Development of a Framework for Static Aeroelastic Analysisof Flexible Wings including Viscous Flow Effects

David Pina Brandão

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisors: Prof. Afzal SulemanProf. Erasmo Carrera

Examination Committee

Chairperson: Prof. Fernando José Parracho LauSupervisor: Prof. Afzal SulemanMember of the Committee: Dr. José Lobo do Vale

December 2015

ii

Acknowledgments

I would like to start by expressing my gratitude to my supervisor at Instituto Superior Tecnico, Professor

Afzal Suleman for the possibility of working in this project and the opportunity to join the CfAR Team at

the University of Victoria.

I would also like to thank my supervisor at Politecnico di Torino, Professor Erasmo Carrera, for taking

the time to follow the development of this thesis.

Also, a very special thanks to Mario Bras, both personally and academically, for all the indispensable

assistance he gave during the first days after my arrival at Victoria and for having closely followed the

development of the work for this thesis. The always pertinent and insightful suggestions he made when

any difficulty was encountered were essential for the conclusion of this work.

I would like to show my appreciation to my colleagues who moved as well to Victoria for easing the

time spent there. I would also like to thank Jose Fernandes for all the support in the later phases of the

development of this thesis. Also to Bruno Albuquerque for breaking the monotony along the final weeks

of work.

To my girlfriend Margarida I would like to thank for always supporting me and cheering me up, even

from thousands of kilometers away.

Last but not least, I would like to express my gratitude to my parents, Oscar and Salome, to my

brother Diogo and sister Ines for their unconditional support and care along this journey and to all my

friends for all the good moments shared during the course of the Degree.

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iv

Resumo

Acompanhando a crescente exigencia da industria por aeronaves mais eficientes, a interacao entre os

comportamentos estrutural e aerodinamico revestem-se hoje em dia de grande importancia. Existe, es-

pecialmente em configuracoes inovadoras, um interesse crescente em ter capacidade de prever as car-

acterısticas aeroelasticas desse tipo de aeronaves ainda em fases de desenvolvimento iniciais. Aplicar

modelos de alta-fidelidade, como e o caso de abordagens baseadas no acoplamento de CFD e CSD, e

ainda hoje computacionalmente exigente. Modelos de baixa fidelidade, embora apresentem tempos de

computacao curtos, desprezam frequentemente efeitos viscosos. Neste trabalho e assim implementado

um procedimento expedito para a avaliacao das caracterısticas aeroelasticas estaticas de asas flexıveis

em voo subsonico com inclusao de efeitos viscosos. As cargas aerodinamicas sao avaliadas atraves de

um procedimento de Interacao Viscosa-Invıscida utilizando um codigo de paineis 3D acoplado com um

codigo de Camada Limıte 2D. A velocidade na superfıcie e obtida para o problema invıscido, que e de

seguida usada como condicao de fronteira para resolver um sistema de equacoes derivadas do integral

de quantidade de movimento de Von Karman ao longo de seccoes bidimensionais da superfıcie. A

espessura de Camada Limite e de seguida tida em conta num novo calculo do escoamento invıscido.

Apos a convergencia da solucao aerodinamica, as cargas de superfıcie sao aplicadas para analise es-

trutural estatica. E dado seguimento ao procedimento iterativo atraves da actualizacao da geometria

deformada no modelo aerodinamico para efectuar nova analise aerodinamica. O efeito do escoamento

viscoso no comportamento aeroelastico e de seguida avaliado e os resultados aerodinamicos com-

parados com software comercialmente disponıvel. As vantagens do metodo aqui implementado sao

finalmente discutidas.

Palavras-chave: Aeroelasticidade, Interacao Viscos-Invıscida, Camada Limite, Metodo de

Paineis

v

vi

Abstract

As the aviation industry demands more efficient aircraft designs, the interaction between structural and

aerodynamic behaviors is nowadays of great importance . Especially in novel designs there is an in-

creased interest in accurately predicting the aeroelastic characteristics of such aircraft still in the early

stages of development. Employing high fidelity models, such as coupled CFD and CSD approaches,

is still today computationally expensive. Low fidelity models, with fast turnaround times, frequently dis-

miss viscous flow effects. In this work, an expedite procedure for the evaluation of the static aeroelastic

characteristics of flexible wings in subsonic flight with inclusion of viscous flow effects is implemented.

The aerodynamic loads are evaluated through a Viscous-Inviscid Interaction procedure using an open-

source 3D Panel Code coupled with a 2D Boundary Layer solver. Surface velocity is obtained for the

inviscid problem, which is then used as boundary condition for solving a set of equations derived from

the Von Karman momentum integral along surface sections. Boundary Layer thickness is then taken

into account in a new computation of the inviscid flow. After the aerodynamic solution converges, the

surface loads are used as input for static structural analysis. The iterative procedure is carried on by

updating the displaced geometry for new aerodynamic analysis. This simplified aerodynamic model is

therefore able to account for viscous flow effects, namely friction drag and displacement effects. The

effect of viscous flow on aeroelastic behavior is assessed and aerodynamic results are compared with

commercially available software. The advantages of the method here implemented are then discussed.

Keywords: Aeroelasticity, Viscous-Inviscid Interaction, Integral Boundary Layer, Panel Method

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Aeroelasticity 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Aeroelastic phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Computational Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Interaction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Interface models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Fluid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.4 Structural models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Aerodynamic models 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Inviscid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 Basic flow solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.3 Panel Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Viscous Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.1 Effects of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.3 Integral Boundary Layer Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

ix

3.4 Viscous-Inviscid Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Implementation 39

4.1 2D Aerodynamic solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 2D Panel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.2 Boundary Layer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.3 2D Viscous-Inviscid Interaction procedure . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 3D Aerodynamic module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 3D Panel method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.2 3D Viscous-Inviscid interaction Procedure . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Structural module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Aeroelastic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Results 47

5.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.1.1 Wing geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.1.2 Flight Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1.3 Wing Aerodynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.4 Wing Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Aerodynamic Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Aeroelastic Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.1 Aeroelastic Solution for Case #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.2 Aeroelastic Solution for Case #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Conclusions 65

6.1 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Bibliography 69

x

List of Tables

5.1 Wing geometry parameters (Adapted from [43]). . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Flow properties at the designed Cruise Flight condition. . . . . . . . . . . . . . . . . . . . 48

5.3 Flow properties at the modified Cruise Flight condition. . . . . . . . . . . . . . . . . . . . . 48

5.4 Definition of the two test cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.5 Spar thicknesses by zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.6 Wing Skin thickness by zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.7 Rib thicknesses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.8 Aluminium 7075-T651 mechanical properties. . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.9 Comparison of aerodynamic coefficient results between StarCCM+ CFD solution, the VII

code implemented and APAME inviscid solution. . . . . . . . . . . . . . . . . . . . . . . . 57

5.10 Comparison of Nomalized CPU computational times for the solution obtained with Star-

CCM+, the VII code implemented and APAME inviscid code. . . . . . . . . . . . . . . . . 57

5.11 Summary of the results obtained for the Aeroelastic analysis of Case #1. . . . . . . . . . 60

5.12 Summary of the results obtained for the Aeroelastic analysis of Case #2. . . . . . . . . . 63

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List of Figures

1.1 Failure of Langley’s Aerodrome in December of 1903 due to Aeroelastic Divergence,

showing the failure of the front wing [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 The Collar Diagram, repesenting the three Aeroelastic Forces and their interactions [11]. . 6

2.2 One-Way Aeroelastic Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Two-Way Aeroelastic Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Transfer of Loads (left to right) and Displacements (right to left) between Aerodynamic

and Structural meshes through the use of an Intemediate grid. (Adapated from [22]) . . . 12

2.5 Wing box segment modelled as a beam element. [29] . . . . . . . . . . . . . . . . . . . . 14

2.6 Different wing structural models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Several degrees of approximation of the General Governing Navier-Stokes equations and

their origin [30]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Example of discretized aircraft geometry for use with a Panel Method. (adapted from [31]) 22

3.3 Potential flow over closed body and definitions.[31] . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Representation of the Kutta condition on a doublet panel method.[31] . . . . . . . . . . . 24

3.5 Examples of prescribed wake shapes and their effect on the resultant aerodynamic coef-

ficients. [31] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 Panel local Reference System and neighbor panels for velocity computation. [31] . . . . . 26

3.7 Projection of the Resultant Force Coefficients on the Body and Aerodynamic Reference

Systems [35] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.8 Classical VII method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.9 Direct VII method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.10 Inverse VII method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.11 Semi-inverse VII method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 Locations of the Boundary Layer and Inviscid variables along the discretized airfoil. (Adapted

from [24]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Evolution of the Boundary Layer parameters obtained from the 2D VII code implemented

and from XFOIL for a flow with AOA = 5 deg and Re = 106 over a NACA 0012 airfoil. . . . 42

4.3 Aerodynamic Module computation flowchart. . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Aeroelastic Framework Modules flowchart. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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5.1 Three view wing geometry sketch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Lift versus Angle of Attack for the flight condition being studied, obtained with the APAME

Panel Method code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3 Final Wing Aerodynamic model with the skin discretized in 5200 panels. . . . . . . . . . . 50

5.4 Convergence of the aerodynamic results with increasing panel number both for Inviscid

and Viscous solution, for the two test cases. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.5 Computational time for executing 10 VII iterations with varying number of panels. . . . . . 52

5.6 Wing structure Finite Element Model: Spars (longitudinally) and Ribs (transverselly). . . . 52

5.7 Wing structure Finite Element Model: Wing Box (Spars + center Skin section) and Ribs

(transverselly). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.8 Convergence of the aerodynamic results with the iteration number for both test cases. . . 55

5.9 Evolution of the Relative Error of the aerodynamic coefficients with the iteration number

for both test cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.10 Wing geometry modeled on StarCCM+ for CFD aerodynamic analysis. . . . . . . . . . . . 56

5.11 Evolution of x, y and z maximum displacements with iteration number for Inviscid Aerostru-

cural analysis for Case #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.12 Evolution of x, y and z maximum displacements with iteration number for Viscous Aerostru-

cural analysis for Case #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.13 Detail of the deformed wing shape after converged Viscous Aerostructural Solution is

obtained for Case #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.14 Convergence of the aerodynamic results for the flexible geometry with the iteration num-

ber for Case #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.15 Evolution of x, y and z maximum displacements with iteration number for Inviscid Aerostru-

cural analysis for Case #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.16 Evolution of x, y and z maximum displacements with iteration number for Viscous Aerostru-

cural analysis for Case #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.17 Deformed Wing shape after converged Viscous Aerostructural Solution is obtained for

Case #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.18 Convergence of the aerodynamic results for the flexible geometry with the iteration num-

ber for Case #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xiv

Nomenclature

Acronyms

AIC Aerodynamic Influence Coefficients

CAE Computational Aeroelasticity

CFD Computational Fluid Dynamics

CSD Computational Structural Dynamics

DNS Direct Numerical Simulation

FEA Finite Element Analysis

FSI Fluid-Strucuture Interaction

RHS Right-Hand Side

TSL Thin Shear Layer

VII Viscous-Inviscid Interaction

Greek symbols

α Angle of attack.

δ∗ Displacement thickness.

δ∗∗ Density thickness.

δij Kronecker delta function.

κ Thermal conductivity coefficient.

µ Molecular viscosity coefficient.

ν Doublet strength.

ω∗ Relaxation parameter.

Φ Velocity Potential.

φ Normal modes of the structure.

xv

ρ Density.

σ Source strength.

τ Stresses.

θ Momentum thickness.

θ∗ Kinetic Energy thickness.

ζ Vorticity.

Roman symbols

[C] Damping Coefficient Matrix

[F ] Generalized Force Vector

[M ] Mass Coefficient Matrix

[M ] Stiffness Coefficient Matrix

q Fluid Velocity Vector

CD Coefficient of drag.

Cf Coefficient of skin friction.

CL Coefficient of lift.

CM Coefficient of moment.

H Shape parameters.

M Mach speed.

q Generalized Displacement Vector

r Distance

t Time.

w Structural Displacement Vector

p Pressure.

u Velocity vector.

u, v, w Velocity Cartesian components.

Subscripts

∞ Free-stream condition.

ξ, ν 2D components.

xvi

a Average between two stations.

b Boundary.

e Exterior.

EQ Equilibrium condition.

i, j, k Computational indexes.

l,m, n Cartesian components in the local reference system.

n Normal component.

tr Transition.

x, y, z Cartesian components.

ref Reference condition.

Superscripts

(n) Iteration number.

BL Boundary Layer solver.

PM Panel Method.

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xviii

Chapter 1

Introduction

1.1 Motivation

Designing an aircraft has always been a challenging task due to the influence of a broad range of

disciplines, each with its challenges and constraints. In the beginning of the aviation era this difficulty

was not only related to the little knowledge available at the time about each of the fields involved, but

also about how they influenced each other mutually.

The field of Aeroelasticity studies this type of coupled phenomena, both of static and dynamic na-

tures, resulting from the interaction between the aerodynamic loads on a flying body and the deformation

they induce in its structure, control system and propulsion, as defined in [1].

In fact, one of the main problems preventing successful flights before the Wright brothers’ was pre-

cisely related to the interaction between these two features, Aerodynamics and Structural Dynamics.

After a failed first flight attempt of his Aerodrome aircraft in October 1903 due to a malfunction with the

catapult system, Samuel P. Langley tried again to fly his aircraft on December the same year [2]. This

attempt, only a few days before the Wright brothers famous successful first sustained flight, also ended

abruptly right after the aircraft left the catapult as its wings were torn apart (Figure 1.1), allegedly due to

a phenomenon know as structural static divergence [2]. This occurs when the airflow around an airfoil

or a wing induces a torsional moment on the wing structure beyond it’s yield strength [3], resulting in a

catastrophic failure.

After the first few years of aviation, this kind of problems were overcome and thought of as having

a lesser importance, as sufficient wing torsional stiffness and wing warping control, combined with low

flight speeds minimized the effects of such interactions [2]. However, with the beginning of World War

I, the appearance of larger, more powerful and faster aircraft made aeroelastic problems arise, this time

in their dynamic form. The most relevant aeroelastic phenomenon falling into this category is known as

flutter, resulting from the interaction between unsteady aerodynamic forces and the aircraft’s structure

and control surfaces. It is characterized as an ”oscillatory structural instabillity” [2] and, if not contained,

can lead to catastrophic structural failure.

Since then, in the age of modern aviation, the importance of the aeroelastic phenomena steadily

1

Figure 1.1: Failure of Langley’s Aerodrome in December of 1903 due to Aeroelastic Divergence, showingthe failure of the front wing [4].

rose as speeds increased to transonic and supersonic ranges. Nowadays these effects have a critical

influence on aircraft design due to the growing focus on fuel efficiency. Novel greener designs ranging

from High Aspect Ratio wings to Joined Wing configurations with better aerodynamic performance also

show highly complex static and dynamic aeroelastic coupling. These designs, now possible through the

employment of advanced structural solutions such as the use of composite materials and high strength

alloys, also have the downside of resulting in structures with greater flexibility because the increase in

strength is not accompanied by an increase in the modulus of elasticity [1], which magnify the influ-

ence of such effects (such as large wing deformations) and the importance of including Aerostructural

considerations right from the first design stages [5] .

Correctly predicting such effects and introducing design solutions right from the first stages of devel-

opment of a new aircraft can therefore bring multiple advantages. Reduction of development costs and

time, as well improved efficiency, safety and comfort are enabled by the achievement of better optimized

designs, for example through Multidisplinary Optimization (MDO) approaches that take into account

aeroelastic considerations [6]. Nowadays, the field of Computational Aeroelasticity (CAE) is in charge

of the prediction of such effects. The models it employs range from very simple and expedite models

presenting low fidelity results to highly complex but computationally heavy models able to deal with a

larger range of flight conditions and phenomena. Although the choice of high fidelity models seems ob-

vious as computational power availability increases, the use of such models in MDO approaches is still

very limited by the computational times required to obtain an high-fidelity aerodynamic and aeroelastic

solution.

Several authors and commercial packages present aeroelastic solutions across various levels of

fidelity. However, the effects of Viscous Flows are usually only taken into account in high-fidelity ap-

proaches, limiting it’s use to later design phases due to the computational and modeling effort needed.

The consideration of the presence of a boundary layer usually thickens the effective airfoil and reduces

it’s camber, reducing lift. Not considering Viscous Flow effects also means dismissing the shear loads on

the skin. Very few authors attempted to include such effects in low-fidelity aeroelastic models, leaving

2

a gap in the range of model fidelity. This work attempts to fill that gap by including the effects of the

presence of a boundary layer on a low fidelity aerodynamic inviscid model.

1.2 Objectives

The main objective proposed for this work is to implement a static aeroelastic analysis framework based

in a low-fidelity expedite aerodynamic model and include viscous flow effects through the implementation

of a Boundary Layer model.

In support to the main objective it is also intended to begin with a review of the state-of-the-art in the

fields of static aeroelastic, aerodynamic and structural modeling. The performance of the new tool is to

be assessed in comparative studies.

Taking into account a study case, the influence of viscous phenomena in static aeroelastic behaviour

is also to be discussed through a comparison of the results obtained by including and neglecting such

effects.

1.3 Thesis Outline

In Chapter 1 the motivation for this work are presented and the objectives established.

In Chapter 2, an introduction to the field of aeroelasticity is briefly done. It is firstly viewed from a

conceptual point-of-view and the aeroelastic problem if defined. Specifically for Static Aeroelasticity, the

main challenges and effects this type of interaction creates to aircraft design and operation are pre-

sented. In the last section of this chapter an overview of the field of Computational Aeroelasticity (CAE)

is done. It’s evolution over the years is presented, as well as the challenges it poses and the different

models used, their advantages, constraints and drawbacks. This will support the the implementation

subsequently made in following chapters.

In Chapter 3 the most relevant aerodynamic models for the work being developed are presented and

derived from the Governing Fluid flow equations. This includes the development of the Panel Method

and in parallel the Boundary Layer approximations. The interaction methods between the two models

are then explored in the Viscous-Inviscid Interaction Section, and the advantages and limitations of each

are presented.

In Chapter 4 the models previously presented are gradually implemented. First the 2D Aerodynamic

model implementation is described. Details about the implementation of the 2D Panel Method are

discussed, as well as for the Boundary Layer code and the Viscous-Inviscid Interaction Procedure. The

final 2D Aerodynamic solver is then run for a test case and compared with results from the literature.

The discussion is then taken to the development and working structure of the 3D Aerodynamic module

to be used in the final Aeroelastic Framework. The modifications done to the APAME 3D Panel Code

to allow for implementation in the Viscous-Inviscid Interaction Procedure are presented. The coupling

procedure used to provide the interaction with the 2D Boundary Layer solver is presented. The structural

solver being used is then briefly described, as its main component is a commercially available software.

3

Finally, the implementation of the Final Aeroelastic Framework is discussed, as well as the file structure

and all the needed routines to perform the interaction between the several modules and external solvers.

In Chapter 5 the results obtained with the developed codes are presented. First, the application case

is described. It’s geometry and flow conditions are defined and convergence studies for the discretized

geometry are done to ensure the final model is adequate. Then the results obtained by the Aeroelastic

Module here developed are compared with a CFD solution from a commercial solver. With the verifi-

cations made, next the final solution for the Static Aeroelastic problem previously defined is obtained

from the Aeroelastic Framework, both for an Inviscid and Viscous flow assumption. The results are

commented and the effects added by the presence of the Boundary Layer evaluated.

Lastly, in Chapter 6, a summary of the work developed and the final conclusions are collected.

Suggestions for future developments are enumerated.

4

Chapter 2

Aeroelasticity

In this section a brief introduction to the field of Aeroelasticity is given, starting from the definition of the

phenomena it studies, as well as typical problems it raises to aircraft design and operation.

The general aeroelastic problem formulation is presented. An overview over the models’ evolution

used is presented, as well as the varied kinds of approaches regarding fidelity, coupling and application.

The state-of-the-art regarding aeroelasticity models used in the industry and for research purposes

is elaborated. Development guidelines for future models are presented taking into account the field

experts.

2.1 Introduction

One of first authors to show interest in aeroelastic oscillations was Ludwig Prandtl (1875- 1953). Walter

Birnbaum, one of Prandtl’s students, first published a thesis entitled “The two-dimensional problem of

the flapping wing.” in 1922 where wing flutter was described for first time as a structural dynamic stability

problem [2]. During that and the following decade several authors, such as Collar investigated into such

effects pointing the causes for their occurrence and improving the understanding of such phenomena

. Advances in unsteady aerodynamics during that period also helped improving the comprehension in

this field.

A work considered to lay the basis for theoretical aeroeslaticity for the following decades cited in [7]

was published by Theodorsen in 1934 [8]. An analytical method for the evaluation of aeroelastic effects

in 2D airfoils and the prediction of flutter in airfoils was there presented.

Although the field of aeroelasticity has always been more connected to the aerospace industry, like

airframes and turbomachinery, where weight, efficiency and safety restrictions push towards cutting

edge solutions, other areas are benefiting from the knowledge acquired. For example, in bio-mechanics,

arteries can be studied as flexible structures which interact with an unsteady fluid flow, blood [9]. Civil

Engineering applications such as long span bridges also experience aeroelastic effects, with the most

notable example being the collapse of the Tacoma-Narrows bridge in 1940 due to the occurrence of

flutter.

5

Aeroelasticity is nowadays a well established field, concerned with the study of the interaction be-

tween three forces: Aerodynamic, Inertial and Elastic [10]. The three disciplines involved (Fluid Me-

chanics, Structural Dynamics and Structural Mechanics) were firstly represented together by Collar in

the 1940’s using a triangle diagram, Figure 2.1, named after him [6].

Figure 2.1: The Collar Diagram, repesenting the three Aeroelastic Forces and their interactions [11].

This diagram can still be further analyzed, as the wide variety of aeroelastic phenomena does not

always necessarily involve all three disciplines. Dowell [3] pairs these three disciplines in different ways

to discover other important fields that, according to the author, may be interpreted as particular cases

within the broader field of Aeroelasticity:

• Flight Mechanics, resulting from the interaction between Structural and Fluid Dynamics;

• Structural Vibrations, resulting from the interaction between Structural Dynamics and Mechanics;

• Static Aeroelasticity, resulting from the interaction between Structural Mechanics and Steady Aero-

dynamics

In Aerospace, aeroelastic phenomena are not only seen in aircraft structures, but also in other

branches, which may couple aeroelastic effects with other phenomena. In turbomachinery, for exam-

ple, aeroelastic behavior is very important and, because of the high temperatures present, a new variant

of the field called aerothermoelasticity appears [10]. With heavy centrifugal, gas and thermal loads

applied to the blades not only the design of the blades have to account for the deformed operating

condition, but also to vibration problems that may arise.

As a composite of several disciplines, the aeroelastic problem formulation requires inputs from the

fields involved. Formally, their interaction can be summarized in the Generalized Equations of Motion

2.1 and 2.2 [7].

[M ] {q(t)}+ [C] {q(t)}+ [K] {q(t)} = {F (t)} (2.1)

{w(x, y, z)} =

Nmodes∑i=1

qi(t) {φi(x, y, z)} (2.2)

6

In equation 2.1, on the left side, q(t) represents the generalized displacement vector and [M ],[C]

and [K] are, respectively, the Mass, Damping and Stiffness matrices. On the right side the generalized

force vector F (t) relates the aerodynamic and inertial loads with structural dynamics. In Equation 2.2,

w(x, y, z, t) is the structural displacement vector and φi(t) represents the normal modes of the structure

[7, 9]. This formulation contains in itself the concepts and physics of every aeroelastic problem without

restraining the use of specific aerodynamic and structural models. This way several combinations of

models with varying fidelity and characteristics can be used according to the development stage of an

aircraft, the flying conditions, computational constraints or the phenomenon to be studied.

2.2 Aeroelastic phenomena

The interaction between the several intervening disciplines is made evident by the appearance of phe-

nomena resultant from that coupling. These can be categorized as Static, meaning that the effects are

not time dependent, and Dynamic, where the unsteady characteristics of the airflow, structure and flight

lead to oscillatory responses.

Static aeroelastic effects can affect aircraft performance in several ways [2]. The most evident one

occurs because of the structural deformation which affects the wings aerodynamic shape through bend-

ing and, in turn, modifies the loads distribution when compared to a rigid body. This affects mostly lifting

surfaces which means that, e.g, the lift distribution will be affected, as well as the total lift generated.

Another effect of the structural deformation is related to the flight dynamics. As the lifting surfaces

deform, the new shapes dictate a new aircraft behavior. This happens for example when a torsional load

is applied to a wing. This torsion affects the angle of attack of each section and therefore effectively

alters the stability derivatives of an aircraft, as well as its trim condition. Other effects compromise

the maneuverability of an aircraft resulting from a reduction or even inversion of the control surfaces

effectiveness. The more extreme cases is called Control System Reversal [1]. Inside this category is, for

example, the Aileron Reversal phenomenon which occurs for speeds above a certain limit (called control

reversal speed) for which a deflection of the control surfaces induces a deformation on the structure that

cancels the effects intended by applying the deflection.

However, one of the most important static aeroelastic effects is know as Divergence. As referred

in the previous chapter, this effect was responsible for many failures in the early days of aviation. It

occurs at the Divergence Speed for which the aerodynamic loads induce a torsional moment on the

wing structure beyond it’s yield strength [3], resulting in a catastrophic failure. It affects lifting structures

in general and, in its most common form, it is due to a combination a low torsional stiffness with a positive

feedback between the torsional moment generated by the aerodynamic loads which in turn deforms the

structure to increase the angle of attack and, thus, increasing even more the torsion. This phenomenon

is counteracted by the structure stiffness for low speeds but, as flow velocity increases, the structure

may no longer offer sufficient stiffness resulting in a deformation that becomes unstable [1].

Aeroelastic phenomena can also take a dynamic form. The dynamic response of the structure to

quick changes in the flight conditions, such as wind gusts or sudden application of controls, must also

7

be studied as such effects can lead to structural failure. One of the most known effects, flutter, is another

kind of dynamic aeroelastic phenomenon. It is related to the unstable dynamic structural response to

the flow perturbations and occurs at a specific speed called flutter speed. If not corrected it can lead to

a catastrophic structural failure.

The disturbed flow leaving from several parts of the aircraft can also be a source of aeroelastic

problems. This kind of interaction is called Buffeting and is most severe when doing maneuvers that

place the tail surfaces inside the wing and engine wakes.

Nevertheless, one of the first considerations to be made within structural design is static aeroelastic

phenomena. As an aircraft goes through a flight, each of the mission phases can be characterized by

the maneuvers being executed, which translate into Aerodynamic and Inertial loads at various configura-

tions and flight conditions. Before studying dynamic aeroelastic phenomena it is important to determine

the static characteristics of the aircraft during such stages. The deformed wing shape, for example, is

a crucial information to have during design, as displacements and torsion affect lift, drag and moments

generated. In fact, the usual operation configuration will be the deformed one so the structural and aero-

dynamic design optimization has to be done for this condition, as well as preventing static aeroelastic

phenomena such as divergence. This work will therefore focus in modeling static aeroelastic phenomena

only.

Regarding the formulation of these phenomena, Equation 2.1 will further simplify for the static condi-

tions as can be seen in Equation 2.3.

[K] {q} = {F} (2.3)

It is possible to observe that the formulation for the static aeroelastic phenomena keeps the generality

offered by the unsteady formulation. Aerodynamic and Structural models can subsequently be employed

in order to solve for the structural displacements.

2.3 Computational Aeroelasticity

In the beginning of the field, wind tunnel and flight testing supported with theoretical and empirical meth-

ods were the tools used for aeroelastic analysis. As computers became available, the first computational

models were implemented from adapted of versions of the theoretical methods [6].

As stated before, the formulation of the aeroelastic problem opens the possibility to make combina-

tions of different Aerodynamic, Structural and Dynamics models. Therefore, with the appearance of the

fields of Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD), aeroe-

lastic modeling also felt improvements. In fact, these events opened the way for the creation of the

field of Computational Aeroelasticity (CAE), which specifically refers to the coupling of different aerody-

namic and structural models, varying in capabilities, formulation, approximations, fidelity and computa-

tion times, for the study of Fluid/Structure Interaction (FSI) problems [6]. This allows the user to tailor

the model and it’s approximations to the purposes and conditions of each study and achieving models

8

with variable fidelity while keeping the same aeroelastic formulation [7].

At the time, the early methods were very limited by computational power, so linear structural and

fluid models were initially employed. With the improvements of computational power and in numerical

methods, the Aerodynamic and Structural models became more complex and the use of CAE started

to become more frequent not only in academic simulation and analysis but also in the industry, as an

integrated part of the design process.

In this section a review of the field of Computational Aeroelasticity is made with focus on Static

Aeroelastic modeling. Specificities of Dynamic Aeroelastic modeling such as time-stepping techniques

or modal analysis are not discussed in this work.

2.3.1 Interaction methods

CAE, being a Fluid/Structure Interaction (FSI) problem, requires the Aerodynamic and Structural dis-

cretized models to be mechanically coupled. As previously seen in Equation 2.1, this implies that Aero-

dynamic Loads computed by the Fluid Dynamics solver must be transfered to the Structural solver. If

the computation is stopped at this point a One-Way Coupled model is being employed, meaning that

the aerodynamic model is not updated to the deformed configuration. This implies that, as structures

become more flexible, a considerable error can be present in the results as the Aerodynamic and Struc-

tural geometries do not coincide. Commercial Finite Element Analysis (FEA) widely used throughout the

industry such as MSC Nastran and NX Nastran are capable of executing this kind of analysis natively

using very simple aerodynamic models, being however limited to One-Way Coupling (Figure 2.2).

?Geometry and Flow Conditions

Aerodynamic Solver

?Aerodynamic Loads

Structural Solver

?Deformed Geometry

Figure 2.2: One-Way Aeroelastic Coupling.

In order to obtain a properly converged solution, the Structural displacements have to be taken into

account to update the Aerodynamic geometry after a deformed configuration is obtained. This kind of

model is what is called a Two-Way Coupling procedure (Figure 2.3). This way an iterative procedure can

be used to update both models which, over the course of the computations, become equivalent. After

convergence, the calculated aerodynamic loads match the deformed structure shape.

Another question that can be raised regarding the model coupling is related to how independent the

solvers and formulations of each discipline are in relation to the other. Kamakoti [9] sorts the models

into three categories, according to how tightly connected the solution procedures are.

The Fully-Coupled Models occupy the top of this scale. In this kind of approach, the problem’s

9

?

Geometry and Flow Conditions

Aerodynamic Solver

?Aerodynamic Loads

Strucutural Solver

Deformed Geometry

- �� ��?

Deformed Geometry

Figure 2.3: Two-Way Aeroelastic Coupling.

formulation is developed so that the aerodynamic and structural governing equations are solved together.

Although this avoids the iterative procedure between the aerodynamic and structural solutions, the size

of the problems is limited to relatively small ones, being only applied in 2D cases. Big differences in the

stiffness of aerodynamic and structural matrices rise from the difference in the reference systems used

for each discipline, Eulerian description for the fluid problem and Lagrangian for the structural one, which

limits the scale of the problem [9]. Some developed works include the acceleration efforts done for the

aeroelastic computation of a fully coupled Navier-Stokes Solver done by Obayashi and Guruswamy [12]

on the ENSAERO code. More recent works by Kennedy and Martins [5] and by James, Kennedy and

Martins [13] successfully implement a Fully-Coupled Model for Multidisciplinary Optimization purposes

where the aeroelastic system is solved with a Newton-Krylov iterative method.

On the opposite side of the scale appear the Loosely-Coupled Models. In this case each solution is

obtained in separated solvers and an iterative procedure between the two disciplines is executed until

convergence is achieved [14]. The main advantage of using such a method is the possibility to use

existent, proven solvers in each of the disciplines. On the other hand, the need of an interface for the

data transfer, usually through output and input files, leads to large computation times. Also, because

of the communication limitations between solver, Aerodyamic meshes may have to be generated from

the begining at each deformed geometry configuration if a discretized volume domain is being used,

resulting in a time-consuming procedure. Therefore, application of this type of coupling may present

limited efficiency when trying to model cases with strong nonlinearities [9]. Severall authors successfully

implemented to this approach. Love et al. [15] applied such an approach, coupling the NASTRAN

commercial solver with a Lockheed CFD code, SPLITFLOW, to analyze a fighter jet during a pull-up

maneuver. Also using NASTRAN and several CFD codes, Heinrich et al. [16] performed aeroelastic

analysis of a wide-body airliner using AMANDA Simulation Environment. Wang and Lin [17] combined

Fluent CFD aerodynamic solver with ABAQUS structural solver to analyze static aeroelastic behaviour

of a generic wing. Huixue et al. [18] dealt with the high number of aerostructural iterations sometimes

needed to achieve convergence when employing this type of scheme with CFD packages by successfully

implementing and acceleration technique and applying it to the analysis of a nonlinear static structural

analysis of a High Aspect Ratio wing. A 2/3 reduction on the required iterations to convergence and

10

computational time when compared of the not accelerated figures was achieved.

As a compromise between the former methods, the Closely-Coupled Models emerge as the most

frequently used kind [9]. As in the Loosely-Coupled Model, the solution is obtained from an iterative

procedure and, while the Aerodynamic and Structural equations are still solved separately, both solvers

are implemented into one single package. While this results in the loss of some flexibility comparing

to Loosely-Coupled Models, it also eliminates the need for an interface tool and/or files, significantly

improving information exchange between solvers [14]. Cavagna et al. [19] present a method for solving

dynamic aeroelastic problems within FLUENT CFD solver through the implementation of an user-defined

plugin containing the modal information of the wing structure being analyzed. The structural modes were

obtained beforehand using the NASTRAN Finite Element Analysis code.

2.3.2 Interface models

The task of exchanging loads and displacements between the Aerodynamic and Structural models can

be relatively simple if the discretized geometries of both models coincide but, frequently, this does not

happen. The reasons for these differences between the models can be attributed to the coupling of

models that are of different fidelity, for example, when coupling a Lifting Line aerodynamic model to

a FEM structural model, or simply because of the different refinement level or resolution between the

discretized aerodynamic and structural models.

Several load and displacement transfer methods were developed. These can fall into two categories,

according to Kennedy [5]. The first one is Direct transfer, where aerodynamic loads are directly trans-

fered to the structure model. With wetted aerodynamic surface represented in both models, the transfer

can be done, for example, by interpolating the data in that same geometric surface, as shown in Figure

2.4. Such an approach was followed in [20]. If not only the geometrical surfaces match but also the dis-

cretized surface (mesh), then no interpolation method is needed and the loads and nodal displacements

can be directly imposed on the corresponding elements and nodes. If the aerodynamic surface is not

represented in the structural model, then rigid links can be used to transfer the loads and displacements.

This was the method implemented by Kennedy et al. [5] and James et al. [13]. However, this last method

requires the verification of the consistency of the forces and conservativeness of the energy transfer by

checking, for example, if the work done by the loads before and after the transfer are the same.

When this occurs, the need to do displacement extrapolation can prove to be difficult when using

Direct Methods [5].

A second method is can be employed when geometry gaps exist between the models. The alternative

consists in evaluating the contribution of the load distribution by integrating it along an intermediate

auxiliary surface or the aerodynamic surface itself. The resultant equivalent forces and moments are

then transfered to the structure. Examples where the equivalent loads can be computed include the

loads coming from a full CFD analysis to a beam or torsion box structural model of a wing, where

the wing skin is not present in the structural model while the aerodynamic solution is obtained for it.

Or, similarly, when a lifting line model is coupled with a full FEM wing model. If the transfer of loads

11

and displacements is done recurring to the same ”geometric representation of the fluid and structure

surface model” [6] independently of the transfer direction, conservation will be assured, according to the

Principle of Reciprocity [21]. This usually exempts the need for any verification as this type of methods

usually comply with this requirement.

Figure 2.4: Transfer of Loads (left to right) and Displacements (right to left) between Aerodynamic andStructural meshes through the use of an Intemediate grid. (Adapated from [22])

2.3.3 Fluid models

In what respects the methods used to model the Aerodynamics of aircraft, Bartels [6] suggests three

major fidelity levels can be identified: Low, Moderate and High Fidelity.

Low Fidelity aerodynamic methods used to model lifting bodies include Panel Methods, Doublet

and Vortex Lattice Methods (DLM and VLM), Lifting Line and Lifting Surface methods, among others.

These compute the inviscid flow quickly as they do not require to discretize the entire fluid volume but

rather only the aerodynamic surfaces. The geometry can then be easily modeled or imported from CAD

models. These linear models are still today frequently used in the industry coupled with FEM structural

models, mainly in early design phases and optimization codes due to the quick turnaround times and

relatively accurate results. This possibility to quickly obtain a solution, enabled by the reduced complexity

of the linearized fluid-equations used, is biggest advantage that these models present, even with the

improvements seen in non-linear methods’ turnaround times since the 1990’s [6]. However this kind

of methods are not capable of modeling flows with prevalent viscous interactions nor transonic flows by

themselves, as they are characterized by nonlinear phenomena such as Boundary Layer separation and

Shock Waves [14]. Kennedy [5] reports computational times that are topped at 522 seconds for obtaining

a converged aeroelastic solution when using a Panel Method aerodynamic solver with a 16 processor

computer, which is much faster than CFD based solutions. The commercially available software ZAERO

does several kinds of static and transient aeroelastic analysis with low fidelity aerodynamic models based

in Panel Methods. NX NASTRAN and MSC NASTRAN implement a Vortex Lattice Method (VLM) as the

aerodynamic model, although the analysis is limited to One-Way Coupling. Mark Drela’s ASWING also

does this type of analysis using a Lifting Line aerodynamic model. Kennedy and Martins [5] and James,

Kennedy and Martins [13] also implement a Low Fidelity aerodynamic methods based on a 3D inviscid

12

panel method, correcting the estimated Drag with empirical relations.

CFD based approaches, on the other hand, offer better accuracy and are capable of dealing with

complex geometries and modeling several kinds of non-linear aerodynamic phenomena, depending on

the Aerodynamic models used. These approaches fall inside the High fidelity models. These usually

solve a version of the non-linear Navier-Stokes equations like the Reynolds-averaged Navier-Stokes

(RANS) over the full discretized flow domain (structured and not) coupled with turbulence models. How-

ever, such models require the discretization of the entire fluid volume surrounding the structure, which

is aggravated by the aeroelastic interactions that require to regenerate the aerodynamic volume mesh

at each aerostructural iteration due to the structural displacements that occur. Additionally, with a more

complex discretization process, appropriate modeling relies significantly on the user expertise.

Raveh [14] cites the works of Love et al. that report 3 to 4 days of computational time on a mid 90’s 16

processor HP V250 supercomputer for maneuver load analysis when employing CFD based flow solvers

and a 35 hours computational time to simulate a 4 second maneuver in an late 90’s 32-processor SGI

Origin 2000 computer. Kennedy, et al. [5], in a more recent paper, still points out the lengthy processes

involved with the use of CFD aerodynamic models are prohibitive for use in early design stages, although

being well adapted for design refinement. In a work from 2008, Huixue, et al. [18] report 72 hours of

computational time on an HP XW6400 workstation with a 3 GHz Xeon 5160 CPU and approximately 20

aerostructural iterations to achieve convergence of wing static nonlinear aeroelastic analysis resorting

to an Inviscid Euler CFD aerodynamic model. Implementation of an acceleration technique allowed to

cut this time to 24 hours.

Still within the CFD based approaches, Moderate Fidelity models are usually a simplified or lin-

earized version of the Navier-Stokes or Euler equations solved in the entire flow domain. Included in this

category are also other variants explored by several authors [6] such as a combination of an unsteady

linearized model with a full steady flow in support to the former. Reducing computational costs up to an

order of magnitude when compared with the nonlinear counterpart this method was considered sufficient

in many cases [9].

Another kind of models that can be considered to present Moderate Fidelity are the ones based in

Viscous-Inviscid Interaction (VII) procedures which couple inviscid and viscous approximations to the

flow field equations to get the final solution. Commercially available software packages like ZEUS use

an Euler equation solver for the inviscid part of the flow coupled with a steady boundary-layer equation

to include viscous effects. The use of this kind of models for aeroelastic analysis is however not very

frequent. In purely aerodynamic problems this is also the approach followed by several authors. Drela in

[23, 24] uses the same approach of coupling an Euler CFD Inviscid code with Integral Boundary Layer

solutions along the surface. Regarding VII models based on Panel Methods, one of the earliest codes

to implement such model was VSAERO in 1987 [25]. There a 3D Panel Method was coupled with 2D

Boundary Layer Equations solved along streamlines on the surface wing. It was however limited to

incompressible flows and small separation regions. In 1997, Milewsky [26] presented a VII procedure

using a low-order 3D Panel Method coupled with 3D Boundary Layer equations for application in the

naval industry. This more generic formulation of the Boundary Layer presented good results regarding

13

the estimation of Loads and Boundary Layer quantities. More recently, Stalewski et al. [27] implemented

an inviscid model solved using a Boundary Element Method coupled with 2D Boundary Layer Equations

in the frame of the European Union 6th Framework Program for a light general aviation aircraft. In

general the aerodynamic characteristics calculated were in line with experimental results, although the

developed code had the tendency to over-predict the CLmax and αCLmax . When accounting for viscous

flow effects over the fuselage, all these codes either dismissed them or assumed the axisimetry of the

flow. Pereira, et al. [28] implemented a 3D Panel Method coupled with a 3D Boundary Layer code

capable of dealing with transverse flows, demonstrating a correct prediction the position of longitudinal

vortices around the nose of a streamlined train.

Regarding the future investment in aeroelastic models, Schuster [7] defended that the development

of all three fidelity levels have its place in aircraft design and should target not only the use in analysis

tools but, most significantly, the full integration in the design tools. He adds that for use within the industry

these methods shall be of simple use, robust, efficient and sufficiently generic.

2.3.4 Structural models

The model classification on fidelity can be extended to account for the structural models’ fidelity as well,

Equivalent Beam approaches to Finite Element Models (FEM) with 2D and 3D elements [9].

The simplest CSD wing structural models are built with 1D Beam elements with a defined cross-

section to match the wing box or the spar section. This is the approach followed by Guo [29], where the

wing structural model is a set of beam elements disposed spanwise (Figure 2.5).

Figure 2.5: Wing box segment modelled as a beam element. [29]

James et al. [13] implements a wing box model with solid hexahedral elements, as presented in

Figure 2.6.

Kennedy and Martins [5] implemented a more detailed model of a wing box including the spars, top

and lower skin and ribs, as shown in Figure 2.6. The whole structure is modeled with shell elements.

Modeling the structure as closely to the real one as possible has several advantages when trying

to predict subtle effects. Also, when performing Design Optimization a well defined structure allows to

test several real configurations. Very important in aeroelastic analysis and design optimization is also

correctly accounting for the weight distribution along the lifting surfaces. Therefore, Kennedy and Martins

[5] include in their implementation an enhanced weight model.

14

(a) Wing box modelled as 3D Solid Elements. [13]. (b) Wing box internal structure modeled with2D shell Elements. [5]

Figure 2.6: Different wing structural models.

Aeroelastic phenomena frequently assume non-linear behavior and nowadays even more, consider-

ing novel geometries and materials. Therefore, many of the presented works used non-linear structural

models which are able to best model large displacements, as opposed to linear formulations.

15

16

Chapter 3

Aerodynamic models

In this chapter the Fluid Flow governing equations are presented. During the derivation of further rela-

tions, first for Inviscid Flow and then for Thin Shear Layer Flows, the assumptions made are explained

and the phenomena being modeled are discussed.

Finally, a Viscous-Inviscid Interaction procedure is chosen. The method picked for modeling the

Inviscid flow is a 3D Panel Method code, APAME. Modifications to allow for the VII procedure to be done

are presented. For the Viscous Boundary Layer, integral relations based on the Von Karman momentum

integral equation are employed, along with a Lag-Entrainment for upstream history effects method and

empirical closure relations for both Laminar and Turbulent flows.

3.1 Introduction

Based on conservation principles, fluid flow is governed by three Fluid Mechanics equations [30]:

• the continuity equation, based on the conservation of mass;

• the momentum equation, an extension of Newton’s 2nd law;

• energy, coming from the 1st law of Thermodynamics.

The detailed steps and operations for developing them are presented in various aerodynamics and

fluid mechanics books [30, 31, 32].

The continuity equation can be derived from the concept that the net mass flow passing through a

surface surrounding a volume has to be equal to the variation of mass inside the volume. This means

that if the net flow is 0 (the amount of mass entering the volume equals the one exiting), then, by the

conservation of mass, the mass inside the volume is constant. Therefore, the differential form of the

Continuity Equation, Equation 3.1 is presented.

∂ρ

∂t+∇ · ρq = 0, (3.1)

where ρ is the fluid density, t is time and q is the fluid velocity vector.

17

The momentum equation, Equation 3.2, is, as exposed before, an extension of Newton’s 2nd law,

which states that ”The time rate of change of momentum of a body equals the net force exerted on it”

[30].

ρDqiDt

= ρfi +∂τij∂xj

, with (i = 1, 2, 3), (3.2)

where DDt = ∂

∂t + q · ∇ is the material derivative which introduces a non-linearity in the second term,

τij are the stress components and fi is the component of the mass force per unit mass second in the

i direction. Taking the assumption of a Newtonian fluid, the velocity field can be used to determine the

stress components using the relation of Equation 3.3:

τij =

(−p− 2

3µ∂qk∂xk

)δij + µ

(∂qi∂xj

+∂qj∂xi

), (3.3)

where δij is the Kronecker delta function, µ is the viscosity coefficient, p is the pressure and k a dummy

variable summed from 1 to 3 [31]. Substituting equation 3.3 into 3.2, the Navier-Stokes equations (3.4)

are obtained.

ρ

(∂qi∂t

+ q · ∇qi)

= ρfi −∂

∂xi

(p+

2

3µ∇ · q

)+

∂

∂xiµ

(∂qi∂xj

+∂qj∂xi

)(i = 1, 2, 3), (3.4)

which evidences the equilibrium between the fluid acceleration terms on the left and the forces exerted

in the fluid.

Lastly, equation 3.5 enforces the Conservation of Energy which states that the variation of a system’s

internal energy equals the heat added to the volume minus the work done by it. There are several forms

for this equation, here is presented one of them from [30].

Dρ(e+ 1

2q2)

Dt= ∇ · (K∇T )− divw, (3.5)

where e is the internal energy per unit mass, K is the coefficient of thermal conductivity, w is the vector

of work associated with each control volume face and −divw is the rate of work done on the system,

represented as W , which can be obtained as in Equation 3.6.

W = −divw =∂

∂x(uτxx + vτxy) +

∂

∂x(uτyx + vτyy) , (3.6)

where u and v are the velocity components in a 2D Reference system.

These equations form a non-linear system with few analytical solutions available. Therefore, Direct

Numerical Simulation (DNS) methods have been developed to solve them directly. However, this kind

of solution still requires considerable computational resources and is therefore limited to very small

problems and research purposes [5, 14]. Trying to overcome this problem several approximations to

these equations were made, as shown in Figure 3.1.

In this work, both branches there presented, inviscid flows on the left and viscous on the right, are

18

All the equations presented in this section provide physical models of classes of flows that,

under the right circumstances, are completely adequate to obtain an accurate representation of

the flow. Many, many other approximate flow models have been proposed. Those presented in

this section represent by far the majority of methods currently used. In recent times, numerous

versions of the Navier-Stokes equations (taken here to include the time-averaged Reynolds equa-

tions to be discussed in Chap. 10) have also been used. These equations will be discussed as ap-

propriate in subsequent chapters. Figure 2-10 given below summarizes the connection between

the various flowfield models.

Figure 2-10. Connection between various approximations to the governing equations.

report typos and errors to W.H. Mason Fluid Mechanics Foundations 2-37

Thursday, January 16, 1997

General Governing EquationsNavier-Stokes Equations

Newtonian fluid, compressible, viscous, unsteady, heat-conducting

inviscid flowassumption

treat turbulence viaReynolds averaging and

turbulence model

restrict viscous effectsto gradients

normal to bodies, directional bias

introduce Prandtl BL assumption• pressure is const. across layer• leading viscous term only

Boundary Layer Eqns.

Reynolds Equations(sometimes called N-S)

Euler Equations

Thin Layer N-S Eqns.

note: aeros 1. drop body force terms2. use divergence form

• onset flow uniform• shocks are weak (Mn<1.25)

Potential or FULL Potential Eqn.(Gas Dynamics Equation)

Irrotational FlowV=∇Φ

incompressible flow

Laplace's Eqn.

small disturbance approx

sub/super & trans, incl.P-G & TSDE Eqns.

(includes integral egn.representation)

All the equations presented in this section provide physical models of classes of flows that,

under the right circumstances, are completely adequate to obtain an accurate representation of

the flow. Many, many other approximate flow models have been proposed. Those presented in

this section represent by far the majority of methods currently used. In recent times, numerous

versions of the Navier-Stokes equations (taken here to include the time-averaged Reynolds equa-

tions to be discussed in Chap. 10) have also been used. These equations will be discussed as ap-

propriate in subsequent chapters. Figure 2-10 given below summarizes the connection between

the various flowfield models.

Figure 2-10. Connection between various approximations to the governing equations.

report typos and errors to W.H. Mason Fluid Mechanics Foundations 2-37

Thursday, January 16, 1997

General Governing EquationsNavier-Stokes Equations

Newtonian fluid, compressible, viscous, unsteady, heat-conducting

inviscid flowassumption

treat turbulence viaReynolds averaging and

turbulence model

restrict viscous effectsto gradients

normal to bodies, directional bias

introduce Prandtl BL assumption• pressure is const. across layer• leading viscous term only

Boundary Layer Eqns.

Reynolds Equations(sometimes called N-S)

Euler Equations

Thin Layer N-S Eqns.

note: aeros 1. drop body force terms2. use divergence form

• onset flow uniform• shocks are weak (Mn<1.25)

Potential or FULL Potential Eqn.(Gas Dynamics Equation)

Irrotational FlowV=∇Φ

incompressible flow

Laplace's Eqn.

small disturbance approx

sub/super & trans, incl.P-G & TSDE Eqns.

(includes integral egn.representation)

Figure 3.1: Several degrees of approximation of the General Governing Navier-Stokes equations andtheir origin [30].

going to be followed and coupled with a VII procedure. This enhances the computational efficiency of

the developed code while not dismissing all of the effects of viscosity. The derivation of both an Inviscid

Panel Method formulation and a Boundary Layer Integral model are presented in the following sections.

3.2 Inviscid flow

3.2.1 Governing Equations

Next, the Governing equations for the Inviscid Flow are derived following [31]. Taking the incompressible

flow approximation, the Continuity Equation 3.1 simplifies to Equation 3.7.

∇ · q = 0. (3.7)

If the flow is irrotational within a region, then the velocity at an arbitrary point can be obtained from

the gradient of the velocity potential, Φ, as show in Equation 3.8,

q = ∇Φ, (3.8)

which, when substituted into the continuity equation (3.7) leads to what is know as Laplace’s equation

(3.9):

19

∇ · q = ∇2Φ = 0, (3.9)

This linear differential equation is an extension of the continuity equation for an incompressible,

inviscid and irrotational fluid.

For a solution to be obtained, appropriate boundary conditions still have to be enforced in all surfaces

and at the infinity. If the viscosity had not been neglected, the usual boundary condition at the surface

would be the no-slip condition, q = 0. However, in the inviscid case, this condition can only be enforced

in the normal direction to the surface, if we consider a body-fixed reference frame. Respecting the

condition at infinity, the velocity disturbance q created by the body movement in relation to the fluid must

decay to zero with increasing distance to the body. Equations 3.10 and 3.11 translate, respectively,

these conditions, where q∞ is the free flow undisturbed velocity far away from the body.

q · n = ∇Φ · n = 0 at the body surface, (3.10)

q− q∞ = ∇Φ− q∞ → 0 far from the body. (3.11)

With the solution for the potential (and consequently also the velocity) obtained from the Laplace

equation, pressure can be obtained recalling the Navier-Stokes Equation (3.4). Making the same as-

sumption of the inviscid flow of an incompressible fluid, the Euler momentum equation (3.12) can be

obtained as follows.

∂q

∂t+ q · ∇q = f − ∇p

ρ, (3.12)

Writing the convective acceleration term by introducing the concept of vorticity, ζ, the Euler momen-

tum equation becomes Equation 3.13.

q · ∇q = ∇q2

2− q× ζ. (3.13)

Taking into account that the vorticity is zero in an irrotational flow, ζ = 0, rewriting the derivative of

the velocity q in order to the potential Φ and defining the body force f as a conservative force, as gravity

aligned with the z axis, for example, the Bernoulli equation (Equation 3.14) is obtained

∇(gz +

p

ρ+q2

2+∂Φ

∂t

)= 0, (3.14)

3.2.2 Basic flow solutions

Elementary analytic solutions to the Laplace equation (3.9) can be established, as derived and pre-

sented in [31]. These will be useful when developing the inviscid aerodynamic models because of the

superposition property enabled by the linear nature of such flow fields. This way, the elementary solu-

tions presented next can be systematically disposed as needed to account for generic body geometries.

The Point Source is one of the basic solutions to the Laplace equation. This singularity creates a

20

flow field in the radial direction starting at its origin, whose induced velocity decays with 1/r2. Equation

3.15 describes the potential created by a source with strength σ at the distance r from its position.

Φ = − σ

4πr(3.15)

Another basic solution is the Point Doublet, which can be obtained by combining a source and a sink

letting the distance between them approach 0. This results in a directional field pointed in the direction

of n and with strength µ, as shown in Equation 3.16.

Φ =µ

4πn · ∇(

1

r) (3.16)

Polynomials showing a linear relation of the velocity potential with position are also solution, repre-

senting the potential of an uniform free-stream flow in a specified direction. Equation 3.17 illustrates

such a solution in a Cartesian coordinate system,

Φ = Ax+By + Zz, (3.17)

where A, B and C are constants that can be interpreted as the free-stream velocity in the x, y and z

directions, respectively.

This type of reasoning can be done not only to punctual singularities, but also to singularity distribu-

tions along lines, useful for example in describing the flow around a rotating cylinder and surfaces, or

along sheets which, in the case of a vortex sheet positioned close to a surface could, for example, be

used to enforce a no-slip boundary condition at a surface, although ignoring the velocity field inside the

Boundary Layer [32].

Using the superposition of such elementary singularities several methods have been developed that

allow to model the inviscid irrotational flow around a generic solid body shape. Some of the most used

methods specifically for the case of 3D Finite Wings are discussed next.

The simplest method available is the Lifting Line model. In this case, the wing is modeled by a single

connected vortex filament positioned along the aerodynamic center of the wing with continuous variable

intensity, producing a continuous vortex sheet from the trailing edge [32]. However, this type of model

cannot account for variations of circulation along the airfoil or wing chord, neither account for dihedral or

sweep.

In response to this, a Lifting Surface model can be achieved by distributing connected vortex along

a curved or chord surface and not only along the span. If these are distributed along the discretized

surface of a wing, a VLM is being employed. However, this kind of model does not account for airfoil

thickness and can only be used for small angles of attack.

The following logical step would consist in discretization the actual body or wing surface into panels

and distributing the singularities in each panel. This is the concept behind Panel Methods, both in

2D (for airfoils) and 3D (for wings and arbitrary bodies). This allows to model complex shapes with a

straightforward procedure. This kind of method will be used to model inviscid flows in this work due to

21

its generality and ease of application to existing generic geometries. It is also a method that can fairly

easily be integrated in a VII procedure. Figure 3.2 shows an example of a discretized aircraft geometry

for use with a Panel Method.

Figure 3.2: Example of discretized aircraft geometry for use with a Panel Method. (adapted from [31])

3.2.3 Panel Methods

As exposed before, Equation 3.7 is a possible description of an inviscid and irrotational fluid flow. Panel

Methods are numerical tools for finding a solution to this equation by superimposing elementary flows

created by singularities disposed in space, as exposed in the previous subsection. This way, and be-

cause of the linear behavior of a flow with such assumptions, complex geometries can be easily modeled

by disposing singularity flows over the airfoil or body surface, for example.

There are several kinds of Panel Methods, depending on the type of body being modeled (lifting or

non-lifting), combinations of types of singularities used (vortex, source and/or doublets) and on the order

of the singularity strengths used (constant, linear or higher orders). Two types of boundary conditions

are also available, Newman (zero normal velocity at the surface) and Dirichelet (constant potential inside

the surface).

Another essential point regard modeling the wake, which is the way to enforce the Kutta condition

in the trailing edge. Several wake models can also be used to take into account the direction the wake

takes after the trailing edge as well as wake relaxation and roll-up [32], as its geometry varies with

increasing distance to the body.

The method used in this work is the Constant-Strength Combined Source and Doublet method pre-

sented in [31], either in 2D and 3D. It was chosen because it represents a good agreement between

complexity, generality and computational time, as well as facilitating a posterior Viscous-Inviscid Interac-

tion procedure by the modification of the source strengths.

Taking this into account, a solution for the Laplace equation (3.9) can be built. The representation of

the main definitions needed is done in Figure 3.3.

22

Figure 3.3: Potential flow over closed body and definitions.[31]

For a distribution of source, σ, and doublet, µ, singularity strengths along surface Sb and the free

stream potential Φ∞ = u∞x+ u∞y + u∞z, the total potential Φ∗ at any point (x, y, z) in the domain V is

given by Equation 3.18.

Φ∗(x, y, z) =1

4π

∫body+wake

µn · ∇(

1

r

)dS − 1

4π

∫body

σ

(1

r

)dS + Φ∞, (3.18)

Sources are associated with modeling the airfoil thickness effect and therefore are going to be dis-

tributed only along the airfoil surface, while Doublets create the circulation that provides lift and will be

distributed not only along the body surface but also along the trailing edge wake to give continuity to the

vorticity present at the trailing edge. The wake is modeled with panels as well and must extend along

several wing chords behind the trailing edge for good approximation [31].

The chosen boundary condition to be applied at the surface was the Dirichlet Boundary Condition

which enforces the interior potential Φ∗i to be set to 0 or any constant value. By enforcing Φ∗i (x, y, z) =

Φ∞, Equation 3.18 simplifies and can equivalently reduced to the relation presented in Equation 3.19.

1

4π

∫body+wake

µn · ∇(

1

r

)dS − 1

4π

∫body

σ

(1

r

)dS = 0, (3.19)

This assumption also sets the source strength values along the wing surface (Equation 3.20), leaving

the only doublet strengths to be discovered.

σ = n ·Q∞ (3.20)

The wake doublet distribution strength, is dictated by the Kutta Condition at the Trailing Edge. The

condition dictates that the double distribution strength in the wake, µw, must be constant and equal to

the strength at the trailing edge. As shown in Figure 3.4, this condition translates into Equation 3.21.

µw = µu − µl, (3.21)

where µu and µl are respectively the doublet strengths at the last upper and lower panels along the

trailing edge.

Regarding the shape of the wake, represented by a thin doublet sheet, the wake panels should be

aligned with the flow [31]. However, as velocity distribution is not known beforehand either the wake is

assumed to be fixed and aligned in a specific direction or an iterative procedure is required to align the

23

Figure 3.4: Representation of the Kutta condition on a doublet panel method.[31]

wake panels. Figure 3.5 shows two fixed rectilinear wake shapes, a in the direction of the external flow

and b in the direction of the upper trailing edge panel, and a converged wake shape convected by the

flow over an iterative procedure, c. The resultant Lift coefficient CL and Drag coefficient CD for a finite

wing with an Aspect Ratio of 1.5 taken from [31] show how the aerodynamic coefficients vary with the

different assumed shapes, with the best approximation with experimental results being Wake shape c.

In the same reference the author considers that in most cases the assumption that the wake leaves the

trailing edge along the median line between the upper and lower trailing edge panels and aligns with the

outer flow as the distance from the trailing edge increases is sufficient.

Figure 3.5: Examples of prescribed wake shapes and their effect on the resultant aerodynamic coeffi-cients. [31]

The body can now be divided into N surface panels and NW wake panels. Taking Equation 3.19 and

introducing the discretization of the surface in panels, it gives origin to Equation (3.22).

N∑k=1

1

4π

∫body panel

µn · ∇(

1

r

)dS +

NW∑j=1

1

4π

∫wake panel

µn · ∇(

1

r

)dS −

N∑k=1

1

4π

∫body panel

σ

(1

r

)dS = 0,

(3.22)

24

The integral terms in Equation 3.22 define the influence of one specific singularity distribution dis-

posed along a given panel in any point inside V . By summing the influence of all panels it is possible

therefore to determine the potential in any point due to the entire surface and wake. However, as dou-

blet strengths of each panel are still unknown, the boundary condition has to be applied as described at

each internal point in order to correctly define the geometry seen by the flow. This is enforced at every

Collocation Point, positioned at the center of each panel that defines the body surface, slightly deviated

towards the inside of the airfoil/wing.

Employing constant-strength singularity panels, the singularity strengths µ and σ can be taken out-

side of the integrals in Equation 3.22 and the remaining integral terms become only dependent on the

geometry of the panel and the position of the point being evaluated, giving origin to the Influence Coef-

ficient matrices Ck, Cl and Bk. Equation 3.23 must then be solved for each Collocation Point.

N∑k=1

Ckµk +

NW∑l=1

Clµl +

N∑k=1

Bkσk = 0 (3.23)

This way, zero-order (constant) singularity strength distributions simplifies the implementation and

solution while good results can still be achieved if a relatively high number of panels is used as compared

to higher order methods [31].

Equation 3.23 can still be arranged taking into account that the source strengths are all know and

wake doublet strength distribution can be obtained of the surface doublets, in order to facilitate the

solution procedure for the doublet strengths. Defining a new Influence Coefficient matrix Ak,

Ak =

Ck if the panel does not belong to the Trailing Edge

Ck ± Ct if the panel belongs to the Trailing Edge, (3.24)

where Ct is the influence coefficient of the first wake panels, the simplified Equation 3.25 can be ob-

tained.

N∑k=1

Akµk = −N∑k=1

Bkσk (3.25)

As stated before, this equation has to be evaluated at each of the N collocation points, originating a

system of N equations with N unknowns that can now be solved (3.26).

a11 a12 ... a1N

a21 a22 ... a2N...

......

aN1 aN2 ... aNN

µ1

µ2

...

µN

=

b11 b12 ... b1N

b21 b22 ... b2N...

......

bN1 bN2 ... bNN

σ1

σ2...

σN

(3.26)

Generalizing for all kinds of panel methods and adapting from the nomenclature used in [33], this

system of equations can be represented as

[AIC] {λ} = {RHS} , (3.27)

25

where [AIC] is the Aerodynamic Influence Coefficients matrix, {λ} is the vector of the unknown singu-

larity strengths and {RHS} is the Right-Hand Side containing the known parameters, in this case the

influence coefficients multiplied by the source strengths obtained from the boundary conditions.

There is only one step missing to allow solving the system, as influence coefficients are still defined

in a generic way. As derived in [31], the velocity potential evaluated at an arbitrary point P = (x, z),

generated by a 2D rectilinear panel with a constant strength source distribution is given by

Φ =σ

4π

{(x− x1) ln[(x− x1)2 + z2]− (x− x2) ln[(x− x2)2 + z2] + 2z

(tan−1

z

x− x2− tan−1

z

x− x1

)},

(3.28)

and with a constant strength doublet distribution

Φ =−µ2π

(tan−1

z

x− x2− tan−1

z

x− x1

), (3.29)

Similarly, for a 3D body, the velocity potential can be evaluated at an arbitrary point P = (x, y, z), gen-

erated by a quadrilateral rectilinear panel with a constant strength source distribution and with a constant

strength doublet distribution. The equations used are presented in [31, 34] and are not presented here

not to lengthen the discussion.

With the solution to the problem available, the aerodynamic parameters can be obtained by integra-

tion of the pressure distribution along the surface following the procedure exposed next, following the

nomenclature presented in [31].12.5 First-Order Potential-Based Panel Methods 357

Figure 12.25 Nomenclature used for the differentiation of the velocity potential for local tangentialvelocity calculations.

solvers are used so that only one row of the matrix occupies the computer memory duringthe solution.

f. Computation of Velocities, Pressures, and LoadsOne of the advantages of the velocity potential formulation is that the computation

of the surface velocity components and pressures is determinable by the local properties ofthe solution (velocity potential in this case). The perturbation velocity components on thesurface of a panel can be obtained by Eqs. (9.26), in the tangential directions:

ql = −∂µ

∂l, qm = −

∂µ

∂m(12.37)

and in the normal direction:

qn = σ (12.37a)

where l,m are the local tangential coordinates (see Fig. 12.25). For example, the perturbationvelocity component in the l direction can be formulated (e.g., by using central differences)as

ql =12 l

(µl−1 − µl+1) (12.38)

where l is the panel length in the l direction. In most cases the panels do not have equalsizes and instead of this simple formula, a more elaborate one can be used (sometimes onlythe term l is modified). The total velocity at collocation point k is the sum of the freestream plus the perturbation velocity:

Qk = (Q∞l , Q∞m , Q∞n )k + (ql , qm, qn)k (12.39)

where lk,mk, nk are the local panel coordinate directions (shown in Fig. 12.25) and of coursethe total normal velocity component on the surface is zero. The pressure coefficient cannow be computed for each panel using Eq. (4.53):

Cpk = 1−Q2k

Q2∞

(12.40)

The contribution of this element to the aerodynamic loads Fk is

Fk = −Cpk

12ρQ2

∞

Sknk (12.41)

12.5 First-Order Potential-Based Panel Methods 357

Figure 12.25 Nomenclature used for the differentiation of the velocity potential for local tangentialvelocity calculations.

solvers are used so that only one row of the matrix occupies the computer memory duringthe solution.

f. Computation of Velocities, Pressures, and LoadsOne of the advantages of the velocity potential formulation is that the computation

of the surface velocity components and pressures is determinable by the local properties ofthe solution (velocity potential in this case). The perturbation velocity components on thesurface of a panel can be obtained by Eqs. (9.26), in the tangential directions:

ql = −∂µ

∂l, qm = −

∂µ

∂m(12.37)

and in the normal direction:

qn = σ (12.37a)

where l,m are the local tangential coordinates (see Fig. 12.25). For example, the perturbationvelocity component in the l direction can be formulated (e.g., by using central differences)as

ql =12 l

(µl−1 − µl+1) (12.38)

where l is the panel length in the l direction. In most cases the panels do not have equalsizes and instead of this simple formula, a more elaborate one can be used (sometimes onlythe term l is modified). The total velocity at collocation point k is the sum of the freestream plus the perturbation velocity:

Qk = (Q∞l , Q∞m , Q∞n )k + (ql , qm, qn)k (12.39)

where lk,mk, nk are the local panel coordinate directions (shown in Fig. 12.25) and of coursethe total normal velocity component on the surface is zero. The pressure coefficient cannow be computed for each panel using Eq. (4.53):

Cpk = 1−Q2k

Q2∞

(12.40)

The contribution of this element to the aerodynamic loads Fk is

Fk = −Cpk

12ρQ2

∞

Sknk (12.41)

Figure 3.6: Panel local Reference System and neighbor panels for velocity computation. [31]

The surface velocities can be directly computed from the singularity strength distribution. In the

local panel Reference System for the 3D case, as represented in Figure 3.6, the perturbation velocity

components can be calculated from the derivatives of the doublet strength in each tangential direction (l

and m) and the source strength in the surface normal direction (n), as shown in Equations 3.30.

ql = −∂µ∂l

, qm = − ∂µ∂m

, qn = σ. (3.30)

When applying central differences to compute the derivatives in Equations 3.30,the pertubation ve-

26

locities can be approximated by Equations 3.31.

ql =1

2∆l(µl−1 − µl+1) qm =

1

2∆m(µm−1 − µm+1) (3.31)

The total velocity at a given k panel collocation point in local coordinates (l,m, n)k can then be

calculated by summing the perturbation velocity components with the free stream velocity components,

as shown in Equation 3.32.

Qk = Q∞(l,m,n)k+ q

(l,m,n)k(3.32)

With the velocity now known, the Pressure Coefficient Cp at each panel collocation point k can be

calculated from Equation 3.33.

Cpk = 1− Q2k

Q2∞

(3.33)

As it can be seen from the previous discussion, compressibility was not taken into account in the

Inviscid Flow model. In fact, in Equation 3.12, the assumption of incompressibility was taken. Therefore,

to account for low-speed compressibility, the Prandtl-Glauert Rule (Equation 3.34) is used for Mach

speeds up to 0.6 [31]. This correction takes the incompressible solution CpM∞=0and divides it by a

compressibility factor.

CpM∞>0=CpM∞=0

β, (3.34)

where the compressibility factor β is defined as

β =√

1−M2∞. (3.35)

From this result, the resultant Aerodynamic Force generated by each panel within the body surface

is given by Equation 3.36

∆Fk = −1

2ρQ2∞SkCpknk, (3.36)

where Sk is the Panel Area and nk is the Normal vector to the panel k.

Summing the contributions from all the wing surface panels in the Body Reference system, the

Resultant Aerodynamic Force , R can be obtained. With this information it is then possible to obtain the

forces in the Aerodynamic Reference System (aligned with the Free Flow, see Figure 3.7) to obtain the

Lift and Drag forces, L and D (Equations 3.37b and 3.37a), and the Moment around the leading edge,

M0 (Equation 3.37c).

D = FZ cos(α)− FX sin(α), (3.37a)

L = FX cos(α) + FZ sin(α), (3.37b)

27

M0 = −N∑k=1

∆Fk(x0k cos γk), (3.37c)

where α is the Angle of Attack of the wing relative to the free flow and x0k is the distance between the

wing/airfoil leading edge to the center of the panel k along the body XB axis, and γk is the angle the k

panel does with XB .

Figure 3.7: Projection of the Resultant Force Coefficients on the Body and Aerodynamic ReferenceSystems [35]

It is important to note that the resultant Drag obtained by this method does not take into account

the contribution of viscous effects, such as skin friction, taking only into account the pressure drag. In

fact, when analyzing a 2D Geometry, e.g. an airfoil, this value will be zero for an inviscid flow, which

is a problem known as the d’Alembert paradox. In a 3D Geometry the Drag obtained by this method

corresponds to the induced drag,Di, resultant from the creation of trailing vortices. This Drag component

can be reduced by increasing the Aspect Ratio of a wing as a result of the better distribution of lift along

the span, reducing the vortex generating pressure gradients that occur along the trailing edge and, more

significantly, at the wingtips.

The Total Wing Drag, D, can therefore be evaluated as the sum of the Pressure and Friction compo-

nents 3.38. The Calculation of the Friction Drag will be explained in the following sections regarding the

Viscous Flow models used.

D = Di +D0 (3.38)

Taking the usual adimensionalizations, the Lift, Drag and Moment Coefficients, Cl, Cd and CM0

(Equations 3.39b, 3.39a and 3.39c ), can then be obtained.

CD =D

12ρQ

2∞S

, (3.39a)

CL =L

12ρQ

2∞S

. (3.39b)

CM0 =M0

12ρQ

2∞cS

. (3.39c)

28

where S is the wing planform surface area and c is the Mean Aerodynamic Chord.

Summarizing, the methodology used to obtain a solution for the flow around and airfoil or 3D surface

is as follows:

1. Discretize the body surfaces into panels;

2. Create source and doublet distributions along panels with unknown strength λ;

3. Determine the Influence-Coefficient expressions describing the influence of each panel’s singular-

ities distribution in the potential at each of the other panels;

4. Apply Boundary Conditions at each Collocation Point using the influence coefficient matrix [AIC],

building the following system of equations: [AIC]{λ} = {RHS};

5. Solve the system for the unknown singularity strengths {λ} and obtain velocity and potential distri-

bution;

6. Compute panel pressures from velocity;

7. Obtain aerodynamic forces and moments through the integration of the pressure distribution.

3.3 Viscous Flow

3.3.1 Effects of viscosity

Before the discovery of the barely visible Boundary Layer by Prandtl, the d’Alembert Paradox stated

that in an incompressible, inviscid and infinite flow, a 2D body presents no drag. The appearance of

the boundary layer due to the no-slip condition and its consequences are a consequence of the viscous

effects in the flow around bodies. It’s not only the source of Drag 2D bodies but also a significant source

in 3D bodies. Its existence also affects the other aerodynamic forces and loads. The displacement effect

of the Boundary Layer on the external flow is known for reducing camber in airfoils and increasing it’s

apparent thickness to the external flow. Boundary Layer separation which may induce stall in aircraft

wings when severe, is also a viscous effect.

All the described effects affect the flight characteristics of aircraft and similarly any kind of interaction

between such effects and the aeroelastic response of the structure may also be relevant, even when

steady analysis is being done.

3.3.2 Governing equations

Recalling the Navier-Stokes equations (3.4) and the continuity equation (3.1), this system of equations

fully describes the flow field at constant properties [32], with the unknowns being the flow velocity, q,

and the pressure, p.

As stated in previous sections, there is only a reduced number of solutions known to this problem and,

as so, different approximations to these equations can be made according to the fidelity and type of flow

29

being studied. One of these approximations, the Thin Shear Layer (TSL) Approximations, also known

as Boundary Layer Approximations, are designed to model the fluid behavior near walls, wakes, mixing

layers or jets [36]. This kind of flows usually present strong transversal velocity gradients and occur within

a very thin region when compared with its length, i.e. the boundary layer thickness, δ, is several orders

of magnitude smaller than the length it occupies, L. Using these assumptions, a dimensional analysis

can be performed to determine the relative scale of the various terms in the Navier-Stokes equations

(3.4), as exposed by several authors [31, 36, 32]. As exposed there, along with the assumptions already

presented, the most important one regards the pressure across the boundary layer thickness, which

is taken as constant. Therefore, pressure gradients along the surface normal are neglected and the

pressure at the wall is assumed to be the pressure acting on the Boundary Layer edge.

This type of approach was first presented in 1904 by Prandtl in his revolutionary paper, Fluid Flow in

Very Little Friction. The resulting equations for a steady 2D flow field are presented next and are known

as the Thin Shear Layer or Prandtl Boundary Layer equations and are here expressed in a local ξ and η

reference system, respectively in the local streamwise and wall normal directions [23, 24, 37]. Equation

3.40 derives from the continuity,

∂(ρu)

∂ξ+∂(ρv)

∂η= 0. (3.40)

Equation 3.41a derives from the momentum conservation,

ρu∂u

∂ξ+ ρv

∂u

∂η= ρeue

∂ue∂ξ

+∂τ

∂η, (3.41a)

where τ is given by the sum of the total shear plus Reynolds stresses,

τ = µ∂u

∂η− ρu′v′. (3.41b)

Equation 3.42a derives from enthalpy conservation,

ρu∂ht∂ξ

+ ρv∂ht∂η

=∂Q

∂η, (3.42a)

where the enthalpy flux Q is given by

Q =µ

Pr

∂ht∂η

+ µ

(1− 1

Pr

)µ∂u

∂η− ρh′v′, (3.42b)

and Pr is the Prandtl Number. The overlined components in the previous equations represent the

Reynolds Stresses

Other approximations could have been taken. Taking for example the well known assumption of

negligible vertical pressure gradients in the Thin Shear Layer equations in its first-order form, these

usually present a reasonable assumption due to the small Boundary Layer thickness. Other authors

[38] have developed other kind of Integral Boundary Layer equations with different assumptions in which

the vertical pressure gradient is not neglected, among other approximations. According to the author,

30

that higher-order version of the equations helps correctly predicting Boundary Layer separation, among

other advantages.

3.3.3 Integral Boundary Layer Method

Direct solutions to the TSL equations for the specific cases of flat-plate and stagnation point boundary

layers were discovered by Blasius and Hiemenz by employing the Similarity Assumption. By definition,

a flow is considered similar if ”the velocity profile at any streamwise station, scaled by the velocity in the

outer flow, can be represented as a function of a suitably scaled transverse coordinate” [31]. However,

this kind of assumption can be restraining, as the velocity profile along the boundary layer length can

change significantly [31] and in many cases it may not even be of interest, as is the case of design

problems where only global effects such as boundary layer displacement and wall shear stresses are

relevant.

For this kind of analysis the flow parameters inside the boundary layer can be obtained by integrating

the Prandtl equations at each streamwise station for all field, as done for a 3D Boundary Layer around

non-axissimetric non-lifting bodies by Pereira and Andre [28]. Another way to do this is using an integral

form of the boundary layer equation, such as the Von Karman momentum integral used by Drela [23, 24,

37], which is the approach followed in this work. Simmilar approches were also followed by Stalewski

and Sznajder [27], Lock [38] and in VSAERO [25].

As stated before, the Von Karman equation 3.43 appears from the integration of the Thin Shear layer

momentum equation 3.41a.

ξ

θ

dθ

dξ=ξ

θ

Cf2−(δ∗

θ+ 2−M2

e

)ξ

ue

duedξ

, (3.43)

where δ∗ and θ are the displacement and momentum thicknesses and Cf is the skin friction coefficient,

defined next in Equations 3.44a through 3.44c.

δ∗ =

∫ ∞0

(1− ρu

ρue

)dη (3.44a)

θ =

∫ ∞0

(1− u

ue

)ρu

ρuedη (3.44b)

Cf =2

ρeu2eτw (3.44c)

By defining a few more integral parameters, density thickness δ∗∗, kinetic energy thickness θ∗ and

dissipation coefficient CD, given respectively by

δ∗∗ =

∫ ∞0

(1− ρ

ρe

)u

uedη, (3.45a)

31

θ∗ =

∫ ∞0

(1−

(u

ue

)2)ρu

ρuedη, (3.45b)

CD =2

ρeu3e

∫ ∞0

τ∂u

∂ηdη, (3.45c)

and shape parameters H, H∗ and H∗∗,

H =δ∗

θ, (3.46a)

H∗ =θ∗

θ, (3.46b)

H∗∗ =δ∗∗

θ, (3.46c)

the shape parameter equation (3.47) can then be obtained by multiplying by u and integrating.

ξ

H∗dH∗

dξ=ξ

θ

2CDH∗− ξ

θ

Cf2−(

2H∗∗

H∗+ 1−H

)ξ

ue

duedξ

(3.47)

The Von Karman equation (3.43) can be rewritten as in Equation 3.48.

ξ

θ

dθ

dξ=ξ

θ

Cf2−(H + 2−M2

e

) ξue

duedξ

. (3.48)

As the system constituted by Equations 3.47 and 3.43 still possesses more than 2 independent

variables, closure relations will have to be included, introducing assumptions depending on the kind o

flow (Laminar, Turbulent or Transition). By defining δ and θ as dependent variables, the remaining ones,

H∗, H∗∗, Cf and CD, will have to be modeled with such relations.

Before introducing the Closure Relations there is the need to define one more parameter, the kine-

matic shape parameter, Hk, which will serve as input for the closure relations, while accounting for

compressibility, relating H and the Mach number, M . The relation present next (3.49) was proposed by

Whitfield and used in [23, 24, 37].

Hk =H − 0.290M2

e

1 + 0.113M2e

, (3.49)

For Laminar Flow, taking into account the Falkner-Skan one-parameter profiles, Drela [39] obtains

the expressions bellow.

H∗k =

1.515 + 0.076 (4−Hk)2Hk

for Hk < 4

1.515 + 0.040 (Hk−4)2Hk

for Hk > 4

(3.50a)

H∗ =H∗k + 0.028M2

e

1 + 0.014M2e

(3.50b)

32

ReθCf2

=

−0.067 + 0.01977 (7.4−Hk)2Hk−1 for Hk < 7.4

−0.067 + 0.022(

1− 1.4Hk−6

)2for Hk > 7.4

(3.51)

Reθ2CDH∗

=

0.207 + 0.00205(4−Hk)5.5 for Hk < 4

0.207− 0.003 (Hk − 4)2 for Hk > 4

(3.52)

Also from Whitfield, an expression for the density thickness parameter H∗∗ is presented next (3.53),

which will be used for both turbulent and laminar flows, due to its limited effect in transonic flows and

negligibility in subsonic ones.

H∗∗ =

(0.064

Hk − 0.8+ 0.251

)M2e , (3.53)

As for turbulent flow, the non-adequacy of a one-parameter velocity profile family due to the oc-

currence of a two-layer structure and to the persistence of the Reynolds stresses in the Momentum

Equation (Equations 3.41a and 3.41b) led to the necessity of applying a skin friction coefficient relation

as presented in [23], specifically in this case the one developed by Swafford and presented in Equation

3.54.

FcCf = 0.3e−1.33Hk[log10

(ReθFc

)]−1.74−0.31Hk+ 0.00011

[tanh

(4− Hk

0.875

)− 1

], (3.54)

with

Fc = (1 + 0.2M2e )

12 . (3.55)

Regarding the energy thickness shape parameter H∗ in turbulent flow, Drela [23] used Swafford’s

expression for the velocity profile to derive the closure relation presente in Equation 3.56.

H∗k =

1.505 + 4

Reθ+(

0.165− 1.6√Reθ

)(H0−Hk)1.6

Hkfor Hk < H0

1.505 + 4Reθ

+ (Hk −H0)2

[0.04Hk

+ 0.007 log(Reθ)(Hk−H0+

4ln(Reθ)

)2

]for Hk > H0

(3.56)

with,

H0 = 3 +400

Reθ, (3.57)

As for CD, Clauser used an approach which refers to an equilibrium boundary layer concept a a

way to establish a parallelism with laminar flow Falkner-Skan profiles. As developed in [23, 24], the

dissipation coefficient comprehends the sum of two contributions, one from a wall layer and another

from a wake layer (Equation 3.58 ).

CD =Cf2Us + Cτ (1− Us), (3.58)

where Cτ represents the magnitude of the shear stresses in the wake and the normalized wall slip

33

velocity Us is given by Equation 3.59.

Us =H∗

2

(1− 4

3

Hk − 1

H

), (3.59)

Cτ is obtained by solving a rate equation (3.60) derived from Green’s Lag Entrainment method used

in [23, 24] which introduces an history effect into the boundary layer that models the lagged response of

the wake layer to the local conditions [23].

δ

Cτ

dCτdξ

= 4.2(CτEQ1/2 − C1/2

τ ), (3.60)

where the nominal boundary-layer thickness δ and the equilibrium shear stress coefficient CτEQ are

given by:

δ = θ

(3.15 +

1.72

Hk − 1

)+ δ∗, (3.61)

CτEQ = H∗0.015

1− Us(Hk − 1)3

H2kH

(3.62)

During the computation of the laminar stages as described above, there is still the need to detect the

point where transition occurs. To do so, the evaluation of the amplification of small disturbances is going

to be made using the e9 method [23, 24]. The amplification of small 2D disturbances is the immediate

precursor to the onset of transition [32] and, in this case, this is assumed to occur when the amplitude

of the disturbances grows by a factor of e9, i.e. when n = 9. This is achieved by solving Equation 3.63

along with Equations 3.47 and 3.48.

dn

dξ(Hk, θ) =

dn

dReθ(Hk)

m(Hk) + 1

2l(Hk)

1

θ, (3.63)

where m(Hk) and l(Hk) are the empirical relations,

l(Hk) =6.54Hk − 14.07

H2k

, (3.64)

m(Hk) =

(0.058

(Hk − 4)2

Hk − 1− 0.068

)1

l(Hk), (3.65)

Regarding the computation of Boundary Layer parameters during the transition phase (or in the

station where transition is detected to occur) this is simply done by doing a weighted average of the

solutions for laminar an turbulent flows due to the similarity of both solutions [23]. Therefore, as an

example, H∗ is calculated as follows:

H∗ = (1− γtr)H∗laminar + γtrH∗turbulent, (3.66)

where the weighting factor, γtr, is defined as

γtr =ni + 9dndξ i

1

ξi − ξi−1. (3.67)

34

For the computation of the parameters on the wake, behind the trailing edge, the relations used until

here are still valid. The wake is assumed to always be turbulent and the condition of Cf = 0 is enforced.

Regarding the estimation of Drag, the contribution of the Boundary Layer is accounted in two different

forms. Firstly, the displacement effect along the wing surface and wake alters their effective shape to

the Inviscid portion of the flow, therefore altering the pressure distribution and indirectly altering the

aerodynamic coefficients obtained by the inviscid assumptions. The second way it is taken into account

is through the integration of the shear stress along the skin, τw. The friction Drag can therefore be

estimated as the sum of the contributions on each station’s panel k, along the flow direction, as shown

in Equation 3.68,

D0 = −N∑k=1

τwSk cos(γk + α), (3.68)

where γk is the angle the k panel does with XB (see Figure 3.7).

This kind of approach, involving the integration of pressure and shear along the wing is known as

the Near Field approximation [30]. Although being a relatively simple approach, it requires a good

precision when executing the integration. Also it is not ideal when trying to relate the results to perform

aerodynamic analysis, as the specific sources of drag are not outlined.

3.4 Viscous-Inviscid Interaction

It is frequently assumed that Viscous Flow effects vanish in external flows with the increase in distance

from a surface [32, 38]. This external region presents velocity gradients sufficiently small so that viscous

effects become negligible in comparison with the inviscid ones. On the other hand, on thin layers close

to a surface or inside a wake viscous effects are not negligible due to the existence of predominant

velocity gradients.

These viscous flows can be modeled by Prandtl’s Boundary Layer Equations [40], however, the

inviscid external region and the viscous boundary layer cannot be independently solved, as they interact

with each other. The evolution of the boundary layer is affected by the external pressure distribution

(and consequently velocity), as well as the opposite is also true. The boundary layer displacement

effect, created by the momentum deficit that occurs in result to the non-slip boundary condition, creates

a displacement body to the eyes of the inviscid flow, which smooths and thickens the original geometry.

With this mutual interaction between flows and taking into account the exposed difficulties in solving

higher-order flow models for complex geometries, Viscous-Inviscid Interaction techniques appeared by

coupling Inviscid models for the external flows with Viscous models for the boundary layer and wake.

There are several ways to do this.

The first one is the Classical approach, where the flow is solved using the inviscid model and then

the surface velocity is used as input for the boundary layer solver. The boundary layer parameters are

calculated but no update to the geometry which is subject to the inviscid calculation is done.

The Direct Method takes the Classical one and establishes an iterative procedure which is able to

35

?geometry�

���Inviscid Solver

?ue�

���Boundary Layer Solver

?δ∗

Figure 3.8: Classical VII method.

account for second order effects by updating the geometry of the body with the boundary layer displace-

ment thickness, δ∗ calculated in the previous iteration [41]. However, when separation occurs the model

exhibits a singular solution known as the Goldstein Singularity [42], not being able to correctly model

such flows [40, 41]. This occurs mainly at the trailing edge where the strong adverse gradients tend to

cause separation.

?

geometry+

��

��Inviscid Solver

?ue�

���Boundary Layer Solver

δ∗

- �� ��?

δ∗

Figure 3.9: Direct VII method.

A solution to this problem is solving the boundary layer flow with prescribed displacement thickness

instead of external velocity. This was firstly successfully accomplished in the late 60’s [40], when a

computation past the separation point was achieved. In an Inverse Method, not only the boundary layer

is solved inversely, but also the inviscid flow, which creates difficulties when trying to achieve this with

Euler codes, for example [40].

Therefore, a Semi-Inverse Method was proposed by Le Balleur which is used in [40]. In this method

the boundary layer is solved through an inverse procedure while the inviscid flow is solved in a direct

way. With the generation of 2 output velocities, one form the boundary layer solution and the other from

the inviscid one, a relation has to be used to relate them and refresh the displacement thickness used

as input for the next iteration.

Another kind of approach is a Quasi-simultaneous method, where a simple approximation to the

inviscid flow, called Interaction Law, is solved together with the boundary layer equations, which is up-

36

��

��Inviscid Solver

6ue�

���Boundary Layer Solver

6δ∗

δ∗

Figure 3.10: Inverse VII method.

��

��Inviscid Solver-

uPMe��

��δ∗(n) = δ∗(n−1) + ω

{uBLe − uPMe

}�δ∗

��

��Boundary Layer Solver-

uBLe

Figure 3.11: Semi-inverse VII method.

dated with the results from the more accurate inviscid model. According to Veldman [40] this procedure

accelerates the convergence and bypasses the Goldstein singularity problem.

Drela [23] uses a Fully-Simultaneous approach in which both systems of equations are solved si-

multaneously, which is usually preferable when a integral approach is used but impracticable when both

viscous and inviscid flow are modeled using a full field formulation [40].

When updating the geometry with the boundary layer displacement, two approaches are possible.

The first one involves altering the actual geometry of the body by adding the boundary layer displace-

ments calculated to the nodal positions, for example. However, this approach is not commonly used

because it implies eventually changing computational grids. The most common option is including tran-

spiration velocity normal to the body surface on the inviscid model. This is the most used option.

The transpiration velocity w can be obtained from the evolution of the inviscid velocity on the edge of

the boundary layer, ue, and the displacement thickness δ∗ as shown by Equation 3.69.

w(x, 0) =d(ueδ

∗)

dx, (3.69)

This velocity can be then included in a Panel Method formulation by simply subtracting w from the

source strength σ at each station, displacing the inviscid flow in the amount dictated by the momentum

deficit caused by the presence of a boundary layer.

37

Regarding the estimation of total Drag, it can be separated into two major components as previously

referred, the Pressure Di and Friction D0 components. These are computed respectively by the Panel

Method and Boundary Layer method and just need to be added in order to obtain the total Drag force

acting on the body.

38

Chapter 4

Implementation

In this chapter, details regarding the implemented solutions for each module of the Framework are going

to be given, as well as the coupling procedures used between models.

First the implementation of the 2D panel method code used as a base to develop the 2D Boundary

layer code is presented. Both codes are compared with commercial solutions or results given by other

authors.

4.1 2D Aerodynamic solver

4.1.1 2D Panel Method

As presented in Section 3.2.3, a 2D Panel Method with Constant Doublet and Source distributions was

implemented in MATLAB. The code accepts as input the discretized geometry of a 2D Body (an airfoil,

for example) and the direction of the far-field flow and outputs the surface pressure, velocity and all other

derived forces and aerodynamic coefficients.

This implementation closely follows some examples presented in [31] and the results were compared

to assure the obtained figures were as expected.

4.1.2 Boundary Layer model

Using the implemented 2D Panel code as a base, the boundary layer model presented in Section 3.3.3,

a 2D Integral Boundary Layer solver based in the Von Karman Equation was implemented in MATLAB.

In its Direct form, the code accepts as input the discretized geometry of a 2D Body (an airfoil, for

example), the surface velocity distribution and the flow conditions such as free-stream velocity, density,

dynamic and kinematic viscosities, Mach number and the direction of the far-field flow. It outputs the

surface pressure, velocity and all other derived forces and aerodynamic coefficients. It computes as well

several Boundary layer integral parameters specified along the discretized surface, including the dis-

placement thickness, δ∗, and the friction coefficient, Cf . Figure 4.1 shows the location of the Boundary

Layer and Inviscid variables along the panels (or boundary layer stations).

39

Figure 4.1: Locations of the Boundary Layer and Inviscid variables along the discretized airfoil. (Adaptedfrom [24])

In order to be used in a computational method, Equations 3.48, 3.47, 3.63 and 3.60 have to be

discretized. The derivatives containing the dependent variables θ, δ∗ and n were approximated through

logarithmic differencing which, according to [23], minimizes discretization errors near the leading edge,

where the relation ∆ξ/ξ assumes large values.

In the discretized form, Equation 3.48 becomes Equation 4.1

ln(θ2/θ1)

ln(ξ2/ξ1)− ξaθa

Cf2

+(Ha + 2−M2

ea

) ln(ue2/ue1)

ln(ξ2/ξ1)= 0, (4.1)

where the subscript a represents the average of parameters between the two stations, in this case

stations 1 and 2. Cf is calculated for the average conditions Hka , Reθa and Mea .

The Shape Parameter Equation 3.47 becomes Equation 4.2.

ln(H∗2/H∗1 )

ln(ξ2/ξ1)+ξaθa

(Cf2− CD

)+

(2H∗∗

H∗a+ 1−Ha

)ln(ue2/ue1)

ln(ξ2/ξ1)= 0, (4.2)

where CD = 2CDH∗ and, as before, Cf , CD and H∗∗ are calculated from the average conditions.

The lag equation 3.60 is discretized differently so as not to make it ”spacially stiff” [23], originating

Equation 4.3.

2δ2

C1/2τ2

C1/2τ2 − C

1/2τ1

ξ2 − ξ1− 4.2(CτEQ2

1/2 − C1/2τ2 ) = 0. (4.3)

The amplification equation used to detect transition, Equation 3.63, becomes Equation 4.4.

n2 − n1ξ2 − ξ1

− dn

dξ(Hk2)

m(Hk2) + 1

2l(Hk2)

1

θ2= 0 (4.4)

With the discretized equations now defined, it is now possible to explain the computation procedure.

Firstly the inputs from the 2D Panel Code are collected. The velocity distribution is analyzed to find

the stagnation point near the leading edge. From that position, the velocity field is separated into two,

corresponding to the upper and lower surfaces boundary layers. After the separating and rearranging all

the variables, the boundary layer calculation can start. First the upper boundary layer is evaluated. The

40

first station is estimated using the Thwaites method (as presented in [32]) due to the lack of information

to use the method previously described. The first station is always assumed to be laminar. The compu-

tation continues to the following stations with the equations presented in Section 3.3.3, until the trailing

edge is reached. There, as described previously, the no-slip condition no longer applies and, therefore,

Cf is set to 0. If transition did not occur while over the surface, transition is forced in the second wake

section as laminar wakes usually do not occur in aerodynamic flows of interest [23] due to the velocity

inflections present at the trailing edge. The same procedure is applied to the lower boundary layer.

In the Inverse form, instead of an imposed outer velocity condition at each station this formulation

uses an imposed displacement thickness δ∗ distribution. The equations are therefore solved for external

velocity at each station which is used later in Inverse and Semi-Inverse procedures as explained in

Section 3.4. Nevertheless, an inverse calculation requires at least that one Direct iteration was done

previously to obtain a physically relevant first input for the thickness displacement.

4.1.3 2D Viscous-Inviscid Interaction procedure

With both the inviscid and boundary layer solvers implemented, the Viscous-Inviscid Interaction proce-

dures were implemented. From the procedures described in Section 3.4, the Direct and the Semi-Inverse

were chosen for implementation, the first one for it’s simplicity and the second one for being reported to

achieve better results when dealing with separation while still keeping the generality by maintaining the

solvers separated. This way, any solver for both inviscid and boundary layer flows could be posteriorly

implemented without much modifications.

For the Direct method, the method presented in Section 3.4 is followed. The computation starts with

the calculation of the velocity distribution from the Panel Method and that velocity is then used for the

Direct Boundary Layer solver. After the Boundary Layer solution is obtained, the Transpiration Velocity

is calculated as described in Section 3.4 from the displacement thickness evolution along the boundary

layer and the external velocity along the surface from the previous iteration. In the next iteration this

Transpiration Velocity is included in the source strength distribution to account for the displacement

effect. The procedure continues this way iteratively.

This procedure is, however, limited, as discussed previously. The computation tends to diverge with

an increasing number of iterations due to the Goldstein Singularity at the trailing edge.

The Semi-Inverse method was therefore also implemented. The computation starts with a Direct

solution. From there, a relaxation formula 4.5 was implemented, as presented by Veldman [40].

δ∗(n) = δ∗(n−1) + ω{uBLe − uPMe

}, (4.5)

where the superscript n denotes the current station, ω is a relaxation parameter and uBLe and uPMe are

the velocity distributions obtained respectively from the Inverse Boundary Layer solver and the Panel

Method.

This approach did not output the expected results which turned to be less accurate and as unstable

as the Direct procedure, possibly due to implementation or numerical problems. Therefore it was decided

41

to proceed with the Direct procedure which, although only being able to provide a weak interaction, can

already model some boundary layer effects.

In Figure 4.2 are presented the obtained results for a NACA 0012 profile at a 5 degree angle of attack

in a flow with a Reynolds number of 106. These results are compared with the ones obtained by using the

XFOIL code [37], presented in [31], which reflect the results possible to obtain with a Fully-Simultaneous

approach.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.2 0.4 0.6 0.8 1 1.2

H

δ*

, C

f

x/c

δ* (XFOIL)

δ* (Direct)

Cf (XFOIL)

Cf (Direct)

H (XFOIL)

H (Direct)

Figure 4.2: Evolution of the Boundary Layer parameters obtained from the 2D VII code implementedand from XFOIL for a flow with AOA = 5 deg and Re = 106 over a NACA 0012 airfoil.

The obtained results present a considerable error in the development of the boundary layer parame-

ters when compared to the bibliography results. This is justified by the fact that only one VII interaction is

being done for the implemented direct code due to a divergent behavior with the iteration number. How-

ever, the same tendencies are observed in both cases. Cf presents a maximum error of around 60%,

underestimating the actual value and δ∗ is overestimated by a maximum error of 90%. Nevertheless,

as will be seen in the following sections, when implemented in a 3D Geometry the method is stabilized.

This is possibly occurring due to the fact that at each iteration the results are reassembled into the 3D

geometry and the Inviscid solver is run for this 3D geometry. This way, divergent tendencies on certain

Boundary Layer sections are smoothed on the spanwise direction by the neighbor sections, allowing for

more iterations to be run and, consequently, significantly reducing the errors committed.

4.2 3D Aerodynamic module

4.2.1 3D Panel method

The 3D Panel Method code used for obtaining the inviscid solution of the flow around the 3D wing was

APAME, developed by Daniel Filkovic [34]. This code, written in FORTRAN, allows the user to import an

already discretized surface model from NASTRAN, ABAQUS or other Structural and CFD commercial

solvers output and input files and generating a Panel Model for Static Aerodynamic Analysis. It uses the

methodology described in Section 3.2.3. This software is includes by a Graphical User Interface (GUI)

to allow for easy manipulation of geometry, flow conditions and wake generation, and a solver.

42

In this work the GUI is used once, only when a new geometry or different discretization is to be

evaluated, in the beginning of the implementation of a wing model for the Aeroelastic framework. This

is required because there is the need to appropriately add the wake geometry to the model and gen-

erate the APAME solver input file that is going to be read, interpreted and modified by the developed

Framework codes along the iterative Aerodynamic and Aeroelastic solution processes.

In order to enable a VII procedure to be done, some modifications had to be implemented to the

source code in order to include effects of Boundary Layer displacement. To do this, the modification

included a portion of code to read a file with the Transpiration Velocity, calculated as described in Section

3.4. These values are then included in the source strength distribution at each panel. This way the

boundary displacement effect is taken into account when solving the inviscid problem and consequently

when computing velocity and pressure distributions, as well as in the calculation of the resultant Lift,

Moment and Induced Drag.

One problem that was encountered when implementing the VII procedure was that the system being

solved is not sufficiently generic to allow for implementation of the interaction procedure in the wake, as

the source distribution vector implemented in APAME does not include the wake panels, due to the fact

that it is usually 0 in the wake. Modifying the code to accept the inclusion of the Transpiration Velocity

would be work intensive so no boundary layer displacement is taken into account in the wake in the

3D VII code. This is believed to result in an overestimation of the lift coefficient and have effects in the

boundary layer evolution, i.e., the effects of the presence of the boundary layer may be underestimated.

4.2.2 3D Viscous-Inviscid interaction Procedure

The VII procedure for the 3D model is similar in concept to the Direct method for the 2D case. However

some approaches had to be adapted.

First, the geometry must be imported and interpreted. The file created by the APAME GUI to be

used by the APAME SOLVER is read and interpreted by a routine implemented in MATLAB. This way

the geometry and the flow conditions are imported into this environment. The geometric model imported

into APAME GUI was obtained from a NASTRAN structural analysis file.

After the geometry was imported it is then interpreted. A routine implemented in MATLAB first finds

the 3D trailing edge and then, for each trailing edge panel, assembles the correspondent streamwise

section. These sections are then treated as 2D airfoils by the 2D Boundary Layer Solver. After the

boundary layer is evaluated for all sections, the results are reassembled and passed to the 3D aero-

dynamic model. A file containing the Transpiration velocity values for each of the 3D surface panels is

then created. The APAME Solver is once again called to solve the inviscid problem but his time with the

displacement correction.

As results are obtained, APAME integrates the pressure loads over the wing surface and outputs the

aerodynamic coefficients. Velocity distribution at the wing surface is also imported back to MATLAB and

is used in the following iteration for a new Boundary Layer solution. This iterative procedure is repeated

until convergence. Figure 4.3 summarizes the calculation procedure implemented in this module.

43

?

Aerodynamic Analysis Definition (APAME ”.inp” file)

?

?

APAME Inviscid Flow Solver

?Aerodynamic Loads and Coefficients (”.res” file)

Read 3D Velocity field

Interpret Geometry and Results

Organize results into 2D Streamwise sections

?

Velocity distribution on upper and lower surfaceBoundary Layers over 2D Wing Sections

2D Boundary Layer Equations (Repeat for each BL section)

First station

Thwaites method

See [32].

Laminar Flow stations (n < 9)

Governing Equations:

Von Karman (Eq. 3.48)Shape Param. (Eq. 3.47)Amplification (Eq. 3.63)

Closure Equations:

H∗k (Eq. 3.50a)H∗ (Eq. 3.50b)Cf (Eq. 3.51)CD (Eq. 3.52)

Turbulent Flow stations

Governing Equations:

Von Karman (Eq. 3.48)Shape Param. (Eq. 3.47)

Lag Entrainment (Eq. 3.60)

Closure Equations:

H∗k (Eq. 3.56)H∗ (Eq. 3.50b)Cf (Eq. 3.54)CD (Eq. 3.58)

?δ∗ over the Wing Sections

Calculate Transpiration Velocity (EQ. 3.69)

Organize BL parameters to 3D

vn over Wing Surface panels

-

?��

��- Boundary Layer parameters along streamwise sections

- Aerodynamic Coefficients and Loads

Figure 4.3: Aerodynamic Module computation flowchart.

4.3 Structural module

By dividing a complex geometry into several small simpler parts, know as Finite Elements, the Finite

Element Method is a numerical method used for finding an approximate solution to partial differential

equations.

Although the method can be used in many fields, including Aerodynamics or Heat Transfer, its use in

Structural Mechanics is by far the most consensual, as very good results can be obtained in structural

analysis with relatively low computational requirements. With various formulations and element types

available, it is the standard for Structural Analysis in the industry across all engineering fields.

Several commercial software packages are available today implementing this kind of methods for

several fidelity levels and offering embedded Multidisciplinary Analysis. Siemmens NX and MSC Pa-

tran/Nastran both use NASTRAN as the structural solver. NASTRAN is an acronym for NASA Structural

44

Analysis, a FEM Solver developed in the 1960’s as a way to uniformize the use of structural solvers

through its engineering centers. Other solutions like Ansys, Abaqus and others use proprietary solvers.

The Structural module is responsible for calculating the solution to the static structure problem. Its

main component is NX NASTRAN, a commercial Finite Element Analysis tool that solves the struc-

tural linear system consisting of the wing structure fixed at the root and loaded with several load cases

corresponding to different aerodynamic loads.

First, the aerodynamic loads originated from the Aerodynamic Module are read. The aerodynamic

pressure and the shear stresses are organized and passed to the NASTRAN structural analysis file,

replacing any existent load cases. NX NASTRAN is then called and the results output file is generated

containing, among other parameters, the nodal displacements.

4.4 Aeroelastic framework

In this section the implementation of the Aeroelastic framework, which is the main objective of this work,

is presented. It was built around two main modules, the Aerodynamic Solver and the Structural Solver,

presented in detail in previous sections.

The first step was to allow that the geometry is imported. To do so, the APAME Graphical User

Interface (GUI) is used, as it already includes routines to import the discretized geometry from the

analysis files used by NASTRAN, ABAQUS and other commercially available software. It is also during

this step that the wake panels are added to the geometry, as well as the flow conditions and reference

parameters are set.

From that point on, the file (*.inp) created by the APAME GUI to be used by the APAME SOLVER

is read and interpreted by a routine implemented in MATLAB. This way the geometry and the flow

conditions are imported into this environment.

At this point, the Aerodynamic solver takes the geometry and executes the iterative procedure de-

scribed in Section 4.2. When the Aerodynamic solution is obtained, the aerodynamic loads, due to both

the aerodynamic pressure and the shear stresses, are imported and included in the original NASTRAN

structural analysis file. As the aerodynamic and structural meshes coincide at the wing surface due

to having been imported from the same file, the loads are assigned back to each individual panel and

the new structural analysis file is generated. Next, the framework calls NASTRAN to solve the static

structural analysis as indicated by the new file.

Once the solution is achieved and the NASTRAN structural analysis output file is generated, the

framework executes a routine to read and interpret the results. The surface node displacements are

taken by the framework and the geometry is updated to the new deformed configuration by adding the

displacements in X, Y and Z to the respective original node positions. The wake position is recalculated

by taking the nodal displacements at the wing trailing edge.

This new geometry is then exported back to the APAME SOLVER input file and the cycle is repeated

until convergence is reached.

The entire process is summarized in the flowchart of Figure 4.4.

45

'

&

$

%- Structural and Aerodynamic Geometry Definition

(NASTRAN ”.dat” file)

- Flight Conditions;

- Reference values

?

APAME GUI

?

Aerodynamic Analysis Definition (APAME ”.inp” file)

?

?

AERODYNAMIC MODULE

?Aerodynamic Loads

LOAD TRANSFER (Aero to Struct)

?Structural Analysis Definition (NASTRAN ”.dat” file)

?

STRUCTURAL MODULE

?Nodal Displacements Results (NASTRAN ”.f04” file)

LOAD TRANSFER (Struct to Aero)

Deformed Aerodynamic Analysis Definition (APAME ”.inp” file)

-

?��

��- Displaced Geometry

- Aerodynamic Coefficients and Loads

Figure 4.4: Aeroelastic Framework Modules flowchart.

46

Chapter 5

Results

5.1 Problem definition

5.1.1 Wing geometry

The wing geometry to be studied is a modified reference wing for the NOVEMOR project of the 7th EU

Framework. It was modified in [43] to present an higher aspect ratio than the original wing in order to

achieve a more flexible design that could better demonstrate the effects of aeroelastic behaviour. Figure

5.1 represents a Three view sketch of the wing geometry.

(a) Top view

(b) Front view (c) Side view

Figure 5.1: Three view wing geometry sketch.

47

The wing geometric parameters are presented in Table 5.1. The semi-wing is composed of two

sections, an inboard section, between the root and the break, and an outboard section, between the

break and the wingtip.

Table 5.1: Wing geometry parameters (Adapted from [43]).Geometric Properties Symbol Value Units

Wingspan b 36.33 mSpan root to break br−b 7.13 m

Span break to tip bb−t 11.03 mRoot chord cr 5.71 m

Break chord cb 3.01 mTip chord ct 1.3 m

Mean Aeordynamic Chord M.A.C. 3.02 mSweep angle Λ 25 deg

Dihedral angle Γ 4.5 degTake-off Weight WTO 58000 kg

Wing planform area S 113.482 m2

5.1.2 Flight Conditions

The flight conditions presented in [43] are shown in Table 5.2. At Mach velocity of 0.78, these conditions

position the aircraft in a high subsonic flight range, where compressibility effects are predominant.

Table 5.2: Flow properties at the designed Cruise Flight condition.Flow properties (original

M 0.78h 38000 ftρ 0.332 kg/m2

Q∞ 230.15 m/s

Such effects cannot be accurately predicted with the simple Prandtl-Glauert compressibility correc-

tion employed in the developed code. As was previously referred, Katz and Plotkin [9] suggest the

correction to be used up to Mach 0.6. Therefore, to be sure that the Test Case conditions were within the

range of applicability of the implemented models, two new Test Cases were defined for Mach 0.5. Table

5.3 describes the flow properties at the modified Cruise Flight condition. Altitude h had to be lowered

because the trimmed, leveled flight condition was not possible to be achieved at the new Mach speed

for the initial cruise altitude.

Table 5.3: Flow properties at the modified Cruise Flight condition.Flow properties (modified)

M 0.5h 20000 ftρ 0.653 kg/m2

Q∞ 158.02 m/s

The first test case corresponds to the trimmed flight condition of this aircraft, disregarding the pitching

48

moment equilibrium, as only the wing and not the stabilizers are being modeled. This condition was

determined by obtaining the curve of Cl vs. α of the wing at the new flight speed and altitude. Running

the APAME aerodynamic code on the wing for several α, the graph presented in Figure 5.2, along with

linear fitting equations, allowed to estimate the angle of attack relative to the wing’s x axis that generates

the necessary lift for cruise flight condition 5.2.

y = 534.17x - 6.792

-50

0

50

100

150

200

0 0.1 0.2 0.3 0.4

Lift

(m

t)

Alpha (rad)

Figure 5.2: Lift versus Angle of Attack for the flight condition being studied, obtained with the APAMEPanel Method code.

With this information, for the aircraft weight of 58000kg the angle of attack for trim condition was

calculated to be 6.95o.

As for the second test case, the same altitude and Mach speed were assumed, changing only the

angle of attack to 3.5o. This was considered because at the conditions of the first test case, the high

figure for the angle of attack was considered to be prone to induce considerable areas of separated flow.

This way, the implemented aerodynamic code can be tested in two different conditions and convergence

and error can be evaluated for each case.

Table 5.4 summarizes the test cases used to verify the implemented solution. The results presented

in the following sections where produced for these two cases.

Table 5.4: Definition of the two test cases.Test case Mach α

#1 0.5 6.95o

#2 0.5 3.5o

5.1.3 Wing Aerodynamic model

The aerodynamic wing model, shown in Figure 5.3, consists of the wing skin geometry, discretized in

a structured way with a refinement bias towards the leading edge and the trailing edge, as required for

better panel method accuracy. To achieve this final discretized geometry, NX FEMAP was used to apply

the mapped mesh.

The surface is discretized in 130 stations along the span and 40 panels for each station. This number

49

Figure 5.3: Final Wing Aerodynamic model with the skin discretized in 5200 panels.

of panels was considered sufficient after the realization of the convergence studies for both test cases,

regarding the Inviscid and the VII solutions, as will be presented next and shown in Figure 5.4.

For the convergence under inviscid conditions, the results are represented in orange. The inviscid

solver APAME was run for 5 different fidelity discretized geometries, ranging from 1040 to 5200 panels,

having the later been chosen as the final configuration. The variation in number of panels represents

only a variation in the number of spanwise stations, from 26 to 130. The number of panels per station,

i.e. in streamwise direction, was kept constant at 40 stations as it was more closely related to the

constrains of the 2D Boundary Layer code. For both angles of attack it is possible to see that for the

geometry discretized in 5200 panels the results for all 3 aerodynamic coefficients can be considered

converged, which means that for the inviscid analysis, the angle of attack had a negligible influence in

the convergence.

The same procedure was done for the VII code, whose results are represented in full line for 10

VII iterations per refinement step for assuring convergence with each mesh. It was seen that a good

compromise between computational time and convergence of results could be achieved for 5200 panels

for both cases. However, as can be seen in 5.4, the convergence of the aerodynamic coefficients with

the number of panels used is slower for the higher angle of attack, meaning that the possible presence

of Boundary Layer separation along the upper wing surface would probably require a further refined

geometry for better result accuracy, especially in the case of the Moment Coefficient Cm.

For the inviscid case, the difference in computational time ranged from around 10 seconds for the

least number of panels to around 40 seconds with the most detailed geometry. Figure 5.5 shows the

computational time is significantly increased when obtaining the VII code and presents an almost linear

relation with the model’s number of panels. Several factors contribute to a significant increase in the

time needed to obtain a VII solution:

• Firstly, the VII procedure here implemented is an iterative one, which involves solving the both the

Inviscid system and the Boundary Layer equations several times until convergence is achieved.

Each cycle is a VII iteration;

• Secondly, as a way to simplify the implementation of aerostructural interaction procedure, the struc-

tural mesh of the wing skin is coincident with the aerodynamic and the boundary layer is currently

50

0.235

0.24

0.245

0.25

0.255

0.26

0.265

0.27

0.275

500 1500 2500 3500 4500 5500

Lift C

oeff

icie

nt

Number of panels

VII code

Inviscid code

(a) Convergence of the lift coefficient for α = 3.5o.

0.52

0.54

0.56

0.58

0.6

0.62

0.64

500 1500 2500 3500 4500 5500

Lift C

oeff

icie

nt

Number of panels

VII code

Inviscid code

(b) Convergence of the lift coefficient for α = 6.95o.

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

500 1500 2500 3500 4500 5500

Dra

g C

oeff

icie

nt

Number of panels

VII code

Inviscid code

(c) Convergence of the drag coefficient for α = 3.5o.

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

500 1500 2500 3500 4500 5500

Dra

g C

oeff

icie

nt

Number of panels

VII code

Inviscid code

(d) Convergence of the drag coefficient for α = 6.95o.

-0.345

-0.34

-0.335

-0.33

-0.325

-0.32

-0.315

-0.31

500 1500 2500 3500 4500 5500

Mo

me

nt

Co

eff

icie

nt

Number of panels

VII code

Inviscid code

(e) Convergence of the moment coefficient for α = 3.5o.

-0.87

-0.85

-0.83

-0.81

-0.79

-0.77

-0.75

-0.73

500 1500 2500 3500 4500 5500

Mo

me

nt

Co

eff

icie

nt

Number of panels

VII code

Inviscid code

(f) Convergence of the moment coefficient for α = 6.95o.

Figure 5.4: Convergence of the aerodynamic results with increasing panel number both for Inviscid andViscous solution, for the two test cases.

being computed in every section. Being that the Boundary layer computation takes around 50%

of the total time used for obtaining a converged aerodynamic solution, by using an interpolation

procedure, this calculation could be lightened at the account of some loss in spanwise fidelity. This

argument was not developed in this work;

• Lastly, as an external software was used to solve the Inviscid flow, the parameters and geometry

have to be passed through file creation and reading processes. For example, the procedure to

read and interpret the results from the inviscid solution was observed to take around 15% of the

total time used for obtaining the converged aerodynamic solution.

Another interesting point is that the computational time for the higher angle of attack is significantly

51

higher than for the low angle of attack, although the same discretization refinement and number of VII

iterations are kept. This may result from an higher difficulty to solve the boundary layer equations,

needing and increased number of iterations within the fsolve Matlab function used.

Nevertheless, this computational time does not represent the actual time spent computing the aero-

dynamic solution when the VII code is integrated in the aerostrucutural framework. As will be seen in

the next chapters the number of VII iterations for achieving a converged solution ranges from as low a 4

VII iterations for the lowest angle of attack up to 8 for the higher angle of attack, potentially cutting these

computational times in half for the best scenario.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 1000 2000 3000 4000 5000 6000

Com

puta

tional tim

e (

s)

Number of panels

α = 3.5º

α = 6.95º

Figure 5.5: Computational time for executing 10 VII iterations with varying number of panels.

5.1.4 Wing Structural model

The wing geometry was firstly modeled in Solidworks and the imported into NX FEMAP. The model

structural design was based in conventional wing structure configurations, consisting of 3 basic types of

parts: Skin, Spars and Ribs. The first two are shown in Figure 5.6.

Figure 5.6: Wing structure Finite Element Model: Spars (longitudinally) and Ribs (transverselly).

The spars are two long longitudinal elements whose main function is to support the bending and

torsion moments generated by the loads the wing is subjected to. Although usually modeled as beam

elements with a defined cross-section, in this work they were modeled as CQUAD4 plate elements for

52

all structural elements in this work. The element thicknesses used in this model are detailed in Table

5.5.

Table 5.5: Spar thicknesses by zone.Spar Thicknes Position Vertical Alignment

Front spar 4.3 mm 25% chord Global Z axisBack spar 4.3 mm 75% chord Global Z axis

Another component is the skin which is the surface that will be in contact with the flow. It is usually

few millimeters thick and it’s main structural function consists in passing the Aerodynamic loads to the

wing structure. However, when the skin in between the front and back spars are combined with the

spars, these elements create aa structural element called Wing Box or Torsion Box, shown in Figure 5.7.

Figure 5.7: Wing structure Finite Element Model: Wing Box (Spars + center Skin section) and Ribs(transverselly).

This macro element works as an approximately rectangular section hollow beam and is responsible

for supporting both bending and torsion loads in the wing. Taking special care when modeling this

section of the skin, modeled as being thicker than the rest of the skin (see Table 5.6), gives the torsion

box the desired torsional stiffness.

Table 5.6: Wing Skin thickness by zone.Wing Skin Thickness

Leading and Trailing edges: Inboard 2.3 mmLeading and Trailing edges: Outboard 1.4 mm

Torsion box: Inboard 3.5 mmTorsion box: Outboard 2.5 mm

The ribs are the components placed transversely to the spars and have the function of maintaining

the airfoil shape, helping to support the wing box section and transferring the aerodynamic loads from the

skin to the spars. These have variable thickness along the wingspan, which were modeled as described

in Table 5.7.

Regarding the material used, all elements were modeled as Aluminium 7075-T651 [44]. Although

nowadays in many aircraft parts this type material is being substituted by composite materials, aluminium

53

Table 5.7: Rib thicknesses.Rib number Thickness

1 to 6 5 mm7 to 14 3 mm

alloys are still frequently used for application in such structural elements in the aeronautical industry due

to its good compromise between weight, strength, stiffness and cost.

Table 5.8: Aluminium 7075-T651 mechanical properties.Material Aluminium 7075-T651

Density ρ 2.810 kg/m3

Modulus of Elasticity E 7.17E+10 PaShear Modulus G 2.69E+10 Pa

Poisson Ratio ν 0.33Tensile Yeald Strength σy 503000000 Pa

Shear Strength Shear 331000000 PaThermal Conductivity K 130 W/mK

Specific Heat Capacity 0.96 J/goC

This structural model was verified to support the Cruise Loads. This was done by checking if the Von

Mises stresses in such condition reached the yield stress of the used aluminium alloy at any position in

the structure. Although these are not the only constraints or verifications to be done when wing structural

design is done, the main scope of this work focuses on the Aeroelastic interaction rather than Structural

design, and so it was considered that this kind of assumption was sufficient to evaluate the aeroelastic

framework’s performance in comparative studies.

5.2 Aerodynamic Analysis Results

With the Aerodynamic model now geometrically defined and appropriately discretized, it is now possible

to take some results from a VII Aerodynamic analysis with the Aerodynamic Module here implemented

for the two cases conditions previously defined . Figure 5.8 shows the convergence of the aerodynamic

coefficients with the iteration number.

It is evident that, regarding the case where α = 3.5o, from the 5th iteration the result can be consid-

ered converged, only demonstrating small oscillations related to the Direct VII procedure implemented.

From then on, the results always oscillate bellow 1%, as can be seen in Figure 5.9. For the other test

case, where α = 6.95o, as already observed in the previous subsections, the convergence of results is

not as good with the increase of the iteration number, with oscillations around 10%. This may also be

due to the incapacity of the implemented model to deal with boundary layer separation that may occur

at such angles of attack.

The obtained solutions can now be compared to the inviscid solution obtained with APAME for each

flow condition and to the solution obtained from CFD analysis of the geometry for the same conditions

as well. The used geometry is presented in Figure

54

0.256

0.258

0.26

0.262

0.264

0.266

0.268

0.27

0.272

0.274

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Lift

Coe

ffic

ien

t

Iteration Number

(a) Convergence of the lift coefficient with α = 3.5o.

0.52

0.54

0.56

0.58

0.6

0.62

0.64

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Lift

Co

eff

icie

nt

Iteration Number

(b) Convergence of the lift coefficient with α = 6.95o.

0.01

0.0105

0.011

0.0115

0.012

0.0125

0.013

0.0135

0.014

0.0145

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Dra

g C

oe

ffic

ien

t

Iteration Number

(c) Convergence of the drag coefficient with α = 3.5o.

0.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Dra

g C

oe

ffic

ien

t

Iteration Number

(d) Convergence of the drag coefficient with α = 6.95o.

-0.34

-0.335

-0.33

-0.325

-0.32

-0.315

-0.31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Mom

en

t C

oeff

icie

nt

Iteration Number

(e) Convergence of the moment coefficient with α = 3.5o.

-0.86

-0.84

-0.82

-0.8

-0.78

-0.76

-0.74

-0.72

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Mom

en

t C

oeff

icie

nt

Iteration Number

(f) Convergence of the moment coefficient with α = 6.95o.

Figure 5.8: Convergence of the aerodynamic results with the iteration number for both test cases.

These results were obtained through the implementation of the geometry and flow conditions in the

CFD commercial package StarCCM+. The model employed was a Reynolds Averaged Navier-Stokes

(RANS) model with an SST (Menter’s Shear Stress Transport) k−ω turbulence model. The comparison

of results from the VII code, CFD and APAME are presented in Table 5.9.

First of all, it is possible to observe that the best results obtained with the VII code were for Case

#2, with the lower angle of attack. For this case every aerodynamic coefficient was calculated to an

error smaller than 3% when compared to the CFD solution. This is a very good approximation, taking

into account that the computational effort was almost 50 times lower for the VII code computation when

compared with the CFD analysis (see Table 5.10). Drag was particularly well estimated, with an error of

−0.8%, meaning that the VII code slightly underestimated the Drag for this case.

55

0.001%

0.010%

0.100%

1.000%

10.000%

100.000%

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Re

lative

Err

or

(lo

g s

ca

le)

Iteration Number

Cl error

Cd error

Cm error

(a) Relative error of the aerodynamic coefficients for α = 3.5o.

0.001%

0.010%

0.100%

1.000%

10.000%

100.000%

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Re

lative

Err

or

(log

sca

le)

Iteration Number

Cl error

Cd error

Cm error

(b) Relative error of the aerodynamic coefficients for α = 6.95o.

Figure 5.9: Evolution of the Relative Error of the aerodynamic coefficients with the iteration number forboth test cases.

Figure 5.10: Wing geometry modeled on StarCCM+ for CFD aerodynamic analysis.

The method that presents consistently bigger errors accross all aerodynamic coefficients for this

case is the inviscid Panel Code, APAME. This was also expected partly because the shear loads on

the wing skin are not taken into account in inviscid computations, especially for the estimation of Drag,

which is underestimated presenting an error of −46.4%. Inviscid methods are only able to account for

56

Table 5.9: Comparison of aerodynamic coefficient results between StarCCM+ CFD solution, the VII codeimplemented and APAME inviscid solution.

Case #1: APAME VII code StarCCM+ Error APAME Error VII code

CL 0.6206 0.5667 / 0.5844 0.5614 10.5% 0.9%/4.1%CD 0.0123 0.0171 / 0.0185 0.0233 −46.4% −20.6% / −26.8%Cm0

−0.8457 −0.7886 / −0.7625 −0.7554 12% 0.9%/4.4%

Case #2: APAME VII code StarCCM+ Error APAME Error VII code

CL 0.272 0.258 0.252 8.2% 2.7%CD 0.00524 0.01385 0.01397 −62.5% −0.8%Cm0

−0.3387 −0.31679 −0.31019 9.2% 2.1%

Table 5.10: Comparison of Nomalized CPU computational times for the solution obtained with Star-CCM+, the VII code implemented and APAME inviscid code.

CPU Time (hours)

APAME VII code StarCCM+

CASE #1 0.022 2.15 114.67CASE #2 0.022 2.326 112.4

induced drag.

The Lift Coefficient is most overestimated by the inviscid solver APAME as was expected due to the

displacement effect of the Boundary Layer not being considered, with an error of 8.2% for Case #2. The

VII code here implemented presents an error of 2.7% when compared with StarCCM+ solution, standing

in between the CFD and the Panel Method solutions.

Looking now at the results obtained for CASE #1, the errors are visibly bigger, both for the APAME

and VII codes. Nevertheless, all aerodynamic coefficients calculated with the VII code fell within the

results from APAME and the CFD analysis. Lift was calculated to a maximum error of 4.1% and Moment

to a maximum error of 4.4%. On the other hand, the Drag computation presented significantly bigger

errors, topped at −26.8%. This difference is justified by the fact that the VII code is still lower in fidelity

when compared to a CFD approach, which is capable of modeling with better fidelity boundary layer

separation effects that may occur at such an high angles of attack, as well as accounting for other

phenomena only possible to model when solving the higher order RANS equations in a volume domain.

However, the error committed by the VII method is still less than half than the Drag estimated with the

inviscid code, which presented a result underestimated by 62.5%.

As intended, the VII code is able to successfully account for the Boundary Layer displacement and

Shear Stress effects with a good agreement to the results obtained by a CFD code, especially for low

angles of attack.

57

5.3 Aeroelastic Analysis Results

With the aerodynamic module verified, this section focuses on the Aeroelastic Analysis results obtained

for both the inviscid flow solution and the VII code aerodynamic model.

Using the Aeroelastic coupling procedure defined in the precedent chapters, the solution for both

Test Cases is presented for both the assumptions of inviscid flow and the VII code developed.

5.3.1 Aeroelastic Solution for Case #1

The maximum displacements presented in Figure 5.11 were obtained for running the conditions of Case

#1 through 30 aerostructural iterations on the developed code, using the inviscid Panel Method aerody-

namic code. It is possible to observe that around 10 aerostructural iterations with one Inviscid Aerody-

namic computation per Aerostructural iteration were needed for the solution to converge in regard to the

maximum displacement present in the structure. The solution for the maximum displacement in each

axis was obtained and the most prominent deflection occurred in the Z axis, which not only along the

general direction of actuation of the Lift force, but also corresponds to the least stiff direction of the wing

structure. In the x direction the direction of the displacement is negative, i.e., against the flow direction.

This occurs due to the aerodynamic moment that the wing is subjected to which induces torsion. In this

case, the torsion pitches down the sections along the wing, resulting in a rotation of the trailing edge,

more prominent at the wingtip, which in result pushes the trailing edge in the negative X direction.

-0.082

0.068

2.181

0.857

1.335

1.343

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

1 4 7 10 13 16 19 22 25 28

Maxim

um

Dis

pla

ce

me

nt

(m)

Iteration Number

Displacement X

Displacement Y

Displacement Z

Figure 5.11: Evolution of x, y and z maximum displacements with iteration number for Inviscid Aerostru-cural analysis for Case #1.

In Figure 5.12 the same parameters are presented, but now for the viscous analysis, which takes

into account the not only the displacement effect of the boundary layer on the surface pressures but

also the applied shear stresses. For this analysis, 30 aerostructural iterations were run, each containing

7 VII iterations. As seen in the Aerodynnamic Results section, the aerodynamic solution for Case #1

does not fully converge, oscillating arround a specific value. The number of iterations needed to achieve

58

convergence in the Aeroelastic analysis has therefore increased to around 20 aerostructural iterations.

-0.077

0.066

2.181

0.857

1.264

1.284

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

1 4 7 10 13 16 19 22 25 28

Ma

xim

um

Dis

pla

ce

me

nt

(m)

Iteration Number

Displacement X

Displacement Y

Displacement Z

Figure 5.12: Evolution of x, y and z maximum displacements with iteration number for Viscous Aerostru-cural analysis for Case #1.

The same tendencies are observed for the Viscous case. However the displacements are less promi-

nent, with a reduction, for example, in the maximum displacement in the Z direction in the order of 6 cm.

This is primarily a result of the reduction of lift due to the presence of the boundary layer. In Figure 5.13

the deformed wing shape for this case is represented.

Figure 5.13: Detail of the deformed wing shape after converged Viscous Aerostructural Solution isobtained for Case #1.

In Figure 5.14 the evolution of the Aerodynamic coefficients along the iterations is presented. The

results converge for around 20 aerostructural equations as also seen for the displacements, but oscillate

around a well defined value. For these coefficients the effects of the boundary layer are evident.

Comparing the results between the Inviscid and Viscous assumptions, it is possible to see that the

effect of the Boundary Layer is not negligible. Table 5.11 compares the most important results for both

Viscous and Inviscid solvers. In the Lift and Moment coefficients, differences of −4.69% and −5.65%

in relation to the Inviscid solution are significant for the estimation of an aircraft performance. As seen

59

0.45

0.47

0.49

0.51

0.53

0.55

0.57

0.59

0.61

0.63

1 4 7 10 13 16 19 22 25 28

Lift C

oeff

icie

nt

Iteration Number

Inviscid

Viscous

(a) Lift coefficient evolution with the aerostructural iterations.

0.01

0.012

0.014

0.016

0.018

0.02

0.022

1 4 7 10 13 16 19 22 25 28

Dra

g C

oeff

icie

nt

Iteration Number

Inviscid

Viscous

(b) Drag coefficient evolution with the aerostructural iterations.

-0.9

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

1 4 7 10 13 16 19 22 25 28

Mom

ent C

oeff

icie

nt

Iteration Number

Inviscid

Viscous

(c) Moment coefficient evolution with the aerostructural itera-tions.

-2.5

-2

-1.5

-1

-0.5

0

1 4 7 10 13 16 19 22 25 28

Win

gtip

An

gle

of

Att

ack (

deg

)

Iteration Number

Inviscid

Viscous

(d) Wingtip section angle of attack variation with the aerostruc-tural iterations.

Figure 5.14: Convergence of the aerodynamic results for the flexible geometry with the iteration numberfor Case #1.

before, Drag is also poorly predicted by inviscid solvers.

Table 5.11: Summary of the results obtained for the Aeroelastic analysis of Case #1.

Inviscid Viscous Difference

Max. displacement (m) 1.343 1.28354 -4.43%Wingtip torsion (o) -2.56874 -2.38851 -7.02%

Cl 0.53809 0.51284 -4.69%Cd 0.0125 0.01907 52.54%Cm -0.6897 -0.6505 -5.68%

CPU time (hours) 1.25 10.7988 763.90%

Regarding the computational effort, the inviscid solution had the fastest turnout time, taking 1.25

hours to complete the calculation in normalized CPU time, actual computational elapsed time multiplied

by the number of CPU cores being used. This is a relatively high time when comparing this type of

solutions employed by other frameworks, however this occurs because the generation, reading and

interpretation of files took significant time. Therefore, by implementing a closely coupled solution, this

figure can be significantly improved. The Viscous solution took 10.8 hours which is a big increase when

comparing to the inviscid solution. However this is still faster than CFD based solver, as the convergence

for one aerodynamic solution may take just as much time or more, as seen in the previous section.

60

5.3.2 Aeroelastic Solution for Case #2

Next are presented the results for the second case, α = 3.5o. For the inviscid solver, the displacements

(Figure 5.15) are significantly lower than the ones seen for Case #1 as in the new condition the lift being

generated was reduced. Convergence of the displacements takes 10 iterations, as previously.

0.440

0.091

0.214

0.216

0.012

-0.008

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 4 7 10 13 16 19 22 25 28

Maxim

um

Dis

pla

cem

ent

(m)

Iteration Number

Displacement X

Displacement Y

Displacement Z

Figure 5.15: Evolution of x, y and z maximum displacements with iteration number for Inviscid Aerostru-cural analysis for Case #2.

For the Viscous case, the displacements converge faster than seen for Case #1, for around 10

aerostructural iterations (Figure 5.16), with 7 VII iterations each. This is due to the lower angle of attack

imposed, for which separation phenomena is less likely to occur. Therefore, the Boundary Layer solver

is more stable which makes the VII aerodynamic solution converge faster and to a greater extent.

-0.005

0.011

0.440

0.091

0.184

0.182

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 4 7 10 13 16 19 22 25 28

Maxim

um

Dis

pla

cem

ent

(m)

Iteration Number

Displacement X

Displacement Y

Displacement Z

Figure 5.16: Evolution of x, y and z maximum displacements with iteration number for Viscous Aerostru-cural analysis for Case #2.

Despite the difference in the absolute values of the parameters being calculated, the same tenden-

cies observed for Case #1 are also seen for the present case. Figure 5.17 shows a representation of

61

the deformed wing geometry under the conditions of the present test case.

Figure 5.17: Deformed Wing shape after converged Viscous Aerostructural Solution is obtained for Case#2.

As seen before for the displacements, the aerodynamic coefficients also converge in a reduced num-

ber of aerostructural iterations for these conditions. Figure 5.18 shows the evolution of the aerodynamic

coefficients with the iteration number, where it is again possible to observe the different results obtained

for the inviscid an viscous analyses.

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0.265

0.27

0.275

1 4 7 10 13 16 19 22 25 28

Lift C

oeff

icie

nt

Iteration Number

Inviscid

Viscous

(a) Lift coefficient evolution with the aerostructural iterations.

0.002

0.004

0.006

0.008

0.01

0.012

0.014

1 4 7 10 13 16 19 22 25 28

Dra

g C

oeff

icie

nt

Iteration Number

Inviscid

Viscous

(b) Drag coefficient evolution with the aerostructural iterations.

-0.35

-0.34

-0.33

-0.32

-0.31

-0.3

-0.29

-0.28

-0.27

-0.26

1 4 7 10 13 16 19 22 25 28

Mo

me

nt C

oe

ffic

ien

t

Iteration Number

Inviscid

Viscous

(c) Moment coefficient evolution with the aerostructural itera-tions.

-3.1

-3

-2.9

-2.8

-2.7

-2.6

-2.5

-2.4

1 4 7 10 13 16 19 22 25 28

Win

gtip

An

gle

of

Att

ack (

deg

)

Iteration Number

Inviscid

Viscous

(d) Wingtip section angle of attack variation with the aerostruc-tural iterations.

Figure 5.18: Convergence of the aerodynamic results for the flexible geometry with the iteration numberfor Case #2.

62

Table 5.12 summarizes the results obtained for Case #2. The differences between the inviscid and

viscous results have an increased importance, both for the aerodynamic coefficients, but also for the

deflections, which present differences above 10%. The difference in the drag estimation between these

two flow approaches has more than doubled when compared to Case #1. This supports the idea that

viscous effects have a significant impact in the static aeroelastic response of an aircraft along different

flight conditions.

Table 5.12: Summary of the results obtained for the Aeroelastic analysis of Case #2.

Inviscid Viscous Difference

Max. displacement (m) 0.216 0.182 -16%Wingtip torsion (o) -0.834 -0.746 -11%

Cl 0.250 0.237 -5%Cd 0.005 0.013 137%Cm -0.297 -0.276 -7%

CPU time 1.672 10.403 522%

As seen for Case # 1, regarding the computational effort, the inviscid solution had the fastest turnout

time, taking 1.67 hours to complete the calculation in normalized CPU time. The Viscous solution took

10.4 hours. As before, although the computational time has increased when compared to the inviscid

solution, so has the accuracy of the model as proved in the previous section. Nevertheless this code is

still significantly faster than a CFD based solution taking as the cases reported in [18, 14].

63

64

Chapter 6

Conclusions

In this work, first an analysis of the state-of-the-art in Aeroelastic, Aerodynamic and Structural modeling

was done. Employing high fidelity aerodynamic models, such as coupled CFD and CSD approaches, is

still today computationally expensive. Low fidelity models, with fast turnaround times, frequently dismiss

viscous flow effects. Regarding Aeroelastic models, the Fluid-Structure Interaction problem was eval-

uated. The Structural model using FEM are nowadays widespread as design and analysis tools over

the industry. Although there have been several advances in CFD efficiency and the industry tends to

high fidelity modeling, several authors defend that development of all kinds of models from Low to High

Fidelity should be pursued as different Aerostructural Analysis needs occur in different contexts.

Taking these considerations into account, a moderate fidelity procedure for the evaluation of the static

aeroelastic characteristics of flexible wings in subsonic flight with inclusion of viscous flow effects was

implemented.

The aerodynamic loads were evaluated through a Viscous-Inviscid Interaction procedure using an

open-source 3-D Panel Code coupled with a 2-D Boundary Layer solver. Surface velocity was obtained

for the inviscid problem, which is then used as boundary condition for solving a set of equations derived

from the Von Karman momentum integral along surface sections. Boundary Layer thickness was then

taken into account in a new computation of the inviscid flow.

After the aerodynamic solution converged, the surface loads were used as input for static structural

analysis. The iterative procedure was carried on by updating the displaced geometry for new aerody-

namic analysis. This simplified aerodynamic model is therefore able to account for viscous flow effects,

namely friction drag and displacement effects.

A 2D test case was run as a benchmark for the VII 2D Boundary Layer code implemented. Results

showed the same tendencies with the ones obtained from a publicly available software solution but with

a considerable error, mainly due to the fact of a Direct VII procedure being implemented.

For the 3D case, first a benchmark of the Aerodynamic VII procedure was done, showing good

agreement with a commercially available CFD solver, both in the Inviscid and Viscous cases, with the

VII code presenting errors of less than 5% in the estimation of aerodynamic coefficients for the higher

angle of attack and of less than 3% for the lower angle of attack. The biggest advantage of the VII code

65

over the inviscid one is however in the estimation of Drag, which reduces the errors bigger than 45%

presented by the inviscid procedure to less than 1% in the case of the lower angle of attack.

As for the Aerostructural solution, the final test configuration was run, both for the Inviscid and Vis-

cous cases. Convergence occurred at a faster rate for the lower angle of attack case. Structural dis-

placements and aerodynamic coefficients were obtained for the deformed configurations. For the higher

angle of attack, the maximum vertical displacement observed was 1.34 meters for the Inviscid computa-

tion, which can be compared to the 1.28 meters obtained for the viscous solution. The inclusion of the

boundary layer resulted in a reduction of 4.43% in the maximum displacement, which contributed to the

4.69% reduction in Lift. Cd was the parameter that showed a greater difference of more than 50%, as

was already expected from the aerodynamic analysis. For the lower angle of attack the differences in

the displacements reached 16% when compared to the inviscid solution. The aerodynamic coefficients

also showed bigger differences.

In conclusion, the method here implemented has the advantage of being up to 10 times faster than

CFD-based codes due to the simplified models being used, but still accounting for the main viscous

effects that affect static aeroelastic phenomena. Although the effect of viscosity is not very perceivable

in the maximum displacements, their effect is still considerable, especially if aerodynamic coefficients

are to be obtained, especially for Drag estimation.

6.1 Future Works

As this work represents the beginning of the development of an Aeroelastic Framework from scratch

with very limited time, there are several points where the implemented solutions can be improved.

Starting from the Aerodynamic model, more specifically the Panel Method, it would be very important

to implement a 3D Panel Method code from scratch. This can be easily done and would significantly

improve the computational times by saving time in writing and interpreting files to and from the external

solver. Other reason this would be beneficial is related to the fact that no boundary layer displacement

effect could be applied to the wake geometry, as previously explained. Ultimately it would also allow

to implement Quasi-Simultaneous VII approaches, which could also improve computational times and

results convergence when reaching separation conditions.

As for the Boundary Layer code, other 2D formulations and implementations could be tested, both

simpler and more complex (as in [38]). The sections used for the Boundary Layer calculation can also

be obtained in different ways such as following the flow stream lines or a combination of streamwise

and spanwise boundary layer calculation by separating the flow velocity into those two components. 3D

Boundary Layer equations could also be employed as in [28, 26].

Regarding the VII procedure, a Fully-Simultaneous approach would preferably be implemented thus

avoiding the iterative Viscous-Inviscid interaction and cutting in computational costs.

As for the Aeroelastic Framework, the implementation of a more generic load and displacement

transfer procedure for passing the loads from the aerodynamic computational domain to the structural

domain would allow to have more flexibility when developing the structural and aerodynamic models

66

which currently have to match at the wing surface.

The possibility to include also non-lifting bodies is already supported by APAME but cannot be cur-

rently interpreted by the MATLAB code developed. Therefore, allowing such functionality in an auto-

mated and user-friendly way would enable to calculate full aircraft configurations. Support to control

surfaces modeling would also be advantageous.

67

68

Bibliography

[1] R. L. Bisplinghoff and H. Ashley. Principles of Aeroelasticity. Dover Publications, Inc., 1st edition,

2013. ISBN:978-0-486-78316-1.

[2] T. A. Weisshaar. Aircraft aeroelastic design and analysis. Online, https://engineering.purdue.

edu/AAE/Academics/Courses/aae556/2010/ClassnotesChapter1, 2009. Accessed on October

1st, 2015.

[3] E. H. Dowell. A Modern Course in Aeroelasticity. Springer, 5th edition, 2015. ISBN:978-3-319-

09453-3.

[4] Langley aerodrome in flight, 1903. Historic Images of the Smithsonian, Online, http://

siris-sihistory.si.edu/ipac20/ipac.jsp?uri=full=3100001%7E!10312!0&term=. Accessed

on October 4th, 2015.

[5] G. Kennedy and J. Martins. A parallel aerostructural optimization framework for aircraft design

studies. Structural and Multidisciplinary Optimization, 2014.

[6] R. E. Bartels and A. I. Sayma. Computational aeroelastic modelling of airframes and turbomachin-

ery: progress and challenges. Philosophical transactions. Series A, Mathematical, physical, and

engineering sciences, 365(1859):2469–2499, 2007.

[7] D. M. Schuster, D. D. Liu, and L. J. Huttsell. Computational aeroelasticity: Success, progress,

challenge. AIAA Dynamics Specialists Conference, 2003.

[8] T. Theodorsen. General theory of aerodynamic instability and the mechanism of flutter. NACA

Report Number 496, National Advisory Committee for Aeronautics, May 1934.

[9] R. Kamakoti and W. Shyy. Fluid-structure interaction for aeroelastic applications. Progress in

Aerospace Sciences, 40:535–558, 2004.

[10] J. G. Marshall. A review of aeroelasticity methods with emphasis on turbomachinery applications.

Journal of Fluids and Structures, 10:237–267, 1996.

[11] Aeroelasticity in turbomachines: Introduction to aeroelasticity. Online, 1urlhttp://gtt.epfl.ch/page-

63561-fr.html, December 2014. Accessed on October 11th, 2015.

69

[12] S. Obayashi and G. P. Guruswamy. Convergence acceleration of a navier-stokes solver for efficient

static aeroelastic computations. AIAA Journal, 33(6):1134–1141, 1995.

[13] K. A. James, G. J. Kennedy, and J. R. Martins. A review of numerical fluids/structures interface

methods for computations using high-fidelity equations. Computers and Structures, 134(1):31–41,

2014.

[14] D. E. Raveh. Computational-fluid-dynamics-based aeroelastic analysis and structural design opti-

mization—a researcher’s perspective. Computer Methods in Applied Mechanics and Engineering,

194(40):3453–3471, 2005.

[15] M. Love, T. D. L. Garza, E. Charlton, and D. Egle. Computational aeroelasticity in high performance

aircraft flight loads. In 22nd International Congress of Aeronautical Sciences, 2000.

[16] R. Heinrich, R. Ahrem, G. Gunther, H. P. Kersken, W. R. Kruger, and J. Neumann. Aeroelastic

computation using the amanda simulation environment. In CEAS, Conference on Multidisciplinary

Aircraft Design and Optimization, 2001.

[17] Y. W. Wang and Y. W. Lin. Combination of cfd and csd packages for fluid-structure interaction.

Journal of Hydrodynamics, 20(6):756–761, 2008.

[18] D. Huixue, Y. Zhichun, and L. Yi. Accelerated loosely-coupled cfd/csd method for nonlinear static

aeroelasticity analysis. Aerospace Science and Technology, 14(4):250–258, 2010.

[19] L. Cavagna, G. Quaranta, S. Ricci, and A. Scotti. Computational aeroelasticity with cfd models.

Bolletino del CILEA, 107:45–51, 2007.

[20] C. Farhat and N. Maman. Matching fluid and strucuture meshes for aeroelastic computations: A

parallel approach. Computers and Structures, 54(4):779–785, 1995.

[21] C. Farhat, M. Lesoinne, and P. LeTallec. Load and motion transfer algorithms for fluid/structure

interaction problems with non-matching discrete interfaces : Momentum and energy conservation,

optimal discretization and application to aeroelasticity. Computer Methods in Applied Mechanics

and Engineering, 157:995–1114, 1998.

[22] G. P. Guruswamy. A review of numerical fluids/structures interface methods for computations using

high-fidelity equations. Computers and Structures, 80(1):31–41, 2002.

[23] M. Drela. Two-dimensional Transonic Aerodynamic Design and Analysis using the Euler Equations.

PhD thesis, Massachusetts Institute of Technology, 1985.

[24] M. Drela and M. B. Giles. Viscous-inviscid analysis of transonic and low reynolds number airfoils.

AIAA Journal, 25(10):1347–1385, 1987.

[25] B. Maskew. Program vsaero: A computer program for calculating the non-linear aerodynamic

characteristics of arbitrary configurations. User’s manual, NASA, 1982.

70

[26] W. M. Milewski. Three-dimensional viscous flow computations using the integral boundary layer

equations simultaneously coupled with a low order panel method. PhD thesis, Massachusetts

Institute of Technology, June 1997.

[27] W. Stalewski and J. Sznajder. Application of a panel method with viscous-inviscid interaction for

the determination od the aerodynamic characteristics of cesar baseline aircraft. Transactions of the

Institute of Aviation, 207:63–83, 2013.

[28] I. Pereira and J. M. C. S. Andre. A semi-analytical model of the 3d boundary layer over the stream-

lined nose of a train. Wind Engineering and Industrial Aerodynamics journal, 119:78–88, 2013.

[29] S. Guo. Aeroelastic optimization of an aerobatic aircraft wing structure. Aerospace Science and

Technology, 11(2):396–404, 2007.

[30] W. Mason. Applied computational aerodynamics text/notes. Online, http://www.dept.aoe.vt.

edu/~mason. Accessed on August 1st, 2015.

[31] J. Katz and A. Plotkin. Low-Speed Aerodynamics. Cambridge University Press, 2nd edition, 2001.

ISBN:978-0-521-66552-0.

[32] V. de Brederode. Fundamentos de Aerodinamica Incompressıvel. Author’s edition, 1st edition,

1997. ISBN:972-97402-0-8.

[33] L. L. Erickson. Panel methods: An introduction. NASA Technical Paper 2995, 1990.

[34] D. Filkovic. Graduate work. Master’s thesis, Faculty of Mechanical Engineering and Naval Archi-

tecture, University of Zagreb, 2008.

[35] The aerodynamic center. Online, http://www.aerostudents.com/files/flightDynamics/

theAerodynamicCenter.pdf. Accessed on October 2nd, 2015.

[36] Thin shear layer approximation. Online, http://web.mit.edu/16.110/www/tsl_equations.html.

Accessed on October 2nd, 2015.

[37] M. Drela. Xfoil: An analysis and design system for low reynolds number airfoils. In Low Reynolds

Number Aerodynamics. Springer-Verlag, 1989.

[38] R. C. Lock and B. R. Williams. Viscous-inviscid interactions in external aerodynamics. Progress in

Aerospace Sciences, 24(2):51–171, 1987.

[39] M. Drela. A new transformation and integration scheme for the compressible boundary layer equa-

tions, and solution behavior at separation. Master’s thesis, Massachusetts Institute of Technology,

1983.

[40] A. E. P. Veldman. Viscous-inviscid interaction: Prandtl’s boundary layer challenged by golstein’s

singularity. Technical report, Institute of Mathematics and Computing Science, University of Gronin-

gen, 2004.

71

[41] P. Y. Lagree. Interactive boundary layer [ ibl ] or inviscid-viscous interactions [ ivi or vii ]. Technical

report, Institut Jean Le Rond d’Alembert, Universite Pierre et Marie Curie, 2014.

[42] S. Goldstein. On laminar boundary-layer flow near a position of separation. The Quarterly Journal

of Mechanics and Applied Mathematics, 1(1):43–69, 1947.

[43] C. Spada. Aeroelastic analysis of nonlinear high aspect ratio wings. Master’s thesis, Instituto

Superior Tecnico, October 2014.

[44] Aluminum 7075-t6; 7075-t651. Online, http://asm.matweb.com/search/SpecificMaterial.asp?

bassnum=MA7075T6. Accessed on August 9th, 2015.

72