Tip vortex cavitation and diffused vorticity of propeller ...

Tip vortex cavitation and diffusedvorticity of propeller profiles: amodelling approachInvestigation of an implemented TVI model, and implementation and

investigation of a DVHmodel

Undersökning av en implementerad TVI modell, och implementering och

undersökning av en DVH-modell.

Lukas Lundin

Faculty of Health, Science, and Technology

Master thesis

30 hp (ECTS)

Supervisor: Adrian Muntean (KAU), Björn Schröder (Rolls-Royce)

Examiner: Lars Johansson

Date: 30 oktober 2017

Serial number N/a

Abstract

To predict fluid properties and interactions is an important task for the industry. Itis plagued, however, by being close to impossible to predict analytically. Hence, it is cus-tomary to turn to numerical solutions. This in itself comes with many different methodsand approaches suitable for different needs. This work focuses on two methods: Tip VortexIndex (TVI) and Diffused Vortex Hydrodynamics (DVH). TVI is a method to predict whena marine propeller will experience cavitation of tip vortices and is based on calculationsfrom a Boundary Element Method (BEM). DVH is a particle method for simulating thecirculation of a fluid in two dimensions and three dimensions. The aim is to investigatean implemented TVI model based on MPUF-3A for different marine propeller series, withdifferent sub-designs for a total of 28 unique propellers, and implement the DVH methodand test it for 3 different bodies. The results of this thesis show that the implemented TVImodel is non-functional for the 28 different propellers, but the DVH method is successfullyimplemented and able to handle 2 different bodies.

Keywords: Vortex dynamics, Cavitation, TVI, DVH, Von Kármán Vortex StreetPACS: 47.32.C-, 47.55.dp,MSC 2010: 76D17, 76M23, 76B10

Sammanfattning

Att förutspå fluid egenskaper och interaktioner är en viktig uppgift för industrin. Detplågas dock av att vara näst intil omöjligt att förutspå analytiskt. Det är därför vanligt attvända sig till numeriska lösningar. Detta kommer i sig med många olika metoder och till-vägagångssätt som passar olika behov. Detta arbete fokuserar på två metoder: Tip VortexIndex (TVI) och Diffused Vortex Hydrodynamics (DVH). TVI är en metod för att förut-säga när en marin propeller kommer att uppleva kavitation av spetsvirvlar och baseras påberäkningar från en Boundary Element Method (BEM). DVH är en partikelmetod för attsimulera cirkulationen i fluid i två dimensioner och tre dimensioner. Syftet är att undersökaen implementerad TVI-modell baserad på MPUF-3A för olika marina propellerserier, medolika underdesigner, för totalt 28 unika propellrar, och implementera DVH-metoden ochtesta den för 3 olika kroppar. Resultaten av denna avhandling visar att den implemente-rade TVI-modellen är icke-funktionell för de 28 olika propellrarna, men DVH-metoden ärframgångsrikt implementerad och kan hantera 2 olika kroppar

Nyckelord: Virveldynamik, Cavitering, TVI, DVH, Von Kármán virvelgata

Acknowledgements

While there are many and big thanks that could go out to all those that have helped me getto this point, I will try to keep it short. I would like to thank Rolls-Royce for providing mewith the opportunity to conduct this thesis in conjunction with them. I would also like tothank my supervisor from Rolls-Royce, Björn Schröder, for giving me free reigns concerning mywork. Furthermore, a special thanks goes out to my supervisor at Karlstad University, AdrianMuntean, for his constant insight and ferocious optimism. Lastly, a special thanks goes out tomy fellow students that resided in 21E200-corridor at Karlstad University for providing supportand company throughout my education as well as outside of it.

Nomenclature

Abbreviations

BEM Boundary Element Method

FMM Fast Multipole Method

fs Full scale

ms Model scale

PS Pressure side

RPD Regular Point Distribution

RPM Rotations Per Minute

SS Suction side

VLM Vortex Lattice Method

Quantities

ω Vorticity vector

τ Tangential vector

g Gravitational acceleration

n Normal vector

u Fluid velocity

η Efficiency

Γ Circulation

γ Circulation contribution

µ Dynamic viscocity

ν Kinematic viscosity

ϕ Scalar potential

ρ Density

σ Cavitation number (Propeller theory), Source and Sink (DVH)

θP Pitch angle

θS Skew angle

θSP Propeller skew angle

ξ Error estimate

D Propeller diameter

Fn Froude number

J Dimensionless speed

K Dimensionless coefficient

L Length

n Rotational number

P Pitch

p Pressure

PD Power

Q Torque

T Thrust

t Time

V Velocity

w Wake factor

x Distance

Co Courant number

Re Reynolds number

Symbols

∆ Laplace operator

∇ Gradient

· Scalar productDDt Material derivative

× Cross product

Contents

1 Introduction 1

2 Theory 22.1 Propeller Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Camber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.3 Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.4 Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.5 Rake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.6 Wake Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Fluid Dynamics: A Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Fluids Interaction with Solids . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 Navier-Stokes Equation and Vorticity Equation . . . . . . . . . . . . . . . 52.2.4 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Propeller Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Dimensionless Propeller Quantities . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.3 Operation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.4 Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.5 Vortex Lattice Method (VLM) . . . . . . . . . . . . . . . . . . . . . . . . 102.3.6 Boundary Element Method (BEM) . . . . . . . . . . . . . . . . . . . . . . 112.3.7 Model Scale versus Full Scale . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Computional Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 Essential Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 MPUF-3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.3 XFOIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.4 Tip Vortex Index (TVI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.5 Regular Point Distribution (RPD) . . . . . . . . . . . . . . . . . . . . . . 142.4.6 Diffused Vortex Hydrodynamics (DVH) . . . . . . . . . . . . . . . . . . . 15

2.5 Things Not Included . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Specific Goal and Approach 20

4 Results and Discussion 204.1 TVI Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Blade Series 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1.2 Blade Series 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1.3 Blade Series 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.4 Blade Series 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.5 Blade Series 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.6 Parameter Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.7 TVI Discussion Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 DVH Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.1 RPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.2 RPD Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 DVH Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.4 DVH summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Conclusion 53

6 Future work 546.1 TVI Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 DVH Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7 References 55

A Appendix 57

List of Figures 57

1 Introduction

Predicting fundamental properties of propeller blades is essential for the process of constructingpropellers for ships. As the demands for performance, efficiency, hull interaction, cavitation, etc.vary vastly depending on the ship in question, there is no standard propeller that can be assigned.For this reason it is of interest to develop continuously new methods to anticipate the behaviourof marine propellers. This usually belongs to the field of Computational Fluid Dynamics (CFD).

The project took place both at Karlstad University and at Rolls-Royce Hydrodynamic Re-search Centre and we therefore benefited from a multidisciplinary collaboration. The first taskof the thesis was to evaluate a method used to predict cavitation inception speed for a propeller.The second task was to implement and evaluate a vortex particle simulation without remeshing.Both these tasks target at the understanding of vortex formation induced by the propeller motion.

Cavitation is a phenomenon where the water vaporises inside a fluid and then collapses. Thereare several ways cavitation can happen due to a propeller and there are also several reasons whyit is preferable to avoid it. Typically, for industrial propellers, tip vortex cavitation occurs firstand thus for cavity inception it is what is most interesting to model. This is the reason why westudy here a Tip Vortex Index (TVI) method.

Simulation of flows around bodies is a fundamental cornerstone in predicting properties for pro-pellers. Hence, it is of general interest to evaluate constantly new methods. Thus for the task ofimplementing and evaluating a vortex particle simulation without remeshing, the chosen modelwas Diffused Vortex Hydrodynamics (DVH). It is worth noting that DVH is a two dimensionalmethod and is primarily aimed to evaluate the circulation generated by bodies. Thus it may beless suited for immediate industrial use. But since it is a new method which has strong potentialto be developed further to cope with complex three dimensional structures, it is of interest toinvestigate from both academic and industrial points of view.

This report will present enough material that even a reader that is inexperienced in the fieldof fluid dynamics will grasp the methods, results, and conclusions based on the theory presentedhere along with the references. But a reader with experience in fluid dynamics may simply justturn to the primary sources for the methods and go directly to the results (section 2). Ourprimary resources are then for the TVI method [1], and for the DVH method [2, 3].

Our main contributions in this thesis are the cavitation plots and predictions for the TVI method,and the plots generated by the implemented DVH method. These contributions are presented insection 4.

The layout of this thesis is as follows.Section 2 is the theory section. It brings up relevant information from both fluid dynamics

and more detailed, propeller theory. Following that the theories for the specific models for thiswork are presented, the TVI and DVH, as well as the Regular Point Distribution (RPD).

Section 3 contains Specific Goal and Approach. This section entails what this thesis aims toaccomplish along with what methods that are used to reach the goals..

Section 4 contains our Results and Discussions. Here the primary results from the TVIsimulations and DVH simulations are presented along with the respective discussion, we includethere also a more general discussion of the methodology.

Section 5 is the Conclusion section, where the outcome of the work is briefly summarised.We close with section 6, the Future Work section, where the outlook for the current TVI and

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DVH models is presented.

2 Theory

To cover all the relevant information and developments for propeller theory and fluid simulationsis far beyond the scope of this thesis. Instead focus will be on providing an adequate descriptionto understand over-arcing results, and providing sources for further reading for the interestedreader. The following sections cover some basic concepts and formulations for propellers and fluiddynamics, before covering a computational fluid dynamics approach for the models of interest,Tip Vortex Index (TVI) and Diffused Vortex Hydrodynamics (DVH).

2.1 Propeller GeometryIt is of interest for the reader to be familiar with the different definitions for propeller geometryto fully appreciate the coming theory. The basic propeller consists of a hub (sometimes referredto as boss) which is typically of cylindrical form, on which the propeller blades are seated. Thepropeller blades are either in one continues piece with the hub (monobloc propeller), or the bladesare bolted on the hub. The propeller blades themselves can vary widely in shape, depending onwhat operational conditions they are designed for. As an example of a propeller, see Figure 1.

Figure 1: Image of a full scale propeller. Published with permission from Rolls-Royce.

However, there exist geometric definitions that are used to describe the blades. While theyare not always defined the same way, the gist of them remains the same. The coming section isbased on [4, 5].

2

2.1.1 Chord

If one were to slice a propeller orthogonal to the radial direction, one would get a section of thepropeller blade which looks similar to an aerofoil. The straight distance from the front mostpoint (leading edge) to the back most point (trailing edge) of the aerofoil is called the chord lineand is denoted c. See Figure 2 for an illustration.

Chord length (c)

Camber

Figure 2: Sketch of a typical aerofoil, where chord and camber are illustrated.

2.1.2 Camber

The camber line is the line that follows the middle of the geometric thickness of the aerofoil. Themaximum distance between the camber line and the chord line, measured orthogonally to thechord line, is called the camber, sometimes denoted as fM . See Figure 2 for an illustration.

2.1.3 Pitch

Pitch is a measurement on how the propeller blades are angled against the hub. It is easiest toexplain it in terms of an already known term, which is screw thread. Imagine that the propelleris lying half in mud, or that it is cutting through a cylinder. The distance that propeller travelsas it completes a full rotation (360◦) would then be called pitch, often denoted P . It can also beof interest to know the actual angle, which then is

θP = arctan(

P

πD

), (1)

where D is the diameter of the blade in metres. This technically describes the pitch of the tip ofthe propeller. For a radius 0 < r < R, with R = D/2, one has

θP (r) = arctan(

P

π(r/R)D

). (2)

It is also not uncommon to talk about a dimensionless pitch which is P/D. The definition ofpitch presented here is simplified, but is sufficient for this thesis.

2.1.4 Skew

Skew is a measurement of how the propeller sweeps. This can be defined in two different ways.Propeller skew angle, θSP , is determined as the angle between two lines that are drawn from thehub centre to the rightmost and leftmost points of the shaft centre line. The skew angle, θS(x),is a radial dependent concept. It is the angle between a line drawn from the hub to the shaftcentre line set at the seating of the hub and to the line made from the hub to the shaft centreline. See Figure 3 for illustrations of these definitions.

3

y

z

r

φS(x)

x = r/R

θSP

Figure 3: Sketch of a propeller blade seated on the hub. The illustrated angles are the skewangle (θS(x)) and propeller skew angle (θSP ).

It is also possible to look at skew as a distance instead of an angle. One does this bymultiplying the angle with the radius, i.e. θSP (x)r.

2.1.5 Rake

Rake is a bit more technical than skew, and is in fact not quite separated from it. Rake can beseen as a measurement for how much the blade sections are offset from a plane through the hub.This can then be for a specific radius or as a total. It can be broken down into two separatepieces, generator-induced rake and skew-induced rake. A more specific definition can be foundin [4].

2.1.6 Wake Field

Wake field is not actually a geometric property of the propeller, but it is still relevant to introduceit. It describes how the flow of a fluid is changed after an object has travelled through it. Thisis relevant because propellers tend to sit on the rear end of the hull of a ship. This means thatthe actual flow experienced by the propeller will be different from the flow experienced by theship. Thus one tends to differentiate between the ship’s speed, which as it sounds is the speed atwhich the ship travels, and the speed of advance which is the speed experienced by the propellerin terms of the inflow of the fluid.

2.2 Fluid Dynamics: A Brief IntroductionWhile fluid dynamics is a field that is enormous in both depth and width, it is not necessary tounderstand all of it to appreciate this work. Hence, the following section introduces the mostrelevant concepts for this work.

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2.2.1 Reynolds Number

Reynolds number is an important dimensionless quantity in fluid dynamics. In essence, it cap-tures the relationship between inertial forces and viscous forces, and is defined as

Re = uL

ν. (3)

In equation (3) u is the fluids speed, L is the characteristic length, and ν is the kinematic viscositywhich is the ratio of the dynamic viscosity and the density, i.e. ν = µ

ρ . It is used, for instance,to quickly determine whether the flow is turbulent (higher Reynolds number) or laminar (lowerReynold’s number). For further reading, see [6].

2.2.2 Fluids Interaction with Solids

A common boundary condition assumed is that when a fluid is in contact with a solid, the velocityof that part of the fluid relative to the solid is zero. This is called the no-slip condition, i.e.

u · τ = 0, (4)

where τ is the tangential vector. It is worth noting that this boundary condition typically alreadyassumes the impermeability condition (at least in the discussion of solid bodies), i.e.

u · n = 0, (5)

where n is the normal vector. At times this is called the free-slip condition, as this alone allowsfor the fluid to travel along the bodies surface without restriction.

From no-slip and impermeability conditions follows the boundary layer thickness, which is thedistance from the body where the fluid is considered to have regained free-stream velocity.

Other bodily interactions includes drag and lift. Drag is then the resistance experienced bythe body as it travels through the fluid. Lift is the force experienced orthogonal to the drag.

For more information see e.g. [3, 4, 6].

2.2.3 Navier-Stokes Equation and Vorticity Equation

Navier-Stokes equation for incompressible and isothermal fluids is found to be (see e.g. [6])

∂u

∂t+ (u · ∇)u = −1

ρ∇p + g + ν∇2u, (6)

where g is the gravitational acceleration, and p is the pressure. Navier-Stokes allows for evaluationof the fluid velocity, but can be unwieldy at times. One possible approach is then to turn to thevorticity equation. The vorticity is defined as the rotation of the flow velocity, i.e.

ω = ∇ × u. (7)

Thus if one knows the flow, one can know the vorticity. However, it may be that it is beneficialto look more directly at the vorticity, i.e. solve immediately for the vorticity. One can derive the

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vorticity equation from Navier-Stokes equation. This is done by applying ∇× from the left side,and through identifying vector calculus identities, one will eventually end up with the expression

∂ω

∂t+ (u · ∇)ω = (ω · ∇)u + ∇ × g + ν∇2ω. (8)

For two dimensional flows, the first term on the right side is zero. For an in depth description,we refer the reader to [7, 8].

2.2.4 Circulation

Circulation is a measurement of the velocity of a fluid in a closed loop, or the sum of vorticityin an area. Circulation can thus be calculated as

Γ =∮

∂S

u · dl, (9)

where ∂S is the boundary of the surface S which the circulation is calculated around, and dl isa infinitesimal length element along said boundary. It follows then, from Stokes theorem, thatthe circulation can also be calculated as

Γ =∫∫

S

ω · dS. (10)

Noticeable connected to circulation is Kelvin’s Circulation Theorem (see e.g. [4, 7]), which statesthat the circulation around a closed curve in an inviscid fluid remains constant, i.e.

DΓDt

= 0, (11)

where DDt is the material derivative.

2.3 Propeller TheoryThe theory of how a propeller works has been in development for many years. It can be arguedthat it is still far from being understood in full detail. In this section we briefly mention somehistorical background and we also give a glimpse on more modern theories.

2.3.1 Dimensionless Propeller Quantities

It can be convenient to work with dimensionless quantities as it makes more sense when present-ing several quantities at the time and not to mention to draw parallels between phenomena fordifferent propellers. As such, the following section presents some of the commonly used dimen-sionless quantities.

Dimensionless speed JA (sometimes called advance coefficient) is defined as

JA = VA

nD, (12)

where VA is the speed of advance experienced by the propeller in metres per second, and n isthe rotations per seconds of the propeller.

6

Dimensionless speed JS is defined asJS = VS

nD, (13)

where VS is the speed of advance of the ship in metres per second. The relationship between JA

and JS isJA = JS(1 − w), (14)

where w is a dimensionless wake dependent factor.

Dimensionless torque KQ is defined as

KQ = Q

ρn2D5 , (15)

where Q is the torque in newton metres, and ρ is the density of the fluid in kilograms per cubicmetres.

Dimensionless thrust KT is defined as

KT = T

ρn2D4 , (16)

where T is the thrust exerted on the blade in newtons.

Propeller efficiency in open water ηo is defined as

ηo = TVS

PD. (17)

where PD is the power being delivered to the propeller, and the product TVS is the power neededto drive the ship at that speed.

Cavitation number σ is defined asσ = p − pv

12 ρV 2

A

, (18)

where p is the pressure in undisturbed fluid, while pv is the vapour pressure of the fluid. It ispossible to come across this number but with an extra factor 2. It is also possible to define thecavitation based on propeller rotational speed, which is

σ = p − pv12 ρn2D2 , (19)

though this is not the definition used in this work. The cavitation number is the ratio betweenstatic pressure head and dynamic pressure head, and characterizes the cativation potential of afluid.

Froude number Fn is defined asFn = VS√

gLLW, (20)

where g is the gravitational acceleration in metres per second squared, and LLW is the length ofthe ship at the waterline. The Froude number is like a non-dimensional speed, which should bekept below a certain limit depending on hull shape to avoid excessive resistance.

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2.3.2 Cavitation

In the context of propellers, cavitation is the phenomenon when parts of a liquid experiences lowenough pressure to become vapour, while still being inside the fluid. i.e. it creates a cavity inthe liquid filled with vapour. This is not to be confused with the phenomenon of dragging downair-bubbles into the vicinity of the propeller, though this still plays partly a role in cavitation.The exact point when cavitation starts is not only pressure dependent, but also relies on thequality of the water, i.e. nuclei content. Also, when running cavity tests, it is up to the observerto decide whether cavitation has started or not. Depending on the propeller geometry, differentparts of the blade will start to cavitate at different points. The different types of cavitation arebubble, sheet, cloud, hub vortex, and tip vortex cavitation. There are in essence three reasonswhy one would like to avoid cavitation:

• Cavitation can damage the propeller and the hull of the ship,

• Cavitation Requires energy to produce and as such lowers the efficiency of the ship,

• Cavitation can be extremely noisy.

Since tip vortex cavitation typically is the first to occur it is also usually the most interestingeffect to investigate, in terms of inception. For further reading, we refer the reader to [4]. Figure4 features a propeller where the tip vortices cavitate, which are seen as the bubbles emanatingfrom the tip of the propeller blades and which trails behind the propeller. Also visible is the toppropeller blade which experiences sheet cavitation as well as tip vortex cavitation.

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Figure 4: An image featuring a propeller which has tip vortex cavitation. This is seen as thefilament that stretches from the propeller blade tips backwards into the wake. Also seen is slightsheet cavitation on the topmost propeller blade. Published with permission from Rolls-Royce.

Tip vortices are caused by blade tip loading, and depending on how it is shaped this loadingcan then be from the suction side of the propeller or the pressure side. This then leads to the factthat one can separate tip vortex cavitation into two different classes suction side and pressure side.For suction side tip vortices the cavitation number decreases with increasing dimensionless speed,while for pressure side tip vortices the cavitation number increases with increasing dimensionlessspeed. Thus there will be an intersection between them and it will resemble a bucket, and istherefore sometimes referred to as the cavitation bucket. Generally, the aim is then to havethe operation curve in the middle of this bucket to avoid premature cavitation, where operationcurve describes then the actual dimensionless speed and associated cavitation number for a givenship. An example of this can be seen in Figure 5. For further reading, we refer the reader to [9].

9

σ(C

avitationnumber)

Js (Dimensionless speed)

Operation curve

Suction sidePressure side

Figure 5: Illustration of how a typical cavitation bucket may appear.

2.3.3 Operation Conditions

For a set blade geometry, there will be a given efficiency and ship speed for a set effect deliveredto the propeller, such that the relationship between JS , JA, KQ, KT , ηo, σ, Fn is fixed. This isthen called design condition or, when plotted against each other, operation curve. Much can besaid on this subject, but will not be brought up here. An interested reader can find more in e.g.[5].

2.3.4 Momentum Theory

One of the earliest theories that was developed to explain how a propeller works is the momentumtheory. It relies on three assumptions :

(i) The propeller experiences no frictional drag, i.e. it works in an ideal fluid and thus losesno energy from friction.

(ii) The propeller can be replaced by an infinitely thin disc.

(iii) The propeller does not cause rotation of the flow when it generates thrust.

Assumptions (i)-(iii) want to suggest a model where the propeller shape does not matter. Lateron, the third assumption was finally removed. While this model is rudimentary, it still predictstwo important things concerning propellers. Firstly, the acceleration of the flow takes even partbefore and after the propeller. Secondly, there is an upper bound to the efficiency of a propeller.This was, however, far from enough to describe how a propeller works and there have been manydevelopments and many theories after momentum theory. Two prominent theories are VortexLattice Method (VLM) and Boundary Element Method (BEM). See [4] for details.

2.3.5 Vortex Lattice Method (VLM)

Vortex Lattice Method (VLM) is derived from lifting surface methods, which in essence arepotential methods, i.e. one solves ∆ϕ = 0, where ϕ is the scalar potential for a conservative

10

vector field. In lifting surface methods, the blade is reduced to a surface which is representedby the chord lines. On this surface then one places vortices and sources to represent the fullblade. In VLM, these vortices and sources are instead replaced by sets of straight lines withconstant values, where only the end points are required to lay on the chord surface. The sourcesare effectively known as they are independent of time, and the distribution is known from thestart. The vortex distribution, however, not known and have to be solved for, with regard to theboundary condition for the flow. The wake is then described as the blade with lines describingthe vortices (see e.g. [10]).

2.3.6 Boundary Element Method (BEM)

Boundary Element Method (BEM) is basically a panel method (which also is a potential method),and attempts to overcome errors that occur, in lifting surface models, where the blades are moreclosely packed near the hub. The blade is modelled with quadrilateral panels on the surface, whereeach panel has a constant source distribution. The vorticity, however, is allowed to vary over thesepanels and must be continuous over the edges. The wake is represented with similar quadrilateralpanes with assigned vorticity. Essentially, BEM solves numerically simplified equations of fluidmechanics formulated as integral equations.

2.3.7 Model Scale versus Full Scale

Cavitation tests are done on model propellers, which usually are significantly smaller than thefull scale propellers. This means that the cavitation phenomenon will be different as well. Asmentioned in Section 2.3.2, for cavitation inception, the tip vortex cavitation is of most interest.Hence, only the scaling for that will be brought up. The relationship for cavitation numberbetween full scale and model scale is

σfs

σms=

(Refs

Rems

)m

, (21)

where m is the McCormick factor, which typically spans between 0.25 to 0.40. Usually the highervalues of m are favoured based on experience (See [4]), though here are some that have a morerigorous motivation (see [11]).

2.4 Computional Fluid DynamicsTo preform realistic analyses of propellers, one thus turns to computers. To solve numericallythe partial differential equations of fluid mechanics, there are many different methods that aremeant to capture different aspects or work under certain time-scales and dimensions.

2.4.1 Essential Concepts

Just as for the introduction of fluid dynamic theory, there are some concepts that one shouldbe aware of when one approaches the numerical treatment of it. Below is an overview of therelevant ones.

Courant number Co in one dimension is defined as

Co = u∆t

∆x, (22)

where ∆t is the time step in seconds, and ∆x is the spatial step or spatial resolution in metres.The Courant number is a way of measuring how fast information or particles travel through a

11

simulation. If the Courant number is higher than one, it means that the information or particlesare travelling distances greater than the spatial resolution during a time step. For some methods(e.g. explicit Euler methods as seen in [12]) this can have disastrous effects on the accuracy ofthe simulation, see, for instance, [13].

Numerical solution for spatial positioning is a problem that come up in all manner of simula-tions where one predicts position based on velocity, i.e.

dr(t)dt

= u(r, t), r(t0) = r0. (23)

There are several possible solutions to this issue, and among the most popular is the Runge-Kuttamethods. These methods can either be explicit or implicit, and vary in numerical complexity.First order explicit Runge-Kutta is also called Euler method, and is the simplest solution toequation (23), and is what will be used in this paper. However, one of the more popular methodsis the fourth order explicit Runge-Kutta. A detailed description of the Runge-Kutta methodscan be found in, e.g. [14].

2.4.2 MPUF-3A

PUF-3 is a program that is based on VLM that is used to calculate the flow through propellers.Over time it was coupled with BEM, and thus renamed MPUF-3A. This was the tool that laidthe foundations for the investigation of tip vortex cavity inception. A detailed explanation onhow MPUF-3A works can be found in [15].

2.4.3 XFOIL

XFOIL is a two-dimensional flow solver that is used for aerofoils. It can work either as a directsolver or as an inverse solver. As a direct solver it is given a aerofoil and calculates the set data(e.g. draft coefficients, lift coefficients, etc), while as an inverse solver it takes these wanted dataparameters as an input and tries to generate an aerofoil which fulfils them. See [16, 17] for moreinformation.

2.4.4 Tip Vortex Index (TVI)

The Tip Vortex Index is a method that is used to predict at which speed a tip vortex startsto cavitate. It is based on a potential method, specifically on a foundation via the boundaryelement method (BEM). One regards the wake behind the propeller, which is modelled as atwo-dimensional sheet, and how it rolls-up. Then the pressure distribution and vaporisation ofthe tip vortex core is evaluated. This is all caused by vorticity leaving the trailing edge of thepropeller blade as it passes through the fluid. The theory is based on [1, 15, 18], e.g. The ideais to model the vortex after the wake. One way is to follow the Rankine vortex model, wherethe inner core of the vortex is treated as a rigid body. However, this predicts a faulty pressuredrop in the vortex. For that reason, one may instead use the Hamel-Oseen vortex model, whichoriginates from the Navier-Stokes equations. The pressure contribution of the vortex is describedas

pvortex(r) = p∞ − cpρu2

τ

2(r > 0), (24)

12

where r is the radius measured outwards from the vortex centre, p∞ is the freestream pressure,cp is the pressure coefficient, and uτ is the tangential velocity of the fluid in relation to the vortexcentre. For the Hamel-Oseen model, the pressure coefficient is defined as

cp(r) =(

1 − e− r2

R2c

)R2

c

r2 + 2(

Ei(

− r2

R2c

)− Ei

(−2 r2

R2c

)), (25)

where Rc is the radius of the vortex core, and Ei is the function defined as

Ei(r) = −∫ ∞

−r

e−x

xdx. (26)

The tangential velocity, as described in Hamel-Oseen model, is defined as

uτ (r) =(

1 − e− r2

R2c

)Γ

2πr. (27)

The vortex core radius Rc is approximated to be about 85% of the boundary layer thickness,which is the sum of the boundary layer thickness of the suction side and of the pressure side ofthe propeller. There are different ways to approximate this quantity. One way is

δ = 0.154Re− 17 c. (28)

The size of the vortex core will also grow with time as it travels behind the propeller, such that

Rc(t) =√

Rc(0)2 + 4νt, (t > 0). (29)

The cavity inception is when the centre of the vortex core reaches vaporization pressure, suchthat with equation (18), and pvortex(0) + p∞ = pv. Consequently, one find here

σi = −pvortex(0)ρ2 V 2

∞. (30)

However, due to equations (24) and (27) this expression goes to p∞/ ρ2 V 2

∞. Instead, one expressesthe vortex pressure contribution for the Rankine-Vortex model as done in [19], which is

pvortex(r) = p∞ − cp(r)ρ

2Γ2

4π2R2c

. (31)

Thus equation (30) becomes

σi = − p∞ρ2 V 2

∞+

cp(0) ρ2

Γ2

4π2R2c

ρ2 V 2

∞. (32)

While this may seem straightforward enough, the numerical evaluation of the wake is difficult torealize accurately from which the vortex radius Rc and circulation Γ are taken, as the vorticesleaving the tip is in the form of a helix that grows in size as it trails behind the propellers. Onethen turns to the wake panels generated from the BEM method. The centre of the vortex ismodelled to be at the tip of the outermost panel. From there one looks to the curvature of thepanels, and if it is to low, then the panels are not considered to be part of the vortex. Similarlyone can see how the distance from the panels to the vortex core is increasing, and if a panelsdistance has increased more than a certain factor compared to the two previous panels, then itwill not contribute to the vortex. A detailed overview of how the wake panels are evaluated canbe found in e.g. [1, 15, 18].

13

2.4.5 Regular Point Distribution (RPD)

In simulations where interactions are proportional to distances between grid and body points,the regularity1 of the points can affect the simulation results. Hence it is of interest to have away to build a grid in which all points are equidistant in relationship to each other and presentbodies. This grid construction can be achieved by introducing a method used in Smooth-ParticleHydrodynamics (SPH), called Regular Point Distribution (RPD). For more details, we refer thereader to [2, 20]. The idea is to introduce forces between the particles and the bodies to acceleratethe particles away from each other. On top of this strategy, one introduces a damping factor toensure the convergence of the solution. In this framework, we use a Wendland kernel which isdefined as

Wh(ri, rj) =

7

64π

(2 − |ri − rj |

h

)4 (1 + 2 |ri − rj |

h

), if r ≤ Rh,

0 else,

(33)

where h > 0 is a small regularization parameter. Typically one has Rh = 2h, but care isto be taken when choosing h as it is recommended for a particle to have no more than sixneighbouring particles within a Rh radius. Let us introduce a quantity that will allow us tograsp how equidistant the particles are. This quantity for particle i is defined by

Zi =∑

j

Wh(ri, rj)Vj . (34)

In equation (34), Vj is the volume that each particle occupies. It can be set as the total volumeof the fluid domain divided by the total amount of particles. Preferably, the changes in thisquantity will signify how close one is to having the particles being equidistant, i.e. when thegradient of Zi is the zero vector,

∇Zi =∑

j

∇iWh(ri, rj)Vj . (35)

To account for the presence of bodies, one could represent them with a set of points that aretreated exactly like the grid points, with the exception that one would not allow them to accel-erate, i.e. we introduce the body as a set of frozen points. This would then dictate that onehas the same resolution of points on the body as the grid. A way to work around this fact is tointroduce a force that scales with the distance between the points defining the body,

Fi =Nb∑j

Wh(ri, rj)nj∆s, (36)

where nj is the normal vector for the point rj on the body, ∆s is the distance between the pointsdefining the body, and Nb is the amount of body points. The governing equations then for theRPD scheme is then

dui

dt= −β (∇Zi + Fi) − ζui,

dri

dt= ui,

(37)

1The regularity issue is connected to how equidistant the spatial distrubution of points are.

14

where β = 2 p0ρ0

with p0 and ρ0 being the starting pressure and starting density of the particles.However, in light of this being just grid points, both p0 and ρ0 are set as 1. The coefficient ζ hereis a scaling factor for the damping term which is the velocity itself, and is defined as ζ = α

√β

V1/d

0where V0 is the same as Vi in this case, i.e. the volume per particle (in this case the same as Vi),and α is a tunable parameter, but here set as 10−3. Equation (37) need initial conditions forthe position and speed before solving, i.e. ui(0) and ri(0). The velocity is set as zero (ui(0)).The particle positioning is, however, spread according to a Cartesian grid at time zero. The timestep for this type of simulations is suggested to be

∆t = CoV1/d

0√β

, (38)

with a Courant number Co = 1.

2.4.6 Diffused Vortex Hydrodynamics (DVH)

Diffused Vortex Hydrodynamics (DVH) is a vortex method. Our main references for this methodare: [2, 3]. The interested reader can also get insight into the operator splitting approach in [21].Vortex methods can be separated by evaluating how the following three matters are treated:

• The discretization of the vortex field;

• The evaluation of the velocity field;

• The solution to the viscous diffusion step.

The DVH method is meshfree, i.e. there is no a priori connection between each point. Insteadinteractions between points are considered. Thus the vortex field is represented by a set of vortexparticles. The vortex field is evaluated using a Fast Multipole Method (FMM) (see e.g. [22]),and the diffusion step is a superposition of the solution to the heat equation. The DVH methodrelies on two separate steps:

(1) the advection step, where the vorticity is allowed to travel with the flow of the fluid whichis seen as inviscid,

(2) the diffusion step, where the fluid is seen as viscous and stationary and the vorticity is allowedto diffuse.

The governing equations then for the vorticity for the advection step can be derived from equation(8) under the assumption of negligible gravitational forces and inviscid fluid in two dimensions.This results in

Dω(r, t)Dt

= 0,

dr

dt= u(r, t),

ω(r, t) = ∇ × u(r, t).

(39)

The diffusion equation can be derived from equation (8), when it is assumed that changes ingravitational acceleration are small and that the fluid velocity is zero. Then the vorticity equationbecomes

dω(r, t)dt

= ν∇2ω(r, t). (40)

15

For computational purposes equation (40) must then be discretized in a suitable way. First, onegenerates a set of points that will represent vortex particles, preferably evenly distributed overthe simulation zone, e.g. via RPD. The velocity can be broken down into three contributing fac-tors: the free stream velocity (u∞), the vorticity-induced velocity (uω), and the bodily-inducedvelocities (u′), i.e.

u = u∞ + uω + u′ (41)

To calculate the vorticity for each point one can use a reverse version of equation (10), whichafter discretization ends up as

ω(r, t) =Np∑j

Γ(rj , t)δϵ(r − rj), (42)

where Np is the total amount of vortex points, seeing that as ϵ → 0 we are led to δϵ(r − rj) → 0such that it is an approximation to a Dirac-delta function. The velocity contribution from thevortices is seen here as

uω(r, t) =∫∫

S

K(r, r∗)ω(r∗, t)dS∗, (43)

where K(r, r∗) here is the free-space Green’s function for two dimensions. Thus from equations(42) and (43) we see that the discretized contribution to velocity from vorticity is

uω(r, t) =Np∑j

Γ(rj , t)Kϵ(r, rj), (44)

where Kϵ(r, rj) is a mollified kernel. The interaction with a body, however, is slightly morecomplex. The bodily contribution to the velocity is modelled as two separate contributions: (i)from sinks and sources on the body surface, and (ii) from the circulation on the body surface.As such, the contribution looks like

u′(r, t) =∫

∂Sb

K(r, r∗)γ(r∗, t)dr∗ −∫

∂Sb

e3 × K(r, r∗)σ(r∗, t)dr∗, (45)

where Sb is the surface of the body, γ(r, t) is the circulation on the body surface, σ(r, t) is thesource and sink on the body surface, i.e. in the DVH context σ(r, t) is not the cavitation number,and e3 is a the unit vector pointing straight out of the two dimensional plane, such that e3×just means that one rotates the vector π/2 radii clockwise. The discretized equation then lookslike

u′(r, t) =Nb∑j

Kϵ(r, rj)γ(rj , t)∆s −Nb∑j

e3 × Kϵ(r, rj)σ(rj , t)∆s, (46)

where Nb is the amount of body points, and ∆s is the distance between the body points. Thusone needs a way to evaluate σ(rj , t) and γ(rj , t). Note that γ(rj , t) can be defined as

γ(rj , t) = Γ(0) − Γ(t)Nb∆s

, j ∈ [1, Nb] (47)

where Γ(t) =∑Np

i Γ(ri, t), i.e. total sum of circulation for the system. In two-space dimensions,one can define a complex velocity

q(z) = ux(x, y) − iuy(x, y), (48)

16

where ux(x, y) and uy(x, y) just are the components of u′(r, t) in the given directions, andi =

√−1 (and not a summation index!). One can then process equation (45) through a substantial

amount of complex analysis and then eventually arrive at

[un] = (∆sR(F)) [γ] −(

12I + ∆sJ(F)

)[σ] , (49)

[uτ ] = (∆sR(F)) [σ] +(

12I + ∆sJ(F)

)[γ] − ∆s

2π

[dσ

ds

], (50)

where [ ] indicates that it is a column vector of length Nb, where the value at place j is the valuefor body point j, R and J are the real and imaginary parts respectively, ds means to differentiatealong the body, i.e. ds is to be treated as a curve-linear infinitesimal element. Furthermore, I isthe unit matrix (N2

b in size), while F is a complex matrix defined as

Fi,j = − 12π

τ

zj − zi, for i ̸= j, i, j ∈ [1, Nb],

(d2z/ds2)2(dz/ds)

, for i = j, i, j ∈ [1, Nb],(51)

where τ = τx + iτy. It follows then that from equation (5) that one can know σ from equation(49), i.e. (

12I + ∆sJ(F)

)[σ] = (∆sR(F)) [γ] + u∞ · n + uω · n. (52)

Thus the source and sink contributions are known, and consequently, one can fully evaluatethe velocities. To enforce the no slip condition, one generates circulation on the body surfaceproportional to the tangential velocity, such that the new circulation becomes

ΓNew(ri) = uτ (ri)∆s, i ∈ [1, Nb]. (53)

Here uτ can be evaluated from equation (50) as

[uτ ] = (∆sR(F)) [σ] +(

12I + ∆sJ(F)

)[γ] − ∆s

2π

[dσ

ds

]+ u∞ · n + uω · τ . (54)

This way the bodily contribution to the flow is described completely. Hence the advection stepequations for the discrete vortex particles are

DΓ(ri, t)Dt

= 0,

dri

dt= u(ri, t),

u(ri, t) = u∞ +Np∑j

Γ(rj , t)Kϵ(ri, rj)

+Nb∑k

[Kϵ(ri, rk)γ(rk, t) − e3 × Kϵ(ri, rk)σ(rk, t)]∆s

(55)

The kernel function Kϵ(r1, r2) can be expressed as Kϵ(r), where r = r1 − r2. The explicitexpression for Kϵ(r) is:

Kϵ(r) = −e3 × r

2π

[|r|4+3ϵ2|r|2+4ϵ4

(ϵ2 + |r|2)3

]. (56)

17

The diffusion of a single vortex particle (i) can be expressed as a solution to the heat equation(40) as

ω(r, t + ∆td) =

Γi1

4πν∆tdexp

(−|r − ri|2

4ν∆td

), if |r − ri|< Rd,

0, else.(57)

From equation (57), the diffusion of the circulation of a single vortex particle (i) can be expressedas

Γ(rk, t + ∆td) =

Γi1

4πν∆tdexp

(−|rk − ri|2

4ν∆td

)(∆r)2, if |rk − ri|< Rd,

0 else.(58)

There exists, however, the issue that more of this circulation will be accounted for if there aremany other particles, or some particles are too close to each other. For this very reason, arenormalization of the circulation is required. The accurate (renormalized) circulation is thus

Γ(rk, t + ∆td)′ = Γi∑l

Γ(rl, t + ∆td)Γ(rk, t + ∆td). (59)

The error mode in the conservation of circulation over the diffusion step can be evaluated by

ξ =∣∣∣∣Γ(t + ∆td) − Γ(t)

Γ(t)

∣∣∣∣ = exp(

− R2d

4ν∆td

). (60)

Thus, for a given error tolerance ξ > 0, one finds that the time step for diffusion is

∆td = − R2d

4ν ln ξ. (61)

The advection time step is defined through the Courant number. With the interpretation that uis the set velocity of the incoming fluid and ∆x is the resolution of the vorticity grid, one findsthat the advection time step, through equation (22) is given by

∆ta = Co ∆r

|u∞|. (62)

At this point, most of the basic theory behind the model is presented, though there still aredetails that need to be brought up. For instance, the diffusion in the presence of a body willnot follow equation (58) exactly. If a body is closer than the diffusion range, i.e. a body pointsis within radius Rd of the point of diffusion, of a particle, then a mirror point is created insidethe body. The position of the mirror point is denoted rim

i . The mirror points will also have anequal diffusion radius, and where the two radii overlap, there will be an extra contribution tothe diffusion. This then looks like

Γ(rk, t + ∆td) =

Γi(∆r)2

4πν∆td

[exp

(−|rk − ri|2

4ν∆td

)+

exp(

−|rk − rimi |2

4ν∆td

) ], if |rk − ri|< Rd and |rk − rim

i |< Rd,

Γi(∆r)2

4πν∆tdexp

(−|rk − ri|2

4ν∆td

), if |rk − ri|< Rd and |rk − rim

i |> Rd,

0, else.(63)

18

An illustration of this can be seen in Figure 6a. Furthermore, a "line of sight" argument is added.If the body somehow obscures a straight line from the particle for diffusion outwards to thecircumference, then that section will not be included in the diffusion. An illustration of thiseffect is seen in Figure 6b.

r i

rim

i

(a) Flat body

r i

rim

i

(b) Body with edge

Figure 6: Sketch illustrating how the diffusion will account for the presence of a body, whetherit is considered flat or it has an edge.

More corrections to the theory are technically needed. The advection time step and diffusiontime step need to be cast in an integer ratio. Specifically, for low Reynolds number simulationswe should have ∆ta < ∆td such that several advection steps can be preformed before anotherdiffusion step is done. For high Reynolds numbers, the relationship should be reversed. Moreover,there exists the possibility of having overlapping vortex point distributions with different spatialresolutions, which in turn will require the use of different time steps. Then it is also importantthat the time steps are all in integer ratios to each other. This is to ensure that the diffusion timeand advection time adds up, and it also enlightening to consider that the suggested time stepsdescribed in this section are not hard rules to follow, but guidelines for how far one can push them.Not described here either is the Fast Multipole Method (FMM), which is based around treatinggroups of vortices as a single one with combined vorticity when calculating vortex interactions.In essence, this will not improve the physical results of the simulation, just the simulation time.

2.5 Things Not IncludedAs mentioned earlier in this text, the field of fluid dynamics and propeller theory is far and wide,and to think that this paper will contain everything that will be of relevance is a grave error.Thus to expand the view, there are some suggested topics presented here.

The Kutta condition: For a steady potential flow, it sets a condition on how the flow leavesan aerofoil, see e.g. [4, 8].

Helmholtz vortex theorems: For a three dimensional fluid, it described the motion near vortexfilaments and is related to Kelvin’s Circulation Theorem, see e.g. [23].

Hamel-Oseen vortex model: A vortex model that comes from a solution to the Navier-Stokesequation, see e.g. [24].

Rankine vortex model: A vortex model where the centre is seen as a rigid body, see e.g. [25].Reynolds-averaged NavierStokes (RANS): RANS equations describes the flow of a fluid with

time-average. For unsteady flows this is called URANS, see e.g. [26].

19

Large Eddy Simulations (LES): Method for describing turbulent flow, where one applies timeand spatial averaging to reduce complexity, see e.g. [27].

While this thesis is focused on vortices, not a lot of focus has been to describe in detail howvortices affect the flow or how to describe the vortices themselves. Examples of where this canbe found would then be [28, 29]

3 Specific Goal and Approach

For this work it is important to have a balance between what the industry (Rolls-Royce) wantsand what is academically feasible for a master thesis. Hence, the set goals are

1. Validation of TVI for 5 blade series

2. Validate and examine suitable input for TVI.

3. Implement TVI for pressure side tip vortex cavitation inception.

4. Prediction Propeller Blade for Performance (PBfP) advanced designs based on TVI.

5. Implementation of DVH for 2D-geometries.

6. Validation of DVH for experimental values or existing theoretical values.

The approach was

1. Validation of TVI for different blade designs was performed by comparing the CIS frommeasured values with the TVI output from MPUF-3A.

2. Validation and examination of suitable input for TVI was preformed by varying the res-olution and varying the ship speed as well as the propeller rotational speed, and thencomparing the results.

3. The approach for implementation of pressure side tip vortex cavity inception was to studythe theory behind TVI and evaluate how well the implemented code for TVI fulfilled it,then correct the code.

4. Prediction for new blades was done by assuming a linear relationship for the CIS from theTVI model and using the scaling factors for a similar blades.

5. Implementation of DVH was done in MATLAB.

6. Validation of DVH was done by running simulations over two simple geometries (circle andsquare), as well as over a NACA profile.

4 Results and Discussion

In this section we present both our our TVI and DVH results.

20

4.1 TVI ResultsThe following sections contains the results from the cavity inception, where calc stands for thecalculated values through the TVI model, while meas stands for measured values from tests donein a water tunnel. Five blade series were investigated with a grand total of 27 designs. Thecalculated values were all achieved by having fixed rotations per minute (RPM) for the propeller,while varying the ship speed, unless stated otherwise. The measured values were achieved byvarying the RPM while having a fixed velocity on the water in the test tunnel for model scales.The full scales values are then achieved from model scale values by scaling them. All blade serieshave 5 blades for the propeller. Within a blade series, all the designs have the same diameter.Also, within a blade series, all the designs have the same hull. Unless stated otherwise, thecalculations are done with a resolution of 30 chord-wise panels and 27 span-wise panels. TheTVI model was applied to the output of MPUF-3A simulations, also in combination with XFOILanalysis. For the coming sections there exists abbreviations, which are: fs (full scale), ms (modelscale), PS (Pressure side), SS (Suction Side).

21

4.1.1 Blade Series 1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

1

2

3

4

5

6

JA (Dimensionless speed)

σ (C

avita

tion

num

ber)

Dimensionless speed vs Cavitation number (blade series 1, design 1)

Fit meas SS (fs)Meas data SS(fs)Fit meas PS (fs)Meas PS (fs)Fit meas SS (ms)Meas SS (ms)Fit meas PS (ms)Meas PS (ms)Fit calc SS (fs)Calc SS (fs)

(a) Design 1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

1

2

3

4

5

6

7


σ (C

avita

tion

num

ber)



(b) Design 2

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

1

2

3

4

5

6


σ (C

avita

tion

num

ber)



(c) Design 3

Figure 7: Figures over the cavity inception results for blade series 1 with 3 different designs.

Figure 7 gives an overview of how the calculated data for blade series 1 stands in relationship withthe measured data, for 3 designs. For Figure 7a, the calculated data points has an appearanceof being linear, though the tilt is not correct. Figure 7b has the calculated data points deviatingfrom the fit, while the fit itself resembles the actual suction side cavitation line. The misfit,however, indicate that is not reliable. Figure 7c is similar to Figure 7b in terms of the calculateddata points, but worse of in all aspects. Furthermore, the leftmost calculated point is lower thanthe coming point, and about the same level as the point after that. This indicates that theimplemented TVI model not only misses the actual cavitation inception point, but also does notcapture the general physical trend.

22

0.92 0.94 0.96 0.98 1 1.02 1.04 1.063

3.5

4

4.5

5

5.5

6


σ (C

avita

tion

num

ber)

Dimensionless speed vs Cavitation number (blade series 1, meas)

Design 1Design 2Design 3

(a) Measured values

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.152.5

3

3.5

4

4.5

5

5.5

6

6.5

7


σ (C

avita

tion

num

ber)

Dimensionless speed vs Cavitation number (blade series 1, calc)


(b) Calculated values

Figure 8: Figures over all the calculated values and the measured values for the suction sidecavity inception on blade series 1.

Figure 8 gives an overview of how the suction side cavitation inception compare between thedesigns, both for the measured data as well as calculated. In Figure 8a there is a clear distinctionbetween the different designs. From the calculated data one would then expect to see a similardistinction, whether the numerical values are correct or not. This is not observed. Though thereis an inversion point, the relationship between the cavitation lines are not correct on either side.The calculated data does not provide accurate numerical data, nor capture general trends of thecavitation inception for blade series 1.


0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

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(a) Design 1

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(b) Design 2

23

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

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(c) Design 3

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(d) Design 4

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.30

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(e) Design 5

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(f) Design 6

24

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.30

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(g) Design 7

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(h) Design 8

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(i) Design 9

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(j) Design 10

Figure 9: Figures over the cavitation results for blade series 2 with 10 different designs.

Figure 9 gives an overview of how the calculated data for blade series 2 stands in relationshipwith the measured data, for 10 designs. It would be redundant to go through the same kind ofarguments as was done for blade series 1. There are, however, unique features for blade series 2that are worth commenting on. These being design 4 and 6. It is very noticeable in Figure 10cthat the obtained values for design 4 and 6 overshoot the other values by a fair margin. Thoughsome overestimation depending on pitch is predicted in [1].

25

0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.142

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Design 1Design 2Design 3Design 4Design 5Design 6Design 7Design 8Design 9Design 10

(a) Measured values

0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.252

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(c) Calculated values, full view


26


0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.30

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(a) Design 1

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(b) Design 2

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(c) Design 3

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(d) Design 4


Figure 11 gives an overview of how the calculated data for blade series 3 stands in relationshipwith the measured data, for 4 designs. As stated in for blade series 2, it would be redundant togo through the same kind of arguments as was done for blade series 1. Blade series 3 is distinctfrom blade series 2 in that there are no unique features that makes it stand out. Hence, nofurther comments will be made here about it.

27

1 1.05 1.1 1.153

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Design 1Design 2Design 3Design 4

(a) Measured values

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Design 1Design 2Design 3Design 4




0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

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(a) Design 1

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(b) Design 2

28

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(c) Design 3

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(d) Design 4

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(e) Design 5

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(f) Design 6


Figure 13 gives an overview of how the calculated data for blade series 4 stands in relationshipwith the measured data, for 6 designs. Once again, it would be redundant to go through thesame kind of arguments as was done for blade series 1. There are similarities with blade series2, however, in terms of overshooting the other calculated values for some designs. Specificallydesigns 3, 4, and 5 for blade series 4 as seen in Figure 14. Worth noting is how far above theyare, as they are overshooting the other values by a factor 10. As mentioned, there is someoverestimation to be expected for increased pitch as stated in [1], though the question remains ifit is reasonable for it to be this extreme. Also of interest is the instability experienced by designs3 and 4, especially prominent in Figure 14c. The fact is that it could be completely missed ifone would have picked certain values for JA, which calls into question all other results as it isnot certain that this is what has happened elsewhere.

29

0.94 0.96 0.98 1 1.02 1.04 1.06 1.084

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Design 1Design 2Design 3Design 4Design 5Design 6

(a) Measured values

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0.9 0.95 1 1.05 1.1 1.15 1.2 1.250

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(c) Calculated values, full view


30


0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

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σ (C

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Dimensionless speed vs Cavitation number (blade series 5, design 1 PD1.25)

Fit meas SS (fs)Meas SS(fs)Fit meas SS (ms)Meas SS (ms)Fit calc SS (fs)Calc SS (fs)

(a) Design 1, Pitch variation 1

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

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Dimensionless speed vs Cavitation number (blade series 5, design 1 PD1.405)

Fit meas SS (fs)Meas SS(fs)Fit meas SS (ms)Meas SS (ms)Fit calc SS (fs)Calc SS (fs)

(b) Design 1, Pitch variation 2

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.30

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Dimensionless speed vs Cavitation number (blade series 5, design 2 Mode 1)


(c) Design 2, mode 1

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(d) Design 2, mode 2

Figure 15: Figures over the cavitation results for blade series 5 with 2 different designs. Theimages (a) and (b) are of the same design but different pitches, while images (c) and (d) are ofthe same design, but different variations of RPM and ship speed to vary the dimensionless speed.

Figure 15 gives an overview of how the calculated data for blade series 5 stands in relationshipwith the measured data, for 2 designs, one of them is tested for two different pitches, while theother has two different modes (i.e. same blade but different operation points, which means thatthe Rotations Per Minute is varied between them). Noticeable is that Figures 15a and 15b isthat there is no pressure side cavitation, though this is just because there was no available datafor that. The fact that the different pitches give different cavity inception points is not surprisingand their relationship to the measured values fall under the same kind of arguments that areused for blade series 1. More interesting is the fact that for Figures 15c and 15d the same JA

gives different cavity inception points. This is then quite contrary to what the aim for TVI modeland contrary to how the cavitation bucket in general works. The issue of having the same JA

but different cavity inception points is also touched upon in the coming section of ParameterVariation (section 4.1.6). Of some interest in Figure 17 is the calculated values for design 5,mode 1, which is the only curve which remotely looks like it might turn into a cavitation bucket.

31

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

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Fit calc SS (fs)Calc SS (fs)Predicted fs SSPredicted ms SS

(a) Design 3, mode 1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

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(b) Design 3, mode 2

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(c) Design 4, mode 1

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(d) Design 4, mode 2

Figure 16: Figures over two designs with different modes and the predicted cavity inception lines.

Figure 16 contains the calculated data points for cavity inception for blade series 5, designs3 and 4. Also included are the predicted cavity inception lines. The prediction functions byhaving calculated how the tilt and shift looks like for blade series 5, design 2, and then underthe assumption that it is the same for designs 3 and 4, been applied. However, in light of howpoor the results have been for this TVI model over all, no data is presented on the specifics ofthe tilt and shift. There is no reason why one would expect these predictions to be accurate.

32

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.23

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(a) Measured values

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Design 1Design 2Design 3, mode 1Design 3, mode 2Design 4, mode 1Design 4, mode 2Design 5, mode 1Design 5, mode 2



4.1.6 Parameter Variation

Below one are comparisons for the TVI results for different inputs. Figure 18 contains variationsof rotations per minute (RPM) and their equivalent ship velocity for the same dimensionlessspeed. Figure 19 contains results for different blade resolutions.

0.9 0.95 1 1.05 1.1 1.15 1.2 1.252

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(a) Blade 2, design 2

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VRpm

(b) Blade 5, design 2, mode 1

Figure 18: Figures over the RPM variations vs. the Advance speed variation for TVI.

For a given dimensionless speed, there should be a given cavitation number for inception.The calculated values seen in Figure 18 do not show this behaviour. A very simple explanationis that equation (30) is adapted for ship speed, and not propeller speed. It is interesting to seein Figure 18a that the RPM curve seemingly follows the trend of the ship speed curve. Thoughone should not be too surprised by it as the panels describing the wake will still be affected by

33

changes in RPM. Thus it seems curious that the RPM curve in Figure 18b is basically flat. Itis possible that for speeds close to the design condition the RPM variations does not contributesignificantly, though more data is needed for such a statement.

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Res 30x27Res 20x18

(a) Blade 4, design 3

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Dimensionless speed vs Cavitation number (blade series 5, design 2, mode 1)

Res 30x27Res 20x18

(b) Blade 5, design 2, mode 1

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Dimensionless speed vs Cavitation number (blade series 5, design 2, mode 2)

Res 30x27Res 20x18

(c) Blade 5, design 2, mode 2

Figure 19: Figures over resolution variations for TVI.

In Figure 19 the results of variation of resolution of the blade can be seen. First thing to bementioned is that the lower resolution in Figure 19a does not capture the abnormal behaviourof the higher resolution. Though towards the higher dimensionless speeds the cavity inceptionfor the different resolutions seem to converge. This is something that is seen in Figures 19b and19c as well. However, not a lot of conclusions should be drawn from Figure 19c due to lackof data points. A common theme for Figure 19 is that the lower resolution results generallyoverestimates the cavity inception, in comparison with the higher resolution.

4.1.7 TVI Discussion Summary

The TVI results are not deemed successful, as there is barely any consistency in the relationshipbetween the calculated values and the measured values. Hence, this section will be devoted todiscussing the possible reasons for this.

34

An observant reader may notice that the x-axis is not JA in the original reference [1], butKT . This is irrelevant as it is only a scaling factor in between and will not change the generalshape of the curves. However, a more valid point of investigation would be that of the wakedifference between Boundary Element Methods (BEM) and Vortex Lattice Methods (VLM),since the original reference is based on BEM, while the implemented method here is based onVLM. In essence, if the wakes panels are similar enough then the evaluation of the wake may stillbe valid, up to a few adjustments, otherwise it needs to be completely re-evaluated. Not deeplyinvestigate either in this report is how MPUF-3A works on a detailed level, nor XFOIL. Sincethe TVI code is based on those, it is natural to assume that if there are differences in BEM andVLM wake, then this will matter, and even if there is not, then it is still of interest to investigate.

The Parameter Variation does not give much besides confirming that the variation of shipspeed is reasonable in the implemented model, instead of the variation of Rotations Per Minute(RPM), and that the resolution is of importance the stronger the suction side tip vortex is.

The lack of pressure side tip vortex cavitation is mysterious, and it may not be completelyunrelated to the fact that the whole model seem to be unable to capture the cavity phenomenonaccurately for the suction side either.

Furthermore, it is curious that there is a mixture of vortex models for the TVI theory. Onecan argue that we just borrow more physical results from one model to improve the other, butthen the question remains why one have not turned to the second model for the rest of the theory.The simple answer could be there does not exist an equivalent description for this phenomenon.

4.2 DVH ResultsThis section contains the RPD simulations and the following DVH simulations based on it.

4.2.1 RPD

The following section contains the results and discussions for the RPD for three different shapes:a Circle, a Square, and a NACA profile.

35

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(c) iteration 83

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(d) iteration 100

36

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(e) iteration 126

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(f) iteration 200

Figure 20: Images of the RPD for a Circle in a Square. The lines refer to the boundariesexperiences by the particles. The images are for the starting points, and then some differentsteps in the iteration, namely: 50, 83, 100, 126, 200.

In Figure 20 we see the RPD for a circle in a square. The starting points are laid out ina Cartesian grid as a starting point and are then allowed to readjust according to the RPDtheory. The total amount of iterations were 200, and then the evaluation was cancelled. Thusthe gradient in equation (35) was never the zero vector, and the points never reached theirequilibrium position. There are two areas where the points are too close to the body, left mostof the body, and bottom most. Also, there are two areas where the points are too far away fromthe body, right most, and top most. Figure 20b has improved these conditions, though therestill are a few points that are undesirably uneven in reference to the body. Figure 20c has adistribution of points that are most equidistant to the body, in comparision to the other images,despite the particles not being in their equilibrium position. Further in the RPD iterations onefinds that the points even enter the body. Hence truncating can be dangerous if one does notindividually evaluate the resulting grid. The alternative to truncating is, of course, to let theRPD simulation to run to an end, though this is not necessary for the DVH scheme, as it merelyrequires the points to be equidistant to the body.

37

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(c) iteration 83

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(d) iteration 100

Figure 21: Images of the RPD for a Square in a Square. The lines refer to the boundariesexperiences by the particles. The images are for the starting points, and then some differentsteps in the iteration, namely: 50, 83, 100.

In Figure 21 we see the RPD for a square in a square. The starting points are laid out on aCartesian grid as a starting point and are then allowed to readjust according to the RPD theory.The total amount of iterations are 100, and then the evaluation was cancelled. Thus the gradientin equation (35) was never the zero vector, and so the points never reached their equilibriumposition. The original grid has two sides where the points are in essence generated on the body,while two sides have an acceptable distance between the points and the body. An easy solutioncould be to simply remove the points generated on the body, but this is more a chance event

38

due to the size of the simulation area, the size of the body, and the grid resolution. Hence it isstill of interest to run the RPD simulation. Figure 21c is the image where the points are mostequidistant from the body.

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(b) iteration 50

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(c) iteration 83

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(d) iteration 100

Figure 22: Images of the RPD for a NACA profile in a Square. The lines refer to the boundariesexperiences by the particles. The images are for the starting points, and then some differentsteps in the iteration, namely: 50, 83, 100.

In Figure 22 we see the RPD for a NACA profile in a square. The starting points are laidout in a Cartesian grid as a starting point and are then allowed to readjust according to theRPD theory. The total amount of iterations were 100, and then the evaluation was cancel. Thus

39

the gradient in equation (35) was never the zero vector, and the points never reached theirequilibrium position. This NACA profile is a special case where one of the dimensions of thebody is in the same range as the spacing of the grid, which leads to a unwanted consequence;points close to the middle of the body will feel the effects of the other side, i.e. get pulled towardsthe body. Also, the edges of the body will have a high-density force. The fact that some pointsget forced outside of the set boundary is alarming, as it tells us that the forces of the boundary isto o weak in comparison to how the particles are accelerated by the bodily force. It is, however,not of great consequence though, since what matters is the regularity of points around the body.These few points outside the set simulation zone will not cause issues, though could be solvedby having a bigger simulation zone. The fact points end up inside the body is of consequence,however. The RPD simulation does not give a significantly better grid at these truncations.

4.2.2 RPD Summary

The RPD simulation is successful in rearranging an existing Cartesian grid around a body to agrid where the closest points to the body is more equidistant than the Cartesian grid, under theassumption the proper truncation is chosen. The simulations were never allowed to finish sincean even distribution around the body is most important and also since there is a time-factor tothis, and as such no conclusion can be made concerning the finial position of the grid. There areconcerns, however, during the simulations as points of the grid are able to enter the body, hencethe requirement to evaluate the different truncations. This could be solved by exaggerating theforce that represents the body, or perhaps by using another kernel. Furthermore, the constantsgiven in Section 2.4.5 about the RPD could possibly be adjusted. An interesting side note is thatthere does not seem to be anything that precludes this theory from being applicable in higherdimensions.

4.2.3 DVH Simulation

The following section contains the results and discussion for the DVH for two different shapes:a Circle, and a Square. For the set of reference parameters for the simulations, see Table 1.Theimages shown in this section will consist of the original generated grid, and the new particlepositioning along with the density of circulation. The density of circulation is gained by creatinga grid and then summing up the contributing circulation from particles in the specific sections andthen dividing by the grid resolution. The grid resolution is 1.01 ∗ ∆r, meaning that occasionallythere will be areas which will have two vortex particles, instead of one, and sometimes zero.Furthermore, for all DVH simulations, the body resolution is the same as the grid resolution.The free stream velocity, u∞, is only ever in x-direction.

Table 1: Different parameters for the DVH simulations.

Shape Re u∞ L Co ∆ta ∆td ξ ∆rCircle 112 0.4 0.25 0.88 0.022 0.022 10−9 0.01Square 112 0.4 0.25 0.88 0.022 0.022 10−9 0.01

Figure 23 shows the original position for the vortex particles for the DVH simulation for acircle. The grid up until 0.75 m in x-direction was generated with RPD and truncated at 500iterations. The other part after 0.75 m is a regular Cartesian grid. Along the right side of thegrid there is a column of points where the circulation is forcefully set to zero after each DVHiteration.

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Figure 23: Image over the starting grid for the DVH simulation of a Circle.

The grid in Figure 23 was not picked because it was the best out of 500 iterations of the RPDsimulation, but rather after 500 iterations we stopped and deemed that the point distributionwas sufficient. As expected from Figure 20, most of the changes for the grid are at the cornerand close to the body.

Figure 24 shows the start of the DVH simulation for a Circle, while Figure 25 shows the DVHsimulation once the start up period is over and a stable phenomenon is generated.

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Figure 24a shows the particle positioning and the circulation generated after the first timestep. As expected the maximum circulation is generated on the sides of the object where thetangent of the body is parallel to the flow, and least on the sides where the tangent normal isnormal to the flow. This is expected since this is how we implemented the no-slip condition.What is not expected is the fact that some vortex particles violate the impermeability conditionof the body, which is also seen in Figure 24b. One could blame this on the fact that the simulationis just getting started and has not reached a point where it correctly captures the physical modelyet. The rest of Figure 24 does not suffered this issue, nor does Figure 25, which supports theclaim.

Figure 24b also show the beginning of the two first vortices being formed. These vortices arenot the ones showed in Figure 24c, bur rather those generated afterwards. The first two vorticeshave dispersed, but in turn caused the backside (the side towards the positive x-direction) toreverse the sign on the circulation generation. This is in good agreement with the low Reynoldnumber DVH simulations done in [2]. Figure 25 shows the continued generation of vortices anddevelopment of the wake.

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Figure 25: Plots of DVH simulation for a Circle once it has stabilized.

Figure 25 shows the continued simulation from Figure 24, and is not itself of great interest dueto how little of the wake that is simulated. It shows how the vortices are continually generatedand alternate in strength. But the most important feature of Figure 25 is that it clearly showsthat it is stable under a longer period than just the start up phase. If more of the wake wassimulated, it should have shown the formation of a Von Kármán vortex street (see e.g. [7, 30]),which is also shown for a circle in [2].

Figure 26 shows the original position for the vortex particles for the DVH simulation for asquare. The grid up until 0.75 m in x-direction was generated with RPD and truncated at 500iterations. The other part after 0.75 m is a regular Cartesian grid. Along the right side of thegrid there is a column of points where the circulation is forcefully set to zero after each DVHiteration.

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Figure 26: Image over the starting grid for the DVH simulation of a Square.

The grid in Figure 26 was not picked because it was the best out of 500 iterations of the RPDsimulation, but rather after 500 iterations we stopped and deemed that the point distributionwas sufficient. As expected from Figure 21, most of the changes for the grid are at the corner ofthe grid and close to the body, specifically at the corners of the body.

Figure 27 shows the start of the DVH simulation for a Square, while Figure 28 shows the DVHsimulation once the start up period is over and a stable phenomenon is generated.

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Figure 27: Plots over the start of DVH simulation for a Square.

Figure 27a shows the particle positioning and the circulation generated after the first timestep. As expected the maximum circulation is generated on the sides of the object where thetangent of the body is parallel to the flow (i.e. top and bottom of the square), while there is nongenerated on the sides where the normal of the body is normal to the flow (left and right of thesquare). This is expected since this is how we implemented the no-slip condition. What is notexpected is the fact that some vortex particles violate the impermeability condition of the body,which in fact happens in all plots in Figures 27 and 28, except for Figure 28b. This can not beblamed on the fact that the simulation is just getting started as it is a repeated behaviour. Itis possible that the RPD grid is not having enough distance from the body, which then possiblycould cause this issue. Also seen seen in both Figure 27 and 28 on the circulation representationis that there appears to be some unevenness along the body boundaries. This is just because theRPD grid is somewhat irregular, and thus the representation of the circulation will suffer smallanomalies, as mentioned before.

Figure 27b also show the beginning of the two first vortices being formed. These vortices arethe ones showed in Figure 27c. Since the backside (the side towards the positive x-direction) isgenerated along with the first vortices. The change in circulation is in good agreement with thelow Reynold number DVH simulations done in [2]. Figure 28 shows the continued generation ofvortices and development of the wake.

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Figure 28: Plots of DVH simulation for a Square once it has stabilized.

Figure 28 shows the continued simulation from Figure 27, and is not itself of great interestdue to how little of the wake that is simulated. It shows how the vortices is continually generatedand alternate is strength. Again, most important of Figure 28 is that it clearly shows that it isstable under a longer period than just the start up phase. While it is worrying that the vortexparticles repeatedly violate the impermeability condition, it does not seem to affect the stabilityof the simulation. Again if more of the wake was simulated, it should have shown the formationof a Von.Karman vortex street is also shown for a square in [2].

4.2.4 DVH summary

The results from the DVH simulations are the very least promising in terms of stability duringlonger simulation periods. Though one can argue about the accuracy of the results. A comparisonat length is not quite viable, due to how short the wake is, and also due to how the circulationis modelled to leave the simulation zone. For now it seems like the simulations close to the bodymatches that of other works. This is with one exception, the actual sign of the circulation. Fromthe compared work it is clear that the sign is reversed, and it can also be realized by a simpleobservation on how fluids interact with objects. With the boundary layer (caused by the no-slipcondition) the fluid will start to rotate and the rotation will be towards the body. By definitioncounter-clockwise rotation leads to positive circulation, and vice versa. Thus from observing theresults we know that this is not what happens.

The difference between the circle and the square can be noted as three things: the generationvortices, the spread of generated vortices, and the violation of the impermeability condition.

Concerning the generation of vortices, one should note that right before the vortices leave

52

the body, there is the appearance of circulation of the opposite sign. While this effect is barelypronounced for the circle, it is very clear towards the end of the sides of the square. For thespread of vortices, it appears that the square releases vortices of different signs at a higher rate,or rather, further apart in the y-direction. The wake behind a circle and a square in this caseshould not deviate much in that sense, hence it could be that the interaction between the vorticeswill form a similar pattern later on, but this close behind the body have not yet. For the violationof the impermeability condition, it is seen that for the circle this happens at the start of thesimulation and eventually is corrected, but for the square this is a repeated behaviour.

Noticeable left out from the DVH results is that of the NACA profile. The issues we had withthe RPD was the same that we had with the DVH, that one of the dimensions for the NACAprofile is to close to the size of the grid resolution. Attempts to overcome this problem was triedby increasing the size of the NACA profile, and to increase the grid resolution. Increasing thegrid resolution immediately lead to Matlab running out of memory. Increasing the size of theNACA profile was at first unsuccessful, but after resizing the grid zones the simulations could goon without MATLAB suffering memory issues. However, there were two problems that remained,one being that the reduction in simulation zone size meant that very little would actually begained from the simulation, and the other being that the memory usage still exceeded the RAM(16 GB) such that swap files were written to the hard-drive instead which together with themassive amount of calculations would slow the simulations down to a halt. Hence it was aconcious decision to leave the NACA profile out of the DVH results. It is, however, possible torun simulations for it, by simply having a high resolution grid around a small NACA profile, andaround that a lower resolution grid.

Some remarks should be done concerning the theory for the DVH model- First and foremostis that the interpretation is not quite straightforward, i.g. how to set the tangents and evaluatethe changes along the body. The approach that was successful was to interpret the counter-clockwise line around the body as the positive abscissa. This then makes sense if one regardsthe origin from calculus where line integrals around bodies is evaluated as being positive inthe counter-clockwise orientation. Furthermore, the code that was built for this thesis has thetotal sum for the generated circulation forcefully set to zero, i.e. Kelvin’s Circulation Theoremis forcefully applied. This is not contained in the DVH theory at all, since the momentarycirculation generated on the body when solving for the source distribution is constant (and inthis case, always zero). The reason for applying Kelvin’s Circulation Theorem this way, however,is to ensure stability, as the simulations without it quickly failed.

5 Conclusion

In conclusion the implemented TVI method does not function as intended as the values are notonly faulty, but the general shape of the resulting cavitation bucket is not right, even when com-pared to other TVI results from other blades. The actual cause for these issues is still unknown.The two possibilities seem to be that either the implementation of the method is faulty, or itis not applicable to the MPUF-3A software. The goals set for TVI concerning the validation,examination, implementation, and prediction were thus not fulfilled.

The implemented DVH method is successful in capturing a hydrodynamic phenomenon alreadystudied for a circle and a square, hence it adds nothing new, but confirms that the implementa-tion works, with the side issue of generating circulation with the wrong sign. Thus the set goalsfor DVH in concerning implementation and validation were fulfilled.

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6 Future work

From this conclusion there are several future possibilities to investigare.

6.1 TVI OutlookPossible future aims for this TVI code would be:

• Deeper investigation into MPUF-3A to see if the wake-description really can be said to beclose enough to that of a BEM to be applicable,

• Deeper investigation into the TVI code to see if it was applied correctly,

• Re-evaluation of the panel interpretation for the vortex,

• Investigation of the choice vortex modelling for the TVI method.

6.2 DVH OutlookPossible future aims for this DVH code would be:

• Implementation of the FMM,

• Re-evaluation of the implemented code to find the cause for the sign fault in the circulation,as well as to lift the requirement of conserved circulation.

• Implementation in 3D, which means investigating another way of evaluating the sourcesand circulation of the body,

• Implementation of solid particles to travel along with the generated flow.

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7 References

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[2] E. Rossi. 2D-vorticity genesis and dynamics studied through particle methods. PhD thesis,Universita di, Roma Sapienza, 2014.

[3] E. Rossi, A. Colagrossi, B. Bouscasse, and G. Graziani. The diffused vortex hydrodynamicsmethod. Communications in Computational Physics, 18(2):351–379, 2015.

[4] J. Carlton. Marine Propellers and Propulsion. Butterworth-Heinemann, 2011.

[5] J. Tornblad. Marine Propellers and Propulsion of Ships. Marine Laboratory, KaMeWa,1987.

[6] A. Cengel Yunus and J. M. Cimbala. Fluid Mechanics: Fundamentals and Applications.International Edition, McGraw Hill Publication, 2006.

[7] G.K. Batchelor. An Introduction to Fluid dynamics. Cambridge Mathematical Library.Cambridge University Press, 2000.

[8] R. Temam and A. Miranville. Mathematical Modeling in Continuum Mechanics. CambridgeUniversity Press, 2005.

[9] G. Kuiper. New developments around sheet and tip vortex cavitation on ships propellers.http://resolver. caltech. edu/cav2001: lecture. 007, 2001.

[10] F. Meng, H. Schwarze, F. Vorpahl, and M. Strobel. A free wake vortex lattice model forvertical axis wind turbines: Modeling, verification and validation. In Journal of Physics:Conference Series, volume 555.

[11] CT. Hsiao and GL. Chahine. Scaling of tip vortex cavitation inception for a marine openpropeller. In 27th Symposium on Naval Hydrodynamics, Seoul, Korea, pages 5–10, 2008.

[12] M.T. Heath. Scientific Computing: an Introductory Survey.

[13] F. X. Giraldo. ”Time-integrators” Lecture Notes. Naval Postgraduate School.

[14] J. HE. Cartwright and O. Piro. The dynamics of Runge–Kutta methods. InternationalJournal of Bifurcation and Chaos, 2(03):427–449, 1992.

[15] L. He. Numerical simulation of unsteady rotor/stator interaction and application to pro-peller/rudder combination. PhD thesis, The University of Texas at Austin, Austin, 2010.

[16] M. Drela. Xfoil: An analysis and design system for low Reynolds number airfoils. In LowReynolds Number Aerodynamics, pages 1–12. Springer, 1989.

[17] XFOIL 6.9 User Primer. http://web.mit.edu/drela/Public/web/xfoil/xfoil_doc.txt.Accessed: 2017-09-28.

[18] H. Lee. Modeling of unsteady wake alignment and developed tip vortex cavitation. PhDthesis, The University of Texas at Austin, Austin, 2002.

[19] R. EA. Arndt. Cavitation in vortical flows. Annual Review of Fluid Mechanics, 34(1):143–175, 2002.

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[20] A. Colagrossi, B. Bouscasse, M. Antuono, and S. Marrone. Particle packing algorithm forSPH schemes. Computer Physics Communications, 183(8):1641–1653, 2012.

[21] A. Colagrossi, G. Graziani, and M. Pulvirenti. Particles for fluids: SPH versus vortexmethods. Mathematics and Mechanics of Complex Systems, 2(1):45–70, 2013.

[22] E. Darve. The fast multipole method: numerical implementation. Journal of ComputationalPhysics, 160(1):195–240, 2000.

[23] H. von Helmholtz. Lxiii. on integrals of the hydrodynamical equations, which express vortex-motion. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science,33(226):485–512, 1867.

[24] P.G. Saffman. Vortex Dynamics. Cambridge Monographs on Mechanics. Cambridge Uni-versity Press, 1995.

[25] D. B. Giaiotti and F. Stel. The Rankine vortex model, Lecture Notes. University of Trieste,2006.

[26] G. Alfonsi. Reynolds-averaged Navier–Stokes equations for turbulence modeling. AppliedMechanics Reviews, 62(4):040802, 2009.

[27] T. JR. Hughes, L. Mazzei, and K. E. Jansen. Large eddy simulation and the variationalmultiscale method. Computing and Visualization in Science, 3(1):47–59, 2000.

[28] S. Kawada. Calculation of induced velocity by helical vortices and its application to propellertheory. title Report of Aeronautical Research Institute, Tokyo Imperial University, 14(172):2,1939.

[29] Y. Fukumoto and VL. Okulov. The velocity field induced by a helical vortex tube. Physicsof Fluids, 17(10):107101, 2005.

[30] G. Birkhoff. Formation of vortex streets. Journal of Applied Physics, 24(1):98–103, 1953.

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A Appendix

List of Figures

1 Image of a full scale propeller. Published with permission from Rolls-Royce. . . . 22 Sketch of a typical aerofoil, where chord and camber are illustrated. . . . . . . . 33 Sketch of a propeller blade seated on the hub. The illustrated angles are the skew

angle (θS(x)) and propeller skew angle (θSP ). . . . . . . . . . . . . . . . . . . . . 44 An image featuring a propeller which has tip vortex cavitation. This is seen as

the filament that stretches from the propeller blade tips backwards into the wake.Also seen is slight sheet cavitation on the topmost propeller blade. Published withpermission from Rolls-Royce. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Illustration of how a typical cavitation bucket may appear. . . . . . . . . . . . . 106 Sketch illustrating how the diffusion will account for the presence of a body,

whether it is considered flat or it has an edge. . . . . . . . . . . . . . . . . . . . . 197 Figures over the cavity inception results for blade series 1 with 3 different designs. 228 Figures over all the calculated values and the measured values for the suction side

cavity inception on blade series 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Figures over the cavitation results for blade series 2 with 10 different designs. . . 2510 Figures over all the calculated values and the measured values for the suction side

cavity inception on blade series 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611 Figures over the cavitation results for blade series 3 with 10 different designs. . . 2712 Figures over all the calculated values and the measured values for the suction side

cavity inception on blade series 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2813 Figures over the cavitation results for blade series 4 with 6 different designs. . . . 2914 Figures over all the calculated values and the measured values for the suction side

cavity inception on blade series 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015 Figures over the cavitation results for blade series 5 with 2 different designs. The

images (a) and (b) are of the same design but different pitches, while images (c)and (d) are of the same design, but different variations of RPM and ship speed tovary the dimensionless speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

16 Figures over two designs with different modes and the predicted cavity inceptionlines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

17 Figures over all the calculated values and the measured values for the suction sidecavity inception on blade series 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

18 Figures over the RPM variations vs. the Advance speed variation for TVI. . . . . 3319 Figures over resolution variations for TVI. . . . . . . . . . . . . . . . . . . . . . . 3420 Images of the RPD for a Circle in a Square. The lines refer to the boundaries

experiences by the particles. The images are for the starting points, and thensome different steps in the iteration, namely: 50, 83, 100, 126, 200. . . . . . . . . 37

21 Images of the RPD for a Square in a Square. The lines refer to the boundariesexperiences by the particles. The images are for the starting points, and thensome different steps in the iteration, namely: 50, 83, 100. . . . . . . . . . . . . . 38

22 Images of the RPD for a NACA profile in a Square. The lines refer to the bound-aries experiences by the particles. The images are for the starting points, and thensome different steps in the iteration, namely: 50, 83, 100. . . . . . . . . . . . . . 39

23 Image over the starting grid for the DVH simulation of a Circle. . . . . . . . . . 4124 Plots over the start of DVH simulation for a Circle. . . . . . . . . . . . . . . . . . 44

57

25 Plots of DVH simulation for a Circle once it has stabilized. . . . . . . . . . . . . 4626 Image over the starting grid for the DVH simulation of a Square. . . . . . . . . . 4727 Plots over the start of DVH simulation for a Square. . . . . . . . . . . . . . . . . 5028 Plots of DVH simulation for a Square once it has stabilized. . . . . . . . . . . . . 52

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