D 2.13E+01
inviscid Cl 6.48188E-01 Cd 2.22572E-
02 inviscid Cl 6.81143E-01 Cd 2.07861E-
02
total Cl 6.48442E-01 Cd 3.46102E-
02 total Cl 6.81125E-01 Cd 3.19422E-
02
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force -2.67E+00 4.02E+01 2.09E-01 Pressure Force -1.18E+01 1.68E+02 8.86E-01
Viscous Force 6.00E-01 6.90E-02 1.81E-03 Viscous Force 2.17E+00 1.98E-01 6.55E-04
Total Force -2.07E+00 4.03E+01 2.10E-01 Total Force -9.58E+00 1.68E+02 8.87E-01
Pressure Torque -3.27E+01 -
2.02E+00 2.94E+00 Pressure Torque -1.36E+02 -
8.89E+00 1.23E+01
Viscous Torque -5.78E-02 5.20E-01 -4.00E-03 Viscous Torque -1.65E-01 1.87E+00 -1.88E-02
Total Torque -3.27E+01 -
1.50E+00 2.93E+00 Total Torque -1.36E+02 -
7.02E+00 1.23E+01
inviscid L 4.02940E+01 D 1.55E+00 L/D 2.60E+01 inviscid L 1.68129E+02 D 5.86E+00 L/D 2.87E+01
total L 4.02998E+01 D 2.15E+00 L/D 1.87E+01 total L 1.68101E+02 D 8.03E+00 L/D 2.09E+01
inviscid Cl 8.16897E-01 Cd 3.13761E-
02 inviscid Cl 8.52138E-01 Cd 2.96820E-
02
total Cl 8.17016E-01 Cd 4.36298E-
02 total Cl 8.51997E-01 Cd 4.07145E-
02
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force -4.43E+00 4.70E+01 2.78E-01 Pressure Force -1.96E+01 1.99E+02 1.11E+00
Viscous Force 5.71E-01 8.31E-02 8.30E-04 Viscous Force 2.10E+00 2.40E-01 8.25E-03
Total Force -3.86E+00 4.71E+01 2.79E-01 Total Force -1.75E+01 1.99E+02 1.12E+00
Pressure Torque -3.83E+01 -
3.41E+00 3.30E+00 Pressure Torque -1.62E+02 -
1.50E+01 1.40E+01
Viscous Torque -6.93E-02 4.99E-01 -2.80E-03 Viscous Torque -1.99E-01 1.82E+00 -1.60E-02
Total Torque -3.84E+01 -
2.92E+00 3.29E+00 Total Torque -1.62E+02 -
1.32E+01 1.40E+01
inviscid L 4.71677E+01 D 2.16E+00 L/D 2.19E+01 inviscid L 1.99332E+02 D 8.25E+00 L/D 2.42E+01
total L 4.71704E+01 D 2.74E+00 L/D 1.72E+01 total L 1.99277E+02 D 1.04E+01 L/D 1.92E+01
inviscid Cl 9.56251E-01 Cd 4.37536E-
02 inviscid Cl 1.01028E+00 Cd 4.18185E-
02
total Cl 9.56306E-01 Cd 5.54493E-
02 total Cl 1.01001E+00 Cd 5.25277E-
02
angle: 10 angle: 10
Type X Y Z Type X Y Z
Pressure Force -6.51E+00 5.40E+01 3.53E-01 Pressure Force -2.82E+01 2.25E+02 1.41E+00
Viscous Force 5.26E-01 9.48E-02 1.23E-03 Viscous Force 1.95E+00 2.80E-01 1.04E-02
Total Force -5.99E+00 5.40E+01 3.54E-01 Total Force -2.63E+01 2.25E+02 1.42E+00
Pressure Torque -4.41E+01 -
5.08E+00 3.70E+00 Pressure Torque -1.84E+02 -
2.19E+01 1.54E+01
Viscous Torque -7.89E-02 4.66E-01 -1.44E-03 Viscous Torque -2.30E-01 1.71E+00 -1.12E-02
Total Torque -4.42E+01 -
4.61E+00 3.70E+00 Total Torque -1.84E+02 -
2.02E+01 1.54E+01
inviscid L 5.42634E+01 D 2.95E+00 L/D 1.84E+01 inviscid L 2.26608E+02 D 1.13E+01 L/D 2.00E+01
total L 5.42656E+01 D 3.49E+00 L/D 1.56E+01 total L 2.26544E+02 D 1.33E+01 L/D 1.70E+01
inviscid Cl 1.10010E+00 Cd 5.98962E-
02 inviscid Cl 1.14853E+00 Cd 5.73582E-
02
total Cl 1.10015E+00 Cd 7.07245E-
02 total Cl 1.14821E+00 Cd 6.73477E-
02
131
angle: 12 angle: 12
Type X Y Z Type X Y Z
Pressure Force -8.00E+00 5.77E+01 4.23E-01 Pressure Force -3.54E+01 2.40E+02 1.76E+00
Viscous Force 4.63E-01 1.03E-01 1.70E-03 Viscous Force 1.72E+00 3.19E-01 2.65E-03
Total Force -7.54E+00 5.78E+01 4.25E-01 Total Force -3.37E+01 2.40E+02 1.77E+00
Pressure Torque -4.76E+01 -
6.33E+00 3.92E+00 Pressure Torque -1.98E+02 -
2.80E+01 1.61E+01
Viscous Torque -8.60E-02 4.17E-01 4.35E-04 Viscous Torque -2.62E-01 1.54E+00 -3.39E-03
Total Torque -4.77E+01 -
5.92E+00 3.92E+00 Total Torque -1.98E+02 -
2.64E+01 1.61E+01
inviscid L 5.80894E+01 D 4.17E+00 L/D 1.39E+01 inviscid L 2.42268E+02 D 1.53E+01 L/D 1.59E+01
total L 5.80939E+01 D 4.64E+00 L/D 1.25E+01 total L 2.42224E+02 D 1.70E+01 L/D 1.42E+01
inviscid Cl 1.17767E+00 Cd 8.45381E-
02 inviscid Cl 1.22790E+00 Cd 7.74252E-
02
total Cl 1.17776E+00 Cd 9.41537E-
02 total Cl 1.22768E+00 Cd 8.62746E-
02
132
1 meter halfwingspan, Wortman Fx 65-137 airfoil
wortman fx 65-137 velocity 15 m/s wortman velocity 30 m/s
angle: -6 angle: -6
Type X Y Z Type X Y Z
Pressure Force 9.56E-01 2.15E+00 3.05E-01 Pressure Force 3.71E+00 1.10E+01 1.21E+00
Viscous Force 3.99E-01 -2.25E-02 1.96E-04 Viscous Force 1.48E+00 -6.94E-02 8.60E-04
Total Force 1.35E+00 2.13E+00 3.05E-01 Total Force 5.19E+00 1.10E+01 1.21E+00
Pressure Torque -1.06E+00 4.69E-01 2.42E+00 Pressure Torque -5.35E+00 1.82E+00 1.02E+01
Viscous Torque 9.29E-03 2.03E-01 -1.26E-02 Viscous Torque 2.86E-02 7.52E-01 -4.58E-02
Total Torque -1.05E+00 6.73E-01 2.40E+00 Total Torque -5.32E+00 2.57E+00 1.01E+01
inviscid L 2.23803E+00 D 7.26E-01 L/D 3.08E+00 inviscid L 1.13559E+01 D 2.53E+00 L/D 4.49E+00
total L 2.25732E+00 D 1.12E+00 L/D 2.01E+00 total L 1.14421E+01 D 4.01E+00 L/D 2.85E+00
inviscid Cl 4.53726E-02 Cd 1.47150E-02 inviscid Cl 5.75556E-02 Cd 1.28329E-02
total Cl 4.57636E-02 Cd 2.27996E-02 total Cl 5.79927E-02 Cd 2.03366E-02
angle: -4 angle: -4
Type X Y Z Type X Y Z
Pressure Force 1.32E+00 9.34E+00 2.56E-01 Pressure Force 5.22E+00 3.97E+01 1.02E+00
Viscous Force 4.32E-01 -8.85E-03 -1.03E-05 Viscous Force 1.58E+00 -2.88E-02 -3.01E-04
Total Force 1.75E+00 9.33E+00 2.56E-01 Total Force 6.80E+00 3.97E+01 1.02E+00
Pressure Torque -4.32E+00 6.40E-01 3.10E+00 Pressure Torque -1.83E+01 2.53E+00 1.28E+01
Viscous Torque 2.74E-03 2.18E-01 -1.24E-02 Viscous Torque 9.20E-03 7.96E-01 -4.45E-02
Total Torque -4.31E+00 8.58E-01 3.08E+00 Total Torque -1.83E+01 3.33E+00 1.28E+01
inviscid L 9.40625E+00 D 6.63E-01 L/D 1.42E+01 inviscid L 3.99571E+01 D 2.43E+00 L/D 1.64E+01
total L 9.42760E+00 D 1.09E+00 L/D 8.61E+00 total L 4.00385E+01 D 4.01E+00 L/D 9.98E+00
inviscid Cl 1.90697E-01 Cd 1.34366E-02 inviscid Cl 2.02517E-01 Cd 1.23357E-02
total Cl 1.91130E-01 Cd 2.21858E-02 total Cl 2.02929E-01 Cd 2.03390E-02
angle: -2 angle: -2
Type X Y Z Type X Y Z
Pressure Force 1.39E+00 1.65E+01 2.37E-01 Pressure Force 5.59E+00 6.92E+01 9.49E-01
Viscous Force 4.60E-01 4.47E-03 2.98E-04 Viscous Force 1.68E+00 1.18E-02 5.45E-04
Total Force 1.85E+00 1.65E+01 2.37E-01 Total Force 7.26E+00 6.92E+01 9.49E-01
Pressure Torque -7.59E+00 6.88E-01 3.76E+00 Pressure Torque -3.17E+01 2.76E+00 1.56E+01
Viscous Torque -3.57E-03 2.32E-01 -1.22E-02 Viscous Torque -9.97E-03 8.43E-01 -4.55E-02
Total Torque -7.59E+00 9.19E-01 3.75E+00 Total Torque -3.17E+01 3.60E+00 1.55E+01
inviscid L 1.65805E+01 D 8.14E-01 L/D 2.04E+01 inviscid L 6.93278E+01 D 3.17E+00 L/D 2.19E+01
total L 1.66015E+01 D 1.27E+00 L/D 1.30E+01 total L 6.93983E+01 D 4.84E+00 L/D 1.43E+01
inviscid Cl 3.36144E-01 Cd 1.64994E-02 inviscid Cl 3.51378E-01 Cd 1.60577E-02
total Cl 3.36570E-01 Cd 2.58119E-02 total Cl 3.51735E-01 Cd 2.45460E-02
angle: 0 angle: 0
Type X Y Z Type X Y Z
Pressure Force 1.14E+00 2.37E+01 2.46E-01 Pressure Force 4.55E+00 9.75E+01 9.87E-01
Viscous Force 4.72E-01 1.70E-02 1.30E-03 Viscous Force 1.72E+00 4.96E-02 3.53E-03
Total Force 1.62E+00 2.37E+01 2.47E-01 Total Force 6.28E+00 9.76E+01 9.91E-01
Pressure Torque -1.08E+01 5.96E-01 4.39E+00 Pressure Torque -4.47E+01 2.38E+00 1.81E+01
Viscous Torque -9.49E-03 2.38E-01 -1.18E-02 Viscous Torque -2.77E-02 8.65E-01 -4.46E-02
Total Torque -1.09E+01 8.34E-01 4.38E+00 Total Torque -4.47E+01 3.24E+00 1.80E+01
inviscid L 2.36510E+01 D 1.14E+00 L/D 2.07E+01 inviscid L 9.75390E+01 D 4.55E+00 L/D 2.14E+01
total L 2.36680E+01 D 1.62E+00 L/D 1.46E+01 total L 9.75890E+01 D 6.28E+00 L/D 1.56E+01
inviscid Cl 4.79487E-01 Cd 2.31888E-02 inviscid Cl 4.94363E-01 Cd 2.30818E-02
total Cl 4.79832E-01 Cd 3.27598E-02 total Cl 4.94616E-01 Cd 3.18080E-02
133
angle: 2 angle: 2
Type X Y Z Type X Y Z
Pressure Force 5.68E-01 3.06E+01 2.90E-01 Pressure Force 2.18E+00 1.25E+02 1.16E+00
Viscous Force 4.84E-01 2.95E-02 2.61E-03 Viscous Force 1.77E+00 8.59E-02 7.35E-03
Total Force 1.05E+00 3.07E+01 2.92E-01 Total Force 3.96E+00 1.25E+02 1.17E+00
Pressure Torque -1.41E+01 3.62E-01 5.01E+00 Pressure Torque -5.75E+01 1.41E+00 2.05E+01
Viscous Torque -1.52E-02 2.44E-01 -1.17E-02 Viscous Torque -4.45E-02 8.93E-01 -4.53E-02
Total Torque -1.41E+01 6.07E-01 4.99E+00 Total Torque -5.75E+01 2.31E+00 2.04E+01
inviscid L 3.06055E+01 D 1.64E+00 L/D 1.87E+01 inviscid L 1.25178E+02 D 6.55E+00 L/D 1.91E+01
total L 3.06186E+01 D 2.12E+00 L/D 1.44E+01 total L 1.25206E+02 D 8.33E+00 L/D 1.50E+01
inviscid Cl 6.20479E-01 Cd 3.31843E-02 inviscid Cl 6.34445E-01 Cd 3.32141E-02
total Cl 6.20744E-01 Cd 4.30135E-02 total Cl 6.34587E-01 Cd 4.22183E-02
angle: 4 angle: 4
Type X Y Z Type X Y Z
Pressure Force -3.39E-01 3.73E+01 3.63E-01 Pressure Force -1.52E+00 1.53E+02 1.47E+00
Viscous Force 4.95E-01 4.07E-02 4.12E-03 Viscous Force 1.79E+00 1.22E-01 1.20E-02
Total Force 1.56E-01 3.74E+01 3.67E-01 Total Force 2.72E-01 1.53E+02 1.48E+00
Pressure Torque -1.72E+01 -1.92E-02 5.57E+00 Pressure Torque -7.02E+01 -1.35E-01 2.28E+01
Viscous Torque -2.04E-02 2.51E-01 -1.18E-02 Viscous Torque -6.08E-02 9.04E-01 -4.51E-02
Total Torque -1.72E+01 2.31E-01 5.56E+00 Total Torque -7.03E+01 7.69E-01 2.28E+01
inviscid L 3.72777E+01 D 2.27E+00 L/D 1.64E+01 inviscid L 1.52604E+02 D 9.15E+00 L/D 1.67E+01
total L 3.72841E+01 D 2.76E+00 L/D 1.35E+01 total L 1.52598E+02 D 1.09E+01 L/D 1.39E+01
inviscid Cl 7.55747E-01 Cd 4.59540E-02 inviscid Cl 7.73450E-01 Cd 4.63662E-02
total Cl 7.55876E-01 Cd 5.60191E-02 total Cl 7.73423E-01 Cd 5.54665E-02
angle: 6 angle: 6
Type X Y Z Type X Y Z
Pressure Force -1.57E+00 4.40E+01 4.73E-01 Pressure Force -6.55E+00 1.80E+02 1.92E+00
Viscous Force 4.94E-01 5.23E-02 6.03E-03 Viscous Force 1.79E+00 1.58E-01 1.74E-02
Total Force -1.08E+00 4.41E+01 4.79E-01 Total Force -4.75E+00 1.80E+02 1.94E+00
Pressure Torque -2.03E+01 -5.43E-01 6.15E+00 Pressure Torque -8.29E+01 -
2.27E+00 2.51E+01
Viscous Torque -2.56E-02 2.52E-01 -1.15E-02 Viscous Torque -7.66E-02 9.09E-01 -4.47E-02
Total Torque -2.03E+01 -2.91E-01 6.13E+00 Total Torque -8.29E+01 -
1.36E+00 2.51E+01
inviscid L 4.39709E+01 D 3.04E+00 L/D 1.45E+01 inviscid L 1.79559E+02 D 1.23E+01 L/D 1.46E+01
total L 4.39709E+01 D 3.54E+00 L/D 1.24E+01 total L 1.79531E+02 D 1.41E+01 L/D 1.27E+01
inviscid Cl 8.91441E-01 Cd 6.16773E-02 inviscid Cl 9.10069E-01 Cd 6.22959E-02
total Cl 8.91441E-01 Cd 7.17558E-02 total Cl 9.09925E-01 Cd 7.14200E-02
angle: 8 angle: 8
Type X Y Z Type X Y Z
Pressure Force -3.11E+00 5.05E+01 6.14E-01 Pressure Force -1.29E+01 2.06E+02 2.50E+00
Viscous Force 4.90E-01 6.38E-02 8.09E-03 Viscous Force 1.79E+00 1.95E-01 2.36E-02
Total Force -2.62E+00 5.06E+01 6.22E-01 Total Force -1.11E+01 2.06E+02 2.52E+00
Pressure Torque -2.34E+01 -
1.21E+00 6.70E+00 Pressure Torque -9.53E+01 -
4.98E+00 2.74E+01
Viscous Torque -3.04E-02 2.52E-01 -1.12E-02 Viscous Torque -9.22E-02 9.11E-01 -4.40E-02
Total Torque -2.34E+01 -9.55E-01 6.69E+00 Total Torque -9.54E+01 -
4.07E+00 2.73E+01
inviscid L 5.04815E+01 D 3.95E+00 L/D 1.28E+01 inviscid L 2.05974E+02 D 1.60E+01 L/D 1.29E+01
total L 5.04768E+01 D 4.44E+00 L/D 1.14E+01 total L 2.05914E+02 D 1.77E+01 L/D 1.16E+01
inviscid Cl 1.02343E+00 Cd 8.00864E-02 inviscid Cl 1.04395E+00 Cd 8.08622E-02
total Cl 1.02334E+00 Cd 9.00942E-02 total Cl 1.04364E+00 Cd 8.99603E-02
134
velocity 15 m/s
naca 0012 halfwingspan 1m;15 m/s naca 0012 halfwingspan 1.25m;15 m/s naca 0012 halfwingspan 1.5m;15 m/s naca 0012 halfwingspan 1.75m;15 m/s
alfa Cl Cd Cl/Cd Cl Cd Cl/Cd Cl Cd Cl/Cd Cl Cd Cl/Cd
1 0,0500 0,0109 4,6046 0,0651 0,0133 4,8971 0,0799 0,0157 5,0784
2 0,0993 0,0115 8,6171 0,1290 0,0140 9,1845 0,1587 0,0166 9,5780 0,1872 0,0192 9,7521
4 0,1975 0,0141 14,0203 0,2560 0,0170 15,0389 0,3149 0,0200 15,7755 0,3715 0,0231 16,0542
6 0,2947 0,0185 15,9146 0,3815 0,0222 17,2032 0,4691 0,0258 18,1512 0,5534 0,0297 18,6028
8 0,3895 0,0249 15,6523 0,5032 0,0297 16,9595 0,6178 0,0345 17,9320 0,7305 0,0394 18,5226
10 0,4786 0,0334 14,3126 0,6169 0,0399 15,4649 0,7562 0,0464 16,3055 0,8920 0,0531 16,7993
12 0,5605 0,0449 12,4721 0,7206 0,0541 13,3130 0,8802 0,0635 13,8690 1,0401 0,0732 14,2127
velocity 30 m/s
alfa naca 0012 halfwingspan 1m;30 m/s naca 0012 halfwingspan 1.25m;30 m/s naca 0012 halfwingspan 1.5m;30 m/s naca 0012 halfwingspan 1.75m;30 m/s
Cl Cd Cl/Cd Cl Cd Cl/Cd Cl Cd Cl/Cd Cl Cd Cl/Cd
1 0,0505 0,0097 5,2306 0,0657 0,0118 5,5699 0,0807 0,0140 5,7751
2 0,1006 0,0103 9,7497 0,1306 0,0126 10,4045 0,1607 0,0148 10,8391 0,1891 0,0173 10,9219
4 0,2000 0,0129 15,4461 0,2592 0,0156 16,6164 0,3189 0,0183 17,4462 0,3762 0,0211 17,7963
6 0,2986 0,0174 17,1427 0,3866 0,0208 18,6010 0,4754 0,0242 19,6684 0,5586 0,0280 19,9221
8 0,3948 0,0238 16,6097 0,5104 0,0282 18,0847 0,6268 0,0327 19,1801 0,7356 0,0378 19,4722
10 0,4860 0,0321 15,1194 0,6265 0,0382 16,4220 0,7681 0,0442 17,3906 0,9000 0,0507 17,7412
12 0,5690 0,0429 13,2742 0,7318 0,0511 14,3101 0,8951 0,0596 15,0122 1,0532 0,0683 15,4252
velocity 15 m/s
eppler434 halfwingspan 1m;15 m/s eppler434 halfwingspan 1.25m;15 m/s eppler434 halfwingspan 1.5m;15 m/s eppler434 halfwingspan 1.75m;15 m/s
alfa Cl Cd Cl/Cd Cl Cd Cl/Cd Cl Cd Cl/Cd Cl Cd Cl/Cd
-4 -0,0024 0,0156 -0,1559 -0,0034 0,0192 -0,1747 -0,0018 0,0231 -0,0769 -0,0134 0,0270 -0,4961
-2 0,0847 0,0135 6,2669 0,1090 0,0165 6,6223 0,1296 0,0195 6,6410 0,1392 0,0231 6,0236
2 0,2741 0,0173 15,8666 0,3535 0,0205 17,2173 0,4244 0,0237 17,8738 0,4787 0,0278 17,2053
4 0,3702 0,0221 16,7606 0,4772 0,0260 18,3542 0,5732 0,0298 19,2422 0,6484 0,0346 18,7356
6 0,4637 0,0285 16,2407 0,5969 0,0333 17,8989 0,7162 0,0379 18,8776 0,8170 0,0436 18,7261
8 0,5524 0,0366 15,0724 0,7098 0,0427 16,6364 0,8483 0,0484 17,5330 0,9563 0,0554 17,2465
10 0,6324 0,0465 13,5911 0,8097 0,0543 14,9138 0,9591 0,0619 15,5044 1,1002 0,0707 15,5555
12 0,6893 0,0585 11,7913 0,8735 0,0687 12,7124 1,1778 0,0942 12,5089
velocity 30 m/s
eppler434 halfwingspan 1m;30 m/s eppler434 halfwingspan 1.25m;30m/s eppler434 halfwingspan 1.5m;30 m/s eppler434 halfwingspan 1.75m;30 m/s
alfa Cl Cd Cl/Cd Cl Cd Cl/Cd Cl Cd Cl/Cd Cl Cd Cl/Cd
-4 0,0006 0,0141 0,0422 0,0006 0,0174 0,0357 0,0034 0,0209 0,1636 -0,0062 0,0242 -0,2560
-2 0,0898 0,0122 7,3697 0,1156 0,0148 7,8311 0,1389 0,0175 7,9241 0,1525 0,0208 7,3441
2 0,2829 0,0161 17,6249 0,3650 0,0190 19,2373 0,4399 0,0219 20,1046 0,5044 0,0255 19,7860
4 0,3812 0,0210 18,1854 0,4916 0,0245 20,0519 0,5926 0,0280 21,1509 0,6811 0,0319 21,3237
6 0,4776 0,0275 17,3366 0,6151 0,0320 19,2466 0,7409 0,0363 20,4213 0,8520 0,0407 20,9262
8 0,5694 0,0357 15,9301 0,7318 0,0413 17,7236 0,8783 0,0467 18,8086 1,0100 0,0525 19,2281
10 0,6525 0,0455 14,3322 0,8354 0,0526 15,8867 0,9953 0,0595 16,7322 1,1482 0,0673 17,0489
12 0,7171 0,0571 12,5565 0,9106 0,0665 13,6954 1,2277 0,0863 14,2299
135
Annex III
Optimum wing configuration data
Next it is presented the best wing configuration data in each condition and the transition points from one configuration to another.
136
Optimum configuration naca hairfoil
Curva eppler 0012 final
Cl Cd Cl Cd
velocity 15 m/s
1 -0,0500 0,0109
velocity 15 m/s
1 -0,0024 0,0156
1 0,0500 0,0109 1 0,0847 0,0135
1 0,0993 0,0115 1 0,2741 0,0173
1 0,1975 0,0141 1--2 0,3400 0,0200
1--2 0,2900 0,0178 1--2 0,3535 0,0205
2--3 0,3800 0,0222 2--3 0,4400 0,0240
3 0,4691 0,0258 3 0,5732 0,0298
3--4 0,5534 0,0297 3--4 0,7000 0,0360
0,7305 0,0394 0,8170 0,0436
0,8920 0,0531 0,9563 0,0554
1,0401 0,0732 1,1002 0,0707
1,0797 0,1136 1,1778 0,0942
velocity 30 m/s
1 0,0505 0,0097
velocity 30 m/s
curva eppler 0012 final
1 0,1006 0,0103 1 0,0006 0,0141
1 0,2000 0,0129 1 0,0898 0,0122
1--2 0,2700 0,0159 1 0,2829 0,0161
2--3 0,3560 0,0198 1--2 0,3400 0,0184
3 0,4754 0,0242 2 0,3650 0,0190
3--4 0,5150 0,0262 2--3 0,4400 0,0220
0,5586 0,0280 2--3 0,5926 0,0280
0,7356 0,0378 3--4 0,6200 0,0295
0,9000 0,0507 4 0,8520 0,0407
1,0532 0,0683 4 1,0100 0,0525
1,1744 0,1000 4 1,1482 0,0673
4 1,2277 0,0863
137
Optimum Wing 15 m/s
Optimum Wing 30 m/s
naca 0012 -0,0500 0,0109 naca 0012; halfwingspan 1m 0,0505 0,0097
naca 0012; halfwingspan 1m
naca 0012 0,0500 0,0109 0,1006 0,0103
naca 0012 0,0993 0,0115 0,2000 0,0129
transição 0,2840 0,0174 eppler 434; halfwingspan 1.25 m 0,2600 0,0155
eppler 434; halfwingspan 1 m
eppler 434 0,3400 0,0200 0,2829 0,0161
eppler 434 0,3535 0,0205 0,3400 0,0184 eppler 434; halfwingspan 1.25 m
eppler 434 0,4600 0,0250 eppler 434; halfwingspan 1.5 m 0,3650 0,0190
eppler 434 0,5732 0,0298 0,4400 0,0220 eppler 434; halfwingspan 1.5 m
eppler 434 0,7000 0,0360 eppler 434; halfwingspan 1.75 m 0,5926 0,0280
eppler 434 0,8170 0,0436 0,6200 0,0295 eppler 434; halfwingspan 1.75 m
eppler 434 0,9563 0,0554 0,8520 0,0407
eppler 434 1,1002 0,0707 1,0100 0,0525
eppler 434 1,1778 0,0942 1,1482 0,0673 1,2277 0,0863
138
Annex IV
Chosen model data comparison between undeformed and deformed wing
The following data refer to the comparison between the undeformed wing and the deformed wing of the chosen mode.
139
angle: 8 Deformed wing Location Type X Y Z
asa Pressure Force -4,8092 48,3810 -0,3833 Viscous Force 0,8341 0,0505 0,0004 Total Force -3,9751 48,4310 -0,3829 Pressure Torque -39,2740 -3,6415 3,4709 Viscous Torque -0,0413 0,7143 -0,0177 Total Torque -39,3150 -2,9272 3,4532 inviscido L 48,5795 D 1,9709 L/D 2,46E+01
total L 48,5129 D 2,8039 L/D 1,73E+01
invisido Cl 0,9849 Cd 0,0400
total Cl 0,9835 Cd 0,0568
angle: 8 Undeformed
Type X Y Z
Pressure Force -4,4273 47,0090 0,2781
Viscous Force 0,5709 0,0831 0,0008
Total Force -3,8564 47,0920 0,2789
Pressure Torque -38,3420 -3,4147 3,2956
Viscous Torque -0,0693 0,4993 -0,0028
Total Torque -38,4110 -2,9154 3,2928
inviscid L 47,1677 D 2,1582 L/D 2,19E+01
total L 47,1704 D 2,7351 L/D 1,72E+01
inviscid Cl 0,9563 Cd 0,0438
total Cl 0,9563 Cd 0,0554
Eppler 434 halfwingspan 1.75m;15 m/s; aoa 8
Lift (N) Drag (N)
Undeformed wing 47,1704 2,7351
Deformed wing 48,5129 2,8039
Variation 2,85% 2,52%
140
angle: 6 Deformed wing Location Type X Y Z
asa Pressure Force -3,8282 58,5250 0,4603 Viscous Force 1,3160 0,0421 -0,0009 Total Force -2,5122 58,5670 0,4595 Pressure Torque -26,8690 -1,6451 3,1880 Viscous Torque -0,0200 0,6603 -0,0030 Total Torque -26,8890 -0,9848 3,1850 inviscido L 58,6045 D 2,3103 L/D 2,54E+01
total L 58,5088 D 3,6235 L/D 1,61E+01
invisido Cl 0,2970 Cd 0,0117
total Cl 0,2965 Cd 0,0184
angle: 6 Undeformed
Type X Y Z
Pressure Force -3,8734 58,7500 -0,3965
Viscous Force 1,3270 0,0466 -0,0004
Total Force -2,5464 58,7970 -0,3969
Pressure Torque -26,9600 -1,6093 3,3780
Viscous Torque -0,0212 0,6570 -0,0076
Total Torque -26,9810 -0,9523 3,3704
inviscid L 58,8330 D 2,2889 L/D 2,57E+01
total L 58,7411 D 3,6135 L/D 1,63E+01
inviscid Cl 0,2982 Cd 0,0116
total Cl 0,2977 Cd 0,0183
Naca 0012 halfwingspan 1m;30 m/s; aoa 6
Lift (N) Drag (N)
Undeformed wing 58,7411 3,6135
Deformed wing 58,5088 3,6235
Variation -0,40% 0,28%
141
angle: 8 Deformed wing Location Type X Y Z
Interface Pressure Force -14,8230 146,8000 -6,2473 Viscous Force 1,9334 0,1973 0,0091 Total Force -12,8900 147,0000 -6,2382 Pressure Torque -119,1100 -11,0350 8,9198 Viscous Torque -0,1575 1,6510 -0,0589 Total Torque -119,2700 -9,3840 8,8609 inviscid L 147,4343 D 5,7519 L/D 25,6324
total L 147,3633 D 7,6939 L/D 19,1533
invisid Cl 0,7473 Cd 0,0292
total Cl 0,7469 Cd 0,0390
angle: 8 Undeformed wing Type X Y Z
Pressure Force -14,6830 144,6200 0,7201
Viscous Force 1,8649 0,1416 0,0039
Total Force -12,8180 144,7600 0,7240
Pressure Torque -117,0000 -11,4440 8,0708
Viscous Torque -0,1168 1,6197 0,0026
Total Torque -117,1200 -9,8243 8,0734
inviscid L 145,2560 D 5,5871 L/D 25,9984
total L 145,1351 D 7,4534 L/D 19,4722
inviscid Cl 0,7362 Cd 0,0283
total Cl 0,7356 Cd 0,0378
Naca 0012 halfwingspan 1.75 m;30 m/s; aoa 8
Lift (N) Drag (N)
Undeformed wing 145,1351 7,4534
Deformed wing 147,3633 7,6939
Variation 1,54% 3,23%
142
Annex V
Morphing wing performance improvement data
Next it is presented the performance improvement of the morphing wing when compared to the Antex‐X2 wing.
143
Possible increase in aircraft weight vs Aircraft speed
Aircartf speed (m/s) Possible increase in aircraft weigh (N)
Velocity Naca Eppler Optim Naca Eppler Optim
halfwing halfwing halfwing wing wing wing
13 -2,86 -2,86 -5,72 -5,72
14 2,39 2,39 4,78 4,78
15 7,40 7,40 14,80 14,80
16 11,73 11,73 23,46 23,46
17 9,93 14,46 14,46 19,86 28,92 28,92
18 11,69 14,69 14,69 23,38 29,38 29,38
19 12,94 14,22 14,22 25,87 28,45 28,45
20 13,62 14,52 14,52 27,23 29,04 29,04
22 14,07 16,68 16,68 28,14 33,36 33,36
24 15,44 19,94 19,94 30,88 39,87 39,87
26 18,80 24,11 24,11 37,60 48,21 48,21
28 24,03 29,41 29,41 48,06 58,81 58,81
30 30,86 36,05 36,05 61,73 72,11 72,11
40 83,80 83,80 167,59
50 161,49 161,49 322,98
Aircraft speed function Drag reduction
Aircartf speed (m/s) Drag reduction (N)
Velocity Naca Eppler Optim Naca Eppler Optim
halfwing halfwing halfwing wing wing wing
13
14 0,80 1,04 1,04 1,60 2,08 2,08
15 0,97 1,45 1,45 1,93 2,91 2,91
16 1,05 1,26 1,26 2,10 2,52 2,52
17 1,02 1,08 1,08 2,03 2,16 2,16
18 0,92 0,97 0,97 1,84 1,95 1,95
19 0,82 0,91 0,91 1,64 1,83 1,83
20 0,73 0,87 0,87 1,47 1,75 1,75
22 0,66 0,82 0,82 1,32 1,63 1,63
24 0,71 0,80 0,80 1,42 1,60 1,60
26 0,86 0,85 0,86 1,72 1,70 1,72
28 1,09 0,97 1,09 2,17 1,95 2,17
30 1,37 1,17 1,37 2,73 2,33 2,73
40 3,32 2,80 3,32 6,63 5,61 6,63
50 5,92 4,98 5,92 11,84 9,96 11,84