Aerodynamic Optimization of Horizontal AxisWind Turbines Using the Lifting Line Theory

Gonçalo dos Santos Sousa

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisors: Prof. José Alberto Caiado Falcão de CamposDr. João Manuel Ribeiro da Costa Baltazar

Examination Committee

Chairperson: Prof. Carlos Frederico Neves Bettencourt da SilvaSupervisor: Prof. José Alberto Caiado Falcão de Campos

Member of the Committee: Prof. Luís Manuel de Carvalho Gato

November 2018

ii

Acknowledgments

While I am aware that this is atypical I could not, in good conscience, start by thanking anyone other

than my family. In particular, I want to thank my parents, Carminda and Luıs Sousa, for during this

last five years they were the ones who endured the most, working up to 16 hours a day, weekends

and holidays included, to give me an opportunity that they did not have in their time. For this, I will be

eternally grateful and I can only hope that, one day, I will be able to repay their kindness and make up

for the time we lost.1

Then, I would like to thank my supervisors, Professors Jose Falcao de Campos and Joao Baltazar,

for giving me the opportunity to work in a field that I love and for always being available to help me.

Last, but in no way least, I want to thank my friends, especially Miguel, Francisco and Marta, for

putting up with me for so long and making life fun.

1Embora esteja ciente de que isto e atıpico, nao poderia, em boa consciencia, comecar por agradecer a alguem que nao aminha famılia. Em particular, quero agradecer aos meus pais, Carminda e Luıs Sousa, pois durante estes ultimos cinco anosforam eles os que mais sofreram, trabalhando ate 16 horas por dia, fins de semana e feriados incluıdos, para me darem umaoportunidade que nao tiveram no seu tempo. Por isto, estarei eternamente grato e so posso esperar que, um dia, eu consigaretribuir a sua generosidade e compensar o tempo que perdemos.

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Abstract

The purpose of this work is to continue the development of a previously written lifting line code,

extending its use to the aerodynamic design of wind turbines and introducing a new method of optimiza-

tion - the Lagrange multiplier method - as an alternative to the classical one. The theory behind it is

presented, as well as the adjustments that were made to implement it.

As for the obtained results, they indicate that, for the same conditions, the Lagrange multiplier method

always outperforms the classical optimization, being the dissemblance between them more pronounced

for higher loads. It is also shown that, in terms of local variables, the difference between the methods is

concentrated in the vicinity of the blade root.

The inclusion of a hub model is also assessed, which brings improvements to the turbine perfor-

mance as well, with the only clear effect on the distributions being the increase of circulation near the

root.

Parametric studies are carried out regarding two inputs: the tip speed ratio, where we see the balance

between the viscous and kinetic energy losses come into play, and the drag-to-lift ratio, which is shown

to have a significant influence in the performance of the turbine, but only a minor one in the aerodynamic

design.

Lastly, a wake alignment scheme is also tested using several combinations of alignment sections.

The resulting wake geometries are analysed, as well as the behaviour of local variables both at the

lifting line and at the alignment sections.

Keywords: Lifting line theory; Wind turbine; Lagrange multiplier method; Wake alignment.

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Resumo

O objectivo deste trabalho e continuar o desenvolvimento de um codigo de linha sustentadora ja

existente, expandindo os seus usos para o projecto aerodinamico de turbinas eolicas e introduzindo um

novo metodo de optimizacao - o metodo do multiplicador de Lagrange - como alternativa ao classico.

A teoria em que se baseia o metodo e apresentada, bem como os ajustes que foram feitos para a sua

implementacao.

Quanto aos resultados, estes indicam que, quando comparados nas mesmas condicoes, o metodo

do multiplicador de Lagrange supera sempre a optimizacao classica, sendo a divergencia dos dois

crescente com a carga. Mostra-se tambem que, em termos de variaveis locais, a diferenca entre os

metodos esta concentrada na vizinhanca da raız da pa.

A inclusao de um modelo do cubo e igualmente avaliada, a qual tambem traz melhorias para o

desempenho da turbina, sendo o unico efeito notorio nas distribuicoes o aumento da circulacao perto

da raız.

Sao tambem realizados estudos parametricos sobre dois inputs: o parametro adimensional de ve-

locidade periferica, onde vemos o equilıbrio entre as perdas viscosas e de energia cinetica a entrar

em jogo, e a razao resistencia/sustentacao que, embora mostre ter um impacto significativo sobre o

desempenho, afecta pouco o projecto aerodinamico.

Por fim, um esquema de alinhamento de esteira tambem e testado usando varias combinacoes

de seccoes de alinhamento. As geometrias das esteiras resultantes sao analisadas, assim como o

comportamento das variaveis locais, tanto na linha sustentadora como nas seccoes de alinhamento.

Palavras-chave: Teoria da linha sustentadora; Turbina eolica; Metodo do multiplicador de

Lagrange; Alinhamento de esteira.

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Contents

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

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

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

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

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

1 Introduction 1

1.1 State of The Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory 7

2.1 Introduction to the Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 System of Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Forces and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Implementation 15

3.1 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Induced Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.3 Hub Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Without Wake Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 With Wake Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Discretization of the Lifting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.2 Discretization of the Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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4 Results 31

4.1 Results Without Wake Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.2 Classical Optimization vs. Lagrange Multiplier Method . . . . . . . . . . . . . . . . 32

4.1.3 Hub Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.4 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Results with Wake Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Conclusions 49

References 51

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List of Tables

3.1 Input for the convergence analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Input for the tests without wake alignment. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Input for the tests with wake alignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Power coefficient for different combinations of alignment sections. . . . . . . . . . . . . . 44

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List of Figures

1.1 Original drawing of Lanchester, showing the vortex shedding at the tip of the wing. (From

[2].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Example of a wind farm composed of several horizontal axis turbines. . . . . . . . . . . . 2

1.3 Statistical data showing the growth of renewables and wind energy in particular. (From [5].) 2

2.1 System of vortices for a single blade. (Adapted from [32].) . . . . . . . . . . . . . . . . . . 8

2.2 Reference frames and velocity triangle disregarding the effect of the system of vortices.

(Adapted from [32].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Triangles of velocities and forces. (From [28].) . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Discretization of the lifting line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Alternatives for modelling the wake geometry. (Only the wake of one blade is shown.) . . 16

3.3 A trailing vortex and it’s image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Flowchart of the computation procedure without wake alignment. . . . . . . . . . . . . . . 23

3.5 Flowchart of the computation procedure with wake alignment. (LL stands for ”lifting line”.) 25

3.6 Convergence of the power coefficient with the increase of the number of lifting line seg-

ments, M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7 Influence of the number of lifting line segments, M , on some local variables. . . . . . . . . 28

3.8 Convergence of the power coefficient with the increase of the number of stream-wise

segments per revolution - Nt - and the axial length of the wake - xuw. . . . . . . . . . . . 29

4.1 Evolution of the power coefficient with the load for both classical and Lagrange multiplier

optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Comparison of the classical optimization with the Lagrange multiplier method for the same

flow conditions and optimum CT0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Optimum distribution of induced aerodynamic pitch using the Lagrange multiplier method

for several values of CT0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Evolution of the power coefficient with the load for the cases with and without hub correction. 35

4.5 Effect of the hub correction on the design for the same flow conditions and optimum CT0 . 36

4.6 Effect of the tip speed ratio on the power coefficient. . . . . . . . . . . . . . . . . . . . . . 37

4.7 Effect of the tip speed ratio on the design for the same flow conditions and optimum CT0 . 39

4.8 Effect of the drag-to-lift ratio on the power coefficient. . . . . . . . . . . . . . . . . . . . . . 40

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4.9 Effect of drag-to-lift ratio on the design for the same flow conditions and optimum CT0. . . 41

4.10 Wake geometries for some of the tested combinations of alignment sections. (Only the

wake of one blade is shown.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.11 Effect of the alignment on the design for fixed CT0. . . . . . . . . . . . . . . . . . . . . . . 45

4.12 Distributions of local variables at the alignment sections when the wake is aligned at

x/R = {0} - Case A, x/R = {0, 1} - Case B, and x/R = {0, 0.25, 1} - Case C. (The line

x/R = 0.25 for Case B was obtained by interpolation.) . . . . . . . . . . . . . . . . . . . . 46

4.13 Distribution of induced axial velocity when the wake is aligned at x/R = {0, 0.2, 1}, show-

ing the detail of the tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

xiv

Nomenclature

Greek symbols

α Angle of attack.

β Undisturbed aerodynamic pitch angle.

βi Induced aerodynamic pitch angle.

~Γ Velocity circulation.

~γ Intensity of the trailing vortices.

ε Drag-to-lift ratio.

εN Numerical tolerance for the aerodynamic pitch angle.

εW Numerical tolerance for the wake dimensionless pitch and tangential velocity.

θk Angular position of the kth lifting line.

κN Under-relaxation factor for the aerodynamic pitch angle.

κW Under-relaxation factor for the wake dimensionless pitch and tangential velocity.

λ Tip speed ratio.

ν Kinematic viscosity of the fluid.

ρ Volumetric mass density of the fluid.

ψ Blade pitch angle.

~ω Angular velocity of the rotor.

Roman symbols

c Section chord.

CD Drag coefficient.

CL Lift coefficient.

CP Power coefficient.

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CT Axial force coefficient.

CT0Imposed axial force coefficient.

Ca,tij Axial and tangential influence coefficients matrices.

D Drag force per unit span.

~e Unit vector.

H Auxiliary function used in the Lagrange Multiplier Method.

` Constant; Lagrange multiplier.

~L Lift force by unit span.

Lk Lifting line k.

M Number of segments in which the lifting line is discretized.

N Number of sections in which a trailing vortex is discretized.

ns Number of wake alignment sections.

Nt Number of equal stream-wise segments per revolution used in the wake discretization.

p Dimensionless wake pitch.

Q Torque.

~R Vector that goes from the integration point to the point where the induced velocity is being com-

puted.

R Rotor radius.

rh Hub radius.

ri Radial position of control point i.

rj Radial position of end point j.

s Stream-wise direction.

Sk Vortex sheet shed from lifting line k.

T Axial force.

U Velocity of the incoming flow.

~V Total velocity.

~V∞ Undisturbed velocity.

~v Velocity induced by the entire system of vortices.

xvi

V Magnitude of projection of the total velocity vector on the blade cross section.

~vk Velocity induced by the kth lifting line and it’s sheet of trailing vortices.

x, r, θ Cylindrical coordinates in the rotating reference frame.

xfw Axial position of the far wake section.

xuw Axial position of the ultimate section, where the wake is truncated.

Z Number of rotor blades.

Subscripts

a Projection on the axial direction.

r Projection on the radial direction.

t Projection on the tangential direction.

x Projection on the direction of x.

y Projection on the direction of y.

z Projection on the direction of z.

Superscripts

′ Relative to the image vortex.

∗ Non-dimensional quantity.

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Chapter 1

Introduction

It was Frederick W. Lanchester (1878-1946), chief engineer and general manager of The Lanchester

Motor Company, that took the first steps in the development of the lifting line theory [1].

Having been publishing on the circulation theory of flight since 1894, it was in 1907 that he realized

that if a finite wing in motion creates a circulation around itself, than it must behave as a vortex [2].

At this time, Helmholtz had already published his three famous theorems on vortex dynamics [3], and

so Lanchester knew that this vortex could not simply vanish at the tip of the wing. His solution was to

suggest that this vortex, bound to the wing, would continue in the wake as a free vortex being shed from

the tip, as depicted in Figure 1.1.

Figure 1.1: Original drawing of Lanchester, showing the vortex shedding at the tip of the wing. (From[2].)

A more rigorous mathematical formulation of this model was presented later, in 1918, by the German

scientist Ludwig Prandtl [4] (1875-1953), to whom is usually credited the conception of the lifting line

theory.

From that point until today, the theory has seen many improvements and variations according to its

vast spectrum of uses - one can find applications of it on the design and analysis of nearly any lifting

surface, even rotating turbo-machinery, like propellers and turbines. It is on the latter that this work will

be focused, more precisely, in the design of horizontal axis wind turbines.

1

Figure 1.2: Example of a wind farm composed of several horizontal axis turbines.

The purpose of these machines, that look like the ones shown in Figure 1.2, is to partially harness

the winds kinetic energy (a virtually infinite source) and convert it into electricity. It all starts in the

rotor blades, whose geometry is such that the motion of the wind around them creates a torque on the

rotor, making it rotate. Then, in the nacelle (the case behind the hub of the rotor), there is an electrical

generator that converts this rotation into electricity and, in most cases, we can also find a gearbox that

allows the rotor shaft and the generator shaft to rotate at different angular velocities. Note that all of this

sits on top of a tower, which allows the rotor to be subjected to greater wind speeds, as it puts it in a

higher region of the atmospheric boundary layer.

At this point one might ask how significant these machines are. The fact is that, with the population

boom, the energy demand has been sharply rising in the last decades. This, along with the increasing

environmental awareness, has caused the demand for renewables to explode (Figure 1.3), and, conse-

quently, the wind energy industry is becoming more important every day. When we check the numbers,

in its 2018 Statistical Review of World Energy, BP determined that in 2017 49% of the global growth in

power generation came from renewables, from which more than half was due to wind energy [5] [6]. We

can thus say that improving the design of wind turbines is not only a very interesting academic exercise

but also a meaningful contribution to the worlds economical and environmental sustainability.

(a) Evolution of the global shares (%) of primary energy. (b) Evolution of the globally installed wind gen-eration capacity (GW) in recent years.

Figure 1.3: Statistical data showing the growth of renewables and wind energy in particular. (From [5].)

2

1.1 State of The Art

Having already presented the first steps taken by Lanchester [2] in 1907 and Prandtl in 1918 [4], it

is now time to dwell on the journey of this theory from that point on and discuss how the design of wind

turbines with this model evolved over the years.

Interestingly, it all started by the design of propellers. The first important development was made

by Betz in 1919 [7], soon after Prandtl’s publication, when he presented the conditions to what we now

call the “Classical Optimization”. This provided a first basis for the optimization of propellers but relied

on several limiting assumptions, like having a light load and absence of viscous forces. The problem

of finding the circulation distribution that would satisfy Betz’s conditions was then solved in 1929 by

Goldstein [8], considering an hubless propeller.

The next big leap took place in 1952 when Lerbs [9], based on the work of Kawada [10], came up

with analytical expressions for the computation of the velocities induced by the system of vortices (on

which, later, Morgan and Wrench [11] also worked). This allowed feats like the analysis at non-optimum

conditions and the design of moderately loaded propellers, broadening the applicability of the model.

The adaptation for wind turbines was made in 1986 by Maekawa [12], who presented a design

method based on Betz’s and Goldstein’s developments.

The theory then evolved both with propellers and turbines, with some authors, like Kerwin [13], fo-

cusing more on the aerodynamic design problem, and others, like Coney [14], focusing more on the

geometrical design problem.

A very important step forward in the model was the inclusion of viscous drag, which was probably

first applied in propellers by Adkins and Liebeck [15] in 1994. Chattot [16] included it in the design

of wind turbines in 2003 and, in the same publication, he used a vortex lattice model to discretize the

wake, allowing him to compute the induced velocities using numerical integration of the Biot-Savart law

(Sharpe [17] was also important in this matter).

This discretization also opened doors to wake alignment schemes that allowed the modelling of a

wake that is aligned with the local velocity in several sections instead of just the lifting line, acquiring

thus a variable geometry. Some examples of this can be found in the works of Aran [18], Kinnas [19]

and Diniz [20].

The improvement of the optimization problem has also been addressed through the years - we have

had, for example, the introduction of a variational approach as an alternative to the classical optimization,

by authors like Yim [21] and Kerwin et al. [13].

In modern times, another key contributor to this theory has been Epps who, while developing his

open source software OpenProp [22] [23] explored new wake models [24], new solvers and a lifting-

line/momentum-theory hybrid approach [25].

Last, but not least, we must mention that this thesis relies on the work of a research group that

has also been actively contributing to the theory, mainly under the supervision of Falcao de Campos

and Baltazar. Among other feats, the group developed a lifting line code that was first implemented by

Duarte [26] [27] in 1997 and then successively improved by many contributors to this date. In 2010,

3

Machado [28] [29], included the effect of the hub in a design routine for marine turbines. Caldeira [30],

in 2014, implemented a source model to simulate the effects of drag, specially in stall conditions, and

used it in the analysis of wind turbines. The last contribution comes from Melo [31] [32], who in 2016

continued the analysis of wind turbines and included a wake alignment scheme that also required the

development of a discretization method for the wake and a new way of computing induced velocities.

1.2 Objectives

The purpose of this work is to continue the development of the lifting line code mentioned in the

previous section - the idea is essentially to take the last version, developed by Melo for the analysis of

wind turbines, and make the necessary adjustments for it to solve the aerodynamic design problem.

In addition, an alternative optimization method is introduced in the code - the Lagrange Multiplier

Method - and compared to the classical optimization. Using this new method, we assess the influence

of several inputs in the aerodynamic design, testing the effect of including/removing the hub correction

and carrying out parametric studies to see the influence of the tip speed ratio and the drag-to-lift ratio on

the results.

Lastly, the alignment scheme that was recently implemented by Melo is also adapted to the design

problem and we take some preliminary conclusions about its performance and the effect it has on the

aerodynamic design itself.

1.3 Thesis Outline

The outline of this thesis is as follows:

• We start in Chapter 2 by presenting the theoretical basis of the lifting line theory applied to wind

turbines. Generally, we define the system of vortices that models each blade and discuss the

velocities and forces involved and the relations between them, while gradually introducing the

typical non-dimensional variables used in this field.

• In Chapter 3 we move to the implementation, i.e., we see how the theory is adapted to have

practical use. We start by presenting the numerical model, where we explore the discretization

of the aforementioned system of vortices and a couple of methods for computing the induced

velocities. Here, we also address some modelling details, like how the hub is included and how

the optimization is performed, both in the classical way and in the freshly implemented Lagrange

multiplier method. Then, we discuss the computational procedure step by step, for both the case

with and without wake alignment. This chapter ends by presenting the results of the convergence

analysis, where we get an idea of the impact of the discretization on the results.

• Chapter 4 is where we present the results of all the tests mentioned in Section 1.2. In the first

part we deal with the simulations without the wake alignment scheme, starting by comparing the

4

classical optimization with the Lagrange multiplier method and then moving on to show the hub

influence and the results of the parametric studies for the tip speed ratio and the drag-to-lift ratio.

Then, in the second part, we move on to the assessment of the alignment scheme, discussing the

effect of the alignment sections on the wake geometry, the performance and the distributions of

local variables.

• We wrap it up in Chapter 5 by summarizing our conclusions.

5

6

Chapter 2

Theory

2.1 Introduction to the Lifting Line Theory

Frederick Lanchester (1868–1946), Martin Kutta (1867-1944) and Nikolai Joukowsky (1847–1921)

were the founding fathers of the circulation theory of flight [1]. This theory established that the lift by unit

span, ~L, produced by an infinite body of constant cross-section is proportional to the velocity circulation

around a contour that encloses that body, namely

~L = −ρ~V∞ × ~Γ (2.1)

where ρ is the volumetric mass density of the fluid, ~Γ is the velocity circulation and ~V∞ is the velocity of

the undisturbed flow - this is known as the Kutta-Joukowsky theorem [33].

Despite it’s usefulness for bi-dimensional studies, the founding assumption of infinite span rendered

this theory very limited on the study of actual lifting bodies and so the lifting line theory appeared a few

years later (see Chapter 1) as a way to account for the finite span, making the leap from two-dimensional

to three-dimensional problems.

To understand the significance of the span being finite, let us start by recalling that the flow around an

airfoil creates a pressure differential (generally, the average pressure is greater at the lower surface than

at the upper surface) which is roughly proportional to the lift produced. Now, when we have a finite lifting

body, the main difference is that at the tip we have a short-cut between these surfaces with different

average pressures, and so a secondary flow appears around it as the fluid will tend to move from the

higher pressure surface to the lower pressure surface. In equilibrium, the pressure differential will then

be continuously decreasing along the span until reaching the tip, where it will have to be zero. With this

established, it is clear that both the lift and the circulation will also have to be zero at the tip, as all three

are related [33].

Now, the question of how the lifting line theory models this behaviour is not a simple one, and so we

will divide the answer through several sections along this chapter.

7

2.2 System of Vortices

With the arguments from the previous section in mind, one can now start by constructing the system

of vortices that is at the base of the lifting line theory:

• As lift is related to circulation (recall Equation 2.1) and circulation can be introduced in a potential

flow model as a vortex [33], the lifting body is modelled as a bound vortex that goes from root to

tip - this is called the lifting line. It is important to note that this vortex has a continuously varying

intensity along its span (we just discussed how the circulation goes to zero at the tip).

• As circulation must be conserved in space (Helmholtz’s second theorem [3]), this variation of

intensity of the bound vortex entails that a sheet of free vortices must be shed along the wake of

the lifting body, convecting the circulation that is gained or lost at the lifting line - these are called

the trailing vortices.

In our particular case the lifting bodies are the blades of the wind turbine rotor, and so our system of

vortices is comprised of a radial lifting line for each blade, with coordinates

rh ≤ r ≤ R and θk =2π(k − 1)

Zfor k = 1, · · · , Z. (2.2)

where rh is the hub radius, R is the rotor radius and Z is the number of blades, and their respective wakes

of trailing vortices (Figure 2.1), whose geometry is more complex and will be discussed in Section 3.1.1.

In accordance to the theory, the lifting lines have a continuously varying intensity along it’s span -

Γ(r) - and the trailing vortices have a local intensity γ = dΓ(r)dr , proportional to that variation.

Figure 2.1: System of vortices for a single blade. (Adapted from [32].)

8

2.3 Velocity Field

Without considering the vortices yet, on a reference frame that is fixed to the rotor (and thus, rotating)

the velocity field is~V (x, r, θ) = ( U , 0 , ωr ) (2.3)

where U is the velocity of the incoming flow (the wind, which we are considering uniform and aligned

with the axis of the rotor) and ω is the angular velocity of the rotor - this is shown in Figure 2.2.

Figure 2.2: Reference frames and velocity triangle disregarding the effect of the system of vortices.(Adapted from [32].)

Now, when we consider the presence of our system of vortices, we must account for their influence

on the velocity field. The velocity induced by the lifting line k and it’s sheet of trailing vortices can be

computed in any point in space using the Biot-Savart law:

~vk (x, y, z) =1

4π

∫Lk

~Γ× ~R

R3dl +

1

4π

∫Sk

~γ × ~R

R3dS (2.4)

where ~R is the vector that goes from the integration point (on the lifting line when we integrate in Lk, or

on it’s vortex sheet when we integrate in Sk) to the point (x, y, z) where we are computing the induced ve-

locity (a deduction of this expression can be found in Melo’s thesis [31], which is based on Sparenberg’s

book [34]). A time-saving remark to make is that when we only need to compute the induced velocities

over the lifting lines, the first integral can be omitted - in fact, as the lifting lines are axisymmetrically

distributed around the rotor, their influence on the induced velocities cancel each other and so we only

need to consider the effect of the trailing vortices (a formal demonstration of this can be found in [11]).

Accounting for all the lifting lines and sheet vortices is as simple as summing up the influence of each

one:

~v (x, y, z) =

Z∑k=1

~vk(x, y, z) (2.5)

9

Note that in the majority of cases all the blades of the rotor are geometrically identical and under

similar loads, and so it is sagacious to confine the calculations to a single blade and assume equivalent

results for the others.

Now, in accordance to the typical convention found in literature, the components of these induced

velocities can be written as positive quantities in cylindrical coordinates:

va =− vx (2.6a)

vr =vy cos θ + vz sin θ (2.6b)

vt =−vy sin θ + vz cos θ (2.6c)

where a stands for axial, r for radial (sometimes written s, for span-wise) and t for tangential.

Adding these to the original velocity field (Equation 2.3) we get the total velocity field

~V (x, r, θ) = (U − va, vr, ωr + vt) (2.7)

2.4 Forces and Angles

One of the main objectives of aerodynamics is to determine the forces involved in a solid/fluid in-

teraction, and this is what we will address now. Let’s start by rewriting the Kutta-Joukowsky theorem

(Equation 2.1) for the three-dimensional case [35]:

~L = −ρ~V × ~Γ (2.8)

This theorem gives us the lift force per unit span (the only one that is actually modelled by the lifting

line theory) which, by definition, is the component of the resulting force that is perpendicular to the

incoming flow. From this equation we can immediately take a couple of conclusions:

• As the circulation vector ~Γ is aligned with the lifting line, the cross product ~vr × ~Γ is zero. This

means that the radial components of the induced velocities do not contribute to the lifting force.

• As the wake is force-free, the cross product ~V × ~γ must also be zero, and so the trailing vortices

have to be aligned with the local velocity. This is one of the biggest complications we have in this

model because the presence of the trailing vortices will affect the velocity field, which in turn will

affect the trailing vortices, making the problem non-linear.

From the first conclusion we can now write that the magnitude of the lifting force per unit span is

given by

L = ρV Γ (2.9)

where V is the magnitude of the total velocity projected on the blade cross section, i.e., not including the

10

component vr. Drawing a velocity triangle (Figure 2.3) we get that

V =

√(U − va)

2+ (ωr + vt)

2 (2.10)

Figure 2.3: Triangles of velocities and forces. (From [28].)

Before proceeding with our force analysis, let’s point out some angles that appear in the velocity

triangle, as they are quite relevant to this model:

• β - The undisturbed aerodynamic pitch angle, which is the angle between the undisturbed velocity

(Equation 2.3) and the tangential direction, and it is given by

tanβ =U

ωr=

1

λr∗(2.11)

where λ is the tip speed ratio, defined as the ratio of the velocity of the blade tip to the velocity of

the incoming flow

λ =ωR

U(2.12)

and r∗ is the non-dimensional radial position, given by r∗ = r/R. 1

• βi - The induced aerodynamic pitch angle, whose definition is similar to β but includes the effects

of the axial and tangential induced velocities, and so it is given by

tanβi =U − vaωr + vt

=1− v∗aλr∗ + v∗t

(2.13)

where v∗a,t = va,t/U . Note that βi is always lower than β.

• α - The angle of attack, which is the angle between the section chord line and the velocity projected

on the blade cross section.

1From this point on, we will be introducing several non-dimensional quantities, as these broaden the applicability of our resultsand facilitate the design process (for example, in an early design stage it is easier to specify that the hub radius will be 15% of therotor radius than to give it an exact dimension).

11

• ψ - The blade pitch angle, which is the geometrical angle between the blade chord line and the

tangential direction. This is related to the previous angles by

ψ = βi − α (2.14)

Continuing with Figure 2.3, but now focusing on the forces, a few more definitions appear. Note

that these are all projections of the resulting force in particular directions, namely, L - Lift (per unit

span) - is the projection in the direction perpendicular to ~V , D - Drag (per unit span) - in the direction

of ~V , T - Axial force (in a propeller this would be thrust) - in the axial direction, Q/r - Circumferential

force (the component that contributes to the torque Q) - in the tangential direction. Continuing with the

nondimensionalization, the lift, drag, axial force and power coefficients are respectively defined by

CL =L

12ρV

2c=

2Γ

V c=

2Γ∗

V ∗c∗(2.15)

CD =D

12ρV

2c(2.16)

CT =T

12ρU

2πR2(2.17)

CP =P

12ρU

3πR2=

ωQ12ρU

3πR2(2.18)

Note that in Equation 2.15 we used the Kutta-Joukowsky theorem from Equation 2.9 and introduced

the non-dimensional variables Γ∗ = Γ/(UR), c∗ = c/R (where c is the section chord) and V ∗ = V/U =

=

√(1− v∗a)

2+ (λr∗ + v∗t )

2.

Starting by the lift and drag coefficients, CL and CD, they depend on the airfoil that is used and are

usually determined experimentally or using numerical methods. Either way, it is common to assume

that, for a given airfoil, they vary only with the angle of attack, α, and the Reynolds number, defined as

Re = V c/ν, being ν the kinematic viscosity of the fluid. The drag-to-lift ratio, D/L = CD/CL, is a very

important parameter and is represented in this work by the letter ε.

As for the axial force coefficient, CT , one can begin by writing something analogous to Equation 2.9

(remember that L is the lift per unit span):

dT

dr= ρ (ωr + vt) Γ(r) (2.19)

Integrating along the span for the Z blades and introducing the viscous effects with ε [16] we get

T = ρZ

R∫rh

(ωr + vt) Γ(r) (1 + ε tanβi) dr (2.20)

Substituting in Equation 2.17 and using the non-dimensional variables we have defined, we finally

12

get

CT =2Z

π

∫ 1

r∗h

(λr∗ + v∗t ) Γ∗ (1 + ε tanβi) dr∗ (2.21)

Finally, treating the power coefficient, CP , in an analogous way, we can write

CP =2Zλ

π

∫ 1

r∗h

(1− v∗a)Γ∗(

1− ε

tanβi

)r∗dr∗ (2.22)

13

14

Chapter 3

Implementation

3.1 Numerical Model

In this section we will see how we adapted the theory discussed in Section 2 to make it implementable

in a computational code. From this point on, all variables are non-dimensional, and so we will omit the

superscript ∗.

3.1.1 Discretization

Lifting Line

In Section 2.2 we saw that the lifting line is a finite vortex whose intensity is continuously varying from

hub to tip. Unfortunately, when we adapt this to the numerical model we can only mimic this behaviour in

an approximate fashion - in fact, we discretize the lifting line in M consecutive vortex segments that are

assumed to have a constant intensity Γi, which means that the intensity of the lifting line will be varying

by steps, instead of continuously. To the points in the center of these segments we call control points -

ri - and to the points that bound them we call end points - rj (see Figure 3.1).

0

𝑟1 = 𝑟ℎ

𝑟2

𝑟3

𝑟𝑀+1 = 𝑅

ഥ𝑟1

ഥ𝑟2

𝑦

Γ1

Γ2

𝛾1 = Γ1

𝛾2 = Γ2 − Γ1

𝛾𝑀+1 = −Γ𝑀

𝛾3 = Γ3 − Γ2

Figure 3.1: Discretization of the lifting line.

15

The way in which we distribute this points over the lifting line greatly influences the convergence

(this is shown, for example, in [31]) and so we choose to concentrate them in the regions where greater

gradients of circulation are expected:

• If the hub is not included in the model, we concentrate the points both at the hub and the tip (where

circulation drops to zero), using a cosine distribution [36]:

ri =1

2(1 + rh)− 1

2(1− rh) cos

(π (i− 1/2)

M

), i = 1, . . . , M (3.1)

rj =1

2(1 + rh)− 1

2(1− rh) cos

(π (j − 1)

M

), j = 1, . . . , M + 1. (3.2)

• If the hub is included, we focus the points at the tip, using an half-cosine distribution (this is similar

to what Aran & Kinnas [37] use):

ri = rh − (1− rh) cos

(π/2 (i− 1/2)

M+π

2

), i = 1, · · · , M (3.3)

rj = rh − (1− rh) cos

(π/2 (j − 1)

M+π

2

), j = 1, · · · , M + 1. (3.4)

Wake

The wake also suffers from the discretization - what was supposed to be a (continuous) vortex sheet

becomes a finite number of M + 1 concentrated vortices being shed at the end points (rj), whose

intensity is given by the change of circulation at the lifting line:

γ1 = Γ1 ; γj = Γj − Γj−1 for j = 2, · · · ,M ; γM+1 = −ΓM (3.5)

(a) Infinite helicoidal wake. (b) Discretized wake with varying geometry.

Figure 3.2: Alternatives for modelling the wake geometry. (Only the wake of one blade is shown.)

16

As for the geometry of these vortices, if we do not use an alignment scheme they are simply infinite

vortices with constant pitch (equal to the one computed at the lifting line), having thus a perfect helicoidal

geometry. The resulting wake will look like the one in Figure 3.2a.

However, when we use an alignment scheme, as the goal is to get a wake that is aligned with the

local velocity at all points, it will inevitably have a varying geometry and so each individual vortex of the

wake must also be discretized. In this particular alignment scheme, introduced by Baltazar et. al [38]

for propellers and adapted by Melo [31] [32] to turbines, each vortex of the wake is discretized in N

consecutive linear segments, whose endpoints have constant radial position (i.e., the expansion of the

wake is neglected). This number of segments, N , depends on two input parameters: Nt - the number

of equal stream-wise segments per revolution (in Figure 3.2b, for example, Nt = 30), and xuw - the axial

position of the ultimate section, where the wake is truncated (i.e., the axial length of the wake).

The alignment itself starts by choosing the axial coordinates of ns sections of alignment, which are

preferably set near the rotor as that is the region where the variation of the wake geometry is more

pronounced.

From the lifting line to the last section of alignment - named the far wake section, xfw - we call the

transition wake. Here, the induced velocities are computed at each of the ns sections for each of the

M + 1 vortices (i.e., at ns × (M + 1) points) and linearly interpolated between them. The geometry of

the vortices is then defined by recursively computing the coordinates of the end points of each segment

- for vortex j, the coordinates of endpoint n+ 1 are given by

xj,n+1 = xj,n + Vxj,n∆t = xj,n + (U − vaj,n)

2π/Ntω

∗= xj,n + pj,n

(1 +

vtj,nλrj

)2

Nt(3.6)

θj,n+1 = θj,n +Vtj,nrj

∆t = θj,n +ωrj + vtj,n

rj

2π/Ntω

∗= θj,n +

(1 +

vtj,nλrj

)2π

Nt(3.7)

where p = πr tanβi is a dimensionless wake pitch ( ∗= denotes the point after which the variables are

non-dimensional). Recall that, as we are not considering the wake expansion, rj,n+1 = rj,n.

Finally, from the far wake section - xfw - to the ultimate section - xuw - we have the ultimate wake

region. Here, both vt and p are considered constant and equal to the values they have at xfw.

3.1.2 Induced Velocities

When it comes to the numerical computation of the induced velocities there is more than one option,

but the idea behind them is similar - start with the Biot-Savart law (Equation 2.4) and work it so that the

induced velocities can be written as linear combinations of the circulation, i.e.

va,ti =

M∑j=1

Ca,tijΓj (3.8)

To Ca,tij we call the axial and tangential influence coefficients matrices, and the way they are com-

puted is what distinguishes the different methods. In this work, one of two options will be used depending

on the situation:

17

1. As discussed in Section 3.1.1, when the alignment scheme is not used the wake vortices are purely

helicoidal. For this case we only need to compute the induced velocities at the lifting line, which

can be done using the analytical expressions developed by Lerbs [9] and implemented by Duarte

& Falcao de Campos [27]. This is computationally very efficient and only requires knowledge of

the wake geometry, namely, the number of blades - Z, the position of the control - ri - and end

points - rj - and the induced aerodynamic pitch angle - tanβi.

2. If the alignment scheme is used, we employ a numerical integration routine implemented by Melo

[31]. As explained in her work, when we discretize Equation 2.4 and take into account the relation

between Γi and γj (Equation 3.5) we get that

~v (x, y, z) =

= −Z∑k=1

M∑i=1

1

4π

∫ rji+1

rji

~ekr × ~R

R3dr +

∫Lk

j i+1

~ektji+1× ~R

R3dskji+1

−∫Lk

j i

~ektji× ~R

R3dskji

Γi

(3.9)

For each control point i, the first integral accounts for the influence of the ith segment of the lifting

line and is solved analytically (recall that, as discussed in Section 2.3, it takes the value zero over

the lifting lines themselves). As for the other two, they account for the vortices being shed at the

endpoints of that segment (rji and rji+1 ) and are solved numerically. Computationally, this method

is much more expensive but it imposes no restrictions to the wake geometry.

3.1.3 Hub Model

In section 2.1 we discussed how the flow around the tip of the blades makes the circulation go to

zero. Now, when we move to the root of the blade, it would be somewhat inaccurate to consider a similar

behaviour, as the hub acts like a physical barrier to this secondary flow.

Following the typical method for including solid walls in a potential flow model, we use an image

vortex system to model the hub as an infinite cylindrical wall with radius rh. As described by Kerwin

[39] and depicted in Figure 3.3, for each of the M + 1 trailing vortices we include an image vortex with

symmetrical intensity

γ′j = −γj , (3.10)

radial position

rj′ =

rh2

rj(3.11)

and induced aerodynamic pitch angle

(tanβi)′j = (tanβi)j

rjrj ′

(3.12)

18

Figure 3.3: A trailing vortex and it’s image.

In practice, this is done by computing the influence coefficients matrices of the image vortex system

(using the coordinates and the pitch given by Equations 3.11 and 3.12), Ca,t′ij , and subtracting them to

the ones of the original vortex system, that is:

Ca,tijtotal = Ca,tij − Ca,t

′ij (3.13)

3.1.4 Optimization

Now that we are familiarized with the basics of the model we arrive at the most important part: the

optimization. In a typical lifting line problem, either of design or analysis, what we are looking for is

the optimum circulation distribution over the lifting line - Γ(r). We choose to optimize the circulation

because it is an excellent bridge between the aerodynamic problem and the geometrical problem [25]

- for example, having determined the distributions of circulation and induced velocities over the lifting

line (aerodynamic problem) and chosen a distribution of CL(r), we can use Equations 2.15 and 2.14 to

determine the chord and geometrical pitch distributions.

In the specific case of a wind turbine, the optimum circulation distribution is the one that maximizes

the power coefficient, CP , and in this work we will explore two alternatives for finding it - the classical

optimization and the Lagrange multiplier method.

Classical Optimization

As explained in Kerwin’s book [40], the classical optimization was first developed by Betz in 1919 [7]

and later expanded by Lerbs in 1952 [9]. Without going into much detail, it states that if we assume an

uniform inflow and absence of viscous forces, the loss of kinetic energy is minimized in the far wake when

the ratio of the input to output power becomes independent of the radial coordinate. If we further assume

that va << U and vt << ωr (which is only reasonable for lightly loaded turbines), this is equivalent to

saying that the optimum circulation distribution is obtained when

(tanβi)i(tanβ)i

= constant for i = 1, · · · ,M (3.14)

19

which is known as the ”Lerbs criterion” [39]. Here, we will call this constant ` and, taking into account

the expression for tanβ (Equation 2.11), rewrite the criterion as

ri(tanβi)i = ` for i = 1, · · · ,M (3.15)

Using now the expressions for tanβi (Equation 2.13) and the induced velocities (Equation 3.8), we

can write that:

ri1− vaiλri + vti

− ` = ri − riM∑j=1

CaijΓj − `λri − `M∑j=1

CtijΓj =

M∑j=1

(Caij +

`

riCtij

)Γj + `λ− 1 (3.16)

And so, finally, we arrive at the system of equations that we need to solve to find the optimum

circulation distribution:

M∑j=1

(Caij +

ˆ

riCtij

)Γj + `λ = 1 for i = 1, · · · ,M (3.17)

Note that ` is also an unknown and so, to linearise this system of equations, we use an estimation

of it ( ˆ ) where it multiplies with Γj , which is updated in every iteration using the results of the previous

one.

To close our system (so far we have M equations and M+1 unknowns), we impose a load CT0 using

the discretized version of Equation 2.21:

CT0 =2Z

π

M∑i=1

{(λri + vti) (1 + εi (tanβi)i) ∆riΓi} (3.18)

The optimization procedure consists thus in finding the value of CT0that will maximize CP , satisfying

Equations 3.17 and 3.18 (we will explore this further in Section 3.2).

Lagrange Multiplier Method

The alternative that we are introducing to the classical optimization is the Lagrange multiplier method,

which is a variational approach to the optimization problem that was initially explored by Yim [21] and

Kerwin et al. [13].

The idea behind it is quite simple: in order to find the circulation distribution that will yield maximum

power, CP , for a given axial load, CT0, we define an auxiliary function H = CP + `(CT − CT0

) and

formulate our problem as

∂H

∂Γi= 0 ∧ ∂H

∂`= 0, for i = 1, · · · ,M (3.19)

where ` is called the Lagrange multiplier.

Although this system gives us as many equations as we need (M + 1 equations for M values of

Γi and the value of `), the partial derivatives ∂H∂Γi

require some work to become useful. To start, let us

20

discretize Equation 2.21 for CT and Equation 2.22 for CP :

CT =2Z

π

M∑i=1

{(λri + vti) Γi (1 + εi (tanβi)i) ∆ri} (3.20)

CP =2Zλ

π

M∑i=1

{(1− vai) Γi

(1− εi

(tanβi)i

)ri∆ri

}(3.21)

Now, let us take as an example the inviscid part of the expression for the power coefficient. Recalling

that va,ti =∑Mj=1 Ca,tijΓj (Equation 3.8), we can write:

2Zλ

π

M∑i=1

(1− vai) Γiri∆ri =2Zλ

π

M∑i=1

Γiri∆ri − Γiri∆ri

M∑j=1

CaijΓj

(3.22)

The differentiation in Γi of the first term is immediate, as

∂

∂Γi

{2Zλ

π

M∑i=1

Γiri∆ri

}=

∂

∂Γi

{2Zλ

π(Γ1r1∆r1 + Γ2r2∆r2 + · · ·+ ΓM ¯rM∆rM )

}=

2Zλ

πri∆ri

(3.23)

Tackling the second term requires a bit more care, but it can be done in a similar fashion. Let’s begin

by expanding:

M∑i=1

Γiri∆ri

M∑j=1

CaijΓj

= Γ1r1∆r1 (Ca11Γ1 + Ca12Γ2 + · · ·+ Ca1M ΓM ) +

+ Γ2r2∆r2 (Ca21Γ1 + Ca22Γ2 + · · ·+ Ca2M ΓM ) + · · ·

= Ca11Γ∗2

1 r1∆r1 + Ca12Γ1Γ2r1∆r1 + Ca21Γ2Γ1r2∆r2 + · · ·

(3.24)

If we assume∂Ca,tij

∂Γi= 0 [13], the differentiation in Γ1 takes the form:

∂

∂Γ1

(Ca11Γ∗

2

1 r1∆r1 + Ca12Γ1Γ2r1∆r1 + Ca21Γ2Γ1r2∆r2 + · · ·)

=

= 2Ca11Γ1r1∆r1 + Ca12Γ2r1∆r1 + Ca21Γ2r2∆r2 + · · ·

=

M∑j=1

(Ca1jΓj r1∆r1 + Caj1Γj rj∆rj

) (3.25)

We thus reach the conclusion that

∂

∂Γi

M∑i=1

Γiri∆ri

M∑j=1

CaijΓj

=

M∑j=1

(CaijΓj ri∆ri + CajiΓj rj∆rj

)(3.26)

Taking this into account, and treating all the remaining derivatives analogously, we get our final

21

system of equations:

∂H

∂Γi= 0 ⇔

⇔M∑j=1

−λ(1− εi

(tan βi)i

) (Caij ri∆ri + Caji rj∆rj

)+

+ˆ(1 + εi (tanβi)i)(Ctij∆ri + Ctji∆rj

)Γj

+

+ {λ (1 + εi (tanβi)i) ri∆ri} ` = −λ(

1− εi(tanβi)i

)ri∆ri, for i = 1, · · · ,M

(3.27)

∂H

∂`= 0 ⇔ CT0

=2Z

π

M∑i=1

{(λri + vti) (1 + εi (tanβi)i) ∆riΓi} (3.28)

Note that, just like in the classical optimization, the system of equations is linearised using an esti-

mation of ` where needed and the optimization procedure consists of finding the value of CT0that will

maximize CP , satisfying Equations 3.27 and 3.28.

22

3.2 Computational Procedure

We will now see the algorithm that was implemented (using the FORTRAN language) to solve the

aerodynamic design problem, with and without the wake alignment scheme.

3.2.1 Without Wake Alignment

To better understand the computational procedure that is followed when the wake alignment scheme

is not used, a flowchart is presented in Figure 3.4.

Read input

Discretize lifting line

Initialize variables

Compute influence

coefficients matrices

Compute induced velocities

Solve system of equations

for Γ and ℓ

Compute tan 𝛽𝑖 𝑖

Convergence:

tan 𝛽𝑖 𝑖 and ℓ?

Update

tan 𝛽𝑖 𝑖 and ℓ

Compute 𝐶𝑃

𝐶𝑃 max?

Write output

Increase 𝐶𝑇0

Y

Y

N

N

Figure 3.4: Flowchart of the computation procedure without wake alignment.

The first step is to read the input, that is provided by an editable text file. Here, one can change all

the parameters that are shown in Table 3.1 as well as a few computational details, like the maximum

number of iterations and the relaxation factor.

Then, the lifting line is discretized using Equations 3.1 and 3.2 or 3.3 and 3.4, depending on the

distribution that is chosen.

At this point, all the yet unspecified variables are initialized at zero, except for the aerodynamic pitch

angle, which is set to the value of the undisturbed aerodynamic pitch angle (which is in accordance with

23

the present absence of induced velocities - va,t = 0):

tanβi = tanβ =1

λr(3.29)

Having the wake geometry defined (Z,ri,rj and (tanβi)) we can now compute the initial axial and

tangential influence coefficients matrices - Ca,tij (Section 3.1.2).

The induced velocities va,t then come easily from Equation 3.8, although in this first iteration they will

just be zero, as the circulation is also zero at all points.

Having now everything we need to solve our system of equations (Equations 3.17 and 3.18 or 3.27

and 3.28, depending on the optimization that is used) we get new values of Γ and `.

Using Equation 2.13 we update the aerodynamic pitch angle which, again, is a futile computation in

the first iteration.

Although we have computed everything that we need, the variables at this point are neither converged

or optimized, and so we see two cycles in the flowchart:

• The inner cycle ensures convergence: at the end of each iteration of this cycle, the new values of

(tanβi) and ` are compared with the values obtained in the previous iteration, and we assume that

convergence is achieved when:∣∣∣∣∣ (tanβi)inew − (tanβi)i

previous

(tanβi)inew

∣∣∣∣∣ < εN ∧∣∣∣∣` new − ` previous

` new

∣∣∣∣ < εN (3.30)

where εN is a tolerance set in the input.

It is important to mention that, for numerical stability, when convergence is not achieved we apply

under-relaxation to tanβi before moving on to the next iteration, i.e.:

(tanβi)inext

= κN (tanβi)inew

+ (1− κN ) (tanβi)iprevious (3.31)

where κN is the under-relaxation factor, also defined in the input.

• The outer cycle ensures global load optimization: each time convergence is achieved, the power

coefficient CP is computed and compared with the value of the previous iteration. As we will see in

Chapter 4, the curve CP (CT ) always has a maximum, and so the procedure for finding it consists

in starting with a value of CT0 and incrementing it slowly until reaching the optimum point.

Having ensured both convergence and optimization, the last step is to write all important variables in

output files.

24

3.2.2 With Wake Alignment

Figure 3.5 shows the algorithm that is followed when the wake alignment scheme is used.

Y

Read input

Discretize lifting line

Initialize variables

Compute influence

coefficients matrices at LL

Compute induced

velocities at LL

Solve system of equations

for Γ and ℓ

Compute tan 𝛽𝑖 𝑖

Convergence:

tan 𝛽𝑖 𝑖 and ℓ?

Update

tan 𝛽𝑖 𝑖 and ℓ

Align Wake

Convergence:

𝑝 and 𝑣𝑡?

Y

N

N

Initialize wake

Update wake

geometry

Compute 𝐶𝑃

𝐶𝑃 max?Increase 𝐶𝑇0

Y

N

Write output

Figure 3.5: Flowchart of the computation procedure with wake alignment. (LL stands for ”lifting line”.)

A quick comparison with the previous section shows that several of the steps are unchanged, and so

we will only address the differences.

As discussed in Section 3.1.1, when we use the alignment scheme each individual vortex of the wake

is discretized in several consecutive linear vortex segments. In the beginning of the program we have

thus to initialize the geometry of this system of trailing vortices, which is done using Equations 3.6 and

3.7. Note that at this point vt = 0, tanβi has been initialized with Equation 3.29 and the discretization

parameters (Nt and xuw) and axial coordinates of the alignment sections have been read from the input

file.

The alignment per se appears a few steps later, when convergence is achieved at the lifting line, and

consists in computing influence coefficients matrices at every point of alignment, and then using them

25

to obtain the induced velocities and pitches at these points. Recall that in this case we can not use

Lerbs analytical expressions, and so these matrices are computed numerically from a manipulation of

Equation 3.9.

Analogously to what is done at the lifting line, the convergence of the alignment is assessed by com-

paring the values of p and vt at the wake in consecutive iterations, i.e., we say the wake has converged

when: ∣∣∣∣pj new − pj previous

pj new

∣∣∣∣ < εW ∧

∣∣∣∣∣vtj new − vtj previous

vtjnew

∣∣∣∣∣ < εW (3.32)

where εW is another tolerance set in the input.

When convergence is not achieved, the geometry of the wake is updated (again, with Equations 3.6

and 3.7) using the new values of p and vt after under-relaxation:

pjnext = κW pj

new + (1− κW ) pjprevious (3.33)

vtjnext = κW vtj

new + (1− κW ) vtjprevious (3.34)

with κW being the under-relaxation factor.

As a last remark, notice that the computation of the influence coefficient matrices at the lifting line

was taken out of the convergence cycle for tanβi and ` to save computational time.

26

3.3 Convergence Analysis

Numerical methods are inevitably bound to uncertainties which, according to Eca et al. [41], can be

divided in three components: the round-off error, the iterative error and the discretization error. As the

last one is usually the main contributor, we dedicate this section to evaluate the effects of the discretiza-

tion of our system of vortices on the solutions.

To emulate what would be a typical design problem, we used the conditions listed in Table 3.1 as

input for this convergence analysis (note that, however, the load is being imposed and not optimized).

Number of Blades - Z 3

Hub Radius - rh/R 0.15

Tip Speed Ratio - λ 7

Drag-to-Lift Ratio - ε 0.01

Interpolation Linear

Optimization Lagrange

Imposed Load - CT00.8

Hub Correction Yes

Point Distribution over the Lifting Line Half-Cosine

Tolerance 10−4

Table 3.1: Input for the convergence analysis.

3.3.1 Discretization of the Lifting Line

To assess the convergence of the solution with the discretization of the lifting line, six tests were ran

where every input was kept unchanged (see Table 3.1) except for the number of segments in which the

lifting line was divided - M . In these tests, the wake alignment scheme was not used and so the matrices

Ca,tij were computed using the Lerbs analytical expressions.

In Figure 3.6 we start by evaluating the convergence of CP , as it is a global variable that is affected

by all the values of circulation that come out of the system of equations (recall Equation 3.21) and so it

should be a good overall indicator of convergence.

27

10 20 30 40 50 60M

0.489

0.4895

0.49

0.4905

0.491

0.4915

0.492

CP

Figure 3.6: Convergence of the power coefficient with the increase of the number of lifting line segments,M .

First, we note that CP (M) is monotonically decreasing, which tells us that it is unlikely that we

reached a point where the discretization error is no longer dominant (note that both the round-off error

and the iterative error increase with the discretization [41]). Then, the fact that the difference between

consecutive points is diminishing (for the first two it is around 36 times higher than for the last two) lets

us induce that the program is indeed converging to a specific value. It is also important to take into

account the scale of the CP axis - in the last points we are dealing with differences in the order of 10−5,

which even goes beyond the tolerance that was used. Overall, these are indicators of good convergence

behaviour and allow us to assume that there are no major mistakes in the code.

0.2 0.4 0.6 0.8 1r/R

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

!/(

UR

)

M=10M=20M=30M=40M=50M=60

0.95 0.96 0.97

0.075

0.08

0.085

0.09

(a) Circulation.

0.2 0.4 0.6 0.8 1r/R

0.086

0.088

0.09

0.092

0.094

0.096

r/R

*tan

(-i)

M=10M=20M=30M=40M=50M=60

0.95 10.0952

0.0954

0.0956

0.0958

0.096

(b) Induced aerodynamic pitch.

Figure 3.7: Influence of the number of lifting line segments, M , on some local variables.

28

As for Figures 3.7a and 3.7b, they show the influence of the discretization on some local variables.

The circulation distributions (which are what comes out of the system of equations) are practically over-

lapping from hub to tip and so they do not tell us much, apart from the fact that the increase of M allows

us to get results for points closer to the extremes. In the aerodynamic pitch, however, we can see a clear

difference from 10 to 20 elements and from 20 to 30, but after that the differences are negligible (again

in the order of 10−5). 30 to 40 elements should thus be range after which the extra computational costs

become futile.

3.3.2 Discretization of the Wake

In Section 3.1.1 we discussed the wake model that is used with the alignment scheme and its dis-

cretization inputs: Nt - the number of equal stream-wise segments per revolution and xuw - the axial

length of the wake. To evaluate the performance of the model and the weight of these inputs on the

discretization error, 12 tests were conducted in which the wake was aligned at the lifting line using all

combinations of xuw/R = {10, 25, 50} and Nt = {10, 100, 500, 1000}. The lifting line was discretized in

M = 30 elements and everything else was kept unchanged from the previous tests (see Table 3.1).

101 102 103

Nt

0.43

0.44

0.45

0.46

0.47

0.48

0.49

CP

xuw

/R=10

xuw

/R=25

xuw

/R=50

Lerbs

Figure 3.8: Convergence of the power coefficient with the increase of the number of stream-wise seg-ments per revolution - Nt - and the axial length of the wake - xuw.

The first conclusion to be taken from the results (Figure 3.8) is that the wake model is working well, as

it is converging to a value close to the one obtained for a wake of infinite and perfectly helicoidal vortices

(which was computed using Lerbs analytical expressions). Secondly, in the range tested, it seems to

be more efficient to increase Nt than to increase xuw - for example, using (Nt, xuw) = (100; 50) takes

practically the same time to compute as using (Nt, xuw) = (500; 10) but the achieved results are around

three times farther from the Lerbs line.

There is, however, a less predictable result - Melo [31] performed similar tests for a classical analysis

problem and saw monotonic convergence for both Nt and xuw, which here does not always hold true

when we move from xuw = 25 to xuw = 50. Note, however, that we are solving a design problem and

29

using the Lagrange multiplier method, which allows the wake pitch to be variable on the radial direction

(this will be seen in Section 4.1.2), and so our conditions are quite different from Melo’s and we can

not really draw conclusions from this comparison. It is also worth mentioning that in these tests we are

reaching wakes with close to three million points separated by distances that can reach the order of

10−9, and so it is possible that we are entering the range where round-off errors start playing a relevant

role.

30

Chapter 4

Results

4.1 Results Without Wake Alignment

4.1.1 Input

The computations without wake alignment were carried out using the following inputs, except when

stated otherwise (the contents between brackets are subject to change in some sections, depending on

the tests that are being performed):

Number of Blades - Z 3

Hub Radius - rhR 0.15

Tip Speed Ratio - λ (7)

Drag-to-Lift Ratio - ε (0)

Interpolation Linear

Optimization (Lagrange)

Hub Correction (Yes)

Point Distribution over the Lifting Line (Half-Cosine)

Table 4.1: Input for the tests without wake alignment.

These conditions are very similar to the ones used for the convergence analysis (Section 3.3), but

two important differences should be pointed out:

1. Most tests consider the inviscid case, i.e., ε = 0. This is a common practice in literature when a

model is being tested and not actually used for a specific design problem, as the blade section is

not defined at this point and the inclusion of drag is generally irrelevant for most of the comparisons

that are being performed, resulting in a futile increase of computational cost.

2. True optimization is performed, i.e., the load, CT0, is no longer imposed but rather optimized,

resulting in the highest possible CP for the model in use.

31

As for the discretization and the numerical tolerance, recall that in Section 3.1.2 we mentioned that,

when the wake alignment scheme is not used, the Ca,tij matrices can be determined using the Lerbs

expressions and so computational time is considerably reduced. Taking this fact into account, and the

conclusions from the convergence analysis (Section 3.3.1), we decided to carry the following tests using

M = 40 elements over the lifting line and a tolerance of 10−4.

4.1.2 Classical Optimization vs. Lagrange Multiplier Method

Let us start by the comparison between the classical optimization and the freshly implemented La-

grange multiplier method.

0.6 0.65 0.7 0.75 0.8 0.85 0.9C

T0

0.46

0.47

0.48

0.49

0.5

0.51

0.52

CP

(0.805, 0.52024)

(0.813, 0.52367)

ClassicalLagrange

Figure 4.1: Evolution of the power coefficient with the load for both classical and Lagrange multiplieroptimization.

In Figure 4.1 we can see how the power coefficient, CP , varies with the axial force coefficient, CT ,

for both methods of optimization. These are global variables and so they give us an overview of the

performance of these methods.

The first immediate conclusion is that the Lagrange multiplier method outperforms the classical opti-

mization in the entirety of the tested range. In terms of the optimum point, marked with ”×” in the plots,

we see an increase of 3.43 · 10−3 which, although small, is a welcome improvement, considering that

the difference in computational time is negligible. It is also interesting to note that the difference between

the curves is continuously increasing with the value of CT , starting as low as 5.05 · 10−4 and reaching

1.54 · 10−2. Recall that in Section 3.1.4 we mentioned that the assumptions used during the deduction

of the classical optimization are only reasonable for lightly loaded turbines, so this is not a big surprise.

32

0.2 0.4 0.6 0.8 1r/R

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

!/(

UR

)

ClassicalLagrange

(a) Circulation.

0.2 0.4 0.6 0.8 1r/R

0.086

0.087

0.088

0.089

0.09

0.091

0.092

0.093

r/R

*tan

(-i)

ClassicalLagrange

(b) Induced aerodynamic pitch.

0.2 0.4 0.6 0.8 1r/R

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

v a/U

ClassicalLagrange

(c) Induced axial velocity.

0.2 0.4 0.6 0.8 1r/R

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

v t/U

ClassicalLagrange

(d) Induced tangential velocity.

Figure 4.2: Comparison of the classical optimization with the Lagrange multiplier method for the sameflow conditions and optimum CT0

.

Moving now to local variables, Figures 4.2a to 4.2d compare the results for the optimum points of

both methods.

Starting with the circulation distribution (which, again, is what comes out of the system of equations),

we can now see why an half-cosine distribution was chosen. Like we expected from the theory, at the

hub we have a finite value of circulation and at the tip it falls to zero. What was not so easy to predict,

however, is that this fall would be concentrated only in the last 1/5 of the blade - this intense gradient is

the true reason why we concentrate more points at the tip.

Comparing now the two methods of optimization, the most noticeable difference appears in the in-

duced aerodynamic pitch (Figure 4.2b). Recall that when we developed the system of equations for the

classical optimization the starting point was imposing a constant pitch over the lifting line, and so the

results for that method are consistent with the theory. When we move to the Lagrange multiplier method

33

we impose no restrictions to the pitch, and this extra degree of freedom reveals itself as a continuously

increasing pitch from hub to tip.

However, and taking into account the scale of the ordinate axis, we can see that the dissemblance of

these curves is negligible near the tip. Interestingly, the same applies to all the other plots, so this seems

to be worth exploring. In 2003, Ribeiro & Falcao de Campos [42] compared methods of optimization

similar to the ones that we are studying here and concluded that if we take the system of equations

for the Lagrange multiplier method (Equation 3.27), set ε = 0 and remove the terms that multiply by

Ca,tji we get exactly the system of equations of the classical optimization. Referring again to Section

3.1.4, we already know that the classical method assumes absence of viscous forces, so having to set

ε = 0 is not astonishing. The big difference between the methods is thus hidden in the presence or

absence of the Ca,tji matrices, which must have vanished in the classical optimization during one of

the approximations that were made. In fact, in 1986, Andersen [43] solved the optimization problem

numerically without some of the approximations used by Betz and showed that the differences in the

results were only noticeable near the blade root (this is discussed in [42]).

It is also worth mentioning that this difference in the aerodynamic pitches seems to come more from

a correction of va than vt - recall that, according to Equation 2.13, the pitch decreases with both of them.

Lastly, mixing the conclusions that we drew from Figures 4.1 and 4.2, we can infer that the differences

that appear near the hub on the local variables should lessen for lighter loads. A quick test shows that is

exactly what happens - in Figure 4.3 we can see how the optimum distribution of induced aerodynamic

pitch tends to a straight line with the decrease of CT0.

0.2 0.4 0.6 0.8 1r/R

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

r/R

*tan

(-i)

CT

0

=0.1

CT

0

=0.5

CT

0

=0.9

Figure 4.3: Optimum distribution of induced aerodynamic pitch using the Lagrange multiplier method forseveral values of CT0 .

34

4.1.3 Hub Influence

In section 3.1.3 we briefly described how the hub is included in the lifting line model and now it is

time to see how it influences our results.

0.6 0.65 0.7 0.75 0.8 0.85 0.9C

T0

0.46

0.47

0.48

0.49

0.5

0.51

0.52

CP

(0.804, 0.51651)

(0.813, 0.52367)

HublessWith Hub

Figure 4.4: Evolution of the power coefficient with the load for the cases with and without hub correction.

Like before, we start by assessing global variables in Figure 4.4.

Qualitatively, the behaviour is very similar to what we saw in the previous comparison - the inclusion

of the hub yields higher values of CP for the entirety of the range of CT that was tested, reaching the

maximum at a higher value of CT than the hubless case. Moreover, the distance between the curves

is also continuously increasing with CT , but by bigger values, starting at 2.77 · 10−3 and ending at

1.74 · 10−2. The inclusion of the hub seems thus to have a higher influence on CP than the type of

optimization, especially at lower loads.

There is, however, an important thing to remember when reading these results - we are approximating

the hub to an infinite cylinder. This has consequences not only on the behaviour of local variables in the

vicinity of the hub, but also on the aerodynamic forces that act on the turbine - as explained by Epps

[22] [25], in a real (finite) hub, the image vorticity that is shed from it rolls up into an additional vortex

and creates a low pressure zone that generates an additional drag force. As we are not considering this

effect, there might be an added inaccuracy in the performance of the turbine with the hub model.

Looking now at Figures 4.5a to 4.5d we can see the effect of the hub on some local variables (recall

that we are using cosine spacing for the hubless study).

35

0.2 0.4 0.6 0.8 1r/R

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

!/(

UR

)

HublessWith Hub

(a) Circulation.

0.2 0.4 0.6 0.8 1r/R

0.084

0.085

0.086

0.087

0.088

0.089

0.09

0.091

0.092

0.093

r/R

*tan

(-i)

HublessWith Hub

(b) Induced aerodynamic pitch.

0.2 0.4 0.6 0.8 1r/R

0.305

0.31

0.315

0.32

0.325

0.33

0.335

0.34

0.345

0.35

v a/U

HublessWith Hub

(c) Induced axial velocity.

0.2 0.4 0.6 0.8 1r/R

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

v t/U

HublessWith Hub

(d) Induced tangential velocity.

Figure 4.5: Effect of the hub correction on the design for the same flow conditions and optimum CT0 .

Overall, the plots with and without the hub are very similar - the induced tangential velocity is almost

coincident from root to tip and the induced axial velocity shows only a minor difference, which translates

into an also minor difference in the aerodynamic pitch. The only clear dissemblance is in the circulation

distribution near the root - as expected, when the hub is not included the circulation falls to zero (for the

same reason we have at the tip - recall Section 2.1) but when it is, we have a finite value of circulation.

From the resemblance of these curves one might believe that the inclusion of the hub is futile, but

keep in mind that we did see a difference in performance and also that the goal of these models is

ultimately to provide the optimum blade geometry for a given condition. If we look at Equation 2.15 we

can see that the blade chord is proportional to the circulation, which means that if we perform our design

without considering the effect of the hub we may get a blade that vanishes at the root. Another possibility

is to have the lift coefficient going to zero there, which will affect the angle of attack and, consequently,

36

the blade pitch angle. Either way, the output geometry will be considerably different in the vicinity of the

hub (one can see this, for example, in [44]).

4.1.4 Parametric Studies

So far we have been focused on the consequences of some choices that we have to make on the

model itself. In this section, however, we will move to a different kind of exercise - our model will remain

the same and we will see the effect of changing a couple of design parameters.

Tip Speed Ratio

Our first parametric study consists in keeping every input constant except for the tip speed ratio which,

as defined in Equation 2.12, is the ratio between the velocity of the blade tip (which is proportional to the

angular velocity of the rotor) and the velocity of the incoming flow.

4 5 6 7 8 9 106

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

CP

(a) Evolution with the tip speed ratio for both the inviscidand a viscous case.

0.6 0.65 0.7 0.75 0.8 0.85 0.9C

T0

0.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

CP

(0.796, 0.50037)

(0.813, 0.52367)

(0.824, 0.53655)

6=56=76=9

(b) Evolution with CT0 for several tip speed ratios.

Figure 4.6: Effect of the tip speed ratio on the power coefficient.

Starting with Figure 4.6a, we can see how the power coefficient is affected by the tip speed ratio for

both the inviscid and a viscous case.1

In the inviscid case we can see a very well known result - the power coefficient increases monotoni-

cally with the tip speed ratio. Note, however, that the slope of this curve is continuously decreasing. This

means that the benefit of raising λ gets increasingly smaller and, if we were to continue the analysis

to even greater values of λ, we would see that it eventually becomes negligible. Out of curiosity, the

asymptote of this curve for the case of infinite blades is known as the Betz limit and takes the value of

16/27 ( ∼= 0.5926) - theoretically, no wind turbine can ever surpass this value [45].

1For clarification, this is not to see how a given turbine performs under different conditions but rather to assess the performancethat we can expect from turbines that are optimized for different values of tip speed ratio.

37

The question now is why the performance drops at low tip speed ratios. To answer this, we will follow

a simple argument [46] [35] - as the fluid exerts a torque on the rotor, the rotor also exerts a torque on the

fluid and so, because there must be conservation of angular momentum, the wake acquires a rotational

speed (whose direction is opposite to that of the rotor’s). This rotation of the wake is essentially a loss

of kinetic energy and so it reduces the performance of the turbine. Consequently, working with high

torques and low rotational speeds (and thus, low tip-speed ratios) is less efficient than working with low

torques and high rotational speeds.

With this settled, one might ask why turbines are not designed for very high values of λ. The answer

to this is also shown in the Figure that we are analysing (4.6a) - when we include viscous drag in our

design (which is more realistic) the curve of CP (λ) is no longer monotonically increasing because after

a certain point viscous losses start overcoming the reduction of the kinetic energy losses [36]. This

can also be checked mathematically - if we look at Equation 3.21 we can see that the viscous term

multiplies by 1/ tan(βi), which increases with λ. Moving away from aerodynamics for a second, it is

worth mentioning that there are additional impediments for using high values of λ, like the increase of

the generated noise and the higher axial stresses on the blade (due to the centrifugal forces) [47].

Moving to Figure 4.6, we can only add that, with the increase of λ, the maximum of CP is achieved

at an higher value of CT (recall that this is the inviscid case).

Figures 4.7a to 4.7d show us now the effect of the tip speed ratio on the distributions of the local

variables (for the inviscid case). This is a parameter that clearly affects all of them from hub to tip but

the overall shape of the curves is very similar - they mostly seem to be scaled on the ordinate axis.

Starting by the induced tangential velocity, notice how it increases across the whole span when the

tip speed ratio decreases, which is in accordance with the aforementioned increase of the losses of

kinetic energy in the wake rotation.

Focusing now on the aerodynamic pitch, we can see that its value decreases with the tip speed

ratio for the whole span. In Section 2.4 we saw that the aerodynamic pitch is always lower than the

undisturbed aerodynamic pitch which, in turn, is proportional to the inverse of the tip speed ratio (tanβ =

1/λr), so this is exactly the result that we were expecting. Physically, this means that the velocity vector

will be closer to the tangential direction (recall Figure 2.3) and so the loops of the helicoidal wake will be

closer together.

Having concluded that the geometry of the wake is inevitably altered by the tip speed ratio, and

knowing that this geometry is the only input for the computation of the influence coefficients matrices

(Section 3.1.2) which, in turn, affect both the computation of the induced velocities and the system of

equations that is solved for the circulation distribution, it is now intelligible why all the studied variables

are affected by this parameter.

38

0.2 0.4 0.6 0.8 1r/R

0

0.05

0.1

0.15

0.2!

/(U

R)

6=56=76=9

(a) Circulation.

0.2 0.4 0.6 0.8 1r/R

0.07

0.08

0.09

0.1

0.11

0.12

0.13

r/R

*tan

(-i)

6=56=76=9

(b) Induced aerodynamic pitch.

0.2 0.4 0.6 0.8 1r/R

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

v a/U

6=56=76=9

(c) Induced axial velocity.

0.2 0.4 0.6 0.8 1r/R

0.05

0.1

0.15

0.2

v t/U

6=56=76=9

(d) Induced tangential velocity.

Figure 4.7: Effect of the tip speed ratio on the design for the same flow conditions and optimum CT0 .

39

Drag-to-Lift Ratio

Following the discussions of the previous section, and looking at our system of equations, we can

identify the last parameter that we must discuss - the drag-to-lift ratio. With this said, we performed tests

for several values of ε, which we considered to be constant along the span of the rotor blades.

0 0.01 0.02 0.03 0.04 0.050.3

0.35

0.4

0.45

0.5

0.55

CP

(Wilson et. al.)

(a) Evolution with the drag-to-lift ratio for optimum CT0 .

0.6 0.65 0.7 0.75 0.8 0.85 0.9C

T0

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

CP

(0.813, 0.52367)

(0.806, 0.48951)

(0.793, 0.4399)

(b) Evolution with CT0 for several drag-to-lift ratios.

Figure 4.8: Effect of the drag-to-lift ratio on the power coefficient.

Once again, we start with the global variables in Figures 4.8a and 4.8b.

The first one shows a very curious result - when everything else is kept constant, the power coefficient

decreases almost linearly (R2 = 0.9996) with the drag-to-lift ratio. It is not surprising that it decreases,

as intuition or a quick glance at Equation 3.21 would tell us, but the fact that it is linear was not easily

predictable. In fact, most of the authors choose to show viscous effects combined with the tip speed ratio

(in a plot similar to Figure 4.6a), and so we could not find a CP (ε) curve in any publication of the lifting

line method. However, broadening our search to wind turbines in general, there is a curve fit obtained

by Wilson et. al. [48] using the momentum theory that can be rewritten to corroborate our findings [46],

showing a linear dependence with ε:

CP ∼=16

27λ

[λ+

1.32 +(λ−820

)2Z2/3

]−1

− ε 0.57λ2

λ+ 12Z

(4.1)

This is plotted along with our results and, curiously, the maximum deviation is just around 3.76%2.

As for the second Figure, it reaffirms that higher values of maximum CP are usually associated to

higher values of optimum CT0.

2Take into account that this is a fit made for a broad range of cases, more precisely, for 0 ≤ Z ≤ 3, 4 ≤ λ ≤ 20 and0 ≤ ε ≤ 0.04.

40

0.2 0.4 0.6 0.8 1r/R

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

!/(

UR

)

(a) Circulation.

0.2 0.4 0.6 0.8 1r/R

0.086

0.088

0.09

0.092

0.094

0.096

0.098

r/R

*tan

(-i)

(b) Induced aerodynamic pitch.

0.2 0.4 0.6 0.8 1r/R

0.3

0.31

0.32

0.33

0.34

0.35

v a/U

(c) Induced axial velocity.

0.2 0.4 0.6 0.8 1r/R

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

v t/U

(d) Induced tangential velocity.

Figure 4.9: Effect of drag-to-lift ratio on the design for the same flow conditions and optimum CT0 .

Advancing to the local variables, the results are shown in Figures 4.9a to 4.9d.

Starting by the optimum circulation distributions, we can see that they are very similar, with even

some overlapping points. This is in agreement with the findings of several authors, like Chattot [16],

Falcao de Campos [36] and Menendez Aran [37], and tells us that, although it has a strong influence in

the performance of the turbine, viscous drag only causes minor changes in the aerodynamic design.

As for the induced tangential velocities the case is similar, with the curves being almost indistin-

guishable from hub to tip. The true difference is thus in the distributions of induced axial velocity, which

then translates into a difference in the distributions of aerodynamic pitch. Recalling that the classical

optimization, which predicts the optimum aerodynamic pitch to be constant along the span, assumes

absence of viscous forces, it is interesting to note here that the inviscid case is in fact the one where this

distribution is closer to constant.

41

4.2 Results with Wake Alignment

4.2.1 Input

For the tests using the wake alignment scheme, the input was the following:

Number of Blades - Z 3

Hub Radius - rhR 0.15

Tip Speed Ratio - λ 7

Drag-to-Lift Ratio - ε 0

Interpolation Linear

Optimization Lagrange

Imposed Load - CT00.8

Hub Correction Yes

Point Distribution over the Lifting Line Half-Cosine

Table 4.2: Input for the tests with wake alignment.

Recall that in this case the Ca,tij matrices are computed by numerical integration, which takes a

considerably longer amount of time to execute (Section 3.1.2). For this reason, and because we are

only using the results to have an idea of how the alignment scheme is performing, several measures

were taken to reduce computation time:

• Like in the convergence analysis, the load is again imposed and not optimized, which lets us avoid

the outer cycle of the computational procedure (Section 3.2.2) but does not yield a truly optimum

solution.

• In terms of discretization, the number of elements over the lifting line was reduced to M = 30, and

for the wake we used Nt = 100 and xuw = 25.

• The tolerance was alleviated to 10−3.

4.2.2 Results

Other than the inputs listed above, there is a very important decision to make regarding the wake

alignment scheme: the number and location of the alignment sections. As we are still in an exploratory

phase, what we will do in this section is essentially test several possibilities for that input and analyse

the outcome. While this method is essentially trial and error, it is not entirely random - as we mentioned

in Section 3.1.1, the variation of the wake geometry is more pronounced near the lifting line, so this is

the region where we should be experimenting.

42

Wake Geometries

(a) x/R = {0}. (b) x/R = {0, 0.2}.

(c) x/R = {0, 0.5}. (d) x/R = {0, 1}.

(e) x/R = {0, 0.2, 1}. (f) x/R = {0, 0.25, 1}.

(g) x/R = {0, 0.2, 0.5}. (h) x/R = {0, 0.3, 1}.

Figure 4.10: Wake geometries for some of the tested combinations of alignment sections. (Only thewake of one blade is shown.)

43

Let us start by performing a simple visual inspection of some output wake geometries, shown in

Figures 4.10a to 4.10h.

You may recall from Section 3.1.1 that this alignment scheme does not compute the radial compo-

nents of the induced velocities, and so the radial position of the trailing vortices remains constant and

there are two phenomena that we can not truly capture - the wake expansion and roll-up. We can, how-

ever, look for behaviours that hint into those effects. In Figure 4.10c, for example, we can see how, after

a certain point, the wake starts folding in the downstream direction - this gives us some confidence in

the results as that is the direction in which the roll-up should happen.

However, there are also cases where we do not see this happening. If, for instance, we move to

Figure 4.10b we can see almost no difference between it and the unaligned case (Figure 4.10a). This

probably has to do with the fact that the roll-up gets more pronounced as we move downstream and

this is the case where we tested the shortest transition wake, failing thus to capture it. Then, we also

have the clear failures. As noticed by Melo in the analysis problem [31] [32], it is very difficult to achieve

convergence when the alignment sections are too close to each other. Attempts to align the wake

closer to the lifting line, like at x/R = 0.1 or x/R = 0.15, simply did not converge. Other cases, like

x/R = {0, 0.2, 0.5} (Figure 4.10g) and x/R = {0, 0.3, 1} (Figure 4.10h), did eventually converge but

yielded strange wake geometries - in the first one the tip is curving in the direction opposite to the one it

is supposed to and in the second one the tip changes direction four times, looking similar to a ζ .

Performance

The power coefficients that were obtained for all the combinations of alignment sections that con-

verged are presented in Table 4.3.

Recalling that we are only optimizing for a fixed value of CT0 , we can not draw definite conclusions

about the effect on performance. Still, it is interesting to note that the number of alignment sections

seems to have a much stronger effect on CP than their location. Another curious fact is that its value is

decreasing with that number of alignment sections, which is opposite to the trend that Melo found in the

analysis problem.

x/R CP x/R CP

{0} 0.5177 {0, 0.2, 1} 0.4350

{0, 0.2} 0.4601 {0, 0.25, 1} 0.4340

{0, 0.5} 0.4702 {0, 0.3, 1} 0.4230

{0, 1} 0.4720 {0, 0.35, 1} 0.4255

{0, 0.2, 0.5} 0.4309 {0, 0.5, 1} 0.4293

Table 4.3: Power coefficient for different combinations of alignment sections.

44

Local Variables

It is now time to get to the details and explore the distributions of local variables. For this analysis,

we will compare the results of three different choices of alignment sections - x/R = {0}, x/R = {0, 1}

and x/R = {0, 0.25, 1} - as these are all cases that produced plausible wake geometries and relatively

smooth distributions.

So far we have only had to worry about what happens at the lifting line, but now, with the use of the

alignment scheme, it is also pertinent to look at what happens at the alignment sections. Thereby, we

will separate the results for the circulation and the induced axial velocity from the induced aerodynamic

pitch and the tangential induced velocity, as these last two are the ones who figure in the definition of

the wake geometry (Equations 3.6 and 3.7).

0.2 0.4 0.6 0.8 1r/R

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

!/(

UR

)

Aligned at x/R=0Aligned at x/R={0,1}Aligned at x/R={0,0.25,1}

(a) Circulation.

0.2 0.4 0.6 0.8 1r/R

0.3

0.35

0.4

0.45

0.5

0.55

0.6

v a/U

Aligned at x/R=0Aligned at x/R={0,1}Aligned at x/R={0,0.25,1}

(b) Induced axial velocity.

Figure 4.11: Effect of the alignment on the design for fixed CT0 .

Starting by Figure 4.11, we can see that with the addition of alignment sections the circulation distri-

bution only suffers minor changes, while the induced axial velocity increases across the whole span.

Recalling the expression for the power coefficient (Equation 3.21), we see that it is proportional to

(1− vai) Γi, and so we can conclude that the change in CP seen in Table 4.3 is essentially due to

this disparity of va, being both somewhat more pronounced from x/R = {0} to x/R = {0, 1} than from

x/R = {0, 1} to x/R = {0, 0.25, 1}.

Another thing to point out is the behaviour at the tip. As this is a region of very intense gradients and

also where the discretization of the wake vortices is coarser in the streamwise direction (Figure 3.2b),

convergence is often difficult. In this particular case it was not really a problem, as we can only see a

slight oscillation in the distributions and a point out of place, but there are, however, cases like the one

shown in Figure 4.13, where although everything is smooth for 90-95% of the span, the tip is packed

with instability.

45

0.2 0.4 0.6 0.8 1r/R

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

r/R

*tan

(-i)

x/R=0, Case Ax/R=0, Case Bx/R=0, Case Cx/R=0.25, Case Bx/R=0.25, Case Cx/R=1, Case Bx/R=1, Case C

(a) Induced aerodynamic pitch.

0.2 0.4 0.6 0.8 1r/R

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

v t/U

x/R=0, Case Ax/R=0, Case Bx/R=0, Case Cx/R=0.25, Case Bx/R=0.25, Case Cx/R=1, Case Bx/R=1, Case C

(b) Induced tangential velocity.

Figure 4.12: Distributions of local variables at the alignment sections when the wake is aligned at x/R ={0} - Case A, x/R = {0, 1} - Case B, and x/R = {0, 0.25, 1} - Case C. (The line x/R = 0.25 for Case Bwas obtained by interpolation.)

Moving now to Figure 4.12, it allows us to take a few conclusions about what is happening at the

wake.

Starting by the induced aerodynamic pitch, when we compare the curves for x/R = 0 with the ones

for x/R = 1 something that stands out is how the pitch at the tip has opposite directions. When we

assessed the wake geometries we said that the wake should fold in the downstream direction at the

tip - this means that the behaviour of the pitch at the lifting line is wrong but it gradually inverts until

somewhere, between x/R = 0.25 and x/R = 1, it acquires the direction it is supposed to take. This has

been seen by Melo [31] [32] in the analysis problem.

As for the induced tangential velocity, we can see that in all cases it increases as we move down-

46

stream, which is in accordance with what the momentum theory predicts far downstream [32]. This is

also the reason why the pitch decreases in that direction.

0.2 0.4 0.6 0.8 1r/R

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

v a/U

0.9 0.95 10.4

0.6

0.8

1

1.2

Figure 4.13: Distribution of induced axial velocity when the wake is aligned at x/R = {0, 0.2, 1}, showingthe detail of the tip.

47

48

Chapter 5

Conclusions

In this work we adapted the lifting line code developed by Melo [31] to the design problem and

included a new method of optimization - the Lagrange multiplier method. The theory behind it, as well

as the adjustments that were made to implement it, were presented along the way.

As this method, like all numerical methods, is inevitably bound to uncertainties, we started the as-

sessment of the code by performing a convergence analysis. Here, we could see that when the wake

alignment scheme is not used there are clear signs of good convergence behaviour with the increase

of the number of segments in which the lifting line is divided. On the other hand, when the alignment

scheme is used the discretization of the wake also comes into play and so the convergence, while still

good, ceases to be monotonic when we truncate the wake further downstream, i.e., while increasing the

length of the wake the results do not always evolve in a definite direction.

Passing to the results per se, we began by assessing the freshly implemented Lagrange multiplier

method using several tests, in which the wake alignment scheme was not used.

We started by the obvious - the comparison between this method and the classical optimization.

The primal conclusion was that, for the same conditions, the first always outperforms the second, being

the dissemblance between them clearly dependent on the load of the turbine - as we move to higher

loads, and thus further away from the assumptions of the classical optimization, the improvement in

performance becomes more significant, as well as the differences in the distributions of local variables.

Speaking of which, it was also possible to see what behaviour the induced aerodynamic pitch acquires

when we give it some freedom (in the classical optimization it is imposed as constant along the span)

and, like other authors before, we saw how the differences between the distributions are concentrated

in the vicinity of the hub.

The second test consisted in assessing the influence of including a hub model. In terms of local

variables, the cases with and without the hub are almost undistinguishable except for the circulation near

the root, which, as expected, goes to zero when we do not include it but assumes a finite value when

we do. This translates into an higher power coefficient and also provides the basis for the generation of

a blade geometry that is better optimized near the root.

We then moved on to some parametric studies, starting by the tip speed ratio. With this analysis

49

we could perceive that there is an optimum point associated to this variable, i.e., the performance of

the turbine drops both at low and high values of tip speed ratio. It is essentially a question of balance:

when we go to low values we deal with high torques which, by conservation of angular momentum, will

translate into a higher loss of kinetic energy in the rotation of the wake; on the other side, when we go

to high tip speed ratios the viscous losses become increasingly significant, eventually overcoming the

reduction of kinetic energy losses. As for the local variables, all the distributions are dissimilar as a result

of the great impact of this parameter in the wake geometry.

The second parametric study assessed the influence of the drag-to-lift ratio. It terms of performance,

we saw it decreasing linearly with this parameter, which was a curious result. However, in terms of

aerodynamic design it only caused minor changes, which was in agreement with the findings of other

authors.

Lastly, we assessed the wake alignment scheme by testing ten different combinations of alignment

sections. Starting by the wake geometries we could see that, although with this model it would be

impossible to capture the wake expansion or the wake roll up, we do see some of the wakes folding in

the downstream direction, which is a hint of good behaviour. We also saw that putting the alignment

sections to close to each other or setting a transition wake that is very short is generally detrimental to

the results. In terms of performance we could not draw definite conclusion as we were not carrying out

true optimizations, but it did seem like the number of alignment sections had a greater impact then their

location. Moving now to the local variables, we saw how over the lifting line the circulation distributions

were similar but the induced axial velocities increased with the number of alignment sections, causing

a reduction in the power coefficient. It was also seen that the tip of the blade is where convergence is

more difficult to achieve, as this is where the gradients are more intense and the discretization of the

wake vortices is coarser. We also assessed the behaviour of some variables in the alignment sections

and concluded that, with this model, the pitch tends to show an abnormal behaviour at the lifting line

but corrects itself as we move downstream. Furthermore, we also verified that the induced tangential

velocity increases in that direction, in accordance to what is predicted by the momentum theory.

In terms of future work, the wake alignment scheme seems to be the most fertile ground for further

development. One could, for example, include the computation of radial induced velocities and their

effect on the wake geometry, in order to allow the wake to expand and roll-up. It could also be worth ex-

ploring other numerical methods for solving the equations of the design problem, as when this alignment

scheme is used the computational time is dramatically increased.

50

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