performance prediction of a hydrodynamic ocean energy device for sustainable electricity generation

7/28/2019 Performance Prediction of a Hydrodynamic Ocean Energy Device for Sustainable Electricity Generation

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Performance Prediction of a Hydrodynamic OceanEnergy Device for Sustainable Electricity Generation

A Computational Model Using a Hybrid Vortex Method

Context : Master of Science Thesis

Author : Jaap Versteegh

Supervisors : Tom van Terwisga (TU Delft)

Peter Scheijgrond (Ecofys B.V.)

Version : 2005-06-20/Final/rev. 2



Intentionally blank: document is double sided



Contents

Preface....................................................................................................................3Abstract..................................................................................................................4

1. Introduction........................................................................................................5

2. Purpose of Investigation...................................................................................6

2.1. State of the Art..........................................................................................................6

2.2. Present Work............................................................................................................6

3. Hydrodynamic Ocean Energy Device..............................................................7

3.1. Wave energy............................................................................................................7

3.2. Tidal energy................................................................................................. .............73.3. Hydrodynamic rotor..................................................................................................7

3.4. Flow Description.......................................................................................................93.4.1. Wave Flow...............................................................................................................93.4.2. Regular Waves........................................................................................................93.4.3. Irregular Waves.....................................................................................................10

3.5. Rotor in Waves.......................................................................................................11

4. Computational Model......................................................................................12

4.1. Potential Flow.........................................................................................................124.1.1. Laplace's Equation................................................................................................12 4.1.2. Vortices..................................................................................................................13

4.1.3. Vortex in Three Dimensions..................................................................................13

4.2. Vortex Theory of Lift...............................................................................................14

4.3. Lift in Three Dimensions: Lifting Line.....................................................................16

4.4. Hybrid Vortex Model...............................................................................................174.4.1. Limitations.............................................................................................................18

4.5. Stability Issues........................................................................................................18

4.6. Dynamics of Two-Dimensional Lift and Drag.........................................................204.6.1. Added Mass...........................................................................................................20 4.6.2. Dynamic Stall.........................................................................................................214.6.3. Combined effect....................................................................................................22

4.7. Rotor modeling.......................................................................................................22

4.8. Free Surface.......................................................................................................... .24

5. Computer Implementation: “WaveVort”........................................................25

5.1. Program Layout......................................................................................................25

5.2. Execution flow.........................................................................................................26

5.3. Input/Output............................................................................................................28

5.4. Performance...........................................................................................................285.4.1. Code......................................................................................................................28 5.4.2. Compilers..............................................................................................................29

1/48



6. Program Verification and Validation..............................................................31

6.1. Three Dimensional Wing........................................................................................316.1.1. Case Input.............................................................................................................31

6.1.2. Test Without Wake................................................................................................316.1.3. Wake Test.............................................................................................................32 6.1.4. Case Results.........................................................................................................32

6.2. Wells In Uniform Flow............................................................................................336.2.1. Case Input.............................................................................................................336.2.2. Case results...........................................................................................................33

6.3. Wells in Waves.......................................................................................................35

6.4. Darrieus in Waves..................................................................................................35

6.5. Wave Rotor.............................................................................................................35

7. Design Considerations....................................................................................36

7.1. Design Parameters.................................................................................................367.2. Rotor shape............................................................................................................36

7.2.1. Design Objective...................................................................................................36 7.2.2. Load Distribution and Angle of Attack...................................................................36 7.2.3. Optimal Shape and Solidity...................................................................................38 7.2.4. Full scale...............................................................................................................397.2.5. Conclusion.............................................................................................................40

7.3. Extrapolating model tests.......................................................................................407.3.1. Extrapolation Rules...............................................................................................40 7.3.2. Reynolds Scaling Effects.......................................................................................41

7.4. Economy of scale ?................................................................................................42

8. Conclusions.....................................................................................................43

9. Recommendations...........................................................................................44

Symbols................................................................................................................45

Index.....................................................................................................................46

References...........................................................................................................47

Quotes..................................................................................................................48

2/48



Preface

Before you lies a Master's thesis, the result of three quarters of a year's work studyingEcofys' Wave Rotor, testing it and finding ways of getting to a proper performance

estimation for it. It has also been an excellent opportunity for me to finally master C++,something that had been on my wish-list for too long, like the Master's degree itself !

Though the nature of the subject implies a rather technical explanation of things, I have

attempted to be descriptive in parts where a reader who is not familiar with fluid dynamicsmight otherwise get lost.

Renewable energy has become more important than ever. Working on ways to develop

elegant and efficient solutions in this area is joyful and rewarding. It has been a pleasureto do this project.

Acknowledgments

My profound thanks go to Professor Tom van Terwisga for accepting me at the SectionShip Hydromechanics and his encouraging remarks, and to Henk de Koning-Gans for technical hints. I owe gratitude to Professor Gerard van Bussel and to Carlos Ferreira

from the department of Wind Energy for monitoring the project and giving good advice.

From Ecofys, I wish to thank Peter Scheijgrond for providing the opportunity to do thiswork and his enthusiasm in the project. Anton Schaap and Bart-Jan Sustronk gave

valuable suggestions.

Further thanks go first to my mother for being critical and believing in me nonetheless andmy father for his never ending patience with my childhood engineering quests and

questions. See where it got me :). Thanks to Eva for love and patience, to friends out

there for being interested and to Pietje for inspiring company during late hours ;)

Finally I wish to thank all people involved in free software development (free as in

freedom) and especially Prahbu Ramachandran for the lovely visualization tool “Mayavi”and Naba Kumar for “Anjuta”, the C++ IDE. Software from India is coming ! Thisdocument was written using OpenOffice.org. Better than MS Word, but I guess I will haveto learn TeX after all :p.

Jaap Versteegh

Delft, spring 2005

3/48



Abstract

This Master's thesis reports on the results of investigations into performance predictionsfor hydrodynamic ocean wave energy devices, in particular the “Wave Rotor” developed

by Ecofys B.V., Utrecht.

A computer model is presented that simulates the “Wave Rotor” motions, considering allmajor forces that act on the device. The hydrodynamic forces are determined with ahybrid vortex model. This model computes local forces on the rotor blades by combiningwell known two dimensional wing section data with local three dimensional velocities.

Three dimensional velocities are taken to be the sum of the undisturbed flow anddisturbances caused by presence of the rotor. The calculation of the rotor disturbances,

i.e. the induced velocities, is the main task of the computer model. Interaction with thefree surface is not modeled at this time. This is required for a more accurate prediction for rotor that have a significant impact on the wave surface i.e. large and solid rotors. Itsdevelopment is therefore the number one recommendation.

The applicability and reliability of the method that has been developed is demonstrated bycomparisons with data from literature and model tests conducted at the NaREC testingfacilities at Blyth (UK). The method is suitable for simulating wave rotor motions of themodel rotor and main performance quantities with reasonable accuracy, provided thatvalid input data is used.

For the designer a rotor shape is suggested and some considerations that may be usefulfor wave energy rotor development.

The following recommendations are made based on insight gained during this work:

• Further develop and validate the current computational model. The results indicate that

this is a good direction, but the model can be improved by adding for example:- Free surface interaction.

- Improved wake modeling.

• Further develop the combination with tidal stream effects. The present work focuses

on wave energy but a rotor can combine this with tidal energy.

• Test blade sections used for model testing at NaREC for (dynamic !) 2D performance.

This helps validation of computational methods using two dimensional wing sectiondata and is vital for extrapolation of model tests to full scale.

• Build a model at larger scale that suffers less from Reynolds scaling effects than the

10th scale model.

• Investigate blade profiles with a camber that matches the rotor blade path curvature.

• Investigate dynamic rotor loading models that optimize performance in irregular waves.During model tests the performance in irregular waves was exceptionally poor. Thesimple linear loading that was applied during these test was certainly doing much harmhere. The rotor only performs well when running close to optimal rpm at all times. Thecomputer model developed in this project can help evaluate loading models.

• Investigate anti-fouling methods. Real-life rotors require a hydrodynamically smooth

surface to operate properly. Marine fouling can be a severe problem, especially for

objects that aren't moving. The feasibility of keeping a rotor spinning at all times hasyet to be determined. Also, considering the nature of the device, the application of environmentally harmful anti fouling seems problematic.

4/48



1. Introduction

This report contains the results from the development of a tool for the power prediction of hydrodynamic ocean energy devices. A hydrodynamic ocean energy device generates

electricity from water surface waves or tidal currents by means of submerged wingprofiles that generate hydrodynamic lift from water velocities, much in the same way that

blades of a wind turbine generate electricity from the velocities of the air in wind. Thefocus of this investigation is on wave energy.

The combination of wave flow and a hydrodynamic turbine is relatively new and therefore

the application of existing flow calculation and performance prediction tools is nonarbitrary. Hence a custom tool was developed based on lifting line theory that models liftgeneration of a wing by means of a vortex line. This method was preferred over “streamtube” models, which are common in wind turbine analysis, because of the unsteady

nature of the problem.The hybrid vortex method that has been developed is simple and robust and gives aperformance prediction that already agrees well with currently available experimental data

of a model scale rotor. It does not yet contain free surface interaction.

The report starts off with the research purpose followed by a description of ahydrodynamic wave energy device. The relevance of development of wave energy and

the wave energy generation concept is discussed. This is followed by a discussion of flowphenomena and the potential flow method describing these phenomena.

The middle part of the report describes the performance prediction method chosen and a

description of its implementation: “WaveVort”. A list of verification and validation casesprovides an insight into the reliability and applicability of the tool.

The investigations associated with the development of the prediction method lead to

views regarding rotor design that are presented as design issues. Finally, a summary of achievements and recommendations for further investigations is given.

Introduction 5/48



2. Purpose of Investigation

“Ecofys B.V.”, Utrecht is developing the “Wave Rotor” in cooperation with “EngineeringFirm Eric A. Rossen”, Denmark. This rotor is a hydrodynamic ocean energy device

indented to generate electricity from ocean waves and tidal currents.

2.1. State of the Art

At present, a 10th scale model prototype of the “Wave Rotor” exists. Two series of model

test have been conducted with this model [11][9]. The UK based consultancy firm Halcrowhas issued a report [10] discussing feasibility of a full scale device including a

performance prediction using spreadsheet based calculations. The results from thesecalculations and the model tests do not yet provide a very consistent picture of the rotor'sperformance.

2.2. Present Work

The present work aims to develop a performance prediction method for an arbitrarilyshaped rotor in various conditions with an improved accuracy. This accuracy is importantfor

• Analyzing, supporting and extrapolating model test results

• Providing reliable data for feasibility studies

• Improving rotor design

In a definition study for this work [17], several options were considered for obtaining aperformance prediction, including:

1) Using existing Boundary Element/Panel flow simulation software.

2) Using existing Finite Element Navier Stokes software.

3) Developing a Multiple Stream tube Method.

4) Developing a Vortex Method.

The fourth option was considered a good option when regarding

• Affordability

• Complexity

• Extensibility

• Expected Accuracy

• Chance of Success

Purpose of Investigation 6/48



3. Hydrodynamic Ocean Energy Device

The Kyoto treaty and general concern about climate changes create a new market for renewable energy; see [17] for a discussion. Ocean energy is a possible source of

renewable energy. Two types of ocean energy are distinguished here: surface wave andtidal energy.

The hydrodynamic ocean energy device under consideration operates on the principle of

fluid dynamic lift in the same way that a wind turbine does. It is driven by water velocitiesassociated with either wind generated surface waves or tidal currents. This 'dual

operation' option is an advantage of a hydrodynamic device over other types of waveenergy devices [17]. The main focus of this investigation lies with wave energy.

3.1. Wave energy

Energy is contained in and transported by wind generated surface waves. Thetransported energy or 'wave power' can have considerable density as can be seen fromfig. 1. Comparing this density to typical power density values for photo-voltaic energy (0.2kW/m2) and wind energy (0.3 kW/m2) -- mind the dimensional difference --, one can

conclude that extraction of wave energy could be economically feasible.

Global average wave energy density (kW per meter wave front) (fig. 1)

The availability of Atlantic shoreline and dislike of wind turbines that disturb the localenvironment has especially got the UK interested in developing wave energy technology.

3.2. Tidal energy

Tidal energy is particularly interesting in places with high velocity tidal currents, of whichthere are only so many in the world. For regular tidal currents, the power density might

not be very interesting, though this could change when combined with wave energy. Thetidal energy is then an 'extra'. An advantage of tidal energy is its good predictability. Seevarious references (e.g. [23]) on the internet for information on tidal power.

3.3. Hydrodynamic rotor

A hydrodynamic rotor operates on the principle of hydrodynamic lift. The rotor has anumber of blades with a wing shaped profile that produce a lifting force.

Hydrodynamic Ocean Energy Device 7/48



Fluid dynamic lift and drag (fig. 2)

The lifting force L is perpendicular to the incident flow U . When the drag force D is

sufficiently small, the resulting fluid dynamic force R on the blades will have a

forwardly direction driving the blade like a sail driving a sailing yacht.

Forces on rotor blades for arbitrary velocity (fig. 3)

The combination of driving forces of the blades produces a torque about the shaft of therotor that can be used to drive a generator.

Ecofys' Wave Rotor (fig. 4)


U

L

D

R

L

L

L

U

r

ωR

U

ωR .r

ωR .r

ωR .r

U

U



Ecofys has designed a “Wave Rotor” that is a combination of a darrieus rotor (V-blades)and a wells rotor (Horizontal blades). See fig. 4. The darrieus is an omni directional rotor,which means it works regardless of the flow direction. This can be deducted from the fact

that in fig. 3, the velocity U can have any direction, without affecting the principle of operation. The wells blades are propelled by vertical flow, either upward or downward.The blades have a symmetrical NACA0015 profile and by default a zero pitch angle withrespect to the rotor circle tangential.

3.4. Flow Description

The “Wave Rotor” is primarily intended to operate on the flow associated with windgenerated waves. The uniform flow associated with tidal currents will also contribute to its

power output, but since wave flow is the most complex, this report focuses on that. See

also §7.2 for considerations regarding currents.

3.4.1. Wave Flow

The flow in ocean waves is rather arbitrary because the waves themselves are arbitrary in

length and height. The complex surface pattern of a sea can be decomposed into regular wave components with a sinusoidal shape [8][19]. For a specific sea state this set of

components make up the wave spectrum (§3.4.3).

3.4.2. Regular Waves

The flow properties for a regular wave component can be determined using linear potential flow theory (§4.1). Since the boundary condition for the free surface is applied at

z=0 and not at the actual position of the free surface, this theory is only accurate for long

crested waves with a relatively small steepness. For regular sea states, the steepness issufficiently low to get accurate results.

The following quantities are obtained by applying linear potential theory [8].

Wave surface elevation:

=a coskx− t (eq. 1)

Horizontal and vertical velocity:

u=a cosh k h z

sinh khcoskx− t (eq. 2)

v=a

sinh k h z

sinh kh sinkx− t (eq. 3)

Water particles are moving in circles, or ellipses when the water is shallow, that decreasein size further away from the water surface. At the surface the radius of the circle is equal

to the wave amplitude. Hence the wave velocity is also defined as

V w=a (eq. 4)

This is the velocity that is used in for example the TSR (Tip Speed Ratio) of a rotor.

The relation between the wave number k and the wave frequency ω, is called the

dispersion relation:

2=kg tanh kh (eq. 5)




It states that the wave length is approximately proportional to the square of the waveperiod.

Water particle motion (fig. 5)

3.4.3. Irregular Waves

Real sea surface conditions are irregular. Changing wind velocity and direction causeirregular waves. This irregular surface can be decomposed into singular components by

means of Fourier transformation, transforming the time signal of the water surfaceelevation to the frequency domain [19], thus creating a wave spectrum. For reasons that

will not be discussed here, this transformation is usually done on the wave power rather

than the elevation level t S . This creates a wave energy spectrum that

effectively removes phase information from the signal.For specific sea areas and weather conditions, the resulting spectrum will have a more or less unique shape making this spectrum a good quantifier for the sea state.

From analysis of many data series, different models for typical shapes of spectralfunctions have been developed. For example the ITTC Two Parameter Spectrum and theJONSWAP spectrum. The latter being specific for the North Sea and other waters with

limited water depth and wave development reach.

Example of ITTC 2 parameter spectrum (H1/3 = 4m, T=8s) (fig. 6)




The amplitude of a specific wave component can be obtained from the spectral value by

a= 2S (eq. 6)

For simulation purposes time signals are relevant. Since the characteristics of the sea ata specific location will usually be provided as a spectrum, this requires a regeneration of atime signal from the spectral function. This can be done by:

t = ∑=min

=minn⋅

a , ⋅cos t (eq. 7)

The phase angle ε is not stored in the spectrum so it has to be randomly chosen. See [19]

for more on irregular waves.

3.5. Rotor in Waves

The flow pattern around a rotor in waves is very complex.

Ecofys' “Wave Rotor” at the testing site of NaREC at Blyth (fig. 7)

These are some general observations that can be made.

• The water particles lose speed because of the rotor blades blocking their path.

• The free surface is deformed by the presence of the rotor, because of the pressures

that it induces.

• Turbulence generated by the blades does not wash downstream as with wind turbines,but remain in the rotor area; the rotor blades pass each other's wakes all the time.

• The blades piercing the surface cause surface waves and spray.




4. Computational Model

Several ways to capture the most important aspects of the flow as described in §3.5 havebeen investigated. See [17] for a survey. The main options were a “multiple stream tube”aka. “blade element momentum -BEM” method and a vortex method. Panel methods andfinite element methods are certainly feasible but generally too complex to be used in thedesign stage unless maybe adapted to the specifics of the problem, which is aconsiderable task. I chose a vortex model over the multiple stream tube method because

of the unsteady nature of the problem. The definition of stream tubes in a rotating flowseems awkward and hard to implement.

4.1. Potential Flow

Potential flow refers to a mathematical method for solving flow problems by choosing the

flow velocity to be the derivative of a potential. This potential is a scalar function in space-time and usually called Φ. Φ is defined such that the velocity is:

U =∇ or u , v , w =∂

∂ x,

∂

∂ y,

∂

∂ z (eq. 8)

4.1.1. Laplace's Equation

When using a potential for describing a flow, the first requirement is that the velocitiesresulting from the potential satisfy the continuity equation, which states that inside a

control volume no fluid can be created or destroyed.

Consider an infinitesimal control volume:

Flow continuity (fig. 8)The sum of flow fluxes through each of the volume boundaries has to be zero:

udyvdx−u∂ u

∂ xdxdy−v

∂ v

∂ ydydx=0

∂ u

∂ x

∂ v

∂ y=0 (eq. 9)

Or in three dimensions:

∂ u

∂ x

∂ v

∂ y

∂ w

∂ z =0 (eq. 10)

Computational Model 12/48

dx

dyu

∂u

∂ xdx

v∂v

∂ ydy

v

u



Substituting Φ from eq. 8 into this equation yields:

∂2

∂ x2

∂2

∂ y2

∂2

∂ z 2

=0 or =0 (eq. 11)

This is called Laplace's equation. A nice feature of this equation is that it is linear, whichmeans that an addition of two solutions of the equation is itself also a solution. Thismakes it possible to construct complex solutions from a number of simpler solutions. Thisproperty is used by almost all methods based on potential flow, like vortex and panelmethods.

4.1.2. Vortices

Potential flow cannot include rotation, which means viscous effects like turbulence cannot be accounted for and also that there can be no vorticity within the flow. A potential

flow field is conservative, which means that no energy can be extracted from it, in the

same way that you cannot extract energy from a stationary gravitational or electrical field.

The question is then: how can potential flow serve a problem with a rotor that is supposed

to extract energy from the flow? The answer is a special kind of solution of Laplace'sequation called a singularity. A singularity satisfies Laplace's equation everywhere except

for one point. The singularity that is of interest for this investigation is the vortex. Itrepresents a circulation of flow about the singular point. The potential associated with a

two dimensional vortex located at (0,0) is:

x , y =−

2arctan − y

x (eq. 12)

Γ is the circulation when going around the singular point. Fig. 9

Vortex potential and velocities (fig. 9)

Observe the discontinuity along the positive x-axis in fig. 9. This shows that the vortex isnot quite a proper solution of Laplace's equation, which it is not intended to be, because it

needs to represent circulation. The step at the jump is exactly Γ .

4.1.3. Vortex in Three Dimensions

A vortex in three dimensions is similar to a vortex in two dimensions, except that it isstretched out along a line.




Three-Dimensional vortex line (fig. 10)

The velocity induced at a point P by a vortex line is given by the Biot-Savard law [6]:

U P =

4∫ dL× R

∥ R∥3(eq. 13)

Here R is the distance vector pointing from the infinitesimal segment vector dL to the

point P . For a straight vortex line this integral reduces to [6]:

U P =

4 R1× R2

∥ R1× R2∥2L⋅ R1

∥ R1∥−

R2

∥ R2∥ (eq. 14)

Note that a vortex line cannot end or start in the flow. It has to be closed or terminate at adomain boundary. For a rotor problem a domain boundary is typically the bottom or thefree surface. This results from the definition of vorticity as:

=∇×U (eq. 15)

By definition, the divergence of γ is zero, as any divergence of a curl is, meaning any

vorticity going into an arbitrary control volume must come out again. If you don't force this

by means of continuing the vortex line, the surrounding flow field will no longer be vorticityfree and irrotational and therefore no longer be a potential flow.

4.2. Vortex Theory of Lift

For understanding the vortex theory of lift we will first consider what lift is and how it isgenerated. As for so many things, this is easiest explained by Newton's second law for anobject.

F =mA (eq. 16)

Or more generally formulated for a system:

F = d mU dt

(eq. 17)

The force exerted on a system is equivalent to the rate of change of momentum of that

system. Note that U is a vector.

By placing an airfoil in a flow at an angle of incidence, the flow will be deflected andtherefore the velocity will change direction.


x

y

z P

R

dL



Lift and change of momentum (fig. 11)

The force exerted by the wing on the flow is proportional to dU , while the force exerted on

the wing is opposing this force (action = -reaction). The part of F that is perpendicular to

U in is called lift. The rest is called lift induced drag.

Remains to explain why the flow is deflected by the airfoil. Theoretically it would be

perfectly possible for it to just go around and continue its course. A potential flow (§4.1)solution gives exactly that result (fig. 12a).

Potential flow without and with circulation (fig. 12)

However the flow in fig. 12a is non-realistic. The fluid goes around the sharp trailing edgeundergoing an infinite acceleration. Due to viscous effects, this will not happen in realityand the flow will separate from the wing at the trailing edge. This is called the Kuttacondition. In order to 'fix' this in the potential solution, a circulation needs to be added by

means of a vortex that 'washes' the aft stagnation point down to the trailing edge. Thestrength of the required vortex is:

=4 U R sin (eq. 18)

R is the radius of the circle that is mapped to the airfoil section using a conformal

mapping (Karman-Trefftz). This radius is approximately ¼ c, one quarter of the airfoilchord. The lifting force on the airfoil can be determined by integration of the pressurearound it. For the airfoil in fig. 12a it is 0 ! and for fig. 12b it is:

L= U (eq. 19)


Uin

F

Uout

dU

L

Di

a b



The lift is proportional to the circulation around the airfoil. By combining eq. 18 and eq. 19

and the R ~ ¼ c relation, we get the rather well known:

L≈ c U

2

sin(eq. 20)

4.3. Lift in Three Dimensions: Lifting Line

When the theory from §4.2 is extended to three dimensions, the (point) vortex becomes a

vortex line. This vortex line can not stop at the tips of the wing (§4.1.3). Since vortex linesin free flow move with the fluid, the vortex line has to be extended along a streamline.

Lifting line (fig. 13)

The trailing vortices will induce non uniform velocities at the lifting line causing the lift tochange over the span of the wing, so a more realistic lifting line will consist of more than

one vortex line. In the limiting case where U is considered a continuous function along the

span of the wing, the discrete vortices transform to a continuous sheet of vorticity.

The correct position of the trailing vortex lines can only be determined by an iterative

procedure since the position of the streamlines on which they are located is a function of the position of the vortices themselves. This iterative procedure is called wake relaxationand can be very expensive in terms of computation time. Sometimes good results can beobtained by estimating the position of the wake based on knowledge of the flow.

When the incoming flow U is non-steady -- i.e. varying with time --, the vorticity will alsovary. The vortex lines associated with the change of vorticity also have to be closed.When the variation of vortex strength -- i.e. the blade loading – is considered in discretesteps, a lattice of vortex rings is shed by the lifting line as shown in fig. 14.

Vortex shedding in unsteady flow (fig. 14)

The shed vortex lines move with the flow like the trailing vortex lines.


U

closed at ∞

L

Γ

Γ

stream- and vortex linewing

U(t)

closed at ∞

Γ(t)

ΔΓ

Γ(t) + ΔΓ

ΔΓ = Γ(t + Δt) -Γ(t)Γ(t) + ΔΓ

Uwake

Δt



4.4. Hybrid Vortex Model

The vortex model applied in this investigation is essentially the lifting model of fig. 14 with

a series of lifting line elements per wing section.

Hybrid vortex model mesh (fig. 15)

However, a real airfoil is not a lifting line but a lifting surface, with a certain chord length.For the sake of simplicity, the airfoil is still modeled using a single vortex. The strength of

the vortex can then be determined in two ways.

• 1: Apply a boundary condition on the airfoil surface

The boundary condition states that the velocity normal to the profile chord is zero. The

location of the vortex on the wing and the location of the point where the boundarycondition will be met, have to be chosen. The boundary condition will be implemented onthe airfoil chord which is a good approximation for thin airfoils.

Vortex and collocation point location (fig. 16)

The vortex is placed at the aerodynamic center, which is located at ¼ c. This can bedetermined from the pressure distribution in fig. 12b. This ensures that the point of actionof the lifting force is the same for the lifting line and the airfoil. The position of the

collocation point (x) is chosen such that for 2D conditions the vortex strength that meetsthe boundary condition also meets the Kutta condition as determined by eq. 18.

= U c sin &

2 a c=U sin a=0.5c (eq. 21)

• 2: Use 2D experimental lift data

An alternative approach is to use eq. 19 right away to determine the vorticity from 2D

experimental lift data.

=L2 D

U in

(eq. 22)


Uin

α

Γ

n

0.25c a = 0.50c

Ui

Γi

Γi

ΔΓi

Γi- ΔΓ

i

Ui+1

Γi+1

Γi+1

Γi+1

- ΔΓi+1

ΔΓi+1

Γi+1

-Γi

Γi+1

-Γi-ΔΓ

i+1+ΔΓ

i



The steps to determine the vorticity are:

Block diagram for hybrid vortex method (fig. 17)

Advantages for the second method are

• Simplicity, no system of equation has to be solved.

• Built-in simple stall model, 2D experimental data contains stall

Disadvantages

• Effect of chord length on 3 dimensionality of flow is ignored. i.e. the influence of

neighboring element vortices on the collocation point velocity.

• Instabilities might occur. See §4.5.

Even apart from the fact that it is unclear if method 1 will handle three dimensional flowover the blade any better, a well designed rotor probably has no submerged tips and hasslender blades, so 3 dimensionality of the flow should not be a big issue.

Stability issues can be dealt with using a high frequency filter (§4.5).

Considering these facts, method 2 was chosen for implementation and designated

“Hybrid Vortex Model”

4.4.1. Limitations

The hybrid vortex model is based on potential flow. This means:

• No turbulent wake effect due to stall are modeled.

• No boundary layer and viscous wake vorticity dissipation are modeled.

A single lifting line is used. This is only valid for

• High aspect ratio or 'slender' blades

4.5. Stability Issues

Consider the model described in §4.4. The mesh model from fig. 15 can be implemented

using doublet panels – a constant strength doublet panel is equivalent to a vortex ring, the

difference between two panel doublet strengths ( μ ) is the vortex strength at the common

edge.


Determine velocityat element by

adding undisturbedflow and velocityinduced by wake

vortices

Determine lift fromlocal 2D velocity and

angle of attack

L

α

Γ



For the sake of simplicity we will consider a steady uniformly loaded wing section.

Error progression in mesh (fig. 18)

At a certain point in time an error ε is present on the panel that is greyed in fig. 18. In

order for the solution to remain stable, the error has to decay with time. The vortex

strength along the rim of the panel is ε. The induced velocity ui due to the error can becalculated using the Biot-Savart law (eq. 13).

ui=

h

4U t

U t

h

U t 21

4h

2

(eq. 23)

The effect of ui on the loading μ for the next time step can be determined using eq. 18

i1=−c

h

4U t

U t

h

U t 21

4h

2

(eq. 24)

The demand that the error decays is:

∣i1−∣ (eq. 25)

Combining eq. 24 and eq. 25 and introducing an aspect ratio for the panel:

c

2 h

1

4 aa

a

21

4

1 with a=U t

h(eq. 26)

This means that the panel sizes cannot be smaller than the chord for practical aspect

ratios. This is inconvenient. The amplification of c/h can be countered by implementing an

exponentially moving average on the solution that has an amplification of h/c.

i1=1−h

c ih

ci1

(eq. 27)

This has a damping effect on the development of the wake. However, the filter length is

ac. For reasonable panel aspect ratios (a < 2), this filter length is sufficiently small to get

accurate results.


UΔt

h

ui

P

μμ

μ

μ

μ

μ+ε



Note that for small aspect ratios (a << 1), the U Δt part becomes dominant in eq. 24, so

the filter length should never be smaller than c/2. It is a bit of a hack, but it works.

4.6. Dynamics of Two-Dimensional Lift and Drag

The present model uses two dimensional hydrodynamic coefficients evaluated for thelocal Reynolds number and angle of attack. Two dimensional coefficients for many wingprofiles are available from numerous experiments for steady flow conditions. However,the rotation of the rotor and the rotary movement of water particles in waves cause the

flow to be not quite steady. The velocity and angle of attack on the rotor blades vary withtime.

This has a two-fold effect on the lift and drag of the blade profile.

• The fluid surrounding the blades moves along with the blades, effectively being added

mass.

• Dynamic stall occurs. Among other effects, the stall process is delayed when the angleof attack varies.

The degree of 'unsteadiness' of the flow is often indicated by the reduced frequency ,which is the ratio of the profile chord and the wave length of the unsteady flow, times π.

k =c

2U (eq. 28)

For the rotor this reduced frequency is in the order of 0.05-0.1.

4.6.1. Added Mass

The fluid mass that is added to the heaving or pitching motion of a flat plate can be

determined by potential flow analysis. It is equivalent to the fluid contained in a circle witha diameter equal to the length of the blade.

ma=c

2

4

(eq. 29)

The downward (i.e. perpendicular to the incoming flow) velocity of this added mass at the

flat plate is approximately U . The reactive upward force (i.e. lift !) according to

Newton's second law is then the added mass times the downward acceleration i.e. the

change of downward velocity.

La

=ma

⋅d U

dt

(eq. 30)

Or dimensionless

CLa= c

2U 2

d U

dt (eq. 31)

When taking into account that U̇ ≪U ̇ , this can be further simplified to:

CLa= c

2U ̇ (eq. 32)




4.6.2. Dynamic Stall

Stall is a complex phenomenon and dynamic stall even more so. It will not be discussedhere in detail. See e.g. [12] and [13] for more information. Two main effects can beobserved.

• The flow separation starting from the trailing edge is delayed with respect to the angle

of attack.

• A leading edge separation can occur, which creates a vortex that is being shed. This

vortex can cause a lift deficiency even after the angle of attack has returned into the'non-stalling' region.

From model testing it was clear that the Wave Rotor was operating at angles of attack inthe stalling region (10-18 degrees). This is particularly true for the inner part of the wellsrotor.

The following graphs from [14] shows that the delayed stalling has a significant effect on

the coefficients with respect to the static/steady curve.

NACA0015 static and 13 ± 5 degrees k=0.04 Re=2E6 k=0.10 (fig. 19)

For the rotor the range of angles of attack is 0±15 degrees rather than 13±5, meaning

that fluxion α will be even larger than in these figures, so some sort of dynamic stall

modeling will be required.

For the present project a simplified version of the Beddoes Leishman model for dynamicstall was chosen [15][16]; simplified in the sense that leading edge separation will not betaken into account, only separation from the trailing edge. The model suggests a fixedrelation between the static lift coefficient and the static position of the separation point.

Position of separation point (fig. 20)




Again analyzing the potential flow around a flat plate, the lift as a function of theseparation point position can be determined to be (Kirchhoff flow):

CL st = dCLd 1 f 2

(eq. 33)

Now we assume that the lift coefficient for purely attached flow is linear:

CLatt =dCL

d (eq. 34)

The position of the separation point can then be expressed as a function of α when a

static CL curve is provided.

f =2 CL

CLatt

−12

(eq. 35)

This relation is only valid for CLatt 4CL where f = 0 -> full separation. This is

inconvenient for computational purposes and can be circumvented by rewriting CL to:

CL st =CLatt f CL fs 1− f (eq. 36)

With

CL fs=CL−CLatt f

1− f (eq. 37)

being the lift with fully separated flow at any angle of attack.

In dynamic stall, the position of the separation point is delayed with respect to the angle of

attack. Therefore a dynamic separation point is introduced that is coupled to the angle of attack by means of a time constant that has to be determined empirically.

˙ f dyn=f − f dyn

T f

(eq. 38)

4.6.3. Combined effect

The dynamic lift can now be evaluated from eq. 36 and eq. 32.

CLdyn=CLatt f dynCL fs 1− f dyn c

2U ̇ (eq. 39)

The drag also lags behind. This is implemented as a correction to the static drag.

CDdyn=CD ACD⋅CL−Cl st (eq. 40)

With ACD an empirical constant.

4.7. Rotor modeling

The wake -- i.e. the trailing and shed vortices – of the blades of a rotor in waves remain in

the rotor area, as already mentioned in §3.5. This presents a problem, because these

vortices will accumulate in time. When wake relaxation is applied (§4.3), the vortex lines

will roll up into each other and the computations will rapidly run out of hand.




4.8. Free Surface

The rotor operates just under the free surface of water. In order to arrive at a proper

potential flow solution, the boundary condition at the free surface should be satisfied. Theboundary condition at the free surface consists of two parts: a dynamic boundarycondition and a kinematic boundary condition [6][8].

The dynamic boundary condition accounts for the fact that the dynamic pressure of theflow at the surface must equal the atmospheric pressure. In linearized form this conditionyields:

∂

∂ t =− g (eq. 41)

The kinematic boundary condition states that the velocity of the free surface is equal tothe velocity of the flow at the free surface. In linearized form:

∂∂ z

= ∂∂ t

(eq. 42)

Equations 41 and 42 can be combined into:

∂

∂ z =−

1

g

∂2

∂ t 2

(eq. 43)

The potential of the undisturbed waves satisfies these boundary conditions by its very

definition. However, the potential of the trailing vortices of the rotor blades does not.

In order to satisfy the free surface boundary condition, a singularity distribution needs tobe placed on or above the free surface. The strenghts of the singularities have to be

solved for such that the boundary conditions at the free surface are satisfied. This ishowever beyond the scope of this project.

In the current model the free surface is either ignored or implemented as a solid surface.

The latter can be done by mirroring all vortices with respect to the z = 0 plane. Thisensures:

∂

∂ z =0 (eq. 44)

This satifies the kinematic boundary of the undisturbed flow.

Some of the validation results [18] with wave lengths in the order of twice the rotor diameter indicate that this simplification is a problem because the rotor starts radiatingsurface waves. This is due to the size of the rotor disturbance creating a surface wavewith a frequency approaching the incident wave frequency. Further development of the

free surface interaction is therefore desired.




5. Computer Implementation: “WaveVort”

An implementation of the model presented in §4 has been developed using C++. Thischapter provides a global description of the software. Some UML-like diagrams [22][27]should provide a good insight into the overall structure. The reader is referred to thesoftware package itself for a more code-level description of things.

5.1. Program Layout

The software is split up in several packages or modules. Each module provides a certainfunctionality. Though the modules are not physically separated in terms of C++ libraries,

interdependency between files of the modules was reduced to a minimum, whichimproves maintainability and provides for separate testing of module functionality.

Package diagram (fig. 23)

Fig. 23 show a overview of the program modules, a short description of the modules'functionality and the files that make up the module. The arrows indicate dependencies.

Fig. 24 shows a class diagram for the most important modules. Closed arrows indicate ageneralization and open arrows an association.

Computer Implementation: “WaveVort” 25/48



Class diagram (fig. 24)

5.2. Execution flow

The main program flow of WaveVort is rather simple. The program takes input, runs a

loop with time steps and produces output as can be seen in the global activity diagram fig.25. More than 90% percent of the program's time is spent calculating the induced

velocities from the vortex mesh by evaluating the Biot Savard law (eq. 14).

Once the velocity on an element has been determined, the force on an element isevaluated by first looking up the aerodynamic coefficients, through bilinear interpolation in

the user provided table, and then applying dynamic effects (§ 4.6) when requested.




Activity diagram (fig. 25)

The implementation of the time loop for a rotor object is shown in fig. 26

Time loop sequence diagram (fig. 26)




5.3. Input/Output

The WaveVort input module accepts input from either the console or a file. When input is

entered from the console, the program provides a short description of the value expected.The main input categories are

• General values (density, viscosity etc)

• Geometry (nodes, elements, and element directions)

• Rotor specific values (Rotor rotational inertia, mesh size etc)

• Simulator settings (time step, number of steps etc)

• 2D aerodynamic data for the blade profile

Output can be generated at different times during the simulation. This geometry basedoutput has an ASCII style VTK format [25], that can be visualized with MayaVi [24]. Thisoutput can also be generated in Los Alamos' GMV format [26]. A list of main values at

each time step and overall statistics are generated in whitespace separated text files thatcan be further processed with a spreadsheet program.

5.4. Performance

WaveVort is computationally expensive, meaning that it takes lots of CPU time. This CPU

time is spent on evaluating the effect of each of the vortices on the wake mesh for eachelement of the object (wing or rotor). Typically this is an O(N3) operation, where N is the

number of elements used to model the object. This O(N3) operation of evaluating theinduced velocities has to be done for each time step.

5.4.1. Code

WaveVort was not developed with performance in mind. Especially on modern CPU's likethe Pentium 4 and Itanium that can hurt. These CPU's use caching, pipelining,

vectorization and parallelization for maximum performance allowing them to execute evenmore than one instruction per clock cycle. However, these features can only be exploitedwhen the code is properly optimized for them.

A removal of obvious sloppiness in the WaveVort code improved performance 5 fold.Further improvements can only be gained by adjusting the design of the application.

An example of a typical WaveVort calculation: running a 30 second simulation at a time

step of 0.01 seconds with a rotor model that has three blades with 32 elements each anda wake mesh with 120 elements along the rotor diameter. The number of vortex influence

evaluations amounts to 30⋅100⋅3⋅32⋅32⋅120=1.1⋅109

Each vortex influence evaluation takes

Addition 11

Subtraction 15

Multiplication 32

Division 4

Square Root 2

Total 64




So roughly 64 FLOPS (FLoating point OPerationS) per evaluation. This amounts to 71GFLOPS for the whole simulation. Luckily this is no problem even for a desktop PC thesedays.

The next paragraph shows what a compiler can do to improve performance.

5.4.2. Compilers

Some research was conducted regarding the performance of several compilers. Fivecompilers are compared.

• The GNU C++ compiler (3.3.5) on Linux and Windows (3.3.3 – cygwin) -GCC

• The Intel C++ compiler (8.1) on Linux and Windows - ICC

• The Microsoft C++ compiler (7.1) on Windows - MSVC71

• The Microsoft C++ compiler (6.0) on Windows -MSVC6

• The Borland C++ compiler (5.5) on Windows - BCC

The GNU C++ compiler was also tested with different settings to see their influence.

The tests were conducted on a Pentium 4 2.8C. This processor supports HT(Hyper Threading) which provides parallel execution. The effect of this was tested by running twoapplication instances at the same time. Furthermore the effect of using SSE2 instructionswas evaluated. The SSE2 instruction set contains various vectorized floating pointoperations, that should theoretically provide about double execution speed.

Apart from the Intel compiler on Windows and the old Microsoft Visual Studio 6 compiler,all these compilers are freely available. The Intel compiler on Linux is intended for non-commercial development only.

Results running a benchmark input file:

Compiler comparison (fig. 27)


Compiler Parameters Exec. Time (sGCC – Linux -O3 -march=pentium4 -mfpmath=sse -msse2 21.73

Intel – Linux -cxxlib-icc -xN -O3 -ipo 25.56Intel – Windows /O3 /Qipo /QxN /GR /GX 24.11

Microsoft VC7.1 /G7 /Ox /GR /GX 26.19GCC – Windows (cyg) -O3 -march=pentium4 -mfpmath=sse -msse2 34.59

Microsoft VC6 /G6 /O2 /GR /GX 53.62Borland -P -6 -O2 -OS -ff 61.87

GCC – Linux Intel – Linux Intel –Windows

MicrosoftVC7.1

GCC –Windows

MicrosoftVC6

Borland

0

5

10

15

20

25

30

35

40

45

50

55

60

65

Compiler Comparison Exec Time (s)

lower is better



As can be seen the differences between the relatively new compilers is marginal. GCC onwindows pays the price for not being native to the platform. The older VC6 and BCC55compilers don't have the advanced optimizations and awareness of modern CPU's

causing them to produce a 2-3 times slower application.The effect of SIMD (SSE2) [28]and HT has been tested with GCC on Linux:

Processor feature comparison (fig. 28)

As can be seen from fig. 28, switching to SSE2 helps, but not much. This emphasizes

that specialized features like these only work well when the code is designed to use themin the proper way. Hyper Threading can boost performance by up to 20%. Maximum

performance is achieved, not surprisingly, when one process is running 387 code whilethe other uses SSE2 instructions, effectively using different parts of the CPU at the same

time. WaveVort is very well parallelizable (the order of interactions in fig. 26 can beswapped as well as the loops vectorized), so doing vortex influence evaluation in multiplethreads could be beneficial on Hyper Threading CPU's and even more on multiprocessor systems or dual/multi core CPU's like AMD's new Opteron.


Mode Parameters Exec. Time (s)Optimized -O3 23.19Pentium4 -O3 -march=pentium4 22.70SSE2 -O3 -march=pentium4 -mfpmath=sse -msse2 21.73SSE2 – HT -O3 -march=pentium4 -mfpmath=sse -msse2 19.53387/SSE2 – HT -O3 -march=pentium4 / -mfpmath=sse -msse2 17.13

Optimized Pentium4 SSE2 SSE2 – HT 387/SSE2 – HT

0

2.5

5

7.5

10

12.5

15

17.5

20

22.5

25

Processor Features Comparison Exec Time (s)

lower is better



6. Program Verification and Validation

This chapter contains the results from calculations performed with WaveVort compared totheoretical, experimental, and other computational results. The first case serves as a

verification case to test the proper implementation of the intended functionality. Further verification is done in test modules within the code, because code verification is easiest at

the module or unit level.

The other cases provide an insight into the accuracy of the program. The effect of variation of input parameters is demonstrated for these cases, which can help to

understand the their relevance.

6.1. Three Dimensional Wing

This case is intended for verification of the doublet panel/vortex wake. Lift and drag are

determined for a simple straight wing with an aspect ratio of 8.

6.1.1. Case Input

A straight wing:

• span : 1 m

• chord : 0.125 m

• profile : NACA0015, Sheldhal Data [3]

• AR : 8

Flow:

• flow density : 1025 kg/m^3 (seawater)

• flow kin visc : 1.28e-6 m^2/s

Simulation:

time : 10 s

time step : 0.04 s

statistics : last 2 seconds

6.1.2. Test Without Wake

The purpose of this test is to check the Reynolds dependent airfoil performanceimplementation

The airfoil was split up into 8 elements, no wake• uniform flow : 3 m/s

• uniform flow : 4 m/s

• angle of attack : 0.1 rad (5.73 °)

• angle of attack : 0.25 rad (14.33 °)

Results:

WaveVort (L/D) 3 m/s 4 m/s Manual (L/D) 3 m/s 4 m/s

0.1 rad 359/7.7 646/12.4 0.1 rad 360/7.7 N 646/12.4

0.25 rad 318/35.9 751/29.6 0.25 rad 319/35.5N 755/29.5

Program Verification and Validation 31/48



6.1.3. Wake Test

The purpose of this test is to check the implementation of the doublet panel wake.

• The airfoil was split up into 8, 16, and 32 elements.• uniform flow : 3 m/s

• angle of attack: 0.1 rad (5.73 °)

• trailing wake panels: on

Results:

WaveVort (L/D) 8 elements 16 elements 32 elements

293/12.4 287/12.5 284/12.6

Loading on straight wing with 32 elements at t = 0.8s (fig. 29)

Manual results can be obtained by taking the lifting line result from [7]:

∂

∂Cl =10

20

A(eq. 45)

In which A is the (effective) aspect ratio of the wing. The ratio of lift coefficients for a two

dimensional wing and a wing with an aspect ratio of 8 is thus:

Cl A=8

Cl A=∞

=0.8 (eq. 46)

After applying this to the manual result from §6.1.2, the lift is 287 N. The induced angle of

attack is (1 – 0.8) * 0.1 = 0.02 making the induced drag 0.02 * 287 = 5.74 N. Together with the frictional drag, the total drag amounts to 13.4 N.

6.1.4. Case Results

All results agree with the manual calculations with respect to value and sign indicatingthat the basic functionality of the program has been correctly implemented.




6.2. Wells In Uniform Flow

A test case was set up to compare results with the “Propsi” program. A BEM code from

the Wind Energy Department of the Delft University of Technology.

6.2.1. Case Input

Rotor:

• Number of blades : 3

• Profile : NACA 0015, Sheldahl, Reynolds 3.6E5 only, [3]

• Chord : 0.125 m

• Inner radius : 0.5 m

• Outer radius : 1.0 m

• Blade span : 0.5 m

• Rotary velocity : 3.1415927 rad/s• Pitch : 0

• Elem. per blade : 18

Element distribution (fig. 30)

Conditions:• Density : 1025 kg/m^3

• Kin. viscosity : 1.28e-6 m^2/s

• Velocity : 0.3 m/s, downward, negative z direction.

Simulation:

• Length of run : 8 s

• Fixed wake flowing downstream with undisturbed flow (0.3 m/s) and a fixed wake

flowing downstream at 0.15 (m/s), the latter being a 'manual iteration' on the wakeposition from the induced velocity of the 0.3 m/s simulation.

6.2.2. Case resultsFig. 31 Shows the loading distribution along the blade span in terms of angle of attack.The error in the wake position when no wake relaxation is applied is considerable,causing an error of about 25% in the induced velocities. A second run with an adaptedwake position – the wake flows downstream with undisturbed flow minus the meaninduced velocity from the first run – gives results almost similar to those produced by

“Propsi”.

The torque produced according to Propsi and the 'adapted wake' simulation is about zero,so the conditions chosen approximately resemble the free running speed of the rotor.




Comparison of angle of attack along blade span with Propsi (TU Delft Wind Energy) (fig. 31)

Fig. 32 shows the doublet strength on the fixed wake along with the angle of attack on the

blades and the wake induced velocities (vectors) in three dimensions.

Wake and induced velocity visualization (fig. 32)




6.3. Wells in Waves

This paragraph is located in [18], due to confidentiallity of the experimental data.

6.4. Darrieus in Waves


6.5. Wave Rotor





7. Design Considerations

During the course of the research done, some topics surfaced that might of interest to adesigner. This chapter sheds some light on these topic.

7.1. Design Parameters

The following design parameters can be identified for a rotor.

• r/l w : ratio of rotor radius and wave length

• S : Solidity of the rotor

• n : Number of blades

• c(z), r(z) : Shape of the rotor – chord and radius – as a function of depth

Optimization for all of these parameters can be done. Because of the absence of free

surface modeling, WaveVort can at present only help with the last two.

7.2. Rotor shape

The optimum rotor shape of a rotor from a hydrodynamic point of view has yet to bedetermined. The purpose of the work done during this project has in part been to help

improve the hydrodynamic design. This paragraph contains a suggestion for a rotor shape with some calculation results to back it up.

7.2.1. Design Objective

The goal set for this design suggestion is to determine the blade shape for a rotor in

terms of two functions.1) z(r/R)/R, The relative z or depth coordinate of a blade section as a function of the radial

position.

2) c(r/R)/R, The relative chord length of a blade section as a function of it's radial position.

The following assumptions are made for an optimal design, in order to have someguidelines and directions.

• A rotor blade will perform best when evenly loaded.

• A rotor blade will perform best at a specific angle of attack, close to but below the stall

angle.

7.2.2. Load Distribution and Angle of Attack

The assumptions for the design objective are both based on the suggestion that a bladewill perform best when its forward tangential force is maximized. The forward tangentialforce is:

T = L⋅sin− D⋅cos (eq. 47)

The first assumption means minimization of D by minimizing the induced drag, which

maximizes T . The second assumption means optimization of Lα /D, which also maximizes

T .

At first I will disregard induced velocities, which leads to:

Design Considerations 36/48



=arctan U

r (eq. 48)

and

L=U 22

r 2c (eq. 49)

The adjustable or 'design' parameters are the ratio of wave length (l ) and rotor radius ( R),

which I will call q, the desired maximum angle of attack (α ), and the overall rotor solidity

(S ).

S =n C

2 R(eq. 50)

Where C is the blade chord at z = 0 or r/R = 1.

The wave height is normalized such that U surface = 1 and for the sake of the argument the

water is assumed deep. Rewriting eq. 48 yields:

z / R=

ln r R q2

(eq. 51)

and eq. 49:

c / R=2 S

n r

R −2

(eq. 52)

Rotor shape not considering induced velocities (fig. 33)




Due to induced velocities the incident velocity U will be smaller then the undisturbed

velocity ekz. These induced velocities will be dependent on the local solidity of the rotor,

which increases towards the shaft of the rotor. The effect is that the optimal z increases

and the optimal chord decreases towards the center. This will be investigated usingWaveVort in the next paragraph.

7.2.3. Optimal Shape and Solidity

Based on the shape in §7.2.2 an initial guess is made for a shape of a rotor with q=10:

z / R=a0 ln r

R with a0=1 (eq. 53)

and

c / R=

b0b1

r

R

S with b0=2 b1=−1

There are no fundamentals to these functions, they are just a choice. This shape is testedwith WaveVort for a 2m diameter rotor with 10m long waves.

Impression of possible rotor shape (fig. 34)

Rotors with a solidity of 0.07 and 0.093 (3 and 4 blades) were put into WaveVort:

Cp's for rotor design suggestion (fig. 35)




These Cp's are exceptionally high. Also for even more blades, the Cp will further increase. This is not realistic and probably due to the absence of free surface interaction

(§4.8).

The following graphs show the angle of attack and load distribution on the blades for acouple of rotor positions and wave phases.

Loads and angle of attack (fig. 36)

These graphs show that the guess for the rotor shape wasn't a very bad one in terms of

the assumptions for the design objective (§7.2.1)

7.2.4. Full scale

As a prelude to the next paragraph (§7.3), a simulation was also done for a full size rotor.

Rotor diameter : 20m

Wave period : 8s




Full scale rotor Cp for design (fig. 37)

7.2.5. ConclusionThis is a rather crude and simple approach but it gives an indication of what a rotor shape

design process might look like and how WaveVort can fit into this design process. Atpresent it seems that WaveVort does not yet predict the performance of high solidityrotors well so it is still not suitable for determining an optimal rotor solidity.

The full scale calculation shows the rotor's sensitivity to changes in friction.

The ideal shape for uniform tidal flow is an H-darrieus. A possibly good shape for awave/tidal rotor might be an average between the suggested design and an H-darrieus

depending on the velocity ratio of tidal current and wave.

7.3. Extrapolating model testsModel testing and extrapolation may only be partly of concern to the designer. However, a

good understanding of these scaling laws is crucial in the design process as will be

further discussed in §7.4

7.3.1. Extrapolation Rules

The basis for testing wave energy devices is, not surprisingly, scaling of the waves. The

waves are scaled with equal geometry (Froude Scaling). The wave height and length arelinearly proportional to the scale of the model.

The dispersion relation, which states the relation between wave length and speed, for

deep water waves is:

2=k ⋅ g (eq. 54)

k is the wave number 2/ and thus inversely proportional to the scale. Since

varying g is practically impossible, this means that the wave frequency will be proportional

to the inverse square root of the scale. And therefore wave period and time in general arescaled according to the square root of the scale.

The density of the water can also not be scaled under normal circumstances whichmeans that mass of the water is scaled with the third power of the scale.

This means for scaling of other physical quantities: ( s is scale)




Quantity Unit Scaling

Length m s

Time t √sVelocity m/s √s

Force kg.m/s2 s3

Power kg.m2 /s3 s3√s

This scaling is called Froude Scaling because the non dimensional Froude number

V / g ⋅l remains constant while scaling.

7.3.2. Reynolds Scaling Effects

Under normal conditions it's not possible to keep both the Reynold's number and the

Froude number constant when scaling. The Reynold's number is a non-dimensionalnumber representing the ratio between inertial and viscous forces. It's defined as

V ⋅l / . Where V is velocity, v the fluid's kinematic viscosity, and l the so-called

characteristic length, which is defined for different flow phenomena and is usually the'running' length of the fluid, but sometimes another dimension like the diameter of the

pipe in pipe flow.

When applying the scaling rules from the previous chapter it is clear that the Reynold'snumber does not remain constant, but scales according to s√s.This is considering the fact

that the kinematic viscosity can not be scaled for practical applications.

For the Wave Rotor device which uses airfoil lift to generate energy, the (down) scaling of the Reynold's number can have a considerable effect.

Typically a real-life rotor will operate at a Reynolds number of around 5E6. This gives a

Re of 1.5E5 for a 1 to 10 scale model. The effect of this is twofold:

1. The Reynold's number has an effect in the skin frictional part of the drag coefficient.

2. The Reynold's number has an effect on flow separation.

These effects can be observed from fig. 38 and fig. 39 and will generally mean that the

performance will improve with increased Reynolds number because of better L/D ratios.




Reynold's scaling of minimum drag (fig. 38)

Reynold's scaling of maximum lift (fig. 39)

7.4. Economy of scale ?

Economy of scale is the law that is typically associated with decreased cost per unit whenincreasing production scale. With the advent of 5MW (!) wind turbines, it appears that thislaw applies to wind turbines. Apparently capital and maintenance costs per kWh decrease

with increased rotor size. Considering the rotor power is proportional to its swept area andthe swept area is proportional to the square of the rotor diameter, it appears possible tobuild and maintain a rotor that is twice as large for less than four times the cost.

Wave energy rotors are a bit different in this respect. Wave power density has a unit of

kW/m, indicating that the maximum available wave power for a rotor will be approximately

proportional to the diameter rather than to the square of it. For small rotors the power coefficient will increase with size, so the square power output is still satisfied but only up

to a certain point. So “the bigger, the better”s will not apply to wave energy rotors. It ismore likely that an optimal ratio of typical wave length versus rotor size can be found andthat economy of scale should come from numbers rather than from size.




8. Conclusions

The goal of this investigation has been to develop a performance prediction for ahydrodynamic ocean energy device. This goal has partly been achieved. A hybrid vortex

model has been developed and implemented using the C++ programming language. Theimplementation is called WaveVort.

Validation case results indicate that the program is useful for predicting power output for

the model rotor with a reasonable accuracy. Cp values and optimal TSR values areestimated to high with respect to the test results. This might partly be explained by

parasitic drag on the model and partly by the lack of damping from the free surface in thecomputations. For rotors that are very solid or large with respect to the wave length, theinteraction with the free surface will have to be modeled to get accurate results.

Free surface interaction can be added to WaveVort with a time dependent evaluation of

the free surface elevation and singularity distribution (either source or doublet) on a panelmesh located at z = 0. The whole model could also be taken to the frequency domain. A

diffraction and radiation problem can then be formulated for a complex potential with the

frequency of the incident wave [21]. The rotor envelope can be divided into doublet panelsin the same way as in this work. The boundary condition for these panel is a pressure jump. The pressure jump can be determined by actuator disk theory as a result of rotor solidity, rate of revolution and the normal velocity on the panel.

The investigation has given an improved insight into parameters that govern the operationof a rotor, like the Reynolds number and dynamic stall. This insight has lead to someuseful hints for improvement and implementation of the current design.

Many issues require further investigation, development and testing, including the

possibility to simulate the combination of waves and tidal current flow, surface waveradiation and dynamic stall. The current work has been specifically set up to

accommodate future extensions that address these issues.

Conclusions 43/48



9. Recommendations

Apart from providing some answers, researching things always raises many morequestions. Possible actions are provided here that follow up on the questions that were

raised during this investigation.

• Further develop the current computational model. The results indicate that this is a

good direction, but the model should be improved by adding free surface interaction.

• Further validate the assumption about the rotor wake position by for example

improved wake modeling by some form a wake relaxation.

• Further develop the combination with tidal stream effects. The present work focuses

on wave energy but a rotor can combine this with tidal energy. Full wake relaxation ?

• Test blade sections used for model testing at NaREC for (dynamic !) 2D performance.

This helps validation of computational methods using two dimensional wing sectiondata and is vital for extrapolation of model tests to full scale.

• Build a model at larger scale that suffers less from Reynolds scaling effects than the

10th scale model.

• Investigate blade profiles with a camber that matches the rotor blade path curvature.

• Investigate dynamic rotor loading models that optimize performance in irregular waves.

During model tests the performance in irregular waves was exceptionally poor. Thesimple linear loading that was applied during these test was certainly doing much harmhere. The rotor only performs well when running close to optimal rpm at all times. Thecomputer model developed in this project can help evaluate loading models.

• Investigate anti-fouling methods. Real-life rotors require a hydrodynamically smooth

surface to operate properly. Marine fouling can be a severe problem, especially for

objects that aren't moving. The feasibility of keeping a rotor spinning at all times hasyet to be determined. Also, considering the nature of the device, the application of

environmentally harmful anti fouling seems problematic.

• Create a more user friendly interface to WaveVort that makes the program more

accessible. As accessible as is responsible at least.

Recommendations 44/48

current/straightsuggested/cambered



Symbols

k wave number (1/m), reduced frequency (-)c profile chord length (m)

α angle of attack (rad)

t time (s)

ω radial frequency (rad/s)

CL lift coefficient

CD drag coefficient

m mass (kg)

g earth gravitational acceleration (9.81 m/s2)

ρ density (kg/m3)

ν kinematic viscosity (m2/s)

Γ circulation (1/s) μ doublet strength

ε error

l length (m)

r radius (m)

q wave length : rotor diameter ratio (-)

h wave depth (m)

n number of blades (-)

Φ flow potential (m2/s)

φ potential due to singularities (vortices) (m2/s)

ζ wave surface elevation (m)

S ζ wave spectral ordinate, energy of frequency component (m2s/rad)

f relative separation point position, 0: fully separated, 1: fully attached (-)

Vectors:

P point ( x, y, z) (m)

F force (N)

A acceleration (m/s2)

U flow velocity (u, v, w) (m/s)

Subscripts:

a added, due to added mass / amplitude

st stall, due to stalled flow

att attached, due to attached flow

fs full stall, due to fully stalled flow

w of a wave.

r of a rotor.

i at index i / induced, due to vortex induction.

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Index

aerodynamic center............................................................................................................17Beddoes Leishman model..................................................................................................21Biot-Savard law..................................................................................................................14collocation point..................................................................................................................17darrieus rotor........................................................................................................................9drag force.............................................................................................................................8dynamic stall.......................................................................................................................21flow separation...................................................................................................................21Froude number...................................................................................................................41hydrodynamic rotor...............................................................................................................7irregular waves...................................................................................................................10Kutta condition....................................................................................................................15Laplace's Equation.............................................................................................................12lifting force........................................................................................................................ ....8potential flow......................................................................................................................12reduced frequency..............................................................................................................20regular wave........................................................................................................................ .9Reynold's number...............................................................................................................41singularity...........................................................................................................................13stall model..........................................................................................................................18tidal energy.................................................................................................................. .........7vortex..................................................................................................................................13vorticity...............................................................................................................................14wake relaxation..................................................................................................................16wave energy.........................................................................................................................7wave energy spectrum.......................................................................................................10wave flow..............................................................................................................................9wave spectrum...................................................................................................................10wells rotor.............................................................................................................................9

46/48



References

1. Strickland, James H., “The Darrieus Turbine: A Performance Prediction Model UsingMultiple Stream tubes.”, 1975

2. Paraschivoiu, I., ”Wind Turbine Design.”,

3. Sheldahl, R.E., Klimes, P.C., “Aerodynamic Chararcteristics of Seven Symmetrical Airfoil

Sections Through 180-Degree Angle of Attack for Use in Aerodynamic Analysis of

Vertical Axis Wind Turbines., 1981

4. Sharpe, D.J., “Refinements and Developments of the Multiple Streamtube Theory for the

Aerodynamic Performance of Vertical Axis Wind Turbines”

5. Sharpe, D.J., Taylor, D. A., “The aerodynamic performance of the vee type vertical axis

wind turbine”

6. Katz, J., Plotkin, A, “Low-Speed Aerodynamics”, Second Edition,

7. Hoerner, “Fluid Dynamic Lift”.

8. Gerritsma, “Golven”, Dictaat bewegingen sturen en golven 1, mt513

9. Scheijgrond, P.C., “Wave Rotor testing at NaREC”, 2004

10. Halcrow, “Marine Energy Challlenge, Report on Task 1 (Technology Costs) and Task 2

(Performance Analysis) for the Evelop Wave Rotor Device”,

11. Rossen, E.A., Scheijgrond P., “Wave Rotor – Test Report”, 2002

12. McCroskey, W. J., “The Phenomenon of Dynamic Stall” NASA TM-81264

13. Vicente, P.G., Viedma, A., Horn R., “Oscillating Turbulent Flow Over Differenent NACA

Profiles: A Finite Element Approach”, 1999

14. Piziali, R.A., NASA TM-4632, “2-D and 3-D Oscillating Wing Aerodynamics for a Rangeof Angels of Attack Including Stall”, 1994

15. Hansen, M.H., Gaunaa, M., Madsen, H.A., “A Beddoes-Leishman type dynamic stall

model in state-space and indicial formulations”, Risø-R-1354

16. Björck, A., “DYNSTALL:Subroutine Package with a Dyanmic stall model”, FFAP-V-110

17. Versteegh, J.R. ”Performance Assessment of an Ocean Wave Energy Device for

Sustainable Electricity Generation, A Definition Study”, 2005

18. Versteegh, J.R. ”Comparison of WaveVort Results with NaREC measurements,

Validation – Confidential”, 2005

19. Lloyd, A.R.J.M., “SEAKEEPING: Ship behaviour in rough weather”

20. Falnes, J., “Principles for Capture of Energy from Ocean Waves”, Department of Physics,NTNU, N-7034 Trondheim, Norway

21. Lee, C.H., Newman J.N. “Computation of wave effects using the panel method”, 2003

22. Fowler, M., “UML Distilled”

23. http://www.esru.strath.ac.uk/EandE/Web_sites/01-02/RE_info/Tidal%20Power.htm

24. http://mayavi.sourceforge.net/

25. http://www.vtk.org/

26. http://www-xdiv.lanl.gov/XCM/gmv/

27. http://www.omg.org/technology/documents/formal/uml_2.htm

28. http://arstechnica.com/articles/paedia/cpu/simd.ars

47/48



Quotes

Simulation"Nobody trusts a computer simulation, except the guy who did it,and everybody trusts experimental data, except the guy who did it."

Programming

It's multiple choice time...

What is FORTRAN?

a: Between thir- and fivtran.b: What two computers engage in before they interface.c: Ridiculous.

Optimization

More computing sins are committed in the name of efficiency (without necessarily achieving it)than for any other single reason - including blind stupidity.

-- W. A. Wulf

Optimizations always bust things, because all optimizations are, in the long haul, a form of cheating, and cheaters eventually get caught.

-- Larry Wall

Mathematics

As far as the laws of mathematics refer to reality, they are not certain, and as far as they arecertain, they do not refer to reality.

-- Albert Einstein

-- Bill Waterson

People

"And you Marcus, you have given me many things; now I shall give you this good advice. Bemany people. Give up the game of being always Marcus Cocoza. You have worried too much

about Marcus Cocoza, so that you have been really his slave and prisoner. You have notdone anything without first considering how it would affect Marcus Cocoza's happiness andprestige. You were always much afraid that Marcus might do a stupid thing, or be bored. Whatwould it really have mattered ? All over the world people are doing stupid things... I shouldlike you to be easy, your little heart to be light again. You must from now, be more than one,many people, as many as you can think of..."

-- Isak Dinesen (Karen Blixen), The DreamersTaken from "The C++ Programming Language" by Bjarne Stroustrup

Life

"Life ? Don't talk to me about life !"

performance prediction of a hydrodynamic ocean energy device for sustainable electricity generation

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