OSCILLATING FOIL PROPULSION-VERIFICATION AND VALIDATION OF A RANS-BASED NUMERICAL MODEL

Oscillating Foil Propulsion

Verification and Validation of a

RANS-based Numerical Model

A. Dubois

O

SCILL ATING

F

OIL

P

ROPULSION

V

ERIFICATION AND

V

ALIDATION OF A

RANS-

BASED

N

UMERICAL

M

ODEL

by

A. Dubois

in partial fulfilment of the requirements for the degree of

Master of Science

in Marine Technology, Science Track

at the Delft University of Technology,

to be defended publicly on Friday June 20th, 2014 at 10:00 AM.

Student number: 1538101

Supervisor: Prof. dr. ir. T.J.C. van Terwisga

Thesis committee: Prof. dr. ir. R.H.M. Huijsmans, TU Delft

Dr. ir. M.J.B.M. Pourquie, TU Delft

ir. E. van Rietbergen, SpaarnWater BV

P

REFACE

The final hurdle in achieving the degree of Master of Science in Marine Technology (Science Track - Specialisation Ship Hydromechanics) at Delft University of Technology is the completion of a thesis research. This research, spanning over a nine-month period (45 ECTS), is the chance for a student to show what he learned over the coarse of the past years and earn the right to call himself an engineer.

Hereby I would like to extend gratitude to several individuals, who helped and assisted me in finalising this thesis (and my education at Delft University of Technology in general). First of all thanks go out to my main-supervisor Prof. dr. ir. T.J.C. van Terwisga, who pushed me to the limits to produce the presented results. Next thanks go out to Dr. ir. M.J.B.M. Pourquie, who never got tired of aiding me with the many problems encountered in the numerical modelling of the problem faced during the completion of this thesis. Furthermore I want to thank the other commission members, Prof. dr. ir. R.H.M. Huijsmans, ir. E. van Rietbergen and ir. J. Vermeiden for the many fruitful discussions relating to the research and for being part of my graduation committee. Additional thanks go out to AP Moller-Maersk for the providing of the data-set relating to the FINPROP-project.

But the list of people to thank isn’t finished yet. I would also want show my sincerest gratitude and respect to my parents, who made it possible for me to leave Belgium and go study in Delft, in the Netherlands. This goes the same for my two brothers, Gerrit and Roel, who were always there and provided the necessary mental support. The number of friends of Delft who supported me throughout the process, during ups and downs, is too large to mention them all but special thanks go out to David Markey for his contributions in the proof-reading of my thesis. Of course thanks go out to all the rest of them too. I can also not forget my many friends at home in Belgium, who made sure they were always available for mental support and some de-stressing fun. Finally I want to thank her, just for being her and being part of my life.

As a final part of this preface I want to add a motivational text by Paulo Coelho from which I drew a lot of inspiration and motivation during the pursuit of my Master degree and the finalisation of this thesis. In “Life is like a great bicycle race” (published on September, 30th on his blog) Coelho describes life as a great bicycle race, but the word “life” can easily be replaced by “research” or “hard work” in general.

Life is like a Great Bicycle Race

Life is like a great bicycle race, whose aim is to fulfil one’s Personal Legend - that which, according to the ancient alchemists, is our true mission on Earth.

At the start of the race, we’re all together - sharing the camaraderie and enthusiasm. But as the race progresses, the initial joy gives way to the real challenges: tiredness, monotony, doubts about one’s own ability.

We notice that some friends have already given up, deep down in their hearts - they’re still in the race, but only because they can’t stop in the middle of the road. This group keeps growing in number, all of them pedalling away near the support car - also known as Routine - where they chat among themselves, fulfil their obligations, but forget the beauty and challenges along the road.

We eventually distance ourselves from them; and then we are forced to confront loneliness, the surprises of unknown bends in the road, and problems with the bicycle. After a time, when we have fallen off several times, without anyone nearby to help us, we end up asking ourselves whether such an effort is worthwhile.

Yes, of course it is.

A. Dubois Delft, June 10, 2014

C

ONTENTS

2 Oscillating Foil Propulsion 7 2.1 Kinematics of Oscillating Foil Propulsion. . . 7

2.1.1 Heave and Pitch Motion. . . 7

2.1.2 Angle of Attack Kinematics. . . 8

2.2 Design Parameters and Flow Phenomena. . . 9

2.2.1 Vortex Formation . . . 9

2.2.2 From Vortex Formation to Thrust Generation . . . 11

2.2.3 Reynolds Number . . . 11

2.2.4 Strouhal Number . . . 13

2.2.5 Heave Motion - Heave-to-Chord Ratio. . . 15

2.2.6 Pitch Motion - Pitch Amplitude and Pitch Point . . . 15

2.2.7 Phase Angle Heave-Pitch Motion . . . 17

2.2.8 Angle of Attack. . . 17

2.2.9 Three-dimensional Flow Effects . . . 19

2.2.10 Effects of Initial Vorticity and Vorticity Control. . . 20

2.3 Experimental Research . . . 21

2.3.1 Experimental Set-Up. . . 21

2.3.2 Flow Visualisation Experiments . . . 22

2.3.3 Force and Efficiency Experiments . . . 23

2.4 Numerical Research. . . 24

2.4.1 Selection of Numerical Method . . . 24

2.4.2 Two- vs. Three-Dimensional Research. . . 25

2.4.3 Verification and Validation. . . 26

3 The Numerical Model 27 3.1 Introduction to the Numerical Model. . . 27

3.2 Numerical Modelling . . . 27

3.2.1 Formulation of the Governing Equations . . . 28

3.2.2 Turbulence Modelling . . . 29

3.2.3 Discretization in Space and Time . . . 30

3.2.4 Initial and Boundary Conditions. . . 31

3.2.5 Solving the Problem . . . 32

3.3 Methodology . . . 33

3.3.1 Case Selection . . . 33

3.3.2 Parameter Determination . . . 33

3.3.3 Geometry Modelling. . . 34

3.3.4 Meshing / Grid Generation. . . 34

3.3.5 Mesh Motion Definition. . . 37

3.3.6 Solving the Problem . . . 39

3.3.7 Results Processing and Analysis . . . 40 v

4 Verification Study 41

4.1 Set-Up and Objectives Verification Study. . . 41

4.2 Baseline Settings and Results. . . 42

4.2.1 Baseline Settings. . . 43

4.2.2 Baseline Results . . . 43

4.3 Grid Convergence. . . 44

4.4 Time Step Analysis . . . 46

4.4.1 Time Discretization Scheme. . . 47

4.4.2 Time Step Refinement. . . 47

4.4.3 Time Step Enlargement . . . 48

4.5 Flow Transition Model . . . 48

4.6 Additional Solver Parameters. . . 49

4.6.1 Under-Relaxation Factors . . . 50

4.6.2 Number of Iterations. . . 50

4.6.3 Pressure Discretization. . . 51

4.7 C-Grid Size Convergence . . . 52

4.8 Momentum Flux Analysis. . . 53

5 Validation Study 57 5.1 Set-Up and Objectives Validation Study. . . 57

5.2 Strouhal Number Influence. . . 58

5.2.1 Strouhal Number Influence - Comparison 1 . . . 59

5.2.2 Strouhal Number Influence - Comparison 2 . . . 60

5.3 Vortex Formation, Development and Interaction . . . 61

5.3.1 Leading-Edge Vortex Evolution and Trailing-Edge Interaction. . . 62

5.3.2 Leading-Edge Vortex Position . . . 63

5.3.3 Further Vortex Development Visualisation. . . 64

5.4 Angle of Attack Profile Influence . . . 64

5.4.1 Angle of Attack Profile Influence - Comparison 1. . . 65

5.4.2 Angle of Attack Profile Influence - Comparison 2. . . 68

5.4.3 Influence of Angle of Attack Bias. . . 69

5.5 Velocity Profile Development. . . 70

5.6 High Reynolds Number Operation . . . 71

5.7 Other Numerical Models . . . 74

5.7.1 Numerical Model - Comparison 1 . . . 74

5.7.2 Numerical Model - Comparison 2 . . . 75

6 Uncertainty Assessment and Model Evaluation 77 6.1 Uncertainty Assessment . . . 77

6.1.1 Numerical Uncertainty. . . 78

6.1.2 Experimental Uncertainty . . . 79

6.1.3 Input Uncertainty . . . 79

6.1.4 Validation Comparison Error. . . 80

6.1.5 Conclusions Uncertainty Assessment . . . 81

6.2 Model Evaluation. . . 81

6.2.1 Qualitative Model Evaluation for Future Research Possibilities. . . 81

6.2.2 Identification of Key Modelling Features. . . 83

7 Conclusions and Recommendations 85 7.1 Conclusions. . . 85 7.2 Recommendations . . . 86 List of Symbols 87 List of Figures 89 List of Tables 93 Bibliography 95

CONTENTS vii

A Numerical Modelling 99

A.1 Formulation of the Governing Equations . . . 99

A.2 Turbulence Modelling. . . 100

A.3 Discretization in Space and Time. . . 101

A.4 Initial and Boundary Conditions . . . 102

A.5 Solving the Problem. . . 102

B MATLAB®Scripts 105 C FLUENT®Input 119 D Verification Study 123 D.1 Mesh Approach Verification Study . . . 123

D.2 Flow Transition Model Verification . . . 125

D.3 Momentum Flux Analysis. . . 126

E Validation Study 129 E.1 General Case Information. . . 129

E.2 Plots Case-1, Case-2a and Case-2b . . . 133

E.3 Case-3 . . . 142

E.4 Case-4 . . . 145

A

BSTRACT

In history scientists and engineers have constantly been looking into nature for innovative solutions to problems and questions they encounter. This is also the case for the maritime industry and more specific for those interested in developing new types of propulsion, continuously seeking improved efficiency in the chase for lower fuel consumption and reduced emissions. Oscillating foil propulsion is such a development. When investigating fish, it was seen that some fish types use the motions of their tail to behave as an oscillating foil (so-called carangiform mode of propulsion) resulting in a form of unsteady flow thereby generating efficient propulsion. The oscillating behaviour of the foil consists generally of heaving and pitching motions combined with a forward motion resulting in a periodic cycle, which can easily be described mathematically. This oscillatory motion results in the creation of a (time-averaged) velocity profile in the wake behind the foil resembling an unstable jet caused by vortex shedding from the foil. The shed vortices form a so-called reverse von Kármán (vortex) street, resulting in the jet being formed behind the foil. This jet, when controlled correctly and under the correct conditions, will manifest itself as a momentum excess resulting in a thrust force. The shedding, strength and form of the vortices themselves can be controlled through control of the different kinematic parameters involved in order to form an efficient maritime propulsor.

The current research focuses on a RANS-based numerical model to simulate oscillating foil behaviour and its appli-cation as a viable maritime propulsor. The research identified the most important kinematic parameters, being the Strouhal number and the angle of attack profile, and the importance of the vortex formation, timing and interaction. The created understanding of the phenomena behind also form the base for the development of the numerical method-ology (choices made).

The numerical model is developed within ANSYS FLUENT®and uses a dynamic, moving mesh approach to simulate

the foil behaviour. The definition of the motion, and consequent mesh motion, proofed a great modelling challenge but was eventually overcome. The resulting model applies a two-dimensional finite volume RANS-based approach using second-order spatial discretisation schemes, a first-order implicit time discretisation scheme and the k − ω SST turbulence model.

The model is verified using different verification procedures to test a wide range of settings and choices of the numeri-cal approach. The verification study determined the appropriate grid size, time step and that the application of an additional flow transition model is not deemed necessary. It also shows through a momentum flux analysis that the calculated forces make sense. The conclusions of the verification study concerning these settings form the base for further modelling in the validation study.

A validation study is performed over a wide range of different parameters, each relating to a different objective of the numerical model. It is proven that the influence of a variation in Strouhal number, a changing angle of attack and a higher Reynolds numbers can be modelled. Flow visualisation comparisons also proofed the capabilities of the model in accurately visualise (and predict) the relevant flow phenomena. A comparison with other numerical results identified the model to produce similar results compared to other codes and these conclusions can be used for further model improvement and analysis.

The numerical results produced are validated using an uncertainty assessment, in which the comparison between the validation uncertainty and numerical comparison error is shown to be appropriate. The uncertainty assessment quantifies the numerical modelling error as acceptable and proofs the validity of the used modelling approach. The qualitative model evaluation, which follows, shows several interesting fields for further use of the model (as there are hull interaction studies, further improvement of the propulsive efficiency and uses of oscillating foils in other areas). The evaluation is concluded by stating that the used k − ω and dynamic mesh capability are essential parts, required for a RANS-based model to accurately simulate oscillating foil behaviour.

I

NTRODUCTION

In history scientists and engineers have constantly been looking into nature for innovative solutions to problems and questions they encounter. This is also the case for the maritime industry and more specific for those interested in developing new types of propulsion, continuously seeking improved efficiency in the chase for lower fuel consumption and reduced emissions. In order to design and develop new marine propulsion systems it seems a small step to evaluate

how fish (and aquatic mammals) generate efficient propulsion. In [Lighthill,1969] the merit of researching fishlike

propulsion is described accurately by stating that millennia of necessary evolution (for survival purposes, “survival of the fittest” as Darwin described) have inevitably resulted in advanced solutions to generate fast propulsion at high efficiency.

When studying fish and their means of propulsion, [Lighthill,1969] found that in some fish types the motions of the

tail behave as an oscillating foil (so-called carangiform mode of propulsion), resulting in a form of unsteady flow control thereby generating efficient propulsion. The oscillating behaviour of the foil consists generally of heaving and pitching motions combined with a forward motion resulting in a periodic cycle which can easily be described mathematically. In the early 1900’s Knoller and Betz first observed that an oscillating foil created both a lift and a thrust

component as described in [Jones et al.,1998]. The Knoller-Betz effect, as it is called from then onwards, was later

explained theoretically by [von Kármán et al.,1935]. A clear explanation of the physics behind the thrust generation

of the oscillating foil (as described in previously mentioned publication) is given by [Triantafyllou et al.,1991] and

[Triantafyllou et al.,1993]. Essentially thrust is generated by the formation of a (time-averaged) velocity profile in

the wake behind the foil resembling an unstable jet. This jet (actually a momentum excess) is formed under certain conditions by the harmonic oscillation of the foil, which causes vortices to be shed from both the trailing-edge and / or leading-edge of the foil, resulting in the formation of two staggered rows of vortices with opposite sense of rotation. These vortices move downstream forming a so-called reverse von Kármán (vortex) street thus creating the unstable

jet (as seen in figure1, part B). The reverse von Kármán has vortices with opposite sense of rotation compared to the

traditional von Kármán street (seen in figure1, part A) forming behind bluff bodies resulting in a momentum deficit in

the wake, causing drag instead of thrust generation. The generation of these vortex streets was proven conclusively in

several experimental studies by means of visualisation, as mentioned in [Triantafyllou et al.,1991].

Figure 1: Formation of von Kármán and reverse von Kármán street (taken from [Drucker and Lauder,2002])

The principle of oscillation foil propulsion is seen as a viable form of propulsion for maritime applications and more specific in order to propel ships and unmanned underwater vehicles (UUV’s). Although the concept is (and has been) thoroughly researched, both through experiments and numerical simulations, little research has been done towards the actual application of the concept on real life ships. When used as a maritime propulsor, the concept needs to prove the capability to generate high thrust forces combined with a high efficiency and this in such a way that it can compete

with traditional propulsion by screw-propellers. Research by [Yamaguchi and Bose,1994] showed this capability; they showed that in cases where high propeller loading is unavoidable for traditional single screw propellers an oscillating foil propulsor can result in an higher efficiency. The research also points out that a lot of further research is needed in order to investigate the hull interaction of the propulsor and define what proper hull forms (and operating conditions)

are. Other research looked into the possibilities of multi-foil operation [Vermeiden et al.,2012] for more efficient ship

propulsion, while research by [Mattheijssens et al.,2012] investigated the possibilities of propelling UUV’s used for

minesweeping using oscillating foil propulsion. Report Outline

Research into oscillating foil propulsion, and more specifically the verification and validation of a RANS-based

nu-merical model for research into maritime applications, is the topic of this report. In chapter1the framework of

the thesis research is explained by defining the problem and thesis objectives, together with the thesis outline and relevance. An extensive literature study into the concept and mechanics behind oscillating foil propulsion is presented

in chapter2. This literature study serves at defining and developing the numerical model described in chapter3(but

also serves purpose for the analysis of data to be generated later). The numerical model is verified and validated, based on observations made and conclusions drawn during the literature study. The results of the verification and

validation study are explained in chapters4and5. A further analysis of these results is presented in chapter6in the

form of an uncertainty assessment and model evaluation. The report is finalised in chapter7with conclusions and

recommendations concerning the performed research.

The report also contains five appendices. In appendixAmore information concerning numerical modelling in general

is presented. In appendixBandCdifferent scripts (for different software packages used) are presented. Additional

information (more visualisations) of the verification study is found in appendixD, while the same thing is true for the

1

T

HESIS

F

RAMEWORK

In this short first chapter the thesis framework is defined and explained. First of all, section1.1formulates the problem

definition and how this thesis came into existence. In section1.2the objectives of the thesis are formulated after which

the outline, or how these objectives will be realised, is described in section1.3. The final section, section1.4, relates to

the relevance of the research to be performed.

1.1.

P

ROBLEM

D

EFINITION

The maritime sector is a relative conservative sector, in which often technological innovation is limited and new technologies are only scarcely introduced. This due to the large percentage of off-shelf design within the relatively traditional industry in which risks are avoided when possible. The introduction of innovations would cost a lot and only benefit in the long term, which is often not an interesting option for both ship builders and owners. The traditional screw propellor has been the industry standard for many decades now, and although alternatives exist little research is done in this area. Oscillating foil propulsion is such an option, which might proof to increase the propulsive efficiency, increase overall performance and add other benefits as flexibility and ease-of-use. By increasing the performance a significant fuel reduction might be achieved. This can lead to lower operation costs and reduced emissions, which are of course the main objectives of any technological innovation.

To determine the capabilities of oscillating foil propulsion as a usable and viable marine propulsor several options are available. Through experimental and numerical research several aspects of the foil operation can be investigated and even research into design improvements can be realised. In the field several experiments (on different scales) have been performed into developing such a propulsor. These experiments are expensive and will not always result in absolute truths. This is where numerical research might form a cheaper, easier alternative to perform large scale variational studies into oscillation foil propulsion and its performance capabilities. Useful simulations have been performed using a potential flow method (vortex lattice approach) but the discrepancies between these numerical and the experimental results remain relatively large. Investigation into the capabilities of different modelling techniques is necessary, both for better understanding (and improvement) of the experimental procedures as for the development (and initiation) of other numerical models and researches into oscillating foil propulsion. But also to push the technology forward and develop it into a real alternative for the traditional screw-propellor.

1.2.

T

HESIS

O

BJECTIVE

The main objective of this thesis research is the numerical modelling of oscillating foil propulsion, behaviour and performance, using Computational Fluid Dynamics (CFD). The development of a numerical model to accurately model the relevant flow phenomena will intend to determine the capabilities of CFD-research in the field of oscillating foils. A

two-dimensional numerical model will be created using the viscous flow solver ANSYS FLUENT®. Once the modelling

approach is chosen / determined the model can be assessed through verification and validation. In this manner the performance (strengths and weaknesses) of the developed model, both quantitative and qualitative, can be determined. An important additional part to this main objective is the assessment of the modelling approach in general, rather than the tool used. Such an assessment will focus on what makes a numerical approach capable of modelling oscillating

foil behaviour and what key features are necessary to easily judge a (different) tool on its capabilities to do so. It is also attempted to determine what parts of the oscillating foil behaviour (occurring flow phenomena) are essential to accurately model it, and which of these features the tool / numerical approach has to be able to simulate. Based on the analysis and results an attempt should be made in specifying guidelines (a methodology) of some sort for numerical research into oscillating foils. These guidelines should also incorporate recommendations for future research and on possibilities of numerical research into parametric variation studies, as well as investigations into different effects and real-life applications.

The objective can be summarised in short by the title of this thesis: “Oscillating Foil Propulsion - Verification and

Validation of a RANS-based Numerical Model”.

1.3.

T

HESIS

O

UTLINE

The thesis objective, as described in the previous section, will be achieved through different steps. Each of these steps will be discussed in one of the following chapters. First of all an extensive literature study will be performed, attempting to identify the relevant flow phenomena and parameters involved in efficient and usable operation of oscillating foils as maritime propulsors. Based on this literature study an adequate modelling approach can be developed, tested and assessed. The verification of the numerical model is an essential part in understanding the numerical modelling and determining what makes the numerical model function. The validation study that follows aims at determining the “modelling” capabilities of the numerical model (modelling real-life phenomena) and approaches (assumptions and

simplifications) chosen.

The validation of the model is done through comparison of results created by the numerical model with experimental results found in literature. This comparison, and specifically the case selection, is done based on important flow phenomena and properties of oscillating foil propulsion identified during the literature study. The results of the verification and validation studies will be evaluated both quantitative and qualitative. A quantitative evaluation through an uncertainty assessment, in which the differences between the experimental and numerical results are compared based on an analysis of the relevant errors in both results. The qualitative evaluation will focus on determining the strengths and weaknesses of the current modelling approach in general. Thereby it is attempted to identify the essential parts of the approach in accurately modelling oscillating foils.

1.4.

T

HESIS

R

ELEVANCE

The relevance of research is often a point of discussion when appointing time and funds to a certain project or research. The relevance relates to the value a research has for the maritime research community, science in general but also for society in a larger sense. The relevance of research will balance on both scientific and social relevance. Some research might have large direct impacts on the society but have little scientific value, while the opposite might also occur when research has little impact on society but aids to the growth of scientific knowledge.

The scientific relevance in the current research can be found in the fact that more knowledge about the modelling of oscillating foils is needed. An insight in what modelling approaches are needed and what parameters are essential for the accurate simulation of the flow behaviour can proof to be of great value. Some research has been performed using non-commercial, in-house developed codes at universities or research facilities. Viscous flow solvers (Navier-Stokes based solvers), can provide useful tools in further analysis of the concept and understanding more about the mechanics behind oscillating foil propulsion. Through investigation into the possibilities to model the concept using e.g. ANSYS

FLUENT®such knowledge and understanding can be generated.

The social relevance is actually also found partially in the fact that a commercial solver is used. These solvers are readily available for small and large, innovative companies developing real-life applications using oscillating foils. The clear verification, validation and analysis of a commercial solver, and formulation of recommendations concerning solvers in general, can benefit such companies. Using this developed knowledge they can further develop efficient oscillating foil propulsors, aiding at reducing fuel costs and emissions in the maritime industry. Examples of such companies and

researches are the Ofoil concept (developing an oscillating foil propulsor for inland ships, as described in [Ofoil,n.d.])

and FINPROP by AP Moller-Maersk (foil propulsion for cargo vessels using multiple foils, as reported in [Vermeiden

2

O

SCILL ATING

F

OIL

P

ROPULSION

Oscillating foil propulsion is a fairly well researched area and before any new research can be initiated a thorough analysis of the subject (and past research) is needed. In this chapter such an analysis and overview of existing literature

is presented. In section2.1the basic kinematics of oscillating foil propulsion are discussed first. Next an analysis is

made into the effects of these parameters and different occurring flow phenomena, this analysis is presented in section

2.2. In section2.3an overview is given of common practices in experimental research into oscillating foil propulsion,

while in section2.4the same is done for the numerical research side.

2.1.

K

INEMATICS OF

O

SCILLATING

F

OIL

P

ROPULSION

In this section the kinematics involved in oscillating foil propulsion are discussed. In [Kundu et al.,2012] kinematics is

described as “the study of motion without reference to the forces or stresses that produce the motion”, or in other words the study of motion and differential properties such as velocity and acceleration. To properly understand oscillating foil propulsion a good understanding of the kinematics related to the oscillation cycle is key. In this section the kinematics

are described, while section2.2will discuss the consequences of the kinematics (variations of kinematic parameters)

on the flow and thrust generation capabilities of oscillating foil propulsion (and thus investigate the consequences of kinematic choices in terms of forces, moments and efficiencies).

2.1.1.

H

P

M

As described in the introduction the oscillating foil motion consists of heaving and pitching, which are considered the principal motion parameters of the cycle. The resulting kinematics of this motion are more complex than this though and require a proper definition of the heave and pitch motion.

The general idea of the motion is illustrated in figure2.1in which a foil with chord length c and the distance to the pitch

/ pivot point b (Point O, which is also the point where the heave motion is defined) as measured from the leading-edge is defined. The foil is moving with a forward speed U and performing an harmonic heave motion h(t ) and pitch motion

θ(t) (generally) defined as

h(t ) = h0· sin(ωt ) (2.1) θ(t) = θ0· sin(ωt + ψ) (2.2)

with h0the heave amplitude,ω the circular / oscillation frequency (the same frequency for both heave and pitch), θ0

the pitch amplitude andψ the phase angle between pitch and heave (pitch leading heave). Figure2.1clearly illustrates

the different parameters mentioned thus far, but some additional information about other shown variables is useful:

• The heave motion amplitude is closely related to the width of the created wake A, which is (for ease of use)

defined as A = 2 · h0(twice the heave amplitude).

• Another parameter related to the heave motion is the heave velocity ˙h(t ), which will play an important role in the

determination of the net angle of attack as experienced by the foil (mentioned later on). 7

• The pitch angle is defined as the angle between the chord of the foil and the direction of the forward motion (defined by the forward velocity vector U ).

• The wavelength of the foils oscillating cycleλ, defined as λ = 2π/k = 2π/ω.

Figure 2.1: Graphic Overview of Kinematics - Heave and Pitch Motion

A different important remark concerns the term “harmonic” as mentioned with respect to the heave and pitch motion. The behaviour of the heave and pitch is not necessarily harmonic (sinusoidal as defined here) but can also be defined differently. In some cases the behaviour will be varied to adapt to other parameters used (or to achieve different objectives) as main input variables to describe the foils kinematics. The implications and consequences of changing

this behaviour are discussed in the following paragraph and in section2.2, but it can be said that the motions defined

here are the most commonly used.

2.1.2.

A

A

K

When looking closer at figure2.1, a last important unmentioned parameter in the kinematic behaviour of the foil is

missing, namely the time-varying net angle of attack of the foilα(t). The angle of attack of the oscillation foil is visualised

in figure2.2. It is defined as the angle between the chord of the foil and the apparent flow speed V experienced by the

foil, which is the vector sum of the heave velocity ˙h(t ) and the forward speed of the foil U (visualised in the figure).

Figure 2.2: Graphic Overview of Kinematics - Angle of Attack

Figure2.2shows that two important cases arise, resulting from different ratios between the heave and forward foil

velocity. This ratio determines the angle between the apparent flow speed and the forward speed, designated asφ(t).

This angle can either be larger or smaller than the pitch angle, dependent on the aforementioned ratio as well as depend on the pitch amplitude and how the pitch angle (and phase angle) varies compared to this ratio. The two cases determining the angle of attack are defined as follows

α(t) = φ(t) − θ(t) if θ(t) < φ(t) (2.3)

2.2.DESIGNPARAMETERS ANDFLOWPHENOMENA 9 in whichφ(t) is φ(t) = arctan µ_{h(t )}˙ U ¶ (2.4)

Based on (2.3) different motion control options are available. In [Read et al.,2003] the possibility is suggested to fix all

parameters except the pitch amplitude. Further research, mentioned in [Hover et al.,2004], uses direct control of the

angle of attack profile by fixing the pitch motion and determining the needed (nearly harmonic) heave motion for each angle of attack profile. The possibilities, issues and consequences of such choices are discussed in detail in the relevant parts of the next section.

2.2.

D

ESIGN

P

ARAMETERS AND

F

LOW

P

HENOMENA

In the introduction the generation of an efficient thrust generating wake (a reverse von Kármán street) is said to only occur under certain, specific conditions as will be explained in more detail further on in this section. These conditions are found to relate to various design parameters and flow phenomena determining the scale of the generated thrust and the efficiency of propulsion. These parameters and flow phenomena help describe the physical processes behind oscillating foil propulsion and the occurrence of an efficient reverse von Kármán street. The main design parameters and flow phenomena mentioned in literature, are the following:

• Vortex formation; and the importance of flow separation phenomena in the occurrence of vortex shedding.

• From vortex formation to thrust generation; or how the vortex shedding caused by an oscillating foil results in the

generation of a thrust force (under certain conditions).

• Reynolds number; one of the most important dimensionless numbers in fluid mechanics, which will determine

the relative importance of viscous and inertial forces.

• Strouhal number; and the importance of the timing of the vortex shedding during the oscillation cycle.

• Heave-to-chord ratio; and the more general heaving behaviour of the foil.

• Pitch amplitude and pitch point; and the general pitching behaviour of the foil.

• Phase angle between heave and pitch; or how the heave and pitching motions are timed compared to each other.

• Angle of attack properties; considering both the maximum angle of attack as the angle of attack profile during the

oscillation cycle.

• Three-dimensional flow effects; or the three-dimensional character of the vortex formation and its effects on the

flow behaviour

• Effects of initial vorticity and vorticity control; or how the vorticity in the flow can be influenced to benefit

efficiency and / or propulsive capabilities.

The different parameters are discussed in detail (both their significance and influence on the thrust generation and efficiency of propulsion) in the following sub-sections. The analysis of the relevant literature is, to emphasize it again, focussed on finding the parameters needed to generate high propulsive forces combined with high efficiency, as needed

for maritime applications. A useful research in the process of writing this chapter was found in [Platzer et al.,2008], a

literature review referring to a lot of different subjects and relevant researches and thus a useful guideline in writing this chapter.

2.2.1.

V

F

The principle of oscillating foil propulsion is largely based on the formation of a vortex street or vortex formation in the flow behind the foil / object. Therefore it is important to correctly understand the mechanics involved in the formation

of vortices around a foil. The following paragraph is written based on the relevant chapters in [Anderson,2001] and

Impulsively Started Foil Analogy Although it is known that for the oscillating foil viscous effects are of importance and the flow is highly unsteady, it is useful to first analyse the steady, inviscid flow around a fixed foil at a relative small angle of attack. When the flow around such a foil is impulsively started, a region of vorticity is formed at the

trailing-edge, which rolls up and forms a starting vortex (this process is illustrated in figure2.3). The shedding takes

place at a so-called stagnation point (point B in the figure), which is situated somewhere along the suction side of the foil. According to ideal flow theory, Kelvin’s circulation theorem now states that the time rate of change of circulation has to be equal to zero inside a closed domain and thus an equal and counter rotation vortex is formed around the foil. The circulation around the foil will cause the stagnation point to move backwards on the foil and ensures that the flow smoothly leaves the trailing-edge thereby fulfilling the so-called Kutta condition. As the process continues more vorticity is shed, resulting in a stronger starting vortex and thus more circulation occurs around the foil until the stagnation point is located on the trailing-edge. As soon as the Kutta condition is fulfilled no more vorticity is created, the vortex is shed and the steady circulation around the foil results in the generation of lift, as stated by the Kutta-Zhukhovsky lift theorem.

Figure 2.3: Starting Vortex Formation around Impulsively Started Foil (taken from [Kundu et al.,2012])

An important part of this explanation is the understanding that in order to generate lift the presence of viscosity (and thus a boundary layer) is essential. Viscosity is the physical mechanism to achieve the Kutta condition in reality. But viscosity will also have additional effects on the flow behaviour over an oscillating foil, not captured by ideal flow theory, which might be essential in the generation of an efficient thrust producing vortex street. The same holds for the unsteady behaviour and therefore an elaboration of the explanation given in the previous part is necessary.

Including Viscous and Unsteady Flow The previous paragraphs can quite easily be related to the vortex shedding

seen as a result of oscillating foil behaviour. First of all the oscillating foil needs to be seen as a constantly / repeatedly impulsively starting foil. The foils angle of attack changes during the cycle due to the oscillating pitching and heaving motion (as explained by the kinematics). So the foil will shed (start) vortices in a periodic manner, with opposite circulation in the downstroke and upstroke motion of the foil, thereby creating a vortex street. Due to the periodic, constant change of the angle of attack, the flow around the foil does not have the time to become steady. This leads to the expected unsteady behaviour of the flow responsible for the generation of an efficient, thrust-producing reverse

von Kármán street (as already shown in short in the introduction and figure1presented there).

Secondly the viscous flow behaviour is important because of the possibility of leading-edge separation and consequently the occurrence of dynamic stall. When considering a viscous flow and operating at high angles of attack the possibility of leading-edge separation (and vortex formation) becomes real and needs to be accounted for. Different researches (both experimental and numerical) showed the influence and importance of this phenomenon on the performance of oscillating foils. More about these influences and consequences will be discussed in the following sub-sections.

Nomenclature Vortex Formation Since the process of vortex formation, shedding, interaction and destruction is a

fairly complex process, a vocabulary is necessary to accurately describe the processes involved. In [Freymuth,1985]

2.2.DESIGNPARAMETERS ANDFLOWPHENOMENA 11

accurate flow description, judgement and comparison between different flow cases and results from real-life and numerical experiments.

2.2.2.

F

V

F

T

G

In order to understand how the vortex formation caused by an oscillating foil will result in the efficient generation of thrust, some additional explanation is needed. In essence the phenomenon is quite simple, the vortices behind the foil manifest themselves in the form of two staggered rows of vortices with opposite sense of rotation creating a vortex street. The created vortex street will form a (time-averaged) velocity profile in the wake behind the foil as shown

in [Triantafyllou et al.,1991] and [Triantafyllou et al.,1993]. Under certain conditions this velocity profile will occur

as a velocity deficit, causing a drag-like wake behind the foil. Under other conditions this will manifest as a velocity (momentum) excess, resulting in a thrust-producing jet. When the vortex street causes a thrust-producing jet, it is called a reverse von Kármán (vortex) street. Its opposite is called a (regular) von Kármán vortex street (the difference is

visualised in the introduction to this report, in figure1). When and how the flow transitions from a drag wake (velocity

deficit) to a thrust jet (velocity excess) will take place, depends on several parameters and conditions (and will be explained in the following sections).

Figure 2.4: Averaged Stream-Wise Velocity Profile for Different Reduced Frequencies (taken from [Bohl and Koochesfahani,2009])

In [Koochesfahani,1989], experimental observations show that a great amount of control on the wake can be exercised

by controlling the oscillation frequency and amplitude. It is clearly shown that certain combinations of parameters were able to transform the wake profile from a velocity deficit into a velocity excess and thus a thrust-producing jet.

Another extensive research into the jet characteristics of oscillating foils is performed by [S. Lai and Platzer,1999],

accurately determining the transition zone based on a parametric experimental study. An illustration of the evolution of

the averaged stream-wise velocity profile is given in figure2.4for different reduced frequencies (non-dimensional form

of the oscillating frequency, more about this in sub-section2.2.4). Values of the dimensionless stream-wise velocity

uav g/U∞below 1 indicate a velocity deficit and values above 1 are indicative for a velocity excess. Later research,

reported in [Bohl and Koochesfahani,2009], found that the transition from a velocity deficit to a velocity excess does

not necessarily coincides with the change from drag to thrust forces on the foil. Thrust is only achieved when the flow excess is further increased to overcome stream-wise velocity oscillations and the pressure reduction occurring downstream of the foil.

2.2.3.

R

N

In most flows inertial and viscous forces are both important influences on flow behaviour and different phenomena occurring in the flow. A dimensionless parameter relating these two forces can be found in the Reynolds number

Re, probably the most common dimensionless number in the world of fluid mechanics. The Reynolds number thus

The Reynolds number is defined as

Re =ρUL

µ (2.5)

whereρ is the fluid density, L is the relevant / typical length scale involved and µ is the dynamic viscosity of the fluid.

Order of the Reynolds Number The order or size of the Reynolds number is of major importance in judging flow

conditions and flow phenomena. To illustrate the influence of a different order of Reynolds number, the flow past a

cylinder will be used as an analogy (as done in [Kundu et al.,2012]). When considering low Reynolds numbers, the

involvement of small objects, low flow velocities or highly viscous flows is assumed (a single or combination of these parameters will lead to a low Reynolds number). At these numbers (Re < 4) the flow will be symmetric for and aft of the cylinder and the wake behind it will be fully laminar. When increasing the Reynolds number, the symmetry (between the fore an aft part) will disappear and steady vortices form in the wake of the cylinder (Re < 80). An even further increase will lead to unsteady vortex formation and eventually results in a von Kármán vortex street (80 < Re < 200 and higher). To arrive at these higher Reynolds numbers, faster flows, larger object or a lower kinematic viscosity is required (or a combination of these parameters of course).

As the Reynolds number is increased, the flow will transition from laminar to turbulent due to the fact that, at Reynolds

number around 3 · 105(called the critical Reynolds number and this is strongly dependent on the situation and

pa-rameters), the laminar boundary layer starts to become turbulent. This so-called laminar-turbulent transition is a

complicated process and is illustrated in figure2.5(x-axis representing an increasing Reynolds number).

Figure 2.5: Illustration of Laminar-Turbulent Flow Transition (taken from [Frei,n.d.])

In maritime applications, and more specifically when looking at maritime propulsors, the Reynolds numbers involved

are relatively high, in the order of 107and higher. An example of common Reynolds numbers in the shipping industry

can be found in [Hanninen and Mikkola,2006] where a model-scale Reynolds number is 1.0 · 107is used in comparison

with the full-scale Reynolds number of 1.2 · 109. This research (although concerning a different topic than the current

research) shows the importance of considering the appropriate Reynolds number for a specific application.

Reynolds Numbers in Oscillating Foil Propulsion Research In most of the research mentioned in this chapter the

influence of the Reynolds number is not seen as very important nor significant. The different systematic studies use a fixed Reynolds number to investigate the influence of different other parameters (as do the ones mentioned later on in this chapter). An illustration of this trend in the research field is the following statement: “We expect the Re-dependence

to be relatively weak for sufficiently high Re”, as stated in [Wang,2000]. This does not mean that no research into the

influence is done, e.g. [Freymuth,1985] experimentally varies the Reynolds number in an accelerating flow over a foil

from 16 up to 52, 000. This research shows that for higher Reynolds numbers higher order vortex structures appear and the flow becomes more and more dominated by turbulence.

Most research on oscillating foil propulsion uses a fixed Reynolds number based on the chord length of the foil

(Re = ρUc/µ). The Reynolds numbers involved range from 164 in [Von Ellenrieder et al.,2003] up to 11,000 in

[Anderson et al.,1998], 40,000 in [Read et al.,2003] and even up to 100,000 in [Isogai et al.,1999]. The most common

2.2.DESIGNPARAMETERS ANDFLOWPHENOMENA 13

There exists some research with oscillating foils at higher Reynolds numbers (order Re = 105− 106), such as [Lee and

Gerontakos,2004] and [Martinat et al.,2008], but these mainly concern research into the effects of dynamic stall

(elaborated in the next sub-section) and not really into foil performance or propulsive capabilities. The only actual

performance literature found was the case study reported in [Yamaguchi and Bose,1994] and the experimental research

by [Vermeiden et al.,2012]. The first mentioned research compares the foil performance to the performance of a

traditional screw propeller (using linear theory) at Re = 5.0 · 107and concluded that an increase in propulsive efficiency

could be achieved in cases of wide beam ships and this mainly due to the increase in swept area. The second mentioned

research investigated several foil operating configurations (also multi-foil operation) on model scale at Re = 2.5 · 105

and resulted in a performance map and conclusions for different design parameters (which will be referred to later on).

2.2.4.

S

N

In [Triantafyllou et al.,1991] and [Triantafyllou et al.,1993] the importance of the timing of the shedding and growth

of the vortices in the formation of a thrust / drag generating wake is cited. It is found that so-called “preferred frequencies” exist, in which the term “preferred” is dependent on the intended objective of operation. It can either refer to the efficiency of the thrust generation (in terms of input power needed for thrust production), the generation of a sufficiently large thrust force or a combination of both. This can be explained differently by considering the wake of the foil as a tuned amplifier and the preferred frequencies as acting on it as the maximum amplification frequencies. The aforementioned papers investigated and validated this assumption and consequently introduced a new dominant governing parameter, the non-dimensional oscillation frequency or Strouhal number St being

St = f A

U (2.6)

where f is the oscillation frequency of the foil (defined as f = ω/2π).

Optimal Strouhal Number Ranges After “discovering” the Strouhal number as the main governing parameter [

Tri-antafyllou et al.,1993] performed an extensive experimental investigation into what range of Strouhal numbers would

provide optimal efficiency. To further investigate this parameter a wake stability analysis was performed in order to find the maximum spatial amplification of vortical disturbances (find the amplifier frequencies as explained before). This analysis showed that the oscillating frequency causing this maximum spatial amplification is found in a range

of St = 0.25 − 0.35. Numerical research by [Lewin and Haj-Hariri,2003] also mentioned this dependency on the

fre-quency of maximum spatial amplification of the velocity profile / jet. The found range of efficiency was even identified

when investigating different fish and their operational Strouhal ranges. Research by [Anderson et al.,1998] expanded

this range to around 0.4 and showed the existence of two different efficiency peaks, one in the lower Strouhal range corresponding to a relatively low thrust force and one in the higher range (of St = 0.3 − 0.4) corresponding to high

thrust forces as needed for maritime applications (this is illustrated in figure2.6, in which the efficiency and the thrust

coefficient are plotted over a range of Strouhal numbers for a certain combination of kinematic parameters). Another conclusion (of several researches) was that even higher propulsive forces could be reached with a further increase of

the Strouhal number but this at the cost of a reduced propulsive efficiency. Experimental researches by [Read et al.,

2003] and [Hover et al.,2004] confirmed these ranges and they were also found in numerical investigations such as

[Wang,2000] (in which only heaving motion is considered).

The Strouhal number is thus the key parameter in determining the range where a thrust-producing jet is formed as

shown again through visualisation in [Yang and Lee,2006]. The experiments clearly show that by increasing the Strouhal

Number the momentum deficit is transformed into an excess and the formation of the jet occurs. Further increase of the Strouhal number may degrade the formed jet and cause a degraded efficiency, once again showing the importance of operating at the optimal conditions (flow control through Strouhal number).

Dynamic Stall and Leading-Edge Vortex Formation As explained before the Strouhal number plays an important

role in the timing of the shedding and formation of vortices. To better understand the processes involved and explain why the optimal efficiency (and sufficient thrust generation) is found, the visualisation of the flow process is essential.

Extensive flow visualisation results made using DPIV (Digital Particle Image Velocimetry) are presented in [Anderson

et al.,1998]. The visualisations show that when optimal efficiency combined with significant thrust forces is achieved, a

moderately strong leading-edge vortex is formed and then convected towards the trailing-edge. At the trailing-edge this vortex coalesces (merges) with the vorticity shed by the trailing-edge resulting in strong vortices forming an efficient

Figure 2.6: Efficiencies (η) and Thrust Coefficients (Ct) for Different Strouhal Numbers (taken from [Anderson et al.,1998])

phenomenon in which the shear layer separates at the leading-edge and rolls up to form a leading-edge vortex under

unsteady flow conditions is called dynamic stall and is illustrated in figure2.7. Dynamic stall can result in additional

suction and results in performance gain (generated lift), but is also very unstable and detaches easily from the foil,

which in this case has positive consequences (as explained by [AFCAD,n.d.]). Further visualisation cases show that the

timing of dynamic stall vortex occurring (at what point in the oscillation cycle) is of great importance for the efficiency.

Later numerical research by [Guglielmini and Blondeaux,2004] also showed this phenomenon leading to high efficiency

operation. The timing and occurrence of dynamic stall is further influenced by other parameters, as there is the angle of attack (profile), which is discussed in the following relevant sub-sections. A later study by the same authors, found in

[Guglielmini et al.,2004], confirmed this importance by stating that it is essential for accurate numerical efficiency and

thrust prediction to accurately model possible separation from the smooth foil surface.

Figure 2.7: Visualisation of Turbulent Breakdown and Leading Edge Formation, Growth and Separation through Flow Visualisation and Conceptual Sketches over an Oscillating Foil (taken from [Lee and Gerontakos,2004])

Opposing research has also been found. [Isogai et al.,1999] performed numerical simulations which showed a

significant efficiency decrease as soon as flow separation (and leading-edge vortex formation) occurred. [Tuncer

and Kaya,2005] also showed that preventing leading-edge vortex formation by lowering the heave amplitude had

beneficiary effects for the propulsive efficiency. Whether positive or negative effects will occur is clearly dependent on other (unknown) influences. Some additional information about possible vortex interaction effects is given in

2.2.DESIGNPARAMETERS ANDFLOWPHENOMENA 15

whether or not the static stall angle is exceeded during the motion. The maximum angle of attack experienced by the foil during operation will therefore play an important role in the performance (and prediction of the performance) of

this foil. Experimental research on dynamic stall by [Lee and Gerontakos,2004] showed the importance of the reduced

frequency in the occurrence of boundary layer events and significant dynamic stall behaviour. Later numerical research

by [Martinat et al.,2008], using different turbulence models in attempting to predict stall behaviour over a pitching foil,

visualised the stall behaviour and identified differences in the up- and down-stroke motion. More about the occurrence of dynamic stall and the possible consequences of leading- and trailing-edge vorticity will be discussed in sub-section

2.2.8and2.2.10.

Variations on the Strouhal Number In literature the dimensionless expression of the oscillating frequency is not

always the Strouhal number but other variations, illustrating a similar influence / relation, exist. In [Isogai et al.,1999]

the term reduced frequency k is used and defined as k = (2π f c)/2U = (π f c)/U , while [S. Lai and Platzer,1999] and

[Sarkar and Venkatraman,2006] use the same term but define it not based on the semi-chord length but the full chord

length, k = (2π f c)/U . In [Vermeiden et al.,2012] the advance ratio, a more common dimensionless expression of the

velocity common in ship powering calculations, is used and defined as the inverse of the Strouhal number, J = 1/St. So no matter what term is used, it always covers some sort of relation between the forward velocity and the frequency of oscillation. Great care has to be taken in determining how this relation is defined when analysing data in order to properly asses the results (and be able to compare them with other results).

2.2.5.

H

M

- H

-

-C

R

The oscillation behaviour of the foil consists of both heaving and pitching. The heave motion is characterised by the heave amplitude and the oscillation frequency (which is captured by the Strouhal number). The heave amplitude will have a strong correlation to the width of the generated wake and thus play an important role in the amount of generated thrust and efficiency of propulsion. To determine the influence of the heave amplitude, the dimensionless

heave-to-chord ratio h∗is introduced

h∗₌h0

c (2.7)

Heave-to-Chord Ratio for Optimal Efficiency Research by [Yamaguchi and Bose,1994] showed that increasing the

heave amplitude (and thus heave-to-chord ratio) will result in a higher efficiency since the swept area of the foil is larger.

Typical experimentally investigated ranges of the heave-to-chord ratio vary between 0.25 and 0.75 as in [Anderson

et al.,1998] and even up to 1.00 in [Read et al.,2003] and [Hover et al.,2004]. The results of these different experimental

researches shows that higher heave-to-chord ratios result in higher thrust forces due to the larger swept area (can be

seen as raising the Strouhal number at a constant frequency) but at the cost of reduced efficiency. In [Tuncer and Kaya,

2005], where only heaving is considered, it is shown through numerical research that a maximum thrust coefficient is

found due to the lack of pitching. It also confirms the statement that higher heave-to-chord ratios will have negative consequences for the propulsive efficiency. The research helps realising that achieving high propulsive efficiency in combination with sufficient thrust generation depends on a combination of parameters. The heave-to-chord ratio setting is only a single parameter in reaching this goal and will interact with other design parameters.

Heave Motion Profile Besides the heave amplitude the heaving profile is of course of interest. In most cases (research

mentioned in this chapter) the heave profile is considered simply sinusoidal (and thus harmonic) in shape, but this does not have to be the case. Other kinds of profiles and more particularly the introduction of higher harmonics (as

described in [Read et al.,2003] and illustrated in figure2.8) in the heave motion can have beneficiary effects for the

behaviour of the foil. More about this parameter is discussed when the attack profile is discussed (sub-section2.2.8)

since heave profile adjustments are mainly aimed at transforming the angle of attack profile as can be seen in bottom

of figure2.8.

2.2.6.

P

M

- P

A

P

P

Beside heaving the oscillation motion consists of course of a pitching motion. The pitching behaviour will play an important role in the variation of the angle of attack and consequently in how the foil performs. The consequences

of pitch amplitude related to the angle of attack will be discussed in sub-section2.2.8, while the other, more direct

Figure 2.8: Influence of Higher-Harmonics (h.o.t = higher order terms) in Heave Motion on the Angle of Attack Profile (taken from [Read et al.,2003])

Importance of Pitching The pitching of the foil might seem secondary to the generation of the propulsion when

comparing it to the heave motion, but this is far from true. Although research by [Sarkar and Venkatraman,2006]

showed that only pitching will not lead to thrust forces worthy of investigating and using practically as a marine

propulsor. In [Tuncer and Kaya,2005], where only heaving is considered, it is seen that the pitching allows the thrust

force to overcome a certain threshold which could not even be passed by further increases of the heave amplitude. The numerical research also showed that higher efficiencies could be reached with a combination of lower heaving motions and increased pitching amplitudes. At higher loading conditions (higher Strouhal numbers), the positive effects of the pitching on the efficiency become even clearer.

Influence on Dynamic Stall The pitch amplitude also has a major influence on the dynamic stall phenomenon (and

thus the shedding of a leading-edge vortex). Different three-dimensional flow visualisation experiments performed and

reported in [Parker et al.,2003] and [Von Ellenrieder et al.,2003] proved the influence of the pitch amplitude on the

timing of the shedding of a leading-edge vortex (combined of course with the phase angle which will be discussed later on). An increase in pitch amplitude will cause vortices to be shed earlier in the cycle since the dynamic stall angle will be reached earlier (the threshold angle of attack is reached faster). Pitch amplitude variations will thus directly affect

the rate of dynamic stall development. In [Lewin and Haj-Hariri,2003], where only heaving motion is considered, it is

made clear that the pitching motion introduces better flow control and more specific better control of leading-edge formation and its effects.

Pitch Point Location Another important influence on the pitching behaviour is of course the location of the pitch

point from the leading-edge of the foil, which given in its non-dimensionalized form b∗, is defined as

b∗=b

c (2.8)

The location of the pitch point will play a role in determining the forces and moments on the foil, but might also influence the general behaviour of the foil (although little investigation is done in this area). Most experimental research mentioned in this chapter used a pitch point within the range of b∗ = 0.25 − 0.33. Numerical research by

[Sarkar and Venkatraman,2006] using potential flow (discrete vortex technique) on the effects of pure pitch motion

(no heave motion) showed that moving the pitch point towards the leading-edge had the same effect on the thrust force as increasing the pitching frequency (higher Strouhal number). The opposite, moving the pitch point towards the

trailing-edge (higher values of b∗), was also proven.

A different approach to pitching is taken by [Mattheijssens et al.,2012]. In this research the pitch point is placed

around half a chord length upstream of the leading edge. This approach enables the foil to pitch passively due to the hydrodynamic forces acting on the foil, aided by a counteracting spring (otherwise a negative moment would be

generated) with adaptable pretension, visualised in figure2.9. This mechanism thus reduces the complexity of the

2.2.DESIGNPARAMETERS ANDFLOWPHENOMENA 17

Figure 2.9: Visualisation of Passive Pitching Mechanicsm (taken from [Mattheijssens et al.,2012])

2.2.7.

P

A

H

-P

M

The phase angle between heave and pitch motion (also called the phase shift between the motions) might seem a small and unimportant parameter, but it has important consequences on the foil behaviour and operation. The phase angle plays a significant role in determining the instantaneous angle of attack (which is discussed in the next section) and thereby indirectly influencing parameters as timing of trailing-edge vortex shedding, occurrence of leading-edge

separation and the interaction between leading- and trailing-edge vorticity (as reported in [Guglielmini and Blondeaux,

2004]).

Optimal Phase Angle Research into the optimal phase angle is fairly limited and this is most probably due to the

fact that early experimental research by [Anderson et al.,1998] showed that the optimal angle lays in a limited range

around 90 degrees. Later experimental research by [Read et al.,2003] confirmed this and numerical research by [Isogai

et al.,1999], [Tuncer and Kaya,2005] and [Young et al.,2006] showed that the ideal phase angle is located in the 85

to 100 degree range. Phase angles around 90 degrees will result in the highest pitch angle to occur around the mean heave position where the heave velocity is the highest. This will (as can be concluded by close investigation of figure

2.2) reduce the instantaneous angle of attack and benefit the propulsive efficiency, as will be explained in the next

sub-section.

2.2.8.

A

A

The angle of attack is important parameter in the analysis of the performance of any foil. It will in large amounts determine the size of the generated forces (and moments) on the foil and greatly influence the behaviour of the flow passing over the foil. For an oscillating foil the angle of attack is a time-dependent parameter due to the combination of the transverse motion and steady free-stream velocity. The main governing relation for the time-varying angle of attack

can be found by combining (2.3) and (2.4) resulting in (2.9).

α(t) = arctan µ_{h(t )}˙ U ¶ − θ(t ) if θ(t ) < φ(t ) (2.9) α(t) = θ(t) − arctan µ_{h(t )}˙ U ¶ if θ(t) > φ(t)

This new found relation can be used as a starting point for different types / ways of control of the angle of attack profile

(time-varying) but also of the maximum angle of attackαmax. Consequently also other parameters can be controlled by

using this relation, as will be explained in the next paragraphs.

Maximum Angle of Attack The maximum angle of attack can be determined easily for harmonic heaving and pitching

by (2.10) ifψ = 90◦(as stated in [Anderson et al.,1998]). The proposed formula is often practically used since this phase angle is the most commonly used in experimental and numerical research.

αmax= arctan µωh 0 U ¶ − θ0 (2.10)

In [Anderson et al.,1998] it is shown that at small maximum angles of attack the efficiency is fairly poor and the highest

higher angles of attack were tested but turned out not to be beneficiary for the efficiency (although [Schouveiler et al.,

2005] showed that the generated thrust was higher), most certainly due to the higher generated drag of the foil (as also

shown in [Read et al.,2003]).

The maximum angle of attack can also be specified and then used to determine the absolute pitch angle amplitude

needed to guarantee the specified angle of attack profile (of course all other parameters are fixed too). In [Read et al.,

2003] a formulation of this relation is given based on the assumptions thatψ ' 90◦and that the Strouhal Number is

low. The resulting relation, beingθ0= π · St + αmax, shows the resulting pitch amplitude. This relation actually has two

solutions, one relating to drag production and one relating to thrust generation, since the foil can pitch both up and

down. In figure2.10this is shown by visualisation of possible resulting angle of attacks based on a wide range of pitch

amplitudes ranging from −90 to 90 degrees (an angle of attack of 20 degrees is shown to illustrate the two possible solutions at a certain chosen angle). This control mechanism, besides direct control of the maximum angle of attack, may also be used to judge / influence dynamic stall behaviour. The maximum angle of attack itself, as well as when in

the oscillation cycle it occurs, will influence the shedding of leading-edge vortices as [Tuncer and Kaya,2005] showed.

A shift of the peak resulted in improved efficiency (but not of the thrust generation) due to it preventing leading-edge formation. This is once again not always the case since (as earlier mentioned) leading-edge vorticity might also have

positive effects on the propulsive efficiency and thrust generation. In [Tuncer and Kaya,2005] not only the maximum

angle of attack was varied but a mean angle of attack was introduced. The variation of the mean angle of attack showed similar effects since higher set means would result in stronger flow separation, higher amounts of leading-edge vorticity and even the creation of a drag producing wake (at mean angles above 15 degrees).

Figure 2.10: Maximum Angle of Attack for Different Pitch Amplitudes (2 Solutions Problem)

Figure 2.11: Degradation of Angle of Attack Profile due to Increased Strouhal Numbers (taken from [Hover et al.,2004])

Angle of Attack Profiles Another possibility, and perhaps more interesting method, is direct control of the

time-varying angle of attack, as described in [Read et al.,2003] and further investigated in [Hover et al.,2004] and [

Schou-veiler et al.,2005]. The first of the three researches focusses on treating the phenomenon in which large Strouhal

Numbers cause degraded angle of attack profiles (differing from a smooth sinusoidal shape) resulting in reduced thrust

performance as can be seen in figure2.11. Such behaviour can be treated by introducing higher harmonics in the heave

motion thereby smoothening the angle of attack profile. The newly found harmonic behaviour of the angle of attack profile will then again substantially increasing the thrust coefficient. The second research takes things further as certain angle of attack profiles are specified (the specified profiles were a square, sawtooth and traditional cosine profile) for which the necessary heaving motion is then determined. The results showed the possibilities of high thrust generation combined with high efficiency for the cosine profile, while the other profiles managed even higher thrust values even at high Strouhal Numbers. The downside of the alternative profiles is that they pose a threat on the efficiency due to the sharpness of the profiles and rendering them useless for the intended applications in the maritime industry. The third mentioned research performed further visualisation experiments showing vortex pattern changes in the foil wake and their dependence on the angle of attack. When cosine-like angle of attack profiles are prescribed it is seen that two vortices of opposite rotation are shed per oscillation cycle, resulting in an efficient, high thrust-producing reverse von Kármán street. The contrary is also shown, for angle of attack profiles strongly diverting from harmonic behaviour. Here different vortex patterns are shed, creating a time-averaged velocity profile which can be associated with net drag.