A CFD Analysis of Cyclodial Propellers - DiVA portal

A CFD Analysis of Cyclodial Propellers

Author: Fredrik Thelin

LIU-IEI-TEK-A–17/02891—SE

September 2, 2017

Linköpings universitetDepartment of Management and Engineering

Division of Applied Thermodynamics and Fluid MechanicsThesis Work, 30 hp, 2017| LIU-IEI-TEK-A–17/02891—SE


Division of Applied Thermodynamics and Fluid MechanicsThesis Work, 30 hp, 2017| LIU-IEI-TEK-A–17/02891—SE

A CFD Analysis of Cyclodial Propellers

Author: Fredrik Thelin

Supervisors: Petter EkmanLinköpings university

Wei B. JiABB CRC

Examiner: Matts KarlssonLinköpings university

September 2, 2017

Linköpings universitetSE-581 83 Linköping, Sweden

013-28 10 00, www.liu.se

Copyright

The publisher will keep this document online on the Internet – or its possible replacement –from thedate of publication barring exceptional circumstances. The online availability of the document impliespermanent permission for anyone to read, to download, or to print out single copies for his/hers own useand to use it unchanged for non-commercial research and educational purpose. Subsequent transfersof copyright cannot revoke this permission. All other uses of the document are conditional upon theconsent of the copyright owner. The publisher has taken technical and administrative measures toassure authenticity, security and accessibility. According to intellectual property law the author hasthe right to be mentioned when his/her work is accessed as described above and to be protectedagainst infringement. For additional information about the Linköping University Electronic Press andits procedures for publication and for assurance of document integrity, please refer to its www homepage: http://www.ep.liu.se/

Västerås. September 2, 2017©Fredrik Thelin

Abstract

The quest for more efficient machines is always ongoing in the engineering world. This project isno different. ABB are investigating a new type of propeller that seems to offer increased efficiencycompared to normal screw propellers. That is a so called foil wheel propeller. The foil move in acircular pattern with the fluid stream moving in the radial direction of the propeller instead of theaxial as in a screw propeller. If the propeller is placed and modeled correctly it can also be used as athrust vectoring device.

This report focuses on the fluid physics of the foil wheel propeller, or as it is called in this report radialflow propeller. First of all the movements and interactions of the blades must be understood. Both tokeep the efficiency high to compete with screw propellers, but also to foresee any problems that mayoccur with such a new device.

A scaled down version of the propeller have been commissioned by ABB and will be tested in sometime after the work within this report is completed. The effects associated to this will also be analyzed.

The tool to compute the flow physics of the radial flow propeller will be computational fluid dynamics.Computational fluid dynamics uses a numerical method to compute the entire fluid field in spaceand time. The flow around the propeller is highly complex so a detailed analysis is needed if a wellfunctioning control system is to be constructed for instance.

The differences between the downscale and the full-scale are great, even when the non dimensionalcoefficients are considered. The down-scale case will be less efficient, it will be difficulties predictingthe performance of the full-scale since the downscale flow is much less powerful than the full-scale case.

The interaction between the blades has a large effect. There is a strong relation between angle ofattack and the number of blades. The forces that are large change by about 30% so it must definitelybe considered if a model is to be used for a control system.

Acknowledgements

I would like to thank my supervisor Wei Ji at ABB CRC for all of the inputs and help. It have beengood to always have someone available to discuss the different problems and solutions so as to have asecond opinion and insight.

Thanks to Petter Ekman for being my supervisor at Liu. It was really useful getting tips on how to usethe models properly and what thought processes to have. Thank you Zhivco for being my opponentand coming up with some hard and insightful questions on the thesis.

I would like to thank all of the people at ABB CRC and all of my fellow thesis workers there for agood time with people that are passionate about what they do.

Västerås. September 2, 2017Fredrik Thelin

Nomenclature

Orbit The rotational path of the foils around the central axis of the larger spinning wheel.

Abbreviations

Liu Linköpings universityABB CRC Asea Brown Boveri Corporate Research CentreNACA National Advisory Committee for Aeronautics

CFD Computational Fluid DynamicsBL Boundary LayerRFP Radial Flow PropellerUDF User Defined FunctionSST Shear Stress Transport

Letters

θ Rotational position of the foils around the central axis of the larger spinning wheel.D Diameter of the RFPR Radius of the RFPα Blade angle of attack between the blade line and relative velocity vector.αrel The relative velocity vector angle between the chord and x-axis.β Blade pitch angle to the orbit tangentγ Blade pitch angle to the x-axis.λ Advancement ratio. Ratio between rotational velocity and forward velocity.e Eccentricityω Rotational speed.ωfoil Rotational speed of a foil.η Efficiencyρ DensityU∞ Vessel and free-stream velocity.V Y or side velocityS Half span area.c Chord length.M TorqueT Thrust ForceY Force in the y-directiony+ Non dimensional wall distance

Coefficients

CT ThrustCP PowerCY Y forceCM Moment for propeller centreCm Foil moment at 25% chord

September 2, 2017 LIU-IEI-TEK-A–17/02891—SE i

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Theory 52.1 Movement Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Foil Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Non Dimensional Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Fluid Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.4 Flow separation and viscous eddies . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.5 Speed of Sound of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.6 Down wash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Computational Fluid Dynamics (CFD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Sliding Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.4 Capturing Turbulence Behaviours . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.1 FLUENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.2 XFOIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.1 Downscaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.2 Downstream Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.3 Differences between analytical, one foil CFD and four Foil CFD . . . . . . . . . 22

3 Method 233.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Reasoning behind 2D analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Computational Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Foil Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.6 Simulation Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.7 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.8 Time Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.9 Convergence Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.10 Numerical Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.11 Post processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.11.1 Forces and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

iii

CONTENTS

3.11.2 Angle of attack from CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.12 Plots colour scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Results and Discussion 354.1 Foil Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Downscale Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Comparison to Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Conclusion 47

6 Perspectives 49

A Analytical Calculation 1

B Computational Set-up Validation 1B.1 Mesh Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1B.2 Time Step Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B.3 Sample Time Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6B.4 Domain Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

C Transitional SST investigation 1

D Raw Data 1D.1 Full-scale angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

E User Defined Functions 1E.1 Planetary disc rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1E.2 Satellite disc rotation and data extraction . . . . . . . . . . . . . . . . . . . . . . . . . 1

iv LIU-IEI-TEK-A–17/02891—SE September 2, 2017

1 | Introduction

This is a master thesis report by Fredrik Thelin for Linköpings university(Liu) and ABB CorporateResearch(ABB CRC). The thesis will contain an analysis of a radial flow propeller(RFP).

ABB has been looking at new ways of constructing propellers for marine vessels to increase efficiency.At the moment it was found that foil wheel propellers with a cycloidal movement pattern for the foilsis a potential solution for the problem. This concept has been around since the fifty’s but has not beenutilised to a wide extent because of the complexity of the mechanism for such a small improvement.Now, with more sophisticated control systems, Computational Fluid dynamics(CFD) and more demandfor fuel efficiency ABB thinks that the time is now to implement this new technology.

The foil wheel propeller goes under a lot of names depending mostly of the mode of operation. LikeNovel Trochoidal, cycloidal, flapping foil, Voith-Schneider and so on. In this work the device will becalled a cycloidal propeller and the status quoe propellers will be called screw propellers. Essentiallythe operation of the propeller is similar to that of animals with fins or wings. The foil is moved upand down with alternating pitch to create a forward thrust. For the cycloidal propeller the aerofoilsare mounted like in a carousel, orbiting in a circle with the flow moving the radial direction. Thereforethe entire blade will have the same relative velocity so that a large and efficient aspect ratio can beused. In the screw propellers the aspect ratio is limited since the velocity changes along the radius ofthe propeller which may cause cavitation problems on the tips.

The operation of the cycloidal propeller is shown in figure 1.1. The forward speed is U∞, ω and Vω arerotational rate and speed and Vrelative is the relative speed. Much emphasis will be put on analysingthe velocity vectors shown. The relative velocities experienced by the upstream part of the orbit willbe easy to calculate since there is nothing strange in front of the propeller except for the hull. Thedownstream part will be tricky to handle since the upstream foils are turning the flow downwards andcausing it to spin. Therefore the downstream part of the orbit will get the most attention CFD-wise.

1

CHAPTER 1. INTRODUCTION

Vessle Movement Direction

x

y

�

V� Vrelative

U

Figure 1.1: This figure shows the operation of the cycloidal propeller. This example shows thepropeller with four foils. The foils move with an orbit around the centre along the circle. Thefoils can pivot around their quarter chord to achieve the desired angle of attack. Each blade willhave the same pitching pattern so each foil will move between each position with a continuousmotion.

1.1 Background

Radial flow propellers have a certain reseblence to water wheels used in water mills and some steamboats. The idea might have originated form these kinds of devices. Propellers resembling the propellerin this work where the entire device is submerged started to emerge in the beginning of the 20th

century. The mechanism of RFPs is complex, so it was not until the middle of the 20th century thatsome serious work started.

Ficken and Dickenson [4] conducted trials on a cycloidal propeller with a mechanical drive in the 1960s.It shows how complicated the drive system have to be for a mechanical configuration. An efficiencyof about 63% at the same operating conditions as in this work were found. The low efficiency may bedue to the low aspect ratio that was used. Maneuvering behaviours, rotational speeds and number ofblade influences were also investigated.

Sparenberg and DeGraaf [5] optimized the foil motion of a single foil propeller to minimize the kineticenergy of the wake. They came to the conclusion that a near cycloidal pattern achieved this.

Roesler et. al. gives a detailed description of how the radial flow functions in [9]. They go through theimportant parameters and how each of them affect the performance of the propeller. Roeslers secondreport [10] on the subject that includes an experiment of several advancement factors and Reynoldsnumbers will be used for comparison with the results form the tests of this report. The movementpattern is different in the Roesler case. They opted for a sinusoidal pattern instead of the cycloidalbecause of the higher complexity of the cycloidal movement pattern that would make the mechanicallinkage used prohibatly complex.

2 LIU-IEI-TEK-A–17/02891—SE September 2, 2017

CHAPTER 1. INTRODUCTION

A CFD study on RFPs was conducted by Xisto et al. [12]. The RFPs were in this case intended tobe placed on helicopter sized aerial vehicles. A number of operational and geometrical conditions wereexamined.

ABB have selected a number of parameters for the RFP to meet a number of performance parameters.The size was chosen to provide the required thrust, thereby the diameter, chord length, forward speedand rotational rate was decided. The aspect ratio was chosen to get good efficiency which in turndictates the length of the blades.

The main focus of the project as a whole is to provide higher performance at cruise conditions thana screw propeller. RFPs might also have quicker thrust direction change than propellers on variableazimuth pylons.

1.2 Aim

ABB marine will create a model of the propeller that will be tested in the autumn of 2017. A CFDanalysis will be done to validate the test. The hopes are that the CFD analysis will agree with thetest so that the data gathered can be used for various tasks when developing the full scale propeller.The scaled down model might have flow structures that are difference than the full scale model. Thesemight be found with a CFD analysis so that a conclusion about the comparability between the full-and down-scaled models can be drawn.

The inclusion of four blades might cause heavy interaction effects that can not be predicted analytically.If the interactions have a big impact, the analytical model can be adapted so it resembles the resultsfrom the CFD analysis more closely. This is important because the analytical model is used in thecontrol system for the propeller.

1.3 Limitations

As introduced in the background section, a number of parameters were chosen by ABB. They are 4blades, cycloidal motion, blade length and shape, chord length, chord profile, diameter, flow velocity,and advancement ratio(i.e. rotational velocity).

In areas where the Reynolds number goes below turbulent and the risk for heavy flow separation isclear the Reynolds Averaged Navier Stoces(RANS) models does not perform as reliably. Here scaleresolved simulations would be useful to capture the more intricate behaviors of the flow. Scale resolvingsimulations are computationally expensive and may not be viable in this early stage of developmentsince only one node is available on the ABB computational cluster containing 28 cores. This meansthat only RANS models will be used anyway.

ABB prefers the calculation tools to be ANSYS FLUENT version 18.0 and the meshing in ANSAversion 17.0.4.

September 2, 2017 LIU-IEI-TEK-A–17/02891—SE 3

2 | Theory

2.1 Movement Pattern

The advancement factor λ in equation 2.1 describes the relation between the rotational velocity (ωR)of the foil wheel propeller and the forward speed (U∞) of the entire system(or the vessel it is attachedto). It is important for the characteristic of how the power is applied to the fluid.

λ ≡ U∞ωR

(2.1)

The key values for the advancement factor are illustrated in figure 2.1. When the advancement factoris above one the foils will have to turn away from the oncoming flow to have the most efficiency. Thisconfiguration produces the most thrust in relation to the size of the device but is the least effectivesince the flow field is sped up a lot so the jet behind the propeller becomes more powerful than theother cases. The unity case produces the least drag at the places where the foil does not have anyangle of attack but it is still less effective than the case where the advancement factor is below one.When the advancement factor is below one the foils will always point against the direction of theincoming stream. This configuration speeds up the flow the least which aligns to the theory that themost effective device speeds up the flow the least. When taking this to the extreme the propeller wouldhave to be infinetly big and have an infinetly small rotational speed. In reality the foils will themselvesproduce a viscous or zero lift drag and a certain thrust needs to be produced. Therefore there willexist a parabolic function with the best thrust to drag ratio for an advancement factor below one.

<1

=1

>1

Figure 2.1: The advancement factor. When the advancement factor is above one the shape of themovement is in mathematics called a prolate cycloid. When the advancement factor is equal toone it is called a cycloid and when it is above one it is a curtate cycloid.

5

CHAPTER 2. THEORY

2.1.1 Foil Angles

There are a number of angles that are important for the operation of the propeller. Form a fluiddynamics standpoint the angle of attack or α is the most important in combination with the velocitymagnitude. These two combined will create the relative velocity vector in figure 2.2. This velocityvector will decide the lift and drag produced as well as how the viscous effects behave on the foil.

The beta angle in equation 2.2 and figure 2.2 is the pitch angle from the orbit circle tangent describeshow the blade moves along the orbit. The theta angle is the rotational position of the foil and is usedin the beta equation to have a reference for the equation. The eccentricity(e) controls how aggressivethe angles of the foil are.

β = arctane sinθ

1 + e cosθ(2.2)

x

V Vrel

Uγ

αβ

θ

�

αrel

y

Figure 2.2: The relevant angles and velocity vectors to calculate the physical behaviours of theRFP.

The "cycloidal" motion described in equation 2.2 is not symmetrical for the two parts of the orbit. Inthe parts where the foils are moving against the stream direction the angle will be more aggressive andin the section where the foils are moving with the stream the angle is less aggressive This is becausethe foils are more efficient when the incoming velocity is higher, evident by observing the lift to dragcurves of the NACA0015 at high versus low Reynolds numbers. This behavior is best observed whenplotting the rotational speed described in equation 2.3 and the corresponding curve in figure 2.3. θ isthe angular position of the foil on the orbit of the RFP.

dβ

dt= ω

e2 + e sinθ

1 + 2e sinθ + e2(2.3)

The angle of attack is calculated with equation 2.4. Equations 2.5 and 2.6 are needed for equation 2.4and can be deduced from figure 2.2.

α = −αrel − γ (2.4)

Where αrel is the relative velocity angle that the foil experiences that consists of the forward androtational speed that is calculated with equation 2.6. γ is the angle between the chord line and thex-axis and is calculated with equation 2.5.


CHAPTER 2. THEORY

γ =π

2− β − θ (2.5)

Where β is calculated with equation 2.2 and θ is the rotational position.

αrel it the relative velocity angle from the free-stream velocity U∞.

αrel = arctanvyvx

(2.6)

vx and vy are the relative velocity components that make up the total velocity vector. One extravelocity component must be added here. This is the component caused by the constant counter orbitalrotation that the foil has. This will cause a rotational velocity component(VA) on the foil itself.

vx = U∞ − ωR sin θ − VA cos γ (2.7)

vy = ωR cos θ − VA sin γ (2.8)

The foil rotational velocity(VA) contains the foil rotational rate(β) and the length to the distance tothe rotational center of the foil that in this case is 0.25c.

VA = 0.25cβ (2.9)

The angle of attack becomes large in the lower parts of the orbit where the foils are moving alongthe stream. It is caused by the y-velocity being about as large as the x-velocity. This will causeperformance deficiencies since the NACA0015 airfoil has a limited operating window of angles ofattack. The optimum angle of attack is between 8 and 12 degrees for the NACA0015 as described inappendix A. The cycloidal motion tries to combat this difficulty by having different rotational speedsfor the two different parts of the orbit. This is represented by the rotational speed being higher for thefirst 180 degrees and slower for the last part.

0 30 60 90 120 150 180 210 240 270 300 330 360

Angle on orbit

-10

-5

0

5

10

15

20

Ang

le [d

egre

es]

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Rot

atio

nal

spee

d ra

tio [-

]

α

ω foil /U

Figure 2.3: The angle of attack is represented by the blue line and the rotational rate by the red.Notice how the rotational velocity is very different in the lower part of the orbit(θ = −90) thanthe higher(θ = −270). This is caused by the different relative velocity vectors that are causingthe angle of attack that is plotted and creates the cycloidal motion.


CHAPTER 2. THEORY

The foils will have a "flapping" motion much like the one shown in figure 2.4 because of the β inequation 2.2. The picture describes the first half of the orbit while the second half will be the samebut in reverse.

xD

2

2

2

0

Figure 2.4: This is the movement the foil makes around the orbit which will produce a flap-ping motion. The amplitude of the lowest and highest point of the quarter chord will be thediameter(D) of the orbit.

2.2 Non Dimensional Coefficients

The forces, moments and energies involved in the RFP are translated into unit less coefficients so thatobjects with different parameters can be compared clearly. All coefficients are divided by the fluiddensity(ρ), forward velocity(U∞) and reference area(S).

Thrust coefficient:CT =

T12ρU

2∞S

(2.10)

Where T is the thrust of the observed component.

Side(Y)-force coefficient:

CY =Y

12ρU

2∞S

(2.11)

Where Y is the Y-force of the observed component.

Moment coefficient :

CM =M

12ρU

2∞Sc

(2.12)

Where M is the torque of the component. In the moment coefficient a reference torque arm must beintroduced. In this case it is the chord length(c). The same equation is used for the moment of thecenter of the RFP and for a foil. The foil moment coefficient is denoted as Cm.


CHAPTER 2. THEORY

Power coefficient:

CP =Mω

12ρU

3∞Sc

(2.13)

The power coefficient is similar to the moment coefficient. In the power coefficient the rotational rate(ω) have been added to the nominator and another forward velocity(U∞).

Efficiency:

η =CTU∞CP

(2.14)

The efficiency is a measure of the output energy divided by the input energy. In this case the usefulenergy is that of the thrust. The thrust energy consists of the thrust coefficient and forward speed.For the case of the RFP there are two different power coefficients. One for the propeller center axisand for each foil. These must be added together to get the efficiency of the entire system.

2.3 Fluid Physics

The fluid physics are based on Newtonian physics.

2.3.1 Governing Equations

The moving particles within a fluid are governed by the conservation laws of physics. The applicablelaws on a fluid element are:

1. The mass of a fluid is conserved (first law of thermodynamics).

2. The rate of change of momentum equals the sum of the forces on a fluid particle (Newton’ssecond law).

3. The rate of change of energy is equal to the sum of the the rate of heat addition to and the rateof work done on a fluid particle (first law of thermodynamics)

These laws are applied on a macroscopic level on the fluid. The length scale is in the range of micrometers where the intermolecular behaviours are neglected and the fluid is seen as a continuous element.A control volume of a small size can then be created where the in and outflows are specified. Navierand Stokes work on translating this control volume into mathematical functions can be summed upwith the continuity, the three spacial momentum and energy equations. These equations are takenfrom chapter 2 in [1].


CHAPTER 2. THEORY

Equation 2.15 is the unsteady mass conservation equation. Where ρ is the density of the fluid and t isthe time component. The u term describe the volume flow in an element and is present in all equations2.15 to 2.18.

Equations 2.16, 2.17, 2.18 calculates momentum in the x, y, z spatial directions respectively. Whereu, v, w are the x, y, z velocity components, µ is viscosity and p is pressure. SMx , SMy , SMz are sourceterms that controls momentum entering and exiting the system.

∂ρ

∂t+ div(ρu) = 0 (2.15)

∂(ρu)

∂t+ div(ρuu) = −∂p

∂x+ div(µ∇u) + SMx (2.16)

∂(ρv)

∂t+ div(ρvu) = −∂p

∂y+ div(µ∇v) + SMy (2.17)

∂(ρw)

∂t+ div(ρwu) = −∂p

∂z+ div(µ∇w) + SMz (2.18)


CHAPTER 2. THEORY

2.3.2 Fluid Dynamics

A boundary layer(BL) is formed above a solid surface when a fluid is moving over it. The moleculesclosest to the surface will have the same velocity as the surface since they get attached to it by Van derWaals forces. For calculation purposes the surface is stationary and the fluid moving. The moleculesfurther from the surface will then be slowed by the ones closer. As the fluid moves along the surfacethis behavior will grow. The boundary layer is in most cases characterized to end at 0.99U∞.

The inner layer in figure 2.5 in blue is the linear part of the boundary layer. This is dominated byviscous forces and is the only part of a laminar boundary layer and the lower part of a turbulentboundary layer. The orange part is the logarithmic part of the boundary layer. It is dominated byinertial forces and not present in the laminar boundary layer.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

log(y+)

0

5

10

15

20

25

30

35

u+

u+ = y+

u+ = 1/κ ln(Ey+)

Figure 2.5: The non dimensional boundary layer velocity and height. This figure corresponds tothe experimental values in figure 3.11 in [1]

Von Karman’s constant κ = 0.4 and the additive constant E = 9.8 used for the logarithmic scale infigure 2.5.

y+ is a non dimensional parameter to describe the position in the boundary layer in figure 2.5. Theequations to calculate y+ are found in Versteeg [1] and Schlichting [2]. The distance from the wall iscalculated with equation 2.19 and can be reversed if y+ is desired.

y =y+

νuτ(2.19)

where ν is the kinematic viscosity and uτ is the friction velocity in equation 2.20.

uτ =

√τwρ

(2.20)

where τw is the wall bound friction calculated with equation 2.21 and ρ is the density of the fluid.

τw = Cf1

2ρU2∞ (2.21)


CHAPTER 2. THEORY

where U∞ is the free stream velocity and Cf is the friction coefficient calculated with equations 2.22and 2.23. Since the boundary layer can be ether turbulent or laminar, the friction coefficient Cf mustbe different because of the different velocity gradients that are causing the friction.

Laminar:Cf = 0.664Re−1/5 (2.22)

Turbulent:Cf = 0.025Re−1/7 (2.23)

Where Re is the Reynolds number.

2.3.3 Reynolds Number

Reynolds number is the relation between the inertial and viscous forces in a moving liquid. The equationfor Reynolds number is stated in equation 2.24. The inertial forces are placed in the numerator andconsists of the free-stream velocity and the characteristic linear dimension length. The denominatorrepresents the viscous forces that in this form of the equation is the kinetic viscosity. If the viscousforces are dominating, the flow is laminar i.e. the velocity direction is uniform. If the kinetic forces aredominating the flow is "turbulent". The velocity is then chaotic and moving in all spatial directionsand time.

Re=U∞x

ν(2.24)

According to Versteeg [1] page 46 the fluid flow starts to become unsettled at a Reynolds number of91k when moving over a flat plate. "Tollemenin-Schlichting" waves will start to form and from themas the Reynolds number increases turbulent behaviors will form. This will occur until the Reynoldsnumber reaches one million in amplitude. Then the flow is considered to be fully turbulent.

The thickness of the boundary layer is taken from Schlichting [2] p. 140 for laminar and p. 638 forturbulent.

Laminar boundary layer thickness:

δ =5x√Re

(2.25)

Turbulent boundary layer thickness:

δ =(

0.37x

Re

)1/5(2.26)

where x is the length from the beginning of the boundary layer and Re is the Reynolds number.


CHAPTER 2. THEORY

The increase of the Reynolds number has a profound influence on the efficiency of the NACA0015air-foil in figure 2.6. This data is calculated with the XFOIL software and the method is described indetail in appendix A. This range is chosen since it incorporates the mean Reynolds number of the full-and down-scale cases. From zero to one million Re the increase of lift to drag ratio is exponential. Thisis due to the flow being laminar in this region and therefore the viscous forces are dominating so thatthe airflow can not stick properly to the divergent side of the air-foil After the one million Reynoldsnumber mark the flow becomes fully turbulent.

It must be noted that the drag does not have as large impact on the RFP efficiency as comparingCL/CD ratio. The Lift force will also contribute to the drag of the RFP. This graph shows that theNACA0015 is 100% more effective at 9M than 0.5M. If the efficiency is calculated analytically withthe tabulated XFOIL values it can be seen that the efficiency drops by only 14% in appendix A.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Reynolds number ×106

30

40

50

60

70

80

90

100

110

120

130

Cl/C

d

Figure 2.6: The maximum lift over drag of the NACA0015 airfoil for different Reynolds numbers.These values were calculated with XFOIL.

2.3.4 Flow separation and viscous eddies

Described by Schlichting [2] p.131. Flow separation occurs when the boundary layer over a surface isforced to become larger by a geometric divergence. Because the boundary layer has become artificiallylarger the velocity gradient within have been stretched and the inertial energy within it. This makesthe viscous energy ratio greater than the inertial and thus there is not enough centripetal force to keepthe flow attached to the surface. A void is then created where the flow has separated. A low pressurezone is then created that will suck in flow from the separated boundary layer. The separated flow thencontinues to be turned downwards in the divergent direction. If the separated flow eventually hits thewall again, it will be fully sucked in to the void and an "eddy" will be created. If a wall is not found,the separated flow will continue to curve and form a wake of detached eddies. In most applicationexcept eg. fluid mixing, eddies and flow separation are avoided. Reversing the flow direction is anobvious loss of energy. These phenomena must be identified and suppressed or removed as much aspossible if a high efficiensy is to be maintained.

The ways of detecting a separating zone are listed in the previous description and must all be presentfor the instance to be a separation and eddie zone. They are identified as zones with: two paralellflow directions with stationary flow in the middle of them, a sudden pressure gradient, two zonessurrounding an eddie on the wall surface with zero shear.


CHAPTER 2. THEORY

2.3.5 Speed of Sound of Liquids

The speed of sound of a liquid is the measure of the propagation of viscous forces. When the speedis close to, or greater than the speed of sound of the liquid the viscous forces will not have time topropagate and the liquid will instead compress. According to Versteeg [1] chapter 2.2, liquids movingat a slow speed can be assumed to be incompressible. The compressible effects of a fluid starts to takeplace when approaching the speed of sound for that specific fluid. When the highest speed of the fluidis below 30% the speed of sound it can be assumed to have no compressible effects.

2.3.6 Down wash

Down wash occurs in the wake of a wing that is producing lift as described by Silverstein et al. in [3].Newtons second and/or third laws are acting in this situation as the wing is producing an upwardsforce, therefore something must act as the opposite force and be pressed downwards or opposite thelift direction. In the case of wings it is the fluid that is forced away and therefore acts as the impulseobject. This will in turn produce a down wash angle(ε) that is different from the angle of attack(α).These angles are shown in figure 2.7. This is because the down wash is dependent on the curvature ofthe flow that is not solely dependent on the angle of attack but also of the geometry of the foil.

ε

Uα

Vortex Sheet

Blade

Figure 2.7: In this figure it is shown as it would be seen when the RFP is mounted vertically. Theblade have an airflow coming in with a magnitude of U∞ and with an angle of α. This causes thevortex sheet behavior behind the blade that has the angle ε in relation to the blade chord line.The sheet will also bend in on itself at the blade tip because of the pressure difference.

A three dimensional effect is also shown in the lower right corner in figure 2.7. It is a reaction to thefluid that has been forced away where surrounding fluid is filling the void. This is called a wing tipvortex. The start of the wing tip vortex is strongly dependent on the shape of the wing tip itself. Ifthe tip would be flat and end with a plain wing profile, the flow from the high pressure fluid on thelower side would flow to the low pressure side even before the end of the chord. In this case the tipis oval and therefore this effect will occur far back on the tip if at all. If the vortex is formed on thelow pressure side it will produce a low pressure on this side which will increase lift and drag on thisportion of the wing compared to the inwards parts.


CHAPTER 2. THEORY

2.4 Computational Fluid Dynamics (CFD)

2.4.1 Finite Volume Method

Only simple problems like plane-parallel flow can be solved analytically for the Navier-Stokes equations.More complex problems can be solved numerically. This is done by dividing the flow-field in volumeswith a finite size.The flow-field is then divided into a mesh of volume elements.

The governing equations can be written on the general term φ that represents all scalar quantities ofthe Navier Stokes equations as well as the properties added in the finite volume method. This alsoincludes turbulent properties.The governing equations then becomes:

∂(ρφ)

∂t+ div(ρφu) = div(Γ∇φ) + Sθ = 0 (2.27)

Equation 2.27 can then be integrated on a finite volume and become:

∫An.(Γ∇φ)dV +

∫CV

SθdV = 0 (2.28)

Where n is the number of cells and CV is the control volume that is observed.

Equation 2.28 can be implemented in the computational solver in a number of ways. This mainly hasto do with how the properties of one cell propagates to the surrounding cells. The first and simplestscheme present in the FLUENT CFD solver is the first order upwind scheme. The general flow propertyφ and each respective cell is compared to the next cell that is in the velocity vector’s direction. Therebythe observed cells face φf that is facing the the upstream cell will have the same value as the cell centerof the upstream cell.

A method with more precision is the second order upwind scheme. The quantities at the cell centersare then computed using multidimensional linear reconstruction. This basically means that in additionto looking at the upstream cell’s value and setting the face to this value, the gradient of the up-streamcell is also observed and implemented.

Equation 2.28 can be derived in time as well as space for the finite volume method. The generic termφ becomes:

∂φ

∂t= F (φ) (2.29)

The F function incorporates any spatial discretization. Equation 2.29 can them be written on the firstorder numerical discretization in equation 2.30.

φn+1 − φn

∆t= F (φ) (2.30)


CHAPTER 2. THEORY

2.4.2 Mesh

The volume mesh can have a number of different shapes to the volume elements. The most commonkind is the tetrahedral mesh since it has the least problems with keeping the element quality withcomplex geometries. Hexahedral elements are popular because it takes about half as many cells forthe same size of cells. The downside is that hexahedral cells are more difficult to adapt to complexgeometries.

Mesh Quality

The mesh must abide to a number of quality criterion for the numerical solution to be stable. Theseare described in ch. 5.2.2.2 in the FLUENT users manual [18] and are quoted below.

Skewness is a measure of the corner angles of a computational cell. The corner angles can not bebelow a certain value. This is done with a ratio between 0 and 1 that represents an angle between0 and 90 degrees for hexahedrals and 0 to 60 for tetrahedrals. 0 degrees is 1 and 90 degrees is 0.Since the finite volume is used, an actual volume has to be created. Therefore the corner angles of acomputational cell can not be zero. They can also not approach zero. According to the ANSYS usersmanual ch. 5.2.2.2. [18] the highest acceptable value of the skewness criteria is 0.95.

Aspect Ratio is the measure of how similar in length the sides of a cell are. If a cell is stretchedmuch further in one direction the calculated result may be scewed if the flow is not aligned with thecell.

2.4.3 Sliding Mesh

The sliding mesh theory for ANSYS FLUENT can be found in the ANSYS theory guide [19] in chapter3.3. The sliding mesh uses the general conservation equation for dynamic meshes. The only differenceis that the mesh cells does not deform. The equation that is used is shown for the sliding mesh isshown in equation 2.31. Since the pressure based solver is used the density term is constant and canbe neglected.

d

dt

∫V

ρφdV =[(ρφ)n+1 − (ρφ)n]V

∆t(2.31)


CHAPTER 2. THEORY

2.4.4 Capturing Turbulence Behaviours

Turbulent behaviors are manifested all the way to the smallest spatial scale. According to Schlichting[2], the smallest length scales are Kolomogorov scales that are ine billion times smaller than the largeststructures. If the computational regime is in the meters range, this would mean that trillions of cellswould be needed, which is too much for any computer to calculate at the moment.

This is done with Reynolds Averaged Naver Stokes (RANS) equations described in chapter 3 in [1].From experiments it is seen that the velocities have a time fluctuating value from the mean velocities.This can be represented by u′,v′ and w′. The averaged velocities are represented by capital letters U, V,W. These parameters will form Reynolds decomposition that can be written on the form in equation2.32. Equation 2.32 is written for u and the same applies for v and w.

u(x, y, z, t) = U(x, y, z)− u′(x, y, z, t) (2.32)

These terms can then be implemented in the Navier Stoces equations 2.15 to 2.18. After some arith-metic equations 2.33 to 2.36 are produced.

∂ρ

∂t+ div(ρU) = 0 (2.33)

∂(ρU)

∂t+ div(ρUU) = −∂P

∂x+ div(µ∇U) +

[−∂(ρu′2)

∂x− ∂(ρu′v′)

∂y− ∂(ρu′w′)

∂z

]+ SMx (2.34)

∂(ρV )

∂t+ div(ρV U) = −∂P

∂y+ div(µ∇V ) +

[−∂(ρu′v′)

∂x− ∂(ρv′2)

∂y− ∂(ρv′w′)

∂z

]+ SMy (2.35)

∂(ρW )

∂t+ div(ρW U) = −∂P

∂z+ div(µ∇W ) +

[−∂(ρu′v′)

∂x− ∂(ρv′w′)

∂y− ∂(ρw′2)

∂z

]+ SMz (2.36)

Since the extra terms u′,v′ and w′ are added, some extra equations are needed for the problem to besolvable. This is called the closure problem. This can be done in a multitude of ways with varyingquality of the results. The models are always based on some experimental or other methods that in theirnature detaches the fully resolved physics of the Navier Stokes from the solution. This is a reasonabledrawback for most engineering and industrial flows where high flow speeds and large geometry sizesare used. The fact that the turbulence is modeled must always be kept in mind since the solution isonly as good as the model. The closure problem can be solved in a number of different ways. This havebeen the focus of much research in the last fifty years because of the benefits of cheaper computing(compared to directly solving the equations).

Shear Stress Transport (SST)

The SST model is a modified k-ω model which includes a blending with the k-ε model. It tries to reducethe free-stream sensitivity of its counterpart by taking into account the transport of the turbulenceshear stress. It is capable of correctly predicting flow separation regions over smooth surfaces in anopen flow, which the other two models by themselves are not capable of doing.


CHAPTER 2. THEORY

The k-ε model uses two different transport equations to account for the transport (k) and dissipation(ε) of kinetic energy, hence the name. The method supposes that the flow is fully turbulent, thereforelaminar cases can not be simulated with this model. The transport equation used to solve the fluctu-ating components of equations 2.33 to 2.36 are shown in equations 2.37 and 2.38. The equations arewritten in Cartesian tensor form since this is the way they are implemented in the FLUENT software.

∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

[(µ+

µtσk

∂k

∂y

)]+Gk +Gb − ρε− YM + Sk (2.37)

∂

∂t(ρε) +

∂

∂xi(ρεu) =

∂

∂xj

[(µ+

µtσε

∂ε

∂xj

)]+ C1ε

ε

k(Gk + C3εGb) (2.38)

Sk is a user defined source term and the Gb term is for buoyancy. These will not be used in this work.

The Gk term is the so called turbulent production term. This term includes the Reynolds stresses andcan be written as:

Gk = −ρu′iu′j

The k-ε model uses the Bossinesq hypothesis from [15] in equation 2.39.

Gk = µtS2 (2.39)

Where S is the mean rate of strain tensor that can be defined as:

S ≡√

2SijSij (2.40)

The eddy viscosity µt is calculated with the turbulent kinetic energy and turbulent dissipation inequation 2.41.

µt = ρCµk2

ε(2.41)

The rest of the terms are empirical constants:

C1ε = 1.44, C2ε = 1.92, Cµ = 0.09, σk = 1.0, σε = 1.3

Like the k-ε model, the Standard k-ω model also uses two different transport equation to account forthe Reynolds Stresses that appear in the flow. For this method, the turbulent kinetic energy (k) andspecific dissipation rate (ω) are modeled. This model takes into account low-Reynolds number effects,as well as compressibility and shear flow spreading. Without any modifications though, the model issensitive in the free-stream region, making it inappropriate to solve flows that are highly dependenton that region.

The two transport equations are presented in equations 2.42 and 2.42

∂

∂k+

∂

∂kui=

∂

∂xj

(Γk

∂k

∂xj

)+Gk − Yk + Sk (2.42)

∂

∂ω+

∂

∂ωui=

∂

∂xj

(Γω

∂k

∂xj

)+Gω − Yω + Sω (2.43)


CHAPTER 2. THEORY

The k-ω transport equations are similar to the ones in the k-ε model but introduces some differentparameters. Γk and Γω represents the diffusivity of k and ω.

The k-ω model has a model to handle the different regions of the boundary layer in figure 2.5. Theseare so called wall functions that are activated if the resolution of the mesh does not permit the use ofthe standard turbulence models.

The SST model uses a blending function between the k-ε and k-ω models. This is done by limitingthe turbulent viscosity so that the transport of turbulent shear stress can occur between the wall boundflow and the free-stream. The new turbulent viscosity is shown in equation 2.44

µt =ρk

ω

1

max[

1α∗ ,

SF2α1ω

] (2.44)

The SST model puts some requirements on the mesh resolution to get reliable results. Since the SSTdoes not model the logarithmic part of the turbulent boundary layer and only the viscous parts. Thelogarithmic parts must then be resolved numerically. Therefore a a non dimensional wall distancey+ of 1 is recommended by the FLUENT theory guide [19] chapter 4.6.3. Some parts of the viscousboundary layer are then resolved since the viscous boundary is approximately at y+ = 5. This is not tosay that the SST model can not handle meshes with lesser resolution close to the wall. As mentionedpreviously in the k-ω description, the k-ω model has empirical functions to handle the entire boundarylayer. But when using a lower resolution the accuracy for the logarithmic portion will be lowered andsome flow separations or other unforeseen phenomena might not be found.

Transitional SST

The standard SST model is inadequate for flows that have a large laminar to turbulent transition zone.Therefore a more expanded model can be used to obtain more credible results. The transitional SSTmodel is described in chapter 4.6 in the FLUENT theory guide [19]. It adds two additional transportequations to the standard SST model. One for the intermittency and one for the transition onsetcriteria, in terms of momentum-thickness Reynolds number. The laminar part of the boundary layeras well as the transition part can then be modelled.

Aftab et. al. [11] compared the performance of different turbulence models for low Reynolds numberflows over a NACA4415 air-foil. The chord Reynolds number was 120k. The angle of attack was 4 and6 degrees. The main feature to capture in this case was a laminar separation bubble occurring on thetop low pressure face of the foil. It was shown that the transition Shear Stress Transport(SST) modelcaptured this the best compared to experimental data.


CHAPTER 2. THEORY

2.5 Software

There are many softwares available to solve CFD problems. Since the 1990s the field of CFD havegained a lot more attention because of the great increase of computing power.

2.5.1 FLUENT

FLUENT is a commercial CFD solver that is owned by the ANSYS corporation since 2006. FLUENTuses a cell centered approach to solve the governing equations in a finite volume fashion. FLUENTis the most widely used CFD solver at ABB CRC although the ABB concern as a whole uses manydifferent solvers. All CFD simulations within this report uses ANSYS FLUENT version 18.0.

2.5.2 XFOIL

The XFOIL program developed at MIT by Mark Drela [6] is used for calculating aerodynamic propertiesof airfoils. It uses a semi empirical model that combines the panel method for calculating the flowaround an airfoil with experimental data. The XFOIL version used is 6.9.


CHAPTER 2. THEORY

2.6 Hypothesis

The hypothesis for the downscaling and downstream effects.

2.6.1 Downscaling

The Reynolds number is lowered and thereby the separation zone will be moved forward as illustratedin figure 2.8. This will cause increased pressure drag and decreased viscous drag. From calculationswith the XFOIL tool in appendix A it can be estimated that the lift to drag ratio will decrease by 14%when the Reynolds number of the foil goes from 0.5M to 9M. This has been studied to a large extentin the past for the NACA0015, so the exact shape of the velocity gradients will not be obtained fromthe simulations. These will be referred to from external works that should be more accurate.

High Re Low Re

Separation PointSeparation Point

Boundary Layer Boundary Layer

wakewake

U U

Figure 2.8: Two identical symmetrical foils at different Reynolds numbers. The flow separationinitiation point will be suppressed in the high Reynolds number flow compared to the low case.A high Reynolds number in this case is when the flow is far beyond one million. Then the flowcan be counted as turbulent at the leading edge. The lower Reynolds number is below 0.5M. Theboundary layer has less momentum, which means lesser viscous drag, but the lower momentumalso means a greater risk for flow separation.

The most difficult part of this work is to find how the downstream part of the flow is affected by thedecreased Reynolds number. Since the separation point is further up the chord of the foil, the zonebehind the foil that has been affected by it will have more vorticity with a larger are than at the higherReynolds number.

2.6.2 Downstream Behaviour

The flow in the downstream part of the orbit will be affected by the upstream part of the orbit infigure 2.9. This in turn will mean that the angle of attack will not be the same as the one calculatedanalytically where no interaction is present. The foils are moving in the angular direction in relation tothe orbit, therefore the flow will be turned with the rotational direction when the propeller is generatingthrust. This was discussed previously in section 2.3.6 about the down-wash. Therefore the downstreamorbit will have a negative y velocity component in addition to the free flow velocity.


CHAPTER 2. THEORY

Vessle Movement Direction

x

y

�

V� Vrelative

U

Vrelative

Area affected by

upstream

Figure 2.9: The area that is affected by the upstream foils will move with the foil but also havean increased velocity stream-wise. So if the interaction starts at the top pi/2 position and travelsdown to the downstream part of the orbit at a few degrees.

Some works have been conducted on foils interacting with vorticity flows. The foil-wheel will produce areversed von Kármán vortex street, found by Gopalkrishnan et. al. [13]. A von Kármán vortex streetis formed behind round objects like cylinders. A low velocity, low pressure separation zone occursbehind the object with subsequent vortices around the wake where the surrounding flow enters thewake. In this case it will be a jet with flow exiting the the wake into the surrounding flow. So it is ajet with perhaps a bit more erratic vortices.

2.6.3 Differences between analytical, one foil CFD and four Foil CFD

There are a few analytical models existing. Roesler [10] has a detailed potential flow model with a lotof modifiers to include flow separation and upstream interactions and so on. The model to compare toin this case is a model developed at ABB that only takes one foil into consideration.

The case with one foil is to see what the difference is when the other three foils are added. Theinterference and flow speed increase will be monitored.

It is expected that the efficiency will decrease and thrust increase in the multiple foil case. This iscaused by the upstream part of the orbit increasing the flow velocity for the downstream part asdescribed in chapter 2.6.2. The increased velocity will be applied to the normal lift equation whichmeans more lift and more induced drag.

Therefore the single foil configuration is the most effective one. But with one foil there will not be asmuch power produced for the same rotational speed. But if the produced power is the same, then theinduced drag is the same. Then it will only be the viscous and pressure drag that are affecting thepropeller. Perhaps it is the smaller surface area of the single bladed propeller that is the case.


3 | Method

For the CFD solver ANSYS Fluent will be used since it is primarily used at ABB CRC. For the meshingthe mesher in ANSYS workbench is used for the 2D cases ANSA for the 3D case. Only an initial meshwas created for the 3D case because of there being not enough time and is discussed briefly. Thepropeller will first be modelled with one foil in 2D, then all four foils as in the scale model but in 2Dand then lastly in 3D but without a hull. In the test set-up, torque and thrust is measured to quantifyefficiency. The same values will be monitored in the CFD set-up. To tackle the problem with thelowered Reynolds number the velocity along the foils boundary layers will be measured and compared.With this the onset of separation can also be monitored.

3.1 Geometry

The geometry of the propeller was decided beforehand by ABB. The dimensions are presented in figure3.1 in the top and side view. The foil selected is the NACA 0015. The tip shape of the blade is providedby ABB and will not be described in detail in this report.

c

D

0.25c

90°

TOP

0.175c

4c

SIDE

Figure 3.1: The dimensions of the foil wheel from the top and sides.

23

CHAPTER 3. METHOD

3.2 Reasoning behind 2D analysis

A 2D analysis is much cheaper to perform since and entire dimension is removed which removes atleast an order of magnitude to the required amount of cells. A 2D analysis does resolve the majorityof the flow field even in the third dimension. As it can be seen in the side view of figure 3.1 the majorpart of the blades have the flow parallel to the chord direction with a uniform shape in the top plane.In this portion there will only be a minuscule flow in the span wise direction.

The cheaper cost of the 2D analysis is therefore motivated since there were many uncertainties regardingthe resolution and modeling of the RFP. How to simulate the movement was not known. The normalways of simulating an axial propeller and turbo machinery were of little help since the rotation of theRFP is not symmetrical. A moving mesh had to be used. There there are three main methods ofmoving the mesh. Overset mesh, where two or more meshes are super imposed on each other so createone mesh. This type of mesh was only in the beta phase in FLUENT for moving meshes, so it did notwork adequately. The second case is to use a re-meshing method. This would be costly time wise toperform for every 5 time-steps over a total of 10 000 steps. Therefore the sliding mesh type was used,for its ease of use and relatively low computational cost. Because of the interpolation at the interfaces,the double precision and somewhat higher mesh density in this are was used.

The 2d analysis will not find some 3D effects. The blade tip behavior is not found in 2D. It will havedirect effects on the blade it is occurring on. The flow will tend to travel from the high pressure side tothe low pressure side as on the wings of airplanes. This will in turn create a low pressure vortex that cancreate problems with cavitation. This vortex will be affected by the movements of the blade along itsorbit with unforeseen consequences. The vortex will also travel to the blades downstream and inducea rotational component on the tips of those blades. This might also have unforeseen implications thatare eluded to in section 2.3.6 about down wash behaviors and implicitly about wing tip structures.This may necessitate the need for more detailed analysis’s like scale resolving simulations.

The influence of a hull is not simulated in the 2D case while it is in the 3D. In the first test set upthe hull will be simulated by a plate but it will have about the same effect as a boat hull. In the sincethat it will create a boundary layer. The boundary layer will as described in the theory chapter havea velocity gradient. This gradient will interfere with the base of the RFP and decrease the power andthrust gradually to the base of the propeller. The thrust and power are directly linked to the velocityof the flow.


CHAPTER 3. METHOD

3.3 Computational Domain

The computational domain needs to represent the real life conditions that the RFP experiences. Forthe test of the RFP it will be positioned on a carriage and dragged through a water basin. In this caseonly the steady forward speeds are of interest so the accelerating parts will not be modeled. An easyand common practice in CFD is to have the fluid moving and the object being stationary. This savesadditional moving mesh elements. This means that the CFD model have been significantly abstractedfrom the real life test set up. The domain can now be chosen so that it is large enough so that itdoes not influence the results. The chosen dimensions are shown in figure 3.2. The dimensions werechosen from recommendations from previous CFD courses. These differing dimensions aim at cuttingthe number of cells by having the inlet length being slightly smaller than the other two dimensions.Recommended distance to Inlet3-5L, distance to outlet 5-8L and blockage below 5%.

The size of the domain was chosen to be large compared to the size of the RFP. This is so that only onevalidation simulation has to be done to prove that the size of the domain does not have an influenceon the RFP. This validation study is placed in appendix B. The dimensions of the domain that passedthe study are presented in figure 3.2.

D

5D20D

21D

Figure 3.2: The 2D computational domain. The domain is controlled by the diameter of the RFP(D).


CHAPTER 3. METHOD

3.4 Boundary Conditions

The boundary conditions for the 2D case are shown in figure 3.3. They consist of the inlet, outletsymmetry and walls. The inlet boundary condition is a velocity inlet. The velocity of the inlet ischosen to match the parameters stated in the background section to get the correct Reynolds numbers.No information have been provided on the turbulence intensity and viscosity ratio of the test case.Therefore a turbulence intensity of 5% and viscosity ration of 10 was chosen in accordance to theFLUENT user’s manual [18] chapter 7.2.2. This is since the case is considered as an open flow. Thesymmetry boundary condition was chosen for the sides of the domain because the flow is seen asuniform in this section. Since the symmetry boundary condition is far away from the RFP, it shouldnot be affected by it and keep it’s free stream properties. The outlet boundary condition is set toa pressure outlet. The pressure difference is set to zero because the outlet is flowing in to a continuousfree-stream. The wall boundary condition is set on the foils of the RFP. These are set to a no-slipwall so that a boundary layer is created and no flow permitted through the wall. The roughness isunknown so the constant is set to 0.5.

Inlet

Wall

Symmetry

Outlet x

y

Figure 3.3: The boundary conditions.The foils are not to scale for clarity

Fluid properties: The fluid selected was water. The temperature is 20 degrees Celsius which gives theproperties in table 3.1 and are taken from the FLUENT material library.

Table 3.1: The fluid properties used in all simulations.

Density 998.2 [kg/m3]Dynamic viscosity 1.003 [10−3m2/s]


CHAPTER 3. METHOD

3.5 Foil Movement

The foils will have a pitch motion described by equation 2.2. Since the interaction between the upstreamand downstream parts of the orbit are of most importance in this study, the entire motion will haveto be simulated as it is performed in reality. The standard way of simulating propellers by having arotating flow is not adequate since the motion of the blades themselves will not be captured. Thereforethe sliding mesh function in the FLUENT dynamic mesh section will be utilized. This is described inthe FLUENT users guide [18] chapter 10.6. The sliding mesh zones are set up as in figure 3.4.

Disc Interfaces

Disc 1

Domain

Disc 5

Disc 3

Disc 2Disc 4

Figure 3.4: The different sliding mesh zones that are used. In the four foil case there are fivezones. The small zones are rotating in the opposite direction to the big so that the foil is alwayspointing towards the free-stream.

The motion and data gathering from the foils are done with the "user defined function" (UDF) functionin FLUENT. The code is shown in appendix E.


CHAPTER 3. METHOD

3.6 Simulation Cases

The forward velocity for the full scale case is the cruising speed of a water-surface vessel. The rotationalrate is then set from this speed and the advancement factor λ that was described in the theory sectionwith equation 2.1. An advancement factor of 3 was determined in previous sections of the project.

All spatial dimensions are scaled down 13 times in the down-scale case. In the down scale test thepropeller is to be drawn on a carriage in a water basin. The advancement factor is kept constant whilethe forward velocity and rotational rate are altered. The forward speed of the carriage is limited. Themaximum speed is named case 2 and will have 52% of the forward speed of the full scale case. Sincethe forward velocity is only decreased by half and the radius in the denominator of λ equation 2.1 isdecreased 13 times, the rotational rate increases by 7 times. The second down-scale case is case 1. Ithas half the forward speed and rotational rate as case 2. It is treated as the base case since it is lesstaxing on the test equipment.

The Reynolds number is based on the chord of the blades and are presented in table 3.2. The differencein Reynolds number from the full-scale case to down scale 1 is 50 times and for case 2 it is 25 timesbigger.

Table 3.2: The cases to be examined.

Max Re Min ReFull scale 14M 3MDown scale 1 275k 67kDown scale 2 500k 131k

The continuous Reynolds number for one of the foils can be seen in figure 3.5. The Reynolds numberdecreases when the foil is moving with the flow and increases when it moves against the stream. Allof the cases in figure 3.5 have the same proportions because the advancement factor λ is the same.Down-scale case 1 looks different because it is plotted on the same y-axis as case 2. This plot illustratesthat both the magnitude and amplitude are different between all cases.

0 30 60 90 120 150 180 210 240 270 300 330 360

-θ

2

4

6

8

10

12

14×106

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Rey

nold

s N

umbe

r

×105

Full-scaleDown-scale 1Down-scale 2

Figure 3.5: The Reynolds number for the foil chord along one revolution of the full and down-scalemodels. The full-scale case is the blue line with the left y-axis and the downscale cases are redwith the right axis. Here it can be seen that the shape of the curves are almost identical becauseof the same advancement factor. The lower down-scale case has the same proportions as theothers but is squeezed since is is plotted on the same axis as the faster case.


CHAPTER 3. METHOD

The size difference is best illustrated when the full and downscale blades are plotted next to each otherin figure 3.6. The size difference is the largest contributor to the difference in Reynolds number.

Figure 3.6: Comparison between the foil size of the full and down-scale models.


CHAPTER 3. METHOD

3.7 Solution Methods

The pressure based solver can then be used in this case since the highest velocity of the fluid aroundthe RFP is well below 30% the speed of sound in water, as was discussed in section 2.3.5. Accordingto the FLUENT database the speed of sound in water is 1481 m/s while the highest recorded speed ofthe Flow around the RFP is 18.08 m/s, that is 1.22% the speed of sound.

The solution methods are shown in table 3.3. It was recommended in the FLUENT users guide [18]chapter 29.3.1.4 that the Coupled scheme is a good choice if the mesh is poor and the coupling betweenvelocity and pressure needs to be preserved. The mesh in of itself is not bad but the interfaces betweenthe sliding meshes are causing the mesh to become un-coupled at times (from initial tests with onefoil).

Second order upwind was chosen for the pressure and momentum discretization because of the increasedaccuracy compared to the first order. According to the FLUENT users manual [18] chapter 29.2.1. itis highly recommended to use second order accuracy when the flow is not aligned with the mesh. Themesh is always rotation in the RFP case so it will be misaligned in most cases.

The first order upwind scheme was chosen for the turbulent discretizations. Since the turbulent be-haviours in RANS simulations dissipate so quickly, they will never move through a lot of misalignedcells. THis can be seen in the theory section where a lot of damping is applied to the production ofturbulent properties as to contain the local behaviour of them.

The sliding mesh function is hard-coded to use the first order implicit time discretization functionaccording to the FLUENT theory guide [19] ch. 3.3. Therefore the first order discretization must beused for the entire flow. This was tested in the initial validation simulations where artifices on allparameters were seen in the sliding mesh interfaces.

Table 3.3: Settings

Pressure Velocity coupling CoupledGradient Least Squares Cell BasedPressure Second OrderMomentum Second Order UpwindTurbulent Kinetic Energy First Order UpwindSpecific Dissipation Rate First Order UpwindTransient Formulation First Order Implicit

The turbulence model used was SST k-ω that is described in the theory section 2.4.4. Menter [17]suggests that the SST model is suitable for open flows where separation or "adverse pressure gradients"occur over rounded surfaces.

The transition SST model was also tested and a comparison between this and the standard SST modelis presented in appendix C. This model was considered since the Reynolds number of the rfp iscrossing the 97k limit for both down-scale 1 and 2. This will cause the boundary layers to be laminar,transitional and turbulent along the rotation of the RFP. From the test in appendix C it was seenthat the model produces some strange results. Therefore it will not be used directly for the valuecomparison of the different cases. On the other hand, it shows some phenomena that appears in theseslow flows and corresponds to the results of Aftab et. al. [11].

The movement of the RFP is not symmetric in time, the simulation must then be calculated in thetime realm as well as in space. The RANS equations are then calculated in time with the rest of theflow and are then called Unsteady Reynolds Averaged Navier Stokes(URANS) models. The effect isthen that a steady solution is then calculated in each time step with the modelled turbulence.


CHAPTER 3. METHOD

3.8 Time Step

According to the FLUENT theory guide [19] chapter 4.4.3. the URANS should only pick up the largesttime scales. Therefore the time-scale should be in relation to the highest speed and the length of a foil.This time-step was tested initially, but it was found that the simulation became unstable and divergedafter a number of iterations. Therefore some smaller geometrical scales were used. According to areport on the reliability of turbulence models by Spalart [16], it was mentioned that the turbulencemodels can sometimes be related to the thickness of the boundary layer. Since the SST k-ω usesthe omega part close to the wall the boundary layer was chosen as the time scale. For the smallesttime-scale the trailing edge was chosen. The size of this part is not known for the real model so thetrailing edge from the NACA 4 digit aerofoil creator was used resulting in a trailing edge that is c/316long.

The full time step study is presented in appendix B. The time step study was conducted for the fullscale case with four foils. This study resulted in the time step size calculated from the boundary layerthickness to be the most appropriate. This divides one revolution of the RFP in to 2000 steps.

The simulation needs to run until the characteristics of the RFP are stable in time. This study ispresented in appendix B. It was found that five revolutions of the RFP are sufficient to obtain a stableand reoccurring pattern of the RFP’s parameters. This will be used in all subsequent simulations.

The calculations were carried out on ABB’s computer cluster. For this project, one node was availablecontaining 28 cores. Running five revolutions takes about 8 hours.

3.9 Convergence Level

For each time step the parameters in the numerical calculation needs to converge towards a value.This is done by comparing the values of the right hand side of each discretized equation(stated in thetheory section as equation 2.15 to 2.18) to the left side. If the difference is low enough, the calculationis considered converged.

In this case it was the noise of the solution that dictated the selection of convergence point. Thenoise appeared when the time step was lowered. It was noticeable when using the RMS residual of1e-3 for the boundary layer time step. When reducing the RMS limit to 1e-4 the noise was reducedsignificantly. The results from this can be seen in appendix B.


CHAPTER 3. METHOD

3.10 Numerical Mesh

As mentioned in theory section 2.4.1 the fluid zone must be divided into finite volume elements. Thenumerical mesh is constructed so that it can capture the relevant behaviours of the flow field. To makesure that this is done a mesh study must be done.

The main focus in the mesh study is to make sure that the wake and eventual flow separations and highvelocity gradients are captured properly. The foil disc, disc 2 will make a half revolution within disc1. Therefore the entirety of disc 1 must be prepared for the wake of the foil. There is also incomingdisturbed flow from the upstream to the downstream blades in the four foil case, so refinement isneeded for that case. The refinement study was only done for the full scale case and not for the twodown-scale cases. An unstructured quad mesh is used.

The lengths stated for the mesh in table 3.4 are stated in relation to the length of the blade chordlength(c). This is to give a good sense of the size difference of the mesh elements and the geometricalobjects.

Since the SST turbulence model was selected for this analysis, it sets some demands on the meshas described in the theory section. The non dimensional wall distance value(y+) needs to be belowone which dictates the perpendicular cell length closest to the blade surface. Equations 2.19 to 2.23are used to calculate the first layer thickness. The first layer thickness is dependent on the Reynoldsnumber and will become smaller when the Reynolds number increases. The Reynolds number changesalong the entire orbit of the RFP, so the largest Reynolds number will be chosen as the governingvalue. This value can be found in table 3.2 and is 14M for the full-scale case. Implementing this valuein to equations 2.19 to 2.23 gives that the first layer thickness should be 1.71 million times smallerthan the chord length.

According to the FLUENT theory guide [19] chapter 5.2 an aspect ratio below 500 is recommendedfor a quad mesh. The largest element size in the surface parallel direction needs to be 8.55e-04c.

Table 3.4: The settings used for the 2D mesh in ANSYS meshing.

ValueFull-scale Down-scale 2

Mesh type quad quadLargest length c/2 c/2Smallest Length c/8550 c/351Growth rate 1.05 1.05First layer thickness c/(1.71M) c/(0.701M)Surface Layers 20 20Largest feature angle 2 degrees 2 degrees

The mesh study itself is placed in appendix B. The convergence found in the mesh study will decidewith what accuracy all parameters are calculated to. If a value is below this value it will be consideredas not changed. In this work the most important change that will be monitored is the change inefficiency when the RFP is scaled down. From the analytical analysis in appendix A this differencewas found to be about 10%. If the lowest allowable difference is one order of magnitude lower at onepercent, it can safely be said that the difference came from the change in scale. The selected mesh forthe full scale case is presented in figure 3.7. The mesh study resulted in the selection of the secondrefinement as the adequate mesh. This means that the full-scale mesh consists of 500k cells while thedownscale consists of 38k cells. The difference depends on the large Reynolds number difference. Thiscaused some concern for the validity of the mesh study when applied on the downscale. When usingthe same approach for the down-scale as the full-scale the mesh around the chord is divided into 20


CHAPTER 3. METHOD

times less cells around the foil. Then a new mesh study would have been needed for this parameter.Rather than performing a new study, the spatial lengths of the full-scale mesh was scaled down 13times, keeping the same relative resolution shown in table 3.4 for the full-scale.

Figure 3.7: The selected mesh from the mesh dependence study for the full-scale four foil case.This mesh was constructed in relation to a Reynolds number of 14M. It is a 2D mesh comprisedof 455 366 cells.

3.11 Post processing

How some of the data is acquired and what data that is is stated in this section.

3.11.1 Forces and moments

There are a number of important forces and moments in the RFP.

The forces and moments of the foils were obtained with the code in appendix E. The forces in allspatial directions can be obtained with this code for all time-steps.

3.11.2 Angle of attack from CFD

The angle of attack is measured as the angle between the chord line and the relative velocity vector.This vector can be obtained by measuring where the incoming stream strikes perpendicular to thesurface of the air-foil and the pivot point of the foil. In this case the highest pressure was chosen as thepoint where the velocity vector hits the foil. Zero wall shear stress was not useful in all positions sincesome separation and reattachment occurs towards the trailing edge of the foil which will also causezero shear.


CHAPTER 3. METHOD

10.2

0.25c

P=max

Figure 3.8: Angle of attack obtained for the one foil simulation at the -15 degree position.

The blades are moving in a difficult manner to extract the attack angles. An automatic way of findingthe angles from FLUENT was not found. Therefore the process of extracting the attack angles isquite involved and time consuming. It takes about two minutes to obtain one angle, comparing to onerevolution of the RFP that is divided into about two thousand steps. Therefore these graphs are onlycomprised of 10 different angles, which makes this data set a bit rough. Efforts was made to put thehighest concentration of nodes on the curved sections of the data and less on the linear parts.

3.12 Plots colour scheme

The contour plots in this report aims at conveying velocity differences from the free-stream velocity indifferent colours. The areas are divided into green, blue and red. Green stands for increased velocityfrom the free-stream This is beneficial for the propeller since this is what creates the impulse thatpropels the thrust. Blue areas are areas of decreased velocity. This is caused by the viscous effects ofthe flow and can in most cases show the composition of the boundary layer. The blue areas should beas small as possible since they are creating an impulse in the other direction than the wanted thrustdirection. Red zones are where the fluid is moving in the opposite direction than the free-stream.These are the worst zones for the efficiency of the propeller and therefore should be the first ones tobe removed in improvement efforts.


4 | Results and Discussion

The results consists of the foil interaction section, and full-scale to down-scale differences. This orderwas chosen since the full-scale to down-scale results comments relate to the results in the foil interactionsection. All of the results presented are from the 2D analysis since no validated 3D analysis wascompleted.

4.1 Foil Interactions

The difference from using one foil to four is listed in table 4.1. The largest difference is the changein Y-force that is increased two times. The Y-force in the one foil case is small since the movementof the RFP is symmetric in the forces since the foil is moving up and down. In the four foil case anasymmetry is then added that can be seen in figure 4.1 where the Y-force magnitude is a lot higherfor the four foil case after the θ = −190 degree mark which is the downstream part of the RFP. Theupstream part of the RFP at θ = 0 degrees have also changed and become smaller in magnitude forthe four foil case. This adds to the disparity between the one and four blade cases making the meandifference for one revolution even higher.

Table 4.1: Comparison between the one foil case and the four foil case.

Difference [%]Efficiency 3.15CT -16.38CP -18.93CY 228.40CM -18.59Cm 93.46

35

CHAPTER 4. RESULTS AND DISCUSSION

0 30 60 90 120 150 180 210 240 270 300 330 360

- [degrees]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

CY

Four Foils

One Foil

Figure 4.1: The Y-force coefficient of the RFP for one and four blades.

On the downstream thrust peak at the θ = −190 degree position in figure 4.2, the shape has beenaltered significantly as predicted in the hypothesis in section 2.6.2. Caused by the incoming increasedvelocity and altered angle flow. The angle is a bit higher than the one foil case initially, then it goesdown and then increases a lot more.

The upstream peak changes significantly between the one and four foil cases in figure 4.2. This wasnot predicted in the hypothesis. The thrust is lowered by 16% and the curve shifted slightly. This iscaused by a foil travelling behind a previous one that is directing the flow downwards. This causes achange in the angle of attack making the angle more shallow, thereby decreasing the thrust and drag.The efficiency is on the other hand decreased slightly for the angle of attack is now lower than the 13degree optimum.

0 30 60 90 120 150 180 210 240 270 300 330 360

- [degrees]

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CT,C

P

CT Four Blades

CT One Blade

CP

Four Blades

CP

One Blade

Figure 4.2: Power and thrust coefficient comparison between one foil and four foils for the full-scale case. One of the four foils in the four foils case is observed.The maximum Reynolds numberis 14M.

A suction effect is occurring from the downstream blades to the upstream which is causing the loweredthrust on the upstream blade. The low pressure side of the upstream foil has a strong interaction with



the downstream low pressure side. This causes the down-wash angle to decrease which will decreasethe reaction force, as discussed in the down-wash section 2.3.6. The inverse effect is occurring for thedownstream foil causing this to have a slight increase of thrust, as shown in figure 4.2.

This effect is illustrated in figure 4.3a and 4.3b where the rotational positions of the blades are on thepeaks of the graphs in figure 4.2. Figure 4.3b shows the pressures for one foil that is plotted at all fourpositions so that a comparison between interaction and no interaction can be made. The strongestdifference that can be seen between the figures is that the pressure for the top foil is almost symmetricalfor the one foil case where as it is not for the four foil case. This has implications for the Thrust andpower of the foil in figure 4.2 at the θ = −290 degree position. Here both the thrust and requiredpower becomes negative. Looking at figure 4.3a again at this position(top foil), the low pressure sideis facing downwards and backward. This means that these negative forces are produced since the foilis helped to rotate by the downwards force since it is in the rotational direction.

x

y

= -20

= -110

= -190

= -290

(a) Pressure distribution for the 4 foil full scale case.

x

y

= -20

= -110

= -190

= -290

(b) Pressure distribution for 1 foil but at all four positions.

Figure 4.3: A comparison between the pressure distribution of the four foils case (a) and whenonly one foil is present (b). The most noticeable difference can be seen on the top foil where thepressure is evenly distributed on the one foil case while it is not in the four foil case.



The angle of attack of the one foil case is higher that the four foil case for the third and forth quadrantin figure 4.4. This behaviour is reversed in the opposite two quadrants one and two where the angleof attack is higher for the four bladed case. The behavior of the force coefficients in figures 4.2 andpressures in 4.3a and 4.3b are equivalent. The forces and pressures are all coupled together with eachother and ultimately depends on the angle of attack of the foil. The reason for the negative powercoefficients in figure 4.2 is because the attack angles spill over from one quadrant to the other. Forinstance, to produce thrust in the first quadrant the angle of attack should be positive since the foil hasa negative gamma angle. The angle of attack is negative in this quadrant between 270 and 305 degrees.This means that a negative thrust will be produced as can be seen in figure 4.2. This odd angle ofattack also has an effect on the power coefficient in figure 4.2. A negative power coefficient is producedwhich means that power is produced by the RFP. This is caused by the Y-force now pointing downwardinstead of up. The Y-force will then act in the same rotational-direction as the RFP is spinning. Sincethis force is greater then the thrust force that in this case is working against the rotation, the powercoefficient will be negative. The same thing is happening in the third quadrant. Please note that thisdoes not mean that free energy is produced (That would break the first law of thermodynamics). Thisis balanced out by the rest of the orbit where the power gained is not larger than the power input.

Observing the thrust and power coefficients in figure 4.2 at the 220 degree mark, the power coefficient islower than the thrust coefficient. This means that more power is gotten than is input in this area.Thisis because the thrust force is helping the RFP to rotate in this area because the foil is traveling forwardin this area. Why the foil is helped forward so much in this section so that power is gained can beexplained with the θ − π/2 angle. This angle decides which of the Y and Thrust-forces are dominantfor the torque of the RFP.

0 30 60 90 120 150 180 210 240 270 300 330 360

Angle on orbit

-10

-5

0

5

10

15

20

Ang

le o

f atta

ck [d

egre

es]

Analytical4 Foils CFDOne Foil CFD

Figure 4.4: Comparison between one foil and four foils.



The total pressure can explain the hump in figure 4.2 at the θ = −200 degree position. This is thedownstream part of the foil where the downstream foil is hit with the upstream affected flow. In theseries of plots in figure 4.5 the total pessure is shown and how an elevated total pressure or increasedenergy is hitting the downstream foil until it reaches the θ = −114 degree position in figure 4.5d wherethe effect starts to disappear. The dip in power and thrust occurs when the wake exactly hits only thebottom face of the back foil. Or more like when it transfers from top to bottom.

(a) 5 degrees (b) -8 degrees

(c) -21 degrees (d) -34 degrees

Figure 4.5: A series of total pressure coefficient plots. The upstream foil is creating a zone in redbehind it where the total pressure is increased. This means that there have been energy added tothe fluid. When this increased energy zone hits the back foil the total power and indeed efficiencyis increased compared to the one foil case. The blue portions represent where there are losses orenergy taken out of the system.



The wake of the four foil case is a lot more uniform than the one foil case in figure 4.6. The smallerwake of the four foil case in figure 4.6b than the one foil case in 4.6a implies that the four foil caseis more efficient than the one foil case. Less energy is wasted in the four foil case to spread the flowto the sides and is instead focused in one beam. A NASA work [8] on the topic of counter rotatingpropellers where an efficiency increase of 7-14% was found. A mere 3% increase in performance wasfound in this case between one and four foils in table 4.1. This does indicate that it is possible thatthe efficiency can increase with four foils.

(a) One foil.

(b) Four Foils.

Figure 4.6: A comparison of the wake of one foil and four. The four foil wake is a lot morenarrow and uniform. The shape of the up and down-stroke can also be seen in figure (a) as theshort increased velocity section as the up-stroke and the long increased velocity section as thedown-stroke.



4.2 Downscale Study

The single full scale case is compared to the two downscale cases. The results of the two downscalecases are presented in relation to the full-scale case in this entire section.

From table 4.2 it can be seen that the thrust coefficient CT decreases slightly and the drag coefficientincreases significantly for both cases. This causes the 20% decrease in efficiency from table 4.2. Thiscan be related to section 2.3.3 where a clear relation between Reynolds number and efficiency of thefoils is shown. As the Reynolds number goes down, the efficiency also goes down with it.

The Y-force coefficient CY is decreased by 10% which can be related to section 4.1 where the foilinteraction is discussed. In the downscale case the foils on the down-stroke are not as efficient as inthe full-scale. Therefore the back foil will not be so much more powerful than the forward foil. Thismeans that the thrust of the RFP becomes more symmetric and the Y-force coefficient goes down.

Table 4.2: Comparison between the full scale case and the two downscale cases.

Difference [%]High Re Low Re

Efficiency -19.28 -23.47CT Amplitude -20.69 -21.73CP Amplitude -73.86 -89.75CT -2.42 -2.35CP 12.70 15.41CY -7.32 -10.22CM 12.97 15.49Cm 41.18 47.94

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

0.21

0.22

0.23

0.24

0.25

0.26

0.27

CT

Full Scale

Down Scale 1

Down Scale 2

Figure 4.7: The total thrust coefficient for the entire RFP along one revolution. The thrustamplitude decreases with about 20% from the full-scale to the downscale case. The mean thrustcoefficient only decreases by 2% which is almost within the 1% margin where it can be said thatno change is seen.



0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

0.25

0.3

0.35

CP

Full Scale

Down Scale 1

Down Scale 2

Figure 4.8: The total power coefficient for the entire RFP along one revolution. The amplitudeof the power coefficient goes down to almost half of that of the full scale case. The mean valueon the other hand goes up by 20%.

Note that the thrust stays the same at θ = −20 degrees in figure 4.9a while the power increases infigure 4.9b when looking at only one foil. If the reasoning in section 4.1 is used, then both the thrustand drag should decrease if it is the foil interaction is at play. The behavior at θ = −20 degrees canbe explained with the Y-force in figure 4.10 being a bit higher for the two downscale cases. Since theY-force is the only force acting in this position it has a great contribution to the power in figure 4.9b.A similar behavior can be seen at θ = −190, but this time the Y-force is a lot higher for the full-scalebut results in less power needed. This is because the Y-force is helping the RFP with the up-stroke.

The peaks at the θ = −180 degrees in figure 4.9 change significantly due to the downscale effect. Thethrust is in this case reduced when the RFP is scaled down and the power is slightly increased.

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CT

Full Scale

Down Scale 1

Down Scale 2

(a) Thrust coefficient for one of four foils.

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CP

Full Scale

Down Scale 1

Down Scale 2

(b) Power coefficient for one of four foils.

Figure 4.9: Power and thrust coefficients for one of four foils for comparing the full-scale anddownscale cases. At θ = −20 degrees the thrust coefficients are about the same. The powercoefficient at the same place is on the other hand increasing at θ = −180 degrees.



0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

-1.5

-1

-0.5

0

0.5

1

1.5

CY

Full Scale

Down Scale 1

Down Scale 2

Figure 4.10: The Y-force coefficient for one of four foils. The force coefficient is a lot higher for thefull-scale case at the θ = −210 degrees position. This is caused by the more powerful down-washin this case.

A part of the reason for the difference in performance between the full and down scaled cases are shownin figure 4.11. The boundary layer that is shown in blue is 3.5 times bigger in the down-scale 2 casethan in the full-scale case relatively to the chord length. This is due to the viscous forces being muchlarger in relation to the inertial in this case. Therefore the boundary layer is not suppressed as muchand can grow large.

In the downscale case figure 4.11b a recirculation zone marked in red appears. This is also causedby the slow flow and the boundary layer being large as discussed in the theory section about flowseparations and eddies. Durrani [14] found that the trailing edge separation zone behaves exactly likethis with the DDES turbulence model scheme, which uses a RANS model close to walls and a scaleresolving scheme in the free stream. Yarusevych and Sullivan [7] also found these differences when lowReynolds number flows were compared to higher Reynolds numbers. This confirms the hypothesis insection 2.6.1 that the separation zone is bigger in the down-scale case.

(a) Fullscale. (b) Down scale case 2.

Figure 4.11: The trailing edge of one foil at the -180 degree position. The wake on the downscalefoil is a lot larger and has a much larger recirculation zone (red area).



The transitional SST turbulence model that was used in the downscale cases predicts a laminar sepa-ration bubble in figure 4.12. This zone is marked in red and is occurring at the half-chord on the lowerface of the foil.

Figure 4.12: Local separation bubble occurring on the bottom face half chord of the foil. This ishappening at local Reynolds numbers between 100k and 200k with an angle of attack at about 8degrees. In this case the foil is performing a upstroke at θ = −200 degrees. The same occurrenceis happening at θ = −20 degrees.

The separation regions are evident from the pressure coefficient on the foil. The pressure on the lowpressure face at the θ = −20 degree position for the full-scale, down-scale 2 and XFOIL cases areplotted in figure 4.13. All curves have their peak at CP = −4. The full scale case shows a completelycontinuous and smooth pressure distribution with no separations. The downscale case and XFOILcases are showing some deviations compared to the full scale curve.

The XFOIL curve and downscale curves are not following the same pattern as the Aftab article [11].The separation of the XFOIL foil occurs at 0.05 x/c while the FLUENT simulation has two separationsat the 0.5 and 0.75 x/c marks. The divergent zones can be related back to figure 4.12 where the zoneis showing reverse flow in these regions.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x/c

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

CP

Full-scale

Down-scale 2

XFOIL

Figure 4.13: The pressure distribution for the low pressure side of a foil at the θ = −20 degreeposition.



4.2.1 Comparison to Analytical Models

With the RANS models flow interactions can be found that are not possible to do as detailed ana-lytically without creating a highly advanced model. There are some potential flow solutions that areavailable that Roesler uses. They use potential flow theory or vortex lattice methods to compute theinteractions. Might be something to think about for the analytical model of ABB CRC. It is stillnecessary to compare this wake model to CFD and indeed experiments to make sure that they ar all inagreement. So is it valuable to do CFD when we have this wake model? A bit more detail is given inCFD like the smaller turbulent energies coming from one blade to the next. 3D effects are not foundentirely ether with the wake model, like wing tip voracities and the impact on the downstream blades.So each model has it’s place and must be utilized in accordance to it’s strengths and weaknesses.

Comparing to the Roesler report [10]. It shows some resemblance in figure 10 a to the power and thrustplots with the two peaks being different. But not much of the actual behavior is the same since thesettings are different. Comparing the power to efficiency the values are quite similar. The experimentsin Roesler shows an efficiency of about 80% for a thrust coefficient of 0.25. Although this does notprove a huge much since so much is blurred out by all of the values included in the efficiency equation.

The real life tests must be run before anything really concrete can be said about the performance ofthe RFP. It is impossible to monitor the flow-field itself in the test so only the forces can be used.Since the strong correlation exists between the angle of attack and the forces, the equations can bereversed to find what angles are occurring. There is still the problem with the flow having so low power.Roesler made some comments about the scaling problem in the other report [10]. There it was saidthat the speed of the carrage was altered for the same rotational speed so that different advancementfactors were tested. The forces of their RFP were monitored and correlated well with the analyticallycalculated values. It was then concluded that Reynolds independence was found. Roeslers tests weredone for similar Reynolds numbers as in this report, so it seems plausible to choose a similar approachhere. It must be noted that the Roesler analytical model is much more advanced when it comes to foilinteractions than the analytical model used here.

Some time was spent on finding a way of improving the test set up as to increase the similaritiesbetween the down-scale and full-scale cases. Adjusting the rotational speed made the minimum andmean Reynolds number go up for the downscale but never came around the problem that the Reynoldsnumber is below 1M. The advancement factor changed when this was done so that change was testedon the upscale as well. The effect was that the efficiency was lowered significantly for the up-scale butnot so for the downscale. The difference then came down to 10%. The knobs to turn are the forwardvelocity that is coupled by the advancement factor, the medium that the RFP is spinning in and howlarge it is. To bring up the Re by an order of magnitude some kind of petroleum based liquid wouldhave to be used or the downscale would have to be increased in size.


5 | Conclusion

The differences between the downscale and the full-scale are great, eaven when the non dimensionalcoefficients are considered. The down-scale case will be less efficient, it will be difficulties predictingthe performance of the full-scale since the downscale Reynolds number is below one million. The flowseparations are not suppressed as much in the low Reynolds cases compared to the high, so separationscould be observed to be happening a large extent. Roesler [10] managed to find Reynolds independenceunder similar test circumstances. The same methods might be used in the future for this project.

The interaction between the blades has a large effect on the forces of the RFP. There is a strong relationbetween angle of attack and the number of blades. The forces that are large change by about 30% soit must definitely be considered if a model is to be used for a control system. It might even be worthinvesting in some more advanced vortex lattice model that Roesler [10] used.

47

6 | Perspectives

This thesis was only the beginning of the potential work that can be done with CFD for a radial flowpropeller. A three dimensional study is still to be done to get potentially more closely correlated resultswith the real test. When the test is completed the values calculated with CFD can be validated andthe uncertainties about which turbulence model is adequate or even if any are. If no correlation isfound, then a more detailed model would have to be used to find a better correlation. Then one wouldhave to as the question if it is worth putting all that effort in to analysing a down-scaled model whenit is the bigger model that is of interest. Since the flow situation is so much simpler in the full scalecase, it would be better to put more efforts in to this than on the down-scale model.

Only one set of parameters have been tested so there is a very large test matrix that could be done forall of the different situations that a vessel can encounter. Some better parameters might be found froma performance standpoint in the test. When a real world model is present it is easier to perform a largeset of data-points with that than with CFD because of it taking at least and entire working day toperform one test case. In the real life test one experiment could be run about once every hour becausethe water has to become still. CFD would excel over real life tests if geometries would be changed. Likeif the number of blades would change or the shape of them or the diameter of the propeller in relationto the other geometries. As it was seen in the literature study there are many different geometricalparameters to change to give different outcomes in performance.

The movement function is a simple equation that is somewhat arbitrarily constructed. This couldbe improved by attacking the problem from the other direction to optimize the movement patternfrom the optimum forward power. CFD would be of help with this because of all of the difficult flowinteractions occurring with such a coupled system that have been discussed in this report.

49

Bibliography

[1] H. K. Versteeg & W. Malalasekera, An Introduction to Computational Fluid Dynamics the FiniteVolume Method, Second Edition, Pearson Pretice Hall, 2007.

[2] H. Schlichting, Boundary-layer Theory, 6th edition, Mcgraw Hill, 1968.

[3] A. Silverstein, S. Katzoff & W.K. Bullivant, Downwash and Wake Behind Plain and Flapped Air-foils, National Advisory Committee for Aeonautics, Washington, D. C., USA, 1939, Report 651.

[4] N.L. Ficken & Mary C. Dickerson, Experimental Performance and Steering Characteristics of Cy-cloidal Propellers, Department of the Navy Naval Ship Research and Development Center, Wash-ington, D. C., USA, July 1969, Report 2983.

[5] J.A. Sparenberg & R. DeGraaf, On the optimum one-bladed cycloidal ship propeller, Journal ofEngineering Mathematics. Vol 3, No 1., p 1-20, 1969.

[6] M. Drela, H. Youngren, XFOIL 6.9 User Primer, 30 Nov 2001.

[7] S. Yarusevych, P.E. Sullivan,J.G. Kawall, Coherent structures in an airfoil boundary layer and wakeat low Reynolds numbers, PHYSICS OF FLUIDS 18, 044101, 2006.

[8] G. Mitchell, D. Mikkelson, Summary and recent results from the NASA advanced high-speed pro-peller research program, 18th Joint Propulsion Conference. 1982. p. 1119.

[9] B.T.Roesler, M.Fransciquez et al., Design and Analysis of Trochoidal Propulsors Using NonlinearProgramming Optimization Techniques, OMAE2014, San Francisco, California, USA, June 8-132014.

[10] B.T.Roesler, M.L.Kawamura et al., Experimantal Performance of a Novel Trochoidal Propeller,Journal of Ship Research, Vol. 60, No. 1, pp. 48-60, March 2016.

[11] S. M. A. Aftab, A. S. Mohed Rafie et al., Turbulence Model Selection for Low Reynolds NumberFlows, PLOS ONE, 2016.

[12] C. Xisto, J. Leger et al., Parametric Analysis of a Large-scale Cycloidal Rotor in Hovering Con-ditions, Journal of Aerospace Engineering,ISSN 0893-1321, 2016.

[13] R. Gopalkrishnan, M.S.Triantafyllou & D.Barrett, Active Vorticity Control in a Flow Using aFlapping Foil, Journal of Fluid Mechanics, Vol. 274, pp. 1-21, 1994.

[14] N. Durrani & N. Qin, Behavior of Detached-Eddy Simulations for Mild Airfoil Trailing-EdgeSeparation, JOURNAL OF AIRCRAFT, Vol. 48, pp. 193-202, 2011.

[15] J. O. Hinze, Turbulence, McGraw-Hill, 1975.

[16] P. R. Spalart Philosophies and fallacies in turbulence modelling, Progress in Aerospace Sciences,

51

BIBLIOGRAPHY

Vol. 74, pp. 1-15, 2015.

[17] F. Menter, Zonal two equation kw turbulence models for aerodynamic flows, 23rd fluid dynamics,plasmadynamics, and lasers conference. 1993. p. 2906.

[18] ANSYS. Inc, ANSYS Fluent User’s Manual, Southpointe 2600 ANSYS Drive Canonsburg, USA,PA 15317, Release 17.2, April 2015.

[19] ANSYS. Inc, ANSYS Fluent Theory Guide, Southpointe 2600 ANSYS Drive Canonsburg, USA,PA 15317, Release 17.2, April 2015.


A | Analytical Calculation

The analytical case is based around tabular data from the XFOIL program and from the equations forthe RFP by Roesler [9].

The XFOIL program cases are shown in table A.1. There was no convergence at the lower Reynoldsnumbers with high AoA because XFOIL can not model leading edge separation. According to [11]XFOIL over-predicts the lift for low Reynolds numbers so the results for the lower Reynolds numbersmust be viewed with a degree of scepticism.

Table A.1: The data set range from the XFOIL programme.

Re alpha rage50k -14:0.25:14100k -18:0.25:18200k -22:0.25:22500k -22:0.25:221M -22:0.25:225M -22:0.25:2210M -22:0.25:22

Equation 2.2 is for beta

β = −ω e2 + e sin θ

1 + 2e sin θ + e2(A.1)

vx and vy are the relative velocities a foil experiences during one revolution. The U and V terms arethe spatial incoming velocities. The ωR terms are the tangential rotational velocity for the centre ofthe foil. The VA term is the tangential rotational velocity of the leading edge.

vx = U − ωR sin θ − VA cos γ (A.2)

vy = V + ωR cos θ − VA sin γ (A.3)

VA = 0.25cβ (A.4)

VRel =√v2x + v2y (A.5)

1

APPENDIX A. ANALYTICAL CALCULATION

The velocity relative to one foil can then be calculated:

αrel = arctan

(vyvx

)(A.6)

And from that the angle of attack:

α = −(αre + γ) (A.7)

The force and moment coefficients can then be calculated from the angle of attack and the data fromXFOIL.

CT = Clsin(−αre)− Cd cos (−αre) (A.8)

CY = Cl cos (−αre) + Cd sin (−αre) (A.9)

CQ = (CY cosθ − CT sin th)R (A.10)

Equations A.8 and A.10 van be calculated numerically and be implemented in the efficiency equation2.14. The efficiensy is then 95.07 % for the full-scale case and 83.10 % for the down-scale case, this isa difference of 14.59%.


B | Computational Set-up Validation

B.1 Mesh Dependence

The mesh refinement was done by decreasing the global growth rate of the cells. Since the foils andtheir imidiet surrounding area is are the zones of interest, the growth rate decrease will achieve a finermesh at the right places with the least of effort. The cases are presented in table B.1. The number ofcells are expressed compared to the nominal mesh that has the number of cells N. For each refinementthe number of cells are doubled.

Table B.1: The difference between the nominal mesh and the two refined meshes. The values aremean values for one revolution.

Case Growth RateN 1.22N 1.14N 1.02

The quantitative results are presented in table B.2. The first refinement step has differences wellbeyond 1%. The second refinement step gives all values except the amplitudes below one percent.

Table B.2: The difference between the nominal mesh and the two refined meshes. The values aremean values for one revolution.

Difference [%]N to 2N 2N to 4N

Efficiency -0.04 0.05Ct Amplitude 16.47 0.34Cp Amplitude 3.54 0.40Ct 0.14 -0.03Cp -0.64 -0.07Cy 1.71 0.12CM 0.14 -0.07Cm -33.62 -0.01

1

APPENDIX B. COMPUTATIONAL SET-UP VALIDATION

The large change in amplitude form the nominal to the other cases can clearly be seen in figures B.1and B.2. There seems to be a macro period that is occurring that is occurring for the downscale case.The period is a bit longer than one revolution. The causes will be discussed in the results section.

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

0.225

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0.265

CT

N

2N

4N

Figure B.1: The thrust coefficients.

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

0.24

0.25

0.26

0.27

0.28

0.29

0.3

CP

N

2N

4N

Figure B.2: The power coefficients.



The most clear difference that can be seen on a contour plot is the velocity at the trailing edge on theupstroke This is shown in figure B.3 where it can be seen that the wake (blue) of the unrefined caseN is a lot shorter and jagged than the refined case (B.3b). The impact of the growth rate decrease isobvious when comparing the two mesh pictures B.3c and B.3d for this case.

(a) Velocity coefficient for N. (b) Velocity coefficient for 4N.

(c) The N mesh. (d) The 4N mesh.

Figure B.3: The trailing edge for the nominal and finest mesh for a foil at the -225 degree position.



The thrust and Y-forces for one foil in figures B.4 and B.5 shows much smaller differences than thecombined forces of all foils in figures B.1 and B.2. This signifies that two opposite blades has asignificant effect on each other.

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CT

N

2N

4N

Figure B.4: The thrust coefficients.

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

CY

N

2N

4N

Figure B.5: The y-force coefficients.

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

-0.1

-0.05

0

0.05

0.1

Cm

N

2N

4N

Figure B.6: The foil moment coefficients.

0 30 60 90 120 150 180 210 240 270 300 330 360

[degrees]

-0.5

0

0.5

1

1.5

2

CM

N

2N

4N

Figure B.7: The propeller centre moment coefficients.



B.2 Time Step Dependence

The cases for the time step dependence is shown in table B.3. These were chosen from the recommen-dations in [1].

Table B.3: Time step cases. (T is time for one revolution)

Case Time Step0.075c T/460BL T/2000TE T/8070

The results from the time step analysis are shown in table B.4. The first test that compares the halfthickest chord to the boundary layer hight shows that the differences are more than five percent on 5of the most important parameters listed. Parameters like the moment on the foil are expected to varyquite heavily because of the small values of that parameter. The most crucial value that is too largeis the y-force coefficient. This will have a large impact on the propeller since it is about as big as thethrust force and even bigger in some places. The values are still not within one percent difference. Thismay be attributed to the noise that is produced when using URANS models with small time-steps.Comparing the time-steps in B.3 to the ones in [12] where the time step is T/720. The time-step usedin this case is 3 and 10 times smaller than the ones used in [12]. Therefore the BL time-step is chosenas the most suitable

Table B.4: Time step analysis data

Difference [%]2.2e-3 to 8e-4 8e-4 to 2e-4

Efficiency 1.20 -0.28Ct Amplitude -5.31 2.89Cp Amplitude 10.73 7.22Ct 3.57 1.02Cp 3.10 0.98Cy 13.89 0.63CM 2.36 1.07Cm 10.85 2.17



0 50 100 150 200 250 300 350 400

degrees

0.21

0.22

0.23

0.24

0.25

0.26

0.27C

T2.2e-3

8e-4

2e-4

(a) Thrust coefficients.

0 50 100 150 200 250 300 350 400

degrees

0.24

0.25

0.26

0.27

0.28

0.29

0.3

0.31

CP

2.2e-3

8e-4

2e-4

(b) Power coefficients.

Figure B.8: The thrust coefficients to the left and drag to the right for the three time-step cases.The spike that gets amplified at the 110 degree mark is caused by the increased resolution of theseparation occurring on the up-stroke

B.3 Sample Time Length

The full scale simulation was run for 6 rotations to determine how many rotations are needed to achievea reoccurring pattern. It can be seen in figure B.9 that the values at the peaks does not vary by muchbetween each revolution. The largest difference is 0.012%. On the other hand there is a periodicityoccurring along one revolution. All of the blades have a different thrust along the orbit. Since thevalues at the tips of the curve are the same, that means that the periodicity has the same frequencyas the rotation of the RFP.

0 1 2 3 4 5 6

Revolutions

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.3

CT

X: 5.327Y: 0.2605

X: 4.328Y: 0.2605

X: 4.643Y: 0.228

X: 5.642Y: 0.228

X: 3.328Y: 0.2606

X: 2.328Y: 0.2607

X: 1.327Y: 0.2609

X: 3.643Y: 0.228

X: 2.643Y: 0.228

X: 1.643Y: 0.2282

X: 5.796Y: 0.2352

X: 4.796Y: 0.2352

X: 3.796Y: 0.2352

X: 2.796Y: 0.2353

X: 1.796Y: 0.2355

Figure B.9: The thrust coefficient for 6 revolutions. The extreme points does not change signifi-cantly after one revolution. The biggest difference is 0.02% from one peak to another.

In figure B.10 the mean value of the crucial parameters are plotted for one foil. Then the least amountof data smearing is happening for the mean value compared to if the values for all four blades would beanalysed. The foil moment Cm has the greatest difference of -23% that was omitted from the plot forclarity. The foil moment is also the last parameter to be within the 1% difference from the last value



after 5 revolutions. Therefore it can be said that the simulation must be run for at least 5 revolutionsbefore a repeatable pattern is reached.

2 3 4 5 6

Revolutions

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

% D

iffe

ren

ce

to

La

st

Re

v

CT

CP

CY

Cm

Figure B.10: The mean value of the crucial parameters for one foil. The foil moment Cm has thegreatest difference of -23%. This point was omitted from the plot for clarity.



B.4 Domain Size

The first size of the domain was chosen to be large in the nominal analysis so that only one extravalidation simulation is needed. The nominal domain for the test was the finer mesh retained fromthe mesh study with 455 366 cells. The larger domain case with all of the dimensions in figure 3.2except for the diameter of the RFP were doubled ended up with 516 717 cells. That is an increase of60 000 cells. This is a significant size increase of 10% more cells. But the increase is exponential, soif efforts were made to decrease the amount of cells would be below ten thousand which will start tobecome insignificant. The results of the study are shown in table B.5 where all parameters are belowone percent. The two domains are presented in figures B.11 and B.12.

Figure B.11: This is the nominal domain size. It is comprised of 455 366 cells.

Figure B.12: This is the double domain size. It is comprised of 516 717 cells.



Table B.5: Parameter difference between the two domains.

Difference [%]Efficiency 0.20Ct Amplitude 0.19Cp Amplitude -0.74Ct 0.93Cp 0.73Cy 0.33CM 0.81Cm -0.95


C | Transitional SST investigation

The Transitional SST turbulence model was tested for it’s performance against the normal SST. It hassome impact on performance. In figure C.1 it is seen that the thrust and power coefficients are shiftedslightly. This can be seen most clearly at the θ = −20 degrees position where the thrust coefficientis slightly higher and the power coefficient is even more increased. This is not seen in the thrust andpower coefficients in in table D.1 where the thrust coefficient have increased more than the powercoefficient. This is due to the increase of thrust at the θ = −20 degree for the Transition SST withoutmuch change for the other coefficients. These differences in coefficients are due to the increased amountof separation bubbles that are occurring for the transition SST, that can be seen in figure 4.12.

Table C.1: The difference between the normal SST and the transitional SST. Negative meansthat the coefficient have decreased when switching from SST to Transitional SST.

Difference [%]Efficiency 1.10Ct Amplitude -4.09Cp Amplitude -48.22Ct 5.24Cp 4.10Cy -4.73CM 3.98Cm 13.88

0 30 60 90 120 150 180 210 240 270 300 330 360

- [degrees]

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

CT,C

P

CT Transition SST

CT SST

CP

Transition SST

CP

SST

Figure C.1: The power and thrust coefficients for one foil.

In figure C.2 the Transitional SST’s values are behaving a lot more irregularly than the normal SST.

1

APPENDIX C. TRANSITIONAL SST INVESTIGATION

The Transitional SST ran for 6 revolutions and could be considered as macro stable. This may bedue to the irregularity of the laminar separations. Or it could be how the model decides where thetransition point will be and then because of all of the differing velocities this point will move and thenthe values may have been artificially destabilized. To have more confidence in using a model like theTransition SST more studies must be done or a good comparison to a real model must be done.

0 30 60 90 120 150 180 210 240 270 300 330 360

- [degrees]

0.016

0.017

0.018

0.019

0.02

0.021

0.022

0.023

0.024

CT,C

P

CT Transition SST

CT SST

CP

Transition SST

CP

SST

Figure C.2: The power and thrust coefficients for four foils combined.


D | Raw Data

D.1 Full-scale angle of attack

Table D.1: The difference between the normal SST and the transitional SST. Negative meansthat the coefficient have decreased when switching from SST to Transitional SST.

Difference [%]Efficiency 1.10Ct Amplitude -4.09Cp Amplitude -48.22Ct 5.24Cp 4.10Cy -4.73CM 3.98Cm 13.88

1

E | User Defined Functions

E.1 Planetary disc rotation

/**********************************************//* Disc1Rot.c *//* UDF for specifying rotational velocity of *//* planetary disc *//* *//* *//**********************************************/#include "udf.h"

#define omegR /*The angular velocity of the planet disc*/#define R /*The radious of the planet disc*/

DEFINE_CG_MOTION(D1Rot, dt, vel, omega, time, dtime){omega[2] = omegR;return;}

E.2 Satellite disc rotation and data extraction

/**********************************************//* CgMonitorFoil1.c *//* UDF for specifying time varying rotational *//* velocity of Satellite disc 1 and gathering *//* torque data *//* *//**********************************************/#include "udf.h"enum coord {X,Y,Z};

static real moment_centre[ND_3]={0.0,0.0,0.0}; /* Static global vector */static int dyn_thread_id = -1;

#define WALL_ID 108#define omegR /*The angular velocity of the planet disc*/#define R /*The radious of the planet disc*/

1

APPENDIX E. USER DEFINED FUNCTIONS

DEFINE_CG_MOTION(disc_2_motion, dt, cg_vel, cg_omega,time, dtime){double e;

NV_S(cg_vel, =, 0.0);NV_S(cg_omega, =, 0.0);cg_omega[Z] = -omegR*(e*e+e*sin(omegR*time))/(1+ 2*e*sin(omegR*time)+e*e);

NV_V(moment_centre,=,DT_CG(dt)); /* Simple way to set vectors */

if (dyn_thread_id < 0)dyn_thread_id = THREAD_ID(DT_THREAD(dt));}

DEFINE_ON_DEMAND(calculate_forces_and_moments_foil1){Dynamic_Thread *dyn_thread;Domain *domain;Thread *t;real force[ND_3], moment[ND_3];int wall_id;

domain = Get_Domain (ROOT_DOMAIN_ID); /* For multiphase flow, youneed to set the Sub domain */wall_id = WALL_ID; /* Could be set via an RP variable her */

t = Lookup_Thread(domain, wall_id);

#if RP_3DMessage("Calculating forces and moments about (%e %e %e)/n",moment_centre[0],moment_centre[1],moment_centre[2]);#elseMessage("Calculating forces and moments about (%e %e)/n",moment_centre[0],moment_centre[1]);#endif

dyn_thread = THREAD_DT(Lookup_Thread(domain, dyn_thread_id));

if(NNULLP(dyn_thread))NV_V(moment_centre,=,DT_CG(dyn_thread));

#if RP_3DMessage("Current centre (%e %e %e)/n",moment_centre[0],moment_centre[1],moment_centre[2]);#elseMessage("Current centre (%e %e)/n",moment_centre[0],moment_centre[1]);#endif

Compute_Force_And_Moment(domain, t, moment_centre, force, moment, TRUE);

#if RP_2D


APPENDIX E. USER DEFINED FUNCTIONS

force[2] = 0.0;moment[0] = 0.0;moment[1] = 0.0;#endif

#if !RP_NODE{FILE *fp;

fp = fopen("MomentGCfoil1.txt","a");fprintf(fp, "%e %e %e %e %e %e/n", force[X], force[Y], force[Z], moment[X], moment[Y], moment[Z]);fclose(fp);}#endif}

DEFINE_EXECUTE_AT_END(calculate_forces_and_moments_at_end_foil1){calculate_forces_and_moments_foil1();}


FRED

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/LIU

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–17/

0289

1—SE


Division of Applied Thermodynamics and Fluid Mechanics Thesis Work, 30 hp, 2017| LIU-IEI-TEK-A–17/02891—SE

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