ACTA

UNIVERSITATIS

UPSALIENSIS

UPPSALA

2012

Digital Comprehensive Summaries of Uppsala Dissertations

from the Faculty of Science and Technology 998

Fluid Mechanics of Vertical Axis

Turbines

Simulations and Model Development

ANDERS GOUDE

ISSN 1651-6214

ISBN 978-91-554-8539-9

urn:nbn:se:uu:diva-183794

Dissertation presented at Uppsala University to be publicly examined in Polhemssalen,

Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, December 14, 2012 at 13:15 for

the degree of Doctor of Philosophy. The examination will be conducted in English.

Abstract

Goude, A. 2012. Fluid Mechanics of Vertical Axis Turbines: Simulations and Model

Development. Acta Universitatis Upsaliensis. Digital Comprehensive Summaries of

Uppsala Dissertations from the Faculty of Science and Technology 998. 111 pp. Uppsala.

ISBN 978-91-554-8539-9.

Two computationally fast fluid mechanical models for vertical axis turbines are the streamtube

and the vortex model. The streamtube model is the fastest, allowing three-dimensional modeling

of the turbine, but lacks a proper time-dependent description of the flow through the turbine. The

vortex model used is two-dimensional, but gives a more complete time-dependent description of

the flow. Effects of a velocity profile and the inclusion of struts have been investigated with the

streamtube model. Simulations with an inhomogeneous velocity profile predict that the power

coefficient of a vertical axis turbine is relatively insensitive to the velocity profile. For the

struts, structural mechanic loads have been computed and the calculations show that if turbines

are designed for high flow velocities, additional struts are required, reducing the efficiency

for lower flow velocities.Turbines in channels and turbine arrays have been studied with the

vortex model. The channel study shows that smaller channels give higher power coefficients

and convergence is obtained in fewer time steps. Simulations on a turbine array were performed

on five turbines in a row and in a zigzag configuration, where better performance is predicted

for the row configuration. The row configuration was extended to ten turbines and it has been

shown that the turbine spacing needs to be increased if the misalignment in flow direction is

large.A control system for the turbine with only the rotational velocity as input has been studied

using the vortex model coupled with an electrical model. According to simulations, this system

can obtain power coefficients close to the theoretical peak values. This control system study

has been extended to a turbine farm. Individual control of each turbine has been compared to a

less costly control system where all turbines are connected to a mutual DC bus through passive

rectifiers. The individual control performs best for aerodynamically independent turbines, but

for aerodynamically coupled turbines, the results show that a mutual DC bus can be a viable

option.Finally, an implementation of the fast multipole method has been made on a graphics

processing unit (GPU) and the performance gain from this platform is demonstrated.

Keywords: Wind power, Marine current power, Vertical axis turbine, Wind farm, Channel

flow, Simulations, Vortex model, Streamtube model, Control system, Graphics processing

unit, CUDA, Fast multipole method

Anders Goude, Uppsala University, Department of Engineering Sciences, Electricity,

Box 534, SE-751 21 Uppsala, Sweden.

© Anders Goude 2012

ISSN 1651-6214

ISBN 978-91-554-8539-9

urn:nbn:se:uu:diva-183794 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-183794)

To my family

List of papers

This thesis is based on the following papers,which are referred to in the text

by their Roman numerals.

I Goude,A.,Lundin,S.,Leijon,M.,“A parameter study of the inﬂuence

of struts on the performance of a vertical-axis marine current turbine”,

In “Proceedings of the 8th European wave and tidal energy conference,

EWTEC2009”,Uppsala,Sweden,pp.477–483,September 2009.

II Goude,A.,Lalander,E.,Leijon,M.,“Inﬂuence of a varying vertical

velocity proﬁle on turbine efﬁciency for a Vertical Axis Marine Current

Turbine”,In “Proceedings of the 28th International Conference on

Offshore Mechanics and Arctic Engineering,OMAE 2009”,Honolulu,

USA,May 2009.

III Grabbe,M.,Yuen K.,Goude,A.,Lalander,E.,Leijon,M.,“Design of

an experimental setup for hydro-kinetic energy conversion”,

International Journal on Hydropower &Dams,15(5),pp.112–116,

2009.

IV Goude,A.,Ågren,O.,“Simulations of a vertical axis turbine in a

channel”,Submitted to Renewable Energy,October 2012.

V Goude,A.,Ågren,O.,“Numerical simulation of a farmof vertical axis

marine current turbines”,In “Proceedings of the 29th International

Conference on Offshore Mechanics and Arctic Engineering,OMAE

2010”,Shanghai,China,June 2010.

VI Dyachuk,E.,Goude,A.,Lalander,E.,Bernhoff,H.,“Inﬂuence of

incoming ﬂow direction on spacing between vertical axis marine

current turbines placed in a row”,In “Proceedings of the 31th

International Conference on Offshore Mechanics and Arctic

Engineering,OMAE 2012”,Rio de Janeiro,Brazil,July 2012.

VII Goude,A.,Bülow,F.,“Robust VAWT control systemevaluation by

coupled aerodynamic and electrical simulation”,Submitted to

Renewable Energy,September 2012.

VIII Goude,A.,Bülow,F.,“Aerodynamic and electric evaluation of a

VAWT farmcontrol systemwith passive rectiﬁers and mutual

DC-bus”,Submitted to Renewable Energy,November 2012.

IX Goude,A.,Engblom,S.,“Adaptive fast multipole methods on the

GPU”,Journal of Supercomputing,DOI 10.1007/s11227-012-0836-0,

In Press,October 2012.

Reprints were made with permission fromthe publishers.

The author has also contributed to the following paper,not included in the

thesis:

A Yuen,K.,Lundin,S.,Grabbe,M.,Lalander,E.,Goude,A.,Leijon,

M.,“The Söderfors Project:Construction of an Experimental Hydroki-

netic Power Station”,In “Proceedings of the 9th European wave and

tidal energy conference,EWTEC2011”,Southampton,United Kingdom,

September 2011.

Contents

1 Introduction

................................................................................................

14

1.1 Different turbine types

...................................................................

14

1.2 Comparison between wind and marine current turbines

.............

16

1.3 Vertical axis turbine research at Uppsala University

...................

17

1.4 Extended studies within this thesis

...............................................

18

1.5 Outline of the thesis

.......................................................................

18

2 Theory for vertical axis turbines

...............................................................

20

2.1 Basic theory and the Betz limit

.....................................................

20

2.2 Extension to include channels

.......................................................

22

2.3 Theory of lift-based vertical axis turbines

....................................

24

2.4 Angle of attack including ﬂow curvature

.....................................

26

3 Control strategy for vertical axis turbines

................................................

30

3.1 Control of a single turbine

.............................................................

30

3.2 Extension to multiple turbines

.......................................................

36

4 Simulation models

.....................................................................................

37

4.1 Streamtube models

.........................................................................

37

4.1.1 Description of model

.......................................................

39

4.1.2 Including struts

................................................................

41

4.1.3 Obtaining lift and drag coefﬁcients

................................

43

4.1.4 Corrections due to ﬂow curvature

..................................

45

4.1.5 Including ﬂow expansion

................................................

46

4.2 Vortex models

.................................................................................

48

4.2.1 Implementing the turbine

................................................

49

4.2.2 Merging vortices

..............................................................

51

4.2.3 Calculation of velocity

....................................................

51

4.2.4 Numerical evaluation of the velocity ﬁeld

.....................

53

5 Simulation results

......................................................................................

66

5.1 Evaluation of simulation tools

.......................................................

66

5.1.1 Strut modeling

.................................................................

68

5.1.2 Expansion model

.............................................................

69

5.1.3 Tip correction model

.......................................................

70

5.1.4 Curvature modeling

.........................................................

70

5.1.5 Vortex model

....................................................................

73

5.1.6 Concluding remarks about the simulation tools

............

74

5.2 Results from papers

........................................................................

74

5.2.1 The effects of struts

.........................................................

75

5.2.2 The effects of a velocity proﬁle

......................................

77

5.2.3 Design of a turbine for use in a river

..............................

79

5.2.4 Turbines in channels

........................................................

80

5.2.5 Turbines in an array

.........................................................

83

5.2.6 Simulations of control systems

.......................................

87

5.2.7 Control of multiple turbines

............................................

90

6 Conclusions

................................................................................................

95

7 Suggestions for future work

......................................................................

97

8 Summary of papers

....................................................................................

98

9 Errata for papers

......................................................................................

102

10 Acknowledgments

...................................................................................

103

11 Summary in Swedish

...............................................................................

104

References

......................................................................................................

107

Nomenclature

A

m

2

Turbine cross-sectional area

A

∞

m

2

Asymptotic area of streamtube enclosing turbine

A

c

m

2

Cross-sectional area of a channel

A

d

m

2

Area of turbine disc

A

e

m

2

Streamtube area far downstream/center of turbine

AR

−

Aspect ratio of a blade

C

D

−

Drag coefﬁcient

C

Ds

−

Drag coefﬁcient for strut

C

D∞

−

Drag coefﬁcient for inﬁnitely long blade

C

L

−

Lift coefﬁcient

C

Ls

−

Lift coefﬁcient for strut

C

L∞

−

Lift coefﬁcient for inﬁnitely long blade

C

N

−

Normal force coefﬁcient

C

N0

−

Normal force coefﬁcient without curvature corrections

C

Ns

−

Normal force coefﬁcient for strut

C

P

−

Power coefﬁcient

C

P

e

−

Power coefﬁcient equivalent for extracted power

C

Pmax

−

Maximumpower coefﬁcient for a given ﬂowvelocity

C

T

−

Tangential force coefﬁcient

C

Ts

−

Tangential force coefﬁcient for strut

D

m

Turbine diameter

F

−

Velocity correction factor

F

D

N

Drag force

F

D0

N

Drag force at zero angle of attack

F

L

N

Lift force

F

N

N

Normal force

F

Nl

N/m

Normal force per meter

F

R

N

Force in radial direction

F

T

N

Tangential force

F

Ts

N

Tangential force on strut

F

x

N

Aerodynamic force from a blade on ﬂow in a streamtube

F

xs

N

Aerodynamic force from a strut on ﬂowin a streamtube

H

m

Channel height

J

kgm

2

Moment of intertia

L

m

Distance between struts

K

N

Constant to determine lift force

9

Ma

−

Mach number

N

−

Number of particles (FMM)

N

b

−

Number of blades

N

box

−

Number of boxes (FMM)

N

d

−

Number particles per box (FMM)

N

p

−

Number of panels

N

s

−

Number of struts

N

t

−

Number of turbines

N

v

−

Number of vortices

NF

i

−

Set of all boxes in near ﬁeld of box i (FMM)

P

W

Power absorbed by the turbine

P

e

W

Power extracted fromthe turbine

P

e,tot

W

Total power extracted fromthe turbines of a farm

P

tot

W

Power available in ﬂow

R

m

Turbine radius

R

inner

m

Strut inner attachment point

R

outer

m

Strut outer attachment point

Re

−

Reynolds number

T

s

Nm

Torque fromstrut

V

m/s

Flow velocity

V

0

m/s

Flow velocity far upstreamof turbine

V

∞

m/s

Asymptotic ﬂowvelocity

V

abs

m/s

Magnitude of incoming ﬂow velocity at blade position

V

b

m/s

Blade velocity

V

d

m/s

Flow velocity at turbine disc

V

e

m/s

Flow velocity far downstream/center of turbine

V

i

m/s

Vortex velocity

V

r

m/s

Relative ﬂow velocity for a blade (absolute value)

V

rs

m/s

Relative ﬂow velocity for a strut (absolute value)

V

s

m/s

Flow velocity at strut position

V

re f

m/s

Reference ﬂow velocity for estimating angle of attack

V

rel

m/s

Relative ﬂow velocity for blade

V

relz

m/s

Relative ﬂow velocity for blade in its own reference frame

V

s

m/s

Far downstreamvelocity of ﬂowpassing outside turbine

V

s j

m/s

Flow velocity at strut segment

V

ω

m/s

Velocity due to vortices

W

m

2

/s

Complex velocity potential

W

b

m

2

/s

Complex velocity potential for blade velocity

a

−

Axial induction factor

a

i

−

Multipole coefﬁcient i (FMM)

a

s

−

Slope of lift coefﬁcient curve in C

L

/α plot

b

m

Circle radius used for conformal mapping

b

i

−

Local coefﬁcient i (FMM)

10

b

y∞

−

Normalized asymptotic streamtube width

b

ye

−

Normalized streamtube width at the turbine center

b

z∞

−

Normalized asymptotic streamtube height

b

ze

−

Normalized streamtube height at the turbine center

c

m

Blade chord

c

s

m

Strut chord

c

sound

m/s

Speed of sound

c

0

m

Reference chord for struts

g

m/s

2

Gravitational acceleration

h

m

Turbine height

k

1

kgm

2

First control systemconstant

k

2

kgm

2

Second control systemconstant

k

3

kgm

2

Third control systemconstant

k

d1

−

Constant for time estimate of direct evaluation (FMM)

k

d2

−

Constant for time estimate of direct evaluation (FMM)

l

m

Blade length in streamtube

p

−

Number of multipole coefﬁcients (FMM)

p

0

N/m

2

Pressure far upstreamof turbine

p

atm

N/m

2

Atmospheric pressure

p

d1

N/m

2

Pressure directly in front of turbine disc

p

d2

N/m

2

Pressure directly after turbine disc

p

e

N/m

2

Pressure far downstreamof turbine

r

0

m

Box center (FMM)

r

s

m

Radial position on a strut

r

m

Arbitrary position

r

i

m

Vortex position

s

m

Position on blade surface in transformed plane

t

s

Time

t

b

m

Blade thickness

t

d

s

Time estimate for direct evaluation (FMM)

u

−

Interference factor

x

m

Position in the x-direction

x

0

m

Blade attachment point

x

0r

−

Normalized blade attachment point

y

m

Position in the y-direction

y

∞

m

Asymptotic streamtube position in y-direction

y

d

m

Streamtube position at turbine disc in y-direction

y

e

m

Streamtube position far downstreamin y-direction

Δy

m

Streamtube width

z

m

Position on blade surface in the blades reference frame

z

0

m

Position on blade surface in the turbines reference frame

z

b

m

Blade position

Δz

m

Streamtube height

11

Γ

p

m

3

/s

Circulation of three dimensional point vortex

Γ

m

2

/s

Circulation of two dimensional vortex

Δ

m

Cutoff radius used for vortex merging

Ω

rad/s

Turbine rotational velocity

Ω

i

rad/s

Rotational velocity of turbine i

Ω

1

rad/s

First control systemrotational velocity constant

Ω

2

rad/s

Second control systemrotational velocity constant

Ω

3

rad/s

Third control systemrotational velocity constant

α

−

Angle of attack

α

b

−

Corrected angle of attack

α

s

−

Angle of attack for strut

β

−

Direction of incoming wind

δ

−

Blade pitch angle

ε

m

Cutoff radius of Gaussian vortex kernel

η

−

Angle of blade relative to the vertical axis

η

s

−

Angle of strut relative to the horizontal plane

θ

−

Blade azimuthal position shifted 90 degrees

θ

b

−

Blade azimuthal position

λ

−

Tip speed ratio

λ

e

−

Equilibriumtip speed ratio

λ

max

−

The tip speed ratio that gives highest power coefﬁcient

ν

m

2

/s

Kinematic viscosity

ρ

kg/m

3

Density of ﬂuid

σ

N/m

2

Stress

ϕ

−

Angle of relative wind

φ

m

2

/s

Velocity potential

ω

1/s

Vorticity

12

Abbreviations

CPU

Central processing unit

CUDA

Compute uniﬁed device architecture

FEM

Finite element method

FMM

Fast multipole method

FVM

Finite volume method

GPU

Graphics processing unit

L2L

Local to local translation

L2P

Local to particle evaluation

M2L

Multipole to local translation

M2M

Multipole to multipole translation

NACA

National Advisory Committee for Aeronautics

P2M

Particle to multipole initialization

P2P

Particle to particle interaction (direct evaluation)

RANS

Reynolds-averaged Navier-Stokes equations

SIMD

Single instruction multiple data

SSE

Streaming SIMD Extensions

13

1.Introduction

Aturbine is used to convert the energy froma moving ﬂuid into rotational mo-

tion,which in turn can drive an electric generator.The best suited turbine for

this energy conversion depends on the characteristics of the ﬂow.One case,

which is present in gas turbines and traditional hydro power plants,is ﬂowcon-

strained by walls,such as ﬂowwithin a pipe.Here,the turbine cross-sectional

area usually covers the entire ﬂowand the ﬂowhas a large pressure difference

that drives the turbine.Asecond case is the free ﬂow,where no conﬁning walls

are present.This is the case for wind power and often a reasonable approxi-

mation for tidal power in the ocean.In free ﬂow,the ﬂuid can pass around the

turbine and the available energy is the kinetic energy in the ﬂow.This thesis

mainly treat the free ﬂow case,but also a hybrid case where there are conﬁn-

ing walls,but the turbine does not cover the whole cross-sectional area of the

ﬂow,is studied in this thesis.This situation occurs in a river,where the river

cross-sectional shape area usually prevents a turbine from covering the entire

cross-section.

1.1 Different turbine types

The typical turbine design for wind power is a horizontal axis turbine,where

the rotational axis of the turbine is parallel to the ﬂowdirection [1,chapter 1].

In this thesis,however,the vertical axis turbine will be investigated.Here,the

rotational axis is perpendicular to the ﬂow direction.This kind of turbine is

sometimes called “cross-ﬂow turbine”,as the turbine in principle also can be

tilted 90 degrees to have a horizontal axis while still having its rotational axis

perpendicular to the ﬂow.The traditional name “vertical axis turbine” will be

used here,even for situations where the rotational axis is tilted,since this is

the most commonly used name.

There are two different types of vertical axis turbines.The ﬁrst type is

based on the drag force and is often called the Savonius rotor after the Finnish

inventor Sigurd Johannes Savonius,despite that Savonius only patented an im-

provement of older designs [2].This improvement is neither implemented on

all present drag-based turbines.Drag-based devices rely on variation of the

drag coefﬁcient with respect to the orientation of the object.To create a rea-

sonably efﬁcient drag-based turbine,the drag coefﬁcient should be high in one

direction and low in the opposite direction,which gives a torque on the tur-

bine.Drag-based devices achieve lower power coefﬁcients than the lift-based

14

Figure 1.1.Different types of vertical axis turbines.

devices described in the section below[3,chapters 2,7].Another drawback is

that the amount of construction material in drag devices is quite high (as can

be seen in ﬁgure 1.1).This cost is inhibiting the construction of large turbines,

as material usage is proportional to the volume,i.e.the cube of the character-

istic length of the turbine,while the power absorption is proportional to the

cross-sectional area,i.e.the square of the characteristic length.

The second type is the lift-based turbine,which was originally invented by

the French engineer George Jean Marie Darrieus [4] in the 1920’s (approxi-

mately one year after Savonius patented his design).The patent application of

Darrieus covers both the curved blade turbine and the H-rotor (see ﬁgure 1.1),

as well as turbines with varying pitch angle and ducted turbines.It is suggested

in the patent that the designs work both for wind and tidal energy.The aim

of the curved blade design is to reduce the bending stresses in the blades due

to centrifugal forces.The North American company Flowind commercialized

in the 1980s the Darrieus turbine with the curved blade design [5,chapter 1].

During that time,the curved blade turbine was also studied by Sandia National

Laboratories,which is the main reason why much of the published work on

Darrieus turbines is on the curved blade design.This thesis will instead focus

on the straight blade H-rotor design,which currently is in development at Up-

psala University.With recent progress for light materials,composites can be

used in the turbine construction,which reduces centrifugal forces due to the

15

lighter structure.This makes the H-rotor design more feasible.The straight

blade design has the advantages that straight blades are easier to manufacture

and by attaching the blades with struts,it is possible to place the upper bear-

ing much closer to the turbine center,reducing the bending moment on the

axis.In addition,the constant radius of the straight blade design gives a larger

cross-sectional area.Disadvantages compared to the curved blade design are

the addition of extra struts and the higher bending moments due to centrifugal

forces.

The main aerodynamic advantage of vertical axis turbines,compared to

standard horizontal axis turbines,is the independence of ﬂow direction,re-

moving the need for a yaw mechanism.For water ﬂow,an additional advan-

tage is that the cross-sectional area can be more ﬂexibly chosen as both height

and diameter can be varied (and the diameter can vary with height).This can

be useful in shallowwater where a turbine with a large width and small height

can cover a larger area than a horizontal axis turbine,as the cross-sectional

area of a horizontal axis turbine is circular.Disadvantages of the vertical axis

turbines are the lower power coefﬁcients and that the turbines are typically not

self-starting.

The vertical axis turbine can have its generator on the ground,which in

the wind power case simpliﬁes maintenance,tower construction and makes

the weight of the generator less important.This is beneﬁcial for direct driven

generators,which typically have large diameters.The use of direct driven gen-

erators further reduces the number of moving parts in the system.One major

concern for vertical axis turbines is the cyclic blade forces in each revolution,

which leads to torque oscillations and material fatigue.For further compar-

isons between horizontal and vertical axis turbines (and also between curved

and straight blade turbines) see e.g.[6].

1.2 Comparison between wind and marine current

turbines

Even though wind turbines operate in air (gas) while marine current turbines

operate in water (liquid),there are many similarities between the two.Tra-

ditionally,water is considered an incompressible ﬂuid and can therefore be

modeled with the incompressible Navier-Stokes equations.For air,it is typi-

cally expected that compressibility effects can be neglected for Mach numbers

Ma within the range

Ma =

V

rel

c

sound

<0.3.

Here,V

rel

is the relative ﬂow velocity (measured in the blades rest frame) and

c

sound

is the speed of sound in the ﬂuid [7,chapter 9].Note that the major con-

tribution to V

rel

originates fromthe blades’ own motion for lift-based turbines.

16

The speed of the blades in a wind turbine is typically too low for the Mach

number to be above 0.3 and wind turbines can therefore also be modeled with

incompressible aerodynamics.Although both wind and marine turbines can be

studied with the incompressible Navier-Stokes equations,there are still some

characteristic differences.One difference is that for marine current turbines,

there is both a sea bed and a free surface that bounds the ﬂow.Another dif-

ference is the risk for cavitation at too high ﬂow velocities.Cavitation would

modify the ﬂow characteristics and can cause damage to the turbine [8].

The energy absorbed by a turbine is proportional to the ﬂuid density and

the cube of the ﬂow velocity.As the density of water is 800 times higher than

the density of air,comparatively low ﬂuid velocities are adequate for marine

current power generation.For equal cross-sectional area,a wind speed of

10 m/s has the same incoming kinetic power as a water ﬂowspeed of 1.1 m/s.

However,as the forces only are proportional to the square of the ﬂowvelocity,

the marine current turbine experiences approximately 9.3 times higher ﬂuid

mechanical forces than a wind turbine at the same conditions (assuming that

the turbines are identical) and rotates 9.3 times slower.The increased forces

for marine current turbines require both stronger blades and support structure.

Many of the experimental vertical axis turbines for water have used relatively

large blades and thereby low optimal tip speed ratios [9–12].

One important parameter for the effectiveness of the turbine is the Reynolds

number,which for a blade is deﬁned as

Re =

cV

ν

,

where V is the ﬂowvelocity,c is the blade chord and ν is the kinematic viscos-

ity.Ahigher Reynolds number usually decreases the drag losses and increases

the stall angle,which is beneﬁcial for vertical axis turbines.For 20

◦

C,the

kinematic viscosities are 15.1 μm

2

/s for air and 1.00 μm

2

/s for water [13,ap-

pendix A].Under the conditions of equal power extraction mentioned above

(i.e.9.3 times higher ﬂow velocity for the wind turbine),this would give a

63 %higher Reynolds number for the marine current turbine,which is within

the same order of magnitude as the wind turbine.

1.3 Vertical axis turbine research at Uppsala University

At the Division of Electricity at Uppsala University,three vertical axis wind

turbines have been built.The ﬁrst turbine had a cross-sectional area of 6 m

2

and was later followed by a turbine with the cross-sectional area 30 m

2

and the

rated power 12 kW[14–16].This larger turbine is used for most of the experi-

ments.A10 kWturbine for telecomapplications has also been built [17].Fur-

ther,a 200 kWturbine has been constructed by the spin-off company Vertical

Wind AB [18].Additionally,a marine current turbine (described in paper III)

is scheduled to be deployed by the end of 2012.

17

Several simulation tools for turbine simulations have previously been de-

veloped at the division.A two-dimensional inviscid vortex model based on

conformal mappings for the blades has been created by Deglaire et al.[19].In

the turbine implementation,each blade is solved independently [20],allowing

for coupling to an elastic method developed by Bouquerel et al.,see paper IV

in [21].A multibody version for simulating turbines has been developed by

Österberg et al.,see [22] and paper III in [21].Two streamtube models have

also been implemented by Deglaire and Bouquerel.

1.4 Extended studies within this thesis

This thesis focuses on the ﬂuid mechanical modeling of the vertical axis tur-

bine,and two different simulation tools have been developed.The ﬁrst simu-

lation tool uses the streamtube model and the development of this tool started

from the basic streamtube model implemented by Bouquerel,which is based

on the model of Paraschivoiu [3].All additional modeling and code develop-

ment have been developed within this thesis.

The second simulation tool uses a vortex model,and this tool has been

developed from scratch within this work.This model is based on empirical

data for lift and drag coefﬁcients instead of the conformal mapping method

by Deglaire,which is based on inviscid theory.The computational speed is

crucial for the developed vortex model and large efforts have been put into

this.The existing implementation of the fast multipole method by Stefan Eng-

blom[23] has been signiﬁcantly improved and ported to a GPU (paper IX)

Several studies have been carried out with the two simulation models.The

streamtube model has been used to study losses due to struts (paper I),the

effects of a velocity proﬁle (Paper II) and to design a turbine for deployment

in a river (paper III).The more computationally demanding vortex model has

been used to study turbines in channels (paper IV) and turbine arrays (paper V

and VI).The vortex model is also coupled to an electrical model to study con-

trol systems for a single turbine (paper VII) and extended simulations analyze

control systems for a turbine farm (paper VIII).

1.5 Outline of the thesis

After the introduction,theory for vertical axis turbines is presented in chap-

ter 2.This is followed by an introduction to control systems in chapter 3.

The theory and implementation for the simulation models are then presented

in chapter 4,which also includes the GPU implementation of the fast multi-

pole method.The results from the simulations are given in chapter 5,where

the ﬁrst part evaluates the accuracy of the simulation models and the second

part summarizes the results from the articles.The thesis ends by conclusions,

18

suggestions for future work,summary of papers,errata for papers and ac-

knowledgments.

19

2.Theory for vertical axis turbines

Given a cross-sectional area A perpendicular to a homogenous ﬂow of a ﬂuid,

the kinetic power that passes through this area is given by

P

tot

=

1

2

ρAV

3

,(2.1)

where ρ is the density and V is the ﬂow velocity.If the ﬂow is not conﬁned

by any surrounding boundaries,the kinetic power is the available power for a

wind/current turbine.The efﬁciency (i.e.outgoing power divided by incoming

power) would be one possible measure of how good the energy conversion

is.However,adding a turbine will change the velocity and force parts of the

ﬂow to pass outside the turbine area and thereby change the kinetic energy

that passes through this area A.Moreover,some kinetic energy is left in the

ﬂow and can possibly be used later.Therefore,turbine performance is usually

measured with the power coefﬁcient instead,which is deﬁned as

C

P

=

P

1

2

ρAV

3

∞

,(2.2)

where P is the power absorbed by the turbine and V

∞

is the asymptotic up-

stream ﬂow velocity.With this expression,the absorbed power is compared

to the power that would have passed through the cross-sectional area,if the

turbine would be absent,instead of compared to the power that actually passes

through the area.Since this expression is normalized against an expression

that does not change with the turbine characteristics,it is a better measure than

efﬁciency.Improving the power coefﬁcient will give higher power absorption,

which is not always the case with efﬁciency.

2.1 Basic theory and the Betz limit

One of the most basic approximations of a turbine is the one used in the tradi-

tional Betz theory [24],where the turbine is approximated as a single ﬂat disc

with a constant pressure drop over the whole turbine surface.All ﬂowpassing

through the disc is encapsulated in a streamtube that starts far ahead of the

turbine and ends far behind.By making the assumption that the pressure at

both ends of the streamtube is the atmospheric pressure p

atm

,and by using the

20

p

atm

p

atm

V

∞

A

∞

V

d

p

d1

p

d2

V

e

A

e

A

d

Figure 2.1.The streamtube used in the Betz limit derivation.The dashed line shows

the control volume used for momentumconservation.

Bernoulli equation before and after the turbine

p

atm

+

1

2

ρV

2

∞

= p

d1

+

1

2

ρV

2

d

,(2.3)

p

d2

+

1

2

ρV

2

d

= p

atm

+

1

2

ρV

2

e

,(2.4)

combined with continuity

A

∞

V

∞

=A

d

V

d

=A

e

V

e

(2.5)

and momentumconservation for the control volume in ﬁgure 2.1 (marked with

the dashed line)

ρA

e

V

2

e

−ρA

∞

V

2

∞

=A

d

(p

d2

−p

d1

),(2.6)

it is possible to showthat the velocity at the turbine disc V

d

is equal to

V

d

=

V

∞

+V

e

2

.(2.7)

Considering that the Betz theory assumes no losses,the power absorbed by

the turbine is given as the difference between incoming and outgoing power in

the ﬂuid

P =

1

2

ρA

∞

V

3

∞

−

1

2

ρA

e

V

3

e

.(2.8)

If the axial induction factor a,deﬁned as

V

d

=(1−a)V

∞

,(2.9)

is combined with expression with equations (2.5) and (2.8),it can be shown

that the power will reach its maximum value for a = 1/3,and the optimal

power is given by

P =

16

27

·

1

2

ρA

d

V

3

∞

(2.10)

where 16/27 is the traditional Betz limit,limiting the power coefﬁcient to

approximately 59.3 %.

21

p

0

p

0

V

0

V

0

V

0

A

0

A

c

V

d

p

d1

p

d2

V

e

A

e

A

d

p

e

V

s

Figure 2.2.Illustration of a streamtube conﬁning the ﬂow that passes through the

turbine disc for a channel of cross-sectional area A

c

.

2.2 Extension to include channels

The Betz theory assumes no outer boundaries in the system.For a turbine op-

erating in water,it is more common that boundaries are present.One example

is a river,where the ﬂow is limited by the width and depth.The outer walls

prevent the ﬂowfromexpanding,pushing more ﬂowthrough the turbine.This

occurs in a traditional hydro power plant,where the entire ﬂow is forced to

pass through the turbine,which results in much higher power absorption than

the Betz limit [25].

To analyze this case analytically,assume that the ﬂow upstream of the tur-

bine has constant velocity V

0

and pressure p

0

(see ﬁgure 2.2).Note that in

this case,the pressure upstream and the pressure downstream are not equal.

Instead,there will be a drop in pressure,which,for open channel ﬂow,would

correspond to a drop in the surface level.Due to the continuity of the pressure,

the pressure inside the streamtube and outside has to be the same downstream

(p

e

).The cross-sectional area of the channel is A

c

and the cross-sectional area

of the turbine is A

d

.In this case,the Bernoulli equation gives

p

0

+

1

2

ρV

2

0

= p

d1

+

1

2

ρV

2

d

,(2.11)

p

d2

+

1

2

ρV

2

d

= p

e

+

1

2

ρV

2

e

,(2.12)

p

0

+

1

2

ρV

2

0

= p

e

+

1

2

ρV

2

s

,(2.13)

the continuity equation gives

A

0

V

0

=A

d

V

d

=A

e

V

e

,(2.14)

(A

c

−A

0

)V

0

=(A

c

−A

e

)V

s

(2.15)

22

and momentum conservation for a control volume that encloses the entire

channel gives

ρA

e

V

2

e

+ρA

s

V

2

s

−ρAV

2

0

=A

d

(p

d2

−p

d1

) +A

c

(p

0

−p

e

).(2.16)

The velocity at the turbine can be derived from equations (2.11) – (2.16) as

V

d

=

V

e

(V

s

+V

e

)

V

s

+2V

e

−V

0

,(2.17)

which in the free ﬂowlimit (V

s

→V

0

) reduces to equation (2.7).The force on

the turbine becomes

F

x

=A

d

(p

d1

−p

d2

) =

1

2

A

d

V

2

s

−V

2

e

,(2.18)

giving the power as

P =F

x

V

d

=

1

2

A

d

V

e

(V

s

+V

e

)

V

2

s

−V

2

e

V

s

+2V

e

−V

0

.(2.19)

From equation (2.19),it can be found that the highest power absorption is

obtained when V

e

=V

0

/3,which actually is the same as for the free ﬂow.The

highest power is thus given by

P =

16

27

·

1

1−

A

d

A

c

2

·

1

2

ρA

d

V

3

0

(2.20)

and the pressure drop is

p

e

−p

0

=

4

A

d

A

c

3−

A

d

A

c

9

1−

A

d

A

c

2

ρV

2

0

.(2.21)

From these results,it can be seen that the maximum theoretical power co-

efﬁcient increases with the factor (1−A

d

/A

c

)

−2

for a channel.In the limit

A

d

→A

c

,the power coefﬁcient diverges,along with the pressure drop in equa-

tion (2.21).Considering that an inﬁnite drop in pressure is unfeasible,when

the turbine area is almost as large as the channel,the available pressure dif-

ference will start limiting the maximumpower coefﬁcient,which will prevent

inﬁnite energy extraction.This is the case for hydro power turbines,where the

power is limited by the difference in water elevation.

The model above assumes that the cross-section of the channel is constant.

An open channel will have a drop in surface level over the turbine and an

extension of the model to include this drop is given by Whelan et al.[26].

This correction has not been included in the present work,as the model is

used for comparisons with the two-dimensional vortex simulations where no

free surface is modeled.

23

R

θ

b

Ω

x

y

Figure 2.3.Illustration of a vertical axis turbine.

2.3 Theory of lift-based vertical axis turbines

The H-rotor is a lift-based design,which means that the aerodynamic torque is

generated by the lift force of the blades.Therefore,airfoil proﬁles are typically

used for the blades.Airfoils are generally designed to operate at relatively low

angles of attack,where the lift force increases approximately linearly with

the angle of attack and the drag force remains low.When the angle of attack

increases above the stall angle,which for a typical airfoil occurs for angles

around 10 – 15 degrees,the lift force is reduced,and the drag starts to increase

substantially.This work is focused towards blades with a ﬁxed pitch angle.To

keep the angle of attack lowwithout pitching the blades,the blades must move

with a high velocity if the wind comes fromthe side.To illustrate this,assume

that a blade is located at angle θ

b

(see ﬁgure 2.3).Using complex notation,

this gives the blade position

z

b

=Re

iθ

b

(2.22)

and the velocity of the blade is therefore

V

b

=i

˙

θ

b

Re

iθ

b

.(2.23)

The rotational velocity

˙

θ

b

is commonly denoted Ω.The incoming wind V,if

complex,represents wind fromany direction.The blade will nowsee a relative

wind of

V

rel

=V −i ΩRe

iθ

b

.(2.24)

24

V

rel

α

δ

ϕ

V

b

Figure 2.4.Deﬁnitions of angles and velocities.The positive direction for angles is

counter-clockwise,hence α and ϕ are negative for the directions of V

b

and V

rel

in the

ﬁgure.

To obtain the angle of relative wind,rotate V

rel

with the angle ie

−iθ

b

,aligning

the blade motion with the negative real axis,

V

relz

=Vie

−iθ

b

+ΩR.(2.25)

The angle of relative wind will be the argument of this complex number.In

the special case of V being real,the angle of relative wind is

ϕ=arctan

cosθ

b

ΩR

V

+sinθ

b

(2.26)

and the absolute value of the relative velocity is

|V

rel

| =V

ΩR

V

+sinθ

b

2

+(cosθ

b

)

2

.(2.27)

The angle of attack is given by

α=ϕ+δ,(2.28)

where δ is the blade pitch angle (see ﬁgure 2.4).Equation (2.26) shows how

the angle of attack varies during a turbine revolution and how it decreases as

the rotational velocity increases.As an example,ΩR/V =4 gives a maximum

angle of attack of around 14 degrees,approximately where stall begins to oc-

cur.It should be noted that in equations (2.24) – (2.27),V is the ﬂowvelocity

at the blade position.This can be compared with the tip speed ratio,which is

deﬁned as

λ =

ΩR

V

∞

(2.29)

where the asymptotic velocity is used.Due to the energy extracted,the veloc-

ity at the blade will generally be lower than the asymptotic velocity.

25

The turbine torque is calculated from the tangential force,which is given

by

F

T

=F

L

sinϕ−F

D

cosϕ.(2.30)

When the angle of relative wind is low,the approximations sinϕ ≈ ϕ and

cosϕ ≈ 1 can be applied.Assume that the pitch angle is zero,hence α =

ϕ.For symmetric blades,the blade forces can be approximated as F

L

≈Kα,

where K is a constant,and F

D

≈F

D0

,where F

D0

is a constant.The tangential

force can therefore be estimated as

F

T

=Kϕ

2

−F

D0

,(2.31)

showing that when the angle of relative wind decreases,the drag force be-

comes dominating.The conclusion is that for high tip speed ratios,drag will

give a more signiﬁcant contribution,reducing the power coefﬁcient.At too

low tip speed ratios,the turbine will enter stall,where lift decreases and drag

increases,which also should be avoided.For these reasons,the turbine should

be designed to operate with a tip speed ratio close to the stall limit in order to

obtain the highest possible power coefﬁcient.

2.4 Angle of attack including ﬂow curvature

The expression (2.26) is only valid for inﬁnitely small symmetric blades.The

blade performs a rotational motion,which leads to additional curvature effects,

changing the effective angle of attack.To conclude the theory section,a more

proper derivation will be performed using a rotating ﬂat plate instead.

By the use of conformal mappings,a circle can be transformed into a ﬂat

plate with the Joukowski transformation.The s-plane represents the circle

with radius b and the z-plane a ﬂat plate extending between −2b to 2b giving

the blade chord c as c = 4b.The blade coordinates z in its own frame of

reference is given by

z =s +

b

2

s

.(2.32)

Using the same transformation as Deglaire [20],the z

0

plane can be deﬁned as

z

0

=

(z +x

0

)e

−iδ

+iR

e

iθ

=(z +D)e

i(θ−δ)

,(2.33)

where D = x

0

+iRe

iδ

.By assuming that the blade only rotates around the

center,the blade velocity is given by

V

b

= i

˙

θ

(z +x

0

)e

−iδ

+iR

e

iθ

=i Ωz

0

(2.34)

with Ω=

˙

θ.Introduce a complex velocity potential W,with complex conju-

gate

W,such that

d

W

d

z

0

=V,(2.35)

26

hence the potential can be used to calculate the ﬂow velocity V.Assume that

the potential is given on the form

W(s) =V

abs

e

i(−β+θ−δ)

s +V

abs

e

−i(−β+θ−δ)

b

2

s

−

iΓ

2π

log(s) +W

1

(s),(2.36)

where V

abs

e

iβ

is the ﬂow velocity at the blade position.Now,construct a

potential such that

d

W

b

d

z

0

= V

b

,(2.37)

which gives

dW

b

ds

=

V

b

dz

0

ds

=−i Ω

z

0

dz

0

ds

=−i Ω

z +

D

dz

ds

.(2.38)

Note that on the boundary,

z =z.Integrated,this is

W

b

= −i Ω

1

2

z

2

+

Dz

= −i Ω

1

2

s

2

+2b

2

+

b

4

s

2

+

D

s +

b

2

s

.(2.39)

The no-penetration boundary condition states that the stream function should

be constant (possibly time-dependent) on the boundary.Given that the bound-

ary is moving,the condition becomes

Im[W(s)−W

b

(s)] =C.(2.40)

The ﬁrst part of W(s) already fulﬁlls the condition,while W

1

(s) remains to be

determined,hence

Im[W

1

(s) −W

b

(s)] =C.(2.41)

The boundary condition at inﬁnity states

dW

1

ds

s→∞

=0,(2.42)

and on the boundary,

s =

b

2

s

(2.43)

applies.Write equation (2.41) in terms of complex conjugates

iC = W

1

(s)−

W

1

(s)+i Ω

1

2

s

2

+2b

2

+

b

4

s

2

+

1

2

s

2

+2b

2

+

b

4

s

2

+

D

s +

b

2

s

+D

s +

b

2

s

= W

1

(s)−

W

1

(s)+

i Ω

b

4

s

2

+

b

4

s

2

+

b

2

s

b+

b

2

s

D+

b

2

s

D+

D

b

2

s

+2b

2

.(2.44)

27

Note that constants can be excluded from the potential.Therefore,W

1

(s) can

be identiﬁed as

W

1

(s) = −i Ω

b

2

s

D+

D

+

b

4

s

2

= −iΩ

b

2

s

iR

e

iδ

−e

−iδ

+2x

0

+

b

4

s

2

.(2.45)

The Kutta condition [27] states that the velocity has to be ﬁnite at s =b where

ds/dz diverges,giving

dW

ds

s=b

= 0 ⇒

−

iΓ

2π

1

b

= −V

abs

e

i(−β+θ−δ)

+V

abs

e

−i(−β+θ−δ)

−

i Ω

iR

e

iδ

−e

−iδ

+2x

0

+2b

= −V

abs

2i sin(−β+θ−δ) −2i Ω(−Rsinδ+x

0

+b).(2.46)

Use the reference case of a static wing

−

iΓ

2π

1

b

= V

ref

2i sinα⇒

sinα =

−V

abs

sin(−β+θ−δ) −Ω(−Rsinδ+x

0

+b)

V

ref

,(2.47)

and that for small values of x

0

V

re f

≈

(V

abs

cos(θ−β) +ΩR)

2

+V

2

abs

sin

2

(θ−β).(2.48)

Now,redeﬁne x

0

in terms of the chord as x

0

=x

0r

c,which means that x

0r

=

0.25 is the quarter chord position and use that 4b =c

sinα =

−V

abs

sin(θ−β−δ) −Ω

−Rsinδ+x

0r

c+

c

4

V

ref

=

(V

abs

cos(θ−β) +ΩR)sinδ−V

abs

cosδsin(θ−β)

V

ref

−

Ω

x

0r

c+

c

4

V

ref

= sin

δ+arctan

−V

abs

sin(θ−β)

V

0

cos(θ−β) +ΩR

−

Ω

x

0r

c+

c

4

V

ref

.(2.49)

Assuming small angles of attack,one can approximate

arcsin(α+β) ≈ α+β (2.50)

28

which gives the simpliﬁcations

α = δ+arctan

−V

abs

sin(θ−β)

V

abs

cos(θ−β) +ΩR

−

Ωx

0r

c

V

re f

−

Ωc

4V

ref

.(2.51)

Note that at the position θ = 0,the blade is at the position θ

b

= π/2.The

substitution θ =θ

b

−π/2 gives

α=δ+arctan

V

abs

cos(θ

b

−β)

V

abs

sin(θ

b

−β) +ΩR

−

Ωx

0r

c

V

re f

−

Ωc

4V

ref

,(2.52)

which with β = 0 would correspond to equations (2.26) and (2.28),but in-

cludes mounting position x

0r

and ﬂowcurvature.As an example,for a turbine

with chord 0.25 mand radius 3 m(i.e.the experimental turbine in Marsta [16]),

at very high rotational velocities

V

ref

≈ΩR

,the change in angle of attack

due to ﬂow curvature is approximately -1.2 degrees.This gives higher angles

of attack upstreamand lower downstream.

29

3.Control strategy for vertical axis turbines

Optimizing the power from a wind/marine current turbine does not only re-

quire that the turbine is designed with the highest possible power coefﬁcient.

Another important factor is to make sure that the turbine actually runs at the

tip speed ratio associated with the peak power coefﬁcient.Therefore,a con-

trol strategy which keeps the turbine tip speed ratio near this optimal value is

preferable.

Wind turbines in general are either controlled by pitch or stall regulation,

where the most common design today is a horizontal axis turbine with pitch

regulation [28].The advantage with pitch control is that it introduces an ad-

ditional parameter that can be controlled,allowing for a more ﬂexible control

system.Pitch control is mainly used in the region above rated wind speed (see

ﬁgure 3.1) to keep a smoother power and reduce mechanical loads,as the pitch

angle can be changed to reduce the blade forces [1,Chapter 8].Stall regula-

tion,instead,reduces the tip speed ratio,which increases the angle of attack.

This will eventually cause stall,which reduces the lift force and increases the

drag force.This will be most prominent for the tangential force,and thereby

the turbine torque,due to the signiﬁcant increase in the drag force.Pitch con-

trol has been used for vertical axis turbines,mainly to improve performance at

lowtip speed ratios [29,30] where stall is avoided by actively altering the pitch

angle to reduce the angle of attack.The angle of attack oscillates between

positive and negative values as the blade moves between the upstream and

downstream section of the turbine.Hence,reducing the angle of attack with

active pitch requires a change in pitch angle during each revolution.An active

pitch mechanismwould complicate the turbine further.No pitch mechanisms

are included in the turbines studied here to reduce the sources of mechanical

failure.

Without a pitch mechanism,the remaining parameter to control is the ro-

tational velocity,where the turbine power is controlled by regulating the tip

speed ratio and thereby the power coefﬁcient.Even with a pitch mechanism

installed,it is common for horizontal axis wind turbines to use a ﬁxed pitch

angle in the variable rotational speed region illustrated in ﬁgure 3.1 [1,Chap-

ter 8].The following sections will focus on the variable rotational speed re-

gion,where the aimis to maximize the extracted power.

3.1 Control of a single turbine

One way to control a turbine is to perform real time ﬂow velocity measure-

ments and adjust the tip speed ratio to optimal values.However,this would,

30

0

2

4

6

8

10

12

14

16

0

0.5

1

1.5

Fractionofratedpower

Wind speed (m/s)

Cut in

Variable rotational speed

region,constant C

P

Constant

power

Constant

rotational

speed

Rated wind

speed

Figure 3.1.Example of the different control strategies for different ﬂow velocities.

The turbine starts operating at wind speed 3 m/s and operates at optimal power coefﬁ-

cient (approximately constant tip speed ratio) up to rated rotational velocity (wind

speed 10 m/s),where the rotational velocity is kept constant until rated power is

achieved (wind speed 12 m/s).At higher wind speeds,power is kept constant.There

is also a cut-out wind speed where the turbine is stopped (not shown in the ﬁgure).

Rotational velocity

Power

Speedincreases

Turbine stops Stable region

Equilibrium

Pe

P

P

e

Speeddecreases

Speeddecreases

Figure 3.2.Illustration of a control system,where the extracted power only depends

on the rotational velocity.

31

rely on the accuracy of the ﬂow measurements.An alternative approach is

to let the extracted power P

e

be a function only of the rotational velocity of

the turbine.This type of control has been used for horizontal axis wind tur-

bines [31],but due to the low power coefﬁcient at low tip speed ratios for

vertical axis turbines,some care has to be taken when transferring this control

system to a vertical axis turbine to avoid that the turbine ceases to rotate for

low tip speed ratios.

An example of a control system only using the rotational speed as input

parameter is illustrated in ﬁgure 3.2.The angular acceleration

˙

Ωof the turbine

is

˙

Ω=

P−P

e

J Ω

,(3.1)

where J is the moment of inertia of the system.Here,extracted power P

e

in-

cludes both power from the generator and electrical and mechanical losses in

the system.If the extracted power P

e

is less than the turbine power P,the tur-

bine accelerates,while if the extracted power is larger,the rotational velocity

decreases.In Figure 3.2,the turbine would stop if the turbine has too low

rotational velocity as the turbine power at this low rotational velocity is very

limited (P <P

e

).This region where the turbine will stop is characterized by

a low power coefﬁcient at low tip speed ratios,which may apply to a vertical

axis turbine.The existence of such a region depends on the power absorption

characteristics of the turbine and for high enough wind speeds,this region

typically becomes smaller.

Figure 3.2 shows that there will be an equilibrium where extracted power

equals turbine power (P =P

e

).If this equilibrium occurs at the peak of the

power curve in ﬁgure 3.2,maximumenergy is extracted.If the extracted power

is normalized the same way as turbine power,the extracted power coefﬁcient

is

C

P

e

=

P

e

1

2

ρAV

3

∞

.(3.2)

Combining equations (3.1) and (3.2) gives for the equilibrium

where

˙

Ω=0

C

P

e

=C

P

.(3.3)

Equation (3.2) can be written in terms of tip speed ratio and rotational velocity

as

C

P

e

=

P

e

λ

3

1

2

ρA(RΩ)

3

.(3.4)

Denote λ

max

the tip speed ratio with the peak power coefﬁcient C

Pmax

.By

choosing λ

max

as the desired equilibrium,equations (3.3) and (3.4) give

C

Pmax

=

P

e

λ

3

max

1

2

ρA(RΩ)

3

⇒P

e

=

1

2

ρAC

Pmax

RΩ

λ

max

3

=k

2

Ω

3

,(3.5)

32

0

1

2

3

4

5

6

7

8

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Tip speed ratio

Powercoefficient

C at 3 m/s

P

C at 6 m/s

P

C at 12 m/s

P

C

P

e

Figure 3.3.The control strategy A from equation ( 3.5) illustrated for three different

ﬂow velocities.

where the constant k

2

is related to λ

max

and C

Pmax

according to equation (3.5).

Under the conditions that C

Pmax

and λ

max

are constants,maximumpower will

be extracted if the extracted power is chosen to vary with the cube of the rota-

tional velocity and the constant k

2

is chosen according to equation (3.5).This

control strategy will be denoted “strategy A”.With C

Pmax

and λ

max

constant,

equation (3.5) is independent of ﬂow velocity.However,due to the increased

Reynolds number at higher ﬂow velocities,the airfoil performance will in-

crease,as drag is reduced and stall angle is increased.Therefore,the values

for λ

max

and C

Pmax

have a small dependence on the Reynolds number,see

ﬁgure 3.3.The performance of the strategy in equation (3.5) is plotted in ﬁg-

ure 3.3 for three ﬂow velocities (for speciﬁcations on the simulated turbine,

see section 5.2.6).The change in power coefﬁcient with respect to the ﬂow

velocity causes the obtained equilibrium tip speed ratio λ

e

to be slightly dis-

tanced to λ

max

,although the obtained power coefﬁcient is very similar toC

Pmax

(see ﬁgure 3.3).For a ﬂowvelocity of 3 m/s,there is an unstable region for tip

speed ratios below2.2,while for 6 m/s and 12 m/s,the strategy is stable in the

entire interval as long as the turbine power is larger than the mechanical and

electrical losses.Even though the strategy is stable for 6 m/s and 12 m/s,the

difference between turbine power and extracted power is low at low tip speed

ratios,causing a slow acceleration,cf.equation (3.1).

Two modiﬁcations to the devised strategy in equation (3.5) have been made,

with the intention to increase the rotational velocity at low tip speed ratios in

33

0

1

2

3

4

5

6

7

8

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Tip speed ratio

Powercoefficient

C at 3 m/s

P

C at 6 m/s

P

C at 12 m/s

P

C at 3 m/s

P

e

C at 6 m/s

P

e

C at 12 m/s

P

e

Figure 3.4.Control strategy B from equation (3.6) illustrated for three different ﬂow

velocities.

0

1

2

3

4

5

6

7

8

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Tip speed ratio

Powercoefficient

C at 3 m/s

P

C at 6 m/s

P

C at 12 m/s

P

C at 3 m/s

P

e

C at 6 m/s

P

e

C at 12 m/s

P

e

Figure 3.5.Control strategy C from equation (3.7) illustrated for three different ﬂow

velocities.

34

order to create a more stable and faster control strategy.Strategy B is given as

P

e

=0 Ω≤Ω

0

,

P

e

=k

1

Ω

2

(Ω−Ω

0

) Ω>Ω

0

,

(3.6)

where the rotational velocity Ω

0

is required before the strategy starts to extract

energy.This strategy obtains higher differences between extracted and turbine

power for low rotational velocities at the cost of having λ

e

further distanced

from λ

max

.

Strategy C combines a higher torque at low rotational velocities with the

good performance of strategy A by dividing the rotational velocities in three

regions according to

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

P

e

=0 Ω≤Ω

0

,

P

e

=k

1

Ω

2

(Ω−Ω

0

) Ω

0

<Ω≤Ω

1

,

P

e

=k

2

Ω

3

Ω

1

<Ω≤Ω

2

,

P

e

=k

3

Ω

2

(Ω−Ω

2

) +k

2

Ω

2

Ω

2

Ω

2

<Ω,

(3.7)

where

k

1

=

Ω

1

Ω

1

−Ω

0

k

2

due to continuity at Ω=Ω

1

.This control strategy extracts minimum energy

up to Ω

0

,where Ω

0

for both strategy B and C is chosen to correspond to the

rotational velocity at tip speed ratio 4 and ﬂowvelocity 3 m/s.These particular

values are chosen because 3 m/s is considered a reasonable cut-in velocity and

tip speed ratio 4 is the optimal value for the chosen turbine.

The control strategy C is designed to operate according to equation (3.5)

between Ω

1

and Ω

2

.The region between Ω

0

and Ω

1

is the increase from 0 to

the curve in equation (3.5).Hence,with increasing rotational ﬂowvelocity,the

tip speed ratio decreases in the region between Ω

0

and Ω

1

until it has reached

its desired value for high power capture.At rotational velocities above Ω

2

,

the tip speed ratio is intentionally decreased to reduce mechanical loads on

the turbine,which creates a softer transition to the constant speed region in

ﬁgure 3.1.This region is reasonable to implement on a real turbine,but it has

not been studied in detail in this work and for simplicity,k

1

=k

3

was chosen.

The control systemis designed to reach Ω

2

at 9 m/s.This strategy is illustrated

in ﬁgure 3.5,where it is seen that at 3 m/s,λ

e

is higher that the peak value,

causing a small drop in power coefﬁcient,although the difference is smaller

than for strategy B.For 12 m/s,λ

e

is slightly lower than the peak value due

to the intentional decrease in rotational velocity above Ω

2

.Note that the value

for the constant k

1

differs between strategy B and C,while the value for the

constant k

2

is equal for strategy A and C.Simulations of the three control

strategies are found in section 5.2.6.

35

3.2 Extension to multiple turbines

It is common to locate several turbines in close proximity to each other in a

farmconﬁguration,as this can give economical beneﬁts due to synergy effects,

since some parts of the systemcan be commonly used.Therefore,two differ-

ent electrical topologies are suggested,where the second “mutual topology”

is designed to reduce the number of electric components [32].For additional

possible topologies,see e.g.[33–35].

One way to control multiple turbines is to apply the model described in sec-

tion 3.1 for each turbine in the farm.This can be accomplished by using an

individual electric system for each turbine,where each turbine has a passive

diode rectiﬁer and an inverter (“separate topology”).An alternative approach

is to connect all turbines to the same inverter,but to obtain individual control.

This would for a permanent magnet synchronous generator require some addi-

tional electrical components such as an active rectiﬁer or a DC-DC converter.

One simpliﬁcation is to connect all turbines to the same DC-bus with pas-

sive rectiﬁers,which makes the inverter the only parameter to control (“mutual

topology”).This reduces the number of electric components and it is therefore

of interest to study how performance is affected by this simpliﬁcation.

For the mutual topology,the total power extracted from the entire turbine

farm is chosen as

P

e,tot

=

N

t

∑

i=1

P

e

(Ω

i

),(3.8)

where P

e

(Ω

i

) is calculated according to one of the strategies A – C in sec-

tion 3.1.With the total extracted power chosen according to equation (2.1),

the separate and mutual topologies extract the same power if all turbines ex-

perience identical ﬂow velocities.

Other studies have previously been performed for horizontal axis turbines,

with a global control strategy for the entire farm [36,37].Farm effects have

been included in the control system in [38,39] and a method for estimating

the ﬂow velocity within the farm with only rotational velocities and extracted

power as input parameters is presented in [40].

36

4.Simulation models

The ﬂow through a turbine can be simulated by solving Navier-Stokes equa-

tions coupled with the continuity equation.Navier-Stokes equations are valid

for all Newtonian ﬂuids,i.e.both air and water,although in water,cavitation

would require additional modeling.The validity of Navier-Stokes equations

is well established,and the most accurate simulation model would be a direct

solution of these equations.However,Navier-Stokes equations are non-linear

partial differential equations and due to the computational complexity,accu-

rate direct solutions of these equations are numerically very hard to obtain,

even for the two-dimensional case.Considering that the ﬂuid-ﬂow in a large

turbine generally is turbulent,one common simpliﬁcation is to model the tur-

bulence by e.g.Reynolds-averaged Navier-Stokes equations (RANS),which

include additional turbulence models.This has been done by e.g.[41,42].

The most common methods to solve Navier-Stokes equations are the Finite

Element Method (FEM) and the Finite Volume Method (FVM),which are

both based on a mesh over the entire simulated volume.For this reason,

these methods are suited for conﬁned regions.There are commercial FEM

and FVMsimulation software available,but due to the long simulation times,

other methods have been chosen in this work.

Another method for solving Navier-Stokes equations is to use a vortex

method (see section 4.2).This method can also directly solve Navier-Stokes

equations,and is designed for open ﬂows with small boundaries,which is the

case for a vertical axis turbine.Directly solving Navier-Stokes equations with

this method requires much computational time,and has not been done in this

work.An advantage of the vortex method is that there are many simpliﬁca-

tions available,and by including models of blade forces,the simulation times

can be reduced substantially.This simpliﬁcation is used within this work.

One even more simpliﬁed and very fast method is the streamtube model,

which assumes static ﬂow,and does not give a full description of the ﬂow

through the turbine.Due to the high computational efﬁciency,this model can

be used to quickly performmany simulations,and it is one of the models used

in this work,even though the simpliﬁcations of the ﬂowlimit the applicability

of the model.

4.1 Streamtube models

Streamtube models are among the fastest models used for simulating vertical

axis turbines.The ﬁrst application of streamtube models,usually credited to

37

V

∞

V

d

V

e

A

d

R

θ

b

Δθ

Ω

c

Figure 4.1.Illustration of the double multiple streamtube model of Paraschivoiu with-

out ﬂow expansion,where the streamtubes are illustrated by the horizontal lines and

the vertical line in the center illustrates the transition fromthe upstreamto the down-

stream part.Flow velocities are given for the upstream part.It is assumed that the

height of the streamtube Δz is included in the streamtube area A

d

.

Templin [43],used one streamtube for the entire turbine.The disadvantage

of this method is that the ﬂow velocities at the blade positions are constant

throughout the entire turbine.A vertical axis turbine will interact with the

ﬂow two times,one upstreamand one downstream.To overcome the problem

with equal ﬂowvelocity when the blade is up- and downstream,Lapin [44] in-

troduced the double streamtube model,consisting of one streamtube upstream

and another downstream,allowing for lower ﬂow velocities downstream.For

a real turbine,the velocity does not only vary between the upstreamand down-

streampart of the turbine,but could also vary over the entire swept area of the

turbine.To account for this,Strickland introduced the idea of using multiple

streamtubes [45].In this case,the systemhas to be solved for each individual

streamtube.

In Strickland’s implementation,the double streamtube model of Lapin was

not included.Later,both these two ideas were combined into the double

multiple streamtube model.Two different versions of this model were de-

veloped around the same time,one by Paraschivoiu [46] and one by Read and

Sharpe [47].The difference between the models is that the implementation

by Paraschivoiu assumes straight streamlines including only a velocity in the

x-direction (i.e.the direction of the asymptotic ﬂow),while the model by Read

and Sharpe assumes that the streamlines expand linearly through the turbine.

This gives the difference that in the model by Paraschivoiu,all streamtubes are

independent of each other,while in the model of Read and Sharpe,they are

coupled.Both models have shown quite good agreement with experiments for

the curved blade Darrieus turbine [3,48].

38

4.1.1 Description of model

The double multiple streamtube model implementation used here is based on

the implementation of Paraschivoiu [3] and an illustration of this model is

given in ﬁgure 4.1.In the double multiple streamtube model,the turbine is

separated into two discs,one upstreamand one downstream.In the middle of

the turbine,the pressure is assumed to be the same as the asymptotic pressure.

With this assumption,the velocity at the upstream disc will be the average of

the velocity at the middle and the asymptotic velocity,similar to equation (2.7)

in the Betz theory.In the same way,the velocity at the downstream disc be-

comes equal to the average of the velocity at the middle,and the velocity far

behind the turbine.In the implementation by Paraschivoiu,the upstream disc

is solved ﬁrst,independently of the downstream disc.The downstream disc

uses the velocity in the middle of the turbine,calculated when solving the

upstream disc,as input.Except from this calculation of velocity,both discs

are solved in the same way,and most equations will only be presented for the

upstreampart of the system.

This description will followthe same notation as Paraschivoiu for the direc-

tion of the ﬂow velocity.Hence,the ﬂow direction illustrated in ﬁgure 4.1 is

considered positive direction.The deﬁnitions of the blade angle θ

b

and the an-

gle of attack α will still remain the same as in section 2.3.Therefore,negative

angles of attack will be obtained when the blades are upstream,while the de-

scription by Paraschivoiu [3,Chapter 6] has positive angle of attack upstream.

The difference in the deﬁnition of the ﬂow direction is the reason why the

expressions for ﬂow velocity and angle of attack are different in this section,

compared to section 2.3.

The main principle behind streamtube models is the use of momentumcon-

servation in each streamtube.Similar to the Betz derivation,momentum con-

servation gives

ρA

ei

V

2

ei

−ρA

∞i

V

2

∞i

=ρA

di

V

di

(V

ei

−V

∞i

) =F

xi

,(4.1)

where A

∞i

,A

di

and A

ei

are the cross-sectional areas of a streamtube and i

denotes the streamtube index (cf.ﬁgure 2.1).The difference,compared to

the Betz theory,is that in this model,the force F

xi

is calculated from blade

section data.If the streamtubes are discretized to give each streamtube the

same angular distance Δθ and height Δz,the size of each streamtube becomes

A

di

=R

i

ΔzΔθ|cosθ

bi

|.(4.2)

To calculate the force,the ﬁrst step is to obtain the lift and drag coefﬁcients,

which depend on the angle of attack and the Reynolds number.In the model

of Paraschivoiu,it is assumed that the direction of the ﬂow does not change.

This gives the relative velocity as

V

ri

=V

di

R

i

Ω

V

di

−sinθ

bi

2

+cos

2

θ

bi

cos

2

η

i

(4.3)

39

and the angle of attack as

α

i

=ϕ

i

+δ

i

=arctan

cosθ

bi

cosη

i

sinθ

bi

−

R

i

Ω

V

di

+δ

i

,(4.4)

where η is the angle of the blade relative to the vertical axis.

When the coefﬁcients have been determined from the empirical data,the

lift and drag forces are obtained from

F

Li

=

1

2

C

Li

ρc

i

l

i

V

2

ri

,(4.5)

F

Di

=

1

2

C

Di

ρc

i

l

i

V

2

ri

,(4.6)

where c

i

is the chord and l

i

is the blade length in the streamtube.Generally,

it is the normal and tangential forces that are of interest.These forces can be

determined from the corresponding normal and tangential force coefﬁcients

C

Ni

= C

Li

cosϕ

i

+C

Di

sinϕ

i

,(4.7)

C

Ti

= C

Li

sinϕ

i

−C

Di

cosϕ

i

(4.8)

and therefore,the forces are

F

Ni

=

1

2

C

Ni

ρc

i

l

i

V

2

ri

,(4.9)

F

Ti

=

1

2

C

Ti

ρc

i

l

i

V

2

ri

.(4.10)

Note that by this deﬁnition,the normal force is perpendicular to the blade,and

the force in the radial direction is

F

Ri

=F

Ni

cosη

i

.(4.11)

To calculate how the velocity decreases in the streamtube,the forces in the

x-direction (parallel to the ﬂow) are of interest.These are obtained as

F

xi

=F

Ni

cosθ

bi

cosη

i

−F

Ti

sinθ

bi

.(4.12)

Considering that there are N

b

blades on the turbine,and that a streamtube only

have a blade inside it a fraction of the time,the mean force is given by

F

xi

=

N

b

Δθ

2π

(F

Ni

cosθ

bi

cosη

i

−F

Ti

sinθ

bi

).(4.13)

If the length of the blade in a streamtube l

i

=Δz/cosη

i

is inserted into equa-

tions (4.9) and (4.10),equation (4.13) becomes

F

xi

=

N

b

ρc

i

V

2

ri

ΔθΔz

4π

C

Ni

cosθ

bi

−C

Ti

sinθ

bi

cosη

i

.(4.14)

40

V

s

V

di

V

ei

c

s

Δr

s

R

outer

R

inner

θ

b

θ

Streamtube i

Ω

η

s

r

s

Δz

Figure 4.2.Illustration of the parameters used for the strut model.

Combining this with equation (4.1) and (4.2) gives

ρR

i

ΔzΔθ|cosθ

bi

|V

di

(V

ei

−V

∞i

) =

N

b

ρc

i

V

2

ri

ΔθΔz

4π

C

Ni

cosθ

bi

−C

Ti

sinθ

bi

cosη

i

(4.15)

which with the substitutions

V

di

=u

i

V

∞i

(4.16)

and

V

di

=

V

∞i

+V

ei

2

(4.17)

gives

1−

1

u

i

=

N

b

c

i

V

2

ri

8πR

i

|cosθ

bi

|V

2

di

C

Ni

cosθ

bi

−C

Ti

sinθ

bi

cosη

i

.(4.18)

This is the equation to be solved to obtain the interference factors u

i

(deﬁned

in equation (4.16)).Since V

r

,C

N

and C

T

depend on u,this equation has to be

iterated.It should be noted that from equations (4.16) and (4.17),the velocity

in the center is

V

ei

=(2u

i

−1)V

∞i

.(4.19)

4.1.2 Including struts

The above theory only includes the blades,but for an H-rotor,it is necessary

to attach the blades to the main shaft with struts.These struts interact with

the ﬂow,affecting turbine performance.One strut model was implemented by

Moran [49],who derived a correction coefﬁcient for the drag.This is a fast

solution suitable for horizontal struts,but for struts with an angle,a lift force

41

will also be generated.To increase the ﬂexibility of the code,it was chosen to

calculate the forces directly fromthe lift and drag coefﬁcients instead.

To calculate the forces,the relative velocity has to be known.Let r

s

be a

position along the strut

R

inner

<r

s

<R

outer

,(4.20)

where the strut starts at R

inner

and ends at R

outer

,see ﬁgure 4.2.The struts

are discretized into Δr long segments in the radial direction.In the streamtube

model,the velocity is only known at the turbine disc and at the center of the

turbine.The simple approximation that the velocity varies linearly between

these positions gives the velocity V

s j

of strut segment j as

V

s j

=V

ei

+

V

di

−V

ei

R

2

i

−r

2

s j

sin

2

θ

b

r

s j

cosθ

b

.(4.21)

The position of the blade,to which the strut is attached,is given as θ

b

.This

expression involves the velocity in streamtube i,in which the segment is lo-

cated.To determine which streamtube this is,the corresponding angle at the

circumference of the turbine is given by

θ

i

=arcsin

r

s j

R

j

sinθ

b

.(4.22)

In the vertical direction,the position of segment j will be equal to the vertical

position of the corresponding streamtube i.The relative velocity can be calcu-

lated basically in the same way as in equation (4.3),but since the discretization

is performed in the radial direction for the struts,the angle is to be given rel-

ative this direction as well (the reason for not using vertical discretization for

struts is that

F

xi

in equation (4.14) would diverge for horizontal struts).With

the strut angle given as η

s

,the velocity is given by

V

rs j

=V

s j

r

s j

Ω

V

s j

−sinθ

b

2

+cos

2

θ

b

sin

2

η

s j

(4.23)

and the angle attack α

s

for the strut is

α

s j

=arctan

cosθ

b

sinη

s j

sinθ

b

−

r

s j

Ω

V

s j

,(4.24)

where zero pitch angle is assumed for the strut.By deﬁning the chord of the

strut as c

s

and following the same derivation as with equations (4.7) – (4.14),

the expression for the mean force in the ﬂow direction for a section (Δr

s

,Δθ)

of the strut is obtained as

F

xs j

=

N

b

c

s j

ρV

2

rs j

4π

ΔθΔr

s

C

Ns j

cosθ

b

tanη

s j

−C

Ts j

sinθ

b

cosη

s j

,(4.25)

42

where C

Ns j

and C

Ts j

are the normal and tangential force coefﬁcients of the

strut,which are deﬁned in a similar way as for the blades

C

Ns j

= C

Ls j

cosα

s j

+C

Ds j

sinα

s j

,(4.26)

C

Ts j

= C

Ls j

sinα

s j

−C

Ds j

cosα

s j

.(4.27)

Here,C

Ls j

and C

Ds j

are the lift and drag coefﬁcients for the struts.The differ-

ences in equations (4.24) and (4.25),compared to equations (4.4) and (4.14),

originate fromthe different deﬁnition of η

s j

,and that the discretization is done

in the radial direction.

The interference factor u

i

can be calculated according to

1−

1

u

i

=

1

2R

i

|cosθ

b

|V

2

di

ρΔθΔz

F

xi

+

∑

j∈i

F

xs j

(4.28)

where ∑

j∈i

F

xs j

is the sum of the forces from all points (r

si

,θ

b

) that corre-

sponds to streamtube i.Similar expressions can be derived for the downwind

part of the turbine.The torque from the struts can be calculated in a similar

way as for the blades in a curved blade Darrieus turbine

T

s

(θ

b

) =

∑

j

r

s j

F

Ts j

(θ

b

)

cosη

j

,(4.29)

where

F

Ts j

(θ

b

) =

1

2

C

Ts j

(θ

b

)ρc

s j

Δr

s

V

rs j

(θ

b

)

2

.

4.1.3 Obtaining lift and drag coefﬁcients

In the streamtube model,it is necessary to have the lift and drag coefﬁcients

for a given angle of attack.Experimental data can be used to obtain these if

the Reynolds number and angle of attack are known.For several symmetrical

NACA proﬁles,such data can be obtained from [50],which has been used in

this work.The problemwith this kind of data is that it usually is valid for very

long blades and a static angle of attack.For a vertical axis turbine,the blades

will change their angles of attack during the revolution.

When the angle of attack for a blade increases above the stall angle,the ﬂow

will separate fromthe blade surface,which reduces the lift force and increases

the drag.The ﬂow separation takes time to develop and if the blade rapidly

increases its angle of attack,the lift force can be higher than in the static case

for a short period of time.During this time,a vortex starts to form at the

surface of the blade and when this vortex detaches,the lift force is greatly

reduced.This phenomenon is called dynamic stall.When the turbine has a

low tip speed ratio,the angles of attack become high enough for the blades to

experience this in each revolution.This occurs for tip speed ratios around 4.

43

To model dynamic stall,the Gormont model has been used.This model was

originally developed by Gormont [51],later modiﬁed by Massé [52] and ad-

justed by Berg [53].The advantages of this model is that it only requires data

for lift and drag coefﬁcients,ﬂow velocity,blade thickness to chord ratio,an-

gle of attack and rate of change of the angle of attack,making it easy to apply

to any blade,which is the reason why this model was chosen.Other mod-

els have been shown to give better results [3].One example is the Beddoes-

Leishman model [54,55],but it has more parameters that have to be calibrated,

which makes it harder to apply the model to an arbitrary airfoil.

Tip effects

For a blade to obtain a lift force,there has to be circulation around the blade

(cf.Kutta Joukowski lift formula,equation (4.57)).Since the vorticity is di-

vergence free

ω=∇×

V ⇒∇·

∇×

V

=0

,there are vortices generated

from the blade tips with the same circulation as around the blade.These vor-

tices are the source of induced drag,and the corresponding losses.It should be

noted that the curved blade Darrieus turbine does not have any distinct blade

tips (the blades are attached to the main axis).Here,the tip vortices have

to leave the blades more evenly distributed over the length of the blade,and

in the model of Paraschivoiu,no tip losses are applied to curved blade Dar-

rieus turbines.For straight blade turbines,one correction model,derived from

Prandtl’s theory for screwpropellers,used by e.g.Sharpe [48],uses a velocity

correction factor F,which for the upstreamdisc is

F

i

=

arccos

e

−

πa

i

s

i

arccos

e

−

πh

2s

i

,(4.30)

where

s

i

=

πV

ei

N

b

Ω

,(4.31)

a

i

=

h

2

−|z

ai

|,(4.32)

z

ai

is the altitude (with zero deﬁned in the center of the turbine) and h is the

turbine height.A corresponding expression exists for the downstream disc.

The new expressions for velocity and angle of attack are

V

ri

= V

di

R

i

Ω

V

di

−sinθ

bi

2

+F

2

i

cos

2

θ

bi

cos

2

η

i

,(4.33)

α

i

= arctan

F

i

cosθ

bi

cosη

i

sinθ

bi

−

R

i

Ω

V

di

+δ

i

.(4.34)

44

In addition,Paraschivoiu [3] suggests using ﬁnite wing theory as well to cal-

culate induced drag and reduced angle of attack.This is given by

C

Li

=

C

L∞i

1−

a

0i

πAR

i

,(4.35)

a

0i

= 1.8π

1+

0.8t

bi

c

i

,(4.36)

C

D

i

= C

D∞i

+

C

2

Li

πAR

i

,(4.37)

α

bi

= α

i

−

C

Li

πAR

i

,(4.38)

where AR is the aspect ratio,t

b

the blade thickness,C

L∞

and C

D∞

lift and

drag coefﬁcients for an inﬁnitely long blade and α

b

the corrected angle of

attack.Paraschivoiu suggests that equations (4.35) – (4.38) should not be ap-

plied to dynamic stall calculations,but since this would lead to a discontinuity

in the limit when dynamic stall starts to have an effect,the current implemen-

tation uses equations (4.35),(4.36) and (4.38),then calculates the dynamic

stall corrections to the lift coefﬁcient,and then calculates the induced drag

using equation (4.37) with the new lift coefﬁcient.It should be noted that

equations (4.35) – (4.38) are derived using the assumption that the angle of at-

tack is constant,the tip vortices extend along a straight line behind the blade,

and the wing is elliptical,which is not the case for a vertical axis turbine.This

has to be taken into account when evaluating the accuracy of the model for

straight blade turbines.

4.1.4 Corrections due to ﬂow curvature

As was illustrated in section 2.4,there are effects due to ﬂow curvature that

should be taken into account.With the deﬁnition of the velocity as in the

streamtube model (V →−V),the assumption of straight streamlines (β =0),

using that the reference velocity V

ref

is the relative velocity V

r

and using the

velocity at the turbine disc V

d

instead of V

abs

,equation (2.52) gives

α=δ+arctan

V

d

cosθ

b

ΩR−V

d

sinθ

b

−

Ωx

0r

c

V

r

−

Ωc

4V

r

.(4.39)

This is included in the adaptation of Sharp [48],but with the assumptions that

the curvature force only acts in the normal direction and that the mounting

point is in the center of the blade (x

0r

=0).With the assumption

C

N

=a

s

α,(4.40)

the correction to the normal coefﬁcient becomes

C

N

=C

N0

−

a

s

Ωc

4V

r

,(4.41)

45

z

y

a

y∞

a

ye

aze

az∞

bze

1

bz∞

b

y∞

1

b

ye

Figure 4.3.Illustration of the cross-section for a streamtube at the three positions:Far

ahead of turbine (black),at the upstreamdisc (white) and in the center (hatched).

where C

N0

is the original normal force without corrections.

4.1.5 Including ﬂow expansion

In contrast with the model of Read and Sharpe [47],the model of Paraschi-

voiu does not include streamtube expansion in the basic implementation.One

method for implementing the expansion was used by Paraschivoiu [56] where

the streamtubes were allowed to expand,but the direction of the velocity was

not changed.In this implementation,similar to the one by Read and Sharpe,

it was assumed that streamtube expansion only occurs in the horizontal plane,

and that the streamline that intersects the turbine center is unchanged.In this

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